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\begin{document}
\title{\vspace{-2cm}An experimental test of whether \\financial incentives constitute undue inducement \\in decision-making}
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\author{Sandro Ambuehl \\ \normalsize Department of Economics and UBS Center for Economics in Society, \\ \normalsize University of Zurich, \\ \normalsize Bl\"uemlisalpstrasse 10, \\ \normalsize 8006 Z\"urich, Switzerland, \\ \normalsize sandro.ambuehl@econ.uzh.ch.}
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\begin{abstract}
Around the world, laws limit the incentives that can be paid for transactions such as human research participation, egg donation, or gestational surrogacy. A key reason is concerns about `undue inducement'---the influential but empirically untested hypothesis that incentives can cause harm by distorting individual decision-making. Here, I present two experiments ($n=671$ and $n=406$), including one based on a highly visceral transaction (eating insects). I show that incentives cause biased information search---participants offered a higher incentive to comply are more likely to seek encouragement to do so. However, I show theoretically that such behavior is no proof of harmful effects of incentives; it is consistent with Bayesian rationality. Empirically, although a substantial minority of participants make bad decisions, incentives do not magnify them in a way that would suggest allowing a transaction but capping incentives. Under the conditions of this experiment, there was no evidence that higher incentives could undermine welfare for transactions that are permissible at low incentives.
\end{abstract}
\noindent\emph{Keywords: Incentives, Behavioral Welfare Economics, Experiment, Undue Inducement} \\
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\section{Introduction}
\indent Around the world, laws and guidelines limit the incentives that can be paid for transactions such as human research participation, surrogate motherhood, human egg donation, and organ donation. These transactions are often legal if no or limited amounts of money change hands. While there are a number of complex ethical issues at play (e.g. objectification\citep{nussbaum} and distributive justice\citep{beckerElias, grady2005payment}), an important reason for these limits on incentives is the influential, albeit conceptually vague and empirically largely untested notion of \emph{Undue Inducement} \citep{BelmontReport}. It describes the belief that participation incentives can harm participants, because ``something is being offered that is alluring to the point that it clouds rational judgment ... Attention is fixated on the benefit, disallowing proper consideration of the risks'' \citep{sep-informed-consent}. The American Society for Reproductive Medicine, for instance, writes that ``Payments to women providing oocytes should be ... not so substantial that they ... lead donors to discount risks,'' and conjectures that ``the higher the payment, the greater the possibility that women will discount risks'' \citep{financialIncentivesEggs}. The National Bioethics Advisory Commission limits incentives that can be offered for participation in clinical trials based on the assertion that  ``[o]ffers of large sums of money \ldots could lead some prospective participants to enroll ... when it might be against their better judgment'' \citep{NBAC}. Two distinct claims make up the \emph{Undue Inducement Hypothesis} (henceforth: UIH). The \emph{positive part}  (henceforth: UIH-positive) is a behavioral prediction. It ascertains that incentives cause participants to engage in biased information processing and motivated reasoning about the transaction. Indeed, some bioethicists explicitly describe undue inducement as ``a {cognitive} distortion relating to the assessment of risks and benefits'' \citep{Largentetal.Reflections}. The \emph{normative part} (henceforth: UIH-normative) is the \red composite claim \black that these changes cause harm and that some transactions are therefore acceptable only at low but not at high incentives.
This paper empirically tests the cognitive underpinnings of the Undue Inducement Hypothesis. Empirical progress on the issue is crucial. Restrictions are well-advised if incentives really impede decision quality to an extent that outweighs the benefits of the increased payment (including, potentially, benefits to society and other parties). If, however, the effect of incentives on decision quality is minor, then limits on incentives simply amount to underpaying suppliers such as clinical trial participants, surrogate mothers, and organ donors for their valuable contributions. Such underpayment constrains supply and may thus impede medical progress, cause grief to aspiring parents with reproductive health issues, and lead to preventable deaths from kidney failure. While incentives cannot harm rational decision makers, evidence of questionable decision-making is widespread\citep{handbookBehavioral}. Direct empirical evidence on UIH is limited to a small number of case studies and unincentivized surveys concerning clinical trial participation\citep{BentleyThacker, Halpern.undueInducementVignette, SingerCouper, StunkelGrady}. While these provide suggestive evidence that payments do not alter judgments about study risks, they neither study real choices nor perform formal welfare analysis. The literature on motivated reasoning\citep{Kunda} does not directly speak to UIH because it typically considers settings in which the bias benefits rather than harms the decision maker, for instance by allowing him to maintain a positive self-image while dodging costly social obligations\citep{Danaetal.}.
I investigate UIH in laboratory experiments. This is possible even with stakes on the order of dozens or hundreds of dollars for two reasons. First, as I will demonstrate, even these stakes cause the information-biasing that UIH-positive predicts. The ethics literature views such biasing as a key cause of UIH-normative. Second, as the ethics literature clarifies, an offer that is genuinely too good to refuse cannot constitute undue inducement. Undue inducement requires that a subject with undistorted judgment would refuse the offer, but that the incentive warps her judgment in a way that makes her accept it\citep{WerheimerMillerDefinition}. Therefore, undue inducement requires that an offer is high \textit{relative to the incentivized activity}---but not so high that a person with undistorted judgment would accept it. I satisfy this requirement by choosing activities whose downsides are commensurate to the incentive amounts I employ.
While the main contribution of this paper concerns general cognitive mechanisms, college students (my study population) are a typical target demographic for recruiting human egg donors and clinical trial participants.
The paper consists of two experiments and a theoretical model.
Experiment 1 features a setting in which subjects need to evaluate the non-monetary consequences of a transaction in terms of dollars and cents. To give UIH the best chance, I seek a visceral and aversive transaction that is unfamiliar to subjects and that allows for biased information acquisition. Hence, I incentivize subjects to eat whole insects such as mealworms, silkworm pupae, and mole crickets. Some subjects learn that they will receive \$3 in exchange for eating bugs. Others learn that their incentive for the same transaction is \$30. To prevent the incentive from acting as a signal about the discomfort of eating the insects, subjects know of both incentive amounts and that they are randomly assigned to one of them. Next, subjects choose between two videos to help them decide whether to participate in the transaction. They are titled ``Why you may want to try eating insects'' and ``Why you may not want to try eating insects.''
As UIH-positive predicts, I find that a higher incentive increases demand for the encouraging video at the cost of the discouraging video. A comparison to a control condition that precludes access to the videos shows that they causally affect participation decisions. These findings dovetail with the idea that ``payments lead donors to discount risks'' \citep{financialIncentivesEggs}.
UIH-normative predicts that this mechanism links higher incentives to lower welfare. I test this prediction using the \emph{informed consumer paradigm}, a dominant approach in behavioral welfare economics\citep{Bronnenbergetal., Allcottetal.SinTaxes, AllcottKessler}. Initially, subjects decide whether to accept the transaction at the promised incentive based solely on the selected video and a brief verbal description of the insects. After making that decision, subjects receive the items. They must view, touch, and (inadvertently) smell them. Based on this substantial additional information, subjects then reveal the least amount of money they would need to swallow these insects (\emph{reservation price}). For a subject who accepted the transaction, the difference between the incentive she received and this reservation price reveals potential choice mistakes. If she committed to eating them for \$30 but finds, after inspection, that she would need at least \$45 as compensation for her displeasure, she has inflicted harm of $\$(45 - 30) = \$15$ onto herself. Generally, I define \emph{ex-post surplus} as the difference between the reservation price and the incentive if the subject accepted the transaction and zero otherwise. To test UIH-normative formally, I aggregate surplus across subjects. I weight negative values (losses) $\frac{\alpha}{1-\alpha}$ times as much as positive values (gains), for arbitrary values $\alpha \in (0, 1)$. Even though ten to twenty percent of subjects are harmed by their decision to participate, I find that it is never optimal to allow the transaction at the low incentive but prevent it at the high incentive, for any weight on losses---contrary to UIH-normative. (With high weights on losses, the transaction should be prevented altogether.) This result is robust to controlling for the possibility that the initial incentive exerts anchoring effects on reservation prices, and for noisy elicitation of reservation prices.
Experiment 2 uses a broader range of incentive amounts, permits alternative welfare benchmarks, and explores moderators of the effect of incentives on decision quality. Subjects decide whether to accept a given payment of \euro 20 to \euro 80 in exchange for risking to lose \euro 100. After learning the incentive amount, but before committing to a decision, subjects choose between advisors who provide informative but imperfect recommendations about what to do. A `Bold Advisor' likely recommends participation. He always does so if the loss will not materialize, but he may also do so if it will. A `Cautious Advisor' likely recommends non-participation. He always does so if the risk will realize, but may also do so if it will not. Upon observing a recommendation from the chosen advisor, subjects decide whether to participate in the transaction.
UIH-positive predicts that higher incentives increase the preference for the Bold Advisor and that this information biasing causally affects participation decisions. The data confirms both these predictions.
UIH-normative predicts that due to these behavioral changes, higher incentives lower welfare. To measure whether the decision was good given the information the subject had at the time, rather than in hindsight, I use the \emph{reframed decisions paradigm}, the second dominant approach in behavioral welfare economics\citep{ChettyLooneyKroft, AllcottTaubinsky.Lightbulbs, TaubinskyReesJones}. Specifically, I calculate how confident a perfectly rational subject would have been that she will not lose the \euro 100 after receiving the chosen advisor's recommendation. Stage 2 of the experiment presents subjects with a lottery whose probability of losing \euro100 equals a hypothetical perfectly rational agent's confidence, but which---unbeknownst to the subject---parallels the decision she faced in Stage 1 in all other ways. I measure what amount of money the subject considers just as good as that lottery (her \textit{certainty equivalent}). A subject's decision to accept the transaction in stage 1 was a mistake if she would rather lose a given sure amount of money than play the corresponding stage-2 lottery. The magnitude of her certainty equivalent measures the severity of the error.
I find UIH-normative robustly violated. While mistakes are common, they do not substantially vary with the incentive. Depending on the weight assigned to losses, one should therefore permit the transaction at all incentives, or prevent it altoghether. But it is never optimal to allow the transaction at low but not at high incentives. This is the case in treatments that vary the stakes by an order of magnitude, in treatments with a three-month delay between the receipt of the incentive payment and the realization of potential negative consequences, and in treatments in which the risk of losing the \euro100 ranges from 20\% to 80\%.
What explains the simultaneous confirmation of UIH-positive and refutation of UIH-normative in my experiments? To answer this question, I consider how a perfectly rational decision maker would inform herself about the potential consequences of a transaction when doing so is costly. Because such a decision maker will reject any transaction not in her interest, she will violate UIH-normative by construction. Yet, as I show, her information choice bears the hallmarks of UIH-positive: A higher incentive for participation lowers the cost of mistaken participation but increases the cost of mistaken abstention. Hence, rational agents will consult information sources they expect to more likely recommend participation and to less likely  recommend abstention. The model thus shows that UIH-positive does not logically imply UIH-normative. While my experiment participants do not behave perfectly rationally, their behavior is sufficiently rational to defy UIH-normative.
Next to informing the debate on undue inducement, this paper contributes to three strands of academic literature. First, it adds to the literature on \emph{repugnant transactions} which studies third parties' judgments, rather than the effect of incentive payments on the incentivized. \citep{kahnemanKnetschThaler,  Roth.repugnance,  Ambuehletal.Repugnance, Clemens, AmbuehlOckenfelsAERPP, Eliasetal.AER, Stueber, ErkutKubler} Second, this paper contributes to the literature in behavioral public economics\citep{BernheimTaubinsky} by examining the welfare effects of incentive payments---one of the most fundamental economic policy tools.  Third, its welfare analysis of incentive-induced information biasing contributes to the literature on the positive test strategy \citep{Slowiaczeketal.} and motivated reasoning \citep{KlaymanHa, Kunda, BenabouTiroleJEP}.
\section{Results}
\subsection{Experiment 1: Visceral Transaction}\label{insectExperiment}\label{measuredPrefs}
This experiment incentivizes 671 US undergraduate subjects to ingest real insects in exchange for money. This activity is novel and unfamiliar to most subjects. Many find it intensely aversive. Some reported that the experiment was ``stressful'' or that the ``insects were scary,'' while others refused even to touch the containers holding the dead animals.
\paragraph{Design}
As described earlier, subjects are randomly assigned to receiving either \$3 or \$30 in exchange for eating insects, and to select one of two six-minute videos that highlight either the upsides or the downsides of eating insects. To provide more continuous data about information preferences, subjects also select at least four of nine clips grouped in bins of three named ``Reasons for eating insects'', ``Reasons against eating insects'', and ``Other information about eating insects.'' Subjects know that they will not receive any further information and will decide whether to eat the insects based only on a brief verbal description. Thus, they have an incentive to select the videos carefully and to pay attention to their content. With an exogenous 97\% chance, the subject watches the selected video. Otherwise, she watches the selected clips.
Next, subjects reveal the least amount of money for which they would eat each of five different insect species (their \emph{reservation prices}). The procedure makes the decision potentially consequential and renders truthtelling optimal (see methods).
In the \emph{Main Decisions}, which determine each subject's consumption with 80\% probability, subjects then decide whether to eat insects from each of the five species in exchange for the incentive promised in the beginning. These are the decisions whose welfare consequences I seek to evaluate.
At this point, subjects have only seen brief verbal descriptions of the insects and the video they selected. Subjects now receive five containers filled with the insects they may be about to swallow (Figure \ref{bugPics}) and are forced to view, touch, and smell each insect.
\begin{figure}[htb]
\caption{\label{bugPics} Insects eaten by subjects.}
\end{figure}
Based on this substantial additional information, subjects again reveal reservation prices for each species.  I define a subject's \emph{ex-post surplus} from such a decision as the difference between her reservation price and the incentive if the subject accepted the transaction, and zero otherwise. A negative value indicates that Main Decision was mistaken in hindsight. Its magnitude is the severity of the error.
\paragraph{Information choice and participation decisions}
To test UIH-positive, I examine how the incentive affects information choice. Panel A in Figure \ref{insectFig} displays the fraction of subjects opting for the encouraging video (left-hand side) and the number of positive and negative clips they select (right-hand side), both as a function of the incentive amount. The \$30 incentive raises the proportion of subjects who select the encouraging video from 81.2\% to 88.7\%, a 7.5 percentage-points difference \red ($t$-test, $t = 2.09$, $p=0.037$). Preferences for the video clips show a similar effect. The difference between the number of positive and negative clips chosen rises from 0.86 to 1.31 ($t$-test, $t=2.61$, $p = 0.009$) as the incentive increases. \black The chosen number of clips labeled ``other'' remains approximately unchanged \red ($t$-test, $t=-0.03$, $p=0.977$)\black. Incentives bias information acquisition, consistent with UIH-positive.
Potentially harmful effects of incentives can arise only if these changes affect subsequent decisions. Yet, the effect of the incentive on information choice can also be explained by the hypothesis that subjects decide whether to eat the insects before selecting a video and then choose the video that ex-post rationalizes this decision. To disprove this hypothesis, subjects in a control condition complete the experiment without the possibility of selecting or watching any video or clips. I find that the information does affect subsequent choices. Raising the incentive causes participation to increase by 33 percentage points (from 38\% to 71\%; \red $t$-test, $t=8.14$, $p=0.000$\black) when subjects can select a video but by only 23 percentage points (from 36\% to merely 59\%; \red $t$-test, $t=4.51$, $p=0.000$\black) when they cannot, \red though the difference in differences does not reach statistical significance ($t$-test, $t=1.62$, $p=0.105$)\black.
\begin{figure}[htbp!]
\caption{Results from Experiment 1\label{insectFig}\label{insectFigWelfare}}
\end{figure}
\paragraph{Welfare}
Next, I examine how incentives affect welfare, measured as ex-post surplus. This interpretation assumes that handing out the insects conveys substantial information. Indeed, the hand-out causes reservation prices to rise by \$6.68 \red ($t$-test, $t= 12.69$, $p<0.000$) \black for some species and fall by \$1.77 \red ($t$-test, $t=-3.41$, $p=0.001$) \black for others, compared to reservation prices elicited before the hand-out (Supplementary Information A.2). The mean absolute change is \$6.68 \red($t$-test, $t=20.94$, $p = 0.000$)\black. In the \$3-condition, around one-fifth of all choices yield a negative surplus (Figure \ref{insectFigWelfare}.C). This number drops to just under 10\% in the \$30-condition. Moreover, raising the incentive does not increase the magnitude of negative surplusses. Yet, it greatly increases positive surplusses. Thus, there is nearly a stochastic dominance relationship between the surplus distributions across the incentive amounts. Therefore, almost any social welfare function increasing in ex-post surplus will favor the high incentive. No such function will support allowing the transaction at the low incentive but preventing it at the high incentive, contrary to UIH-normative (though planners that heavily weigh ex-post losses prefer preventing the transaction at all incentives).
