Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Population Dynamics in Stochastic Games

joint with J. Flesch, P. Uyttendaele, Maastricht Universityand T. Parthasarathy, Indian Statistical Institute, Chennai

Toulouse, September 12-16, 2011

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Outline

1 Introduction

2 Stochastic Games

3 Evolutionary Games

4 Evolutionary Stochastic Games

5 Concluding Remarks

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

1928, John von Neumann

2-Person Zerosum Games

… a ij (s ) …p (s , i , j )

… a ij (s ), b ij (s ) …p (s , i , j )

j j

i a ij i a ij , b ij

state 1 state s state z

j

i

j

i

state 1 state s state z

Existence of Value and Optimal Strategies

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

1951, John Nash

n-Person Non-Zerosum Games

… a ij (s ) …p (s , i , j )

… a ij (s ), b ij (s ) …p (s , i , j )

j j

i a ij i a ij , b ij

state 1 state s state z

j

i

j

i

state 1 state s state z

Existence of Equilibria

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

1953, Lloyd Shapley

2-Person Zerosum Stochastic Games

… a ij (s ) …p (s , i , j )

… a ij (s ), b ij (s ) …p (s , i , j )

j j

i a ij i a ij , b ij

state 1 state s state z

j

i

j

i

state 1 state s state z

Existence of Value and Optimal Strategies

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

1973, John Maynard Smith and George Price

Evolutionary Games

… a ij (s ) …p (s , i , j )

… a ij (s ), b ij (s ) …p (s , i , j )

j p j

i a ij p i a ij , b ij with b ij = a ji

state z

x j (s )

x i (s )

state 1 state s state z

j

i

state 1 state s

Population of Different Types Playing against Itself.Population Distribution p = (p1,p2, . . . ,pn).Type k has Fitness ekAp in Population p.Concept of Evolutionary Stable Strategies (ESS).

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Question

How to Model a Population Playing a Stochastic Game?

Some Words about Stochastic GamesSome Words about Evolutionary GamesPresentation of a Combined Model

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Question

How to Model a Population Playing a Stochastic Game?Some Words about Stochastic Games

Some Words about Evolutionary GamesPresentation of a Combined Model

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Question

How to Model a Population Playing a Stochastic Game?Some Words about Stochastic GamesSome Words about Evolutionary Games

Presentation of a Combined Model

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Question

How to Model a Population Playing a Stochastic Game?Some Words about Stochastic GamesSome Words about Evolutionary GamesPresentation of a Combined Model

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

The Stochastic Game Model

… a ij (s ) …p (s , i , j )

… a ij (s ), b ij (s ) …p (s , i , j )

j p j

i a ij p i a ij , b ij with a ij = b ji

state z

y j (s )

x i (s )

state 1 state s state z

j

i

state 1 state s

Finitely Many States, Finitely Many Actions for each Player

Payoffs and Transitions at each Stage 1,2,3,4, . . .

Each State can serve as Initial State

Complete Information and Perfect Recall

Discounting or Averaging the Stage Payoffs

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

The Stochastic Game Model

… a ij (s ) …p (s , i , j )

… a ij (s ), b ij (s ) …p (s , i , j )

j p j

i a ij p i a ij , b ij with a ij = b ji

state z

y j (s )

x i (s )

state 1 state s state z

j

i

state 1 state s

Finitely Many States, Finitely Many Actions for each Player

Payoffs and Transitions at each Stage 1,2,3,4, . . .

Each State can serve as Initial State

Complete Information and Perfect Recall

Discounting or Averaging the Stage Payoffs

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

The Stochastic Game Model

… a ij (s ) …p (s , i , j )

… a ij (s ), b ij (s ) …p (s , i , j )

j p j

i a ij p i a ij , b ij with a ij = b ji

state z

y j (s )

x i (s )

state 1 state s state z

j

i

state 1 state s

Finitely Many States, Finitely Many Actions for each Player

Payoffs and Transitions at each Stage 1,2,3,4, . . .

Each State can serve as Initial State

Complete Information and Perfect Recall

Discounting or Averaging the Stage Payoffs

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

The Stochastic Game Model

… a ij (s ) …p (s , i , j )

… a ij (s ), b ij (s ) …p (s , i , j )

j p j

i a ij p i a ij , b ij with a ij = b ji

state z

y j (s )

x i (s )

state 1 state s state z

j

i

state 1 state s

Finitely Many States, Finitely Many Actions for each Player

Payoffs and Transitions at each Stage 1,2,3,4, . . .

