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Mis-judging merit: the effects of adjudication errors in contests

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Abstract

Within contests, adjudication errors imply at the same time the exclusion of a meritorious candidate and the inclusion of a non-meritorious one. We study theoretically how adjudication errors affect bids in all-pay auctions, by disentangling the respective effects of exclusion and inclusion errors, and showing how they interact with the framing of incentives (prize or penalty) under different assumptions on preferences. We test our theoretical predictions with an experiment where we manipulate the presence of exclusion errors, inclusion errors, and the framing of incentives. The experimental evidence indicates that errors of either exclusion or inclusion significantly decrease bids in all-pay auctions relative to a setting without errors, interacting negatively, with no significant difference in the size of their effects. Bid levels are significantly higher in a penalty framing relative to a prize framing, both in the absence of errors and in the presence of adjudication errors.

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Notes

  1. In fact, as argued by Jia et al., (2013), the randomness described by Lazear and Rosen (1981) can be seen as a stochastic foundation for Tullock contest functions, under specific assumptions on the distribution of the noise term (see also Jia 2008).

  2. We can alternatively assume that contest committees are heterogeneous in their commitment to properly judge candidates, and that contest participants are aware of this: while most committees work hard and do not make errors, few shirk and, in order to avoid an adjudication error, select one of two extreme strategies—reward all contestants or none—depending on which outcome is admitted by the contest terms.

  3. Examples of penalties in the workplace are demotion, wage cut, dismissal or unpaid leave of absence.

  4. For a theoretical analysis of optimal prize structures in contests with loss-averse bidders in the absence of adjudication errors see, for example, Mermer (2016).

  5. The assumption that \(p<\frac{1}{2}\) is without loss of generality and guarantees that, in every error setting and for every such p, there exists a unique equilibrium in mixed strategies.

  6. As we assume that bids are drawn from a distribution, we denote the generic bid in treatment T with boldface letter (\(\varvec{b}^T\)) to indicate that it is a random variable, and we use normal letters to denote its possible realizations.

  7. The random merit extension is contained in Appendix B of Baye et al. (2005). From their model we can easily obtain our adjudication error scenario by assuming that their parameter \(\pi\) that indicates the probability of the exogenous shock is equal to \(1-2p\), and that the litigation environment can be described as an all-pay auction where each party pays its litigation cost (i.e., \(\beta =\alpha =1\) in their model).

  8. It is immediate to verify that \(b=\pi\) solves Eq. :1 with \(F^{N}(\pi)=1\).

  9. Notice that for the equilibrium distribution functions in the scenarios with errors we cannot determine the upper bound of their supports. Yet, an inequality relation that holds for all positive bids between any two distribution functions implies a clear ranking of their upper bounds, and allows us to establish a first-order stochastic dominance relation between the two distributions (see for example proof of Proposition 1 in Online Appendix A).

  10. This can be verified by comparing the denominators of the two distribution functions (see Eqs. 2 and 3). The winning utility under exclusion error falls, relative to no error, by the quantity \(p[u(w+\pi -b)+u(w-b)]\) due to the possibility of not obtaining the prize. The losing utility under inclusion error rises by the same quantity, due to the possibility of being awarded the prize.

  11. Notice that receiving the reward in the penalty scheme (i.e., not receiving the penalty) determines the payoff \(w'\), which is equal by construction to the payoff from receiving the reward in the prize scheme (\(w+\pi\)); not receiving the reward in the penalty scheme (i.e., receiving the penalty) determines the payoff \(w'-\pi\), which is equal to the payoff from not receiving the reward in the prize scheme (w).

  12. To see this, it is enough to multiply by \((1-p)\) both the numerator and the denominator of the expression of \(G^N(b)\) and observe that the denominator coincides with that of \(G^I(b)\) while the numerator is smaller.

  13. Under exclusion error, the winning utility is reduced by the quantity \(p[u(\pi -b)+\lambda u(b)]\), as the higher bidder may obtain a loss instead of a sure gain. Under inclusion error, the losing utility increases by the quantity \(p[u(\pi -b)+\lambda u(b)]\), as the lower bidder may obtain a gain instead of a sure loss.

