Belief formation in a signaling game without common prior: an experiment

Abstract

Using belief elicitation, the paper investigates the process of belief formation and evolution in a signaling game in which a common prior is not induced. Both prior and posterior beliefs of Receivers about Senders’ types are elicited, as well as beliefs of Senders about Receivers’ strategies. In the experiment, subjects often start with diffuse uniform beliefs and update them in view of observations. However, the speed of updating is influenced by the strength of initial beliefs. An interesting result is that beliefs about the prior distribution of types are updated slower than posterior beliefs, which incorporate Senders’ strategies. In the medium run, for some specifications of game parameters, this leads to outcomes being significantly different from the outcomes of the game in which a common prior is induced. It is also shown that elicitation of beliefs does not considerably change the pattern of play in this game.

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Notes

  1. 1.

    Decisions from experience, and their differences from decisions from description, have been reviewed and investigated in psychology literature by, among others, Hertwig and Erev (2009), Gonzalez and Dutt (2011) and Ludvig and Spetch (2011). Often an additional complication in those studies is that sampling size is endogenous; in contrast, in our case the sampling size is fixed and known.

  2. 2.

    Chen et al. (2007) also investigated a setting with incomplete information, an auction, with and without induced knowledge of the prior distribution of players’ valuations. They found differences in behavior between these two cases in initial rounds but not in later rounds.

  3. 3.

    Different procedures for eliciting beliefs are compared and reviewed in Palfrey and Wang (2009), Trautmann and van de Kuilen (2015) and Hollard et al. (2016). The quadratic scoring rule appears to be a reasonable compromise between the reliability of a rule and its complexity.

  4. 4.

    There is also an equilibrium in partially mixed strategies, for each value of p. However, these equilibria are unstable under many specifications of adjustment dynamics and indeed are not observed in the data.

  5. 5.

    If risk- and ambiguity-neutral; these assumptions are maintained throughout the discussion, also for the Receiver.

  6. 6.

    The presence of players with a higher level of reasoning than level-1 would accelerate the process but not change the final outcome.

  7. 7.

    The stationarity of the distribution was emphasized in the experiment instructions.

  8. 8.

    Offerman et al. (2009) and Andersen et al. (2012), among others, propose corrections of belief reports to account for different attitudes to risk and ambiguity. These corrections, however, require participants to perform additional tasks. To avoid the additional complexity, and since the focus is on the comparison of beliefs, all likely affected by risk and ambiguity attitudes in a similar manner, no such corrections are applied in this paper.

  9. 9.

    See a sample of the experiment instructions in Section A of Supplementary Materials for more details.

  10. 10.

    Of course, eliciting beliefs more often would have allowed to collect more data; however, this can make the task routine and incentives would have to be smaller. Facing the trade-off between paying less every period or having a higher payment every few periods, the latter option was chosen since it gives the subjects more incentives to take the belief reporting task seriously.

  11. 11.

    One session, in treatment K3, had only 8 participants.

  12. 12.

    Sinn (1980) provides a relatively recent theoretical analysis of it and Binmore et al. (2012) give recent experimental evidence in its favor.

  13. 13.

    Detailed test results for these and other tests in this section are given in Section B.1 of Supplementary Materials.

  14. 14.

    For posterior beliefs in K2 and K3, observations similar to those about posterior beliefs in K1 can be made.

  15. 15.

    This is also evidence that hedging is not a large problem, since it would imply choosing the other action, not the best response.

  16. 16.

    For the game played, the worst-case scenario for the Sender after \(m_{1}\) is to get 15 and after \(m_{2}\) is to get 25. Thus, \(m_{2}\) is always consistent with best response since there always exists a belief that \(m_{1}\) leads to a lower payoff. After \(m_{1}\) the expected payoff is \(15\cdot \Pr (a_{1})+80\cdot (1-\Pr (a_{1}))\) for type \(t_{1}\) and \(80\cdot \Pr (a_{1})+15\cdot (1-\Pr (a_{1}))\) for type \(t_{2}\). The chosen message \(m_{1}\) would be inconsistent with best response if the reported beliefs \(\Pr (a_{1})\) in response to it were more than 11 / 13 for type \(t_{1}\) and less than 2 / 13 for type \(t_{2}\).

  17. 17.

    For N2, there is no statistically significant difference between beliefs in Periods 1 and 36. The test results for this and subsequent subsections are reported in Section B.2 of Supplementary Materials.

