The Weihrauch Degree of Finding Nash Equilibria in Multiplayer Games Tonicha Crook Joint work with Arno Pauly Swansea University 8th April 2020 Tonicha Crook (Swansea University) Multiplayer Games 8th April 2020 1 / 11
Outline Introduction 1 Game Theory 2 Real Number Computabilty 3 Computable Analysis and Weihrauch Reducibility 4 Our Results 5 Tonicha Crook (Swansea University) Multiplayer Games 8th April 2020 2 / 11
Introduction Algorithms to find Nash equilibria have already been investigated if the payoffs in our games are given as integers. However it has not been for real numbers. We will explore how non-computable the task of finding Nash equilbria is using Weihrauch reducibility. For one or two player games, a complete classification has already been obtained but we are addressing the situation with multiplayer games. Tonicha Crook (Swansea University) Multiplayer Games 8th April 2020 3 / 11
Game Theory Nash Equilibirum A strategy vector s ∗ = ( s ∗ 1 . . . s ∗ n ) is a Nash Equilibrium if for each player i ∈ N and each strategy s i ∈ S i , u i ( s ∗ ) ≥ u i ( s i , S ∗ i ) is satisfied. Tonicha Crook (Swansea University) Multiplayer Games 8th April 2020 4 / 11
Game Theory Nash Equilibirum A strategy vector s ∗ = ( s ∗ 1 . . . s ∗ n ) is a Nash Equilibrium if for each player i ∈ N and each strategy s i ∈ S i , u i ( s ∗ ) ≥ u i ( s i , S ∗ i ) is satisfied. Player II A II B II Player I A I 0,1 1,0 1,2 2,3 B I Figure: Simple Game in Strategic Form Tonicha Crook (Swansea University) Multiplayer Games 8th April 2020 4 / 11
Game Theory Nash Equilibirum A strategy vector s ∗ = ( s ∗ 1 . . . s ∗ n ) is a Nash Equilibrium if for each player i ∈ N and each strategy s i ∈ S i , u i ( s ∗ ) ≥ u i ( s i , S ∗ i ) is satisfied. Player II A II B II Player I A I 0,1 1,0 1,2 2,3 B I Figure: Simple Game in Strategic Form Tonicha Crook (Swansea University) Multiplayer Games 8th April 2020 4 / 11
Game Theory Nash Equilibirum A strategy vector s ∗ = ( s ∗ 1 . . . s ∗ n ) is a Nash Equilibrium if for each player i ∈ N and each strategy s i ∈ S i , u i ( s ∗ ) ≥ u i ( s i , S ∗ i ) is satisfied. Player II A II B II Player I A I 0,1 1,0 1,2 2,3 B I Figure: Simple Game in Strategic Form Tonicha Crook (Swansea University) Multiplayer Games 8th April 2020 4 / 11
Game Theory Nash Equilibirum A strategy vector s ∗ = ( s ∗ 1 . . . s ∗ n ) is a Nash Equilibrium if for each player i ∈ N and each strategy s i ∈ S i , u i ( s ∗ ) ≥ u i ( s i , S ∗ i ) is satisfied. Player II A II B II Player I A I 0,1 1,0 1,2 2,3 B I Figure: Simple Game in Strategic Form Tonicha Crook (Swansea University) Multiplayer Games 8th April 2020 4 / 11
Game Theory Nash Equilibirum A strategy vector s ∗ = ( s ∗ 1 . . . s ∗ n ) is a Nash Equilibrium if for each player i ∈ N and each strategy s i ∈ S i , u i ( s ∗ ) ≥ u i ( s i , S ∗ i ) is satisfied. Player II A II B II Player I A I 0,1 1,0 1,2 2,3 B I Figure: Simple Game in Strategic Form Tonicha Crook (Swansea University) Multiplayer Games 8th April 2020 4 / 11
Real Number Computability Computable A function f : ⊆ R → R is computable , if there is a computable function F :: ⊆ � N → � N such that F ( p ) is a decimal expansion of f ( x ) whenever p is a decimal expansion of x . Tonicha Crook (Swansea University) Multiplayer Games 8th April 2020 5 / 11
Real Number Computability Computable A function f : ⊆ R → R is computable , if there is a computable function F :: ⊆ � N → � N such that F ( p ) is a decimal expansion of f ( x ) whenever p is a decimal expansion of x . Computable (final) A function f : ⊆ R → R is computable , if there is a computable function F such that ρ ( F ( p )) = f ( ρ ( p )). Where ρ ( q ) = x is a sequence ( q n ) n ∈ N ∈ Q N representing x ∈ R , if | x − q n | < 2 − n for all n ∈ N . Tonicha Crook (Swansea University) Multiplayer Games 8th April 2020 5 / 11
Computable Analysis and Weihrauch Reducibility Tonicha Crook (Swansea University) Multiplayer Games 8th April 2020 6 / 11
Computable Analysis and Weihrauch Reducibility Represented Spaces A represented space ( X , δ ) is a set X together with a surjective partial function δ : ⊆ N N → X . Tonicha Crook (Swansea University) Multiplayer Games 8th April 2020 6 / 11
