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Mixed Strategies in Simultaneous and Sequential Play of a 2 Player Game

Author

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  • Peter Simmons
Abstract
We take a class of games with two players and two actions which only have mixed strategy Nash equilibria. We show that such games can only have hybrid equilibria if played sequentially with one player moving first. The hybrid equilibrium has the leader p laying a mixed strategy but the follower playing a pure strategy. We apply this result to a game between a debtor and a lender following a loan contract. The debtor can have high or low revenues and has to report his state to the lender. The lender can choose whether or not to undertake a costly audit. With simultaneou s play there is only a mixed strategy Nash equilibrium with random cheating in reports and auditing. With sequential play if the debtor moves first, there is zero auditing and the debtor cheats as much as possible without giving the incentive to audit. We argue that the setting of the game and the valuable first mover advantage of the debtor mean that we should expect the game to be played sequentially with this hybrid outcome. This is important in the context of the loan contract since the hybrid outcome makes the contract renegotiation proof. Alternatively if the timing allowed the lender to move first, then the equilibrium would have the debtor reporting truthfully and the monitor auditing just sufficiently to ensure truthtelling by the debtor. This ha s strong links to the optimal debt contract with no commitment (Mookherjee-Png, 1989 and Jost, 1996). However we argue that the natural timing of events makes the debtor the leader. We then consider other examples and show that the same outcome emerges in matching pennies and in a generic inspection game involving adverse selection in labour markets.

Suggested Citation

  • Peter Simmons, "undated". "Mixed Strategies in Simultaneous and Sequential Play of a 2 Player Game," Discussion Papers 03/07, Department of Economics, University of York.
    • Handle: RePEc:yor:yorken:03/07
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    References listed on IDEAS

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    1. Khalil, Fahad & Parigi, Bruno M, 1998. "Loan Size as a Commitment Device," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 39(1), pages 135-150, February.
    2. Stefan Krasa & Anne P. Villamil, 2000. "Optimal Contracts when Enforcement Is a Decision Variable," Econometrica, Econometric Society, vol. 68(1), pages 119-134, January.
    3. Jost, Peter-Jurgen, 1996. "On the Role of Commitment in a Principal-Agent Relationship with an Informed Principal," Journal of Economic Theory, Elsevier, vol. 68(2), pages 510-530, February.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    Mixed Strategies; Loan Contracts; First Mover Advantage;
    All these keywords.

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games
    • D82 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Asymmetric and Private Information; Mechanism Design

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