Łukasiewicz Games
Enrico Marchioni Michael Wooldridge Institut de Recherche en Dept of Computer Science Informatique de Toulouse University of Oxford, UK Université Paul Sabatier, France [email protected] [email protected]
ABSTRACT ual goal 'i, expressed as a formula of propositional logic. Strate- Boolean games provide a simple, compact, and theoretically attrac- gic concerns in Boolean games arise from the fact that whether i’s tive abstract model for studying multi-agent interactions in settings goal is in fact satisfied will depend in part on the choices made where players will act strategically in an attempt to achieve per- by other players. Players in Boolean games can be understood as sonal goals. A standard critique of Boolean games, however, is representing non-deterministic computer programs, and the over- that the binary nature of goals (satisfied or unsatisfied) inevitably all framework of Boolean games provides an elegant mathematical trivialises the nature of such strategic interactions: a player is as- model through which to investigate issues of strategic interaction sumed to be indifferent between all outcomes that satisfy his goal, in multi-agent systems. and indifferent between all outcomes that do not satisfy his goal. Now, a standard critique of Boolean games is that the binary na- In this paper, we introduce Łukasiewicz Games, which overcome ture of goals (satisfied or unsatisfied) inevitably trivialises the na- this limitation by considering goals to be specified using finitely- ture of strategic interactions. For example, players are assumed to valued Łukasiewicz logics. The significance of this is that formu- be indifferent between all outcomes that satisfy their goal, and are lae of Łukasiewicz logic can express every continuous piecewise indifferent between all outcomes that do not satisfy their goal. This linear polynomial function with integer coefficients over [0, 1]n, assumption is clearly a gross simplification for many situations, thereby allowing goal formulae in Łukasiewicz Games to naturally a concern which led researchers to extend the original Boolean express a much richer range of utility functions. After introducing games model with costs, leading to richer and more realistic pref- the formal framework of Łukasiewicz Games, we present a num- erence structures for agents [11]. While these refinements make it ber of detailed worked examples to illustrate the framework, and possible to model much richer types of interaction, the inherently then investigate some of the properties of Łukasiewicz Games. In dichotomous nature of preferences in Boolean games is surely one particular, after investigating the complexity of decision problems of their most debated features, and the work in the present paper is in Łukasiewicz Games, we give a logical characterisation of the directly motivated by this limitation. existence of Nash equilibria in such games. Specifically, we introduce Łukasiewicz Games, which overcome this limitation of Boolean games by allowing goals to be specified as formulae of finitely-valued Łukasiewicz logics [3]1. The ratio- Categories and Subject Descriptors nale for using Łukasiewicz logic in this way is given by the Mc- I.2.11 [Distributed Artificial Intelligence]: Multiagent Systems Naughton Theorem, which says that every continuous piecewise I.2.4 [Knowledge representation formalisms and methods] linear polynomial function with integer coefficients over [0, 1]n can be expressed as a formula of Łukasiewicz logic in n variables (see, General Terms e.g., [3]). Thus, Łukasiewicz logic provides a natural, compact, for- mally well-defined and expressive logical representation language Theory for payoff functions, allowing much richer preference structures than is easily possible in conventional Boolean games. Keywords The present paper makes three main contributions. First, we Boolean games, Łukasiewicz Games, Łukasiewicz logic provide a formal definition of Łukasiewicz Games. Second, we present a number of detailed worked examples, which illustrate how a range of strategic scenarios can be formalized within this 1. INTRODUCTION framework. In particular, we argue, these scenarios cannot natu- Boolean games provide a simple, compact, and theoretically attrac- rally be formalized using conventional Boolean games. Third, we tive abstract model for studying multi-agent interactions in settings investigate properties of Łukasiewicz Games. We show that despite where players will act strategically in an attempt to achieve private their expressive power, the key decision problems for Łukasiewicz goals [8, 2, 5, 6]. In a Boolean game, each player i exercises unique Games are no more complex than for conventional Boolean games; control over a set of Boolean variables Vi, and will attempt to as- in addition, we give a logical characterisation for the existence of sign values for these variables in such a way as to satisfy an individ- equilibria in Łukasiewicz Games.