Each State can serve as Initial State

Complete Information and Perfect Recall

Discounting or Averaging the Stage Payoffs

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

The Stochastic Game Model

… a ij (s ) …p (s , i , j )

… a ij (s ), b ij (s ) …p (s , i , j )

j p j

i a ij p i a ij , b ij with a ij = b ji

state z

y j (s )

x i (s )

state 1 state s state z

j

i

state 1 state s

Finitely Many States, Finitely Many Actions for each Player

Payoffs and Transitions at each Stage 1,2,3,4, . . .

Each State can serve as Initial State

Complete Information and Perfect Recall

Discounting or Averaging the Stage Payoffs

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

The Stochastic Game Model

… a ij (s ) …p (s , i , j )

… a ij (s ), b ij (s ) …p (s , i , j )

j p j

i a ij p i a ij , b ij with a ij = b ji

state z

y j (s )

x i (s )

state 1 state s state z

j

i

state 1 state s

Finitely Many States, Finitely Many Actions for each Player

Payoffs and Transitions at each Stage 1,2,3,4, . . .

Each State can serve as Initial State

Complete Information and Perfect Recall

Discounting or Averaging the Stage Payoffs

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Some Highlights of Stochastic Game Theory

… a ij (s ) …p (s , i , j )

… a ij (s ), b ij (s ) …p (s , i , j )

j p j

i a ij p i a ij , b ij with a ij = b ji

state z

y j (s )

x i (s )

state 1 state s state z

j

i

state 1 state s

1953, L.S. Shapley:2-Person Zerosum Stopping Stochastic Games - Value

1957, H. Everett / D. Gillette:2-Person Zerosum Undiscounted Stochastic Games

1964, A.M. Fink / M. Takahashi:n-Person β-Discounted Stochastic Games - Equilibria

1981, J.F. Mertens and A. Neyman:2-Person Zerosum Undiscounted Stochastic Games - Value

2000, N. Vieille:2-Person Undiscounted Stochastic Games - Equilibria

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Some Highlights of Stochastic Game Theory

… a ij (s ) …p (s , i , j )

… a ij (s ), b ij (s ) …p (s , i , j )

j p j

i a ij p i a ij , b ij with a ij = b ji

state z

y j (s )

x i (s )

state 1 state s state z

j

i

state 1 state s

1953, L.S. Shapley:2-Person Zerosum Stopping Stochastic Games - Value

1957, H. Everett / D. Gillette:2-Person Zerosum Undiscounted Stochastic Games

1964, A.M. Fink / M. Takahashi:n-Person β-Discounted Stochastic Games - Equilibria

1981, J.F. Mertens and A. Neyman:2-Person Zerosum Undiscounted Stochastic Games - Value

2000, N. Vieille:2-Person Undiscounted Stochastic Games - Equilibria

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Some Highlights of Stochastic Game Theory

… a ij (s ) …p (s , i , j )

… a ij (s ), b ij (s ) …p (s , i , j )

j p j

i a ij p i a ij , b ij with a ij = b ji

state z

y j (s )

x i (s )

state 1 state s state z

j

i

state 1 state s

1953, L.S. Shapley:2-Person Zerosum Stopping Stochastic Games - Value

1957, H. Everett / D. Gillette:2-Person Zerosum Undiscounted Stochastic Games

1964, A.M. Fink / M. Takahashi:n-Person β-Discounted Stochastic Games - Equilibria

1981, J.F. Mertens and A. Neyman:2-Person Zerosum Undiscounted Stochastic Games - Value

2000, N. Vieille:2-Person Undiscounted Stochastic Games - Equilibria

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Some Highlights of Stochastic Game Theory

… a ij (s ) …p (s , i , j )

… a ij (s ), b ij (s ) …p (s , i , j )

j p j

i a ij p i a ij , b ij with a ij = b ji

state z

y j (s )

x i (s )

state 1 state s state z

j

i

state 1 state s

1953, L.S. Shapley:2-Person Zerosum Stopping Stochastic Games - Value

1957, H. Everett / D. Gillette:2-Person Zerosum Undiscounted Stochastic Games

1964, A.M. Fink / M. Takahashi:n-Person β-Discounted Stochastic Games - Equilibria

1981, J.F. Mertens and A. Neyman:2-Person Zerosum Undiscounted Stochastic Games - Value

2000, N. Vieille:2-Person Undiscounted Stochastic Games - Equilibria

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Some Highlights of Stochastic Game Theory

… a ij (s ) …p (s , i , j )

… a ij (s ), b ij (s ) …p (s , i , j )

j p j

i a ij p i a ij , b ij with a ij = b ji

state z

y j (s )

x i (s )

state 1 state s state z

j

i

state 1 state s

1953, L.S. Shapley:2-Person Zerosum Stopping Stochastic Games - Value

1957, H. Everett / D. Gillette:2-Person Zerosum Undiscounted Stochastic Games

1964, A.M. Fink / M. Takahashi:n-Person β-Discounted Stochastic Games - Equilibria

