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IntroductionDefinitions

GamesEvolutionary Game Theory (EGT)

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A Short Introduction to Game Theory

Heiko Hotz

July 13, 2006

Heiko Hotz A Short Introduction to Game Theory
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Table of Contents

1 Introduction

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Definitions3 Games

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Evolutionary Game Theory5 Applications

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Introd ction


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Game Theory - What is it?Game Theory - Where is it applied?

Introduction

Game Theory - What is it?


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Generally, game theory investigates conflict situations, theinteraction between the agents and their decisions.

A game in the sense of game theory is given by a (mostlyfinite) number of players, who interact according to givenrules.

The subject of game theory are situations, where the result for

a player does not only depend on his own decisions, but also

on the behaviour of the other players.


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a player does not only depend on his own decisions, but alsoon the behaviour of the other players.


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a player does not only depend on his own decisions, but alsoon the behaviour of the other players.


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DefinitionsGames

Evolutionary Game Theory (EGT)Applications

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Figure: Johnvon Neumann

Game theory has its historical origin in 1928.By analysing parlour games, John vonNeumann realised very quickly thepracticability of his approaches forthe analysis of economic problems.

In his book Theory of Games andEconomic Behavior, which he wrote together

with Oskar Morgenstern in 1944, he already applied hismathematical theory to economic applications.

The publication of this book is generally seen as the initialpoint of modern game theory.


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Introduction

Game Theory - where is it applied?


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Game Theory - Where is it applied?

When there is competition for a resource to be analysed,game theory can be used either to explain existing behaviouror to improve strategies.

The first is especially applied by sciences which analyselong-term situations, like biology or sociology.

In animality, for example, one can find situations, wherecooperation has developed for the sake of mutual benefits.

The latter is a crucial tool in sciences like economics.Companies use game theory to improve their strategicalsituation in the market.

Many more sciences like sociology, philosophy, psychology andcultural anthropology use game theory as an appropriate tool.


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Normal Form GamesExtensive Form GamesNash Equilibrium

Definitions

Normal Form Games



Normal Form Games


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Normal Form Games

A game in normal form consists of:

1

A finite number of players.2 A strategy set assigned to each player.

3 A payoff function, which assigns a certain payoff to eachplayer depending on his strategy and the strategy of the otherplayers.

Both players choose their strategy simultaneously



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GNormal Form Games
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Extensive Form GamesNash Equilibrium

Payoff Matrix

If the number of players is limited to two and if their sets ofstrategies consist of only a few elements, the outcome of thepayoff function can be represented in a matrix, the so-calledpayoff matrix, which shows the two players, their strategiesand their payoffs:

Player1\Player2 L R

U 3 , 3 0 , 5D 5 , 0 1 , 1



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Payoff Matrix

If the number of players is limited to two and if their sets ofstrategies consist of only a few elements, the outcome of thepayoff function can be represented in a matrix, the so-calledpayoff matrix, which shows the two players, their strategiesand their payoffs:

Player1\Player2 L R

U 3 , 3 0 , 5D 5 , 0 1 , 1



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Definitions

Extensive Form Games




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Extensive Form Games

Contrary to the normal form game, the rules of an extensive formgame are described such that the agents of the game execute theirmoves consecutively.



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Definitions

Nash Equilibrium



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Nash Equilibrium

If each player has chosen a strategy and no player can benefit bychanging his strategy while the other players keep theirs

unchanged, then the current set of strategy choices and thecorresponding payoffs constitute a Nash equilibrium.

Figure:John Nash

John Nash showed in 1950, that every game with a

finite number of players and finite numberof strategies has at least one mixed strategyNash equilibrium.



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Nash Equilibrium

If each player has chosen a strategy and no player can benefit bychanging his strategy while the other players keep theirs

unchanged, then the current set of strategy choices and thecorresponding payoffs constitute a Nash equilibrium.

Figure:John Nash

John Nash showed in 1950, that every game with a

finite number of players and finite numberof strategies has at least one mixed strategyNash equilibrium.



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Nash Equilibrium

In a pure strategies, one strategy is played with probability 1.

