1. Introduction 1.1 What is gametheoretic semantics?


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1 Chapter 18 SOME GAMES LOGIC PLAYS AhtiVeikko Pietarinen Department of Philosophy University of Helsinki 1. Introduction 1.1 What is gametheoretic semantics? Gametheoretic semantics (GTS) is a semantic theory of rational interaction between two imaginary players, who are playing the roles of the Verifier (V ) and the Falsifier (F ). They undertake to show that a logical or naturallanguage formula is true (by the actions of the player with the role V ) or false (by F s actions). This happens in a model with either partially or completely interpreted nonlogical constants, or in a suitable linguistic environment given by collateral actions and the mutually acquired and agreed common ground of players. Formally, GTS agrees with Tarski semantics for traditional firstorder logic on complete models, but otherwise their motivations as well as philosophical repercussions are worlds apart. 1 Semantic games provide a resourceful theory for logical and linguistic analysis. I will focus on three interrelated issues. First, The rules of semantic games are applicable across the logical board. Second, the definition of truth is an outgrowth of a gametheoretic notion of a strategy. Third, depending on the language under evaluation, the semantics make generous use of tools from the general theory of games, sometimes putting gametheoretic notions into a novel perspective. Supported by the Academy of Finland (Dialogical and Game Semantics, project no ), this paper represents first results of future collaboration in the context of the projects Science in Context and Proof of the Maison des Sciences de l Homme du NordPas de Calais, led by Shahid Rahman. 1
2 2 I will largely ignore considerations of natural language here, but it is worth noting that, because of such features as noncompositionality, appeal to rational actions, and proximity to graphical and diagrammatic systems, GTS fares both on logical and linguistic fronts of meaning analysis at least as well as the discourserepresentation theory of Hans Kamp, or compositional dynamic theories of meaning operating more readily on the syntax/semantics than on the semantics/pragmatics interface. 2 In fact, there is a matchless virtue in GTS: its analysis of meaning makes ample use of both the derivational record of past actions and of the multiplicity of possible actions and possible plays not realised in the actual play; all this contributes towards a fulldress contextdependent account of meaning of logical and linguistic expressions and discourse. 1.2 What is IF logic? The other compartment that I will be discussing here is the family of IF ( independencefriendly ) logics, suggested by Hintikka, The term refers to such extensions of traditional logics that accommodate the property of informational independence, which is manifested syntactically by a slash notation and brought out semantically by games of imperfect information. I will outline next the essentials of propositional, firstorder and modal IF logics. 3 I will not be considering any logical properties of these IF logics here, as my concern is restricted on the relationship between GTS and any one of the IF logics that I come to be defining. Propositional IF logic. The propositional fragment of IF logic (L IF ) builds up by: (i) If p PROP, the arity of p is n, and i 1... i n are indices, then p i1...i n and p i1...i n are L IF formulas, (ii) if ϕ and ψ are L IF formulas then ϕ ψ and ϕ ψ are L IF formulas, (iii) if ϕ is an L IF formula then i n ϕ and i n ϕ are L IF formulas, (iv) if ϕ is an L IF  formula then ( i n /U) ϕ is an L IF formula (U is a finite set of indices, i n / U). The notions of free and bound variables are the same as in firstorder logic. In ( i n /U) ϕ the indices on the righthand side of the slash are free. For simplicity, I will omit the clauses for dual prefixes such as ( i n /U). The models for the language will be of the form M = I M, (p M ) p PROP, where I M is a set A with a designated individual a, and each p M is a set of finite sequences of indices from I M. Let us set a = Left and A {a} = Right. The use of quantified indices i n enables us simultaneously to distinguish different tokens of sentential connectives and to rightfully hide choices concerning their values.
