## Abstract

Incentivized methods for eliciting subjective probabilities in economic experiments present the subject with risky choices that encourage truthful reporting. We discuss the most prominent elicitation methods and their underlying assumptions, provide theoretical comparisons and give a new justification for the quadratic scoring rule. On the empirical side, we survey the performance of these elicitation methods in actual experiments, considering also practical issues of implementation such as order effects, hedging, and different ways of presenting probabilities and payment schemes to experimental subjects. We end with a discussion of the trade-offs involved in using incentives for belief elicitation and some guidelines for implementation.

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## Notes

Here we assume that \(\theta \) is uniquely determined given \(X\), so \(\theta =\theta \left( X\right) \). Definitions become a bit more involved when the characteristic of interest is not always uniquely defined, see Sect. 2.7.

Truth-telling applies the concept of incentive compatibility to belief elicitation. Kothiyal et al. (2011) point out that the earliest incentive compatible scoring rules were proposed at least a decade before the first work on mechanism design was published.

\({\mathbf{1}}_{\left\{ r\le c\right\} }=1\) if \(r\le c\) and \(=0\) if \(r>c\). \({\mathbf{1}}_{\left\{ r>c\right\} }\) is defined similarly.

This scheme provide the same incentives as the QSR, and provides the same payoffs if the prize is given with certainty if the event does not occur.

McCarthy (1956) mentions this claim and attributes it to Gleason (unpublished). Since we were unable to locate the latter study, we prove this here for \(n=2\), using the framework of Savage (1971). We search for a rule such that \(Y\left( r\right) =Z\left( 1-r\right) \) for all \(r\), and hence \(J^{\prime }\left( p\right) =Y\left( p\right) -Z\left( p\right) =Y\left( p\right) -Y\left( 1-p\right) \). Since \(J\left( p\right) =Y\left( p\right) p+Y\left( 1-p\right) \left( 1-p\right) \) we obtain \(J^{\prime }\left( p\right) =Y\left( p\right) -Y\left( 1-p\right) +Y^{\prime }\left( p\right) p-Y^{\prime }\left( 1-p\right) \left( 1-p\right) \) and hence \(Y^{\prime }\left( p\right) p=Y^{\prime }\left( 1-p\right) \left( 1-p\right) \) for all \(p\). This implies that \(Y\left( p\right) =a\ln p+b\) for some \(a>0\) and \(b\) and hence \(S\) is an affine transformation of the the logarithmic payment scheme.

Rather than asking for the lowest price that the subject is willing to pay for prospect \(y_E g\), this mechanism can also be implemented by letting subjects complete a menu list of choices between a sure amount \(q\) and the prospect \(y_Eg\), where \(q\) is increasing for each choice. At the end, one decision is randomly selected for payment. The certainty equivalent is the value of \(q\) where the subject switches from the lottery to the sure amount.

The mechanism can also be presented as a scoring rule. Let \(u\left( z\right) \) be the utility of prize \(z\). Then \(S\left( r,1\right) =P\left( Z\le r\right) u\left( y\right) +\int _{r}^{\infty }u\left( z\right) dP_{Z}\left( z\right) \) and \(S\left( r,0\right) =P\left( Z\le r\right) u\left( g\right) +\int _{r}^{\infty }u\left( z\right) dP_{Z}\left( z\right) \) so that \(ES\left( r,X\right) =P\left( Z\le r\right) \left[ pu\left( y\right) +\left( 1-p\right) u\left( g\right) \right] +\int _{r}^{\infty }u\left( z\right) dP_{Z}\left( z\right) \).

If \(F(x)\) can have discontinuities the general definition is \(F(x_1)<\alpha \) for all \(x_1<x\) and \(F(x)\ge \alpha \).

Harrison et al. (2013a), however, show that risk aversion poses less of a problem for the QSR when it is used to elicit the distribution of a continuous event rather than a binary probability.

All definitions above immediately extend to randomized payment schemes, where the payment to the subject is a realization of some random variable. Here \(S:\varTheta \times {\mathcal{X}}\rightarrow \Delta {\mathbb{R}}\) where \(\Delta {\mathbb{R}}\) denotes the set of distributions over \({\mathbb{R}}\).

Again, this method can implemented with a menu list, see Footnote 5. A problem arises when subjects do not have an unique switching threshold. Heinemann et al. (2009) exclude such subjects.

