A Modal Logic for Mixed Strategies

Joshua Sack Wiebe van der Hoek

A Modal Logic for Mixed Strategies

Abstract. Modal logics have proven to be a very successful tool for reasoning about games. However, until now, although logics have been put forward for games in both normal form and games in extensive form, and for games with complete and incomplete information, the focus in the logic community has hitherto been on games with pure strategies. This paper is a first to widen the scope to logics for games that allow mixed strategies. We present a modal logic for games in normal form with mixed strategies, and demonstrate its soundness and strong completeness. Characteristic for our logic is a number of infinite rules. Keywords: Modal Logic, Logics for Games, Mixed Strategies.

1.

Introduction

The recent years have seen a flurry of activities relating modal logics to game theory. A survey is [8] although since its appearance several new developments have started. One of the early interests from modal logic in game theory was through the line of epistemic and doxastic logic, where game theorists formulated the epistemic conditions for Nash equilibrium in [2]. Notions like information partitions and events as used in [2] are easily seen to correspond to S5 accessibility relations and possible worlds, respectively. As remarked in [8], the initial ideas from game theory that the logic community paid attention to were relatively uninteresting for the game theorist — mostly 2-player extensive games of perfect information which are strictly competitive. However, [8] then also observes that more recent work in logic has extended its scope considerably, introducing, e.g., cooperative game theory, imperfect information and games involving more than 2 players. However, as far as we are aware, no matter how rich the modal frameworks, including dynamic notions of belief and incorporating ways to refer to the preferences of agents, have become, all the modal logical work for games until now has focussed on reasoning about games with pure strategies only. That is, the players chose their actions from a, usually finite, set, and

Special Issue: Logic and Games Edited by Thomas ˚ Agotnes

Studia Logica (2014) 102: 339–360

c Springer ∞

2014

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J. Sack, W. van der Hoek

they do not randomize over them. This is a serious limitation: one of the most celebrated theorems of game theory for instance holds that every finite strategic game has a mixed strategy Nash equilibrium (e.g., [6, p. 33]). This result is not true if one restricts himself to only considering pure strategies, as for instance witnessed by the matching pennies game ([6, Example 17.1]). In this paper we introduce a modal logic, Modal Logic for Mixed Strategies (mlms), in which one can reason about mixed strategies in games. mlms is interpreted directly on a given set of mixed strategies of a game, and we provide a proof system that is both sound and strongly complete. The proof system for mlms exploits some non-standard, infinite rules. They cater to the fact that although the sets of pure strategies for each player are assumed to be finite, allowing him to randomize over them presents him with an infinite number of choices. Throughout the paper, we show how to use the logics to reason about the matching pennies game. Our logic mlms, and in particular having a sound and complete axiomatisation, clarifies what is exactly needed in proving game-theoretic results (that are expressible in our language) concerning mixed strategies. It also enables one to compare logics for games with pure strategies with ours: mlms for instance needs a rule with infinitely many premises, which seems inherently linked to the fact that we have mixed strategies. Moreover, linking a fragment of game theory to a well-established framework like modal logic, opens up a suite of possible tools, like theorem provers and model checkers. For instance, there are probabilistic model checkers (like PRISM, [5]) which allow for modal temporal languages, and might be adapted to verify properties involving mixed equilibria in a given game. Having said this, we should also add that our results are only preliminary and limited: our axiom system for instance suggests that one may need to look at sub-systems to get computationally feasible proof systems or model checkers.

2.

Games and mixed strategies

Our work builds on the notion of a normal form game, or game for short, in this paper. Definition 2.1 (Game). A game is a tuple G = hAg, {¶i }i2Ag , {ui }i2Ag i, where Ag is a finite set of players, and for each agent i 2 Ag, we have a set ¶i of pure strategies. We assume an order on Ag and generally use the number in the order to refer to the agents. A pure strategy profile is a tuple

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(º1 , . . . , ºn ), where each ºi 2 ¶i . Let ¶ be the set of all pure strategy profiles. The component ui : ¶ ! R is a utility function, assigning agent i’s payoÆ for each strategy profile. Definition 2.2 (Mixed strategy and strategy profile). Given a finite set ¶i of pure strategies for a player i, a mixed strategy is a probability function æi : ¶i ! [0, 1],

P that is, a function from ¶i to [0, 1] satisfying ºi 2¶i æ(ºi ) = 1. Let ßi be the set of all mixed strategies for player i. We define a mixed strategy profile (or mixed profile for short) as a tuple (æ1 , . . . , æn ) of mixed strategies for each player. Let ß be the set of all mixed strategy profiles. For the semantics we will soon define, we will find it helpful to view a mixed strategy profile as a function rather than a tuple, that is a function æ : ¶ ! [0, 1]. Given a mixed strategy profile (tuple) (æ1 , . . . , æn ), one can define a mixed strategy profile (function) æ by Y æ(º) = æi (ºi ). i2Ag

The original mixed strategy profile (tuple) can be recovered by X æi (ºi ) = æ(Ω).

(1)

Ω2¶,Ωi =ºi

But in general, a function æ : ¶ ! [0, 1] need not be a mixed strategy profile (function), as the derived agent components might not be independent. Traditionally, a function æ : ¶ ! [0, 1] is called a correlated strategy profile, though throughout this paper, we will focus on those that really are mixed strategy profiles. Definition 2.3 (Expected utility). Given a set ¶ of pure strategy profiles, a mixed strategy profile æ : ¶ ! [0, 1], and a utility function ui : ¶ ! R for player i, the expected utility of æ for i is given as an extension of the function ui to the domain ß according to X ui (º)æ(º). (2) ui (æ) = º2¶

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Example 2.1 (Matching Pennies). In the matching pennies game two players each can chose the face of a coin: if both coins show the same face (both Head or both Tail), player a wins, and otherwise player b wins. A diagram for it is given in Figure 1 (left). H T h (+1, °1) (°1, +1) t (°1, +1) (+1, °1)

B S b (2, 1) (0, 0) s (0, 0) (1, 2)

Figure 1. The Matching Pennies Game (left) and BoS (right)

So, here Ag = {a, b}, ¶a = {h, t} (representing ‘a plays head’ and ‘a plays tail’, respectively) and ¶b = {H, T }. So ßa = {æa | æa (h) + æa (t) = 1} and ßb = {æb | æb (H) + æb (T ) = 1}. Also assume that ua (h, H) = ua (t, T ) = 1 and ua (h, T ) = ua (t, H) = °1 and ub (º) = °ua (º) for all pure strategy profiles º. Now take æa 2 ßa such that æa (h) = 1/3, and æb 2 ßb such that æb (H) = 1/4. For simplicity, we can represent ßi by the interval [0, 1], where si 2 [0, 1] denotes the probability that i plays Hi , that is si = æi (Hi ). Then for the mixed strategy profile æ = (sa , sb ) = (1/3, 1/4) we have ua (æ) = 1/4 · 1/3 ° 1/4 · 2/3 ° 3/4 · 1/3 + 3/4 · 2/3 = 2/12 = 1/6 and

ub (æ) = °1/4 · 1/3 + 1/4 · 2/3 + 3/4 · 1/3 ° 3/4 · 2/3 = °2/12 = °1/6.