There are two potential concerns about this analysis. First, the incentive might directly affect ex-post reservation prices, for instance, through anchoring \citep{coherentArbitrariness}. Second, the videos themselves may distort reservation prices, for example, because they are tendentious. I simultaneously address these concerns by assuming that true reservation prices in each treatment follow the distribution revealed by subjects who could not watch a video and were given the \$3 incentive. I assign the subject with the highest reservation price in the (\$30, Video) treatment the highest reservation price observed in the (\$3, No Video) treatment. I perform a similar match for all other reservation price ranks. Based on these counterfactual reservation prices but using the subjects' actual participation decisions, I calculate counterfactual surplus. This measure incorporates the effect of incentives and information on participation decisions but excludes such effects for reservation prices. Figure \ref{insectFigWelfare}.D displays the resulting surplus distribution. Raising the incentive to \$30 increases not only welfare gains but also losses.
Therefore, the higher incentive is no longer unambiguously preferable. Yet, the increase in losses is not \emph{per se} sufficient for UIH-normative. The reason is that UIH-normative requires two conditions to hold simultaneously. First, the weight on losses must be sufficiently \emph{high} to render preventing the transaction at the high incentive optimal (total welfare at the high incentive is lower than that from preventing the transaction). Second, the weight on losses must be sufficiently \emph{low} to make participating at the low incentive acceptable (total welfare at the low incentive exceeds that from preventing the transaction). Amongst the class of welfare functions that place weight $\alpha$ on losses and weight $(1-\alpha)$ on gains, the first condition requires $\alpha \geq \underline{\alpha} = g(30)/\big(g(30) - l(30) \big)$ and the second requires $\alpha \leq \overline{\alpha} = g(3)/\big(g(3) - l(3) \big)$, where $g(m)$ and $l(m)$ denote total surplus gains and losses at incentive $m$ for all subjects who participate in the transaction, respectively (see methods). The specific values of these bounds depend on the empirical distribution of gains and losses. Critically, it is possible that the lower bound $\underline{\alpha}$ exceeds the upper bound $\overline{\alpha}$. If so, it is not optimal to allow the transaction at capped incentives, no matter the weight on those who lose from the transaction.
Table \ref{welfareMain} shows the estimates of the bounds $\underline{\alpha}$, $\overline{\alpha}$, and the difference between them, depending on whether I use imputed reservation prices (columns 2 and 4), and on whether I account for the fact that reservation prices might be elicited with noise, which would cause an incorrect assignment to gains and losses for some observations (columns 3 and 4, see Methods). In each case, the difference between $\underline{\alpha}$ and $\overline{\alpha}$ is large, statistically highly significant, and precludes the existence of a weight on losses consistent with UIH-normative. These results apply separately for each insect species (Supplementary Information A.3). UIH-normative is violated throughout.
\subsection{Experiment 2: Choosing Between Advisors} \label{explicitInfo}\label{explicitInfoExperiment}\label{experiment1}
The previous experiment leaves open two questions. First, will the effect of participation incentives differ if we measure decision quality without the benefit of hindsight but based on the subject's information at the time of decision? Second, are there cases in which UIH-normative is more likely satisfied, such as when incentives for a transaction are paid immediately, but potential downsides only realize with a delay?
\paragraph{Design}
The experiment consists of two stages that each proceed in multiple rounds. 58 German student subjects completed this experiment, in addition to 348 who completed one of the extensions.
In each round of the first stage, a subject, endowed with \euro 110, decides whether to risk losing \euro 100 with a 50\% chance. A subject who agrees to take the risk receives an incentive of \euro $m$, where $m \in \{20, 30, 70, 80\}$ varies across treatments. I vary whether the subject may or may not keep that payment in case the loss materializes.
Crucially, before the subject decides whether to take the risk, but after she learns the amount $m$ she will receive in exchange for participation, she chooses between two computerized advisors (Figure \ref{infoChoiceExplicit}). They have imperfect foresight about whether the loss will occur. The `Bold Advisor' likely recommends taking the risk. He always recommends it if no loss will occur. With a 50\% chance, he also---erroneously---recommends taking the risk when it will cause a loss. The `Cautious Advisor' likely recommends rejecting the gamble. He always recommends rejection if the loss will materialize. With a 50\% change, he also---erroneously---recommends rejection if no loss will occur. UIH-positive predicts that subjects will more often opt for the Bold Advisor the higher $m$. Subjects make 18 decisions such as this, with one decision from the entire experiment randomly selected to determine the subjects' study payment (see methods).
\begin{figure}[htbp!]
\caption{Information choice in Experiment 2. \label{infoChoiceExplicit}}
\end{figure}
According to UIH-normative, a higher incentive causes worse decision-making. It may happen, for instance, that a \euro 80 incentive causes a subject to choose the Bold Advisor and take his recommendation at face value, unaware that the positive recommendation implies merely a $\frac{2}{3}$ success probability. (By Bayes' law, $P($ success $|$ positive recommendation from bold advisor) $ = \frac{0.5\cdot 1}{0.5\cdot 1 + 0.5\cdot 0.5}=\frac{2}{3}$.) If so, she will participate too often, against her own interest, if she is risk-averse.
I measure the presence and severity of such mistakes according to the subjects' own preferences. To this end, the second stage of the experiment presents subjects with a series of lotteries. Subjects reveal the certain amount of money they consider just as good as the lottery (the \emph{certainty equivalent}), elicited in a way that ensures truth-telling (see methods), for each lottery. Unbeknownst to the subject, each lottery corresponds to a decision from the first stage. For instance, consider a subject who, in the first stage, faced incentive $m=80$, which she would lose in case the risk materialized and who observed a positive recommendation from the Bold Advisor. The corresponding stage-2 decision is a win \euro 80 / lose \euro 100 gamble with a $\frac{2}{3}$ success probability, described as such. If the subject accepted the risk in stage 1 but reveals a negative certainty equivalent in stage 2, her stage-1 decision was mistaken, and its magnitude is its cost to the subject.  Were it positive, the certainty equivalent would show how much the subject profited from the opportunity to participate in the transaction at the given incentive.
As supporting evidence, each round of the first stage also elicits subjective beliefs about whether the loss will materialize, after the subject has seen the advisor's recommendation. Rational behavior requires that mean beliefs do not depend on the incentive amount (law of iterated expectations).
\begin{figure}[htbp]
\caption{\label{figureExplicit} Results from Experiment 2.}
\end{figure}
\paragraph{Information choice}
Consistent with UIH-positive, an increase in the incentive from \euro 30 or below to \euro 70 or above increases the choice of the Bold Advisor from 29.7\% to 47.4\% \red ($t$-test, $t=3.84$, $p=0.000$; Figure \ref{figureExplicit}A) \black. This effect substantially exceeds that of a condition in which subjects choose an advisor before knowing the incentive they will face \red($t$-test, $t=2.98$, $p=0.004$)\black. The change in advisor preference also affects subjects' beliefs that the gamble will pay off. If the incentive amount is \euro 30 or lower, mean subjective posteriors are 48.9\%, which is statistically indistinguishable from the Bayesian expected posterior of 50\% \red ($t$-test, $t = -0.49$, $p=0.627$)\black. If the incentive amount increases to \euro 70 or more, mean subjective posteriors increase by 7.7 percentage points \red($t$-test, $t=2.39$, $p=0.020$)\black, to 56.6\% \red(exceeds 50\%; $t$-test, $t=3.06$, $p=0.003$)\black. This effect is inconsistent with rational inference and suggests potentially harmful effects of higher incentives on welfare.
\paragraph{Welfare}
Panel B shows the effect of incentives on welfare. For incentives \euro 30 or lower, just under 20\% of decisions are mistaken. For incentives \euro 70 or more, the fraction is slightly higher. Yet, the magnitude of positive certainty equivalents increases greatly.
Because raising the incentive is not unambiguously good, I test UIH-normative formally. As in Experiment 1, I estimate the maximal weight one may place on losses while still deeming the low incentive acceptable as $\overline{\alpha} = \frac{g(20)}{g(20) + l(20)}$ and the minimal weight one must place on losses to deem the high incentive unacceptable as $\underline{\alpha} = \frac{g(80)}{g(80) + l(80)}$. Here, a subject's ex-ante gain $g$ or loss $l$ from a transaction she accepted in stage 1 equals her certainty equivalent for the corresponding lottery elicited in stage 2. If she rejected the transaction, the value is zero. I use seemingly unrelated regression to estimate mean gains and losses at \euro20 and \euro80 using all four incentive amounts (see methods).
I find that regardless of whether the incentive payment is lost if the risk materializes, the lower bound $\underline{\alpha}$ exceeds the upper bound $\overline{\alpha}$ by more than 50\% (Table \ref{welfareExp2}). Hence, regardless of the weight on losses from the transaction, it is never optimal to permit the transaction at \euro20 but not at \euro80. If the risk entails losing the incentive payment, preventing the transaction at either incentive is already prescribed for a lower weight on losses, but UIH-normative remains violated.
These results continue to hold when I replace the ex-ante welfare benchmark with a benchmark that measures whether decisions were good in hindsight, and when I control for noise in certainty equivalents (Supplementary Information B).
I replicate these results in an experiment with a lower overall stake size with 348 subjects (see methods \ref{methodsExp2}). I include treatments to address the following two concerns.
First, incentives may cause bad decision-making when the risk materializes much later than the incentive payment. In the corresponding treatment, subjects receive incentive payments immediately, but potential losses take effect only with a three-month delay (discounted from a fixed payment they would otherwise receive at that date). Second, I vary the prior risk of loss between 20\% and 80\%. This treatment will raise error rates, for example, for individuals subject to base rate neglect \citep{KahnemanTversky1974}. If such difficulties interact with the incentive amount, UIH-normative might be satisfied.
In these extensions, UIH-normative is robustly violated (Supplementary Information B). While some treatments increase choice mistakes, these changes do not strongly interact with the incentive amount. While such cases present a stronger argument for preventing the transaction altogether, they do not support capping incentives.
A separate experiment (Supplementary Information C) tests whether UIH-normative is more likely satisfied if subjects search through a large amount of information rather than observing a single recommendation from an advisor. There, too, choices satisfy UIH-positive but violate UIH-normative. In fact, higher incentives increase efforts to determine whether participation is the right choice.
\subsection{Effect of incentives on perfectly rational agents}\label{theory} \label{model}\label{rationalInfoChoice}\label{theorySetting}
To understand the simultaneous confirmation of UIH-positive and refutation of UIH-normative in the preceding experiments, consider an entirely rational decision maker who decides what information to consult before choosing whether to participate (formal model in Supplementary Information D). Because rational agents do not make mistakes, UIH-normative is violated by construction. Yet, this agent's information choice behavior will conform to UIH-positive.
The reason is intuitive. A decision maker who is constrained in how much information he can consult can make two types of mistakes. First, she may participate, even though she would have abstained under complete information. Second, she may abstain, even though she would have participated under complete information. A rational decision maker will select information sources that help her make more expensive mistakes less often. Incentives for participation change the costs of the two mistakes. With a low incentive, she has little to gain from participating but potentially much to lose. Therefore, she will rationally target her information search towards sources that help prevent mistaken participation, even if this comes at the cost of abstaining by mistake more often (the discouraging video and the Cautious Advisor in experiments 1 and 2, respectively). By contrast, if the incentive is high, she will target her search toward sources more likely to prevent mistaken abstention, even if that comes at the cost of participating by mistake more often (the encouraging video and the Bold Advisor). Mechanically, these information sources will more often recommend participation than those she seeks out under low incentives. Therefore, the rational decision maker's behavior is consistent with UIH-positive.
\section{Discussion}\label{conclusion}
Overall, this paper shows that incentives exert the behavioral effects the undue inducement literature hypothesizes. Yet, from a consequentialist point of view, these effects do not justify that literature's normative conclusions. A formal model of costly information acquisition provides a unifying explanation.
Given the potentially high costs of preventing voluntary transactions, \red experiments paralleling these reported here should be conducted in the field. Unless their results differ drastically from the current ones, \black the rules and guidelines restricting incentives due to undue inducement concerns \red should be reconsidered. \black
It is important to distinguish my results from two related intuitions. First, some authors worry that incentives would disproportionately attract the poor. Concerns about inequality are logically distinct from concerns with undue inducement. The latter deems high incentives problematic \emph{per se}; it applies even to a hypothetical world \emph{without} any inequality\citep{EmanuelNonsenseOnStilts}. I theoretically and experimentally examine the relation between inequality and incentives elsewhere\citep{AmbuehlStewartOckenfels}. A second concern is the (empirically partly disproved)\citep{TishlerBartholomae, Siddiqueetal.,CamposMercadeEtAl,Schneideretal.SideEffects} hypothesis that incentives decrease the supply of willing participants by lowering altruistic participation\citep{Titmuss, FreyJegen}. Concerns about supplied quantities are distinct from concerns about the welfare of the incentivized. UIH only addresses the latter.
Future research should test the robustness of my findings in other contexts, with different subject pools, and with transactions currently subject to undue-inducement-based regulation, such as female egg donation or gestational surrogacy. These cases might recruit psychological mechanisms not present in the current study, such as choking under pressure\citep{LargeStakesBigMistakes}. To address non-consequentialist concepts such as autonomy and consent, future research should also extend the empirical study of undue inducement beyond the welfarist framework used here.
\newpage
\section{Methods}
\subsection{Experiment 1}\label{methodsExp1}
\label{analysisInsects} \label{prelimAnalysis} \label{preliminary}
This research complies with all relevant ethical regulations. It was approved by Stanford University's Non-Medical IRB. Informed consent was obtained from each participant. All participants earned a completion payment of \$35, with the exception of 5 participants who reneged on their decision to eat insects. Those subjects' completion payment was reduced to \$15. In addition, each subject received payment in exchange for ingesting insects if they had decided to do so. Of the total of 671 subjects, 313 were women, 353 were men, and 5 were nonbinary. Mean age was 21.56 years.
\paragraph{Additional design information.}
To increase statistical power, each subject made choices about five different `food items' in each stage of the experiment: 2 house crickets, 5 large mealworms, 3 silkworm pupae, 2 mole crickets, 2 field crickets (see \ref{bugPics}). Even in countries such as China and Mexico, many people do not practice insect eating. Instead, it is concentrated within particular regions and communities. In my data, Asians and Hispanics are just as unwilling to eat insects as white Americans.
Subjects are not allowed to watch both videos. This design choice increases the chance of finding harmful effects of incentives. It reflects the fact that learning about transactions subject to undue inducement concerns typically costs significant effort and time, which constrains the amount of information that can be processed.
For each item subjects reveal their reservation price on a single page with 21 binary choices between the alternatives ``Get \$$p$. In exchange, eat the food item'' and ``Do not participate in this transaction'' where $p\in \{$1, 2, 3, 4, 6, 8, 10, 12.5, 15, 17.5, 20, 22.5, 25, 27.5, 30, 33, 36, 39, 44, 50, 60$\}$. (The amount \$3 was not included in the decision lists for the first 79 subjects.) Subjects click on the line at which they prefer to switch from refusing the transaction to accepting it. The remaining choices are filled in automatically.
To force subjects to view, touch, and smell each insect, within each container they sit atop a folded strip of paper with a code. Subjects must remove each strip and enter the code into the computer. During the handing out of the insects, subjects complete filler tasks during the handing out of the insects (an extended Cognitive Response Scale \citep{CRT2}, and sets D and E of Raven's standard progressive matrices\citep{Raven}).
\paragraph{Incentives}
Each subject makes multiple decisions. To incentivize truthful preference revelation, subjects learn that exactly one of all decisions will be randomly chosen for implementation at the end of the experiment. That decision entirely determines a subject's payment and consumption of insects.
