Open problems in repeated games with finite automata Abraham Neyman - PowerPoint PPT Presentation

Open problems in repeated games with finite automata Abraham Neyman Jerusalem, May 23, 2011 subject, date – p. 1/13

Two-person zero-sum games subject, date – p. 2/13

Two-person zero-sum games Quantify the advantage of larger automata in repeated games. subject, date – p. 2/13

Two-person zero-sum games Quantify the advantage of larger automata in repeated games. G T ( k 1 , k 2 ) is the repeated game where subject, date – p. 2/13

Two-person zero-sum games Quantify the advantage of larger automata in repeated games. G T ( k 1 , k 2 ) is the repeated game where a stage game G is repeated T ≤ ∞ times, player i is confined to play a strategy that is implementable by a deterministic automaton of size k i ≤ ∞ , the repeated game payoff is the average per-stage payoff. subject, date – p. 2/13

Two-person zero-sum games Quantify the advantage of larger automata in repeated games. G T ( k 1 , k 2 ) is the repeated game where a stage game G is repeated T ≤ ∞ times, player i is confined to play a strategy that is implementable by a deterministic automaton of size k i ≤ ∞ , the repeated game payoff is the average per-stage payoff. Explicitly, what is the asymptotic behavior of VAL G T ( k 1 , k 2 ) as T, k 1 , k 2 → ∞ . subject, date – p. 2/13

Two-person zero-sum games Quantify the advantage of larger automata in repeated games. G T ( k 1 , k 2 ) is the repeated game where a stage game G is repeated T ≤ ∞ times, player i is confined to play a strategy that is implementable by a deterministic automaton of size k i ≤ ∞ , the repeated game payoff is the average per-stage payoff. Explicitly, what is the asymptotic behavior of VAL G T ( k 1 , k 2 ) as T, k 1 , k 2 → ∞ . subject, date – p. 2/13

Two-person zero-sum games Quantify the advantage of larger automata in repeated games. G T ( k 1 , k 2 ) is the repeated game where a stage game G is repeated T ≤ ∞ times, player i is confined to play a strategy that is implementable by a deterministic automaton of size k i ≤ ∞ , the repeated game payoff is the average per-stage payoff. Explicitly, what is the asymptotic behavior of VAL G T ( k 1 , k 2 ) as T, k 1 , k 2 → ∞ . subject, date – p. 2/13

Two-person zero-sum games Quantify the advantage of larger automata in repeated games. G T ( k 1 , k 2 ) is the repeated game where a stage game G is repeated T ≤ ∞ times, player i is confined to play a strategy that is implementable by a deterministic automaton of size k i ≤ ∞ , the repeated game payoff is the average per-stage payoff. Explicitly, what is the asymptotic behavior of VAL G T ( k 1 , k 2 ) as T, k 1 , k 2 → ∞ . subject, date – p. 2/13

Non-zero-sum games subject, date – p. 3/13

Non-zero-sum games Quantify the equilibrium payoff of non-zero-sum repeated games with large automata subject, date – p. 3/13

Non-zero-sum games Quantify the equilibrium payoff of non-zero-sum repeated games with large automata Explicitly, what is the asymptotic behavior of the equilibrium payoffs of G T ( k 1 , k 2 , . . . , k n ) as T, k i → ∞ subject, date – p. 3/13

Non-zero-sum games Quantify the equilibrium payoff of non-zero-sum repeated games with large automata Explicitly, what is the asymptotic behavior of the equilibrium payoffs of G T ( k 1 , k 2 , . . . , k n ) as T, k i → ∞ subject, date – p. 3/13

Non-zero-sum games Quantify the equilibrium payoff of non-zero-sum repeated games with large automata Explicitly, what is the asymptotic behavior of the equilibrium payoffs of G T ( k 1 , k 2 , . . . , k n ) as T, k i → ∞ subject, date – p. 3/13

Non-zero-sum games Quantify the equilibrium payoff of non-zero-sum repeated games with large automata Explicitly, what is the asymptotic behavior of the equilibrium payoffs of G T ( k 1 , k 2 , . . . , k n ) as T, k i → ∞ subject, date – p. 3/13

Non-zero-sum games Quantify the equilibrium payoff of non-zero-sum repeated games with large automata Explicitly, what is the asymptotic behavior of the equilibrium payoffs of G T ( k 1 , k 2 , . . . , k n ) as T, k i → ∞ subject, date – p. 3/13

Non-zero-sum games Quantify the equilibrium payoff of non-zero-sum repeated games with large automata Explicitly, what is the asymptotic behavior of the equilibrium payoffs of G T ( k 1 , k 2 , . . . , k n ) as T, k i → ∞ subject, date – p. 3/13

