Compositional Game Theory: Building Complex Games from Simple Ones, Slides of Game Theory

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Compositional Game Theory

Neil Ghani

and Julian Hedges, Viktor Winschel, Philipp Zahn

MSP group, The Scottish Free State

Overview

Question:

What is Game Theory?

How to make decisions in eg, finance, scheduling algorithms

Use Nash equilibria in non-cooperative games

Claim:

Game theory is too concrete

Uses non-structural, reductive measures, eg payo

ff

matrices.

Category theory turns meta-structure into actual structure.

Compositionality:

Operators build big games from small games

Lift results about parts of a game to the whole games

Better mathematics and better software for games.

Neil Ghani

Bourget du Lac, July 4, 2017

One Player Games

Defn:

A

basic game

consists of

A set of actions

A

the player can take, and a set

U

of utilities

A function

f

: A

U

assigning to each action, a

utility

Defn:

Eq Optimal actions/equilibria for a simple game are

A, U, f

argmax

f = { a ∈ A | ( ∀ a ′ ∈ A )

f a

f a

′ }

Question:

Is this definition correct for a game?

f : A 1 × A 2 → U 1 ×

U

2

Neil Ghani

Bourget du Lac, July 4, 2017

The Prisoners Dilemma

Motivation:

Two academics face a choice

the authorities who seek evidence to discipline the academics.Each is under pressure to report bad behaviour of the other to

They can cooperate with each other, or defect

A

C, D

Utilities are given by

f

: A

(^) ×

A

Z

×

Z

f (^) ( C, C

f (^) ( D, C

f (^) ( C, D

f (^) ( D, D

Conclusion:

The best strategy for each player is to defect!

cation, no future cost for bad behaviour etc.Rather depressing for utopians! Assumptions: no communi-

Neil Ghani

Bourget du Lac, July 4, 2017

5

Compositionality

Key Idea:

Nash equilibria are given as primitive.

derived from equilibria for simpler gamesThis is not a compositional definition as the definition is not

It is simply postulated as reasonable, justified empirically.

Question:

Is there no operator which combines two 1-player

games into a 2-player game?

the component games.And defines the equilibria of the derived game via those of

Remark:

Of course this is di

ffi

cult as optimal moves for one

into a networked collection of games.game may not remain optimal when that game is incorporated

Neil Ghani

Bourget du Lac, July 4, 2017

From Games to Utility Free Games

Defn:

A

utility-free game

consists of

A set

A

of moves, a set

U

of utilities and an equilibria function

E

A

U

P

A

The set of utility-free games with actions

Y

and utilities

U

is written

UF

A U

Key Idea:

These games leave the utility function abstract

The equilibria is given for

every

potential utility function

And its not always

argmax

, eg El Farrol bar game

Neil Ghani

Bourget du Lac, July 4, 2017

8

Part II: Complex Games

Motivation

Motivation:

Simple games possess limited structure, and hence

support limited operators

More operators

more compositionality

Lets develop a more complex model!

Example:

Lets place a bet

I have a bank balance.

I might have di

ff erent strategies.

makerThese factors decide on my bet which I give to the book-

bet. When the event is finished, he returns my winningsThe bookmaker has a variety of strategies to deal with my

Neil Ghani

Bourget du Lac, July 4, 2017

Definition of an Open Game

Defn

An open game

G

X, S

Y, R

) is defined by

A set

of strategies

A play function

P : Σ × X → Y

A coutility function

C

×

X

×

R

S

An equilibrium function

E : X × ( Y → R ) → P Σ

Example:

Prisoners Dilemma

PD

M

×

M, Z

×

Z

where

M

C, D

Two rounds of prisoners dilemma?

Neil Ghani

Bourget du Lac, July 4, 2017

13

Variations on a Defintion

Via Lenses:

A

lens

L

X, S

Y, R

) is a map

f

: X

Y

and

g : X × R → S

An open game

G

X, S

Y, R

) is a set

and for each

σ

∈

Σ

A lens

G

σ

: (

X, S

Y, R

A predicte

E

σ

⊆

X, S

×

Y, R

Via

Interaction

Structures

and

Indexed Containers

The

algebra becomes easier if we use dependent types:

S

C R → Y → Σ → X

Neil Ghani

Bourget du Lac, July 4, 2017

14

Compositionality of Open Games II: A Monoidal Category

Abstraction:

Now we can define a monoidal category of open

games

Objects are pairs of sets (

X, S

Morphisms (

X, S

Y, R

) are open games

Composition:

This requires composition. Given open games

G

X, S

Y, R

and

H

Y, R

Z, T

construct an open game

H

G

X, S

Z, T

Neil Ghani

Bourget du Lac, July 4, 2017

Conclusions

Achievements:

A new model of game theory

New paradigms — Compositionality

New concepts — Coutility

New Techniques — String diagrams

Future Work:

Much more to do

More operators, more categories, more algorithms

Translate into better software

Applications: smart contracts, energy grids, blockchains

Neil Ghani

Bourget du Lac, July 4, 2017