Abstract
We discuss the issues that arise in modeling the notion of common belief of rationality in epistemic models of dynamic games, in particular at the level of interpretation of strategies. A strategy in a dynamic game is defined as a function that associates with every information set a choice at that information set. Implicit in this definition is a set of counterfactual statements concerning what a player would do at information sets that are not reached, or a belief revision policy concerning behavior at information sets that are ruled out by the initial beliefs. We discuss the role of both objective and subjective counterfactuals in attempting to flesh out the interpretation of strategies in epistemic models of dynamic games.
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- 1.
Evolutionary game theory has been applied not only to the analysis of animal and insect behavior but also to studying the “most successful strategies” for tumor and cancer cells (see, for example, [32]).
- 2.
On the other hand, in a situation of incomplete information at least one player lacks knowledge of some of the aspects of the game, such as the preferences of her opponents, or the actions available to them, or the possible outcomes, etc.
- 3.
- 4.
The notion of rationality in dynamic games is also discussed in [41].
- 5.
- 6.
Thus \(\mathcal {B}_{i}\) can also be viewed as a function from \(\varOmega \) into \( 2^{\varOmega }\) (the power set of \(\varOmega \)). Such functions are called possibility correspondences (or information functions) in the game-theoretic literature.
- 7.
For more details see the survey in [8].
- 8.
In modal logic belief operators are defined as syntactic operators on formulas. Given a (multi-agent) Kripke structure, a model based on it is obtained by associating with every state an assignment of truth value to every atomic formula (equivalently, by associating with every atomic formula the set of states where the formula is true). Given an arbitrary formula \(\phi \), one then stipulates that, at a state \(\omega \), the formula \(B_{i}\phi \) (interpreted as ‘agent i believes that \(\phi \)’) is true if and only if \(\phi \) is true at every state \(\omega ^{\prime } \in \mathcal {B}_{i}(\omega )\) (that is, \(\mathcal {B}_{i}(\omega )\) is a subset of the truth set of \(\phi \)). If event E is the truth set of formula \(\phi \) then the event \(\mathbb {B}_{i}E\) is the truth set of the formula \(B_{i}\phi \).
- 9.
That is, \(\omega ^{\prime }\in \mathcal {B}^{*}(\omega )\) if and only if there is a sequence \(\left\langle \omega _{1},...,\omega _{m}\right\rangle \) in \(\varOmega \) and a sequence \(\left\langle j_{1},...,j_{m-1}\right\rangle \) in N such that (1) \(\omega _{1}=\omega \), (2) \(\omega _{m}=\omega ^{\prime }\) and (3) for all \(k=1,...,m-1\), \(\omega _{k+1}\in \mathcal {B} _{j_{k}}(\omega _{k})\).
- 10.
As can be seen from Fig. 1, the common belief relation \(\mathcal {B}^{*}\) is not necessarily euclidean, despite the fact that the \(\mathcal {B}_{i}\)’s are euclidean. In other words, in general, the notion of common belief does not satisfy negative introspection (although it does satisfy positive introspection). It is shown in [24] that negative introspection of common belief holds if and only if no agent has erroneous beliefs about what is commonly believed.
- 11.
This is a local version of knowledge (defined as true belief) which is compatible with the existence of other states where some or all players have erroneous beliefs (see [23], in particular Definition 2 on page 9 and the example of Fig. 2 on page 6). Note that philosophical objections have been raised to defining knowledge as true belief; for a discussion of this issue see, for example, [52].
- 12.
Strategic-form games can also be used to represent situations where players move sequentially, rather than simultaneously. This is because, as discussed later, strategies in such games are defined as complete, contingent plans of action. However, the choice of a strategy in a dynamic game is thought of as being made before the game begins and thus the strategic-form representation of a dynamic game can be viewed as a simultaneous game where all the players choose their strategies simultaneously before the game is played.
- 13.
