TR-2010010: Robust Knowledge and Rationality

Synthese, 2009

Aumann has proved that common knowledge of substantive rationality implies the backward induction solution in games of perfect information. Stalnaker has proved that it does not. The jury is still out concerning the epistemic conditions for backward induction, the "oldest idea in game theory" (Aumann, 1995, p. 635). and take conflicting positions in the debate: the former claims that common "knowledge" of "rationality" in a game of perfect information entails the backwardinduction solution; the latter that it does not. 1 Of course there is nothing wrong with any of their relevant formal proofs, but rather, as pointed out by Halpern , there are differences between their interpretations of the notions of knowledge, belief, strategy and rationality. Moreover, as pointed out by ) and others, the reasoning underlying the backward induction method seems to give rise to a fundamental paradox: in order even to start the reasoning, a player assumes that (common knowledge of, or some form of common belief in) "rationality" holds at all the last decision nodes (and so the obviously irrational leaves are eliminated); but then, in the next reasoning step (going backward along the tree), some of these (last) decision nodes are eliminated, as being incompatible with (common belief in) "rationality"! Hence, the assumption behind the previous reasoning step is now undermined: the reasoning player can now see, that if those decision nodes that are now declared "irrational" were ever to be reached, then the only way that this could happen is if (common belief in) "rationality" failed. Hence, she was wrong to assume (common belief in) "rationality" when she was reasoning about the choices made at those last decision nodes. This whole line of arguing seems to undermine itself!

Notre Dame Journal of Formal Logic, 2008

We develop a logical system that captures two different interpreta- tions of what extensive games model, and we apply this to a long-standing debate in game theory between those who defend the claim that common knowledge of rationality leads to backward induction or subgame perfect (Nash) equilib- ria and those who reject this claim. We show that a defense of the claim à la Aumann (1995) rests on a conception of extensive game playing as a one-shot event in combination with a principle of rationality that is incompatible with it, while a rejection of the claim à la Reny (1988) assumes a temporally extended, many-moment interpretation of extensive games in combination with implausi- ble belief revision policies. In addition, the logical system provides an original inductive and implicit axiomatization of rationality in extensive games based on relations of dominance rather than the usual direct axiomatization of rationality as maximization of expected utility.

fil.lu.se

Abstract: In this paper, we propose a backward induction argument that does not rest on the assumption of common knowledge of rationality. Nor is certainty of rationality assumed. Instead, each player is taken to assign a certain probability to the hypothesis that all ...

2021

The dominant theories of rational decision making assume what we will call logical omniscience. That is, they assume that when facing a decision problem, an agent can perform all relevant computations and determine the truth value of all relevant logical/mathematical claims. This assumption is unrealistic when, for example, we offer bets on remote digits of π or Goldbach’s conjecture; or when an agent faces a computationally intractable planning problem. Furthermore, the assumption of logical omniscience creates contradictions in cases where the environment can contain descriptions of the agent itself. Importantly, strategic interactions as studied in game theory are decision problems in which a rational agent is predicted by its environment (the other players). In this paper, we develop a theory of rational decision making that does not assume logical omniscience. We consider agents who repeatedly face decision problems (including ones like betting on Goldbach’s conjecture or games...

gtcenter.org

We propose a logical system in which a notion of the structure of a game is formally defined and the meaning of sequential rationality is formulated. We provide a set of decision criteria which, given sufficiently high order of mutual belief of the game structure and of every player following these criteria, entails Backward Induction decisions in generic perfect information games. We say that a player is rational if the player follows these criteria in his/her decisions. The set of mutual beliefs is also necessary, in the sense that any mutual belief of lower order can not entail the Backward Induction decisions. These conditions are determined by the length of the game structure, and they are never involved with common belief. Moreover, we give a set of epistemic conditions for subgame perfect equilibria for any perfect information game, which requires every player follow these decision criteria and there be mutual belief of the the equilibrium strategy and of the game structure.

2013

We investigate the extension of backward-induction to von Neumann extensive games (where information sets have a synchronous structure) and provide an epistemic characterization of it. Extensions of the idea of backward-induction were proposed by Penta (2009) and later by Perea (2013), who also provided an epistemic characterization in terms of the notion of common belief in future rationality. The epistemic characterization we propose, although differently formulated, is conceptually the same as Perea's and so is the generalization of backward induction. The novelty of this contribution lies in the epistemic models that we use, which are dynamic, behavioral models where strategies play no role and the only beliefs that are specified are the actual beliefs of the players at the time of choice. Thus our analysis is free of (objective or subjective) counterfactuals.