Appears in: Alessio Lomuscio, Paul Scerri, Ana Bazzan, and 1 Michael Huhns (eds.), Proceedings of the 13th International Con- In the remainder of the paper, when we refer to Łukasiewicz logic, ference on Autonomous Agents and Multiagent Systems (AAMAS it should be understood that we are referring specifically to finitely- 2014), May 5-9, 2014, Paris, France. valued Łukasiewicz logics, as distinct from the many other exten- Copyright c 2014, International Foundation for Autonomous Agents and sions of Łukasiewicz logic that have been investigated in the liter- Multiagent Systems (www.ifaamas.org). All rights reserved. ature [3].
837 2. PRELIMINARY DEFINITIONS 3. ŁUKASIEWICZ GAMES Since Łukasiewicz logic is fundamental to our present work, but We now introduce the framework of Łukasiewicz games. First, let is not as widely known as the classical logics that underpin con- V = p1,...,pm be a finite set of propositional variables, as ventional Boolean games, we begin by introducing the concepts of above.{ Our games} are populated by a finite set of players P = Łukasiewicz logic that are used in the remainder of the paper. P1,...,Pn (also referred to as “agents”). Note that throughout { } The language of Łukasiewicz logic Ł[3] is built from a countable this paper, we assume that P = n. Each player Pi controls a | | set of variables V = p1,p2,... , the binary connective “ ”, and subset of propositional variables Vi V, so that the sets Vi form a { } ! ✓ the truth constant 0 (for falsity). Further connectives are defined as partition of V. The fact that player Pi is in control of the set Vi of follows: propositional variables means that Pi has the unique ability within the game to choose values for the variables in Vi. It is assumed that ' is ' 0¯, ' is '&(' ), 1 k 1 ¬ ! ^ ! variables take values from the set Lk = 0, ,..., , 1 . '& is (' ), ' is ((' ) ), { k k } ¬ !¬ _ ! ! A strategy for an agent Pi is a function si : Vi Lk, which ' is ( '& ), ' is (' )&( '), ! ' is '¬&¬ , ¬ d('$, ) is ('! ). ! corresponds to a valuation of the variables controlled by Pi.A ¬ ¬ $ strategy profile is a collection of strategies (s1,...,sn), one for We ofter write n' as an abbreviation for ' ', with n 1. each player. Every strategy profile directly corresponds to a valua- ··· tion function e : V Lk and vice versa; we find it convenient to n ! Let Form denote the set of formulae of Łukasiewicz logic. A abuse notation a little by treating strategy profiles as valuations and valuation, e, is a mapping e : V [0|, 1],{z which} assigns to all valuations as strategy profiles. ! propositional variables a value from the real unit interval; we re- As in conventional Boolean games, we assume that each player e(0) = 0 c quire that . The semantics of Łukasiewicz logic are then is associated with a Łk-formula 'i, with propositional variables defined, with a small abuse of notation, by extending the valuation from V, whose valuation is interpreted as the payoff function for e to complex formulae. Although strictly speaking we only need state the rule for the “ ” operator (as we can define the remaining player Pi. That is, the player Pi seeks a valuation e that maximises ! the value of the corresponding function f' . Of course, not all the operators in terms of this operator and 0), we present the complete i ruleset in the interest of clarity: variables in 'i will in general be under Pi’s control and, conse- quently, the utility Pi obtains by playing a certain strategy (i.e., e(' )=min(1 e(')+e( ), 1) choosing a certain variable assignment) also potentially depends in e!