A preference relation over a set S is a binary relation \(\succsim \) on S which is complete or connected (for all \(s,s^{\prime }\in S\), either \( s\succsim s^{\prime }\) or \(s^{\prime }\succsim s\), or both) and transitive (for all \(s,s^{\prime },s^{\prime \prime }\in S\), if \(s\succsim s^{\prime }\) and \(s^{\prime }\succsim s^{\prime \prime }\) then \(s\succsim s^{\prime \prime }\)). We write \(s\succ s^{\prime }\) as a short-hand for \(s\succsim s^{\prime }\) and \(s^{\prime }\not \succsim s\) and we write \(s\sim s^{\prime }\) as a short-hand for \(s\succsim s^{\prime }\) and \(s^{\prime }\succsim s\). The interpretation of \(s\succsim _{i}s^{\prime }\) is that player i considers s to be at least as good as \(s^{\prime }\), while \(s\succ _{i}s^{\prime }\) means that player i prefers s to \(s^{\prime }\) and \(s\sim _{i}s^{\prime } \) means that she is indifferent between s and \(s^{\prime }\). The interpretation is that there is a set Z of possible outcomes over which every player has a preference relation. An outcome function \( o:S\rightarrow Z\) associates an outcome with every strategy profile, so that the preference relation over Z induces a preference relation over S.
- 14.
Cardinal utility functions are also called Bernoulli utility functions or von Neumann-Morgenstern utility functions.
- 15.
That is, \(\forall \omega \in \varOmega \), \(\forall \omega ^{\prime }\in \mathcal {B}^{*}(\omega )\), \(\omega ^{\prime }\in \mathcal {B}_{1}(\omega ^{\prime })\) and \(\omega ^{\prime }\in \mathcal {B}_{2}(\omega ^{\prime })\).
- 16.
- 17.
- 18.
When \(\mathcal {E}\) coincides with \(2^{\varOmega } \backslash \varnothing \), Condition 4 implies that, for every \(\omega \in \varOmega \), there exists a complete and transitive “closeness to \(\omega \)” binary relation \(\preceq _{\omega }\) on \(\varOmega \) such that \(f(\omega ,E) = \{ \omega ^{\prime } \in E : \omega ^{\prime } \preceq _{\omega } x, \forall x \in E \} \) (see Theorem 2.2 in [54]) thus justifying the interpretation suggested above: \(\omega _{1} \preceq _{\omega } \omega _{2}\) is interpreted as ‘state \(\omega _{1}\) is closer to \(\omega \) than state \(\omega _{2}\) is’ and \(f(\omega ,E)\) is the set of states in E that are closest to \(\omega \).
- 19.
As remarked in Footnote 17, both authors use the less general definition of selection function where \(f:\varOmega \times \mathcal {E}\rightarrow \varOmega \), that is, for every state \(\omega \) and event E, there is a unique state closest to \(\omega \) where E is true.
- 20.
For this reason, some authors (see, for example, [40]), instead of using strategies, use the weaker notion of “plan of action” introduced by [44]. A plan of action for a player only contains choices that are not ruled out by his earlier choices. For example, the possible plans of action for Player 1 in the game of Fig. 3 are \(a_{1}, (a_{2},d_{1})\) and \((a_{2},d_{2})\). However, most of the issues raised below apply also to plans of action. The reason for this is that a choice of player i at a later decision history of his may be counterfactual at a state because of the choices of other players (which prevent that history from being reached).
- 21.
This interpretation of strategies has in fact been put forward in the literature for the case of mixed strategies (which we do not consider in this chapter, given our non-probabilistic approach): see, for example, [6] and the references given there in Footnote 7.
- 22.
[45] was the first to propose models of perfect-information games where states are described not in terms of strategies but in terms of terminal histories.
- 23.
Note that, if at state \(\omega \) player i believes that history h will not be reached (\(\forall \omega ^{\prime }\in \mathcal {B}_{i}(\omega ) \), \(\omega ^{\prime }\notin [h]\)) then \(\mathcal {B}_{i}(\omega )\subseteq \lnot [h]\subseteq [h]\rightarrow [ha]\), so that \(\omega \in \mathbb {B}_{i}\left( [h]\rightarrow [ha]\right) \) and therefore (11) is trivially satisfied (even if \(\omega \in [h])\).
- 24.
On the other hand, we have not represented the fact that \(f(\alpha ,\{\alpha ,\delta \}) = \{ \alpha \}\), which follows from point 3 of Definition 4 (since \(\alpha \in \{\alpha ,\delta \}\)) and the fact that \(f(\delta ,\{\alpha ,\delta \}) = \{ \delta \}\), which also follows from point 3 of Definition 4. We have also omitted other values of the selection function f, which are not relevant for the discussion below.
- 25.