3.

Language and semantics

We now define a language that does not include mixed strategy profiles directly, but which explicitly involves the values that are rational linear combinations of mixed strategy values. With uncountably many strategy profiles, there will be uncountably many such values, making the language uncountable. To ensure a countable language, we could enforce the mixed strategies be rational valued probabilities, but we choose a less restrictive e an arbitrary subset of the set and more flexible route, and involve a set ß, ß of mixed strategy profiles. Assuming that the set of agents and the set of pure strategy profiles are both finite, the language will be countable if and e is. only if ß e µ ß of mixed Given a game G = hAg, {¶i }i2Ag , {ui }i2Ag i and a set ß e strategy profiles, we define Z(ß) µ R to be the set of rational linear combie as follows: nations of values of mixed strategies in ß X e = {a | 9æ 2 ß. e 8º 2 ¶. 9rº 2 Q. a = Z(ß) rº æ(º)}. (3) º2¶


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e are rational, then Z(ß) e = Q. If all mixed strategy values in ß We next define our language that is syntactically similar to one in [4] in that linear combinations are involved, but diÆerent in that here the input to the probability symbol P ranges over pure strategy profiles, while in [4] they range over propositional formulas. (The reason for this diÆerence is that we view pure strategy profiles as individual outcomes in a sample space, while in [4] atomic propositions are interpreted as events.) Linear combinations allow us to reason naturally about conditions such as additivity, which is central to probability. e of mixed stratDefinition 3.1 (Language Lms ). Given a game G and set ß e with formulas and egy profiles, we define a two-sorted language Lms (G, ß), terms. The terms will make it easier to reason about linear combinations of probabilities, which will be helpful in our proof system. Given Ag, define the set of Modality Indices ModIn(Ag) = {G, @i , Ai , =i , | i 2 Ag, G µ Ag}. Moreover, for every º 2 ¶, we assume to have a syntactic represenation º 2 ¶ for it. The formulas are given by the following Bachus-Naur form: ' ::= t ∏ a | ¬' | ' ^ ' | [?]' e and ? 2 ModIn(Ag), and terms t are given by where a 2 Z(ß) t ::= rP(º) | t + t

where r 2 Q and º 2 ¶. The modality [G] can be read as “any mixed strategy profile with alternative strategies for those not in G.” For example, [Ag \ {i}] is read as “any mixed strategy profile with an alternative strategy for i,” and [;] is read that “for any mixed strategy profile” (note that [;] is a universal modal operator i.e., [;]' means that ' is true in all states of the model). We consider that each player knows her own strategy, and hence [G] can also be viewed as distributed knowledge for those in a set G of players. The modality [@i ] is read as “for any profile with a greater payoÆ for i” (a reading more loyal to the direction of the inequality would be “for any profile such that the current profile yields less payoÆ for agent i”), and similarly for other inequality or equality modalities. As we will see in Section 3.1, the semantics will specify an “actual” mixed strategy profile, say æ. Then a probability term P(º) represents the value æ(º). As we will see from the table of abbreviations below and Proposition 3.1 that follows the definition of the semantics, the abbreviation Pi (ºi ) represents the value æi (ºi ), the abbreviation ui represents the value

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ui (æ), and the abbreviations ø and øi represent the assertions that ø and øi are the actual mixed strategy profile æ and mixed strategy æi , respectively. Abbreviations are given in the table below (and in thePparagraph that n follows), where t0 is some given term, c 2 Q, and t = k=1 qk P(ºk ) = q1 P(º1 ) + · · · + qn P(ºn ), with each ºk 2 ¶ a pure strategy profile. false ct t∑q tq t=q t∏s t=s

def

= def = def = def = def = def = def = def =

(t0 ∏ 0) ^ ¬(t0 ∏ 0) Pn k=1 cqk P(ºk ) °t ∏ °q ¬(t ∏ q) ¬(t ∑ q) t∑q^t∏q t°s∏0 t°s∏0^s°t∏0

[i]' [[G]' [vi ]' [wi ]' Ui ' ø ui Pi (ºi ) øi

def

= = def = def = def = def = def = def = def = def

[{i}]' V i2G [i]' [=i ]' ^ [@i ]' [=i ]' ^ [Ai ]' [=i ]' ^ [Ai ]' ^ [@i ]' V P(º) = ø (º) Pº2¶ ui (º)P(º) Pº2¶ {P(Ω) | Ω 2 ¶, Ωi = ºi } V ºi 2¶i (Pi (ºi ) = øi (ºi ))

def

We furthermore involve the standard propositional abbreviations '_√ = def def ¬(¬' ^ ¬√), ' ! √ = ¬' _ √, and ' $ √ = (' ! √) ^ (√ ! '). We also def have the duals of the modal operators: h?i' = ¬[?]¬' for ? 2 ModIn(Ag). def We additionally define Ei ' = ¬Ui ¬'. We will later see that Ui and Uj are equivalent for every two agents (meaning that Ui ' $ Uj ' holds). Although the subscript is still syntactically relevant, we may drop it when the emphasis is on the truth rather than the structure of the formula. Note that all the abbreviations above are for formulas, except for ct, ui , and Pi (ºi ), which are terms. The first subscript of Pi (ºi ) indicates that the term represents an individual mixed strategy for player i, and that the argument of Pi must be an individual pure strategy for player i. The second subscript is just to emphasize that ºi is a pure strategy for i, and we could replace it with any other notation we use for an individual’s strategy (note that subscripts on strategies do not always indicate agents, such as when we consider linear combinations of probability terms). Example 3.1 (Matching pennies, ctd). Examples of term formulas for the matching pennies example are 7 · Pa (h) + 3 · Pa (t) ∏ 8, expressing that seven times the probability that player a chooses h plus three times the probability that player a chooses h is at least eight. A more complicated formulas is [a]([@b ]Pa (h) ∏ 0.7 ! [@a ]Pb (H) ∏ 0.6), saying that given player a’s current commitment, if b prefers that a chooses a strategy to play h with probability at least 0.7, then player a prefers that b chooses H with probability at least 0.6.

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3.1.