Subjects know that their decision whether to eat insects in exchange for the initially promised amount will determine the outcome with 80\% probability. The first and second reservation price elicitation each determines the subject's outcome with a 7\% chance. With the remaining 6\% chance, the subjects' outcome is determined according to a final stage that comes as a surprise to the subject in which she predicts other subjects' reservation prices. To minimize social motives such as trying to impress other participants, subjects know that all insects will be consumed in a visually secluded space in the presence of only the experimenter, who ensures that the subject consumes the animals in their entirety.
Subjects make some consumption decisions before seeing the actual insects and thus may be unpleasantly surprised. To ensure that participants do not accept transactions expecting to renege on their choice, any subject who does not honor her commitment to eat the insects in exchange for money foregoes the promised amount and loses an additional \$20, discounted from a \$35 completion payment.
\paragraph{Data collection}
Each of the 671 subjects participated in one of 39 computerized sessions run on Qualtrics lasting about 2.5 hours each in early summer 2015 at the Ohio State University (499 subjects), Stanford University (110 subjects), and the University of Michigan (62 subjects), recruited using the laboratories' experimental economics participant databases. 271 subjects participated in the No Video condition (136 and 135 with \$3 and \$30 incentives, respectively), and 400 participated in the Video condition (197 and 203 with \$3 and \$30 incentives, respectively). Each session involved both payment conditions, but either all or none of the subjects in a session were in the Video condition and did not make decisions about field crickets. Subjects at Stanford in the Video condition (79 subjects) did not select video clips. Subjects knew that the Stanford IRB had approved the experiment and that all insects are produced for human consumption.
Five subjects (0.8\%) refused to follow through when one of their decisions to eat an insect was selected for implementation and paid the \$20 penalty. Of these subjects, all were in the \$30-condition, four were in the Video condition, and three had seen the encouraging video. Randomization into treatments was successful. Of 24 $F$-tests for differences in subjects' predetermined characteristics across the four treatments, only one is significant at the 5\% level (Supplementary Information A.1).
\paragraph{Analysis}
The calculation of $\underline{\alpha}$ and $\overline{\alpha}$ is based on the following Bergson-Samuelson social welfare function: $W_{\alpha} = \sum_{i=1}^n f_{\alpha}(w_i)$ with $f_{\alpha}(w) = \alpha$ if $w < 0$ and $f_{\alpha}(w) = (1-\alpha)$ if $w \geq 0$. This function is utilitarian if $\alpha = \frac{1}{2}$. The pair consisting of mean gain $g(m) = \frac{1}{n}\sum_{i=1}^n\max\big(w_i(m), 0\big)$, and mean loss $l(m) = \frac{1}{n}\sum_{i=1}^n\min\big(w_i(m),0\big)$ are a sufficient statistic for this class of welfare functions. The transaction is acceptable at incentive $m$ if welfare from permitting it, $(1-\alpha)g(m) + \alpha l(m)$ exceeds welfare from preventing it, which is zero. Rearranging this inequality at $m=3$ yields $\alpha \leq \frac{g(3)}{g(3)-l(3)} = \underline{\alpha}$. Rearranging the inequality with the opposite sign at $m=30$ yields $\alpha \geq \frac{g(3)}{g(3)-l(3)} = \overline{\alpha}$, as used in the main text.
Elicited reservation prices are right-censored at \$60. To calculate $\underline{\alpha}$ and $\overline{\alpha}$ in Table \ref{welfareMain}, I account for the censoring by fitting a lognormal distribution to the empirical distribution of reservation prices that exceed the median. I then replace the censored observations with the implied expected value. The predicted mean reservation prices for censored observations are \$107 and \$126 in the low and high incentive conditions, respectively.
To average out elicitation noise in columns 3 and 4 of Table \ref{welfareMain}, I predict each subject's reservation price for species $r$ using her choices for all remaining species. I use ridge regression with a penalty term selected using 10-fold cross-validation \citep{Hastieetal.ESL}. I re-calculate the implied bounds $\underline{\alpha}$ and $\overline{\alpha}$ based on these predictions.
\subsection{Experiment 2}\label{methodsExp2}
This research complies with all relevant ethical regulations. It was approved by the IRB of the Economics Department of the University of Zurich. Informed consent was obtained on screen.
Of the 58 subjects who completed this experiment, 38 were women and 20 were men, with a mean age of 27.05. Of the 348 subjects who completed the Extension Experiment, 208 were women, and 140 were men, with a mean age of 26.63.
\paragraph{Study structure}
Treatments vary along three dimensions, administered within-subject. (i) Incentive amount $m \in \{20, 30, 70, 80\}$. (ii) Information choice. In the \emph{Incentive First} condition subjects learn the incentive before selecting an advisor. The \emph{Advisor Choice First} condition asks subjects to choose an advisor \emph{before} learning the incentive amount, so they cannot tailor information choice to the specific incentive they will face. (iii) Downside. In the \emph{Limited downside} condition, a subject receives the participation payment $m$ even if the loss materializes. In the \emph{Large Downside} condition, a subject receives the participation payment $m$ only if the risk does not materialize.
In the first stage of the study, each subject completes 18 rounds in random order. 16 rounds present her with all combinations of (i) the four incentive amounts, (ii) the Incentive First and Advisor Choice First conditions, and (iii) the Limited and Large Downside conditions. The two remaining rounds serve as an attention check. They offer incentive $m=0$ in the Incentive First condition. Subjects do not learn the realization of any gamble in any round.
An elicitation of her posterior belief follows each round. Subjects select one of 12 bins, corresponding to 0\%, 1-10\%, 10-19\%, 20-29\%, \ldots, 80-89\%, 90-99\%, or 100\% certainty that the state in that round was good. Subjects can return from the belief elicitation stage to the betting stage to change their participation decision.
In the second stage of the study, subjects face 26 multiple decision lists consisting of 11 questions of the form ``Which of the following two options do you prefer?'' Alternative 1: ``\textit{Win \euro $m$ with $\gamma \cdot 100$\% chance, lose $L$ with $(1-\gamma)\cdot 100$\% chance}.'' Alternative 2: ``\textit{Receive \euro $c$ for sure}'' (if $c>0$) or ``\textit{Lose \euro $c$ for sure}'' (if $c<0$). Parameters $g, l$, and $q$ are fixed in each round while $p\in\{-85, -75, -50, -25, -15, 0,15, 25, 50, 75, 85\}$ across the eleven questions, in random order.
Each round corresponds to a different vector $(m, L, \gamma)$. For each of the four levels of incentive amounts $m$ and for each of the Limited and Large Loss conditions, the subject completes one decision list in which the success probability $\gamma$ equals the Bayesian posterior from a recommendation to participate by the Bold Advisor (2/3) and another one in which it equals the Bayesian posterior from a recommendation to abstain by the Cautious Advisor (1/3). Subjects also complete decision lists for the posteriors corresponding to a recommendation to abstain by the Bold Advisor (which is zero) and corresponding to a recommendation to participate by the Cautious Advisor (which is one). For each of these degenerate posteriors, subjects complete four lists corresponding to $m=30$ and $m=70$ for each of the Limited and Large Loss conditions.
\paragraph{Randomization}
Within each stage of the study, rounds are randomized at the individual level. The advisors are presented next to each other. For a random half of subjects, the Bold Advisor is displayed on the left, and for the remaining half on the right. Decisions in each list in stage 2 appear in random order, drawn anew in each round.
\paragraph{Framing}  In stage 1 of the experiment, the subject decides whether to accept a `venture' that may either `succeed' or `fail,' in exchange for a `venture participation payment.' If the venture succeeds, she can keep the venture participation payment, and no further consequences occur. If the venture fails, she must `pay damages.' Advisor recommendations read either ``The [type] advisor recommends: Participate in the venture!'' accompanied by a thumbs-up symbol on a green background, or ``The [type] advisor recommends: Don't participate in the venture!'' accompanied by a thumbs-down symbol on a red background.  Each survey round alerts the subject that a new state is drawn using a short animation that requires the subject to click to shuffle and randomly pick an unobserved outcome.
\paragraph{Incentives}
The study payment is entirely determined by a single, randomly drawn decision, which incentivizes truthful revelation of preferences. With a 2/3 chance, that decision is from stage 1. With the remaining 1/3 chance, it is from stage 2. In the case of the former, the payment is determined with an 80\% chance by the decision whether to accept the transaction and whether the loss materializes. With the remaining 20\% chance, it is determined by the belief elicitation, which is incentivized by the probabilistic quadratic scoring rule \citep{HoltSmith}.  Incentive compatibility only requires that subjects know that any single decision could determine their payment, not that subjects remember the implementation probabilities.
\paragraph{Extensions}
Using smaller overall stake sizes, I test whether the timing of consequences and the prior success probability affects conclusions about UIH-normative. Subjects receive a completion payment of \euro15. Subjects risk losing \euro $6$ by accepting a transaction. They face incentive amounts $m \in \{1, 2, 4, 5\}$. The second stage uses certain amounts  $\{$-6, -4.5, -3, -1.5, -0.5, 0, 0.5, 1.5, 3, 4.5, 6$\}$ across 11 questions on each page. For context, the PPP exchange rate in 2020 was \euro 1 = \$1.38. The University of Cologne estimates monthly student living expenses of \euro 832.
To maintain comparability to the main experiment, I replicate that experiment with the lower stake size. In addition to the three within-subject treatments, I administer the following treatments across subjects:
(iv) Prior risk probabilities  $\mu \in \{0.2, 0.5, 0.8\}$. The risk remains constant throughout the study for any given subject.
(v) Delay condition. Subjects receive incentive payments immediately, but potential losses take effect only with a three-month delay. Delayed losses are discounted from a \euro 6 payment that subjects otherwise receive with a three-month delay. The participation payment disbursed to these subjects immediately is \euro 9, thus bringing the total unconditional payment to \euro 15, as is the case for all other subjects. In stage 2 of the study, certain payments are presented as ``\textit{Your immediate payment rises [falls] by \euro $c$ for sure.}'' In the Large Loss condition, the lottery reads ``\textit{With $\gamma \cdot 100$\% chance: your immediate payment rises by \euro $m$, and your delayed payment stays unchanged. With $(1-\gamma)\cdot 100$\% chance: your immediate payment stays unchanged, and you lose \euro $L$ of your delayed payment.}'' In the Limited Loss condition, the text ``\textit{your immediate payment stays unchanged}'' is replaced by ``\textit{your immediate payment rises by \euro $m$}.''
\paragraph{Data collection} I conducted the study online with the Qualtrics research survey tool using the subject pool and procedures of the Cologne Laboratory for Economic Research at the University of Cologne, disbursing payment through PayPal. 58 unique subjects participated on February 7, 2022. In addition. 348 unique subjects in the Regular Stakes condition completed the experiment on November 25-27, 2020. In the contemporaneous consequences condition, 53, 64, and 58 subjects participated with a prior success probability of 20, 50, and 80\%, respectively. The corresponding subject counts for the delayed consequences condition are 64, 50, and 59. Subjects took between 45 and 90 minutes to complete the study. All subjects gave informed consent to participate in this study on screen.
All subjects had to pass two comprehension checks before making decisions. The first check required subjects to correctly mark each of nine statements as true or false. The second check consisted of six such statements. Subjects who answered incorrectly did not receive feedback on which statement was mismarked. Hence, subjects were highly unlikely to pass by chance or trial and error. Subjects were asked to review the instructions until they passed the comprehension checks.
\label{AnalysisExperiment1}
\paragraph{Attrition}
There was no attrition in the main experiment. In the Extension Experiment, the 348 complete responses make up 95.6\% of the 364 surveys started on a machine satisfying the technical requirements (desktop or laptop computers with sufficient screen width).
Attrition is unrelated to treatment. Importantly, the crucial variation of incentive amounts and information order occurs within subject.
\paragraph{Certainty equivalents}
Some subjects choose a certain amount $c$ over a given lottery but choose that lottery over some larger sure amount $c'$. Averaging across all subjects, such non-monotonicities occur in 2.3 of the 26 rounds (median: 2). The corresponding number in the Extension Experiment is 2.9 (median: 2). In case of a nonmonotonicity, I define the subject's certainty equivalent as the lowest certain amount the subject prefers to the lottery.
(Results are qualitatively unchanged if I only include observations that respect monotonicity.)
\paragraph{Estimation of UIH-normative}
To use all four incentive amounts in the formal test of UIH-normative, I estimate the following linear system using seemingly unrelated regression:
\bmath \label{gainsLossesSystem}
\begin{array}{r@{}l}
g_i(m) &{}= \beta_0^G + \beta_1^G m + \epsilon^G_i(m) \\
l_i(m) &{}= \beta_0^L + \beta_1^L m + \epsilon^B_i(m).
\end{array}
\vspace{-0.4cm}
\emath
Here, $\epsilon^G_i(m)$ and $\epsilon^B_i(m)$ are independent individual effects with expectation zero such that $g_i(m)\geq 0 \geq l_i(m)$ for all $i$. I impute $g(m) = \hat{\beta}_0^G + \hat{\beta}_1^G m$ and $l(m) = \hat{\beta}_0^L + \hat{\beta}_1^L m$, where $\hat{\beta}_i^j$ indicates a parameter estimate, to calculate the bounds $\underline{\alpha}$ and $\overline{\alpha}$ using $m=20$ and $m=80$ for the low and high incentive amounts, respectively.
\section{Data availability}
The datasets generated and analyzed during the current study are available in a replication package on the Harvard Dataverse, \url{https://doi.org/10.7910/DVN/3PFZKP}.
\section{Code availability} The replication package includes all Stata code to replicate the statistical analysis. It is available at \url{https://doi.org/10.7910/DVN/3PFZKP}.
\section{Acknowledgments}
I am deeply indebted to my Ph.D. advisors Muriel Niederle, B. Douglas Bernheim, and Alvin E. Roth. I am grateful to Yan Chen, P.J. Healy, and Axel Ockenfels for providing experimental laboratory resources and to Eva K\"upper for excellent research assistance.  Experiment 1 was funded by the Department of Economics at Stanford University. Experiment 2 was funded by the Department of Economics at the University of Zurich. Moreover, this project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 741409); the results reflect the authors' view; the ERC is not responsible for any use that may be made of the information it contains. The funders had no role in study design, data collection and analysis, decision to publish or preparation of the manuscript.
\section{Author Contributions Statement}
S.A. conceived, designed, executed, and analyzed the experiments and wrote the paper.
\section{Competing Interests Statement}
The author declares no competing interests.
\section{Tables}
\begin{table}[htb]
\caption{Tests for UIH-normative in Experiment 1 \label{welfareMain}}
\begin{center}
\begin{tabular}{lcccc}
\toprule
& (1) & (2) & (3) & (4) \\
Alternative welfare benchmark& &  \checkmark&  & \checkmark \\
Noise correction & &  &\checkmark & \checkmark \\
\midrule
Weight on losses \\
\quad min for \$30 unacceptable, $\underline{\alpha}$ &     0.915 &     0.801 &     0.677 &     0.618  \\
& (    0.016) & (    0.034) & (    0.036) & (    0.038)  \\
\quad max for \$3 acceptable, $\overline{\alpha}$ &     0.183 &     0.000 &     0.134 &     0.000  \\
& (    0.046) & (    0.000) & (    0.035) & (    0.000)  \\
UIH-normative satisfied & No & No & No & No \\
$ p$-value $\underline{\alpha} = \overline{\alpha}$ &     0.000 &     0.000 &     0.000 &     0.000  \\
Subjects &       400 &       400 &       400 &       400  \\
Observations &      1921 &      1921 &      1921 &      1921  \\
\bottomrule
\end{tabular}
\end{center}
\footnotesize
\textbf{\textit{Notes:}} UIH-normative requires $\underline{\alpha}\leq\overline{\alpha}$. Columns labeled \emph{noise correction} predict a subject's reservation price for a species of insect using her reservation prices for the remaining four species in a ridge-regression. Columns labeled \emph{alternative welfare benchmark} use reservation prices imputed from the No-Video condition at the \$3 incentive. Standard errors in parentheses, clustered by subject. $p$-values concern two-sided $z$-tests of the null hypothesis that $\underline{\alpha} = \overline{\alpha}$.