1981, J.F. Mertens and A. Neyman:2-Person Zerosum Undiscounted Stochastic Games - Value

2000, N. Vieille:2-Person Undiscounted Stochastic Games - Equilibria

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Some Highlights of Stochastic Game Theory

… a ij (s ) …p (s , i , j )

… a ij (s ), b ij (s ) …p (s , i , j )

j p j

i a ij p i a ij , b ij with a ij = b ji

state z

y j (s )

x i (s )

state 1 state s state z

j

i

state 1 state s

1953, L.S. Shapley:2-Person Zerosum Stopping Stochastic Games - Value

1957, H. Everett / D. Gillette:2-Person Zerosum Undiscounted Stochastic Games

1964, A.M. Fink / M. Takahashi:n-Person β-Discounted Stochastic Games - Equilibria

1981, J.F. Mertens and A. Neyman:2-Person Zerosum Undiscounted Stochastic Games - Value

2000, N. Vieille:2-Person Undiscounted Stochastic Games - Equilibria

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

1973, John Maynard Smith and George Price

Evolutionary Games

… a ij (s ) …p (s , i , j )

… a ij (s ), b ij (s ) …p (s , i , j )

j p j

i a ij p i a ij , b ij with b ij = a ji

state z

x j (s )

x i (s )

state 1 state s state z

j

i

state 1 state s

Population of Different Types Playing against Itself.Population Distribution p = (p1,p2, . . . ,pn).Type k has Fitness ekAp in Population p.Concept of Evolutionary Stable Strategies (ESS).

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

The ESS Concept

Evolutionary Games

… a ij (s ) …p (s , i , j )

… a ij (s ), b ij (s ) …p (s , i , j )

j p j

i a ij p i a ij , b ij with b ij = a ji

state z

x j (s )

x i (s )

state 1 state s state z

j

i

state 1 state s

ESS: Population Distribution p = (p1,p2, . . . ,pn) with

pAp ≥ qAp ∀qIf q 6= p and qAp = pAp, then pAq > qAq

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

The Replicator Dynamic by Taylor and Jonker, 1978

Evolutionary Games

… a ij (s ) …p (s , i , j )

… a ij (s ), b ij (s ) …p (s , i , j )

j p j

i a ij p i a ij , b ij with b ij = a ji

state z

x j (s )

x i (s )

state 1 state s state z

j

i

state 1 state s

Population Development by the Replicator Equation:ṗk = pk (ekAp − pAp)

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Remarks on ESS and Asymptotic Stability

A Static Concept and a Dynamic ProcessESS Not Always ExistsReplicator Dynamic Not Always ConvergesAny ESS is Asymptotically StableLimit Points of Dynamic Not Always ESS

0 6 −4−3 0 5−1 3 0

(Hofbauer and Sigmund, 1998)

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Remarks on ESS and Asymptotic Stability

A Static Concept and a Dynamic Process

ESS Not Always ExistsReplicator Dynamic Not Always ConvergesAny ESS is Asymptotically StableLimit Points of Dynamic Not Always ESS

0 6 −4−3 0 5−1 3 0

(Hofbauer and Sigmund, 1998)

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Remarks on ESS and Asymptotic Stability

A Static Concept and a Dynamic ProcessESS Not Always Exists

Replicator Dynamic Not Always ConvergesAny ESS is Asymptotically StableLimit Points of Dynamic Not Always ESS

0 6 −4−3 0 5−1 3 0

(Hofbauer and Sigmund, 1998)

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Remarks on ESS and Asymptotic Stability

A Static Concept and a Dynamic ProcessESS Not Always ExistsReplicator Dynamic Not Always Converges

Any ESS is Asymptotically StableLimit Points of Dynamic Not Always ESS

0 6 −4−3 0 5−1 3 0

(Hofbauer and Sigmund, 1998)

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Remarks on ESS and Asymptotic Stability

A Static Concept and a Dynamic ProcessESS Not Always ExistsReplicator Dynamic Not Always ConvergesAny ESS is Asymptotically Stable

Limit Points of Dynamic Not Always ESS

0 6 −4−3 0 5−1 3 0

(Hofbauer and Sigmund, 1998)

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Remarks on ESS and Asymptotic Stability

A Static Concept and a Dynamic ProcessESS Not Always ExistsReplicator Dynamic Not Always ConvergesAny ESS is Asymptotically StableLimit Points of Dynamic Not Always ESS

