Equilibrium semantics for languages with im- perfect information - PDF document

Equilibrium semantics for languages with im- perfect information Gabriel Sandu University of Helsinki 1

IF logic and multivalued logic • IF logic is an extension of FOL • IF languages define games of imperfect in- formation • Imperfect information introduces indeter- minacy • To overcome indeterminacy we apply von Neumann’s Minimax Theorem (Aitaj): 1. Equilibrium semantics under mixed strate- gies (Blass and Gurevich 1986, Sevenster, 2006) 2. Equilibrium semantics under behavior strate- gies (Galliani, 2008) 2

IF languages • The sentence ϕ inf ∀ x ∃ y ( ∃ z/ { x } )( x = z ∧ w � = c ) • Lewis sentence ϕ sig ∀ x ∃ z ( ∃ y/ { x } ) { S ( x ) → (Σ( z ) ∧ R ( y ) ∧ y = b ( x )) } • Monty Hall ϕ MH ∀ x ( ∃ y/ { x } ) ∀ z [ x � = z ∧ y � = z → ( ∃ t/ { x } ) x = t ] • Matching Pennies ϕ MP ∀ x ( ∃ y/ { x } ) x = y • Inverted Matching Pennies ϕ IMP ∀ x ( ∃ y/ { x } ) x � = y 3

Extensive IF games • G ( M , s, ϕ ) where ϕ is an IF sentence, M is a model, and s is a partial assignment • These are win-lose 2 player game of imper- fect information • The players are Eloise ( ∃ ) and Abelard ( ∀ ) • An information set δ for player i ∈ {∃ , ∀} is a set of partial plays (nonterminal histories) • A strategy s i is a specification of what ac- tions player i should implement for each information set 4

Example: perfect information • The game G ( M , ϕ ) where ϕ is ∀ x ∃ yx = y and M = { a, b } • Eloise has 2 inform. sets: δ 1 = { a } and δ 2 = { b } • A strategy s ∃ for Eloise has the form s ∃ = ( s ∃ ( δ 1 ) , s ∃ ( δ 2 )) where s ∃ ( δ 1 ) , s ∃ ( δ 2 ) ∈ { a, b } • Abelard has 1 information set: γ 1 = ∅ • A strategy s ∀ for Abelard has the form s ∀ = ( s ∀ ( γ 1 )) 5

Example: imperfect information • The game G ( M , ϕ MP ) , where ϕ MP is ∀ x ( ∃ y/ { x } ) x = y and M = { a, b } • Eloise has one information set δ 1 = { a, b } (which has 2 histories) • Abelard has one information set γ 1 = ∅ • A strategy for Eloise has the form s ∃ = ( s ∃ ( δ 1 )) • A strategy for Abelard is as before. 6

Game-theoretical truth and falsity • For ϕ an IF-formula, M a model and s an assignment in M , we stipulate: = + • M , s | GTS ϕ iff there is a winning strategy for Eloise in G ( M , s, ϕ ) = − • M , s | GTS ϕ iff there is a winning strategy for Abelard in G ( M , s, ϕ ) . 7

Expressive power • Infinity. The sentence ϕ inf ∀ x ∃ y ( ∃ z/ { x } )( x = z ∧ y � = c ) defines (Dedekind) infinity. 8

Model-theoretical properties • We restrict the set of universes to those containing at least 2 objects. • Compactness: An IF theory is satisable if every finite subtheory of it is satisable. • Lowenheim-Skolem property. • Separation property: any two contrary IF sentences can be separated by an elemen- tary class. • Interpolation property: Let ϕ and ψ be contrary IF L -sentences. Then there is an IF L -sentence χ such that ϕ ≡ + χ ψ ≡ + ¬ χ and • Definability of truth. 9