A mixed strategy is a linear combination of at least two purestrategies.

The coefficients denote the probabilities of the pure strategiesto be played.



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Nash Equilibrium





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E l ti G Th (EGT)

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Nash Equilibrium






E ol tionar Game Theor (EGT)



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Nash Equilibrium

Best Response

The best response is the strategy (or strategies)

which produces the most favorable immediate

outcome for the current player, taking other players

strategies as given.



Evolutionary Game Theory (EGT)



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Nash Equilibrium

Localizing a Nash Equilibrium in a Payoff Matrix

Player1\Player2 L RU 3 , 3 0 , 5

D 5 , 0 1 , 1






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Nash Equilibrium



D 5 , 0 1 , 1




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y y ( )Applications

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( )Applications

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Nash Equilibrium


Player1\Player2 L RU 3 , 3 0 , 5D 5 , 0 1 , 1



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Prisoners DilemmaPublic Good GameRock, Paper, Scissors
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Prisoners Dilemma

Two suspects arrested

Police has insufficient evidence for a conviction

If both testify:Both receive 2-year sentence

If both stay silent:

Only a six-month-sentence for a minor charge

If one testifies and the other remains silent:Betrayer goes free, accomplice receives 10-year sentence






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Payoff Matrix of the Prisoners Dilemma

Abbreviations: To testify means to betray the other suspectand thus to defect (D), to remain silent means to cooperate(C) with the other suspect.

We will use positive numbers in the payoff matrix.

Player1\Player2 C D

C 3, 3 0, 5D 5, 0 1, 1




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Payoff Matrix of the Prisoners Dilemma

Abbreviations: To testify means to betray the other suspectand thus to defect (D), to remain silent means to cooperate(C) with the other suspect.

We will use positive numbers in the payoff matrix.

Player1\Player2 C D

C 3, 3 0, 5D 5, 0 1, 1




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Games

Public Good Game




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Public Good Game

A group of 4 people are given $ 200 each.

They can invest an amount of money in a project.

Every $1 invested will yield $ 2, which would be distributed toall group members.

If everyone invested, each would get $ 400.

However, if only one person invested, that sucker wouldtake home a mere $ 100.


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Public Good Game







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Payoff Function of the Public Good Game

This is a game in normal form and therefore has a payoff functionfor each player. The payoff function for, lets say, player 1 is givenby

P1 = 2 (s1 + s2 + s3 + s4)4

s1

=2 (s2 + s3 + s4)

4 0, 5 s1

Every investment s1 of player 1 will diminish his payoff.

Therefore, a rational player will choose the strategy sn = 0,i.e. he will invest no money at all.


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P1 = 2 (s1 + s2 + s3 + s4)4

s1

=2 (s2 + s3 + s4)

4 0, 5 s1




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P1 = 2 (s1 + s2 + s3 + s4)4

s1

=2 (s2 + s3 + s4)

4 0, 5 s1




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P1 = 2 (s1 + s2 + s3 + s4)4

s1

=2 (s2 + s3 + s4)

4 0, 5 s1

Every investment s1 of player 1 will diminish his payoff.Therefore, a rational player will choose the strategy sn = 0,i.e. he will invest no money at all.


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P1 = 2 (s1 + s2 + s3 + s4)4

s1

=2 (s2 + s3 + s4)

4 0, 5 s1

Every investment s1 of player 1 will diminish his payoff.Therefore, a rational player will choose the strategy sn = 0,i.e. he will invest no money at all.


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Games

Rock, Paper, Scissors


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Two players form a symbol (rock, paper or scissors) with theirhands

Rock crushes ScissorsScissors cut Paper

Paper covers Rock


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Paper covers Rock


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R k P S
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Paper covers Rock


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Paper covers Rock


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Why EGT?Evolutionary Stable Strategy (ESS)Hawk-Dove GameThe Replicator Dynamics
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Evolutionary Game

Theory

Why EGT?


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Wh EGT?
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Why EGT?

Games can be played repeatedly, and also with a population ofplayers.

Which populations/strategies are stable?

Is it possible for mutants to invade a given population?

How does cooperation arise and evolove?