The jury is still out concerning the epistemic conditions for backward induction, the "oldest idea in game theory" ([2, p. 635]). Aumann and Stalnaker [31] take contradictory positions in the debate: Aumann claims that common 'knowledge' of 'rationality' in a game of perfect information entails the backward-induction solution; Stalnaker that it does not. 1 Of course there is nothing wrong with any of their relevant formal proofs, but rather, as pointed out by Halpern [22], there are differences between their interpretations of the notions of knowledge, belief, strategy and rationality. Moreover, as pointed out by Binmore , Bonanno [17], Bicchieri [13], Reny [26], Brandenburger [18] and others, the reasoning underlying the backward induction method seems to give rise to a fundamental paradox (the so-called "BI paradox"): in order to even start the reasoning, a player assumes that (common knowledge, or some form of common belief in) Rationality holds at all the last decision nodes (and so the obviously irrational leaves are eliminated); but then, in the next reasoning step (going backward along the tree), some of these (last) decision nodes are eliminated, as being incompatible with (common belief in) Rationality! Hence, the assumption behind the previous reasoning step is now undermined: the reasoning player can now see, that if those decision nodes that are now declared "irrational" were ever to be reached, then the only way that this could happen is if (common belief in) Rationality failed. Hence, she was wrong to assume (common belief in) Rationality when she was reasoning about the choices made at those last decision nodes. This whole line of arguing seems to undermine itself! In this paper we use as a foundation the relatively standard and well-understood setting of Conditional Doxastic Logic (CDL,), and its "dynamic" version (obtained by adding to CDL operators for truthful public announcements [!ϕ]ψ): the logic PAL-CDL, introduced by Johan van Benthem . In fact, we consider a slight ex-1 Others agree with Stalnaker in disagreeing with Aumann: for example, Samet and Reny [26] also put forwards arguments against Aumann's epistemic characterisation of subgame-perfect equilibrium. Section 5 is devoted to a discussion of related literature. tension of this last setting, namely the logic APAL-CDL, obtained by further adding dynamic operators for arbitrary announcements [!]ψ, as in [3]). We use this formalism to capture a novel notion of "dynamic rationality" and to investigate its role in decision problems and games. As usual in these discussions, we take a deterministic stance, assuming that the initial state of the world at the beginning of the game already fully determines the future play, and thus the unique outcome, irrespective of the players' (lack of) knowledge of future moves. We do not, however, require that the state of the world determines what would happen, if that state were not the actual state. That is, we do not need to postulate the existence of any "objective counterfactuals". But instead, we only need "subjective counterfactuals": in the initial state, not only the future of the play is specified, but also the players' beliefs about each other, as well as their conditional beliefs, pre-encoding their possible revisions of belief. The players' conditional beliefs express what one may call their "propensities", or "dispositions", to revise their beliefs in particular ways, if given some particular pieces of new information.

2013

We investigate the extension of backward-induction to von Neumann extensive games (where information sets have a synchronous structure) and provide an epistemic characterization of it. Extensions of the idea of backwardinduction were proposed by Penta (2009) and later by Perea (2013), who also provided an epistemic characterization in terms of the notion of common belief in future rationality. The epistemic characterization we propose, although differently formulated, is conceptually the same as Perea’s and so is the generalization of backward induction. The novelty of this contribution lies in the epistemic models that we use, which are dynamic, behavioral models where strategies play no role and the only beliefs that are specified are the actual beliefs of the players at the time of choice. Thus our analysis is free of (objective or subjective) counterfactuals.

Economic Theory, 1994

Many have objected to the use of the Nash equilibrium (or more generally, Bayesian Nash equilibrium) concept in game theory, and similarly to the use of the rational expectations concept in the theory of competitive markets, on the grounds that the theory assumes too much sophistication and coordination of beliefs on the part of decision-markers. The papers in this volume are among those which study the implications of relaxing those assumptions. * The first author thanks the C. V. Starr Center at New York University for their generosity. 2 Indeed, associate with each play of an agent, i, a type, %. Let #~ denote the ex ante belief of player i so that #i (-[%) denotes the beliefs of player-type z~. The Kalai-Lehrer assumption requires that the true play, induced by collection {#~ (. I%)}~, be absolutely continuous with respect to the beliefs of each agent, #~ (. 1%) for each i. It is straightforward to see that this in turn implies that the set of possible types is either finite or countable, which is violated in the coin-tossing example. See Nyarko (1992) for details.

Topics in Theoretical Economics, 2004

When evaluating the rationality of a player in a game one has to examine counterfactuals such as "what would happen if the player were to do what he does not do?" In this paper I develop a model of a normal form game where counterfactuals of this sort are evaluated as in the philosophical literature (cf. Lewis, 1973; Stalnaker, 1968). According to this method one evaluates a statement like ``what would the player believe if he were to do what he does not do'' at the world that is closest to the actual world where the hypothetical deviation occurs. I show that in this model common knowledge of rationality need not lead to rationalizability. I also present assumptions that allow rationalizability to follow from common knowledge of rationality. These assumptions suggest that rationalizability may not rely on weaker assumptions about belief consistency than Nash equilibrium.