Indeterminacy • Indeterminate sentences on finite models: • ϕ inf ∀ x ∃ y ( ∃ z/ { x } )( x = z ∧ y � = c ) • Lewis sentence ϕ sig ∀ x ∃ z ( ∃ y/ { x } ) { S ( x ) → (Σ( z ) ∧ R ( y ) ∧ y = b ( x )) } • Monty Hall ϕ MH ∀ x ( ∃ y/ { x } ) ∀ z [ x � = z ∧ y � = z → ( ∃ t/ { x } ) x = t ] • Matching Pennies ϕ MP ∀ x ( ∃ y/ { x } ) x = y • Inverted Matching Pennies ϕ IMP ∀ x ( ∃ y/ { x } ) x � = y 10

Strategic IF games: Definition • Let G ( M , ϕ ) be an extensive IF game. • Γ( M , ϕ ) = ( N, ( S i ) i ∈ N , u i ∈ N ) is the strate- gic IF game where: - N = {∃ , ∀} is the set of players - S i is the set of strategies of player i in the extensive G ( M , ϕ ) - u i is the utility function of player i such that u i ( s, t ) = 1 if playing s against t in G ( M , ϕ ) yields a win for player i , and u i ( s, t ) = 0 , otherwise. 11

Example • Let M = { a, b } . The strategic game for ∀ x ∃ yx = y : a b ( a, a ) (1 , 0) (0 , 1) ( a, b ) (1 , 0) (1 , 0) ( b, a ) (0 , 1) (0 , 1) ( b, b ) (0 , 1) (1 , 0) • The strategic games for ∀ x ( ∃ y/ { x } ) x = y and ∀ x ( ∃ y/ { x } ) x � = y : a b a b (1 , 0) (0 , 1) (0 , 1) (1 , 0) a a b (0 , 1) (1 , 0) b (1 , 0) (0 , 1) • There are no equilibria in the last two games 12

Multivalues • Mixed strategy equilibria in strategic IF games • Behavior strategy equilibria in extensive IF games (Galliani) 13

Mixed strategies in strategic IF games • Fix a strategic IF game Γ( M , ϕ ) = ( N, ( S i ) i ∈ N , u i ∈ N • A mixed str. σ i for player i , σ i : S i → [0 , 1] such that � σ i ( s ) = 1 s ∈ S i • σ i is uniform over S ′ i ⊆ S i if it assigns equal probability to all the strategies in S ′ i • Let σ be a mixed str. for ∃ and τ a mixed str. for ∀ . • The expected utility for player i for the strategy profile ( σ, τ ) : � � U i ( σ, τ ) = σ ( s ) τ ( t ) u i ( s, t ) s ∈ S ∃ t ∈ S ∀ 14

Behavior strategies • Fix an extensive IF game G ( M , ϕ ) • Let δ 1 , ..., δ n be the information sets of player ∃ • A pure strategy for pl. ∃ has the form s ∃ = ( s ∃ ( δ 1 ) , ..., s ∃ ( δ n )) where each s ∃ ( δ i ) ∈ A ( δ i ) . • A behavior strategy for pl. ∃ has the form b ∃ = ( p 1 ( δ 1 ) , ..., p n ( δ 2 )) where each p i ( δ i ) is a probability distribu- tion over A ( δ i ) 15

• Let a ∈ A ( δ i ) for some i . Let p i ( a/δ ) denote ( p i ( δ i ))( a ) • We must have: a ∈ A ( δ ) p i ( a/δ ) = 1 �

Example • The extensive game G ( M , ϕ ) where ϕ is ∀ x ( ∃ y/ { x } ) x = y and M is { a, b } • Pl. ∃ has one information set δ 1 = { a, b } • A behavior strategy for ∃ : = ( 1 / 2 a ⊕ 1 / 2 b ) b ∃ • Pl. ∀ has one information set γ 1 = { ∅ } • A behavior strategy for ∀ : b ∀ = ( 1 / 2 a ⊕ 1 / 2 b ) 16

Example continued: expected utility • When the strategy profile ( b ∃ , b ∀ ) is played, each terminal history will receive a proba- bility. • This probability is the product of the prob- abilities of the actions which compose the history. • In the example, each terminal history has probability 1 / 4 . • The expected utility U i ( b ∃ , b ∀ ) : we sum up the probability of each terminal history with the payoff of player i . 17