Computable Analysis and Weihrauch Reducibility Represented Spaces A represented space ( X , δ ) is a set X together with a surjective partial function δ : ⊆ N N → X . Weihrauch Reducibility A multivalued function between represented spaces is Weihrauch reducible to another, if there is an otherwise computable procedure invoking the second multi-valued function as an oracle exactly once that solves the first. Tonicha Crook (Swansea University) Multiplayer Games 8th April 2020 6 / 11
Computable Analysis and Weihrauch Reducibility Represented Spaces A represented space ( X , δ ) is a set X together with a surjective partial function δ : ⊆ N N → X . Weihrauch Reducibility A multivalued function between represented spaces is Weihrauch reducible to another, if there is an otherwise computable procedure invoking the second multi-valued function as an oracle exactly once that solves the first. Weihrauch Degree The Weihrauch degrees are the equivalence classes for Weihrauch reductions. Tonicha Crook (Swansea University) Multiplayer Games 8th April 2020 6 / 11
Computable Analysis and Weihrauch Reducibility Parallel The degree f ∗ represents being allowed to invoke f any finite number of times in parallel . Tonicha Crook (Swansea University) Multiplayer Games 8th April 2020 7 / 11
Computable Analysis and Weihrauch Reducibility Parallel The degree f ∗ represents being allowed to invoke f any finite number of times in parallel . Sequential The degree f ⋄ represents being allowed to invoke f any finite number of times (not specified in advance), where later queries can be computed from previous answers. Tonicha Crook (Swansea University) Multiplayer Games 8th April 2020 7 / 11
Our Results Nash A multivalued function Nash maps finite games in strategic form to some Nash equilibrium. Tonicha Crook (Swansea University) Multiplayer Games 8th April 2020 8 / 11
Our Results Nash A multivalued function Nash maps finite games in strategic form to some Nash equilibrium. All-or-Unique Choice For a represented space X = { 0 , 1 } N , denote a multivalued function AoUC [0 , 1] : ⊆ A ( X ) ⇒ X via { A ∈ A (( X ) | | A | = 1 } ∪ { X } Tonicha Crook (Swansea University) Multiplayer Games 8th April 2020 8 / 11
Our Results Nash A multivalued function Nash maps finite games in strategic form to some Nash equilibrium. All-or-Unique Choice For a represented space X = { 0 , 1 } N , denote a multivalued function AoUC [0 , 1] : ⊆ A ( X ) ⇒ X via { A ∈ A (( X ) | | A | = 1 } ∪ { X } BRoot BRoot ⊆ : R [ X ] ⇒ [0 , 1] map real polynomials to a root in [0 , 1], provided there is one. Tonicha Crook (Swansea University) Multiplayer Games 8th April 2020 8 / 11
Our Results Main Theorem AoUC ∗ [0 , 1] ≤ W Nash ≤ W AoUC ⋄ [0 , 1] Tonicha Crook (Swansea University) Multiplayer Games 8th April 2020 9 / 11
Our Results Main Theorem AoUC ∗ [0 , 1] ≤ W Nash ≤ W AoUC ⋄ [0 , 1] Theorem AoUC ∗ [0 , 1] ≡ W BRoot ∗ Tonicha Crook (Swansea University) Multiplayer Games 8th April 2020 9 / 11
Our Results Corollary Let f : X → Y be a function where Y is computably admissible. Then if f ≤ W Nash then f is already computable. Tonicha Crook (Swansea University) Multiplayer Games 8th April 2020 10 / 11
Our Results Corollary Let f : X → Y be a function where Y is computably admissible. Then if f ≤ W Nash then f is already computable. Corollary Nash is solvable with finitely many mind changes. Tonicha Crook (Swansea University) Multiplayer Games 8th April 2020 10 / 11
Our Results Corollary Let f : X → Y be a function where Y is computably admissible. Then if f ≤ W Nash then f is already computable. Corollary Nash is solvable with finitely many mind changes. Corollary Nash is Las Vegas computable. Tonicha Crook (Swansea University) Multiplayer Games 8th April 2020 10 / 11
Our Results Corollary Let f : X → Y be a function where Y is computably admissible. Then if f ≤ W Nash then f is already computable. Corollary Nash is solvable with finitely many mind changes. Corollary Nash is Las Vegas computable. Corollary Nash is Monte Carlo computable and we can compute a postive lower bound for the success chance from the dimensions of the game. Tonicha Crook (Swansea University) Multiplayer Games 8th April 2020 10 / 11
Thank You Any Questions? Tonicha Crook (Swansea University) Multiplayer Games 8th April 2020 11 / 11
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