Recall that, by Definition 4, since \(\alpha \in [a_{2}]\), \(f(\alpha ,[a_{2}])=\{ \alpha \}\), so that, since \(\alpha \in [a_{2}c_{2}]\) (because \(a_{2}c_{2}\) is a prefix of \(\zeta (\alpha )=a_{2}c_{2}d_{2}\)), \(\alpha \in [a_{2}]\rightrightarrows [a_{2}c_{2}]\). Furthermore, since \(f(\beta ,[a_{2}])=\{ \alpha \}\), \(\beta \in [a_{2}]\rightrightarrows [a_{2}c_{2}]\). There is no other state \(\omega \) where \(f(\omega ,[a_{2}]) \subseteq [a_{2}c_{2}]\). Thus \([a_{2}]\rightrightarrows [a_{2}c_{2}]=\{\alpha ,\beta \}\). The argument for \([a_{2}]\rightrightarrows [a_{2}c_{1}]=\{\gamma ,\delta \}\) is similar.
- 26.
Since \(\mathcal {B}_{2}(\beta )=\{ \gamma \}\) and \(\gamma \in [a_{2}]\rightrightarrows [a_{2}c_{1}]\), \(\beta \in \mathbb {B}_{2}\left( [a_{2}]\rightrightarrows [a_{2}c_{1}]\right) \). Recall that the material conditional ‘if E is the case then F is the case’ is captured by the event \(\lnot E\cup F\), which we denote by \( E\rightarrow F\). Then \([a_{2}]\rightarrow [a_{2}c_{1}]=\{\beta ,\gamma ,\delta \}\) and \([a_{2}]\rightarrow [a_{2}c_{2}]=\{\alpha ,\beta ,\gamma \}\), so that we also have, trivially, that \(\beta \in \mathbb {B}_{2}\left( [a_{2}]\rightarrow [a_{2}c_{1}]\right) \) and \(\beta \in \mathbb {B}_{2}\left( [a_{2}]\rightarrow [a_{2}c_{2}]\right) \).
- 27.
Recall that a game is said to have complete information if the game itself is common knowledge among the players. On the other hand, in a situation of incomplete information at least one player lacks knowledge of some of the aspects of the game, such as the preferences of her opponents, or the actions available to them, or the possible outcomes, etc.
- 28.
As shown above, at state \(\omega \) Player 1 chooses \(a_{2}\); \(f(\omega ,[a_{1}])\) is the set of states closest to \(\omega \) where Player 1 chooses \(a_{1}\); in these states Player 2’s prior beliefs must be the same as at \(\omega \), otherwise by switching from \(a_{2}\) to \(a_{1}\) Player 1 would cause a change in Player 2’s prior beliefs.
- 29.
- 30.
In [28] there is also an objective counterfactual selection function, but it is used only to encode the structure of the game in the syntax.
- 31.
For example, in a perfect-information game one can take \(\mathcal {E} _{i}=\{[h]:h\in D_{i}\}\), that is, the set of propositions of the form “decision history h of player i is reached” or \(\mathcal {E}_{i}=\{[h]:h\in H\}\), the set of propositions corresponding to all histories (in which case \(\mathcal {E}_{i}= \mathcal {E}_{j}\) for any two players i and j).
- 32.
Note that it follows from Condition 3 and seriality of \(\mathcal {B} _{i}\) that, for every \(\omega \in \varOmega \), \(f_{i}(\omega ,\varOmega )= \mathcal {B}_{i}(\omega )\), so that one could simplify the definition of model by dropping the relations \(\mathcal {B}_{i}\) and recovering the initial beliefs from the set \(f_{i}(\omega ,\varOmega )\). We have chosen not to do so in order to maintain continuity in the exposition.
- 33.
- 34.
Equivalently, one can think of \(\mathcal {\rightrightarrows }_{i}\) as a conditional belief operator \(\mathbb {B}_{i}(\cdot |\cdot )\) with the interpretation of \(\mathbb {B}_{i}(F|E)\) as ‘player i believes F given information/supposition E’ (see, for example, [15] who uses the notation \(\mathbb {B}_{i}^{E}(F)\) instead of \(\mathbb {B}_{i}(F|E)\)).
- 35.
The author goes on to say that “The models can be enriched by adding a temporal dimension to represent the dynamics, but doing so requires that the knowledge and belief operators be time indexed...” For a model where the belief operators are indeed time indexed and represent the actual beliefs of the players when actually informed that it is their turn to move, see [20].
- 36.