Example: mixed strategies ⇒ behavior strate- gies • Let ϕ be ∃ x ( ∃ y/ { x } ) x = y and M = { a, b } • In the game G ( M , ϕ ) , ∃ has 2 information sets δ 1 = { ∅ } and δ 2 = { a, b } • ∃ has 4 pure strategies: ( a, a ) , ( a, b ) , ( b, a ) , ( b, b ) • Let σ be the mixed strategy σ ( a, a ) = σ ( b, b ) = 1 / 2 • The behavior strategy induced by σ P ( a/δ 1 ) = P ( b/δ 1 ) = 1 / 2 18

and P ( a/δ 2 ) = P ( b/δ 2 ) = 1 / 2 • However this induces a different probability ( 1 / 4 ) on terminal histories than σ .

Example continued • The mixed str. σ allows ∃ to create a differ- ent probability distribution at each of the nodes of the same information set. • At the left node she chooses a with prob- ability 1; at the right node she chooses a with probability 0 . • A conditional probability on the other side will impose the same probability distribu- tion on both nodes. 19

Mixed strategy equilibria • Let N = {∃ , ∀} and Γ = (( S i ) i ∈ N , ( u i ) i ∈ N ) be a constant sum, strategic game • Let ( σ ∃ , σ ∀ ) be a pair of mixed strategies in Γ . ( σ ∃ , σ ∀ ) is an equilibrium if - for every mixed strategy σ of Eloise: U ∃ ( σ ∃ , σ ∀ ) ≥ U ∃ ( σ, σ ∀ ) - for every mixed strategy σ of Abelard: U ∀ ( σ ∃ , σ ∀ ) ≥ U ∀ ( σ ∃ , σ ) 20

Von Neumann’s Minimax Theorem: equilib- rium semantics • Every finite, constant sum, two-player game has an equilibrium in mixed strategies • Every two such equilibria have the same expected utility • We can talk about the probabilistic value of an IF sentence on a finite model M . • The satisfaction relation | = ε between IF sentences ϕ and models M , with ε such that 0 ≤ ε ≤ 1 defined by: M | = ε ϕ iff the value of the strategic game Γ( M , ϕ ) is ε . 21

Equilibrium semantics: A conservative exten- sion of classical GTS • Conservativity: = + (i) M | GTS ϕ iff M | = 1 ϕ = − (ii) M | GTS ψ iff M | = 0 ϕ . 22

Example • Recall the strategic games Γ( M , ϕ MP ) and Γ( M , ϕ IMP ) , where M = { a, b, c } : a b c a b c (1 , 0) (0 , 1) (0 , 1) (0 , 1) (1 , 0) (1 , 0) a a b (0 , 1) (1 , 0) (0 , 1) b (1 , 0) (0 , 1) (1 , 0) (0 , 1) (1 , 0) (1 , 0) (1 , 0) (1 , 0) (0 , 1) c c • Let σ and τ be uniform probability distri- butions over { a, b, c } . • The pair ( σ, τ ) is an equilibrium in both games. • The value of ϕ MP on M is 1 / 3 and that of ϕ IMP is 2 / 3 . 23

• As the size of M increases, the value of ϕ MP on M asymptotically approaches 0 and that of ϕ IMP asymptotically approaches 1 .

Example (Galliani): the value of the game is different in the two semantics • Let ϕ be ∃ x ( ∃ y/ { x } )( ∀ z/ { x, y } )( x = y ∧ x � = z ) and M = { a, b } • The strategic IF game: a b ( a, a ) (0 , 1) (1 , 0) ( a, b ) (0 , 1) (0 , 1) ( b, a ) (0 , 1) (0 , 1) ( b, b ) (1 , 0) (0 , 1) • The strategies ( a, b ) and ( b, a ) are weakly dominated by ( a, a ) 24