3 Some games logic plays 3 Let us also write i 1 ( i 2 /i 1 ) p i1 i 2 as (p 11 ( / ) p 12 ) (p 21 ( / ) p 22 ). If we wish to represent by restricted quantifiers unbalanced formulas, we would use identities between subformulas to denote coinciding indices. For example, (p 1 ( / ) p 2 ) p 3, rewritten as i 1 ( i 1 /i 2 ) p i1 i 2, p 21 = p 22 is balanced by applying idempotence law in their interpretation, subject to certain qualifications as soon as semantic games are implemented (see the next section). IF logic with quantifiers. IF firstorder logic is created thus: Let Qxψ, Q {, } and φ ψ, {, } be L ωω formulas in the scope of Q 1 x 1... Q n x n, where A = {x 1... x n }. Then the firstorder language L IF ωω is formed by: If B A, then (Qx/B) ψ and φ ( /B) ψ are wffs of L IF ωω. There are options: to require x / W (likewise i n / U etc.) means that the information sets of the corresponding game (to be defined below) are reflexively closed ( Eyes Open ), and to require that all x W are in Var(ϕ) (the recursivelydefined set of variables of ϕ) of any L IF ωωformula ϕ in which W occurs means that the formulas are globally contextindependent (and locally contextdependent), in other words the associated game does not make references to constants that do not occur in it ( Its All in the Game ). If we require that all x W are in BoundVar(ϕ) (the recursivelydefined set of bound variables of ϕ) of any IF formula ϕ in which W occurs, and there are no other free variables than those in W, then ϕ is an IF sentence. 2. Variety of games variety of interpretations and implementations There are differing perspectives as to the reality of the game theoretic component of semantics. My viewpoint is to adopt an implementation in terms of the factual theory of games. 2.1 Semantic games for IF logic The notion of independence may be investigated either from the point of view of Skolem functions or from the point of view of the induced information structures of the correlated games in the sense of game theory. 4 The underlying insight is the same in both cases: the strategies (typically functions) have to be such that they may not only be defined on one previous history of the game, but have to work invariably for multiple such histories, in other words, independently of any particular interpretation of some previous element already encountered.
4 4 Let A j (h), j {V, F } define a set of legitimate actions a i n i=1, n ω (a move) for each nonterminal quasihistory (qhistory) h H Z, from the domain A of the structure A, for each j. A qhistory is a 1... a n A. The structure of the logical formula uniquely determines the order of the elements in qhistories. Given a qhistory h n, j chooses a i A j (h), and the game proceeds to h n+1 := h n a i. A root r has no incoming actions. A play is a finite sequence r, a 1, h 1, a 2..., from which V s as well as F s choices can be singled out. I will dispense with the use of roles and denote players directly by V and F. Further, P : (H Z) {V, F } is a player function assigning to every h H Z a player j in {V, F } whose turn is to move. A pseudoif formula is a subformula ψ of an IF formula ϕ in which W and x W, x Var(ψ). Let Sub(ϕ) be a recursively defined set of pseudosubformulas of an IF formula ϕ. The labelling function L: H Sub(ϕ) assigns to each h H: L( r ) = ϕ; for every terminal history h Z, L(h) is a literal. The components of the game G A = H, L, P, u j jointly satisfy: if L(h) = ϕ and P (h) = V, then h ϕ H, L(h ϕ) = ϕ, P (h ϕ) = F ; if L(h) = ϕ and P (h) = F, then h ϕ H, L(h ϕ) = ϕ, P (h ϕ) = V ; if L(h) = (ψ ( /W ) θ) or L(h) = (ψ ( /W ) θ), then h Left H, h Right H, L(h Left) = ψ, and L(h Right) = θ; if L(h) = (ψ ( /W ) θ), then P (h) = V ; if L(h) = (ψ ( /W ) θ), then P (h) = F ; if L(h) = ( x/w ) ϕ or L(h) = ( x/w ) ϕ, then h a H for every a A ; if L(h) = ( x/w ) ϕ, then P (h) = V ; if L(h) = ( x/w ) ϕ, then P (h) = F. Payoffs are mappings u j (h) {1, 1}, h Z. For every h Z, if L(h) = St 1... t m and (A, g) = St 1... t m, then u V (h) = 1 and u F (h) = 1, and if L(h) = St 1... t m and (A, g) = St 1... t m, then u V (h) = 1 and u F (h) = 1. A history is now qualified to be a prefix of the playsequence with labels terminating at a qhistory h n Z, n > 0. By a history I customarily mean a finite presequence of (L(r), a 1, L(h 1 ), a 2... L(h n )), h n Z. Histories are thus labelled with the subformulas of the formula under evaluation. A set of plays is a game frame. A set of plays with payoffs assigned to the terminating histories gives rise to a game G(ϕ, A, g) with a τ structure A for ϕ L IF ωω and to G(ψ, M) for ψ L IF. A deterministic strategy has for all h H Z and a i A probability f j (h)(a i ) {0, 1}. Actions are prescribed by a deterministic strategy f j : P 1 ({j}) (2 A ), f j (h) A(h). Next, I j is information partition of P 1 ({j}) such that for all h, h Sj i, h a H if and only if h a H, a A. In case W =, the only sets in I j are singletons.