( ')=1 e(' ) part on the choices made by other players. e('&¬ )=max(0 ,e(')+e( ) 1) Łukasiewicz Games can be seen as a generalization of Boolean e(' )=min(1,e(')+e( )) Games [8, 2], since the latter are obviously a special case of the e(' )=max(0,e(') e( )) e(' )=min(e('),e( )) former. However, Łukasiewicz Games also incorporate the notion e(' ^ )=max(e('),e( )) of cost/effort, given that each player’s strategic choice can be seen e(' _ )=1 e(') e( ) as an assignment to each controlled variable carrying an intrinsic e(d('$, )) = e ('|) e( ) | cost. This makes it possible to formalize situations in which agents | | aim at a better tradeoff between the costs of making certain choices A valuation e is a model for a formula ' if e(')=1. A valuation and the resulting payoff. e is a model for a theory T , if e( )=1, for every T . XAMPLE Consider the following example with two players 2 E 1. In this paper, we restrict our attention to finite-valued Łukasiewicz P1 and P2, who control variables x and y, z , respectively. The { } { } logics Łk. In such logics, it is assumed that the domain is a set of payoffs are given by the following formulae: 1 k 1 the following form: Lk = 0, k ,..., k , 1 . The notions of { } '1 := x y z and '2 := x z. valuation and model for Łk are defined analogously just replacing ^¬ ^ [0, 1] by Lk as set of truth values for the logic, that is, valuations In order for P2 to maximize the payoff, it suffices to assign s2(z)= are functions with the signature e : V Lk. It is sometimes use- 1, no matter what P does. Player P , however, has no chance of ! 1 1 ful to introduce constants in addition to 0 that will denote values in maximizing the payoff except by assigning s1(x)=1and hoping c the domain Lk. Specifically, we will denote by Łk the Łukasiewicz that s2(z)=1and s2(y)=0. The latter makes sense, since, logic obtained by adding constants c for every value c Lk.We given s (z)=1, any assignment to y does not change the value 2 2 assume that valuation functions e interpret such constants in the of '2, so P2 has no incentive in assigning any other value to y but natural way: e(c)=c. 0 and raising the costs. Still, P2 knows that P1 needs s1(x)=1 A well-known result by McNaughton established that every con- and consequently can get the maximum payoff by simply keeping tinuous piecewise linear polynomial function with integer coeffi- the costs at minimum with s2(z)=0and s2(y)=0. cients over [0, 1]n (called a McNaughton function) is definable by a formula in Łukasiewicz logic [3]. We now formally define Łukasiewicz games. c In the case of finite-valued Łukasiewicz logics Łk, the functions DEFINITION 1(ŁUKASIEWICZ GAMES). AŁukasiewicz game c defined by a formula are combinations of the restrictions of Mc- on Łk is given by a structure: n Naughton functions over (Lk) and the constant functions for each G c = P, V, Vi , Si , 'i c Lk. Notice that the class of functions definable by Łk-formulas G h { } { } { }i 2 n exactly coincides with the class of all functions f :(Lk) Lk, where: for every n 0.2 In this sense, we can associate to every formula! c n 1. P = P1,...,Pn is a finite set of players. '(p1,...,pn) from Łk a function f' :(Lk) Lk. { } ! Finally, note that the satisfiability problem for finite Łukasiewicz 2. V = p1,...,pm is a finite set of propositional variables logics is no more complex than for classical propositional logic: the taking{ values from} problem is NP-complete [7, 3]. 1 k 1 Lk = 0, ,..., , 1 . 2 k k This result can be obtained by using Lemma 3. ⇢
838 3. Vi V is the set of propositional variables under control of The NON-EMPTINESS problem is the problem of determining for a ✓ player Pi, so that the sets Vi form a partition of V, i.e.: given game whether there exist any pure strategy Nash equilibria G p n for the game. The problems are co-NP-complete and ⌃2-complete V = Vi, and for i = j, Vi Vj = . for conventional Boolean games. And, perhaps surprisingly, we 6 \ ; i=1 have: [
