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Feasibility and individual rationality in two-person Bayesian games

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Abstract

We define feasible, posterior individually rational solutions for two-person Bayesian games with a single informed player. Such a solution can be achieved by direct signalling from the informed player and requires approval of both players after the signal has been sent. Without further assumptions on the Bayesian game, a solution does not necessarily exist. We show that, if the uninformed player has a “uniform punishment strategy” against the informed one, the existence of a solution follows from the existence of Nash equilibrium in infinitely repeated games with lack of information on one side. We also consider the extension of the result when both players have private information.

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Notes

  1. Kalai et al. (2010) and Tennenholtz (2004) consider conditional contracts in games with complete information.

  2. For any finite set E, \(\Delta (E)\) denotes the set of probability distributions over E.

  3. The word “solution” may look too strong and is just used for convenience. As will become clear below, we do not make a list of all properties that a solution should satisfy but concentrate on two of them.

  4. The properties detailed below are well identified in Sorin (2002).

  5. The index 1 in \(f_{1}(\cdot )\) is for “one-shot”.

  6. As for the informed player, the index 1 in \(g_{1}(\cdot )\) is for “one-shot”.

  7. For simplicity, we just refer to “Nash equilibrium,” but given the Bayesian game behind the noncooperative game designed for implementation, we might as well use the terminology “Bayesian Nash equilibrium.” Except in Sect. 4.2, we do not impose any refinement.

  8. We thank the referee for suggesting the game \(\Gamma (p)\), which does not depend on the joint decision mapping \(\chi \).

  9. If \(\chi (s^{\prime })=\chi (s^{\prime \prime })\), \(s^{\prime }\) and \( s^{\prime \prime }\) can be merged into a single signal s, such that \(\mu (s\mid k)=\mu (s^{\prime }\mid k)+\mu (s^{\prime \prime }\mid k)\). Incentive compatibility and individual rationality for player 1 are still satisfied. Individual rationality for player 2 follows from the facts that the posterior \(p_{s}\) is a convex combination of \(p_{s^{\prime }}\) and \( p_{s^{\prime \prime }}\) and that \(g_{1}\) is convex.

  10. As soon as there is a mediator, the revelation principle applies and without loss of generality, player 1’s possible signals are just his possible types.

  11. Nonemptiness of \(\mathcal {M}\left[ B(p)\right] \) can also be checked directly, since a mediated solution is defined by linear inequalities. Myerson starts by defining \(\mathcal {M}\left[ B(p)\right] \) but goes on by imposing further requirements on cooperative solutions.

  12. In the context of \(B_{\infty }(p)\), a correlated strategy in \(\Delta (A)\) is interpreted as a limit frequency of moves.

  13. As illustrated by Example 2, the first inequality can be strict for some \(q\in \Delta (K)\). If player 1 cannot make use of his information, the “uniform punishment strategy” \(\tau ^{*}\) of player 2 is not necessarily the worst punishment of player 2 against player 1.

  14. More precisely, recalling the definition of the concave mapping \(f_{1}\) [i.e., (4)] and Lemma 9, under UPS, \(cavf_{NR}(q)=f_{1}(q)= \sum _{k\in K}q^{k}\underline{u}^{k}\).

  15. The mapping \(g_{1}\) is convex. Hence the value of information for player 2 is positive and there is no way to lower player 2’s individually rational level as in Celik and Peters (2011, 2016).

  16. The proof is still inherited from the repeated games literature: Renault (2000)’s Proposition 4.2—also stated as Theorem 4 in Simon et al. (2008)—is a consequence of the proof of Simon et al. (1995).

  17. Since values are private and utility functions are linear, a player only cares about his own type and the expected distribution over actions.

  18. Koren (1992)’s characterization is more complex in that it makes use of Blackwell’s approachability, i.e., individual rationality is defined as in (11).

  19. The undiscounted infinitely repeated version of the model of Sect. 3.2.2. (namely, lack of information on one side and private values) was studied by Shalev (1994). He showed that all Nash equilibria are payoff equivalent to completely revealing ones [the result that was extended by Koren (1992)] and that Nash equilibria always exist (the reasoning in Sect. 3.2.2 is similar to Shalev (1994)’s one).

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Authors and Affiliations

  1. CEREMADE and LEDa, PSL, Université Paris-Dauphine, Paris, France

    Françoise Forges

  2. Department of Mathematics, Humboldt University Berlin, Berlin, Germany

    Ulrich Horst

  3. LEDa, PSL, Université Paris-Dauphine, Paris, France

    Antoine Salomon

Authors
  1. Françoise Forges
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  2. Ulrich Horst
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  3. Antoine Salomon
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Corresponding author

Correspondence to Françoise Forges.

Additional information

We are grateful to Gorkem Celik, Marie Laclau, Antoine Loeper, Jérô me Renault, Rann Smorodinsky, Bernhard von Stengel, Roland Strausz, an anonymous referee and seminar participants at Vanderbilt University for useful comments. F. Forges acknowledges support from Institut Universitaire de France. U. Horst acknowledges financial support through SFB 649 Economic Risk.

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Forges, F., Horst, U. & Salomon, A. Feasibility and individual rationality in two-person Bayesian games. Int J Game Theory 45, 11–36 (2016). https://doi.org/10.1007/s00182-015-0520-8

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