5 Some games logic plays 5 As to G(ϕ, A, g) and G(φ, M), it is required that if h, h S j i then f j (h) = f j (h ). In the terminology of extensive games, strategies are defined on the components of the information partition, viz. on information sets Sj i. A V partition is n i=1 Si V and an F partition n i=1 Si F. Let h j be a quasihistory produced by f j. Then f j is winning for j if for every h j H, u j (h j ) = 1. A truth (resp. falsity) is defined as an existence of a winning strategy for the player who initiated G(ϕ, A, g) or G(φ, M) as V (resp. F ). There are plenty of histories not on the winning (equilibrium) path and would for that reason be assigned a zero probability. In the definition of truth it suffices to take into account only a subset of strategies that does not lead to such histories. Considerable work has been done in game theory to make solution concepts work for all positions. 2.2 Concurrency vs. sequentiality The above imperfectinformation games are sequential in the sense that there is a leftlinear order of moves from the first component onwards captured by the system of game rules described above. The linear order of moves means that at each position, at most one player makes a choice. Accordingly, games in which more than one player may choose at any position are concurrent. I will call this sense of concurrency informational. It follows from the above definitions that the semantic games for IF logic are in this sense not concurrent. However, an alternative sense of concurrency is that it is informationally independent moves that count as concurrent. Thus, the histories that pass through an information set (see below) prescribe a concurrent move at this information set with respect to the sets through which these histories have passed at a lesser depth. This sense may or may not induce temporal considerations. I will call this sense actional concurrency. The semantic games for IF logic are in this sense concurrent; they involve independent actions. The linear order of moves (sequentiality) thus means that there are singleton information sets at all histories. Such games are associated with the slashfree fragment of IF logic. An IF formula ϕ associated with a truly concurrent game is one in which no variable x n in Q x n or an index i n in Q i n, Q {, } fails to occur in some set W deeper in ϕ. A typical implementation of semantics for an IF formula lies between sequential and truly concurrent game. 2.3 Teams that communicate Different IF formulas give rise to structurally different streams of information. For instance, one may hold that semantic games are ones of
6 6 perfect information in the sense that the two players V and F do not lose knowledge about positions and the history of the game, and in which there is a difference in the communication of information, not between V and F but between members that constitute these two players, viewed as teams with agents M j = {m j 1... mj n}, j {V, F }, n ω. Another point worth noting is that, since we deal with finite formulas, the length of communication and the amount of information transmitted is limited by the length, organisation and type of the components the formula contains. Precisely which information is picked, and thus the content of information that may need to be revealed to others, is left for the players to decide. In team games, the available information concerning past actions is restricted for individual members. I will consider both the cases in which there is communication (coordination) between the members of the team and the cases in which there is no such communication. If the team members are not allowed to communicate, each member chooses a i A(h) independently of others decisions. Player s choice sequence in any particular play is made by M j 1 M j. Why such games? Consider an L IF ωωformula x y( z/x) Sxyz. This is correlated with noncommunicating team games, in which it is assumed that existential quantifiers are explicitly slashed for other existentially