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Information and beliefs in a repeated normal-form game


We study beliefs and actions in a repeated normal-form game. Using a level-k model of limited strategic reasoning and allowing for other-regarding preferences, we classify action and belief choices with regard to their strategic sophistication and study their development over time. In addition to a baseline treatment with common knowledge of the game structure, feedback about actions in the previous period and random matching, we run treatments (i) with fixed matching, (ii) without information about the other player’s payoffs, and (iii) without feedback about previous play. In all treatments with feedback, we observe more strategic play (increasing by 15 percent) and higher-level beliefs (increasing by 18 percent) over time. Without feedback, neither beliefs nor actions reach significantly higher levels of reasoning (with increases of 2 percentage points for actions and 6 percentage points for beliefs). The levels of reasoning reflected in actions and beliefs are highly consistent, but less so for types with lower levels of reasoning.

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  1. If subjects have other-regarding or efficiency-oriented preferences, the game we study has multiple equilibria. However, the level-k model is well suited to make a unique prediction for such games, thereby avoiding a selection problem.

  2. The level-k model has been associated with introspection by Goeree and Holt (2004). For one-shot games, introspection or deductive reasoning is clearly the obvious choice. However, when levels of reasoning are analyzed in repeated interactions, feedback may affect beliefs and may thereby lead to higher-level play.

  3. It is conceivable that experience and observation of past play reduce sophistication relative to play without feedback. In a feedback-free environment, subjects are presumably forced to think about the game and therefore they may acquire simple solution concepts such as iterated dominance or backwards induction. Rick and Weber (2010) demonstrate that subjects are able to acquire and to transfer such concepts to similar games.

  4. Oechssler and Schipper (2006) used a similar setup to study subjects’ ability to learn about the game payoffs. So-called “minimal social situations”, introduced by Sidowski (1957), have been studied widely by psychologists. They differ from our control treatment in that players do not know their own payoffs, nor the other player’s strategy set. Often they are not even aware that they are in a situation with interdependent payoffs.

  5. Hyndman et al. (2012) ran a treatment with incomplete information about the other player’s payoffs, but none of the mentioned papers employs a treatment without feedback.

  6. The unique Nash equilibrium of the stage game is also the unique subgame perfect equilibrium of the repeated game. However, there exist Nash equilibria of the finitely repeated game with fixed matching that are not subgame perfect. For example, playing the Pareto-efficient strategy combination (Bottom, Right) for at most 17 periods and then reverting to the Nash equilibrium (Top, Left) for the remaining three periods is a Nash Equilibrium.

  7. We are grateful to a referee for pointing out that this widely-used model is originally due to Edgeworth. For more recent applications see Anderson et al. (1998), Goeree et al. (2002) or Cox et al. (2007).

  8. While in Stahl and Wilson’s formulation a level-k type best responds to a distribution of lower level types, we use the described alternative formulation, as for example, in Nagel (1995) or Costa-Gomes et al. (2001). The level-k model has been tested and extended by various other studies in the context of one-shot normal-form games (see, among others, Costa-Gomes et al. 2001; Costa-Gomes and Weizsäcker 2008; Rey-Biel 2009; and Camerer et al. 2004). The most common types found in normal-form games are level-1, level-2 and Nash types.

  9. We use the labels L1, \(L2_{\theta>0.15}^{+}\) etc. also in treatment PI even though a priori the subjects are not aware of the other player’s payoffs and other-regarding preferences are less relevant. However, subjects can use their feedback to construct a “subjective game”. Kalai and Lehrer (1993) show that subjective games can converge to an ε-Nash equilibrium of the underlying game.

  10. Upon entering the lab, subjects received written instructions including an understanding test. The experiment only started after all subjects had answered the questions correctly. For a sample of the instructions see the Appendix.

  11. Of course, subjects could infer their belief-elicitation payoff from the game feedback. The main reason for not providing the belief-elicitation payoff was to change as few parameters as possible when going from RM, PI and FM to NF.

  12. Subjects may for example coordinate on some non-equilibrium outcome and become unwilling to move away from it in order to avoid losses in the belief task (lock-in effect) or they may try to hedge with the stated belief against adverse outcomes of the action task within a given period. Blanco et al. (2010) investigate the potential of hedging and find no evidence for it unless the hedging opportunity is very salient. Their hedging-proof procedure either pays the decision or the belief-elicitation task randomly.

