Introducing open games Jules Hedges Joint work with Neil Ghani - - PowerPoint PPT Presentation

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Introducing open games Jules Hedges Joint work with Neil Ghani - - PowerPoint PPT Presentation

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Intro to game theory Compositionality Open games Introducing open games Jules Hedges Joint work with Neil Ghani Viktor Winschel Philipp Zahn University of Oxford February 24, 2017 Intro to game theory Compositionality Open games

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Intro to game theory Compositionality Open games

Introducing open games

Jules Hedges Joint work with Neil Ghani Viktor Winschel Philipp Zahn

University of Oxford

February 24, 2017

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Intro to game theory Compositionality Open games

Motivation: powerful machinery vs. hard problems

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Intro to game theory Compositionality Open games

What game theory is

Mathematical theory of interacting “rational” agents

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Intro to game theory Compositionality Open games

What game theory is

Mathematical theory of interacting “rational” agents Players make observations and then make choices

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Intro to game theory Compositionality Open games

What game theory is

Mathematical theory of interacting “rational” agents Players make observations and then make choices Group choices determine payoffs

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Intro to game theory Compositionality Open games

What game theory is

Mathematical theory of interacting “rational” agents Players make observations and then make choices Group choices determine payoffs “Local view” of rationality: players act to maximise payoff

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Intro to game theory Compositionality Open games

What game theory is

Mathematical theory of interacting “rational” agents Players make observations and then make choices Group choices determine payoffs “Local view” of rationality: players act to maximise payoff “Global view”: equilibrium strategies

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Intro to game theory Compositionality Open games

Example: penalty shootout

a, b ∈ {L, R}

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Intro to game theory Compositionality Open games

Example: penalty shootout

a, b ∈ {L, R} π(a, b) =

  • (+1, −1)

if a = b (−1, +1) if a = b

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Intro to game theory Compositionality Open games

Example: penalty shootout

a, b ∈ {L, R} π(a, b) =

  • (+1, −1)

if a = b (−1, +1) if a = b Unique (probabilistic) equilibrium: a = b = 1

2 |L + 1 2 |R

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Intro to game theory Compositionality Open games

Example: penalty shootout

a, b ∈ {L, R} π(a, b) =

  • (+1, −1)

if a = b (−1, +1) if a = b Unique (probabilistic) equilibrium: a = b = 1

2 |L + 1 2 |R

Nash’s theorem generalises this situation

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Intro to game theory Compositionality Open games

Example: the $$$ auction

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Intro to game theory Compositionality Open games

Game theory has some issues

Well known: equilibrium as behavioural prediction is experimentally falsified (e.g. ultimatum game)

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Intro to game theory Compositionality Open games

Game theory has some issues

Well known: equilibrium as behavioural prediction is experimentally falsified (e.g. ultimatum game) Harsanyi type spaces are accurate but underfit (and mathematically hard!)

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Intro to game theory Compositionality Open games

Game theory has some issues

Well known: equilibrium as behavioural prediction is experimentally falsified (e.g. ultimatum game) Harsanyi type spaces are accurate but underfit (and mathematically hard!) There is no accepted operational theory (or “equilibriating process”) (c.f. evolutionary game theory)

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Intro to game theory Compositionality Open games

Game theory has some issues

Well known: equilibrium as behavioural prediction is experimentally falsified (e.g. ultimatum game) Harsanyi type spaces are accurate but underfit (and mathematically hard!) There is no accepted operational theory (or “equilibriating process”) (c.f. evolutionary game theory) Serious computability/complexity issues (algorithmic game theory)

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Intro to game theory Compositionality Open games

Game theory has some issues

Well known: equilibrium as behavioural prediction is experimentally falsified (e.g. ultimatum game) Harsanyi type spaces are accurate but underfit (and mathematically hard!) There is no accepted operational theory (or “equilibriating process”) (c.f. evolutionary game theory) Serious computability/complexity issues (algorithmic game theory) Ordinary games do not compose/scale

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Intro to game theory Compositionality Open games

The fundamental headache of social science

Beliefs have causal effects

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Intro to game theory Compositionality Open games

What is compositionality?

A theory is compositional if:

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Intro to game theory Compositionality Open games

What is compositionality?

A theory is compositional if: Objects of interest are black boxes with interfaces

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Intro to game theory Compositionality Open games

What is compositionality?

A theory is compositional if: Objects of interest are black boxes with interfaces Objects can be composed without knowing how they are defined

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Intro to game theory Compositionality Open games

What is compositionality?

A theory is compositional if: Objects of interest are black boxes with interfaces Objects can be composed without knowing how they are defined Consequence: can easily change one component in a large structure

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Intro to game theory Compositionality Open games

What is compositionality?

