Abstract
Gupta (2011) has proposed a definition of strategic rationality cast in the framework of his revision theory of truth. His analysis, relative to a class of normal form games in which all players have a strict best reply to all other players’ strategy profiles, shows that game-theoretic concepts (e.g. Nash equilibrium) have revision-theoretic counterparts. We extend Gupta’s approach to deal with normal form games in which players’ may have weak best replies. We do so by adapting intuitions relative to Nash equilibrium refinements (in particular, trembling-hand perfection and properness) to the revision-theoretic framework. We prove that there is a precise equivalence between trembling-hand perfect equilibria in two-player normal games and a revision-theoretic property. We then introduce lexicographic choice of action as a way to represent players’ expectations, which allows our analysis to reach full generality. Finally, we provide an example of the versatility of revision theory as applied to strategic interaction by formalizing a risk-and-compensation procedure of strategic choice in the revision-theoretic framework.
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Notes
In this article, we confine our analysis to normal form games, which are mathematically defined in Section 2. As in [14], we only deal with pure strategies, that is the actions that each players may choose from, without introducing mixed strategies, that is probability distributions on a player’s pure strategies.
As Aumann and Brandenburger [2] have discovered, sufficient epistemic condition for Nash equilibrium to occur are fairly weak in the case of pure strategies. The mere fact of rationality and mutual knowledge of the strategy profile suffice.
For the sake of readibility, we shall get rid of the upper index Γ for the elements of a finite game henceforth, as it will be clear from the context how to supplement it.
Therefore, all game–theoretic concepts occurring henceforth must be understood as applying to pure strategies only.
The reader acquainted with the customary game–theoretic notation (see, for instance, [1, p. 7] should notice that \(s\not =_{i}s^{\prime }\) if and only if \(s^{\prime }=(s_{-i},s^{\prime }_{i})\). We adopt the unconventional formalism here defined to ease readibility when, in further sections, the combinatorics become intricate.
We add an index to the rational predicate R(⋅) because its defining condition depends upon the features of Γ1, and the definition of this notion is different as the features of the game vary. We may get rid of the index whenever the context allows to.
In which u X (x)≥u Y (y) is an abbreviation for (u X (x) = u Y (y)∨u X (x)>u Y (y)).
Notice that, according to part 1 of this definition, ⋅= i ⋅ and ⋅≠ i ⋅ are not the negation of one another in the n–player (with n>2) case (whereas in a two–player game Γ we have that \(s\not =_{i} s^{\prime }\) holds if and only if \(s=_{i}s^{\prime }\) does not hold for every strategy profiles \(s,s^{\prime }\)). In particular, it follows that in part 2.1 it is not redundant to require that s ∗≠ i s ‡ holds since s ∗= i s and \(s^{\dagger }=_{i}s^{\prime }\) do not imply that \(s_{j}^{*}=s_{j}^{\dagger }\) is the case for every j∈P with j≠i, as s ∗≠ i s ‡ does instead. Also, notice that in a quasi–strict game in normal form Γ for no player are there actions which are fully equivalent (i.e., actions which guarantee the same payoff under all possible circumstances).
A strategy \(s^{\prime }_{i}\) is strictly dominated if there exists a strategy s i such that for all s −i ∈S −i , \(u_{i}(s^{\prime }_{i},s_{-i}) < u_{i}(s_{i},s_{-i})\), while a strategy is weakly dominated if \(u_{i}(s^{\prime }_{i},s_{-i}) \leq u_{i}(s_{i},s_{-i})\) for all s −i and \(u_{i}(s^{\prime }_{i},s_{-i}) < u_{i}(s_{i},s_{-i})\) for at least one s −i .
But, of course, the precise game–theoretic definition of proper equilibrium, involving fully mixed strategies and arbitrarily small perturbations of different magnitudes, cannot.
Because b 2 gives B higher utility than b 1 against a 1 but b 1 gives B higher utility than b 2 against a 3.
In the usual game–theoretic notation, we use [s] j to represent profiles s j ,s −j varying over all possible s −j ∈S −j .
That is, for every \(s^{\prime \prime },s^{\prime \prime \prime }\) such that \(s^{\prime \prime }=_{i}s\), \(s^{\prime \prime \prime }=_{i}s^{\prime }\) and \(s^{\prime \prime }\not =_{i}s^{\prime \prime \prime }\), either \(u_{i}(s^{\prime \prime })=u_{i}(s^{\prime \prime \prime })\), or if \(u_{i}(s^{\prime \prime })>u_{i}(s^{\prime \prime \prime })\) then u i (s †)>u i (s #) for some s †,s #∈S, \(s^{\ddagger }=_{i}s^{\prime \prime \prime }\), \(s^\#=_{i}s^{\prime \prime }\), s †≠ i s #, and if \(u_{i}(s^{\prime \prime \prime })>u_{i}(s^{\prime \prime })\) then u i (s †)>u i (s #) for some s †,s #∈S, \(s^{\ddagger }=_{i}s^{\prime \prime }\), \(s^\#=_{i}s^{\prime \prime \prime }\), s †≠ i s #.