4. Si is the strategy set for player i that includes all valuations THEOREM 1. The MEMBERSHIP and NON-EMPTINESS prob- p si : Vi Lk of the propositional variables in Vi, i.e. lems for Łukasiewicz games are co-NP-complete and ⌃2-complete, ! respectively. Si = si si : Vi Lk . { | ! } The proof of this can be adapted from proofs of the corresponding c 5. 'i is a Łk-formula, built from variables in V, whose associ- results for conventional Boolean games: the key point is that strate- ated function gies for Łukasiewicz games are “small witnesses” in the terminol- t ogy of computational complexity. A strategy for a player simply f'i :(Lk) Lk ! consists of a value from the domain Lk for every variable con- corresponds to the payoff function (also called utility func- trolled by that player. Since values in Lk are rational numbers that tion) of Pi, and whose value will be determined by the valu- can be assumed to be expressed in binary, they have a compact rep- ations in S1,...,Sn . resentation. { }
A strategy profile ~s for is a tuple ~s =(s1,...,sn), with each G 4. EXAMPLES si Si being the strategy selection for the corresponding player in 2 . Given a strategy si for Pi, we denote by s i the collection of We now introduce a number of examples to illustrate Łukasiewicz G strategies (s1,...,si 1,si+1,...,sn) not including si, and S i is games, and in particular, we choose examples that we believe can- the set of all s i’s. With an abuse of notation, we use not be easily expressed in the framework of conventional Boolean games (i.e., using classical logic to express goals). f'i (s1,...,sn) and f'i (si,s i) 4.1 Generalized Matching Pennies to denote Pi’s payoff under the strategy profile (s1,...,sn): re- The first game we present is a generalization of Matching Pennies3, call that 'i defines a payoff function f' , and a strategy profile i the classic example of a zero-sum game without a pure strategy (s1,...,sn) corresponds to a valuation e : V Lk. We now introduce the concepts of dominance,! best response and equilibrium. In the original game, two players P1 and P2 both Nash equilibrium adapted to our framework. In what follows, we have a penny and must secretly choose whether to turn it to head always assume we are working with an arbitrary n-player Łukasiewicz or tails, revealing their choices simultaneously. If their choices are game on Łc . the same, then P1 takes both pennies; if they are different, P2 takes G k both. DEFINITION 2(DOMINANCE). Let be a Łukasiewicz game. Now, imagine that both players must perform an action with a G A strategy si Si for player Pi is called strictly dominated (weakly certain cost and are in charge of the variables p1 and p2, respec- 2 dominated) if there exists a strategy si0 Si such that for all tively. P1’s overall strategy is to be as close as possible to P2’s s i S i 2 2 choice. In contrast, P2 wants to keep the greatest possible distance between the choices. The players’ strategy spaces are given by the f'i (si,s i)
839 P2 0, 1/52/53/54/51 0 1, 0 4/5, 1/5 3/5, 2/5 2/5, 3/5 1/5, 4/5 0, 1 1/5 4/5, 1/5 1, 0 4/5, 1/5 3/5, 2/5 2/5, 3/5 1/5, 4/5 2/5 3/5, 2/5 4/5, 1/5 1, 0 4/5, 1/5 3/5, 2/5 2/5, 3/5 P1 3/5 2/5, 3/5 3/5, 2/5 4/5, 1/5 1, 0 4/5, 1/5 3/5, 2/5 4/5 1/5, 4/5 2/5, 3/5 3/5, 2/5 4/5, 1/5 1, 0 4/5, 1/5 1 0, 1 1/5, 4/5 2/5, 3/5 3/5, 2/5 4/5, 1/5 1, 0
Table 1: Generalized Matching Pennies.
parameters. Intuitively, xi is interpreted as the effort i is willing The game is as follows. Two travelers fly back home from a to make in her interaction with others. The payoff ui is heavily trip to a remote island where they bought exactly the same an- influenced by the choice of the agent with the lower effort level. tiques. Unfortunately for them, their luggage gets damaged and Therefore, each player’s payoff depends on the weakest link in the all the items acquired are broken. An agent of the airline promises strategic interaction. The game has m pure strategy Nash equilibria a refund for the inconvenience, but, not knowing the exact value of corresponding to the strategy profiles in which the players select the the objects, she puts forward the