A theory is compositional if: Objects of interest are black boxes with interfaces Objects can be composed without knowing how they are defined Consequence: can easily change one component in a large structure All reasoning is by structural induction on composition

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Intro to game theory Compositionality Open games

Examples of compositional systems

Any serious programming language

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Intro to game theory Compositionality Open games

Examples of compositional systems

Any serious programming language Internal reasoning in a logic, algebraic structure, category, etc.

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Intro to game theory Compositionality Open games

Examples of compositional systems

Any serious programming language Internal reasoning in a logic, algebraic structure, category, etc. Most systems designed by human engineers

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Intro to game theory Compositionality Open games

Examples of compositional systems

Any serious programming language Internal reasoning in a logic, algebraic structure, category, etc. Most systems designed by human engineers Organisations made of people (e.g. a company)

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Intro to game theory Compositionality Open games

Examples of compositional systems

Any serious programming language Internal reasoning in a logic, algebraic structure, category, etc. Most systems designed by human engineers Organisations made of people (e.g. a company) Non-examples: Analytic solution of differential equations

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Intro to game theory Compositionality Open games

Examples of compositional systems

Any serious programming language Internal reasoning in a logic, algebraic structure, category, etc. Most systems designed by human engineers Organisations made of people (e.g. a company) Non-examples: Analytic solution of differential equations Natural (biological) systems

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Intro to game theory Compositionality Open games

Examples of compositional systems

Any serious programming language Internal reasoning in a logic, algebraic structure, category, etc. Most systems designed by human engineers Organisations made of people (e.g. a company) Non-examples: Analytic solution of differential equations Natural (biological) systems Economic systems

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Intro to game theory Compositionality Open games

The compositionality hypothesis

If a theory is compositional, then reasoning with it is scalable

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Intro to game theory Compositionality Open games

The compositionality hypothesis

If a theory is compositional, then reasoning with it is scalable Hypothesis Compositionality is the only way to be scalable

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Intro to game theory Compositionality Open games

Compositionality is delicate

Compositionality is not all-or-nothing

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Intro to game theory Compositionality Open games

Compositionality is delicate

Compositionality is not all-or-nothing In programming languages it is harmed by: Goto Mutable global state Inheritance Type classes a la Haskell

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Intro to game theory Compositionality Open games

Compositionality is delicate

Compositionality is not all-or-nothing In programming languages it is harmed by: Goto Mutable global state Inheritance Type classes a la Haskell More generally: Leaky abstractions Emergent behaviour

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Intro to game theory Compositionality Open games

Compositionality via symmetric monoidal categories (1)

Objects (aka. interfaces, types, systems) X X

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Intro to game theory Compositionality Open games

Compositionality via symmetric monoidal categories (1)

Objects (aka. interfaces, types, systems) X X Morphisms (aka. black boxes, processes) f : X → Y X Y f

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Intro to game theory Compositionality Open games

Compositionality via symmetric monoidal categories (1)

Objects (aka. interfaces, types, systems) X X Morphisms (aka. black boxes, processes) f : X → Y X Y f Compound object X1 ⊗ X2 X1 X2

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Intro to game theory Compositionality Open games

Compositionality via symmetric monoidal categories (1)

Objects (aka. interfaces, types, systems) X X Morphisms (aka. black boxes, processes) f : X → Y X Y f Compound object X1 ⊗ X2 X1 X2 Monoidal product (aka. tensor product, simultaneous/spatial composition) f1 ⊗ f2 : X1 ⊗ X2 → Y1 ⊗ Y2 X1 X2 Y1 Y2 f1 f2

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Intro to game theory Compositionality Open games

Compositionality via symmetric monoidal categories (2)

Categorical composition (aka. sequential/temporal composition) g ◦ f : X → Z X Z f g Y

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Intro to game theory Compositionality Open games

Compositionality via symmetric monoidal categories (2)

Categorical composition (aka. sequential/temporal composition) g ◦ f : X → Z X Z f g Y f : I → X ⊗ Y g : X → Z h : X ⊗ Z → I h ◦ σZ,X ◦ (g ⊗ Y ) ◦ f : I → I f g h X Z Y

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Intro to game theory Compositionality Open games

Open games

A closed game consists of: A set Σ of strategy profiles A best response function B : Σ → P(Σ) An open game G : (X, S) → (Y , R) consists of: A set Σ of strategy profiles A play function PG : Σ × X → Y A coplay function CG : Σ × X × R → S A best response function BG : X × (Y → R) × Σ → P(Σ)

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Intro to game theory Compositionality Open games

Bimatrix game

A1 A2 U X R Y R

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Intro to game theory Compositionality Open games

Perfect information game

A1 A2 U X X R R X Y R

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Intro to game theory Compositionality Open games

Imperfect information

A1 f A2 U X X R R X X ′ Y R