Semantics

e is meant to describe only one structure: the game The language Lms (G, ß) e of mixed strategy profiles. We interpret G together with the restricted set ß e formulas over mixed strategy profiles, using a relation |=(G,ß) between mixed e and formulas in Lms (G, ß). e We eliminate the superscripts strategy profiles ß when they are understood from context, as is done in the following table which defines the relation: æ æ æ æ æ æ æ

e

The semantics (relation |=(G,ß) ) P |= k=1 qk P(ºk ) ∏ r iÆ nk=1 qk æ(ºk ) ∏ r |= ¬' iÆ æ 6|= ' |= ' ^ √ iÆ æ |= ' and æ |= √ |= [G]' iÆ ø |= ' whenever for each i 2 G, æi = øi |= [@i ]' iÆ ø |= ' whenever ui (æ) < ui (ø ) |= [Ai ]' iÆ ø |= ' whenever ui (æ) > ui (ø ) |= [=i ]' iÆ ø |= ' whenever ui (æ) = ui (ø ) Pn

e we have æ |= '. Given a set B of formulas, We write |= ' if for all æ 2 ß, e we write æ |= B precisely when æ |= ' for each ' 2 B. The relation |=(G,ß) can also be extended to a relation between sets of formulas A and formulas ' by the condition A |= ' if and only if for every mixed strategy profile æ, if æ |= √ for all √ 2 A, then æ |= '. Our language is largely multi-modal. But rather than including proposition letters, we involve probability formulas that reflect certain facts about the strategy profile assigned to the state the formula is evaluated in. We add further intuition regarding the probability and utility formula abbreviations with the following proposition. Proposition 3.1. The following relationships hold: a. b. c. d. e.

æ æ æ æ æ

Proof. For Item (a),

|= ø |= øi |= ui ∏ a |= Pi (ºi ) ∏ a |= Ui '

æ |= ø , æ |=

^

iÆ iÆ iÆ iÆ iÆ

æ=ø æi = øi ui (æ) ∏ a æi (ºi ) ∏ a e ø |= ' for all ø 2 ß

P(º) = ø (º)

º2¶

, æ(º) = ø (º) for all º 2 ¶ , æ = ø.

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For Item (c), using Equation (2) in Definition 2.3, we have X æ |= ui ∏ a , æ |= ui (º)P(º) ∏ a ,

X

º2¶

º2¶

ui (º)æ(º) ∏ a , ui (æ) ∏ a.

For Item (d), using Equation (1) on Page 341, we have X æ |= Pi (ºi ) ∏ a , æ |= P(Ω) ∏ a Ω2¶,Ωi =ºi

,

X

Ω2¶,Ωi =ºi

æ(Ω) ∏ a , æi (ºi ) ∏ a

For Item (b), using Item (d), we have ^ æ |= øi , æ |= (Pi (ºi ) = øi (ºi )) ºi =¶i

, æi (ºi ) = øi (ºi ) for all ºi 2 ¶i , æi = øi

Item (e) follows directly from the semantics and the trichotomy of the real numbers (for any two numbers a and b, a < b, a = b, or a > b). Items (a) through (d) of Proposition 3.1 justify the reading that the terms Pi (ºi ) and ui represent the values æi (ºi ) and ui (æ), respectively, and the formulas ø and øi represent the assertions that ø and øi are the actual mixed strategy profile and mixed strategy for i, respectively. Item (e) justifies the notion that U is a “universal modality”, and that Ui ' holds if and only if Uj ' holds. Note that, by definition of [i]', we have that æ |= [i]' holds iÆ ø |= ' whenever æi = øi . The formula [i]' is to be read “if player i sticks to his current mixed strategy, ' is guaranteed, no matter what the other agents do”. Likewise, [G]' means that “if all the players in G stick to their chosen strategy, no matter what the agents outside G decide to do, ' holds”. We then have that hGi' says that “Given the current mixed strategy, the agents outside G can make a choice such that, if the agents within G stick to theirs, ' holds”. If we then define hhGiiÆ ' = h(Ag \ G)i[G]' we have what is known as Æ-eÆectivity, or Æ-ability: the collective G can make a choice such that, no matter what the other agents choose to do, ' holds. This is reminiscent of the truth definition of hhCii g' in atl.

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There is also a weaker notion, that of Ø-eÆectivity, which follows a 89pattern, and ensures that, once the choice of the players outside G is given, those in G can respond so as to ensure ': hhGiiØ ' = [G]h(Ag \ G)i' Moreover, using this apparatus, we can define a notion of control as studied in [9], where ctrl(i, ') = hhiiiÆ ' ^ hhiiiÆ ¬'. For a discussion on the notion of control, Æ and Ø-eÆectivity and atl, see [10]. 3.1.1.

Nash equilibria

e = ß, the set of all mixed strategy For this section, fix a game G and set ß profiles for the game G. A strategy æi is a best response, if i cannot improve her payoÆ by changing her strategy alone. Given a mixed strategy profile, i’s strategy is a best response if for every formula ', we have æ |= ' ! [(Ag \ {i})]hvi i'. Given a specific æ we define bri (æ) ¥ æ ! [(Ag \ {i})]hvi iæ. Then æ is a best response in G if and only if |= bri (æ) if and only if for any mixed strategy profile ø we have that ø |= [;]bri (æ). A Nash equilibrium is a mixed strategy profile, such that everyone’s strategy is a best response. For each æ, define ^ Nash(æ) ¥ bri (æ). i2Ag

So æ is a Nash equilibrium in G if and only if |= Nash(æ) if and only if for any mixed strategy profile ø , it holds that ø |= [;]Nash(æ). Thus the validity of Nash(æ) asserts that æ is a Nash equilibrium, while the satisfiability of [;]Nash(æ) asserts that æ is a Nash equilibrium. Example 3.2 (Matching pennies, ctd). In the matching pennies example, æ = (1/2, 1/2) is the only mixed strategy profile, where Nash(æ) is satisfiable. As is well known in this game, there are no pure strategy Nash equilibria, and the only Nash equilibrium is a mixed strategy Nash equilibrium. Example 3.3 (BoS, p. 34 [6]). The game represented in Figure 1 (right) is also known as the Battle of the Sexes or Bach or Stravinsky. A couple tries to synchronise on a night out and have decided they will either go to Bach or to Stravinsky. Both players prefer going to the same concert over each