Finite automata strategies subject, date – p. 4/13

Finite automata strategies G = � N, A = × i ∈ N A i , g = ( g i ) i ∈ N � – the stage game subject, date – p. 4/13

Finite automata strategies G = � N, A = × i ∈ N A i , g = ( g i ) i ∈ N � – the stage game N – set of players subject, date – p. 4/13

Finite automata strategies G = � N, A = × i ∈ N A i , g = ( g i ) i ∈ N � – the stage game N – set of players A i – stage actions of player i subject, date – p. 4/13

Finite automata strategies G = � N, A = × i ∈ N A i , g = ( g i ) i ∈ N � – the stage game N – set of players A i – stage actions of player i g i : A → R – payoff function of player i subject, date – p. 4/13

Finite automata strategies G = � N, A = × i ∈ N A i , g = ( g i ) i ∈ N � – the stage game N – set of players A i – stage actions of player i g i : A → R – payoff function of player i M = � S, s 1 , α, τ � – an automata of player i subject, date – p. 4/13

Finite automata strategies G = � N, A = × i ∈ N A i , g = ( g i ) i ∈ N � – the stage game N – set of players A i – stage actions of player i g i : A → R – payoff function of player i M = � S, s 1 , α, τ � – an automata of player i S – states of the automaton, s 0 ∈ S – the initial state subject, date – p. 4/13

Finite automata strategies G = � N, A = × i ∈ N A i , g = ( g i ) i ∈ N � – the stage game N – set of players A i – stage actions of player i g i : A → R – payoff function of player i M = � S, s 1 , α, τ � – an automata of player i S – states of the automaton, s 0 ∈ S – the initial state α : S → A i – action function of the automaton M subject, date – p. 4/13

Finite automata strategies G = � N, A = × i ∈ N A i , g = ( g i ) i ∈ N � – the stage game N – set of players A i – stage actions of player i g i : A → R – payoff function of player i M = � S, s 1 , α, τ � – an automata of player i S – states of the automaton, s 0 ∈ S – the initial state α : S → A i – action function of the automaton M τ : S × A → S – the transition function of M subject, date – p. 4/13

Finite automata strategies G = � N, A = × i ∈ N A i , g = ( g i ) i ∈ N � – the stage game N – set of players A i – stage actions of player i g i : A → R – payoff function of player i M = � S, s 1 , α, τ � – an automata of player i S – states of the automaton, s 0 ∈ S – the initial state α : S → A i – action function of the automaton M τ : S × A → S – the transition function of M σ = σ ( M ) – the strategy defined by the automaton subject, date – p. 4/13

Finite automata strategies G = � N, A = × i ∈ N A i , g = ( g i ) i ∈ N � – the stage game N – set of players A i – stage actions of player i g i : A → R – payoff function of player i M = � S, s 1 , α, τ � – an automata of player i S – states of the automaton, s 0 ∈ S – the initial state α : S → A i – action function of the automaton M τ : S × A → S – the transition function of M σ = σ ( M ) – the strategy defined by the automaton σ 1 = f ( s 1 ) , σ ( a 1 , a 2 , . . . , a k ) = f ( s k ) where subject, date – p. 4/13

Finite automata strategies G = � N, A = × i ∈ N A i , g = ( g i ) i ∈ N � – the stage game N – set of players A i – stage actions of player i g i : A → R – payoff function of player i M = � S, s 1 , α, τ � – an automata of player i S – states of the automaton, s 0 ∈ S – the initial state α : S → A i – action function of the automaton M τ : S × A → S – the transition function of M σ = σ ( M ) – the strategy defined by the automaton σ 1 = f ( s 1 ) , σ ( a 1 , a 2 , . . . , a k ) = f ( s k ) where s k = τ ( s k − 1 , a k − 1 ) the state of the automaton before play at stage k subject, date – p. 4/13

Finite automata strategies G = � N, A = × i ∈ N A i , g = ( g i ) i ∈ N � – the stage game N – set of players A i – stage actions of player i g i : A → R – payoff function of player i M = � S, s 1 , α, τ � – an automata of player i S – states of the automaton, s 0 ∈ S – the initial state α : S → A i – action function of the automaton M τ : S × A → S – the transition function of M σ = σ ( M ) – the strategy defined by the automaton σ 1 = f ( s 1 ) , σ ( a 1 , a 2 , . . . , a k ) = f ( s k ) where s k = τ ( s k − 1 , a k − 1 ) the state of the automaton before play at stage k subject, date – p. 4/13

Open Problem - I subject, date – p. 5/13

Open Problem - I I.1 Do the values of G ( k, n k ) := G ∞ ( k, n k ) converge as log n k n k ≥ k → ∞ and lim k →∞ = x ? k subject, date – p. 5/13