quantifiers variables occurring further out in the formula. Otherwise they are dependent, as in x y( z/xy) Sxyz. Such formulas are not the only ones coming together with noncommunication. For instance, in x y( z/y) Sxyz, the V team consists of {m V 1, mv 2 }, in which neither mv 1 nor mv 2 passes on the information they have derived from higher up. Even a twostage game between V and F may need {m 1, m 2 }, as in x (S 1 x ( /x) S 2 x). Here V should not get the value of x, but since it goes with the disjuncts, we take m V 1 to receive this value while being blocked from communicating it to m V 2, who gets to choose at L(h) = (S 1 x ( /x) S 2 x). However, consider also φ = x y z (S 1 x ( /x) S 2 yz). The truth of φ is revealed in terms of extended Skolem normal form: f x y ((S 1 x f(y) = 0) (S 2 yz f(y) 0)). Yet, if we consider move by move what is going on in G(φ, A, g), when individuals have been distributed for x, y after the first two moves by F at r and its successors, does not V get this illicit information by observing the labels attached to histories when the fourth move of disjunction is planned? To circumvent this, I will assume that variables in the labelled subformulas carry no instantiated values, in other words, the players do not observe assignments. Ditto for y in ( z/y) and x in ( /x). Accordingly, semantic games may be viewed
7 Some games logic plays 7 in their extensive forms, because in order to make φ true, V needs to know the value of x when choosing for z, and she needs to know the value of y when making a decision that would lead to either S 1 x or to S 1 yz. To know the values can symbolically be represented only by instantiating constants to corresponding variables, including those on the lefthand side of the slashes, which amounts to a pseudoskolem normal form representing only one particular play of the game with respective instantiations. For in extensive forms, all values, hidden and public, are included in histories. Players may also try to recover the identity of the information set they are at by looking at the available choices also in cases in which there is a possibility that some of the histories within an information set are terminal. Propositionally, an example of this is (p 1 ( / ) p 2 ) p 3. A different implementation is produced if the coordinating player gets to decide which choices are actually put forward among those proposed by his or her agents. Such strategies are twotiered: first, memberspecific strategies delineate actions, and second, F s or V s coordinating strategies pick from the teaminternal, private choice sets so induced. Yet another possibility is that a predetermined set of suboptimal agents proposes the actions, the weighed average of such (possibly randomised) actions being elected as the player s preferred, representative choice. This is related to the concept of bounded rationality popularised in interactive decision theory of agents. Further, it makes the small worlds doctrine, according to which agents are able to preview only fragments of domains and states of the game, better understood. 2.4 Teams that do not communicate If the team members are allowed to communicate, we get games that are associated with such IF formulas in which existentially (universally) quantified variables may depend on other existentially (universally) quantifier variables, as in x y( z/x) Sxyz. The amount of intrateam communication determines the extent to which such dependence is realised. Such incestuous dependence creates channels by which a player may elicit additional information. For instance, although the existentially quantified variable z does not depend on the universally quantified variable x, the second member m V 2 of the V team choosing a value for z gets to hear the value F chose for x via the other member, m V 1, who chose a value for y, the choice of which being dependent on the choice of the value for x, for the sole reason that m V 1 s communication to m V 2 is not specifically blocked.