\end{table}
\begin{table}[htbp!]
\caption{Tests for UIH-normative Experiment 2 \label{welfareExp2}}
\begin{center}
\begin{tabular}{lcc} \toprule
&(1) & (2)  \\
& \multicolumn{2}{c}{Treatment} \\
\cmidrule(r){2-3}
Potential downside & \euro 100 & \euro 100 - incentive \\
\midrule
Weight on losses \\
\quad	min	for	\$30	unacceptable,	$\underline{\alpha}$	&	0.605***	&	0.818***	\\
	&	(	0.068)	&	(	0.066)	\\
\quad	max	for	\$3	acceptable,	$\overline{\alpha}$	&	0.386***	&	0.451***	\\
	&	(	0.106)	&	(	0.120)	\\
$p$-value	$\underline{\alpha}	=	\overline{\alpha}$	&	0.084	&	0.004	\\
UIH-normative	satisfied	&	No	&	No	\\
Subjects	&	58	&	58	\\
Observations	&	232	&	232	\\
\\
\bottomrule
\end{tabular}
\end{center}
\footnotesize \textbf{\textit{Notes:}} UIH-normative requires $\underline{\alpha}\leq\overline{\alpha}$. Reported significance levels concern tests of the hypotheses $\underline{\alpha} = 0$ and $\overline{\alpha} = 1$, respectively. Standard errors in parentheses, clustered by subject. $p$-values concern two-sided $z$-tests of the Null hypothesis that $\overline{\alpha} = \underline{\alpha}$.
\end{table}
\FloatBarrier
\section{Figure legends / captions}
\begin{itemize}
\item Figure \ref{bugPics}: Subjects do \emph{not} see the insects or pictures of them until after making participation decisions. A. House cricket. B. Mole cricket.  C. Field cricket. D. Mealworm. E. Silkworm pupa.
\item Figure \ref{insectFig}: A: Information choice. Left half: fraction of subjects who choose the encouraging video, $400$ independent subjects. Right half: mean number of clips labeled `pro' and `con' chosen, respectively, $321$ observations from 321 independent subjects. B: Fraction of subjects willing to eat the food item for the promised incentive, averaged across species, by incentive and video condition, 3307 observations from 671 independent subjects. C: Cumulative distribution functions of ex-post surplus by incentive, using data from the Video condition. Data are left-censored because reservation prices are right-censored at \$60. D: Replication of C with the distribution of reservation prices imputed from the No-Video treatment with \$3 incentive. Whiskers display 95\% confidence intervals. Standard errors clustered by subject.
\item Figure \ref{infoChoiceExplicit}: Screenshot of subjects' interface for the case $m=80$ which is forfeited in case the loss realizes.
\item Figure \ref{figureExplicit}: A: Fraction of subjects choosing the Bold Advisor depending on whether the incentive amount is known when choosing. 928 observations from 58 independent subjects. Whiskers display 95\% confidence intervals. Standard errors clustered by subject. B. Cumulative distribution functions of ex-ante surplus by incentive.
\end{itemize}
\newpage
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\end{document}


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\begin{document}
\begin{appendix}
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\section{SI Experiment 1}
\subsection{Randomization check}\label{randomizationCheck}\label{additionalAnalysis}
The four treatments are balanced across demographic characteristics. Table \ref{randCheck} displays summary statistics of these variables by treatment. For each variable, the table reports the $p$-value of an $F$-test for differences in the variable's mean value across treatments. Of 24 tests conducted, one is significant at the 5\% level.
\begin{table}[htbp!]
\caption{\label{randCheck} Summary statistics and randomization check.}
\begin{minipage}{\textwidth}
\small
\setlength{\tabcolsep}{3pt}
\begin{tabular}{lccccccc}
\toprule
& (1)& (2) & (3) & (4) & (5) \\
\emph{Treatment condition} \\
\hspace{0.5cm}Incentive & \$30 & \$3 & \$30 & \$3 \\
\hspace{0.5cm}Video & Yes & Yes & No & No \\
\\
Variable &\multicolumn{4}{c}{Mean}& $p$-value \\
\midrule
Male&0.55&0.53&0.54&0.54&0.997&\\
Age&21.43&22.01&21.37&21.30&0.337&\\
\emph{Ethnicity} \\
\hspace{0.5cm}African-American&0.05&0.06&0.07&0.07&0.701&\\
\hspace{0.5cm}Caucasian&0.57&0.51&0.59&0.56&0.280&\\
\hspace{0.5cm}East Asian&0.19&0.26&0.19&0.23&0.224&\\
\hspace{0.5cm}Hispanic&0.07&0.08&0.04&0.04&0.987&\\
\hspace{0.5cm}Indian&0.03&0.04&0.04&0.07&0.557&\\
\hspace{0.5cm}Other&0.08&0.05&0.07&0.04&0.703&\\
Monthly spending in USD &251.72&301.40&289.07&288.42&0.443&\\
Year of study &3.50&3.60&3.61&3.47&0.317&\\
Graduate student&0.13&0.15&0.13&0.05&0.068&\\
\emph{Field of study} \\
\hspace{0.5cm}Arts and humanities&0.16&0.09&0.13&0.11&0.042&\\
\hspace{0.5cm}Business or economics&0.27&0.36&0.34&0.43&0.086&\\
\hspace{0.5cm}Engineering&0.20&0.16&0.11&0.12&0.490&\\
\hspace{0.5cm}Science&0.21&0.23&0.27&0.23&0.466&\\
\hspace{0.5cm}Social science (excluding business and economics)&0.17&0.17&0.15&0.11&0.593&\\
Political orientation &0.50&0.32&0.27&0.09&0.081&\\
Raven's score&14.77&14.76&14.69&14.68&1.000&\\
CRT score&3.76&3.80&3.50&3.22&0.078&\\
\emph{Experience with insects as food (1 = Yes, 0 = No)} \\
\hspace{0.5cm}Has intentionally eaten insects before&0.19&0.22&0.19&0.20&0.715&\\
\hspace{0.5cm}Grown up in culture that practices entomophagy&0.15&0.14&0.13&0.15&0.925\\
\hspace{0.5cm}Grown up eating mostly western foods&0.81&0.73&0.82&0.78&0.057&\\
\hspace{0.5cm}Had a pet that feeds on store-bought insects&0.25&0.25&0.21&0.26&0.684&\\
\hspace{0.5cm}Knew that this study concerns insect eating&0.20&0.30&0.26&0.29&0.231\\
\bottomrule
\end{tabular}
\end{minipage}
\footnotesize
\textbf{\textit{Notes}:} $p$-values in column (5) correspond to an $F(3, N-4)$-test of the joint significance of a regression of the indicated variable on treatment dummies where $N$ is the number of non-missing observations for each listed characteristic. Year of study only includes undergraduate students. Political orientation is measured on a scale of -2 (conservative) to 2 (liberal). Raven's score is measured on a scale of 0 to 24. CRT score indicates the number of correct answers (out of 6) on an extended version of the CRT test\cite{CRT2}.
\end{table}
\FloatBarrier
\subsection{Reservation prices}\label{reservationPricesInsects}
Table \ref{reservationPrices} displays mean reservation prices. Because elicited reservation prices are interval coded and censored above at \$60, I estimate these means using interval regression which accounts for both issues. These means represent substantial subject heterogeneity. Population standard deviations exceed \$20 for each species both before and after the handout of the insects.
\begin{table}[htb!]
\setlength{\tabcolsep}{3pt}
\begin{center}
\caption{Reservation prices for eating insects \label{reservationPrices}}
\begin{tabular}{lcccccc} \toprule
 & (1) & (2) & (3) & (4) & (5) & (6) \\
VARIABLES & \multicolumn{6}{c}{Reservation price} \\
\cmidrule(r){2-7}
Species	& All & 2 house & 5 meal- & 3 silkworm  & 2 mole  & 2 field  \\
		&      & crickets & worms & pupae & crickets & crickets \\ \midrule
\textit{Levels}\\
\quad Before & 23.976*** & 20.283*** & 28.084*** & 23.678*** & 24.039*** & 23.769*** \\
 & (0.847) & (0.844) & (0.927) & (0.889) & (0.901) & (0.953) \\
\quad After & 26.164*** & 23.957*** & 27.447*** & 26.210*** & 30.719*** & 21.995*** \\
  & (0.874) & (0.919) & (0.940) & (0.922) & (0.954) & (0.928) \\
\textit{Difference}
 & 2.188*** & 3.674*** & -0.637 & 2.532*** & 6.680*** & -1.774*** \\
 & (0.342) & (0.484) & (0.494) & (0.470) & (0.526) & (0.520) \\
 &  &  &  &  &  &  \\
Observations & 6,552 & 1,342 & 1,342 & 1,342 & 1,342 & 1,184 \\
Subjects & 671 & 671 & 671 & 671 & 671 & 592 \\ \bottomrule
\end{tabular}
\end{center}
\footnotesize \textit{Notes:} Mean reservation prices, estimated by interval regression. Standard errors in parentheses, clustered by subject. The number of observations in column 6 is smaller because the 79 Stanford subjects who first participated in this experiment were not given any decisions regarding field crickets.
\end{table}
Column 1 of Table \ref{ethnicity} shows the ethnic makeup of the subject population. Columns 2 to 3 examine whether subjects of different ethnicities differ in terms of takeup of the promised offer, as well as reservation prices before and after the handout of the insects, respectively. The only statistically significant effect concerns Indians who have substantially higher reservation prices.
\begin{table}[htb!]
\setlength{\tabcolsep}{3pt}
\begin{center}
\caption{Effect of subjects' ethnicity on participation and reservation prices \label{ethnicity}}
\begin{tabular}{lcccc} \toprule
 & (1) & (2) & (3)  \\
VARIABLES &  Offer accepted & \multicolumn{2}{c}{Reservation price} \\
\cmidrule(r){3-4}
&&before handout& after handout \\ \midrule
 &  &  &  \\
\textit{Ethnicity} \\
\quad Black & 0.035 & 2.098 & 1.957 \\
 & (0.067) & (4.633) & (4.982) \\
\quad Asian & -0.061 & 5.538* & 3.247 \\
 & (0.039) & (2.955) & (3.189) \\
\quad Hispanic & 0.083 & -2.945 & -4.300 \\
 & (0.070) & (4.898) & (4.993) \\
\quad Indian & -0.123 & 15.375** & 17.552** \\
 & (0.081) & (6.529) & (7.062) \\
\quad Other & 0.005 & 1.059 & 2.190 \\
 & (0.074) & (5.170) & (5.988) \\
High incentive & 0.283*** &  &  \\
 & (0.032) \\
Constant & 0.386*** & 24.865*** & 28.988*** \\
 &   (0.028)&  (1.557) &  (1.738)    \\
 &  &  &  \\
Observations & 3,276 & 3,276 & 3,276 \\
 Subjects & 671 & 671 & 671 \\ \bottomrule
\end{tabular}
\end{center}
\footnotesize \textit{Notes:} Column 1 estimated by OLS. Columns 2 and 3 estimated with interval regression. White is the omitted category. Standard errors in parentheses, clustered by subject.
\end{table}
\subsection{Welfare analysis by species}\label{welfareBySpecies}\label{lassoWelfareExperiment2}
The analysis of welfare effects in Section 2.1 pools across species. Table \ref{welfareExp2main} performs parallel analysis separately for each species. Its Panels A to D correspond to columns 1 to 4 of Table 1, respectively. Columns 1 and 2 show mean welfare gains and losses, respectively, at the \$3 incentive. Column 3 shows the maximum weight $\underline{\alpha}$ that can be placed on those who lose from the transaction while still keeping the transaction admissible at \$3. Each panel's estimates are far below the utilitarian weight $\alpha = 0.5$ for each species. Columns 1 and 2 show mean welfare gains and losses, respectively, at the \$30 incentive. Column 6 shows the minimum weight that needs to be placed on those who lose from the transaction to keep the transaction inadmissible at \$30. The estimated thresholds are far above $\underline{\alpha}$ in each case. Hence, the condition $\underline{\alpha}<\overline{\alpha}$ is violated for each species ($t$-tests, $p<0.001$ in each case), contrary to UIH-normative. These results are particularly strong in panels B and D, which use ridge regression to average out elicitation error of reservation prices. In both panels, predicted mean welfare gains at the \$3 incentive are 0 for each species.
\afterpage{
\thispagestyle{empty}
\begin{table}[htp!]