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going to a diÆerent event. However, player m prefers they both go to Bach, while player w prefers both attending the Stravinsky concert. We now follow the reasoning of [6]. Let æ = (æm , æw ) be a Nash equilibrium. If æm (b) = 1 (that is, m goes definitely to Bach) we get an equilibrium for æw (B) = 1 (this corresponds to the pure Nash equilibrium (b, B)); if æm (b) = 0, we get an equilibrium with æw (B) = 0 (corresponding to (s, S)). Now suppose that 0 < æm (b) < 1 and that (æm , æw ) is an equilibrium. Now [6, Lemma 33.2] implies that both b and s are best responses to æw (if x 2 {b, s} were not a best response to æw , then as some pure strategy is a best response, y 2 {b, s} \ {x} is a best response, and m would benefit by transferring some probability of playing x to playing y, meaning (æm , æw ) would not be an equilibrium). But then, player m’s actions b and s must yield the same payoÆ given æw , so we have 2æw (B) = æw (S) implying æw (B) = 13 . Similarly, we obtain æm (b) = 23 , yielding a third equilibrium to be (æm , æw ) with æm (s) = æw (B) = 13 . Note that for this quilibrium 0 is a best response to æ , whereas any æ 0 (æm , æw ), any mixed strategy æm w w is a best response to æm . e includes all mixed strategy profiles, there must It is well known that if ß be a Nash equilibrium. It is also true (and can be checked using linear complementary programming techniques, such as those in [11]) that if the e contains all rational game is 2-player, the utility values are rational, and ß valued mixed strategies, then there exists a rational valued Nash equilibrium.

4.

Proof system

e In this section, we present our proof system, which we denote by mlms(G, ß), e for a given game G and set ß of mixed strategy profiles. The proof system consists of axioms and rules, and we will first present and discuss the axioms, since the rules involve additional notation that we will define right after our discussion about the axioms. e consists of modal schemes and inequality The set of axioms of mlms(G, ß) schemes, shown in Definition 4.1. The modal schemes extend basic propositional logic with normality axioms K for each modality ? 2 ModIn(Ag), reflexivity axioms REF for modalities of the form G and =i , a scheme ED asserting that if the group G sticks to their strategies, the individual members i stick to their strategies, and schemes QL and QG which state that every mixed strategy profile that yields a higher (respectively lower) payoÆ for i yields a higher (resp. lower) payoÆ than any mixed strategy with an equal payoÆ. As committing to a strategy and sharing the same utility

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are both equivalence relations, one might expect transitivity and Euclidean axiom schemes for G and =i . However, these can be proved from the rules that we will soon define. The next block of modal schemes consists of PE, which reflects the setting where players commit to their own strategy, and the schemes UQ, UL, and UG, which reflect the meaning of the the operators =i being concerned with mixed strategy profiles with the same utility, @i as being concerned with mixed strategy profiles with greater utility (hence the interaction between @i , which corresponds to the utility of the actual mixed strategy profile being smaller, and > in the scheme), and Ai with less utility. The last block of modal schemes corresponds relates mixed strategy profiles with these operators, and will be given more explanation immediately after the rules are explained. The use of the universal modality [;] and the fact that we only consider one game, makes these schemes similar to rules. For the inequality schemes, IP, I0, IA, IM, MON, correspond to basic properties of inequalities, and P0 and P1 correspond to basic properties of probability. The scheme NS makes false any expressible mixed strategy e Since in NS, æ must be expressible in the language, profile that is not in ß. e is. If ß e includes all probability functions with NS is a countable scheme if ß e (defined in (3)), then the scheme NS will be empty. function values in Z(ß) e schemes). Definition 4.1 (mlms(G, ß) We define the following Modal Schemes, where ? 2 ModIn(Ag), § 2 {G, =i } and i 2 G. CL K REF PE UQ UL UG SD SQ SL SG

Classical Logic Tautologies [?](' ! √) ! ([?]' ! [?]√) [§]' ! ' ±Pi (ºi ) ∏ q ! [i] ± Pi (ºi ) ∏ q ±ui ∏ q ! [=i ] ± ui ∏ q ui ∏ q ! [@i ]ui > q ui ∑ q ! [Ai ]ui < q V [;](æ ! ') ! [;]( i2G æi ! hGi') [;](æ ! ') ! [;](ui = ui (æ) ! h=i i') [;](æ ! ') ! [;](ui < ui (æ) ! h@i i') [;](æ ! ') ! [;](ui > ui (æ) ! hAi i')

ED QL QG

[i]' ! [G]' [@i ]' ! [=i ][@i ]' [Ai ]' ! [=i ][Ai ]'

On top of that, we have the following Inequality Schemes, where, in IP, j is a permutation of {1, . . . , n}.

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IP I0 IA IM MON P0 P1 NS

Pn

P ∏ q ! nk=1 qjk P(ºjk ) ∏ q, tP∏ q $ t + 0P(ºk+1 )P∏ q n n 0 0 k=1 Pnqk P(ºk ) ∏0 q ^ k=1 qk P(º0 k ) ∏ q ! k=1 (qk + qk )P(ºk ) ∏ (q + q ) t ∏ q $ dt ∏ dq where d > 0 (t ∏ q) ! (t > q 0 ) where q > q 0 P(º) P ∏0 º2¶ P(º) = 1 e ¬æ for each æ 62 ß k=1 qk P(ºk )

In order to express the rules of the proof system, we will benefit from additional abbreviations. We first define the set ModSeq of modality sequences as {(s1 , . . . , sn ) | n 2 N, sk 2 Lms [ ModIn(Ag)} Definition 4.2 (pseudo modalities, [7]). We define the following pseudo modalities, which are (possibly empty) sequences s = () or s = (s1 , . . . , sn ), where each si is a formula or a modality index ? 2 ModIn(Ag). The string h(s)i¡ (mostly written as hsi', if s has more than one element) represents a formula, as follows: h()i' = ' h√, s2 , . . . , sn i' = √ ^ hs2 , . . . , sn i' ha, s2 , . . . , sn i' = hai(hs2 , . . . , sn i') We also define [s]¡ as ¬hsi¬¡. So, for instance h@i , √, =j i' is an abbreviation of h@i i(√ ^ h=j i') while [Ai √, @j ]' is short for [Ai ](√ ! [@j ]'). e characterizes a derivability relation The proof system of mlms(G, ß) e (G, ß) ` between sets of formulas and formulas, which is defined to be the e smallest relation that contains every instance of ; `(G,ß) ' (typically write ten `(G,ß) ') for every axiom ' given by Definition 4.1, every instance of A ` ' when ' 2 A, and is closed under the closure rules given in Table 1. e We typically write ` instead of `(G,ß) when the superscripts are understood e from context. We say that a set of formulas A is mlms(G, ß)-consistent (or consistent for short) if A 0 false (that is, it is not the case that A ` false), e and otherwise mlms(G, ß)-inconsistent (or just inconsistent). We now briefly comment on some of the rules. The rule MONR ensures that any theorem (' such that ` ') is derivable from any set of formulas. One may need to use MONR if the proof of ` ' required the use of, say, NEC.