(15) is implied by (11) whenever player i’s initial beliefs do not rule out h. That is, if \(\omega \in \lnot \mathbb {B}_{i} \lnot [h]\) (equivalently, \(\mathcal {B}_{i}(\omega )\cap [h] \ne \varnothing \)) then, for every \(a\in A(h)\),
$$\begin{aligned} \text {if } \omega \in [ha] \text { then } \omega \in \left( [h] \rightrightarrows _{i} [ha]\right) .&~~~~(F1) \end{aligned}$$In fact, by Condition 3 of Definition 6 (since, by hypothesis, \(\mathcal {B}_{i}(\omega )\cap [h]\ne \varnothing \)),
$$\begin{aligned} f_{i}(\omega ,[h])=\mathcal {B}_{i}(\omega )\cap [h].&~~~~(F2) \end{aligned}$$Let \(a\in A(h)\) be such that \(\omega \in [ha]\). Then, by (11), \(\omega \in \mathbb {B}_{i}([h]\rightarrow [ha])\), that is, \(\mathcal {B}_{i}(\omega )\subseteq \lnot [h]\cup [ha]\) . Thus \(\mathcal {B}_{i}(\omega )\cap [h]\subseteq \left( \lnot [h]\cap [h]\right) \cup \left( [ha]\cap [h]\right) =\varnothing \cup [ha]=[ha]\) (since \([ha]\subseteq [h]\)) and therefore, by (F2), \(f_{i}(\omega ,[h])\subseteq [ha]\), that is, \(\omega \in [h]\rightrightarrows _{i}[ha]\).
- 37.
This is a “local” definition in that it only considers, for every decision history of player i, a change in player i’s choice at that decision history and not also at later decision histories of hers (if any). One could make the definition of rationality more stringent by simultaneously considering changes in the choices at a decision history and subsequent decision histories of the same player (if any).
- 38.
Proof. Suppose that \(\omega \in [ha]\cap \lnot \mathbb {B}_{i} \lnot [h].\) As shown in Footnote 36 (see (F2)),
$$\begin{aligned} \mathcal {B}_{i}(\omega )\cap [h]=f_{i}(\omega ,[h]).&~~~~(G1) \end{aligned}$$Since \([ha]\subseteq [h]\),
$$\begin{aligned} \mathcal {B}_{i}(\omega )\cap [h] \cap [ha] = \mathcal {B}_{i}(\omega )\cap [ha].&~~~~ (G2) \end{aligned}$$As shown in Footnote 36, \(f_{i}(\omega ,[h])\subseteq [ha]\) and, by Condition 1 of Definition 6, \( f_{i}(\omega ,[h])\ne \varnothing \). Thus \(f_{i}(\omega ,[h])\cap [ha]=f_{i}(\omega ,[h])\ne \varnothing .\) Hence, by Condition 4 of Definition 6,
$$\begin{aligned} f_{i}(\omega ,[h])\cap [ha] =f_{i}(\omega ,[ha]).&~~~~(G3) \end{aligned}$$By intersecting both sides of (G1) with [ha] and using (G2) and (G3) we get that \(\mathcal {B}_{i}(\omega )\cap [ha]=f_{i}(\omega ,[ha])\).
- 39.
In fact, common belief of material rationality does not even imply a Nash equilibrium outcome. A Nash equilibrium is a strategy profile satisfying the property that no player can increase her payoff by unilaterally changing her strategy. A Nash equilibrium outcome of a perfect-information game is a terminal history associated with a Nash equilibrium. A backward-induction solution of a perfect-information game can be written as a strategy profile and is always a Nash equilibrium.
- 40.
In Fig. 6, for every terminal history, the top number associated with it is Player 1’s utility and the bottom number is Player 2’s utility. In Fig. 7 we have only represented parts of the functions \(f_{1}\) and \( f_{2}\), namely that \(f_{1}(\gamma , \{ \alpha , \beta , \delta \}) = \{ \delta \}\) and \(f_{2}(\beta , \{ \alpha , \beta , \delta \}) = f_{2}(\gamma , \{ \alpha , \beta , \delta \}) = \{ \alpha \}\) (note that \([a_{1}] = \{ \alpha , \beta , \delta \}\)). Similar examples can be found in [15, 28, 43, 50].
- 41.
For an example of epistemic models of dynamic games where strategies do play a role see [41].
- 42.
In general dynamic games, a strategy specifies a choice for every information set of the player.
- 43.