\setlength{\tabcolsep}{2pt}
\footnotesize
\caption{Welfare effects by species in Experiment 1 \label{welfareExp2main}}
\begin{center}
\begin{tabular}{lcccccccc} \toprule
&(1) & (2) & (3) & (4) & (5) & (6) & (7) & (8)  \\
Variable & $g(\underline{m})$ & $l(\underline{m})$  & $\overline{\alpha}$  & $g(\overline{m})$ & $l(\overline{m})$  & $\underline{\alpha}$ & \multicolumn{2}{c}{UIH-normative} \\
\cmidrule(r){8-9}
&&&&&&&satisfied & $p$-value \\ \midrule
\multicolumn{8}{l}{\textbf{\textit{A. Within-treatment benchmark}}} \\
2 house crickets    &		0.310***	  &  	-1.914**	  &  	0.139***	  &  	13.596***	  &  	-1.699**	  &  	0.889***	  &  	No	 & 	0.000	 \\
&	(0.056) & (0.821) & (0.057) & (0.845) & (0.775) & (0.047) & 		 & 		 \\
5 superworms        &		0.254***	  &  	-1.011**	  &  	0.201***	  &  	10.372***	  &  	-0.325**	  &  	0.970***	  &  	No	 & 	0.000	 \\
&	(0.050) & (0.495) & (0.086) & (0.865) & (0.153) & (0.014) & 		 & 		 \\
3 silkworm pupae    &		0.269***	  &  	-2.691***	  &  	0.091***	  &  	12.248***	  &  	-2.330***	  &  	0.840***	  &  	No	 & 	0.000	 \\
&	(0.051) & (0.903) & (0.033) & (0.847) & (0.872) & (0.053) & 		 & 		 \\
2 mole crickets     &		0.239***	  &  	-4.554***	  &  	0.050***	  &  	11.869***	  &  	-9.202***	  &  	0.563***	  &  	No	 & 	0.000	 \\
&	(0.050) & (1.313) & (0.018) & (0.842) & (2.038) & (0.062) & 		 & 		 \\
2 field crickets    &		0.358***	  &  	-2.803***	  &  	0.113***	  &  	10.570***	  &  	-1.483**	  &  	0.877***	  &  	No	 & 	0.000	 \\
&	(0.060) & (1.032) & (0.042) & (0.833) & (0.679) & (0.051) & 		 & 		 \\
\multicolumn{8}{l}{\textbf{\textit{B. Within-treatment benchmark, predicted by ridge regression}}} \\
2 house crickets    &		0.000		  &  	-2.979***	  &  	0.000***	  &  	13.781***	  &  	-2.321***	  &  	0.856***	  &  	No	 & 	0.000	 \\
&	(0.556) & (0.724) & (0.187) & (0.548) & (0.713) & (0.039) & 		 & 		 \\
5 superworms        &		0.000		  &  	-2.439***	  &  	0.000***	  &  	9.550***	  &  	-0.159		  &  	0.984***	  &  	No	 & 	0.000	 \\
&	(0.479) & (0.322) & (0.196) & (0.472) & (0.317) & (0.032) & 		 & 		 \\
3 silkworm pupae    &		0.000		  &  	-4.409***	  &  	0.000***	  &  	10.945***	  &  	-3.368***	  &  	0.765***	  &  	No	 & 	0.000	 \\
&	(0.497) & (0.836) & (0.113) & (0.489) & (0.823) & (0.046) & 		 & 		 \\
2 mole crickets     &		0.000		  &  	-6.993***	  &  	0.000***	  &  	8.384***	  &  	-5.535***	  &  	0.602***	  &  	No	 & 	0.000	 \\
&	(0.429) & (1.037) & (0.061) & (0.422) & (1.022) & (0.049) & 		 & 		 \\
2 field crickets    &		-0.000		  &  	-3.227***	  &  	-0.000***	  &  	11.204***	  &  	-2.040***	  &  	0.846***	  &  	No	 & 	0.000	 \\
&	(0.589) & (0.715) & (0.182) & (0.580) & (0.704) & (0.046) & 		 & 		 \\
\multicolumn{8}{l}{\textbf{\textit{C. No Video, \$3 incentive benchmark}}}\\
2 house crickets    &		0.363***	  &  	-2.218***	  &  	0.141***	  &  	12.940***	  &  	-5.039***	  &  	0.720***	  &  	No	 & 	0.000	 \\
&	(0.060) & (0.860) & (0.052) & (0.873) & (1.388) & (0.061) & 		 & 		 \\
5 superworms        &		0.312***	  &  	-1.571**	  &  	0.166***	  &  	10.038***	  &  	-3.307***	  &  	0.752***	  &  	No	 & 	0.000	 \\
&	(0.058) & (0.611) & (0.061) & (0.867) & (1.078) & (0.066) & 		 & 		 \\
3 silkworm pupae    &		0.327***	  &  	-3.441***	  &  	0.087***	  &  	11.063***	  &  	-10.169***	  &  	0.521***	  &  	No	 & 	0.000	 \\
&	(0.058) & (1.045) & (0.029) & (0.860) & (2.227) & (0.063) & 		 & 		 \\
2 mole crickets     &		0.266***	  &  	-5.659***	  &  	0.045***	  &  	9.607***	  &  	-31.724***	  &  	0.232***	  &  	No	 & 	0.000	 \\
&	(0.053) & (1.539) & (0.015) & (0.832) & (4.515) & (0.035) & 		 & 		 \\
2 field crickets    &		0.404***	  &  	-1.331***	  &  	0.233***	  &  	12.831***	  &  	-0.251		  &  	0.981***	  &  	No	 & 	0.000	 \\
&	(0.064) & (0.438) & (0.068) & (0.887) & (0.144) & (0.011) & 		 & 		 \\
\multicolumn{8}{l}{\textbf{\textit{D. No Video, \$3 incentive benchmark, predicted by ridge regression}} }\\
2 house crickets    &		-0.000		  &  	-3.827***	  &  	-0.000***	  &  	9.049***	  &  	-4.077***	  &  	0.689***	  &  	No	 & 	0.000	 \\
&	(0.499) & (0.785) & (0.131) & (0.492) & (0.773) & (0.045) & 		 & 		 \\
5 superworms        &		-0.000		  &  	-3.917***	  &  	-0.000***	  &  	5.052***	  &  	-1.821***	  &  	0.735***	  &  	No	 & 	0.000	 \\
&	(0.353) & (0.517) & (0.090) & (0.347) & (0.509) & (0.058) & 		 & 		 \\
3 silkworm pupae    &		0.000		  &  	-5.269***	  &  	0.000***	  &  	7.042***	  &  	-5.446***	  &  	0.564***	  &  	No	 & 	0.000	 \\
&	(0.428) & (0.901) & (0.081) & (0.422) & (0.888) & (0.046) & 		 & 		 \\
2 mole crickets     &		-0.000		  &  	-8.758***	  &  	-0.000***	  &  	3.190***	  &  	-10.344***	  &  	0.236***	  &  	No	 & 	0.000	 \\
&	(0.260) & (1.168) & (0.030) & (0.256) & (1.151) & (0.028) & 		 & 		 \\
2 field crickets    &		0.000		  &  	-2.367***	  &  	0.000***	  &  	10.974***	  &  	-0.130		  &  	0.988***	  &  	No	 & 	0.000	 \\
&	(0.529) & (0.273) & (0.224) & (0.522) & (0.269) & (0.024) & 		 & 		 \\
\bottomrule
\end{tabular}
\end{center}
{\scriptsize \textbf{\textit{Notes:}} Each row corresponds to a separate SUR-regression. Columns 1 and 2 show mean welfare gains and losses, respectively, at the \$3 incentive. Column 3 shows the maximal weight $\overline{\alpha}$ that may be placed on those who lose from the transaction such the transaction is permissible at the \$3 incentive. Columns 4 and 5 show mean welfare gains and losses, respectively, at the \$30 incentive. Column 6 shows the minimal weight $\underline{\alpha}$ that must be placed on those who lose from the transaction such that the transaction is inadmissible at the \$30 incentive. Reported significance levels in columns 3 and 6 reflect the two-sided $z$-tests that $\overline{\alpha} = 1$ and $\underline{\alpha} = 0$, respectively. Column 7 indicates whether $\overline{\alpha} \geq \underline{\alpha}$, a necessary condition for UIH-normative. Column 8 lists $p$-values for two-sided $z$-tests of the Null hypothesis that $\overline{\alpha} =\underline{\alpha}$. Standard errors in parentheses, clustered by subject. Standard errors in columns 3, 6, and 8 are calculated by the Delta method.}
\end{table}
\clearpage
}
One potential critique of Experiment 1 concerns the fact that subjects in this experiment have no opportunity to express whether they derive positive utility from eating insects and if so, by how much. Considering non-monetary benefits from the transaction is especially important in light of the position, common in the undue inducement literature, that individuals should be allowed to participate in transactions only if they are intrinsically motivated \citep{LondonDismal}. To address this point, I estimate the smallest utility benefit that subjects would need to derive from eating insects such that UIH-normative is satisfied ($\underline{\alpha}\leq \overline{\alpha}$). Let $\Delta$ denote the amount by which $g(m)$ needs to increase to satisfy the condition. By random assignment of subjects to incentive conditions, $\Delta$ is independent of $m$. Based on the estimates of columns (1), (3), or (4), UIH-normative cannot be satisfied regardless of potential utility benefits because  $l(3)<l(30)$ and $g(3)<g(30)$ imply that $W_\alpha(3)<W_\alpha(30)$ for any $\Delta \geq 0$ and any $\alpha \in [0,1]$. This argument does not apply, however, to the estimates in column (2), where $l(3)>l(30)$. In this case, the minimum increase required to satisfy UIH-normative is $\Delta = \frac{g(30)l(3)-g(3)l(30)}{l(30)-l(3)} = 7.67$. This average utility benefit must be generated by the 4.3 percent of observations in which a subject agreed to eat the insects for free. Hence in these cases, subjects would need to derive the implausibly high utility of at least \$$7.67/0.043 = \$178.37$ from bug-eating to satisfy UIH-normative.
\FloatBarrier
\newpage
\section{SI Experiment 2}

\paragraph{Replication of Experiment 2 in Extension Experiment}
\label{appendixAdvisorsExperiment}
Figure \ref{appFigureExplicitWelfare} compares results from Experiment 2 and the Extension Experiment. For comparability, I only use data corresponding to the 50\% prior without delayed consequences in this comparison. Panels A and B plot information choice. Panels C and D show posterior beliefs. Panels E and F plot distribution functions of individual welfare. Information choice patterns appear similar across the experiments. So do posterior belief distributions. In both cases, higher incentives cause more optimistic posterior beliefs. The latter effect has two possible causes. First, subjects might have different biases when interpreting information from the Bold Advisor than from the Cautious Advisor. Second, the high incentive might affect how subjects interpret recommendations from a given advisor. Data from the Advisor Choice First treatment suggests that the first mechanism almost entirely drives the increase in posteriors.
Critically, the welfare distributions are also highly similar across the experiment. If anything, choice mistakes increase less when the stakes are larger.
These comparisons indicate that the effects in my experiment are driven by the magnitude of the incentive {relative} to the potential loss, which is approximately constant across the two panels, rather than by the stake size, which varies by an order of magnitude.
\begin{figure}[htb!]
\caption{Comparison Experiment 2 and Extension Experiment.
 \label{appFigureExplicitWelfare}}
\begin{center}
\begin{tabular}{cc}
\multicolumn{2}{c}{Information choice} \\
\midrule
\textbf{A. Experiment 2} &
\textbf{B. Extension Experiment}\\
\includegraphics[width = 0.5\textwidth]{graphs/SIexp2infoChoice} &
\includegraphics[width = 0.5\textwidth]{graphs/infoChoiceExplicitChoiceComparison}\\
\multicolumn{2}{c}{Subjective posterior beliefs} \\
\midrule
\textbf{C. Experiment 2} &
\textbf{D. Extension Experiment}\\
\includegraphics[width = 0.45\textwidth]{graphs/exp2posteriors} &
\includegraphics[width = 0.45\textwidth]{graphs/subjectivePosteriorsCDF} \\
\multicolumn{2}{c}{Welfare} \\
\midrule
\textbf{E. Experiment 2} &
\textbf{F. Extension Experiment}\\
\includegraphics[width = 0.5\textwidth]{graphs/SIexp2welfareCDF.pdf} &
\includegraphics[width = 0.5\textwidth]{graphs/welfareCDFsimpleComparison}\\
\end{tabular}
\end{center}
\footnotesize
\textit{\textbf{Notes:}} Panels on the left replicate the corresponding graphs from the main article. To maintain comparability, the panels on the right show the corresponding graphs from the Extension Experiment when the prior risk is 50\% and consequences are not delayed. Whiskers in Panels A and B display 95\% confidence intervals. Standard errors clustered by subject. Panel A is based on 928 observations from 58 independent subjects. Panel B is based on 5568 observations from 348 independent subjects.
\end{figure}
\paragraph{UIH-normative: Additional results}

I next study the effects of the various treatments on UIH-normative. Panel A of Table \ref{welfareExp2appendixMain} replicates Table 2 from the main text and adds information about mean gains and losses at $\underline{m}=20$ and $\overline{m}=80$. Panel B shows the corresponding results for the Extension Experiment with $\underline{m} = 1$ and $\overline{m} = 5$. Column 7 reveals that UIH-normative is violated in eleven of twelve treatments. In five of them, the violation is statistically significant at the 5\% level. In all these cases, optimal policy demands that the transaction either be allowed only at sufficiently high incentives or not at all, depending on $\alpha$. In the former case, a policy that allows the transaction at low but not high incentives \emph{minimizes} welfare.
While the necessary condition for UIH-normative is satisfied in one treatment (regular stakes, contemporaneous, Large Loss, with a 50\% prior success probability), the test of the hypothesis that $\underline{\alpha} = \overline{\alpha}$ cannot be rejected at any conventional significance level. Even taking the estimates at face value, UIH-normative is satisfied only for welfare weights $\alpha$ in the narrow range from 0.568 to 0.608. Moreover, once I account for noisy elicitation of certainty equivalents, UIH-normative is violated in this condition, too (see Table \ref{welfareExp1lasso}).
How do the welfare effects vary across individual treatments? Column 3 shows that a lower prior success probability leads to substantially tighter upper bounds on the weight that can be placed on losses while keeping the transaction admissible at the low incentive. In fact, with a 20\% prior success probability, welfare losses exceed gains by so much that even a utilitarian ($\alpha = 0.5$) prefers preventing the transaction to allowing it at the lowest incentive no matter the remaining conditions. Column 6 shows that a lower prior success probability also increases losses relative to gains at the highest incentive. The Large Loss condition also tends to raise losses relative to gains, compared to the Limited Loss condition. Timing of potential negative consequences does not exert clear and systematic welfare effects. Importantly, none of these level effects translate into support for UIH-normative.
\label{lassoWelfareExperiment1}
Noisy elicitation of certainty equivalents might bias the estimates in Table \ref{welfareExp2appendixMain} because it causes some misclassification of gains as losses and vice versa. To correct for this possibility, I run ridge regressions with 10-fold cross-validation to predict individuals' certainty equivalents for each of the 26 lotteries they face in stage 2 of the experiment. (The choice of ridge regression is informed by previous research on predicting choice under risk.\citep{NaeckerPeysakhovich}) I run a separate regression for each round of stage 2. As predictors, I include the gain and loss amounts, the success probability, and a treatment indicator for the Delay condition of the current round. I include all two- and three-way interactions between these variables. Additionally, I include the certainty equivalents of all other rounds, as well as gain and loss amounts, success probabilities, and Delay condition indicators. In addition, I include all two-, three-, and four-way interactions between these predictor variables.
Table \ref{welfareExp1lasso} displays the results. The Undue Inducement Condition is violated in all cases but one. The exception occurs in the case of contemporaneous consequences with a prior success probability of 20\% in the Large Loss condition. The range of welfare parameters $\alpha$ for which capping incentives is welfare-optimal is given by $[0.153, 0.177]$, which requires the planner to place more than five times as much weight on gains than on losses. By contrast, the undue inducement literature places more weight on individuals who are harmed. According to such welfare weights, the transaction in the affected row should be prevented at all incentives.
Table \ref{welfareExp2appendixMain} tests UIH-normative in an ex-ante welfare framework. Table \ref{welfareExp1pat} reproduces Table \ref{welfareExp2appendixMain} using ex-post welfare, setting $w_i$ equal to the realized monetary gain or loss for each subject in each trial. UIH-normative is violated in all but one cases. Arguably, this is an even stronger result than that based on ex-ante analysis. The reason is that, in contrast to the ex-ante framework, the ex-post benchmark lets the planner disagree with the subjects' preferences. The violation of UIH-normative according to the ex-post benchmark is particularly noteworthy in the Delay condition, in which the planner imposes judgment not only on risk preferences (by weighting outcomes in different states) but also on time preferences. The violation of UIH-normative in the ex-post framework occurs for a planner who insists that future consequences should not be discounted at all. (In fact, the average subject in the Delay condition is indifferent between a \euro 1 loss incurred with a three-month delay and a \euro 0.83 immediate loss.)