following proposal. Both travelers same values. must privately write down on paper a number between 0 and 100, Notice that in weak-link games, the payoff function’s domain is corresponding to the cost of the antiques. If they both write the a strict subset of the function’s range. Therefore, we introduce a same number, the agent can assume that they are both telling the variant of such games where domain and range coincide in order to truth, so they will both receive exactly that amount minus one unit c make it possible to define a representation in terms of Łk-formulae. (with a minimum of 0). If the travelers write different numbers, the We formalize n-player weak-link games in the following game: one who wrote the lower number, say x, (assumed to be the honest one) will receive x plus a reward of two units (with a maximum of = P, V, Vi , Si , 'i G h { } { } { }i 100). The other player, who is regarded by the agent as dishonest, where: will receive x with a penalty of two units (with a minimum of 0). Payoffs in the Traveler’s Dilemma are defined by the following 1. V = p1,...,pn . function, whose payoff matrix is shown in Table 3: { }
2. Vi = pi . max (x 1, 0) x = y { } f(x, y)= min (x +2 , 100) x 'i(p1,...,pn):= pj pi pj choice and writes 99, while the other player sticks to the original j=1 j=1 ! selection, she can increase her payoff to 100. Under the assumption ^ ^ of common knowledge and rationality, however, the other player is whose associated payoff function is drawn to make the same decision, which leads to writing 99, yield- n n ing a mutual payoff of 98. Still, deviating from this selection is f' =max min xj max xi min xj , 0 , 0 . unilaterally beneficial for each individual, producing again a situa- i j=1 j=1 ✓ ✓ ◆ ◆ tion of coordination between the players’ choices. This reasoning only ends when both players select 0, thus gaining nothing in the Table 2 shows the payoff matrix for P1 for the two-player version process. The strategy profile (0, 0) is the unique pure strategy Nash of the weak-link game with k =7(P2’s payoff is obtained by equilibrium of the game. This, however, clearly clashes with what replacing P1 by P2 and vice versa), where intuition would lead us to accept as a rational choice. It seems un- reasonable that two individuals would follow the above line of rea- '1(p1,p2):=(p1 p2) (p1 (p1 p2)) ^ ^ soning and rationally come to the conclusion that the best solution with payoff function is ending up empty-handed. The Traveler’s Dilemma can be formalized as a Łukasiewicz f' =max(min(x1,x2) max(x1 min(x1,x2), 0), 0). game over Ł100 as follows. Define the following game 1 Notice that in this game there is no strictly dominated strategy, = T1, T2 , p1,p2 , p1 , p2 , '1(p1,p2),'2(p1,p2) , G h{ } { } { } { } { }i while the strategy s1(p1)=0is weakly dominated. where T1 and T2 are the two travelers; p1,p2 is the set of propo- { } 4.3 Traveler’s Dilemma sitional variables, with p1 being controlled by T1 and p2 being con- 5 The Traveler’s Dilemma was introduced by Basu [1] in order to il- trolled by T2; the payoff formulas are defined as follows lustrate the tension between the rational solution suggested by the 5The connective is defined as follows in each finite-valued existence of a Nash Equilibrium, and apparently reasonable behav- Łukasiewicz logic Łk: ior based on intuition. We introduce (a slightly modified variant of) the Dilemma and show how to formalize it as a Łukasiewicz game. ' := (k( ')) ¬ ¬ 840 P2 01/72/73/74/75/76/7 1 0 0 0 0 0 0 0 0 0 1/7 0 1/7 1/7 1/7 1/7 1/7 1/7 1/7 2/7 0 0 2/7 2/7 2/7 2/7 2/7 2/7 3/7 0 0 1/7 3/7 3/7 3/7 3/7 3/7 P1 4/7 0 0 0 2/7 4/7 4/7 4/7 4/7 5/7 0 0 0 1/7 3/7 5/7 5/7 5/7 6/7 0 0 0 0 2/7 4/7 6/7 6/7 1 0 0 0 0 1/7 3/7 5/7 1 Table 2: Weak-Link Game. T2 0123 97 98 99 100 0 0, 0 2, 0 2, 0 2, 0 ··· 2, 0 2, 0 2, 0 2, 0 1 0, 2 0, 0 3, 0 3, 0 ··· 3, 0 3, 0 3, 0 3, 0 2 0, 2 0, 3 1, 1 4, 0 ··· 4, 0 4, 0 4, 0 4, 0 3 0, 2 0, 3 0, 4 2, 2 ··· 5, 0 4, 0 4, 0 4, 0 ··· T1 ...... 