8 8 The strategies in communication games differ from noncommunicative ones in that their input includes the choices of action in the histories by the other members of the team, which is not permitted in the noncommunicative case. 2.5 Two ways of losing information Let be a partial order on the tree structure of extensiveform games. A game satisfies nonabsentmindedness, if for any h, h H, h, h Sj i, i {V, F }: if h h then h = h. Let a depth d(q) of a quantifier and a connective be defined inductively in a standard way. All semantic games for IF formulas ϕ described here satisfy nonabsentmindedness, because every Q has a unique depth d(q), and hence every subformula of ϕ has a unique position in the game given by its labelling. For any two subformulas of ϕ at h, h Sj i, h h and h h. Let Z(h) be a set of plays that pass through any h H, if h becomes a subsequence of any h Z(h). Likewise, let Z(a i ) be a set of plays that pass through an action a i A or a sequence of actions a i n i=1, if a i h Z(a i ). Define a precedence relation < between any two information sets Sj i, Si k I i so that if h, h Sj i Si k such that h h, then Sj i < Sk i. Thus Si j < Sk i says that there exists h Z passing through h and h. If nonabsentmindedness holds then any h Z passes through Sj i or Si k at most once. As before, let P 1 ({i}) be the set of histories where i moves playing a strategy f i. Information set Sj i is relevant for f i, if Sj i P 1 ({i}) is nonempty. Now let Sj i I i. A game has perfect recall 1, if Sj i is relevant for f i implies Sj i P 1 ({i}) for all f i. This says that while players move within their information sets they will have perfect recall in the sense of not forgetting information (or knowledge) that they possess. There is also an alternative way of characterising perfect recall. A game has perfect recall 2, if S j i < Si k implies the existence of a sequence of actions a i n i=1 A available from Sj i such that Z(Si k) Z( ai n i=1 ) (for otherwise there will be wider information sets occurring for a player later on in the game). This says that i does not forget his or her actions. Semantic games G(ϕ, A, g) do not in general satisfy perfect recall i (i = 1, 2), because any L IF ωωformula ϕ that contains a subformula ψ = Q 1 x 1... (Q 2 x n /x 1 ), Q 1, Q 2 = or Q 1, Q 2 = and d(q 1 ) < d(q 2 ), gives rise to a partition in which ψ Sub(ϕ) beginning with Q 2 induces Sk i and η Sub(ϕ) beginning with Q 1 induces S j i, such that Sj i < Si k. Thus a i n i 1 A that i chooses for Q 1x 1 are available from S j i, but then clearly Z( a i n i 1 ) Z(Sk i ). On the other hand, perfect recall 1 de
9 Some games logic plays 9 pends on allowing nonstandard information sets that are not relevant for player s strategies f i. But any such information set violates perfect recall 1, all P 1 ({i}) being L(P 1 ({i})). Syntactically speaking, a formula ϕ in L IF ωω exhibits perfect recall 1, if for any (Q 1 i 1 /U 1 ), (Q 2 i 2 /U 2 ) in ϕ, if either Q 1, Q 2 = or Q 1, Q 2 = and d(q 1 ) < d(q 2 ), then i 1 U 2. It exhibits perfect recall 2, if for any (Qj 1 /U), (Q 1 j 2 /U 1 ), (Q 2 j 1 /U 2 ) in ϕ, if either Q 1, Q 2 = or Q 1, Q 2 = and d(q) < d(q 1 ) < d(q 2 ) and j 1 U 1, then j 1 U 2, j 2 U 1. The former clause is about player forgetting his or her own actions. The latter says that Q 1, Q 2 = or Q 1, Q 2 = give rise to imperfect recall even if one is not independent of the other, provided that players have acquired different information from elements higher up in a formula. One may thus study fragments of IF logic in which imperfect recall does not hold, or holds in some restricted sense. In the latter case we are dealing with aspects of bounded recall Lehrer, 1988, as well as with imperfect monitoring of actions and information transmission. 2.6 Screening vs. signalling Dually with imperfect recall, one may characterise information increase that is seen to happen in x( y/x) z Sxyx. Both imperfect recall and informational increase may naturally be manifested: x( y/x)( z/y) Sxyz. Such learning is similar to what happens in screening games Rasmusen, 1994, in which the first player is uninformed of certain aspects of the game and the second player, being fully informed, can screen his or her actions, for instance via its members. In signalling games, on the other hand, connected to imperfect recall by the team perspective, the informed player moves first and may signal previous features of the game to the subsequent uninformed player. If in the former case the types of the first and the second player are the same, then screening amounts to learning, and likewise, the phenomenon of signalling means, for the two players of the same type, that information is being forgotten. 