\afterpage{
\thispagestyle{empty}
\begin{table}[htb]
\setlength{\tabcolsep}{2pt}
\caption{Tests for UIH-normative in Experiment 2 and Extension Experiment \label{welfareExp2appendixMain}}
\vspace{-0.4cm}
\footnotesize
\begin{center}
\begin{tabular}{lcccccccc} \toprule
&(1) & (2) & (3) & (4) & (5) & (6) & (7) & (8)  \\
Variable & $g(\underline{m})$ & $l(\underline{m})$  & $\overline{\alpha}$  & $g(\overline{m})$ & $l(\overline{m})$  & $\underline{\alpha}$ & \multicolumn{2}{c}{$\underline{\alpha}<\overline{\alpha}$} \\
\cmidrule(r){8-9}
&&&&&&&satisfied & $p$-value \\ \midrule
& \multicolumn{8}{c}{\textbf{A. Experiment 2}} \\
\cmidrule(r){2-9}
\emph{Loss} \\
\quad Limited & 		3.942***	  &  	-4.794***	  &  	0.451***	  &  	19.654***	  &  	-4.366**	  &  	0.818***	  &  	No	 & 	0.004	 \\
&	(1.261) & (1.453) & (0.120) & (2.395) & (1.729) & (0.066) & 		 & 		 \\
\quad Large & 		4.082***	  &  	-6.494***	  &  	0.386***	  &  	16.125***	  &  	-10.532***	  &  	0.605***	  &  	No	 & 	0.084	 \\
&	(1.065) & (2.196) & (0.106) & (2.508) & (2.185) & (0.068) & 		 & 		 \\
\\
& \multicolumn{8}{c}{\textbf{B. Extension Experiment}} \\
\cmidrule(r){2-9}
\emph{Loss} \\
Limited Loss \\
\quad Contemporaneous \\ \qquad Prior success probability\\
\qquad 20\% & 		0.155***	  &  	-0.224***	  &  	0.409***	  &  	0.357***	  &  	-0.290***	  &  	0.552***	  &  	No	 & 	0.395	 \\
&	(0.059) & (0.083) & (0.138) & (0.092) & (0.087) & (0.080) & 		 & 		 \\
\qquad 50\% & 		0.100		  &  	-0.267**	  &  	0.272***	  &  	1.315***	  &  	-0.167***	  &  	0.887***	  &  	No	 & 	0.001	 \\
&	(0.070) & (0.105) & (0.165) & (0.185) & (0.045) & (0.034) & 		 & 		 \\
\qquad 80\% & 		0.267***	  &  	-0.121**	  &  	0.689		  &  	2.468***	  &  	-0.113		  &  	0.956***	  &  	No	 & 	0.062	 \\
&	(0.091) & (0.055) & (0.139) & (0.211) & (0.081) & (0.031) & 		 & 		 \\
\quad Delayed \\ \qquad Prior success probability\\
\qquad 20\% & 		0.000		  &  	-0.207*	  &  	0.000***	  &  	0.831***	  &  	-0.137*	  &  	0.858***	  &  	No	 & 	0.004	 \\
&	(0.057) & (0.109) & (0.276) & (0.179) & (0.073) & (0.074) & 		 & 		 \\
\qquad 50\% & 		0.114*	  &  	-0.362***	  &  	0.239***	  &  	1.526***	  &  	-0.473***	  &  	0.763***	  &  	No	 & 	0.000	 \\
&	(0.069) & (0.123) & (0.134) & (0.222) & (0.174) & (0.077) & 		 & 		 \\
\qquad 80\% & 		0.174**	  &  	-0.080**	  &  	0.685		  &  	2.197***	  &  	-0.406**	  &  	0.844***	  &  	No	 & 	0.318	 \\
&	(0.074) & (0.040) & (0.154) & (0.197) & (0.169) & (0.061) & 		 & 		 \\
Large Loss \\
\quad Contemporaneous \\ \qquad Prior success probability\\
\qquad 20\% & 		0.112*	  &  	-0.361**	  &  	0.237***	  &  	0.359***	  &  	-0.501***	  &  	0.417***	  &  	No	 & 	0.397	 \\
&	(0.064) & (0.170) & (0.143) & (0.121) & (0.134) & (0.114) & 		 & 		 \\
\qquad 50\% & 		0.175***	  &  	-0.113		  &  	0.608		  &  	0.812***	  &  	-0.618***	  &  	0.568***	  &  	Yes	 & 	0.862	 \\
&	(0.063) & (0.088) & (0.213) & (0.159) & (0.149) & (0.081) & 		 & 		 \\
\qquad 80\% & 		0.177**	  &  	-0.119*	  &  	0.597		  &  	1.982***	  &  	-0.242***	  &  	0.891***	  &  	No	 & 	0.126	 \\
&	(0.082) & (0.071) & (0.194) & (0.227) & (0.090) & (0.041) & 		 & 		 \\
\quad Delayed \\ \qquad Prior success probability\\
\qquad 20\% & 		0.069		  &  	-0.444***	  &  	0.134***	  &  	0.324***	  &  	-0.283***	  &  	0.534***	  &  	No	 & 	0.016	 \\
&	(0.046) & (0.161) & (0.091) & (0.108) & (0.096) & (0.128) & 		 & 		 \\
\qquad 50\% & 		0.168**	  &  	-0.237**	  &  	0.415***	  &  	1.054***	  &  	-0.401**	  &  	0.724***	  &  	No	 & 	0.132	 \\
&	(0.077) & (0.119) & (0.176) & (0.190) & (0.172) & (0.100) & 		 & 		 \\
\qquad 80\% & 		0.095		  &  	-0.159**	  &  	0.375***	  &  	1.535***	  &  	-0.447***	  &  	0.774***	  &  	No	 & 	0.043	 \\
&	(0.065) & (0.066) & (0.200) & (0.180) & (0.141) & (0.064) & 		 & 		 \\
\bottomrule
\end{tabular}
\end{center}
\footnotesize \textbf{\textit{Notes:}} Each row corresponds to a separate treatment within the Incentive First condition. Each row reflects a single SUR-regression. Columns 1 and 2 show mean welfare gains and losses, respectively, at the lowest incentive, $\underline{m}$. Column 3 shows the maximal weight $\overline{\alpha}$ that may be placed on those who lose from the transaction such the transaction is permissible at the lowest incentive. Columns 4 and 5 show mean welfare gains and losses, respectively, at the highest incentive, $\overline{m}$. Column 6 shows the minimal weight $\underline{\alpha}$ that must be placed on those who lose from the transaction such that the transaction is inadmissible at the \$30 incentive. Reported significance levels in columns 3 and 6 reflect two-sided $z$-tests of the Null hypothesis that $\overline{\alpha} = 1$ and $\underline{\alpha} = 0$, respectively. Column 7 indicates whether $\overline{\alpha} \geq \underline{\alpha}$, a necessary condition for UIH-normative. Column 8 lists $p$-values for two-sided $z$-tests of the Null hypothesis that $\overline{\alpha} = \underline{\alpha}$. Standard errors in parentheses, clustered by subject. Standard errors in columns 3, 6, and 8 are calculated by the Delta method.
\end{table}
\clearpage
}
\afterpage{
\thispagestyle{empty}
\begin{table}[htb]
\setlength{\tabcolsep}{2pt}
\caption{Tests for UIH-normative in Experiment 2 and Extension Experiment controlling for elicitation noise \label{welfareExp1lasso}}
\vspace{-0.4cm}
\footnotesize
\begin{center}
\begin{tabular}{lcccccccc} \toprule
&(1) & (2) & (3) & (4) & (5) & (6) & (7) & (8)  \\
Variable & $g(\underline{m})$ & $l(\underline{m})$  & $\overline{\alpha}$  & $g(\overline{m})$ & $l(\overline{m})$  & $\underline{\alpha}$ & \multicolumn{2}{c}{$\underline{\alpha}<\overline{\alpha}$} \\
\cmidrule(r){8-9}
&&&&&&&satisfied & $p$-value \\ \midrule
& \multicolumn{8}{c}{\textbf{A. Experiment 2}} \\
\cmidrule(r){2-9}
\emph{Loss} \\
\quad Limited & 		2.808***	  &  	-3.503***	  &  	0.445***	  &  	13.517***	  &  	-0.890**	  &  	0.938***	  &  	No	 & 	0.000	 \\
&	(0.456) & (0.987) & (0.085) & (1.440) & (0.404) & (0.028) & 		 & 		 \\
\quad Large & 		2.135***	  &  	-4.465***	  &  	0.324***	  &  	5.490***	  &  	-6.801***	  &  	0.447***	  &  	No	 & 	0.084	 \\
&	(0.477) & (0.946) & (0.071) & (1.362) & (1.249) & (0.082) & 		 & 		 \\
\\
& \multicolumn{8}{c}{\textbf{B. Extension Experiment}} \\
\cmidrule(r){2-9}
\emph{Loss} \\
Limited Loss \\
\quad Contemporaneous \\ \qquad Prior success probability\\
\qquad 20\% & 		0.084***	  &  	-0.191***	  &  	0.305***	  &  	0.263***	  &  	-0.125**	  &  	0.678***	  &  	No	 & 	0.002	 \\
&	(0.025) & (0.053) & (0.094) & (0.046) & (0.053) & (0.105) & 		 & 		 \\
\qquad 50\% & 		0.108***	  &  	-0.195***	  &  	0.356***	  &  	0.781***	  &  	-0.051**	  &  	0.939***	  &  	No	 & 	0.000	 \\
&	(0.025) & (0.058) & (0.092) & (0.101) & (0.020) & (0.025) & 		 & 		 \\
\qquad 80\% & 		0.258***	  &  	-0.054**	  &  	0.828		  &  	2.177***	  &  	-0.000		  &  	1.000***	  &  	No	 & 	0.011	 \\
&	(0.029) & (0.024) & (0.068) & (0.163) & (0.005) & (0.002) & 		 & 		 \\
\quad Delayed \\ \qquad Prior success probability\\
\qquad 20\% & 		0.040***	  &  	-0.141***	  &  	0.222***	  &  	0.398***	  &  	-0.053*	  &  	0.883***	  &  	No	 & 	0.000	 \\
&	(0.015) & (0.054) & (0.093) & (0.089) & (0.028) & (0.061) & 		 & 		 \\
\qquad 50\% & 		0.146***	  &  	-0.228***	  &  	0.391***	  &  	0.885***	  &  	-0.162		  &  	0.845***	  &  	No	 & 	0.000	 \\
&	(0.031) & (0.087) & (0.109) & (0.118) & (0.132) & (0.112) & 		 & 		 \\
\qquad 80\% & 		0.195***	  &  	-0.123***	  &  	0.614***	  &  	1.563***	  &  	-0.192*	  &  	0.891***	  &  	No	 & 	0.000	 \\
&	(0.027) & (0.044) & (0.099) & (0.145) & (0.110) & (0.059) & 		 & 		 \\
Large Loss \\
\quad Contemporaneous \\ \qquad Prior success probability\\
\qquad 20\% & 		0.057***	  &  	-0.268***	  &  	0.177**	  &  	0.089**	  &  	-0.494***	  &  	0.153**	  &  	Yes	 & 	0.738	 \\
&	(0.020) & (0.081) & (0.071) & (0.042) & (0.115) & (0.069) & 		 & 		 \\
\qquad 50\% & 		0.109***	  &  	-0.207***	  &  	0.345***	  &  	0.281***	  &  	-0.376***	  &  	0.428***	  &  	No	 & 	0.403	 \\
&	(0.026) & (0.061) & (0.092) & (0.065) & (0.094) & (0.092) & 		 & 		 \\
\qquad 80\% & 		0.202***	  &  	-0.065**	  &  	0.757***	  &  	1.176***	  &  	-0.062*	  &  	0.950***	  &  	No	 & 	0.015	 \\
&	(0.033) & (0.027) & (0.088) & (0.122) & (0.035) & (0.029) & 		 & 		 \\
\quad Delayed \\ \qquad Prior success probability\\
\qquad 20\% & 		0.027**	  &  	-0.288***	  &  	0.085**	  &  	0.115**	  &  	-0.398***	  &  	0.224**	  &  	No	 & 	0.054	 \\
&	(0.012) & (0.095) & (0.046) & (0.048) & (0.101) & (0.091) & 		 & 		 \\
\qquad 50\% & 		0.146***	  &  	-0.214***	  &  	0.406***	  &  	0.480***	  &  	-0.320**	  &  	0.600***	  &  	No	 & 	0.073	 \\
&	(0.032) & (0.066) & (0.101) & (0.094) & (0.148) & (0.129) & 		 & 		 \\
\qquad 80\% & 		0.158***	  &  	-0.185***	  &  	0.460***	  &  	0.894***	  &  	-0.154**	  &  	0.853***	  &  	No	 & 	0.000	 \\
&	(0.032) & (0.055) & (0.098) & (0.126) & (0.071) & (0.064) & 		 & 		 \\
\bottomrule
\end{tabular}
\end{center}
\footnotesize \textbf{\textit{Notes:}} Each row corresponds to a separate treatment within the Incentive First condition. Each row reflects a single SUR-regression. Columns 1 and 2 show mean welfare gains and losses, respectively, at the lowest incentive, $\underline{m}$. Column 3 shows the maximal weight $\overline{\alpha}$ that may be placed on those who lose from the transaction such the transaction is permissible at the lowest incentive. Columns 4 and 5 show mean welfare gains and losses, respectively, at the highest incentive, $\overline{m}$. Column 6 shows the minimal weight $\underline{\alpha}$ that must be placed on those who lose from the transaction such that the transaction is inadmissible at the \$30 incentive. Reported significance levels in columns 3 and 6 reflect two-sided $z$-tests of the Null hypothesis that $\overline{\alpha} = 1$ and $\underline{\alpha} = 0$, respectively. Column 7 indicates whether $\overline{\alpha} \geq \underline{\alpha}$, a necessary condition for UIH-normative. Column 8 lists $p$-values for two-sided $z$-tests of the Null hypothesis that $\overline{\alpha} = \underline{\alpha}$. Standard errors in parentheses, clustered by subject. Standard errors in columns 3, 6, and 8 are calculated by the Delta method.
\end{table}
\clearpage
}
\afterpage{
\thispagestyle{empty}
\begin{table}[htb]
\footnotesize
\setlength{\tabcolsep}{2pt}
\caption{Tests for UIH-normative in Experiment 2 and Extension Experiment based on the ex-post welfare benchmark \label{welfareExp1pat}}
\vspace{-0.4cm}
\begin{center}
\begin{tabular}{lcccccccc} \toprule
&(1) & (2) & (3) & (4) & (5) & (6) & (7) & (8)  \\
Variable & $g(\underline{m})$ & $l(\underline{m})$  & $\overline{\alpha}$  & $g(\overline{m})$ & $l(\overline{m})$  & $\underline{\alpha}$ & \multicolumn{2}{c}{$\underline{\alpha}<\overline{\alpha}$} \\
\cmidrule(r){8-9}
&&&&&&&satisfied & $p$-value \\ \midrule
& \multicolumn{8}{c}{\textbf{A. Experiment 2}} \\
\cmidrule(r){2-9}
\emph{Loss} \\
\quad Limited & 		3.876***	  &  	-5.272***	  &  	0.424***	  &  	35.865***	  &  	-6.625***	  &  	0.844***	  &  	No	 & 	0.002	 \\
&	(1.247) & (2.028) & (0.125) & (4.110) & (1.089) & (0.032) & 		 & 		 \\
\quad Large & 		5.871***	  &  	-7.166***	  &  	0.450***	  &  	31.913***	  &  	-17.736***	  &  	0.643***	  &  	No	 & 	0.006	 \\
&	(0.595) & (1.863) & (0.073) & (3.391) & (3.458) & (0.055) & 		 & 		 \\
\\
& \multicolumn{8}{c}{\textbf{B. Extension Experiment}} \\
\cmidrule(r){2-9}
\emph{Loss} \\
Limited Loss \\
\quad Contemporaneous \\ \qquad Prior success probability\\
\qquad 20\% & 		0.106		  &  	-0.517***	  &  	0.170***	  &  	0.644***	  &  	-0.448***	  &  	0.590***	  &  	No	 & 	0.007	 \\
&	(0.067) & (0.176) & (0.109) & (0.156) & (0.078) & (0.073) & 		 & 		 \\
\qquad 50\% & 		0.144*	  &  	-0.402***	  &  	0.264***	  &  	1.981***	  &  	-0.302***	  &  	0.868***	  &  	No	 & 	0.000	 \\
&	(0.079) & (0.149) & (0.133) & (0.257) & (0.065) & (0.035) & 		 & 		 \\
\qquad 80\% & 		0.369***	  &  	-0.124		  &  	0.749		  &  	3.282***	  &  	-0.093**	  &  	0.972***	  &  	No	 & 	0.092	 \\
&	(0.087) & (0.078) & (0.130) & (0.269) & (0.041) & (0.013) & 		 & 		 \\
\quad Delayed \\ \qquad Prior success probability\\
\qquad 20\% & 		-0.025		  &  	-0.397**	  &  	-0.066***	  &  	0.897***	  &  	-0.289***	  &  	0.756***	  &  	No	 & 	0.000	 \\
&	(0.058) & (0.155) & (0.165) & (0.219) & (0.073) & (0.077) & 		 & 		 \\
\qquad 50\% & 		0.219***	  &  	-0.421**	  &  	0.342***	  &  	2.091***	  &  	-0.309***	  &  	0.871***	  &  	No	 & 	0.000	 \\
&	(0.082) & (0.170) & (0.121) & (0.274) & (0.070) & (0.032) & 		 & 		 \\
\qquad 80\% & 		0.190**	  &  	-0.227**	  &  	0.455***	  &  	3.365***	  &  	-0.140***	  &  	0.960***	  &  	No	 & 	0.004	 \\
&	(0.082) & (0.093) & (0.169) & (0.215) & (0.043) & (0.013) & 		 & 		 \\
Large Loss \\
\quad Contemporaneous \\ \qquad Prior success probability\\
\qquad 20\% & 		0.104***	  &  	-0.395***	  &  	0.208***	  &  	0.635***	  &  	-0.982***	  &  	0.393***	  &  	No	 & 	0.095	 \\
&	(0.029) & (0.119) & (0.068) & (0.162) & (0.227) & (0.088) & 		 & 		 \\
\qquad 50\% & 		0.239***	  &  	-0.330***	  &  	0.420***	  &  	1.787***	  &  	-0.779***	  &  	0.696***	  &  	No	 & 	0.008	 \\
&	(0.034) & (0.108) & (0.084) & (0.223) & (0.188) & (0.062) & 		 & 		 \\
\qquad 80\% & 		0.466***	  &  	-0.050*	  &  	0.903		  &  	2.962***	  &  	-0.619***	  &  	0.827***	  &  	Yes	 & 	0.149	 \\
&	(0.049) & (0.029) & (0.054) & (0.253) & (0.181) & (0.047) & 		 & 		 \\
\quad Delayed \\ \qquad Prior success probability\\
\qquad 20\% & 		0.112***	  &  	-0.412***	  &  	0.214***	  &  	0.664***	  &  	-0.979***	  &  	0.404***	  &  	No	 & 	0.026	 \\
&	(0.028) & (0.121) & (0.064) & (0.149) & (0.231) & (0.086) & 		 & 		 \\
\qquad 50\% & 		0.253***	  &  	-0.293***	  &  	0.464***	  &  	2.112***	  &  	-0.896***	  &  	0.702***	  &  	No	 & 	0.017	 \\
&	(0.039) & (0.107) & (0.105) & (0.239) & (0.250) & (0.070) & 		 & 		 \\
\qquad 80\% & 		0.353***	  &  	-0.109**	  &  	0.764		  &  	2.746***	  &  	-0.440***	  &  	0.862***	  &  	No	 & 	0.358	 \\
&	(0.041) & (0.055) & (0.094) & (0.226) & (0.144) & (0.043) & 		 & 		 \\
\bottomrule
\end{tabular}
\end{center}
\footnotesize \textbf{\textit{Notes:}} Each row corresponds to a separate treatment within the Incentive First condition. Each row reflects a single SUR-regression. Columns 1 and 2 show mean welfare gains and losses, respectively, at the lowest incentive, $\underline{m}$. Column 3 shows the maximal weight $\overline{\alpha}$ that may be placed on those who lose from the transaction such the transaction is permissible at the lowest incentive. Columns 4 and 5 show mean welfare gains and losses, respectively, at the highest incentive, $\overline{m}$. Column 6 shows the minimal weight $\underline{\alpha}$ that must be placed on those who lose from the transaction such that the transaction is inadmissible at the \$30 incentive. Reported significance levels in columns 3 and 6 reflect the two-sided $z$-tests of the Null hypothesis that $\overline{\alpha} = 1$ and $\underline{\alpha} = 0$, respectively. Column 7 indicates whether $\overline{\alpha}\geq \underline{\alpha}$, a necessary condition for UIH-normative. Column 8 lists $p$-values for two-sided $z$-tests of the Null hypothesis that $\overline{\alpha}= \underline{\alpha}$. Standard errors in parentheses, clustered by subject. Standard errors in columns 3, 6, and 8 are calculated by the Delta method.