97 0, 2 0, 3 0, 4 0, 5 96, 96 99, 95 99, 95 99, 95 98 0, 2 0, 3 0, 4 0, 5 ··· 95, 99 97, 97 100, 96 100, 96 99 0, 2 0, 3 0, 4 0, 5 ··· 95, 99 96, 100 98, 98 100, 97 100 0, 2 0, 3 0, 4 0, 5 ··· 95, 99 96, 100 97, 100 99, 99 ··· Table 3: Traveler’s Dilemma Payoff Matrix. games. We first need some additional terminology and preliminary 1 results. In particular, we need payoff functions to be defined ex- '1(p1,p2):= (p1 p2) p1 $ ^ 100 _ actly on the same domain and have as inputs not only the same ⇣ ⇣ ⌘⌘2 set of variables but the whole set of variables occurring in a game. (p2 p1) p1 100 , ¬ ! ^ _ This requirement leads to introducing the concept of a normalized 2 ⇣ (p1 p2) ⇣p2 ⌘⌘ ¬ ! ^ 100 Łukasiewicz game. ⇣ ⇣ ⌘⌘ DEFINITION 5(NORMALIZED GAME). AŁukasiewicz game 1 '2(p1,q2):= (p1 p2) p1 100 = P, V, Vi , Si , 'i , $ ^ _ G h { } { } { }i ⇣ ⇣ ⌘⌘2 (p2 p1) p1 100 . where V = p1,...,pm , is called normalized whenever each ¬ ! ^ _ { } ⇣ ⇣ 2 ⌘⌘ payoff function 'i is of the form 'i(p1,...,pm), i.e., all the vari- (p1 p2) p2 100 ¬ ! ^ ables from V occur in each 'i. ⇣ ⇣ ⌘⌘ These objective formulae define the following payoff functions: As mentioned above, not every Łukasiewicz game is normalized, 1 max x1 100 , 0 x1 = x2 since each payoff function might contain a different subset of vari- 2 f'1 (x1,x2)= min x1 + 100 , 1 x1 c 1 e(')=1 LEMMA 1. Let '(p1,...,pw) be a be any Ł -formula. For any e( ')= . k 0 otherwise set of variables q1,...,qv , there exists an equivalent extension ⇢ { } of '(p1,...,pw) in q1,...,qv . { } 841 The above Lemma makes it possible to assume every Łukasiewicz j0 e j (p) =1if and only if e(p)= , 0 k0 game is normalized. To make this precise, we introduce first a no- ✓ k0 ◆ tion of equivalence for games. j j0 where j0 and k0 are coprime and = . k k0 DEFINITION 7(EQUIVALENT GAMES). Let Let qj,k and rj,k denote the quotient and the remainder, respec- tively, of the Euclidean division of k by j. = P, V, Vi , Si , 'i and 0 = P0, V0, V0 , S0 , '0 G h { } { } { }i G h { i} { i} { i}i If e(p)=0, let c be two Łukasiewicz games overŁ . We say that and 0 are equiv- k 0(p):= p. alent whenever: G G ¬ Then 1. P = P0, e(p)=0 iff e( p)=1. 2. V = V0, ¬ 1 If e(p)= k , then let 3. For each i, Vi = Vi0 and Si = Si0 , 1 (p):= d( ((k 1)p),p). 4. (s?,...,s? ) is a NE for if and only if (s?,...,s? ) is a NE k ¬ ¬ 1 n G 1 n for 0. It is easy to check that G e(p)= 1 if and only if e( d( ((k 1)p),p)) = 1. Given Lemma 1 it is straightforward to prove that all Łukasiewicz k ¬ ¬ games have an equivalent normalized counterpart. In fact, LEMMA 2. Every Łukasiewicz game is equivalent to a normal- e( d( ((k 1)p),p)) = 1 (1 (k 1)x) x , ized game. ¬ ¬ | | and From now on, we will tacitly assume each game to be normal- 1 1 (1 (k 1)x) x =1 iff x = . ized. Also, notice that, so far, we have been denoting by | | k j f'i (si,s i) For e(p)= , with j 2, the proof proceeds by induction. For k j and k coprime, let the value of the function f' given the strategy combination (si,s i). i As mentioned above this actually is an abuse of notation since the j (p)= r ,k ( (qj,kp)) , k ( j,k ) ¬ strategy combination (si,s i) corresponds to a specific assignment to all the variables in the game, but for the valuation of f'i only the while, for j and k not coprime, take j0 and k0 coprime such that assignments to the variables actually occurring in f' are taken into j j0 i and k = k and let account. Since every game can be considered normalized, the use 0 j (p)= j0 (p)= r ,k ( (qj0,k0 p)) , of this notation can now be regarded as correct. k ( j ,k 0) k0 0 0 ¬ 5.1 Equilibria and Satisfiability Notice that rj,k In fact, let Vi = p1i ,...,pmi be the set of variables in control LEMMA 3. For every propositional variable p and every valu- of player i, and let { } ation e : p Lk there exists a formula (p) such that mi { }! (↵1 ,...,↵m ) (Lk) . j i i 2 e(p)= if and only if e( (p)) = 1.6 k The formula PROOF. We assume j and k to be coprime. If that is not the case ↵ (p1 ) 1i i then we have that encodes the assignment by player i of the value ↵1 to the variable 6 i Notice that the following proof translates into logical terms the p1i , i.e. algebraic proof of Lemma 19 in [10], whose context and content e( ↵ (p1 )) = 1 e(p1 )=↵1 . are significantly different from those of the present article. 