  18. 18.

    Since there are only a few observations per subject, estimating parameters for each subject is infeasible.

  19. 19.

    Indeed, play did not converge clearly to either of the separating equilibria in treatment N1; see Fig. 6 in Sect. 4.3 for the evolution of the average proportions of each strategy played.

  20. 20.

    Scores are based on all treatments, not only on those with \(p=1/4\). The SSE score for the forgetting model is very similar to that of the baseline model; the score for the initial strength model without forgetting is similar to that of this model with forgetting (see Section B.2 of Supplementary Materials). Thus only the baseline and the full (initial strength and forgetting) scores are reported.

  21. 21.

    It is the predictions of the model with \(\gamma =0.97\) and \(A_\mathrm{Ps}=2.29\) that are also shown in Fig. 4 by dotted lines.

  22. 22.

    Scores are again based on all treatments, not only on those with \(p=1/4\). The SSE score for the forgetting model is very similar to that of the baseline model; the score for the initial strength model without forgetting is similar to that of this model with forgetting (see Section B.2 of Supplementary Materials). Thus only the baseline and the full (initial strength and forgetting) scores are reported.

  23. 23.

    The same results holds for tests based on all periods or on the last eight periods. The data on which the tests are based are given in Section B.3 of Supplementary Materials, also for other tests in this section.

  24. 24.

    The tests are reported in Section B.4 of Supplementary Materials.

References

  1. Andersen, S., Fountain, J., Harrison, G. W., Hole, A. R., & Rutström, E. E. (2012). Inferring beliefs as subjectively uncertain probabilities. Theory and Decision, 73, 161–183.

    Article  Google Scholar 

  2. Anderson, C. M., & Camerer, C. F. (2000). Experience-weighted attraction learning in sender–receiver signaling games. Economic Theory, 16, 689–718.

    Google Scholar 

  3. Armantier, O., & Treich, N. (2013). Eliciting beliefs: Proper scoring rules, incentives, stakes, and hedging. European Economic Review, 62, 17–40.

    Article  Google Scholar 

  4. Binmore, K., Stewart, L., & Voorhoeve, A. (2012). How much ambiguity aversion? Finding indifferences between Ellsberg’s risky and ambiguous bets. Journal of Risk and Uncertainty, 45, 215–238.

    Article  Google Scholar 

  5. Blanco, M., Engelmann, D., Koch, A. K., & Normann, H.-T. (2010). Belief elicitation in experiments: Is there a hedging problem? Experimental Economics, 13, 412–438.

    Article  Google Scholar 

  6. Brandts, J., & Holt, C. A. (1996). Naive Bayesian learning and adjustment to equilibrium in signaling games. Working paper, Instituto de Análisis Económico (CSIC), Barcelona and University of Virginia (unpublished).

  7. Camerer, C. F., & Ho, T. (1999). Experience-weighted attraction learning in normal form games. Econometrica, 67, 827–874.

    Article  Google Scholar 

  8. Chen, Y., Katuščák, P., & Ozdenoren, E. (2007). Sealed bid auctions with ambiguity: Theory and experiments. Journal of Economic Theory, 136, 513–535.

    Article  Google Scholar 

  9. Cheung, Y.-W., & Friedman, D. (1997). Individual learning in normal form games: Some laboratory results. Games and Economic Behavior, 19, 46–76.

    Article  Google Scholar 

  10. Costa-Gomes, M. A., & Weizsäcker, G. (2008). Stated beliefs and play in normal-form games. Review of Economic Studies, 75, 729–765.

    Article  Google Scholar 

  11. Drouvelis, M., Müller, W., & Possajennikov, A. (2012). Signaling without a common prior: Results on experimental equilibrium selection. Games and Economic Behavior, 74, 102–119.

    Article  Google Scholar 

  12. Eichberger, J., Kelsey, D., & Schipper, B. (2008). Granny versus game theorist: Ambiguity in experimental games. Theory and Decision, 64, 333–362.

    Article  Google Scholar 

  13. Fischbacher, U. (2007). z-Tree: Zurich toolbox for ready-made economic experiments. Experimental Economics, 10, 171–178.

    Article  Google Scholar 

  14. Gonzalez, C., & Dutt, V. (2011). Instance-based learning: Integrating sampling and repeated decisions from experience. Psychological Review, 118, 523–551.