Vlek (1973b) points out that even with hypothetical payoffs these mechanisms may still matter because they encourage subjects to think in a particular way and may align the preferences of experimenter and subject. An appropriate feedback rule can clarify what the experimenter really wants to know and avoid wrong interpretations. One example of misinterpretations by the subject comes from the elicitation of confidence intervals. Yaniv and Foster (1995, 1997) show that subjects seem to think that 50 % confidence intervals strike the right balance between accuracy and preciseness even when the requested level of confidence is much larger. This interpretation casts doubts on the widespread interpretation that intervals that are too narrow are a sign of ‘overconfidence’. Krawczyk (2011) shows that using incentives for interval elicitation improves the level of calibration of subjects. Winkler and Murphy (1968) provide a discussion of scoring rules as learning devices.

Hurley and Shogren (2005), for example, argue that their inability to recover induced beliefs with a belief elicitation procedure stems from a failure to induce correct beliefs rather than a failure of the elicitation process.

For example, in the context of public goods games, a deeper understanding of the game may have very different implications for selfish individuals (who would reduce contributions) or altruistic individuals (who would increase contributions). Indeed, in the public good game of Gächter and Renner (2010), the interpretation that elicitation improves understanding rests on the assumption that people are conditional cooperators. Note that in this study, the statistical effect is weak and the results are also consistent with a consensus effect or the use of stated beliefs to justify (selfish) actions.

Most of the literature discussed so far is based on the decision theoretic approach by Savage (1954), where subjective utilities are a primitive concept used in evaluating uncertain prospects. In contrast, psychologists have argued that choices may affect beliefs. A discussion of the merits of these approaches is beyond the scope of this paper and we limit ourselves discussing the empirical effect of

*elicitation*on responses. Costa-Gomes et al. (2012) and Smith (2013) use an instrumental variable approach to identify a causal relationship between beliefs and actions.Jaffray and Karni (1999) present mechanisms that can overcome these problems, which require either additional elicitation tasks, or the payment of very large sums of money to exploit the domain where the utility function is relatively flat.

In Offerman et al. (1996), 50 % of the subjects indicate that they would have reported different beliefs in the absence of incentives, often deviating to an ‘easier’ report.

Note that these conditions apply to most studies testing incentivized elicitation schemes, where belief elicitation is typically the only experimental task and thus receives full attention of the subjects. Therefore, these studies may understate effects of incentives in other, more complex, experimental settings (see the comments in Sonnemans and Offerman 2001).

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## Acknowledgments

We would like to thank Peter Wakker, Theo Offerman, Glenn Harrison, Gerhard Sorger and two anonymous referees for useful comments.

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###
*Proof*

(*Proof of Proposition* 1) We use the characterization of Schervish (1989). To simplify exposition assume that \(\nu \) has no point masses and admits a piecewise continuous density \(f\), hence \(S\left( r,1\right) =S\left( 1,1\right) -\int _{r}^{1}\left( 1-c\right) f\left( c\right) dc\) and \(S\left( r,0\right) =S\left( 0,0\right) -\int _{0}^{r}cf\left( c\right) dc\). Consequently, if \(EX=p\) then

and

So \(f\left( r\right) \) describes the strength of the local incentives to tell the truth for reports that are close to \(r\).

Now note that

Assume now w.l.o.g. that the scoring rule gives payoffs in \(\left[ 0,k\right] \) (i.e. \(\omega _1=0\) and \(\omega _2=k\)). For instance, the quadratic scoring rule would be represented as \(S^{QSR}\left( r,1\right) =k\left( 1-\left( 1-r\right) ^{2}\right) \) and \(S^{QSR}\left( r,0\right) =k\left( 1-r^{2}\right) \). It is easy to show that for the QSR, \(f(r)=2k\). Note that

Comparing (10) and (11) it follows that \(f\equiv 2k\) if \(f\left( c\right) \ge 2k\) for all \(c\).

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Schlag, K.H., Tremewan, J. & van der Weele, J.J. A penny for your thoughts: a survey of methods for eliciting beliefs.
*Exp Econ* **18**, 457–490 (2015). https://doi.org/10.1007/s10683-014-9416-x

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DOI: https://doi.org/10.1007/s10683-014-9416-x