How does biodeversity emerge?


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Why EGT?







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DefinitionsGamesEvolutionary Game Theory (EGT)

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Introduction


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Why EGT?







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EGT New Concepts
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EGT- New Concepts

Evolutionary stable strategy (ESS)

To investigate the stability of populations.Replicator dynamics

To describe the evolution of different species within apopulation over time.


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EGT New Concepts
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EGT- New Concepts




Heiko Hotz A Short Introduction to Game Theory Introduction


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EGT- New Concepts






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EGT New Concepts






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Evolutionary Game

Theory

Evolutionary Stable Strategy



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An ESS is a strategy which, if adopted by a population,cannot be invaded by any competing alternative strategy.

An equilibrium refinement to the Nash equilibrium.



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An ESS is a strategy which, if adopted by a population,cannot be invaded by any competing alternative strategy.

An equilibrium refinement to the Nash equilibrium.



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y gy ( )

Let E(, ) be the payoff for a player playing against aplayer playing .

For to be an Nash equilibrium it is demanded:E(,) E(, )

For to be an ESS it is demanded:1 E(, ) > E(, ) (equlibrium condition), or2

E(, ) = E(, ) and E(,) > E(, ) (stability condition)



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y gy ( )



For to be an ESS it is demanded:1 E(, ) > E(, ) (equlibrium condition), or2 E

(, ) =E

(, ) andE

(,) >E

(, ) (stability condition)



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( )



For to be an ESS it is demanded:1 E(, ) > E(, ) (equlibrium condition), or2 E

(, ) =E

(, ) andE

(,) >E

(, ) (stability condition)


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Evolutionary Game

Theory

Hawk-Dove Game


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Two individuals compete for a resource V.

Each individual follows exactly one of two strategies describedbelow:

Hawk: Initiate aggressive behaviour, not stopping until injuredor until ones opponent backs down.Dove: Retreat immediately if ones opponent initiatesaggressive behaviour.


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Each individual follows exactly one of two strategies describedbelow:

Hawk: Initiate aggressive behaviour, not stopping until injuredor until ones opponent backs down.Dove: Retreat immediately if ones opponent initiatesaggressive behaviour.


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Each individual follows exactly one of two strategies described

below:Hawk: Initiate aggressive behaviour, not stopping until injuredor until ones opponent backs down.Dove: Retreat immediately if ones opponent initiatesaggressive behaviour.


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Each individual follows exactly one of two strategies described

below:Hawk: Initiate aggressive behaviour, not stopping until injuredor until ones opponent backs down.Dove: Retreat immediately if ones opponent initiatesaggressive behaviour.


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We assume

whenever two individuals both initiate aggressive behaviour,conflict eventually results and the two individuals are equally

likely to be injured,the cost of the conflict reduces individual fitness by someconstant value C,

when a hawk meets a dove, the dove immediately retreats andthe hawk obtains the resource, and

when two doves meet the resource is shared equally betweenthem.

Heiko Hot A Short Introd ction to Game Theor Introduction

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We assume





H ik H t A Sh t I t d ti t G Th Introduction

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We assume





H ik H t A Sh t I t d ti t G Th Introduction

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We assume








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Why EGT?

Evolutionary Stable Strategy (ESS)Hawk-Dove GameThe Replicator Dynamics

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We assume








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This leads to the following payoff matrix:

Hawk Dove

Hawk 12 (V C),12 (V C) V, 0

Dove 0,V V/2,V/2

Dove is no stable strategy, sinceV2 = E(D,D) < E(H,D) = V.

Because of E(H,H) = 12 (V C) and E(D,H) = 0, H is an

ESS if V C.

But what if V < C?

Neither H nor D is an ESS.




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Hawk Dove

Hawk 12 (V C),12 (V C) V, 0

Dove 0,V V/2,V/2



ESS if V C.

But what if V < C?





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Hawk Dove

Hawk 12 (V C),12 (V C) V, 0

Dove 0,V V/2,V/2



ESS if V C.

But what if V < C?





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Hawk Dove

Hawk 12 (V C),12 (V C) V, 0

Dove 0,V V/2,V/2



ESS if V C.But what if V < C?





