Łukasiewicz Games Examples Results O N Ł UKASIEWICZ G AMES E NRICO M ARCHIONI Institut de Recherche en Informatique de Toulouse Universit´ e Paul Sabatier, France M ICHAEL W OOLDRIDGE Department of Computer Science University of Oxford, U.K. ManyVal 2013 4-6 September 2013 Prague, Czech Republic
Łukasiewicz Games Examples Results O UTLINE Łukasiewicz Games Basic Definitions Examples Traveler’s Dilemma Results Theorem Best Response Sets Equilibrium Formula Satisfiable Games
Łukasiewicz Games Examples Results O UTLINE Łukasiewicz Games Basic Definitions Examples Traveler’s Dilemma Results Theorem Best Response Sets Equilibrium Formula Satisfiable Games
Łukasiewicz Games Examples Results O VERVIEW ◮ We introduce a compact representation of non-cooperative games based on finite-valued Łukasiewicz logics.
Łukasiewicz Games Examples Results O VERVIEW ◮ We introduce a compact representation of non-cooperative games based on finite-valued Łukasiewicz logics. ◮ Łukasiewicz Games are inspired by, and greatly extend, Boolean games [Herrenstein et al. 2001].
Łukasiewicz Games Examples Results O VERVIEW ◮ We introduce a compact representation of non-cooperative games based on finite-valued Łukasiewicz logics. ◮ Łukasiewicz Games are inspired by, and greatly extend, Boolean games [Herrenstein et al. 2001]. ◮ In Boolean games each individual player strives for the satisfaction of a goal, represented as a classical Boolean formula that encodes her payoff;
Łukasiewicz Games Examples Results O VERVIEW ◮ We introduce a compact representation of non-cooperative games based on finite-valued Łukasiewicz logics. ◮ Łukasiewicz Games are inspired by, and greatly extend, Boolean games [Herrenstein et al. 2001]. ◮ In Boolean games each individual player strives for the satisfaction of a goal, represented as a classical Boolean formula that encodes her payoff; ◮ The actions available to players correspond to valuations that can be made to variables under their control.
Łukasiewicz Games Examples Results O VERVIEW ◮ We introduce a compact representation of non-cooperative games based on finite-valued Łukasiewicz logics. ◮ Łukasiewicz Games are inspired by, and greatly extend, Boolean games [Herrenstein et al. 2001]. ◮ In Boolean games each individual player strives for the satisfaction of a goal, represented as a classical Boolean formula that encodes her payoff; ◮ The actions available to players correspond to valuations that can be made to variables under their control. ◮ The use of Łukasiewicz logics makes it possible to more naturally represent much richer payoff functions for players.
Łukasiewicz Games Examples Results Ł UKASIEWICZ AND G AMES ◮ Classic Game Theory: ◮ Non-cooperative games: ◮ Łukasiewicz Games on Ł c k [M. & Wooldridge] ◮ Constant Sum Łukasiewicz Games on Ł ∞ [Kroupa & Majer] ◮ Cooperative games: MV-coalitions [Kroupa] ◮ Game-Theoretic Semantics: ◮ Dialogue games [Ferm¨ uller, Giles, . . . ] ◮ Evaluation games [Cintula & Majer] ◮ Ulam games [Mundici]
Łukasiewicz Games Examples Results D EFINITION I A Łukasiewicz game G on Ł c k is a tuple G = � P , V , { V i } , { S i } , { φ i }� where:
Łukasiewicz Games Examples Results D EFINITION I A Łukasiewicz game G on Ł c k is a tuple G = � P , V , { V i } , { S i } , { φ i }� where: 1. P = { P 1 , . . . , P n } is a set of players ;
Łukasiewicz Games Examples Results D EFINITION I A Łukasiewicz game G on Ł c k is a tuple G = � P , V , { V i } , { S i } , { φ i }� where: 1. P = { P 1 , . . . , P n } is a set of players ; 2. V = { p 1 , p 2 , . . . } is a finite set of propositional variables;
Łukasiewicz Games Examples Results D EFINITION I A Łukasiewicz game G on Ł c k is a tuple G = � P , V , { V i } , { S i } , { φ i }� where: 1. P = { P 1 , . . . , P n } is a set of players ; 2. V = { p 1 , p 2 , . . . } is a finite set of propositional variables; 3. V i ⊆ V is the set of propositional variables under control of player P i , so that the sets V i form a partition of V .
Łukasiewicz Games Examples Results D EFINITION II 4. S i is the strategy set for player i that includes all valuations s i : V i → L k of the propositional variables in V i , i.e. S i = { s i | s i : V i → L k } .