[20] uses a dynamic framework where the set of “possible worlds” is given by state-instant pairs \((\omega ,t)\). Each state \(\omega \) specifies the entire play of the game (that is, a terminal history) and, for every instant t, \((\omega ,t)\) specifies the history that is reached at that instant (in state \(\omega \)). A player is said to be active at \((\omega ,t)\) if the history reached in state \(\omega \) at date t is a decision history of his. At every state-instant pair \((\omega ,t)\) the beliefs of the active player provide an answer to the question “what will happen if I take action a?”, for every available action a. A player is said to be rational at \((\omega ,t)\) if either he is not active there or the action he ends up taking at state \(\omega \) is optimal given his beliefs at \((\omega ,t)\). Backward induction is characterized in terms of the following event: the first mover (at date 0) (i) is rational and has correct beliefs, (ii) believes that the active player at date 1 is rational and has correct beliefs, (iii) believes that the active player at date 1 believes that the active player at date 2 is rational and has correct beliefs, etc.
- 44.
The focus of this chapter has been on the issue of modeling the notion of rationality and “common recognition” of rationality in dynamic games with perfect information. Alternatively one can use the AGM theory of belief revision to provide foundations for refinements of Nash equilibrium in dynamic games. This is done in [19, 21] where a notion of perfect Bayesian equilibrium is proposed for general dynamic games (thus allowing for imperfect information). Perfect Bayesian equilibria constitute a refinement of subgame-perfect equilibria and are a superset of sequential equilibria. The notion of sequential equilibrium was introduced by [36].
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Acknowledgments
I am grateful to Sonja Smets for presenting this chapter at the Workshop on Modeling Strategic Reasoning (Lorentz Center, Leiden, February 2012) and for offering several constructive comments. I am also grateful to two anonymous reviewers and to the participants in the workshop for many useful comments and suggestions.
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A Summary of Notation
A Summary of Notation
The following table summarizes the notation used in this chapter.
Notation | Interpretation |
|---|---|
\(\varOmega \) | Set of states |
\(\mathcal {B}_{i}\) | Player i’s binary “doxastic accessibility” relation on \(\varOmega \). The interpretation of \(\omega \mathcal {B}_{i} \omega ^{\prime }\) is that at state \(\omega \) player i considers state \( \omega ^{\prime }\) possible: see Definition 1 |
\(\mathcal {B}_{i}(\omega ) =\{\omega ^{\prime }\in \varOmega :\omega \mathcal {B}_{i}\omega ^{\prime }\}\) | Belief set of player i at state \(\omega \) |
\(\mathbb {B}_{i}:2^{\varOmega }\rightarrow 2^{\varOmega }\) | Belief operator of player i. If \(E \subseteq \varOmega \) then \(\mathbb {B}_{i}E\) is the set of states where player i believes E, that is, \(\mathbb {B}_{i}E=\{\omega \in \varOmega :\mathcal {B}_{i}(\omega )\subseteq E \}\) |
\(\mathcal {B}^{*}\) | Common belief relation on the set of states \(\varOmega \) (the transitive closure of the union of the \(\mathcal {B}_{i}\)’s) |
\(\mathbb {B}^{*}:2^{\varOmega }\rightarrow 2^{\varOmega }\) | Common belief operator |
\(\left\langle N,\left\{ S_{i},\succsim _{i}\right\} _{i\in N}\right\rangle \) | Strategic-form game: see Definition 2 |
\(f:\varOmega \times \mathcal {E}\rightarrow 2^{\varOmega } \) | Objective counterfactual selection function. The event \(f(\omega ,E)\) is interpreted as “the set of states closest to \(\omega \) where E is true”: see Definition 4 |
\(E\rightrightarrows F=\{\omega \in \varOmega :f(\omega ,E)\subseteq F\}\) | The interpretation of \(E\rightrightarrows F\) is “the set of states where it is true that if E were the case then F would be the case.” |
\(\left\langle A,H,N,\iota ,\left\{ \succsim _{i}\right\} _{i\in N}\right\rangle \) | Dynamic game with perfect information. See Definition 5 |
\(f_{i}:\varOmega \times \mathcal {E}_{i}\rightarrow 2^{\varOmega }\) | Subjective counterfactual selection function. The event \(f_{i}(\omega ,E)\) is interpreted as the set of states that player i would consider possible, at state \(\omega \), under the supposition that (or if informed that) E is true: see Definition 6 |
\(E\rightrightarrows _{i}F=\{\omega \in \varOmega :f_{i}(\omega ,E)\subseteq F\}\) | The event \(E\rightrightarrows _{i}F\) is interpreted as “the set of states where, according to player i, if E were the case, then F would be true” |
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Bonanno, G. (2015). Reasoning About Strategies and Rational Play in Dynamic 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_2
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