2.7 Concurrent games and Henkin quantifiers The first sense of concurrency, namely that at each position, there may be more than one player choosing, gives rise to games that are associated with formulas with finite, partiallyordered quantifier prefixes of the form x 1... x n y z 1... z m w (for some n, m ω), and are interpreted via informational concurrency. This may happen at r as well as at any h n H Z, and thus the games do not need to form trees. Since on models with pairing functions two rows suffice to represent arbitrary parallel
10 10 orderings Krynicki, 1993, at most two moves exists at each qhistory h of such concurrent games, and the order of these rows is immaterial. 2.8 Remark on scope A logical distinction between informational concurrency and sequentiality is in terms of logical priority. Logical priority between logically active components may be defined as preferences in the evaluation of such components, which in turn is a derivative of the semantic information flow in the structure determined by the syntax of an IF formula. If semantic information flows from a component C 1 to C 2, the former is logically prior to the latter; in other words, the latter is logically dependent of the former. But in case information does not transmit between C 1 and C 2, they are not ordered by priority, and thus C 1 and C 2 are independent of one another in the sense that their strategic evaluation makes no use of the semantic attributes assigned to them provided, of course, that there is no illicit communication. Traditional logic assumes that logical priority goes by the recursively defined subformula relation, which no longer holds in IF extensions. Logical priority need to be distinguished from binding, which encodes the extent to which a quantifier and its quantified variable reaches in assigning the same values to other occurrences of the same variable in the formula Hintikka, This distinction has significant repercussions in graphical (heterogeneous) logics of diagrams, for instance, mandating a move from two to threedimensional spaces in which such graphs are described Pietarinen, 2003d. 2.9 Nonpartitional information structures The information structure defined above is typically taken to be partitional in that its cells are composed of histories closed under equivalence relations. For the purposes of IF logic, this may be too ideal, as φ = (p 1 ( / ) p 2 ) (p 3 p 4 ) witnesses. After F has chosen a conjunct, only if the action was Left will V be able to fully process that she does not know whether continuation occurs in L(h ) = (p 1 ( / ) p 2 ) or L(h) = (p 3 p 4 ). Rather than saying that V s strategy f V is defined on Si V in which h V h, this is implemented by defining three relations Rhh, Rh h and Rhh. Thus Si V contains antisymmetric relations. Such games are still nondetermined, since by setting u V (h 1 ) = u V (h 4 ) = 1, L(h 1 ) = p 1, L(h 4 ) = p 4, u F (h 2 ) = u F (h 3 ) = 1, L(h 2 ) = p 2, L(h 3 ) = p 3 for φ there are no winning strategies for V nor for F. In fact, the same payoff distributions for G(φ, g) and G(φ, g), φ = (p 1 ( / ) p 2 ) (p 3 ( / ) p 4 ), the latter being closed under sym
11 Some games logic plays 11 metric relations, give rise to nondetermined games. The sole difference is that the player whose actions lead to nonpartitional cells has fewer options to prevent the opponent from winning. For the opponent who in planning his or her moves in such cells there is no difference, and if it is costly to process all accessible histories, nonpartitional models would in fact be preferable. The specialty introduced by IF logic is that even the fundamental question of whether a player has available a strategy that gets a nonsingleton information set as an input may depend on actions made in previous parts of the game. It is indeed unrealistic to expect players to fully process the information concerning the reasons that led to their uncertainties. Such uncertainties themselves may in this sense be partially realised. As noted, players may then also lack selfawareness of their own actions by taking information sets to be nonreflexive. To make games even more realistic, players knowing that they will lose information at some point on amounts to selfawareness of their bounded memory resources. This is one of the motivations for introducing players as multiagent teams in the first place. 