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Hawk Dove

Hawk 12 (V C),12 (V C) V, 0

Dove 0,V V/2,V/2








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Hawk Dove

Hawk 12 (V C),12 (V C) V, 0

Dove 0,V V/2,V/2








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Evolutionary Game

Theory

The Replicator Dynamics




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Let us consider now a population consisting of n types, andlet xi(t) be the frequency of type i at some time t.

The state of the population is given by the vector

x(t) = x1(t), . . . , xn(t).The state of the population will change, since some specieswill get higher payoffs and will reproduce better than others,or others adopt the strategy of the species with the betterfitness.

We want to postulate a law of motion for x(t), i.e. we wantto investigate the evolution of the different species over time.




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If individuals meet randomly and then engage in a symmetricgame with payoff matrix A, then (Ax)i is the expected payoff

for an individual of type ixTAx is the average payoff in the population state x.

The evolution of x over time is described by the replicatorequation:

xi = xi[(Ax)i xTAx]




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xi = xi[(Ax)i xTAx]




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xi = xi[(Ax)i xTAx]




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By setting xi = 0, we obtain the evolutionary stable states ofa population.

A population is said to be in an evolutionarily stable state if

its genetic composition is restored by selection after adisturbance, provided the disturbance is not too large.

For V > C, the only evolutionary stable state is a populationconsisting of hawks.

For V < C, a mixed population with a fraction V/C of hawksand a fraction 1 V/C of doves is evolutionary stable.




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Evolution of CooperationBiodiversity


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We have seen, that the logical move in the PrisonersDilemma is to defect

Consider spatially structured populations

Limited local interactions enable cooperators to form clusters

Individuals can outweigh their losses against defectors by gainsfrom interactions within the cluster.




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Example revealed critical phase transitions.

This falls into the universality class of directed percolation onsquare lattices.

Results for different population structures in the PD arediscussed and related to condensed matter physics.




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One of the central aims of ecology is to identify mechanismsthat maintain biodiversity.

Theoretical models have shown that competing species cancoexist if ecological processes such as dispersal, movement,and interaction occur over small spatial scales.

In particular, this may be the case for nontransitivecommunities, that is, those without strict competitive

hierarchies.




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hierarchies.




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hierarchies.




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There exist three strains of Escherichia coli bacteria.

Type A releases toxic colicin and produces, for its ownprotection, an immunity protein.

Type B produces the immunity protein only.

Type C produces neither toxin nor immunity.

The production of the toxic colicin and the immunity proteincauses some higher costs.

Thus B beats A, C beats B and A beats C.




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green: resistant strainred: colicin producing strainblue: sensitive strain




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Most evolutionary game dynamics have been studied in thecontext of infinitely large populations. We expect that finitepopulation size effects will lead to surprising outcomes and

might question the importance of traditional evolutionarystability.

Emerging fields as diverse as metabolic control networkswithin cells and evolutionary psychology, for example, should

benefit from game theory.




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Most evolutionary game dynamics have been studied in thecontext of infinitely large populations. We expect that finitepopulation size effects will lead to surprising outcomes and

might question the importance of traditional evolutionarystability.

Emerging fields as diverse as metabolic control networkswithin cells and evolutionary psychology, for example, should

benefit from game theory.




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C lt al i te etatio s of e licato d a ics ofte ass e
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Cultural interpretations of replicator dynamics often assumethat successful strategies spread by imitation or learning, butthe learning of complicated strategies from behavioralobservations is a nontrivial task that needs specific

investigation.

Interactions among genes, viruses, cells, and humans are ofteninstances of evolutionary games that are amenable toempirical and theoretical investigation.

Game theory is the appropriate tool whenever the success ofan individual depends on others.



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investigation.

Interactions among genes, viruses, cells, and humans are ofteninstances of evolutionary games that are amenable toempirical and theoretical investigation.

Game theory is the appropriate tool whenever the success ofan individual depends on others.



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investigation.Interactions among genes, viruses, cells, and humans are ofteninstances of evolutionary games

heiko hotz spieltheorie vortrag

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