  13. When considering first-period choices, we can compare our results to experiments using one-shot games. Pooling the data over player roles, we observe 51% L1/L1+ behavior in the first period in RM, FM and NF. This is in line with previous studies. For instance, Costa-Gomes et al. (2001) observed a frequency of L1 actions of about 45%, Rey-Biel (2009) found 48% L1 behavior in constant-sum games, whereas Costa-Gomes and Weizsäcker (2008) observed slightly higher rates of about 60%.

  14. All results reported as significant in the paper are based on a 5%-level of significance.

  15. Note that at first sight our results in treatment NF differ somewhat from the results of Rick and Weber (2010) who find significant learning without feedback information. However, in their repeated asymmetric 3×3 game which is closest to the game we use, the absolute changes in action choices are comparable in size to our results.

  16. We run probit regressions with L1-beliefs as the dependent variable and a dummy for player role as the independent variable where we corrected for repeated observations on the individual level. The lowest p-value is found in RM (p=0.167).

  17. Similarly, aggregated comparisons between actions and beliefs (such as the row player’s fractions of Middle choices and L1 beliefs) are consistent with a unique underlying distribution of levels of reasoning (see Figs. 1 and 4). Together with the reported comparisons within the action as well as the belief space, only 4 out of 72 possible tests (5.6%) on the treatment level contradict the level-k predictions.

  18. The only qualitative differences to Table 6 are that the row players’ weight trends of L2+ in RM and \(L4_{\theta\leq .15}^{+}\) in PI are insignificant. However, the non-parametric tests are based on crude classifications of stated beliefs as compared to the mixture model approach in (3) that accounts for the proximity of the stated beliefs to each underlying belief.

  19. If θ changes between the statement of the belief and the action choice within one period, we count this as an inconsistency. We also expect that subjects with a θ near the limits of the intervals for which the predicted action and belief change show more inconsistent behaviour if θ is not stable within periods.

  20. We run a probit regression with an L1-action (\(L3_{\theta\leq.15}^{+}\)-action) indicator variable as the dependent variable and an L1-belief (\(L3_{\theta\leq.15}^{+}\)-belief) indicator variable as the independent variable (standard errors corrected for observational clusters on the individual level), which yields p=0.003 (p<0.001). Fisher’s exact test also rejects the independence of both measures for both player roles at p<0.001. On the treatment level, independence for the row player is rejected in FM and RM, while it is rejected in all treatments for the column player.

  21. We can compare our data to best-response rates found in similar studies by assuming θ=0 and calculating the best-response rate as the fraction of choosing the action with the highest expected payoff given one’s belief as in the literature. This yields an overall best response rate of 63% for our data. In related experiments of normal-form games with constant-sum 2×2 games (Nyarko and Schotter 2002) or with 3×3 constant-sum and variable-sum games (Rey-Biel 2009), best response rates are about 70%, whereas the rates range from 49% to 63% in the games used in Costa-Gomes and Weizsäcker (2008) and Hyndman et al. (2012).

  22. For the exact findings, see the working paper by Fehr et al. (2008). In contrast, Rutstrom and Wilxoc (2009) find that inferred beliefs can be better than stated belief in a matching pennies game, especially when payoffs are highly asymmetric.


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Correspondence to Dietmar Fehr.

Additional information

We are grateful for valuable comments of the editor, Jacob Goeree, and two anonymous referees whose critical comments helped us to improve the paper. We also wish to thank Britt Grosskopf, Kyle Hyndman, Rajiv Sarin, Roberto Weber, Georg Weizsäcker, Axel Werwatz and seminar participants at the Humboldt-Universität zu Berlin, EUI Florence, CRC 649 Workshop 2007, ESA World Meeting 2007, IMEBE 2008, VfS Annual Meeting 2008 and Econometric Society Meetings 2009 for valuable comments. We are indebted to Jana Stöver and Susanne Thiel for their help with running the experiments. Financial support from the Deutsche Forschungsgemeinschaft via CRC 649 Economic Risk is gratefully acknowledged.

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Appendix Sample Instructions (for FM) (PDF 175 kB)

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Danz, D.N., Fehr, D. & Kübler, D. Information and beliefs in a repeated normal-form game. Exp Econ 15, 622–640 (2012).

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  • Beliefs
  • Level-k model
  • Learning

JEL Classification

  • C72
  • C92
  • D83
  • D84