IDEAS home Printed from https://ideas.repec.org/p/ecm/nasm04/625.html
   My bibliography  Save this paper

Growing Strategy Sets in Repeated Games

Author

Listed:
  • Daijiro Okada
  • Abraham Neyman

Abstract

A (pure) strategy in a repeated game is a mapping from histories, or, more generally, signals, to actions. We view the implementation of such a strategy as a computational procedure and attempt to capture in a formal model the following intuition: as the game proceeds, the amount of information (history) to be taken into account becomes large and the \quo{computational burden} becomes increasingly heavy. The number of strategies in repeated games grows double-exponentially with the number of repetitions. This is due to the fact that the number of histories grows exponentially with the number of repetitions and also that we count strategies that map histories into actions in all possible ways. Any model that captures the intuition mentioned above would impose some restriction on the way the set of strategies available at each stage expands. We point out that existing measures of complexity of a strategy, such as the number of states of an automaton that represents the strategy needs to be refined in order to capture the notion of growing strategy space. Thus we propose a general model of repeated game strategies which are implementable by automata with growing number of states with restrictions on the rate of growth. With such model, we revisit some of the past results concerning the repeated games with finite automata whose number of states are bounded by a constant, e.g., Ben-Porath (1993) in the case of two-person infinitely repeated games. In addition, we study an undiscounted infinitely repeated two-person zero-sum game in which the strategy set of player 1, the maximizer, expands \quo{slowly} while there is no restriction on player 2's strategy space. Our main result is that, if the number of strategies available to player 1 at stage $n$ grows subexponentially with $n$, then player 2 has a pure optimal strategy and the value of the game is the maxmin value of the stage game, the lowest payoff that player 1 can guarantee in one-shot game. This result is independent of whether strategies can be implemented by automaton or not. This is a strong result in that an optimal strategy in an infinitely repeated game has, by definition, a property that, for every $c$, it holds player 1's payoff to at most the value plus $c$ after some stage

Suggested Citation

  • Daijiro Okada & Abraham Neyman, 2004. "Growing Strategy Sets in Repeated Games," Econometric Society 2004 North American Summer Meetings 625, Econometric Society.
  • Handle: RePEc:ecm:nasm04:625
    as

    Download full text from publisher

    File URL: http://repec.org/esNASM04/up.28915.1075757972.pdf
    Download Restriction: no

    References listed on IDEAS

    as
    1. Abreu, Dilip & Rubinstein, Ariel, 1988. "The Structure of Nash Equilibrium in Repeated Games with Finite Automata," Econometrica, Econometric Society, vol. 56(6), pages 1259-1281, November.
    2. Gilboa, Itzhak, 1988. "The complexity of computing best-response automata in repeated games," Journal of Economic Theory, Elsevier, vol. 45(2), pages 342-352, August.
    3. Rubinstein, Ariel, 1986. "Finite automata play the repeated prisoner's dilemma," Journal of Economic Theory, Elsevier, vol. 39(1), pages 83-96, June.
    4. Ben-Porath Elchanan, 1993. "Repeated Games with Finite Automata," Journal of Economic Theory, Elsevier, vol. 59(1), pages 17-32, February.
    5. Abraham Neyman & Daijiro Okada, 2000. "Two-person repeated games with finite automata," International Journal of Game Theory, Springer;Game Theory Society, vol. 29(3), pages 309-325.
    Full references (including those not matched with items on IDEAS)

    More about this item

    Keywords

    Repeated Games; Complexity; Entropy;

    JEL classification:

    • C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:ecm:nasm04:625. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Christopher F. Baum). General contact details of provider: http://edirc.repec.org/data/essssea.html .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.