1i i iff i i 842 So, the formula encodes the fact that player i’s payoff does not increase. To sim- plify the notation, we denote the formula (5) by . ↵1 (p1i ) ↵m (pmi ) i ^···^ i Now, define the formula , where each EG encodes player i’s strategy n i=1 mi ~s (Lk) ↵1i ,...,↵mi , 2 P { } is a strategy combination: and the formula n n ↵1 (p1i ) ↵m (pmi ) (1) i i := ↵1 (p1i ) ↵m (pmi ) ^···^ G n i i i=1 E mi "i=1 ^···^ ^ ^ ⇣ ⌘ ~s (Lk) i=1 ⇣ ⌘ 2 WP V n encodes the strategy combination 1i mi q qm 1i 1i mi i i=1 s (L )mi ^···^ ^ # ↵11 ,...,↵m1 ,...,↵1i ,...,↵mi ,...,↵1n ,...,↵mn . i2 k ⇣ ⌘ ⇣ ⌘ { } V V (6) To avoid any possible confusion, notice that for j = i, we might 6 have that mi = mj , since i and j might be in control of a different From the above construction, it is easy to check that actually 6 G number of variables. encodes the existence of equilibria. In fact, is a disjunctionE in- G Now, take, for each player i the set of all strategies dexed by all possible strategy combinations.E The existence of an mi equilibrium requires at least one of the disjuncts to be satisfiable. Si = si si =( 1i ,..., mi ) (Lk) . { | 2 } Each disjunct is a conjunction of formulas encoding the require- Assign to each player i a new set of variables ment that for a given strategy combination and for every player, si 1i mi every change of strategy does not result in any payoff increase. So, V = q ,...,qm , 1i i if any such a disjunct is satisfiable, the related strategy combination mi n o for each si (Lk) . This means that if a player controls mi actually corresponds to a NE. 2 m variables, she has (k +1) i different strategy combinations and is c THEOREM 2. AŁukasiewicz game on Ł admits a Nash Equi- mi k therefore assigned mi (k +1) new variables. G · librium if and only if is satisfiable. Proceeding as above, take the formula EG 1i 5.2 Equilibria and Satisfiable Games q 1i 1i Our purpose now is to show that whenever a game has a Nash ⇣ ⌘ equilibrium, there exists a special kind of game, calledG a satisfiable that encodes the assignment by player i of the value 1i to the 1 game, that is equivalent to . q i variable 1i , so that the formula G DEFINITION 8(SATISFIABLE GAMES). AŁukasiewicz game 1i mi q qm (2) c 1i 1i ^···^ mi i on Łk is called satisfiable if there exists a strategy combination (Gs ,...,s ) such that f (s ,...,s )=1for all i. encodes player i’s strategy⇣ ⌘ ⇣ ⌘ 1 n 'i 1 n The following lemma is an immediate consequence: 1i ,..., mi . { } c Let LEMMA 4. Every satisfiable Łukasiewicz game on Łk admits a Nash Equilibrium. G c 'i(p11 ,...,pm1 ,...,p1i 1 ,...,pmi 1 ,...,p1i ,...,pmi , As said above, we want to show that every game on Łk admits (3) a NE if and only if it is equivalent to a satisfiable game.G In order to p1 ,...,pm ,...p1n ,...,pmn ) i+1 i+1 do so, we need to use some concepts related to the algebraic seman- tics of Łukasiewicz logics and their model theory. Space limitations be player i’s payoff formula, and let prevent us from defining these concepts; we refer the reader to [10]. 