  14. Notice that whether the bid opportunity cost is larger in the exclusion than in the inclusion error depends on whether it holds that \(\lambda u(b) \geqslant u(\pi )-u(\pi -b)\).

  15. Notice that, for non-linear PT preferences, \(L^{N}\) may differ from \(F^N\), the equilibrium distribution function of bids in a prize scheme with EUT bidders obtained from Eq. 1.

  16. Notice that for a linear u(x), instead, the two equilibrium distributions coincide and thus exclusion and inclusion errors are predicted to have the same effects on equilibrium bids.

  17. Notice that this result does not hold for bidders who are risk loving in the gain domain, as for such bidders \(u(b)< u(\pi +b)-u(b)\), and thus the bid opportunity cost is larger under exclusion error.

  18. The occurrence of an error is determined by two coin tosses. Subjects are informed about the realization of the coin tosses after their bidding decision.

  19. Notice that in order to predict and interpret subjects’ behavior in the experiment, we adopt the population interpretation for mixed strategies and do not assume that each subject actually randomizes. In fact, we do not provide subjects with any randomization device that they could use to pick a bid in the support. Thus, individual i’s bid in the support [0, 1000] has to be interpreted as part of a continuous mixed strategy played by the sub-population from which i is drawn.

  20. Payment of a single randomly selected period was chosen in order to avoid wealth and portfolio effects (see Charness et al., 2016; Azrieli et al., 2018).

  21. The task in Dave et al. (2010) measures risk aversion on the basis of a lottery choice within a menu of six lotteries. The risk-aversion indicator takes values from 1 to 6 depending on the choice made in the risk task. The task in Gächter et al. (2022) measures loss aversion on the basis of the gain-loss ratio in risky choices involving mixed gain-loss prospects. The loss-aversion indicator takes values from 1 to 7 depending on the lottery switching point, with higher values corresponding to rejecting lotteries with smaller losses (subjects who made inconsistent choices were excluded). Further details on the tasks and the distributions of risk and loss indicators in our experiment are reported in Online Appendix B.

  22. Notice that we use the indicator of risk aversion as a proxy for decreasing marginal utility in gains of loss-averse bidders, which is appropriate under the assumption that the payoff obtained in the experiment is not taken as a reference point for the lottery choice task.

  23. As in Dave et al. (2010), we classify as risk neutral subjects with a risk-aversion indicator equal to 2. The results in Table 9 are robust to the inclusion of subjects with an indicator equal to 3, which is below the median.

  24. The existence of a cap on bids is one of the arguments provided by Potters et al. (1998) to explain why they do not find strong evidence of overbidding in an all-pay auction setting that shares some key features with ours: number of bidders, matching protocol, and large number of rounds. In every round, subjects receive an endowment that is slightly above the prize value, and this cap may exert a stronger pressure on average bids with respect to Gneezy and Smorodinsky (2006), making Potters et al. (1998) results more similar to ours.

  25. In contrast, in a different context Levitt et al. (2016) find that the framing of educational incentives as gains or losses does not significantly affect effort. Another notable exception is the study of De Quidt et al. (2017).

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Acknowledgements

We gratefully acknowledge financial support from the Einaudi Institute for Economics and Finance (EIEF). We thank for their useful comments Pierpaolo Battigalli, Lorenzo Cappellari, Marco Faravelli, Marco Mantovani, Elena Manzoni, Matteo Rizzolli, seminar participants at the SAET conference, held in Ischia, at the conference Contests: Theory and Evidence held at the University of East Anglia, at the IMEBESS conference, held at LUISS, and two anonymous referees. Any remaining errors are our responsability.

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  1. Department of Economics, University of Insubria, Via Monte Generoso 71, 21100, Varese, Italy

    Astrid Gamba

  2. Department of Economics Management and Statistics, and NeuroMI, University of Milan-Bicocca, Piazza dell’Ateneo Nuovo 1, 20126, Milan, Italy

    Luca Stanca

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Gamba, A., Stanca, L. Mis-judging merit: the effects of adjudication errors in contests. Exp Econ 26, 550–587 (2023). https://doi.org/10.1007/s10683-022-09785-4

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