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Rules A`'!√ A`' (MP) A`√

`' ([?]-NEC) ` [?]'

`' (MONR) A`'

e q 6= p A ` [s](P(º) 6= q) for all q 2 Z(ß), (PR) A ` [s](P(º) = p)

e such that øi = æi for all 2 G ` ø ! ' for all ø 2 ß (DR) ` æ ! [G]' e such that ui (ø ) = ui (æ) ` ø ! ' for all ø 2 ß (QR) ` æ ! [=i ]' e such that ui (ø ) > ui (æ) ` ø ! ' for all ø 2 ß (LR) ` æ ! [@i ]'

e such that ui (ø ) < ui (æ) ` ø ! ' for all ø 2 ß (GR) ` æ ! [Ai ]' e rules. We have ? 2 ModIn(Ag), s 2 ModSeq and æ 2 ß. e Table 1. mlms(G, ß)

For the rule PR, s is a pseudo modality, whose components are either modalities in ModIn(Ag) or formulas. A string of modalities can be thought of as a path in a model whose points are mixed profiles. This follows the standard view of the modalities as taking one from one state to another in a model. The formula components of the pseudo modality allow us to impose extra conditions along the path. Let A be a set of formulas, and suppose that from A it follows that, along all path s, the probability for pure profile º is diÆerent from any q 6= p. Rule PR then allows us to conclude that, given A, at the given paths s, the probability of º must be p. Rule PR is a stronger version of the Archimedean rule (in [12]), which states that if given condition ' it holds that the probability of √ is at least s for all s < r, then given ', the probability of √ is at least r. The rules DR, QR, LR, and GR respectively provide conditions a state should have to guarantee that the formulas [G]', [=i ]', [@i ]', and [Ai ]' respectively to hold in that state. For example, DR asserts that if ' holds for any strategy profile ø that has the same mixed strategy as æ for every agent i 2 G (written ` ø ! '), then [G]' holds in æ (written ` æ ! '). Notice the similarity between these rules and the schemes SD, SQ, SL, and

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SR. These schemes provide conditions for a hGi', h=i i', h@i i', and hAi i' to hold in certain types of mixed strategy profiles. For example, if ' is true for mixed strategy profile æ (written [;](æ ! æ), where the universal modality [;] is used to enforce the condition (æ ! æ) is valid, as the formula might not be evaluated in æ), then h@i i' is true for any ø that yields a lower payoÆ for player i than æ (written [;](ui < ui (æ) ! h@i i')). 4.1.

Provable formulas and consistent sets

e Proposition 4.1. If ° is a maximally mlms(G, ß)-consistent set, then there e is a unique mixed strategy profile æ 2 ß, such that æ 2 °, that is ^

º2¶

P(º) = æ(º) 2 °.

Proof. From Rule PR, we have, for each º, a value pº , such that P(º) = pº 2 °. The axiom scheme MON guarantees pº is unique. Let æ be the unique function that maps º 7! pº . The axiom P0 guarantees that 0 ∑ pº , and the axiom P1, with P the help of the inequality axioms (IP, I0, IA, and IM), guarantees that º2¶ pº = 1. Thus æ is a probability function, and e hence a mixed strategy profile. The scheme NS ensures that it is in ß. The following proposition will be used for the Truth Lemma (Lemma 4.10), a central lemma for the completeness proof. e Proposition 4.2. If ° is a maximally mlms(G, ß)-consistent set, then for all i 2 Ag, there is a unique value qi , such that ui = qi 2 °. V Proof. By Proposition P 4.1, there is a unique æ, such that º2¶ P(º) = æ(º) 2 °. Let qi = º2¶ ui (º)æ(º) = ui (æ). Since the æ is unique by Proposition 4.1, and the function ui isP fixed, the value qi is also unique. Then by the linear inequality axioms, º2¶ ui (º)P(º)qi 2 °, and hence ui = qi 2 °. The following two lemmas will also be used for the Truth Lemma. e Lemma 4.3. If ° is a maximally mlms(G, ß)-consistent set and æ^hGi' 2 °, e then there exists ø 2 ß with æi = øi for each i 2 G, such that hGi('^ø ) 2 °.


353

Proof. Let (æ ^ hGi') 2 °. Let º 2 ¶. Suppose for a contradiction that there is no p such that hGi('^P(º) = p) 2 °. Then for each p, we have that [G]¬(' ^ P(º) = p) 2 °. Hence for each p, we have [G](' ! P(º) 6= p) = ['][G]P(º) 6= p 2 °. By the Rule PR, we obtain that ['][G]P(º) = p 2 ° for each such p. Since hGi' 2 °, we have that [G]? 62 °, and hence ['][G]P(º) = p 2 ° for every p results in a contradiction. Thus there is a p such that hGi(' ^ P(º) = p) 2 °. We can repeat this processVfor each º 2 ¶ until we have a mixed strategy profile ø , such that hGi(' ^ º2¶ P(º) = ø (º)) 2 °. It remains to show that æi = øi for each i 2 G, and this follows from schema ED and PE. e Lemma 4.4. If ° is a maximally mlms(G, ß)-consistent set and ui = qi ^h@i 0 i' 2 °, then there exists q 2 Q, such that q 0 > qi and h@i i(' ^ ui = q 0 ) 2 °.

Proof. Let (ui = qi ^ h@i i') 2 °. By the V same reasoning as given in Lemma 4.3, there is a æ, such that h@i i(' ^ º2¶ P(º) = æ(º)) 2 °. Then using inequality axioms, h@i i(' ^ P P propositional, and modal logic, we obtain, P is a formula º2¶ ui (º)P(º) = º2¶ ui (º)æ(º)) P 2 °. Note that the first abbreviation, while the second gives us a single number. Letting q0 = P 0 º2¶ ui (º)æ(º)), we have that h@i i(' ^ ui = q ) 2 °. It remains to show that q 0 > q, and this follows from the axiom UL. The following lemma will be used in Lemma 4.9, a key step to relating the standard game semantics to a new one used solely for the completeness proof. Lemma 4.5. If æ ^ ' is consistent, then ` æ ! '. Proof. We prove this by induction on the structure of the formula. Probability formulas and Booleans: The (base) cases, where ' is a probability formula t ∏ q, follow from MON and the inequality axioms, since æ specifies completely the probability values of each input º. The Boolean case for conjunction is trivial. The negation case goes as follows. Suppose that 0 æ ! ¬'. Then æ ^ ' is consistent. By the inductive hypothesis, ` æ ! '. Hence æ ^ ¬' is inconsistent. Case [G]: Suppose that æ ^ hGi' were consistent. Then by DR, there is a ø , such that æi = øi for all i 2 G, and ø ^ ' is consistent. By the inductive hypothesis, we have that ` ø ! ', and by SD, we have V V that ` i2G øi ! hGi'. By the inequality axioms, we have that ` æ ! i2G øi and hence by modus ponens, ` æ ! hGi'. The case [=i ]: is similar to the case [G], except the rule QR is used instead of DR (and SQ rather than SD). Case [@i ]: Suppose that æ ^ h@i i' were consistent. Then by LR, there is a ø such that @i (ø ) >@i (æ), and ø ^ ' is consistent. By the inductive