3. Modal extensions 3.1 IF modal logic, propositional case In modal logic, it may easily happen that the domain (the subset U of the set of possible worlds W) from which the players are to pick values is such that A(h) A(h ) for h, h H Z, h, h Sj i. This violates the traditional assumption in imperfectinformation games according to which for any such h, h within the same information set, A(h) = A(h ), and the reason is that otherwise a player could, by virtue of perfect foresight, detect differences in the histories, thus infer his or her actual location, and thus derive some information that was supposed to be hidden. Prima facie, such restrictions will have to go in IF modal logic, because in arbitrary frames of a possibleworlds structure, there may be any composition of accessible worlds from any world that a play has reached. And if so, IF modal logic appears to dispense with perfect foresight, making it to look an idealisation of similar dubious standing as decisionmaker s hyperrationality. Furthermore, this conclusion would, in any case, backed by the simple observation that in perfectinformation games with singleton information sets, no such coincidence of available actions is needed. Let us define φ to be a formula of a propositional modal logic φ := p j ϕ j ϕ ϕ 1 ϕ 2 ϕ 1 ϕ 2 ϕ. An IF modal logic is got
12 12 by letting k j ψ ( k j { k j, k j }, j, k ω) be in Sub(ϕ), and letting A = { k n} be such that all k i A, i ω occur in ϕ and k j occur after k i, j i. Now, if B A, then {( k j /B) ψ, (φ ( /B) ψ)} MLIF. For example, ( 2 2 / 1 1 ) ψ and 1 1 (ϕ ( / 1 1 ) ( 1 1 / 1 1 ) ψ) are wffs of ML IF. However, since the notion of independence may mean different things, to keep its semantic definition general, one may impose further restrictions on the meaning of independence as need arises. Indeed, precisely what does the slash in 1 i ( 2 j / 1 i ) ϕ reflect? Its general meaning is that the choice of a world by V for 2 j has to be made in ignorance of the choices made by F for 1 i. The phrase in ignorance of can be interpreted in several ways. It may mean that (i) there is an equivalence relation (h 1 w 1 ) i (h 2 w 2 ) linking any worlds w 1, w 2 that a player cannot distinguish, and consequently he or she loses track of some of the choices in the past that lead to those worlds, (ii) the choices for several modal operators are actionally concurrent, (iii) the play backtracks to the world from which the worlds chosen for 1 i departed. The general definition needs to dispense with the accessibility relations {ρ 1... ρ n } (defined for each agent i = 1... n, ρ i W W, w 1 [w 0 ] ρi meaning that w 1 is iaccessible from w 0 ), which normally guide player s choices from worlds that the game has reached. Accordingly, player s position in the game traversing the possibleworlds structure and his or her available choices are no longer strictly correlated. When encountering an expression of the form ( i j /W ) ϕ in ψ, in which W is a sequence m n of operators already occurring in ψ, V (likewise F for ( i j /W ) ϕ) chooses w U W. Reasonable condition for the subsethood is that at w, only the worlds chosen prior to reaching w ( regret ) and the immediately available ones w [w] ρj ( no longdistance foresight ) are the candidates to be included in U. When the play backtracks from h H to one of its subsequences h, it only means that the subdomain U is copied from A(h ) to be A(h). This needs no equivalence relations between histories. In case h, h Sj i and A(h) A(h ), to satisfy perfect foresight, those worlds are added from A(h) to A(h ) that do not occur in A(h ) and vice versa. Since these dummy copies are never reached by rational players strategies, they do no harm. 5 The game rules and winning are now such that given a modal model M = W, ρ 1... ρ n, g (g is a valuation of atomic formulas) and a game G(ϕ, M, w), if w g(p), L(h) = p, h Z for p atomic, u V (h) = 1. Otherwise u F (h) = 1. Rules for conjunction and disjunction are usual. As to the sense (i) of sect. 1, if h, h Sj i, then f i(h) = f(h ). As to the sense (ii), an actionally concurrent game is associated with partially
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