\end{table}
\clearpage
}
\FloatBarrier
\newpage
\section{SI Experiment 3}\label{inducedPrefs}\label{dynamicInfoExperiment}\label{appendixBsGsExperiment}
\subsection{Design}
The experiment in this section follows the same design as Experiment 2, but it provides subjects with an information acquisition technology that allows subjects to access a much wider range of information structures than Experiment 2.
Following the paradigm of \citet{DeanNeligh}, subjects obtain information about whether the venture will be successful by examining a picture of 450 randomly arranged letters as in panel A of Figure \ref{inducedPrefsFraming}. Subjects know that if the venture is a success, the image contains 50 letters  $G$ and 40 letters  $B$ (for ``good'' and ``bad'', respectively) and that these numbers are reversed if the venture is a failure. In contrast to Experiment 2, information costs now depend on the subjects' disutility of effort.  Moreover, while subjects in Experiment 2 can, in principle, derive Bayesian posteriors by simple arithmetic, this is not possible in the present experiment. As a result, correct belief updating might be more difficult, worsening choice quality.
Any search strategy induces a probability of participating in the good state and a probability of participating in the bad state, and thus it corresponds to a stochastic information structure. There is no time constraint, so subjects can implement a wide range of search strategies. (Technically, this paradigm is an approximation to the \citep{Wald} sequential sampling framework, in which any pair of Bayes-consistent posterior beliefs $(\gamma_{\mathscr{G}}, \gamma_{\mathscr{B}}) \in [0,1]^2$ can be implemented as the informativeness of individual draws approaches zero.) They can choose the amount of information to gather in the sense that they have access to information structures that can be Blackwell-ordered. They can also choose the kind of information, for instance, by searching longer before accepting the transaction than before rejecting it. In contrast to Experiment 2, symmetric information structures (in which encouraging and discouraging signals are equally likely ex-ante) are available.
I record subjects' information acquisition as follows.  A black panel covers the entire grid of letters, but subjects can use their mouse to move a small hole across the grid through which they can inspect one letter at a time, as in panel B of Figure \ref{inducedPrefsFraming}. I record the sequence of letters the subject observes and use it to calculate Bayesian posteriors. I perform this calculation using the formula $\logit[P(s = G | n, k)] = \logit(\mu) + \log(D)$ where $\logit(x) = \log\left(\frac{x}{1-x}\right)$, where $D = \frac{P(k|n, s=G)}{P(k|n, s=B)} = \frac{(40-k)!}{(50-k)!}\frac{(50 - (n-k))!}{(40-(n-k))!}$ is the likelihood ratio for observing $n$ draws of informative letters ($G$ or $B$) with $k$ letters $G$, as follows directly from the hypergeometric distribution function.
\begin{figure}[]
\caption{\label{inducedPrefsFraming} Presentation of information about the state.}\begin{center}
\begin{tabular}{cc}
\includegraphics[width = 0.48\textwidth]{graphs/lettersYellow} &
\includegraphics[width = 0.48\textwidth]{graphs/spotlight} \\
(A) & (B) \\
\end{tabular}
\end{center}
\footnotesize
\textbf{\textit{Notes:}} Panel A: If the venture is good, the picture has 50 letters $G$ and 40 letters $B$. If the venture is bad, it has 40 letters $G$ and 50 letters $B$. Panel B: The subject moves the mouse to change the location of the hole in the black panel.
\end{figure}
The set of treatments is the same as in the Extension Experiment. I also use the same parameters, with two exceptions. First, all amounts are in USD instead of EUR. Second, the incentive amounts are \$0.5, \$1, \$5, and \$5.5. I use the term \emph{Grid First} to refer to the case in which subjects examine the grid of letters before learning about the incentive amount for the round (corresponding to the Advisor Choice First condition in Experiment 2).
Because searching a grid of letters is more time-consuming than the choice of advisors in Experiment 2, stage 1 of this experiment consists of only 16 rounds, corresponding to 4 incentive levels, the Incentive First and Grid First condition, and the Limited and Large Loss conditions, administered in random order.
To highlight the independence of the rounds, each picture is presented with a different background color. The color assignment is randomized on the individual level. A state is drawn independently for each subject and decision, and a new picture of randomly scrambled letters is generated. Subjects could not use a text editor to automatically count the letters because they were presented in a picture format (HTML5 Canvas).
\paragraph{Elicitation of the ex-ante welfare benchmark}
Stage 2 of the current experiment presents subjects with lotteries corresponding to the decisions the subject faced in each round of stage 1 if she updated her beliefs according to Bayes' law. The subject proceeds through 16 such decisions corresponding to stage 1, as well as through 12 additional decisions for which the success probability is exogenously given by 0.2, 0.5, and 0.8, once for each of the incentive amounts \$1 and \$5 and once for each of the Limited and Large Loss conditions, respectively. A total of 28 rounds are administered in individually randomized order.
The survey automatically calculates the Bayesian posterior depending on the letters the subject has uncovered during her search. If a subject rapidly moves the mouse across the display, letters may be displayed for imperceptibly brief periods. Research shows that subjects cannot identify letters with any accuracy if they are displayed for less than 50 milliseconds.\citep{BundesenHarms} Accordingly, I record a letter as seen only if it is displayed for 50 milliseconds or longer. The Bayesian posteriors subjects encounter in stage 2 of the experiment are calculated based on this benchmark.
\paragraph{Implementation}
I conducted this experiment on the Amazon Mechanical Turk online labor market with a total of 405 subjects on November 9-12, 2020. I required subjects to be located in the US, have at least a 99\% approval rate on previous tasks completed on the platform, and have completed at least 500 tasks on the platform. Each subject receives a \$12 completion payment from which potential losses are discounted. Subjects in the Delay condition receive \$6 of these \$12 with a three-month delay (net of any losses they may incur). In addition, subjects receive a base payment of \$1. Subjects could only participate on a desktop or laptop computer with a screen sufficiently large to display all pictures without scrolling. The median subject completed the experiment in 63.4 minutes; average earnings are \$13.58. Subjects examined each image for a median time of 12 seconds, with 55 seconds at the 75th percentile and 2 seconds at the 25th percentile.
Because the experiment is administered online, attrition is a possible concern. 614 subjects begin the survey on a device that is technically equipped to display the survey. Of these, 70 attrit during the instructions before reaching the first comprehension check. Another 104 do not progress beyond the first comprehension check, and another 11 attrit at or before the second comprehension check. Of these 429 subjects, 405 completed the entire survey. In the instructions, subjects learn whether they will receive some part of the payment with a three-month delay, and they learn the prior success probability they will face. Accordingly, attrition could vary by treatment condition. Amongst the 209 subjects who attrited, 99 were in the Contemporaneous condition, and 110 were in the Delay condition, whereas 75, 81, and 53 faced a 20\%, 50\%, or 80\% prior success probability, respectively. Regressing an indicator for whether a subject attrited on an indicator for the Delay condition, the numerical prior success probability and the interaction between the two have no explanatory power ($p>0.3$ for separate $t$-tests on all coefficients except for the constant term, and $p>0.5$ for the $F$-test of the joint explanatory power of all variables compared to a constant-only model).
\begin{figure}
\caption{\label{figureBsGs}\label{figureElicitedObjective} Results of Experiment 3.}
\begin{center}
\begin{tabular}{cc}
\textbf{A. Information choice} &
\textbf{B. Welfare } \\
\includegraphics[width = 0.48\textwidth]{graphs/diffBsGs} &
\includegraphics[width = 0.48\textwidth]{graphs/welfareCDFexp3} \\
\end{tabular}
\end{center}
\footnotesize
\textbf{\textit{Notes}}: Panel A: Mean difference between the number of Gs and the number of Bs seen when making a decision. Spikes show 95\% confidence intervals. Standard errors are clustered by subject. Based on 6480 observations from 405 independent subjects. Panel B: Ex-ante welfare. For visibility, I group the \$0.5 and \$1 incentives into the low incentive category and the \$5 and \$5.5 incentives into the high incentive category.
\end{figure}
\subsection{Analysis}\label{analysisInducedPrefs}
I first examine the effect of incentives on information acquisition. The undue inducement literature is concerned that higher incentives lead to less informed decision-making. On the contrary, higher incentives lead to increased search in the Incentive First condition (pooling across all remaining conditions), as measured by the number of informative letters (``B'' or ``G'') in the string of letters the subject observed. The magnitude of the effect is substantial. In the case of the \$0.5 incentive, subjects parse strings with an average of 12.4 informative letters. This number nearly doubles to 22.6 informative letters for the case of the \$5.5 incentive ($p<0.001$ for the difference). Hence, on average, higher incentives do not cause less careful decision-making.
Yet, incentives might still cause subjects to alter their information acquisition in a way reminiscent of UIH-positive. UIH-positive predicts that subjects will consult information they more likely expect to recommend participation when the incentive is higher. In the present experiment, a subject who seeks to increase the chance of observing a string of data that implies the risk is less likely to realize than the prior probability (i.e., a string that contains more Gs than Bs), for instance, can achieve by selecting a stronger evidentiary standard before rejecting the transaction than before rejecting it. Specifically, she can accept the transaction as soon as she has seen just a few more letters G than B, but reject only if she has seen many more letters B than G. Previous research investigates these mechanisms formally\citep{MorrisStrack}.
Panel A of Figure \ref{figureElicitedObjective} shows that incentives do exert this effect. I plot how many more Gs than Bs a subject has observed before making a participation decision, separately for acceptance and rejection, by incentive amount. The evidentiary standard subjects require for accepting the transaction is to observe just under two more Gs than Bs regardless of the incentive. The evidentiary standard for rejecting the transaction becomes much stricter when the incentive is high, as one would expect, given that mistakenly rejecting the transaction implies a larger foregone gain when the incentive is higher ($p<0.001$). This behavior is reminiscent of UIH-positive, though the larger amount of information acquired in the high-incentive case does not suggest that higher incentives would decrease decision quality.
To test UIH-normative, I use the same methods as in Experiment 2. Panel B of Figure \ref{figureBsGs} displays the welfare distribution for the Incentive First, Large Loss condition, pooling across the remaining conditions. (Certainty equivalents were not recorded for the case of Limited Loss in the Incentive First condition.) Just as in Experiments 1 and 2, higher incentives greatly increase the average welfare gain and have little, if any, effect on the average welfare loss.
Table \ref{welfareExp3} performs formal tests analogous to Table \ref{welfareExp2appendixMain}. As column 7 shows, UIH-normative is violated throughout.
\begin{table}[htbp]
\caption{Welfare effects in Experiment 3 \label{welfareExp3}}
\footnotesize
\setlength{\tabcolsep}{2pt}
\begin{center}
\begin{tabular}{lcccccccc} \toprule
&(1) & (2) & (3) & (4) & (5) & (6) & (7) & (8)  \\
Variable & $g(\underline{m})$ & $l(\underline{m})$  & $\overline{\alpha}$  & $g(\overline{m})$ & $l(\overline{m})$  & $\underline{\alpha}$ & \multicolumn{2}{c}{UIH-normative} \\
\cmidrule(r){8-9}
&&&&&&&satisfied & $p$-value \\ \midrule
\emph{Loss} \\
Limited (Grid First) \\
\quad Contemporaneous \\ \qquad Prior success probability\\
\qquad 20\% & 		0.018		  &  	-0.500***	  &  	0.035***	  &  	0.359***	  &  	-1.684***	  &  	0.176***	  &  	No	 & 	0.034	 \\
&	(0.017) & (0.144) & (0.032) & (0.107) & (0.234) & (0.054) & 		 & 		 \\
\qquad 50\% & 		0.019		  &  	-0.547***	  &  	0.034***	  &  	0.863***	  &  	-0.801***	  &  	0.518***	  &  	No	 & 	0.000	 \\
&	(0.022) & (0.150) & (0.039) & (0.157) & (0.137) & (0.073) & 		 & 		 \\
\qquad 80\% & 		-0.013		  &  	-0.221**	  &  	-0.064***	  &  	2.257***	  &  	-0.322***	  &  	0.875***	  &  	No	 & 	0.000	 \\
&	(0.033) & (0.090) & (0.170) & (0.238) & (0.110) & (0.044) & 		 & 		 \\
\quad Delayed \\ \qquad Prior success probability\\
\qquad 20\% & 		0.043		  &  	-0.315***	  &  	0.119***	  &  	0.325**	  &  	-0.935***	  &  	0.258***	  &  	No	 & 	0.307	 \\
&	(0.038) & (0.113) & (0.103) & (0.134) & (0.213) & (0.097) & 		 & 		 \\
\qquad 50\% & 		0.020		  &  	-0.273**	  &  	0.070***	  &  	0.955***	  &  	-0.789***	  &  	0.548***	  &  	No	 & 	0.007	 \\
&	(0.044) & (0.119) & (0.143) & (0.155) & (0.162) & (0.074) & 		 & 		 \\
\qquad 80\% & 		0.037		  &  	-0.413***	  &  	0.082***	  &  	2.240***	  &  	-0.371***	  &  	0.858***	  &  	No	 & 	0.000	 \\
&	(0.050) & (0.143) & (0.102) & (0.242) & (0.135) & (0.051) & 		 & 		 \\
Large (Incentive First) \\
\quad Contemporaneous \\ \qquad Prior success probability\\
\qquad 20\% & 		0.021		  &  	-0.462***	  &  	0.044***	  &  	0.408***	  &  	-0.841***	  &  	0.327***	  &  	No	 & 	0.008	 \\
&	(0.024) & (0.148) & (0.050) & (0.126) & (0.180) & (0.089) & 		 & 		 \\
\qquad 50\% & 		0.024		  &  	-0.387***	  &  	0.058***	  &  	0.736***	  &  	-0.763***	  &  	0.491***	  &  	No	 & 	0.000	 \\
&	(0.035) & (0.116) & (0.083) & (0.143) & (0.157) & (0.078) & 		 & 		 \\
\qquad 80\% & 		0.036		  &  	-0.261**	  &  	0.122***	  &  	2.100***	  &  	-0.314**	  &  	0.870***	  &  	No	 & 	0.000	 \\
&	(0.057) & (0.106) & (0.175) & (0.263) & (0.124) & (0.051) & 		 & 		 \\
\quad Delayed \\ \qquad Prior success probability\\
\qquad 20\% & 		0.032		  &  	-0.668***	  &  	0.046***	  &  	0.482***	  &  	-0.661***	  &  	0.422***	  &  	No	 & 	0.000	 \\
&	(0.033) & (0.224) & (0.048) & (0.159) & (0.199) & (0.113) & 		 & 		 \\
\qquad 50\% & 		0.069		  &  	-0.230**	  &  	0.232***	  &  	0.882***	  &  	-0.533***	  &  	0.623***	  &  	No	 & 	0.009	 \\
&	(0.050) & (0.094) & (0.150) & (0.166) & (0.146) & (0.086) & 		 & 		 \\
\qquad 80\% & 		0.081		  &  	-0.434***	  &  	0.157***	  &  	2.574***	  &  	-0.238**	  &  	0.915***	  &  	No	 & 	0.000	 \\
&	(0.050) & (0.122) & (0.080) & (0.251) & (0.114) & (0.040) & 		 & 		 \\
\bottomrule
\end{tabular}
\end{center}
\footnotesize \textbf{\textit{Notes:}} Each row corresponds to a separate treatment. Each row reflects a single SUR-regression. Columns 1 and 2 show mean welfare gains and losses, respectively, at the lowest incentive, $\underline{m}$. Column 3 shows the maximal weight $\overline{\alpha}$ that may be placed on those who lose from the transaction such the transaction is permissible at the lowest incentive. Columns 4 and 5 show mean welfare gains and losses, respectively, at the highest incentive, $\overline{m}$. Column 6 shows the minimal weight $\underline{\alpha}$ that must be placed on those who lose from the transaction such that the transaction is inadmissible at the \$30 incentive. Reported significance levels in columns 3 and 6 reflect two-sided $z$-tests of the Null hypothesis that $\overline{\alpha} = 1$ and $\underline{\alpha} = 0$, respectively. Column 7 indicates whether $\overline{\alpha} \geq \underline{\alpha}$, a necessary condition for UIH-normative. Column 8 lists $p$-values for two-sided $z$-tests of the Null hypothesis that $\overline{\alpha} = \underline{\alpha}$. Standard errors in parentheses, clustered by subject. Standard errors in columns 3, 6, and 8 are calculated by the Delta method.