    Article  Google Scholar 

  15. Greiner, B. (2015). Subject pool recruitment procedures: Organizing experiments with ORSEE. Journal of the Economic Science Association, 1, 114–125.

    Article  Google Scholar 

  16. Harsanyi, J. C. (1967). Games with incomplete information played by “Bayesian” players. Part I. The basic model. Management Science, 14, 159–182.

    Article  Google Scholar 

  17. Hertwig, R., & Erev, I. (2009). The description-experience gap in risky choice. Trends in Cognitive Sciences, 13, 517–523.

    Article  Google Scholar 

  18. Hollard, G., Massoni, S., & Vergnaud, J.-C. (2016). In search of good probability assessors: An experimental comparison of elicitation rules for confidence judgments. Theory and Decision, 80, 363–387.

    Article  Google Scholar 

  19. Hyndman, K., Özbay, E. Y., Schotter, A., & Ehrblatt, W. (2011). Belief formation: An experiment with outside observers. Experimental Economics, 15, 176–203.

    Article  Google Scholar 

  20. Ivanov, A. (2011). Attitudes to ambiguity in one-shot normal form games: An experimental study. Games and Economic Behavior, 71, 366–394.

    Article  Google Scholar 

  21. Kelsey, D., & le Roux, S. (2015). An experimental study on the effect of ambiguity in a coordination game. Theory and Decision, 79, 667–688.

    Article  Google Scholar 

  22. Li, Z., Loomes, G., & Pogrebna, G. (2017). Attitudes to uncertainty in a strategic setting. Economic Journal, 127, 809–826.

  23. Ludvig, E. A., & Spetch, M. L. (2011). Of Black Swans and Tossed coins: Is the description-experience gap in risky choice limited to rare events? PLoS ONE, 6, e20262.

    Article  Google Scholar 

  24. Nickerson, R. S. (2004). Cognition and Chance. Mahwah, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  25. Nyarko, Y., & Schotter, A. (2002). An experimental study of belief learning using elicited beliefs. Econometrica, 70, 971–1005.

    Article  Google Scholar 

  26. Offerman, T., Sonnemans, J., van de Kuilen, G., & Wakker, P. P. (2009). A truth-serum for non-Bayesians: Correcting proper scoring rules for risk attitudes. Review of Economic Studies, 76, 1461–1489.

    Article  Google Scholar 

  27. Palfrey, T. R., & Wang, S. W. (2009). On eliciting beliefs in strategic games. Journal of Economic Behavior & Organization, 71, 98–109.

    Article  Google Scholar 

  28. Rutström, E. E., & Wilcox, N. T. (2009). Stated beliefs versus inferred beliefs: A methodological inquiry and experimental test. Games and Economic Behavior, 67, 616–632.

    Article  Google Scholar 

  29. Sinn, H.-W. (1980). A rehabilitation of the principle of insufficient reason. Quarterly Journal of Economics, 94, 493–506.

    Article  Google Scholar 

  30. Stahl, D. O., & Wilson, P. W. (1994). Experimental evidence of players models of other players. Journal of Economic Behavior & Organization, 25, 309–327.

    Article  Google Scholar 

  31. Stahl, D. O., & Wilson, P. W. (1995). On players models of other players: Theory and experimental evidence. Games and Economic Behavior, 10, 218–254.

    Article  Google Scholar 

  32. Trautmann, S. T., & van de Kuilen, G. (2015). Belief Elicitation: A Horse Race among Truth Serums. Economic Journal, 125, 2116–2135.

    Article  Google Scholar 

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Acknowledgements

I would like to thank the School of Economics, University of Nottingham, for financial support and CeDEx for providing access to the infrastructure to run the experiment. At different stages of the project, the paper benefitted from presentations at various conferences, including Foundations of Utility and Risk (FUR) 2016 conference. I thank the editor of this special issue, Ganna Pogrebna, for providing an opportunity for papers presented at the 2016 FUR conference to be considered for publication. I am grateful to an anonymous referee for comments that led to improvements in the paper and to Maria Montero for suggestions to make the exposition better.

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Correspondence to Alex Possajennikov.

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Possajennikov, A. Belief formation in a signaling game without common prior: an experiment. Theory Decis 84, 483–505 (2018). https://doi.org/10.1007/s11238-017-9614-z

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Keywords

  • Beliefs
  • Signaling
  • Experiment
  • Learning
  • Belief elicitation