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hypothesis, we have that ` ø ! ', and by SL, we have that ` ui < ui (ø ) ! h@i i'. By the inequality axioms, we have that ` æ ! ui < ui (ø ) and hence by modus ponens, ` æ ! h@i i'. Case [Ai ] is similar as [@i ], except we use GR rather than LR. e then ` h;iæ. Proposition 4.6. If æ 2 ß,

Proof.VBy propositional logic ` æ ! æ. By [;]-NEC and SD, we have that ` i2; æi ! h;iæ. As an empty intersection is defined to be true, the desired result immediately follows. e we have Proposition 4.7 (Nash equilibrium). For any æ 2 ß, ^ ^ ^ ` ( æj ! ui ∑ ui (æ)) if and only if ` (æ ! [Ag \ {i}]hvi iæ). i2Ag j6=i

i2Ag

Proof. We first show that ` ui ∑ ui (æ) $ hvi iæ.

(4)

Clearly ` æ ! æ, and hence ` [;](æ ! æ). Then by SQ, modus ponens, and REF, we have ` ui = ui (æ) ! h=i iæ. Similarly, using SL, we have ` ui < ui (æ) ! h@i iæ. Thus by propositional reasoning, we have ` ui ∑ ui (æ) ! hvi iæ. We use similar reasoning to obtain ` ui > ui (æ) ! hAi iæ. By the inequality axioms, we have that ` ui ∑ ui (æ) _ ui ∏ u( æ). Hence by propositional reasoning, we have that ` hvi iæ _ hAi iæ. Thus the contrapositive of ui > ui (æ) ! hAi iæ is hvi iæ ! ui ∑ ui (æ). We next show that ^ ` æ ! [G]' if and only if ` ( æi ! ') (5) i2G

V Suppose that ` æ ! [G]'. Then then by SD, we have that ` i2G æi ! hGi[G]'. We argue that ` hGi[G]' ! '. If not, ` hGi[G]'^¬' is consistent, and belongs to a maximally consistentV set °. By Proposition 4.1, there must be a ø 2V°. By SD, we have that ` i2G øi ! hGiø . By this and modal logic, ` ( i2G øi ^ [G]') ! hGi(ø ^ '). Also, Lemma 4.3 yields that there e such that ∑i = øi for all i 2 G, and hGi(∑ ^ [@i ]') 2 °. By the is a ∑ 2 ß, V V inequality axioms, ` i2G øi = i2G ∑i . Hence hGihGi(ø ^ ') 2 °. Thus ø ^ ' is consistent, and by Lemma 4.5, ` ø ! ', contradicting the fact that ¬' 2 °. Having V established ` hGi[G]' ! ', we use modus ponens to arrive at the desired ` i2G æi ! '.

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V Suppose that ` i2G æi ! '. Then for each ø such that øi = æi for each i 2 G, we have that ` ø ! '. Then by DR, we have that ` æ ! [G]'. Now the following are equivalent: V V 1. ` i2Ag ( j6=i æj ! ui ∑ ui (æ)) V V 2. ` i2Ag ( j6=i æj ! hvi iæ) V 3. ` i2Ag (æ ! [Ag \ {i}]hvi iæ)

The first equivalence follows from (4) and propositional reasoning. The second equivalence follows from (5) and propositional reasoning. Regarding the matching pennies example, the Nash equilibrium ø is where both player give equal likelihood to each of the two pure strategies. The expect utility ui (ø ) for each playerVi 2 {a,Vb} is then 0. The inequality axioms can then be applied to yield ` i2Ag ( j6=i øj ! ui ∑ 0), which by the previous result is equivalent to ø being a Nash equilibrium. 4.2.

Soundness and completeness

e is sound with Theorem 4.8 (Soundness). The proof system mlms(G, ß) e (G, ß) respect to the semantics given by |= (which we refer to as |=, here).

Proof. Scemes REF and ED are valid as seen directly from the definition of the semantics For the sceme QL, suppose æ |= [@i ]'. Then ø |= ' whenever ui (æ) < ui (ø ). Any æ 0 where ui (æ) = ui (æ 0 ) is such that ui (æ 0 ) < ui (ø ), and hence æ 0 |= [@i ]'. As æ 0 was chosen arbitrarily, æ |= [=i ][@i ]'. The scheme QG is similar to QL For PE, suppose that æ P |= ±Pi (ºi ) > q. Expanding the abbreviation of Pi (º ), we have æ |= ± {P(Ω)P| Ω 2 ¶, Ωi = ºi } > q. If øi = æi , Pi then P {ø (Ω) | Ω 2 ¶, Ωi = ºi } = {æ(Ω) | Ω 2 ¶, Ωi = ºi }. Thus ø |= ± {P(Ω) | Ω 2 ¶, Ωi = ºi } > q, whence æ |= [i] ± Pi (ºi ) > q. For UQ, suppose P that æ |= ±ui ∏ q. Expanding the abbreviation of ui , we have P that æ |= ± º2¶ P ui (º)P(º) ∏ q. If ø is such that ui (æ) = ui (ø ), then º2¶ ui (º)æ(º) = º2¶ ui (º)ø (º), and hence ø |= ±

X

º2¶

ui (º)P(º) ∏ q.

