The Weihrauch Degree of Finding Nash Equilibria in Multiplayer Games - - PowerPoint PPT Presentation

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

1

Introduction

2

Game Theory

3

Real Number Computabilty

4

Computable Analysis and Weihrauch Reducibility

5

Our Results

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

• btained but we are addressing the situation with multiplayer games.

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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 si ∈ Si, ui(s∗) ≥ ui(si, 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 si ∈ Si, ui(s∗) ≥ ui(si, S∗

i ) is satisfied. 0,1 1,0 2,3 1,2 AI BI AII BII Player II Player 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 si ∈ Si, ui(s∗) ≥ ui(si, S∗

i ) is satisfied. 0,1 1,0 2,3 1,2 AI BI AII BII Player II Player 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 si ∈ Si, ui(s∗) ≥ ui(si, S∗

i ) is satisfied. 0,1 1,0 2,3 1,2 AI BI AII BII Player II Player 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 si ∈ Si, ui(s∗) ≥ ui(si, S∗

i ) is satisfied. 0,1 1,0 2,3 1,2 AI BI AII BII Player II Player 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 si ∈ Si, ui(s∗) ≥ ui(si, S∗

i ) is satisfied. 0,1 1,0 2,3 1,2 AI BI AII BII Player II Player 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.

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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 (qn)n ∈ N ∈ QN representing x ∈ R, if |x − qn| < 2−n for all n ∈ N.

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Computable Analysis and Weihrauch Reducibility

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Computable Analysis and Weihrauch Reducibility

Represented Spaces

A represented space (X, δ) is a set X together with a surjective partial function δ :⊆ NN → X.

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Computable Analysis and Weihrauch Reducibility

Represented Spaces

A represented space (X, δ) is a set X together with a surjective partial function δ :⊆ NN → 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 :⊆ NN → 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]

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Our Results

Main Theorem

AoUC∗

[0,1] ≤W Nash ≤W AoUC⋄ [0,1]

Theorem

AoUC∗

[0,1] ≡W BRoot∗

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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.

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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.

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Thank You Any Questions?

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