\end{table}
Overall, this experiment demonstrates that the conclusions of Experiment 2 are robust to a more flexible form of information acquisition.
\section{SI Theory}
\paragraph{Setting} A Bayesian expected utility maximizer decides whether or not to participate in a transaction in exchange for a material incentive $m$. The consequences of participation depend on a state of the world $s$, which is good ($s = G$) with prior probability $\mu$, and bad ($s=B$) otherwise. If the agent participates and the state is good, he experiences utility $u_G(m)>0$.  If the agent participates and the state is bad, he experiences disutility $u_B(m)<0$. If the agent abstains, he receives no payment and experiences utility 0. I assume that $u_G(m)$ is strictly increasing in $m$ and that $u_B(m)$ is weakly increasing in $m$, reflecting the fact that higher compensation never decreases a participants' utility in a given state (though, in principle, it may lower his ex-ante expected utility by causing him to participate more often in the bad state).  The functions $u_G(m)$ and $u_B(m)$ encompass the agent's risk preferences.
Before deciding whether to participate in the transaction, the agent chooses an information structure that produces a stochastic signal about the state. (An information structure $I$ is a function $I:\{G, B\} \rightarrow \Delta \mathscr{S}$ that maps states of the world into probability distributions over a space of signals $\mathscr{S}$.)  Critically, I assume that the agent's information processing capacity is limited, so he needs to be selective about the kind of information to consult. Formally, the agent chooses an information structure from a finite set $\mathscr{I}=\{I_1, \ldots, I_n\}$ with $n\geq 2$ which may represent, for instance, advisors with various biases.  For simplicity, I assume each information structure's signal space has two elements, $\mathscr{G}$ and $\mathscr{B}$, which I interpret as a recommendation to participate and a recommendation to abstain, respectively. I let $p_{I}^{s}$ denote the probability that information structure $I$ produces signal $\mathscr{G}$ if the state is $s$.  Without loss of generality, I assume $p_{I}^{G} \geq p_{I}^{B}$ for each information structure $I\in\mathscr{I}$.
I will show that participation incentives change the kind of information a rational agent demands, specifically, the optimal information structure's \emph{bias}, as defined in  \citet{Gentzkowetal.Review}. In my setting, that definition specializes as follows:
\begin{definition}\label{defSkew}
Consider two information structures $I$ and $I'$ and a given prior $\mu$. Then, $I'$ is \emph{biased towards $\mathscr{G}$} relative to $I$ if $P(I' = \mathscr{G}) \geq P(I = \mathscr{G})$. Moreover, $I'$ is \emph{statewise biased towards $\mathscr{G}$} relative to $I$ if $p_{I'}^s \geq p_{I}^s$ for all states $s \in \{G, B\}$.
\end{definition}
\paragraph{Analysis}\label{analysis}
The agent's optimization problem involves two stages. First, the agent selects an information structure. Second, he decides whether to participate based on his observed signal realization. The selected information structure is \emph{instrumental} if, in the second stage, the agent chooses to participate after observing signal realization $\mathscr{G}$ and chooses to abstain after observing signal realization $\mathscr{B}$. The agent also has the opportunity to ignore the signal realization and to either participate or abstain with certainty. I account for this possibility by defining the information structures $I^\mathscr{G} = (1,1)$ and $I^\mathscr{B} = (0,0)$, and letting $\bar{\mathscr{I}} = \mathscr{I} \cup \{I^\mathscr{G}, I^\mathscr{B}\}$. Considering the agent's choice from $\bar{\mathscr{I}}$, I can then assume, without loss of generality, that the agent follows the signal produced by the chosen information structure. Thus, the agent's probability of participating in state $s$ equals the probability of a good signal in that state, $p_I^s$. Because the agent's utility from non-participation is 0, his utility maximization problem in the first stage is
\vspace{-0.4cm}
\bmath \label{maxU}
\max_{I\in\bar{\mathscr{I}}} \mu p_I^G u_G(m) + (1 - \mu) p_I^B u_B(m).
\emath
\vspace{-0.4cm}
The main result of this section shows that Bayes-optimal behavior appears as if the agent engaged in \emph{motivated reasoning}: as the incentive $m$ increases, the agent chooses an information structure with a stronger bias towards $\mathscr{G}$. (Formally, the solution to (\ref{maxU}), which may not be unique, is increasing in $m$ in the strong set order.)  Intuitively, this result arises from a simple cost-benefit calculation. Information helps the agent limit choices that are \emph{ex-post mistaken} in the sense that the chosen action is suboptimal conditional on the state of the world (even though the choice may have been optimal conditional on the information available at the time of decision). There are two possible ex-post mistakes. A \emph{false positive} occurs if the agent participates in the bad state. A \emph{false negative} occurs if the agent abstains in the good state. The agent seeks to minimize the cost of these mistakes, which depends on the incentive amount $m$. A higher incentive decreases the cost of false positives (because the additional money acts as insurance against ex-post undesirable outcomes) and increases the cost of false negatives (because abstention means foregoing a larger amount of money). Accordingly, as the incentive increases, the agent becomes more interested in preventing false negatives and less interested in preventing false positives. The agent achieves this goal by selecting an information structure more strongly biased towards $\mathscr{G}$.
\begin{proposition}\label{mainProp}
Consider incentive amounts $m$ and $m'$ with $m > m'$. Suppose at least one information structure is instrumental at $m$, and at least one information structure is instrumental at $m'$. Let ${I}^*$ and ${I}'^*$ denote the solutions to (\ref{maxU}) at $m$ and $m'$, respectively, and suppose that they are unique and $I^* \neq I'^*$. Then ${I}^*$ is statewise more biased towards $\mathscr{G}$ than ${I}'^*$.
\end{proposition}
\begin{proof}
To prove the claim, I parametrize the constraint, insert it into the objective function, and characterize the optimal information structure. I then show how a change in $m$ changes the optimal information structure.
The objective function (\ref{maxU})  is linear in $(p_G, p_B)$, with indifference curves that are upward-sloping in $(p_G, p_B)$-space. Accordingly, the only candidates for optimal information structures lie on the boundary of the free-disposal convex hull $H$ of the set of information structures. Formally, $H$ is defined as follows. A point $(p_G, p_B)$ is in $H$ if and only if there exists a subset of information structures $(I_1, ..., I_n) \subseteq \tilde{\mathscr{I}}$ with weights $a_1, ..., a_n$ such that $\sum_{k=1}^n a_k p_I^G \geq p_G$ and $\sum_{k=1}^n a_k (1 - p_I^B) \geq (1 - p_B)$.
Let $\mathscr{J}\subseteq\tilde{\mathscr{I}}$ denote the subset of information structures on the boundary of $H$; let $n$ denote the cardinality of $\mathscr{J}$. Enumerate these information structures such that $p_{I_1}^G \leq p_{I_2}^G \leq ...\leq p_{I_n}^G$. Because $H$ is convex, the boundary of $H$ is convex. Formally, $\frac{p_{I_k}^B - p_{I_{k-1}}^B}{p_{I_k}^G - p_{I_{k-1}}^G}$ is an increasing function of $k$ for $2 \leq k \leq n$.
The boundary of $H$ implicitly defines $p_B$ as a function of $p_G$. This function, denoted $b(p_G)$, is piecewise linear and continuous. Accordingly, we can write the objective function in (\ref{maxU}) as $u(p_G; m) = \mu p_H u_G(m) + (1 - \mu) b(p_G) u_B(m)$. A composition of a finite number of continuous, piecewise linear functions, $u(p_G; m)$ is continuous and piecewise linear. For each $k = 2, ... , n$ the slope of $m(p_G; m)$ as a function of $p_G$ over the interval $(p_{I_{k-1}}^G,  p_{I_{k}}^G)$ is given by
\bmathnn
u'(p_G; m) = \mu u_G(m) + (1 - \mu)\frac{p_{I_k}^B - p_{I_{k-1}}^B}{p_{I_k}^G - p_{I_{k-1}}^G}u_B(m)
\emathnn
Because $u_B(m) < 0$ and because $\frac{p_{I_k}^B - p_{I_{k-1}}^B}{p_{I_k}^G - p_{I_{k-1}}^G}$ is increasing, $u'(p_G)$ is decreasing.
Let $k^*$ denote the optimal information structure. If $u'(p_G; m) > 0$ for all $p_G \in (p_{I_1}^G, p_{I_n}^G)$, the optimal information structure is $I_n$, hence $k^* = n$. If $u'(p_G; m) < 0$ for all $p_G \in (p_{I_1}^G, p_{I_n}^G)$, the optimal information structure is $I_1$, hence $k^* = 1$. Otherwise, there exists $\hat{k} = 2, ..., n-1$ such that $u'(p_G; m)$ is weakly positive to the left of $p_{I_{k^*}}^G$ and weakly negative to the right. In that case, $\hat{k}$ is within the set of optimal information structures. If both inequalities are strict, then $k^* = \hat{k}$. If $u'(p_G, m)$ is zero to the left of $p_{I_{k^*}}^G$, then the set of optimal information structures includes $\hat{k} - 1$. If $u'(p_G, m)$ is zero to the right of $p_{I_{k^*}}^G$, then the set of optimal information structures includes $\hat{k} +1$.
How does an increase in $m$ affect the optimal information structure?  Consider $m'$, which, as assumed in the proposition, satisfies $m'>m$. Define $k^*(m')$ as the index $k$ such that $u'(p_G; m')$ is positive to the left of $p_{I_{k^*(m')}}^G$ and negative to the right if such a value exists; otherwise set $k^*(m') = 1$ if $u'(p_G; m')$ is uniformly negative and set $k^*(m') = n$ if $u'(p_G; m')$ is uniformly positive. Then, $k^*(m')$ is the optimal information structure at incentive $m'$. Because $u_G(m)$ and $u_B(m)$ are both increasing in $m$, an increase in $m$ to $m' > m$ increases $u'(p_G;m)$. Accordingly, $k^*(m') \geq k^*(m)$ in the strong set order.
Finally,  the assumption that both $m$ and $m'$ are associated with at least one information structure that is instrumental implies that neither $k^*(m')$ nor $k^*(m)$ are equal to the unconditional participation or abstention information structures $(0,0)$ or $(1,1)$. This completes the proof.  \end{proof}
\begin{figure}[htb!]
\caption{Bayesian information choice \label{infoChoiceTheoryGraph}}
\begin{center}
\includegraphics[width=0.55\textwidth]{graphs/infoStrChoiceNew}
\end{center}
\end{figure}
Figure \ref{infoChoiceTheoryGraph} illustrates Proposition \ref{mainProp} graphically. It plots an example of a set of information structures $\mathscr{I}$ in $(p_G, 1-p_B)$-space. These information structures, together with unconditional participation and abstention information structures $I^\mathscr{G}$ and $I^{\mathscr{B}}$, constitute the agent's choice set $\bar{\mathscr{I}}$. The agent's utility increases to the northeast. By expected utility, the agent's indifference curves are straight lines in $(p_G, 1-p_B)$-space. Therefore, it suffices to consider the (free disposal) convex hull of the agent's opportunity set, shaded in gray in the figure. A higher incentive $m$ increases the benefit of participation in the good state, $p_G$, and (weakly) decreases the benefit of abstention in the bad state, 1 - $p_B$. Accordingly, an increase in $m$ tilts indifference curves counterclockwise. Therefore, the optimal information structure associated with a higher incentive lies further to the northwest. In the figure, the optimal information structure for a given low incentive amount is labeled $A$, and the optimal information structure for a given high incentive amount is labeled $B$. The remaining information structures lie in the interior of the convex hull of the agent's opportunity set. Hence, they will never be chosen.
\paragraph{Welfare}
While a rational agent will never participate in a transaction at an ex-ante welfare loss, boundedly rational individuals may make such mistakes, and higher incentives may increase their frequency, as I show by example. Hence, for non-Bayesians, behavior that satisfies UIH-positive may facilitate UIH-normative. Specifically, consider the family of welfare functions that place weight $\alpha$ on losses and weight $(1-\alpha)$ on gains, along with the set of information structures $\mathscr{I} = \{I^\mathscr{G}, I^\mathscr{B}\}$ with $I^\mathscr{G} = (1, 0.5)$ and $I^\mathscr{B} = (0.5, 0)$ (the parameters of Experiment 2). Hence, $I^\mathscr{G}$ only recommends abstention if the state is bad, and $I^\mathscr{B}$ only recommends participation if the state is good. Consider two incentive levels $\underline{m}$ and $\overline{m}$ with $0<\underline{m}<\overline{m}$. Suppose an agent chooses $I^\mathscr{B}$ in case of incentive $\underline{m}$, chooses $I^\mathscr{G}$ in case of incentive $\overline{m}$ (consistent with UIH-positive), and always follows the signal produced by the chosen information structure, possibly due to excess optimism upon receiving a signal $\mathscr{G}$. In the case of the low incentive, $m = \underline{m}$, this agent never participates in the bad state but sometimes participates in the good state. Accordingly, at $\underline{m}$, mean welfare gains  are positive, $E\big(g(\underline{m})\big)>0$,  and mean welfare losses are zero, $E\big(l(\underline{m})\big)=0$, both ex-ante and ex-post, implying $\overline{\alpha} = 1$. At incentive $\overline{m}$, by contrast, there is a positive probability that the agent participates in the bad state and suffers an ex-post loss. In the case of ex-post analysis, this fact directly implies $E\big(l(\overline{m})\big)<0$, and thus $\underline{\alpha}<1$, so that UIH-normative is satisfied. Notably, this is true even if the agent is Bayesian and chooses information rationally because ex-post welfare analysis permits the planner's weight on losses to differ from the agent's risk attitudes. In the case of ex-ante welfare $E\big(l(\overline{m})\big)<0$ obtains if the subject is non-Bayesian (making it possible to observe the assumed participation strategy) and sufficiently risk-averse (placing sufficient weight on the potential loss). In this case, too, $\underline{\alpha}<1$, so UIH-normative is satisfied.
\newpage
\addcontentsline{toc}{section}{References}
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\end{appendix}
\end{document}