As ø was chosen arbitrarily, æ |= [=i ] ± ui ∏ q. For UL, suppose P that æ |= ui ∏ q. Expanding the abbreviation of ui , we have that æ |= º2¶ ui (º)P(º) ∏ q. If ø is such that ui (æ) < ui (ø ), then

356 P


P P < º2¶ ui (º)ø (º), and hence ø |= º2¶ ui (º)P(º) > q. As ø was chosen arbitrarily, æ |= [@i ] ± ui > q. The proof for UG is similar to the one for UL. For SD, suppose that ∑ ` [;]æ ! '. Then by Proposition 3.1, æ |= æ ^'. V It is then immediate that æ |= hGi', and hence æ |= i2G æi ! hGi'. In V e general, given V ø 2 ß, if øi = æi for each i 2 G, then ø |= i2G æi ^ hii', V and hence ø |= i2G æ ! hGi'. If ø = 6 æ for some i 2 G, then ø |= ¬ i i i i2G æi , V andVhence ø |= i2G æi ! hGi'. By the definition of [;], we have ∑ |= [;] i2G æi ! hGi'. The proofs for SQ, SL, and SG are proved in a similar way to SD. The proofs for IP, I0, IA, IM, P0, P1, and MON use standard arguments (see [3]). For the rule PR, suppose that æ |= A implies that æ |= [s]P(º) 6= q, for all s 2 ModSeq, and q 6= p. Then supposing that æ |= A, we wish to show that æ |= [s]P(º) = p. We show this by induction on the length of s. In particular, our induction hypothesis is that if s has length n, then for any set A of formulas and pure strategy profile º, if A |= [s]P(º) 6= q for all e q 6= p (we only need to check q 2 Z(ß)), then A |= [s]P(º) = p. The base case, where s = () follows immediately from the semantics of the probability formulas. The case where s = (√, s1 , . . . , sn ) is as follows. We assume A |= [s]P(º) 6= q for all q 6= p and that both æ |= A and æ |= √ (for otherwise, æ |= [s]P(º) = p is vacuously true). By the induction hypothesis æ |= [(s1 , . . . , sn )]P(º) = p. The case where s = (G, s1 , . . . , sn ) is as follows. Let s0 = (s1 , . . . , sn ). Suppose first that A |= [G][s0 ]P(º) 6= q for all q 6= p. Let æ be such that æ |= A. Then let ø be such that æi = øi for each i 2 G. Then ø |= [s0 ]P(º) 6= q e 9i 2 G.øi 6= æi }, that is, B is the set of for all q 6= p. Let B = {¬ø | ø 2 ß, e where øi 6= æi for some i 2 G. Then formulas ¬ø for strategy profiles ø 2 ß, 0 B |= [s ]P(º) 6= q for all q 6= p, and we can apply the induction hypothesis, such that B |= [s0 ]P(º) = p. Note that as ø was chosen such that øi = æi for all i 2 G, we have ø |= B by the definition of B. Hence ø |= [s0 ]P(º) = p, and as ø was chosen arbitrarily, æ |= [G][s0 ]P(º) = p, which is what we set out to show. The other modal cases are similar to the case for [G]. e such that øi = æi For DR, suppose that |= ø ! ' or every ø 2 ß for all 2 G. Then by Proposition 3.1, ø |= ' for each such ø , and hence by the definition of the semantics, æ |= [G]'. Again by Proposition 3.1, |= æ ! [G]'. The cases for QR, LR, and GR follow a similar argument as for DR. º2¶ ui (º)æ(º)

357


e is strongly complete with respect to the We will show that mlms(G, ß) e (G,ß)

e

e

semantics |=GS , that is whenever A |=(G,ß) ' we have that A `(G,ß) '. e This is equivalent to showing that for any mlms(G, ß)-consistent set A of e e formulas, there is a mixed strategy profile æ in ß, such that æ |=(G,ß) A. Toward proving this, let • W be the set of maximally consistent sets.

• and æ ˆ : W ! ß assign to each maximally consistent set w 2 W the unique mixed strategy profile consistent with w (existence and uniqueness of this profile is guaranteed by Proposition 4.1). We will sometimes write æ ˆx for æ ˆ (x), thus allowing us to conveniently provide an input to the mixed strategy profile, such as æ ˆx (º) for the probability that the pure strategy profile º is played at x. However, we will typically stick with the notation æ ˆ (x) in a context where individual strategies will be involved, such as æ ˆ (x)i for i’s strategy at x. Lemma 4.9. The function æ ˆ is a bijection from the set of maximal consistent e sets to the set of mixed strategy profiles in ß.

Proof. For a maximally consistent set x we have by Proposition 4.1 that e For any æ 2 ß, e by Theorem 4.8 (soundness), the formula æ is æ ˆx 2 ß. consistent, and hence can be extended to a maximal consistent set. Thus æ ˆ is surjective. By Lemma 4.5, one can prove a complete theory from any e Hence this extension to a maximally consistent mixed strategy profile in ß. set is unique. Thus æ ˆ is injective, and hence a bijection. Lemma 4.10 (Truth Lemma). For each formula ' and maximally consistent set x 2 W , e ' 2 x if æ ˆx |=(G,ß) '.

Proof. We prove this by induction on the structure of formulas. The base cases t ∏ q (where t is a term) follow almost directly from Proposition 4.1, which justifies the function æ ˆ . The Boolean cases are standard. Case [G]: First suppose that hGi' 2 x. From Proposition 4.1, we have that there is a mixed strategy profile æ (it is æ ˆ (x)), such that æ ^ hGi' 2 x. By Lemma 4.3, there is a ø , such that æi = øi for each i 2 G and hGi('^ø ) 2 x. Thus ' ^ ø is consistent, and hence there is a maximally consistent set y, such that æ ˆ (y) = ø and '^ø 2 y, hence ' 2 y. By the induction hypothesis, æ ˆy |= '. Since æ ˆ (x)i = æ ˆ (y)i for all i 2 G, we have that æ ˆx |= hGi'. Conversely, suppose that æ ˆx |= hGi'. By Lemma 4.9, there is a maximally consistent set y, such that æ ˆy |= æ ˆ (y) ^ ' and æ ˆ (x)i = æ ˆ (y)i for each

358


V i 2 G; hence by the inequality axioms, i2G æ ˆ (y)i 2 x. Since ˆ (y) and ' V æ are consistent, we have by Lemma 4.5 and scheme SD that ( i2G æ ˆ (y)i ! hGi') 2 x. Thus by modus ponens, hGi' 2 x. Case [@i ]: First suppose that h@i i' 2 x. By Proposition 4.2, we have that there is a q, such that (ui = q)^h@i i' 2 x. Then by Lemma 4.4, we have that there is a q 0 > q such that h@i i(' ^ ui = q 0 ) 2 x. Thus ' ^ (ui = q 0 ) is consistent, and hence must belong to some maximally consistent set y. By the induction hypothesis, we have that æ ˆy |= ', and by a base case 0 0 æ ˆx |= ui = q and æ ˆy |= ui = q . Then, as q > q, we have that æ ˆx |= h@i i'. Conversely, suppose that æ ˆx |= h@i i'. By Lemma 4.9, there is a maximally consistent set y, such that æ ˆy |= æ ˆ y ^ ' and ui (ˆ æx ) < ui (ˆ æy ); hence by Proposition 4.2 and MON, (ui < ui (ˆ æy )) 2 x. Since ' and æ ˆy are consistent, we have by Lemma 4.5 and scheme SL that (ui < ui (ˆ æy ) ! h@i i') 2 x. Thus by modus ponens, h@i i' 2 x. Cases [Ai ] and [=i ]: These are identical in reasoning to the case for @i . Analogs of Lemma 4.4 for Ai and =i can be obtained using the same structure proofs, but with UG and UQ in place of UL. The schemes SG and SQ are involved instead of SL. e is Theorem 4.11 (Strong completeness). The proof system mlms(G, ß) e e in the game G, that is, if A |=(G,ß) strongly complete with respect to the set ß e ', then A `(G,ß) ' for any set A of formulas and any formula '.