7 Take any linearly ordered finite MV-algebra MVk. The first- 1i mi 'i(p11 ,...,pm1 ,...,p1i 1 ,...,pmi 1 ,...,q1 ,...,qmi , order theory Th(MVk) of MVk is the set of first order sentences in i the language , , 0 that hold over MVk. As shown in [10], each p1i+1 ,...,pmi+1 ,...p1n ,...,pmn ) h ¬ i theory Th(MVk) admits quantifier elimination in , , 0 . This (4) h ¬ i means that every formula of Th(MVk) is equivalent to a quantifier- free formula. be the formula obtained from (3) by replacing the variables c Now, take any Łukasiewicz game on Ł , and let ~x i, ~y i denote G k p1 ,...,pm tuples of variables assigned to player i, and let E be the following { i i } G sentence in Th(MVk) (where denotes the Boolean metalanguage with the variables u conjunction and the order relation): q 1i ,...,q mi . 1i mi So, using (3) and (4), then formula o ~x 1,...,~xn ~y 1,...,~yn 9 8 n 'i(~x 1,...,~xi 1,~yi,~xi+1,...,~xn) 1i mi 'i(p11 ,...,pm1 ,...,p1i 1 ,...,pmi 1 ,...,q1 ,...,qmi , . i 0 1 i=1 'i(~x 1,...,~xi 1,~xi,~xi+1,...,~xn) p1i+1 ,...,pmi+1 ,...p1n ,...,pmn ) l ! @ A 'i(p11 ,...,pm1 ,...,p1i 1 ,...,pmi 1 ,...,p1i ,...,pmi , It is then easy to check that: 7 p1i+1 ,...,pmi+1 ,...p1n ,...,xmn ) MV-algebras generalize Boolean algebras and form the algebraic (5) semantics of Łukasiewicz logics [3]. 843 c LEMMA 5. AŁukasiewicz game on Łk admits a Nash Equi- 6. CONCLUSIONS G librium if and only if E holds in Th(MVk). G Boolean games are a rich and natural model for understanding strate- gic behaviour in logic-based goal-oriented multi-agent systems. Al- From Lemma 5, we obtain a characterization of the existence of though they have many desirable features and have attracted much NE in terms of the validity of a first-order formula E . The fact G interest, the dichotomous nature of the preferences in conventional that E is valid in Th(MVk) is equivalent to the definability of G Boolean games severely limits the types of scenarios that can easily a non-empty set that corresponds to the set of equilibria of . By be expressed using Boolean games based on classical (two valued) G quantifier-elimination, such a set is in turn definable by a quantifier- logic. By expressing goal formulae with Łukasiewicz logic, we are free formula (~x 1,...,~xn). In order to show the existence of an able to naturally and compactly express much richer objectives for equivalent satisfiable game, we exploit the fact that this quantifier agents, as we have demonstrated here. c free-formula can be translated into a formula of the logic Łk. This Several questions suggest themselves for future work. For ex- step will be a consequence of the following lemma. ample, it is natural to consider whether we can adapt techniques for theorem proving with Łukasiewicz logic to solving Łukasiewicz LEMMA 6. For every quantifier-free formula (with param- games. In addition, further consideration of the computational com- c eters) of Th(MVk) in the language , , 0 there exists a Ł - plexity of Łukasiewicz games would be of interest. h ¬ i k formula ' such that is satisfiable over MVk if and only if there Acknowledgments. Marchioni acknowledges support of the Marie exists a valuation e such that e(')=1. Curie Intra-European Fellowship NAAMSI (FP7-PEOPLE-2011- IEF). Wooldridge was supported by the ERC Advanced Grant RACE PROOF. Let TermsMV be the set of terms t of Th(MVk) in c (291528). the language of MV-algebras and let Form c be the set of Ł - Łk k formulas. Define a translation ⌧ : TermsMV Form c such Łk that ! 7. REFERENCES [1] K. Basu. The traveler’s dilemma: Paradoxes of rationality in 1. If t is x, then ⌧(t)=p. game theory. American Economic Review, 84(2): 391–395, 1994. 2. If t is j , then ⌧(t)= j . [2] E. Bonzon, M.-C. Lagasquie-Schiex, J. Lang, and k k B. Zanuttini. Boolean games revisited. In Proceedings of the Seventeenth European Conference on Artificial Intelligence 3. If t is t0 t00, then ⌧(t)=⌧(t0) ⌧(t00). (ECAI-2006), pages 265–269, Riva del Garda, Italy, 2006. 4. If t is t0, then ⌧(t)= ⌧(t0). [3] R. Cignoli, I. M. L. D’Ottaviano, and D. Mundici. Algebraic ¬ ¬ Foundations of Many-Valued Reasoning (Trends in Logic Now, every quantifier-free formula of Th(MVk) is a Boolean Volume 7). 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