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Dynamic Models of Rational Deliberation in Games

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Book cover Models of Strategic Reasoning

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8972))

Abstract

There is a growing body of literature that analyzes games in terms of the “process of deliberation” that leads the players to select their component of a rational outcome. Although the details of the various models of deliberation in games are different, they share a common line of thought: The rational outcomes of a game are arrived at through a process in which each player settles on an optimal choice given her evolving beliefs about her own choices and the choices of her opponents. The goal is to describe deliberation in terms of a sequence of belief changes about what the players are doing or what their opponents may be thinking. The central question is: What are the update mechanisms that match different game-theoretic analyses? The general conclusion is that the rational outcomes of a game depend not only on the structure of the game, but also on the players’ initial beliefs, which dynamical rule is being used by the players to update their inclinations (in general, different players may be using different rules), and what exactly is commonly known about the process of deliberation.

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Notes

  1. 1.

    The assumption is that once each player settles on a strategy, this identifies a unique outcome of the game. This is a simplifying assumption that can be dropped if necessary. However, for this chapter, it is simpler to follow standard practice and identify the set of “outcomes” of a game with the set of all tuples of actions.

  2. 2.

    For example, one can assume that each player knows which strategies the other players are going to choose. Robert Aumann and Adam Brandenburger use this assumption to provide an epistemic characterization of the Nash equilibrium [10].

  3. 3.

    Recall that I am restricting attention to finite strategic games.

  4. 4.

    Furthermore, the definitions of strict and weak dominance can be extended so that strategies may be strictly/weakly dominated by mixed strategies. This is important for the epistemic analysis of iterative removal of strictly/weakly dominated strategies. However, for my purposes in this chapter, I can stick with the simpler definition in terms of pure strategies.

  5. 5.

    The reflexive transitive closure of a relation R is the smallest relation \(R^*\) containing R that is reflexive and transitive.

  6. 6.

    Some interesting issues arise here: It is well-known that, unlike with strict dominance, different orders in which weakly dominated strategies are removed can lead to different outcomes. Let us set aside these issues in this chapter.

  7. 7.

    There is also an ex post analysis when all choices are “out in the open,” and the only remaining uncertainties are about what the other players are thinking.

  8. 8.

    Well-foundedness is only needed to ensure that for any set X, Min \(_{\preceq _i}(X)\) is nonempty. This is important only when W is infinite – and there are ways around this in current logics. Moreover, the condition of connectedness can also be lifted, but I use it here for convenience.

  9. 9.

    There are other natural modal operators that can. See [57] for an overview and pointers to the relevant literature.

  10. 10.

    This is the standard interpretation of \(K_i\varphi \) in the game theory literature. Whether this captures any of the many different definitions of knowledge found in the epistemology literature is debatable. A better reading of \(K_i\varphi \) is “given all of the available evidence and everything i has observed, agent i is informed that \(\varphi \) is true”.

  11. 11.

    I assume that the formal definition of common knowledge is well-known to the reader. For more information and pointers to the relevant literature, see [34, 36, 57, 76].

  12. 12.

    The same definition will, of course, hold for epistemic-plausibility and epistemic-probability models.

  13. 13.

    Since beliefs need not be factive, I do not force \(R_G^B\) to be reflexive.

  14. 14.

    However, see [1] for interesting new issues that arise with more than two players.

  15. 15.

    See [72], pgs. 44 – 52 and Chap. 5.

  16. 16.

    See [72], Chap. 4.

  17. 17.

    This game is called the “Battle of the Sexes”. The underlying story is that Ann and Bob are married and are deciding where to go for dinner. Ann would rather eat Indian food than French food, whereas Bob prefers French food to Indian food. They both prefer to eat together rather than separately. The outcome (u, l) is that they go to an Indian restaurant together; (d, r) is the outcome that they go to a French restaurant together; and (u, r) and (d, l) are outcomes where they go to different restaurants.

  18. 18.

    This graph was produced by a python program with an index of caution \(k=25\) and a satisficing value of 0.01. A satisficing value of 0.01 means that the process stops when the covetabilities fall below 0.01. Contact the author for the code for this simulation.

  19. 19.

    The outcome may end in a mixed-strategy Nash equilibrium.

  20. 20.

    If Bob assigns probability 0 to Ann playing d, then the strategies l and r give exactly the same payoffs.

  21. 21.

    The only probability measures such that \(m_1\) maximizes expected utility are the ones that assign probability 1 to Bob playing r.

  22. 22.

    See [56] for an interesting discussion of “picking” and “choosing” in decision theory.

  23. 23.

    Of course, Bob may think it is possible that Ann is irrational, and so she could choose the strictly dominated strategy d. Then, depending on how likely Bob thinks it is that Ann will choose irrationally, l may be the only rational choice for him. In this chapter, we set aside such considerations.

  24. 24.

    The key papers include [8, 9, 16, 39, 75]. See [61] for a complete survey of the literature.

  25. 25.

    Strictly speaking, it is all epistemic formulas. The important point is to not include formulas with the [ ! ] operator in them.

  26. 26.

    These graphs were generated by a python program using a satisficing value of 0.001 and an index of caution of 50. The reason that the simulations stopped before reaching the pure Nash equilibrium is because the simulation is designed so that deliberation ends when the covetabilities fall below the satisficing value.

  27. 27.

    Interested readers are referred to [72] (especially Chap. 7), and [35, 50, 67] for broader discussions.

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  1. Department of Philosophy, University of Maryland, College Park, USA

    Eric Pacuit

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  1. Inst for Logic Language & Computation, University of Amsterdam, Amsterdam, Noord-Holland, The Netherlands

    Johan van Benthem

  2. Indian Statistical Institute, Chennai, Tamil Nadu, India

    Sujata Ghosh

  3. Faculty of Math & Natural Sci, University of Groningen, Groningen, Groningen, The Netherlands

    Rineke Verbrugge

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Pacuit, E. (2015). Dynamic Models of Rational Deliberation in Games. In: van Benthem, J., Ghosh, S., Verbrugge, R. (eds) Models of Strategic Reasoning. Lecture Notes in Computer Science(), vol 8972. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48540-8_1

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