That is to say, neither of the two equilibria is a strict trembling-hand perfect equilibrium in the sense of Okada [19].
We refer the interested reader to our companion paper [10] for these.
In game theory lexicographic probability systems have become a standard tool in the literature on epistemic foundations (see Blume et al. [7] for the seminal paper, Brandenburger et al. [9] for an important application to admissibility, Halpern [16] for the relation with conditional and non-standard probability). In such systems, each player has a collection of probability measures representing her beliefs. Intuitively speaking, the “first” measure in the collection is the most important, followed by the second one, the third one, and so on. For instance if the collection is p={p 1,p 2} and the event under consideration is, say, b 2 the overall probability of b 2 is given by p 1(b 2) + 𝜖 p 2(b 2), where 𝜖 is infinitesimally small. Lexicographic beliefs are useful when, for instance, a player takes another player to be rational, and therefore rules out a certain irrational action by that player, and still wants to entertain the non-zero possibility that the player will choose that irrational action (indeed it is possible to express refinements like trembling-hand perfection and properness in this framework, see Blume et al. [8]). Lexicographic beliefs are similar to conditional probability systems used by Battigalli and Siniscalchi [4] in sequential games, which allow to condition on “zero” probability events since they are given infinitesimally small probability. In our case, we are limiting ourselves to all-or-nothing expectations thus leaving probabilistic measures outside the scope of our current work. Still, representing players’ all-or-nothing expectations through lexicographic choice allows us to introduce epistemic considerations in the framework of revision theory and offers a useful way to “build-in” the consistency of a given epistemic assumption and strategic behavior.
The definition of canonical reordering of a game as it is given here does not make clear that the canonical version of a given game is unique if it exists (for, it might well be possible that the original game features two or more most efficient weak equilibria). It is not difficult to make a slight modification of it in order to achieve uniqueness, for instance by differentiating two equivalent efficient weak equilibria by leftmost occurrence in the given game (see [10] for details).
Loss of equilibria, however, is not the only issue in the revision-theoretic analysis of Γ10 and \(\Gamma _{10}^{\prime }\). The former is regular, while the latter is not. Loss of regularity is not solved by the canonical ordering of actions, as \(\Gamma _{10}^{\prime }\) actually is the canonical version of Γ10. Delving deeper in the issue of regularity shows that we can preserve regularity only by imposing stronger trivializing assumptions (see [10] for details).
Notice that while the restriction to pure strategies in the previous sections allowed us to consider games with ordinal utility only (that is, games in which the payoff only showed the ordering of players’ preferences, and not their intensity), in this section, as we use the idea of regret, utilities are taken to be cardinal (that is, reflecting the actual intensity of preference).
There are many ways to define the risk factor. A natural one is for instance considering a strategy riskier than another if it warrants a higher regret, in absolute terms, than the other action. For models of regret, see [17] or [5]. For examples of the relevance of regret in decision-making, see [13]. In our case, we can define the risk factor in terms of regret by taking the distance between the preference of the (weak) best reply of B to a and the (weak) worse reply to it. For instance, \(Risk_{a}(b):=|\max \{u_{B}(a^{\prime }b):a^{\prime }\not =a\}-\min \{u_{B}(a^{\prime }b):a^{\prime }\not =a\}|\) would indicate for instance the risk factor of b given a.
Of course, one can imagine that players in a game may prefer a different attitude, inspired by prudence, and avoid taking the risk anyway. Due to the character of the section, which mainly serves the purpose of illustrating the versatility of the revision-theoretic framework, we avoid taking a stance in this matter. Just as an illustration of these ideas, the compensation of b to strategy a of A according to the “risky” intuition above, can be defined as \(Comp_{(b,a)}:=\max \{u_{B}(a^{\prime }b)\mid a^{\prime }\not =a\}-Risk_{a}(b)\) and it generalizes to a two-person game where, for instance, player A has n actions and player B m actions instead, by \(Comp_{(b,a)}:=Mean\{u_{B}(a^{\prime }b)\mid a^{\prime }\not =a\}-Risk_{a}(b)\) where M e a n is the arithmetical mean of the preferences.
Instead, this game has the same problems as Γ12 with the old definition if one takes into account the variant of it where \(u^{\prime }_{A}(a_{1}b)=u_{A}(a_{3}b)\), \(u^{\prime }_{A}(a_{2}b)=u_{A}(a_{1}b)\), and \(u^{\prime }_{A}(a_{3}b)=u_{A}(a_{2}b)\) for b∈S B , since an additional fixed point turns out.
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Acknowledgments
The authors would like to thank Anil Gupta and Shawn Standefer for their useful comments on a previous draft of the paper, as well as an anonymous referee, whose suggestions were particularly important to improve it.
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Bruni, R., Sillari, G. A Rational Way of Playing: Revision Theory for Strategic Interaction. J Philos Logic 47, 419–448 (2018). https://doi.org/10.1007/s10992-017-9433-2
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DOI: https://doi.org/10.1007/s10992-017-9433-2