Proof. Using the strategy discussed immediately after the proof of Theorem 4.8, we show that for any consistent set A, there is a mixed strategy e e such that æ |=(G,ß) profile æ in ß, A. Let A be a consistent set of formulas. Then it is contained in some maximally consistent set x. By the truth e lemma (Lemma 4.10), æ ˆ (x) |=(G,ß) °. Recall that Nash(æ) ¥

V

i2Ag (æ

! [Ag \ {i}]hvi iæ).

e Then ß e includes a Nash equilibrium if and Corollary 4.12. Let ø 2 ß. only if the following is true: {¬Nash(æ) | æ 2 ß \ {ø }} ` Nash(ø ).

e does include a Nash equilibrium. We show Proof. First suppose that ß that {¬Nash(æ) | æ 2 ß \ {ø }} |= Nash(ø ), and then apply the strong completeness theorem. First, if {¬Nash(æ) | æ 2 ß \ {ø }} is not valid (that e not equal to ø ), then the implication holds is, there is Nash equilibrium in ß


359

e includes a Nash equilibrium, it vacuously. Otherwise, as we assume that ß must be ø , and hence the implication holds. e does not include a Nash equilibrium. Then {¬Nash(æ) | æ 2 Suppose ß e but ø 6|= Nash(ø ). ß \ {ø }} is satisfied in every mixed strategy profile in ß, Thus {¬Nash(æ) | æ 2 ß \ {ø }} 6|= Nash(ø ).

5.

Conclusion

As far as we know, this paper is the first to propose a logic for mixed strategies in games. The logic we employ is fixed for an arbitrary multi-agent perfect information game in normal form. A fundamental feature of our logic is the interpretation of terms of the form P(º) as values of a locally specified mixed strategy profile. If rather than fixing a utility function for the logic, we introduced terms of the form ui (º) for values of a globally specified utility function, we would add the flexibility for one logic to describe properties of a much larger variety of games. Another extension of our logic could be to involve (as many logics for games already do in a pure strategy setting) epistemic or doxastic operators for players’ knowledge or belief, respectively. Such operators have similar properties to the commitment operators [i], as players are likely to know their own strategies, but provide more flexibility for players to, say, be unaware or mistaken about the strategies of others. Such operators also open the door to expressing in the object language rationality, which is often formulated in terms of what an agent knows or believes. In a setting with epistemic or doxastic operators, we may localize the utility functions by assigning utility functions to states. Then we could have terms of the form ui (º) representing locally specified utility values, similar to the way P(º) represents locally specified mixed profile values. As players may consider to be possible diÆerent states with diÆerent utility, and would hence be uncertain about the payoÆs of the game, such a logic would allow us to reason about incomplete information games. Extensive form games provide a setting where we can analyze what happens as a game unfolds. Dynamic epistemic logic has been used to express the dynamics of a game, by adding to the language formulas of the form h√i', asserting that after √ is truthfully revealed, ' may be true. Here √ could express that a certain move in an extensive form game was played, or indeed, that playe i is rational. A variant of probabilistic dynamic epistemic logic [1] could be useful in expressing such dynamics in extensive game forms that involve mixed strategies. Epistemic and doxastic operators also allow us to analyze imperfect information games.

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Acknowledgements. Joshua Sack was partially supported by the Netherlands Organisation for Scientific Research VIDI project 639.072.904. References [1] Aceto, L., W. van der Hoek, A. Ingolfsdottir, and J. Sack, Sigma algebras in probabilistic epistemic dynamics, in TARK XIII, 2011, pp. 191–199. [2] Aumann, R. J., and A. Brandenburger, Epistemic conditions for Nash equilibrium, Econometrica 63(5):1161–1180, 1995. [3] Fagin, R., and J. Halpern, Reasoning about knowledge and probability, Journal of the ACM 41:340–367, 1994. [4] Fagin, R., J. Y. Halpern, and N. Megiddo, A logic for reasoning about probabilities, Information and Computation 87:78–128, 1990. [5] Kwiatkowska, M., G. Norman, and D. Parker, PRISM 4.0: Verification of probabilistic real-time systems, in G. Gopalakrishnan, and S. Qadeer, (eds.), Proc. CAV’11, vol. 6806 of LNCS, Springer, 2011, pp. 585–591. [6] Osborne, M. J., and A. Rubinstein, A Course in Game Theory, The MIT Press: Cambridge, MA, 1994. [7] Renardel de Lavalette, G., B. Kooi, and R. Verbrugge, Strong completeness for PDL, in P. Balbiani, N. Suzuki, and F. Wolter, (eds.), AiML, 2002, pp. 377–393. [8] van der Hoek, W., and M. Pauly, Modal logic for games and information, in P. Blackburn, J. van Benthem, and F. Wolter, (eds.), Handbook of Modal Logic, Elsevier, Amsterdam, 2006, pp. 1077–1148. [9] van der Hoek, W., D. Walther, and M. Wooldridge, Reasoning about the transfer of control, JAIR 37:437–477, 2010. [10] van der Hoek, W., and M. Wooldridge, Multi-agent systems, in F. van Harmelen, V. Lifschitz, and B. Porter, (eds.), Handbook of Knowledge Representation, Elsevier, 2008, pp. 887–928. [11] Von Stengel, B., Computing equilibria for two-person games, in R.J. Aumann and S. Hart, (eds.), Handbook of Game Theory with Economic Applications, Elsevier, 2002, pp. 1723-1759. [12] Zhou, Ch., Complete deductive systems for probability logic with application to harsanyi type spaces, Ph.D. thesis, Indianapolis, IN, USA, 2007. AAI3278239.

Joshua Sack Institute for Logic, Language and Computation Universiteit van Amsterdam PO Box 94242, 1090GE Amsterdam The Netherlands [email protected]

Wiebe van der Hoek Department of Computer Science University of Liverpool Liverpool L69 7ZF UK [email protected]