Łukasiewicz Games Examples Results D EFINITION II 4. S i is the strategy set for player i that includes all valuations s i : V i → L k of the propositional variables in V i , i.e. S i = { s i | s i : V i → L k } . 5. φ i ( p 1 , . . . , p t ) is an Ł c k -formula, built from variables in V , whose associated function f φ i : ( L k ) t → L k corresponds to the payoff function of P i , and whose value is determined by the valuations in { S 1 , . . . , S n } .
Łukasiewicz Games Examples Results E QUILIBRIA ◮ A tuple ( s 1 , . . . , s n ) , with each s i ∈ S i , is called a strategy combination .
Łukasiewicz Games Examples Results E QUILIBRIA ◮ A tuple ( s 1 , . . . , s n ) , with each s i ∈ S i , is called a strategy combination . ◮ s − i the set of strategies { s 1 , . . . , s i − 1 , s i + 1 , . . . , s n } not including s i .
Łukasiewicz Games Examples Results E QUILIBRIA ◮ A tuple ( s 1 , . . . , s n ) , with each s i ∈ S i , is called a strategy combination . ◮ s − i the set of strategies { s 1 , . . . , s i − 1 , s i + 1 , . . . , s n } not including s i . ◮ The strategy s i for P i is called a best response whenever, fixing s − i , there exists no strategy s ′ i such that f φ i ( s i , s − i ) ≤ f φ i ( s ′ i , s − i ) .
Łukasiewicz Games Examples Results E QUILIBRIA ◮ A tuple ( s 1 , . . . , s n ) , with each s i ∈ S i , is called a strategy combination . ◮ s − i the set of strategies { s 1 , . . . , s i − 1 , s i + 1 , . . . , s n } not including s i . ◮ The strategy s i for P i is called a best response whenever, fixing s − i , there exists no strategy s ′ i such that f φ i ( s i , s − i ) ≤ f φ i ( s ′ i , s − i ) . ◮ A strategy combination ( s ⋆ 1 , . . . , s ⋆ n ) is called a pure strategy Nash Equilibrium whenever s ⋆ i is a best response to s ⋆ − i , for each 1 ≤ i ≤ n .
Łukasiewicz Games Examples Results O UTLINE Łukasiewicz Games Basic Definitions Examples Traveler’s Dilemma Results Theorem Best Response Sets Equilibrium Formula Satisfiable Games
Łukasiewicz Games Examples Results T RAVELER ’ S D ILEMMA I [B ASU 1994] ◮ Two travelers fly back home from a trip to a remote island where they bought exactly the same antiques.
Łukasiewicz Games Examples Results T RAVELER ’ S D ILEMMA I [B ASU 1994] ◮ Two travelers fly back home from a trip to a remote island where they bought exactly the same antiques. ◮ Their luggage gets damaged and all the items acquired are broken.
Łukasiewicz Games Examples Results T RAVELER ’ S D ILEMMA I [B ASU 1994] ◮ Two travelers fly back home from a trip to a remote island where they bought exactly the same antiques. ◮ Their luggage gets damaged and all the items acquired are broken. ◮ The airline promises a refund for the inconvenience
Łukasiewicz Games Examples Results T RAVELER ’ S D ILEMMA I [B ASU 1994] ◮ Two travelers fly back home from a trip to a remote island where they bought exactly the same antiques. ◮ Their luggage gets damaged and all the items acquired are broken. ◮ The airline promises a refund for the inconvenience ◮ Both travelers must write on a piece of paper a number between 0 and 100 corresponding to the cost of the antiques.
Łukasiewicz Games Examples Results T RAVELER ’ S D ILEMMA II [B ASU 1994] ◮ If they both write the same number x , they both receive x − 1.
Łukasiewicz Games Examples Results T RAVELER ’ S D ILEMMA II [B ASU 1994] ◮ If they both write the same number x , they both receive x − 1. ◮ If they write different numbers, say x < y , the one playing x will receive x + 2.
Łukasiewicz Games Examples Results T RAVELER ’ S D ILEMMA II [B ASU 1994] ◮ If they both write the same number x , they both receive x − 1. ◮ If they write different numbers, say x < y , the one playing x will receive x + 2. ◮ The other player will receive x − 2.
Łukasiewicz Games Examples Results T RAVELER ’ S D ILEMMA II [B ASU 1994] ◮ If they both write the same number x , they both receive x − 1. ◮ If they write different numbers, say x < y , the one playing x will receive x + 2. ◮ The other player will receive x − 2. ◮ Travelers’ payoff is given by the functions: max ( x − 1 , 0 ) x = y max ( x − 1 , 0 ) x = y f 1 ( x , y ) = min ( min ( x , y ) + 2 , 100 ) x < y ; f 2 ( x , y ) = min ( min ( x , y ) + 2 , 100 ) y < x . max ( min ( x , y ) − 2 , 0 ) y < x max ( min ( x , y ) − 2 , 0 ) x < y
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