Abstract
In this paper, we use experimental data to study players’ stability in normal-form games where subjects have to report beliefs and choose actions. Subjects saw each of 12 games four times in a regular or isomorphic form spread over two days without feedback. We document a high degree of stability within the same (strategically equivalent) game, although time and changes in the presentation of the game do lead to less stability. To look at stability across different games, we adopt the level-k theory, and show that stability of both beliefs and actions is significantly lower. Finally, we estimate a structural model in which players either apply a consistent level of reasoning across strategically different games, or reasoning levels change from game to game. Our results show that approximately 23% of subjects apply a consistent level of reasoning across the 12 games, but that they assign a low level of sophistication to their opponent. The remaining 77% apply different levels of reasoning to different games. We show that this may be due to subjects being attracted to the action with the highest possible payoff.
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Notes
Although not central to our study, our experimental design also allows us to test the robustness of Costa-Gomes and Weizsäcker (2008) who showed that subjects’ actions and beliefs are inconsistent with each other.
While risk aversion may play a role, we do not think that it leads to biased beliefs since subjects best-respond to uniform beliefs at about the same rate as they do to more extreme beliefs. We discuss this in Online Appendix D.
In contrast, many of the games in Costa-Gomes and Weizsäcker (2008) had, in addition to the unique pure strategy equilibrium, two mixed strategy Nash equilibria. Although an analysis of their data did not find any evidence that subjects played any of the mixed strategy equilibria, we wanted our games to be as “clean” as possible and, therefore, made every effort to ensure that there were no mixed strategy equilibria.
To be sure, we differ from the procedure of Costa-Gomes and Weizsäcker (2008) in that, for each game, subjects first state a belief and then choose an action. They had subjects either first state all beliefs and then choose all actions, or vice-versa. Nevertheless, their null results suggest that this design choice is unlikely to materially affect behavior.
Note that the “other action” is rarely a Nash action, and when it is, it is one of two Nash equilibrium actions in the multiple equilibria games.
We restrict our sample to games with a unique equilibrium and for which the opponent’s \(L_1\) and \(L_2\) actions differ.
We can confirm the same insights parametrically. Specifically, a conditional fixed-effects logistic regression of “choosing a best-response” on dummies for having beliefs near the corners of the simplex and “being close to the uniform beliefs” with a full set of game, day and period dummies. Just as shown in Fig. 3, having strong beliefs toward the \(L_2\) or other action increases the rate of best-response, as does having beliefs which are close to uniform.
Additionally, the raw Spearman correlation between the indexes is 0.1940.
If we did not normalize, then beliefs close to the centre of the simplex would look more stable because the maximum distance beliefs could change is smaller than more extreme beliefs.
Note that the above analysis can be repeated in a regression framework. In the interest of space, we omit these results, but remark that both isomorphic changes in the game frame between consecutive instances and a 7-day delay between consecutive instances are associated with larger normalized Euclidean distance between belief statements.
Of course, it is possible, given potential incentives for hedging, that subjects beliefs and actions could be independent or even negatively related. The fact that the relationship is positive suggests that subjects are using their beliefs as the basis for their actions.
Here, \(h=1,\dots , 48\) denotes the 48 beliefs statements an individual makes during the experiment, and \({\mathcal {G}}_g\) denotes the set of all belief statements that are instances of the same set of strategically equivalent games g.
Using t-tests at the individual level, we find that the differences in beliefs are significantly higher for movers than for stayers (\(p<0.001\) for panels (a) and (b), \(p=0.02\) for panel (c)).
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Acknowledgements
Kyle Hyndman would like to thank Southern Methodist University, where he began working on this project, and the University of Texas at Dallas, where he finished working on the project, for financial support. Antoine Terracol and Jonathan Vaksmann would like to thank the scientific council of the Université du Maine and Paris School of Economics for financial and logistical support. Maxim Frolov is also acknowledged for technical assistance. We also gratefully acknowledge the valuable comments received by Colin Camerer and Andrew Schotter, participants at the Workshop on the Formation and Elicitation of Beliefs in Experiments (Berlin) as well as the 2012 International Economic Science Association Conference.
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Hyndman, K., Terracol, A. & Vaksmann, J. Beliefs and (in)stability in normal-form games. Exp Econ 25, 1146–1172 (2022). https://doi.org/10.1007/s10683-022-09747-w
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DOI: https://doi.org/10.1007/s10683-022-09747-w