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Specifying a game-theoretic extensive form as an abstract 5-ary relation

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  • Peter A. Streufert

    (Western University)

Abstract

This paper specifies an extensive form as a 5-ary relation (that is, as a set of quintuples) which satisfies eight abstract axioms. Each quintuple is understood to list a player, a situation (that is, a name for an information set), a decision node, an action, and a successor node. Accordingly, the axioms are understood to specify abstract relationships between players, situations, nodes, and actions. Such an extensive form is called a “pentaform”. Finally, a “pentaform game” is defined to be a pentaform together with utility functions. To ground this new specification in the literature, the paper defines the concept of a “traditional game” to represent the literature’s many specifications of finite-horizon and infinite-horizon games. The paper’s main result is to construct an intuitive bijection between pentaform games and traditional games. Secondary results concern disaggregating pentaforms by subsets, constructing pentaforms by unions, and initial pentaform applications to Selten subgames and perfect-recall (an extensive application to dynamic programming is in Streufert (Dynamic programming for pure-strategy subgame perfection in an arbitrary game. arXiv:2302.03855v3 , 2023)).

Suggested Citation

  • Peter A. Streufert, 2025. "Specifying a game-theoretic extensive form as an abstract 5-ary relation," International Journal of Game Theory, Springer;Game Theory Society, vol. 54(1), pages 1-53, June.
  • Handle: RePEc:spr:jogath:v:54:y:2025:i:1:d:10.1007_s00182-025-00928-4
    DOI: 10.1007/s00182-025-00928-4
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    References listed on IDEAS

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    1. Kreps, David M & Wilson, Robert, 1982. "Sequential Equilibria," Econometrica, Econometric Society, vol. 50(4), pages 863-894, July.
    2. Peter A. Streufert, 2023. "Dynamic Programming for Pure-Strategy Subgame Perfection in an Arbitrary Game," University of Western Ontario, Departmental Research Report Series 20233, University of Western Ontario, Department of Economics.
    3. Mas-Colell, Andreu & Whinston, Michael D. & Green, Jerry R., 1995. "Microeconomic Theory," OUP Catalogue, Oxford University Press, number 9780195102680, Decembrie.
    4. Carlos Alós-Ferrer & Klaus Ritzberger, 2016. "The Theory of Extensive Form Games," Springer Series in Game Theory, Springer, number 978-3-662-49944-3, June.
    5. J. Jude Kline & Shravan Luckraz, 2016. "Equivalence between graph-based and sequence-based extensive form games," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 4(1), pages 85-94, April.
    6. Ritzberger, Klaus, 2002. "Foundations of Non-Cooperative Game Theory," OUP Catalogue, Oxford University Press, number 9780199247868, Decembrie.
    7. Klaus Ritzberger, 1999. "Recall in extensive form games," International Journal of Game Theory, Springer;Game Theory Society, vol. 28(1), pages 69-87.
    8. Peter A. Streufert, 2019. "Equivalences among five game specifications, including a new specification whose nodes are sets of past choices," International Journal of Game Theory, Springer;Game Theory Society, vol. 48(1), pages 1-32, March.
    9. Carlos Alós-Ferrer & Klaus Ritzberger, 2017. "Characterizations of perfect recall," International Journal of Game Theory, Springer;Game Theory Society, vol. 46(2), pages 311-326, May.
    10. Dockner,Engelbert J. & Jorgensen,Steffen & Long,Ngo Van & Sorger,Gerhard, 2000. "Differential Games in Economics and Management Science," Cambridge Books, Cambridge University Press, number 9780521637329, Enero-Abr.
    11. Hillas, John & Kvasov, Dmitriy, 2020. "Backward induction in games without perfect recall," Games and Economic Behavior, Elsevier, vol. 124(C), pages 207-218.
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    More about this item

    Keywords

    Extensive-form game; Pentaform; Subgame; Perfect recall;
    All these keywords.

    JEL classification:

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

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