0 6 −4−3 0 5−1 3 0

(Hofbauer and Sigmund, 1998)

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Remarks on ESS and Asymptotic Stability

A Static Concept and a Dynamic ProcessESS Not Always ExistsReplicator Dynamic Not Always ConvergesAny ESS is Asymptotically StableLimit Points of Dynamic Not Always ESS

0 6 −4−3 0 5−1 3 0

(Hofbauer and Sigmund, 1998)

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Remarks on ESS and Asymptotic Stability

A Static Concept and a Dynamic ProcessESS Not Always ExistsReplicator Dynamic Not Always ConvergesAny ESS is Asymptotically StableLimit Points of Dynamic Not Always ESS

0 6 −4−3 0 5−1 3 0

(Hofbauer and Sigmund, 1998)Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Assumptions for Evolutionary Stochastic Games… a ij (s ) …

p (s , i , j )

… a ij (s ), b ij (s ) …p (s , i , j )

j p j

i a ij p i a ij , b ij with b ij = a ji

state z

x j (s )

x i (s )

state 1 state s state z

j

i

state 1 state s

Symmetric Payoffs: bij = ajiSymmetric Transitions: p(s, i , j) = p(s, j , i)Unichain Stochastic GameOne Ergodic Set for Any Pair of Stationary StrategiesTypes Correspond to Pure Stationary Strategies

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Assumptions for Evolutionary Stochastic Games… a ij (s ) …

p (s , i , j )

… a ij (s ), b ij (s ) …p (s , i , j )

j p j

i a ij p i a ij , b ij with b ij = a ji

state z

x j (s )

x i (s )

state 1 state s state z

j

i

state 1 state s

Symmetric Payoffs: bij = aji

Symmetric Transitions: p(s, i , j) = p(s, j , i)Unichain Stochastic GameOne Ergodic Set for Any Pair of Stationary StrategiesTypes Correspond to Pure Stationary Strategies

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Assumptions for Evolutionary Stochastic Games… a ij (s ) …

p (s , i , j )

… a ij (s ), b ij (s ) …p (s , i , j )

j p j

i a ij p i a ij , b ij with b ij = a ji

state z

x j (s )

x i (s )

state 1 state s state z

j

i

state 1 state s

Symmetric Payoffs: bij = ajiSymmetric Transitions: p(s, i , j) = p(s, j , i)

Unichain Stochastic GameOne Ergodic Set for Any Pair of Stationary StrategiesTypes Correspond to Pure Stationary Strategies

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Assumptions for Evolutionary Stochastic Games… a ij (s ) …

p (s , i , j )

… a ij (s ), b ij (s ) …p (s , i , j )

j p j

i a ij p i a ij , b ij with b ij = a ji

state z

x j (s )

x i (s )

state 1 state s state z

j

i

state 1 state s

Symmetric Payoffs: bij = ajiSymmetric Transitions: p(s, i , j) = p(s, j , i)Unichain Stochastic Game

One Ergodic Set for Any Pair of Stationary StrategiesTypes Correspond to Pure Stationary Strategies

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Assumptions for Evolutionary Stochastic Games… a ij (s ) …

p (s , i , j )

… a ij (s ), b ij (s ) …p (s , i , j )

j p j

i a ij p i a ij , b ij with b ij = a ji

state z

x j (s )

x i (s )

state 1 state s state z

j

i

state 1 state s

Symmetric Payoffs: bij = ajiSymmetric Transitions: p(s, i , j) = p(s, j , i)Unichain Stochastic GameOne Ergodic Set for Any Pair of Stationary Strategies

Types Correspond to Pure Stationary Strategies

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Assumptions for Evolutionary Stochastic Games… a ij (s ) …

p (s , i , j )

… a ij (s ), b ij (s ) …p (s , i , j )

j p j

i a ij p i a ij , b ij with b ij = a ji

state z

x j (s )

x i (s )

state 1 state s state z

j

i

state 1 state s

Symmetric Payoffs: bij = ajiSymmetric Transitions: p(s, i , j) = p(s, j , i)Unichain Stochastic GameOne Ergodic Set for Any Pair of Stationary StrategiesTypes Correspond to Pure Stationary Strategies

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Assumptions Continued

… a ij (s ) …p (s , i , j )

… a ij (s ), b ij (s ) …p (s , i , j )

j p j

i a ij p i a ij , b ij with b ij = a ji

state z

x j (s )

x i (s )

state 1 state s state z

j

i

state 1 state s

Fitness of Type k in Population x̄ = (x̄1, x̄2, . . . , x̄n)is Average Reward γ(ek , x)where x Stationary Strategy determined by x̄Different Populations can give Same Stationary StrategyStationary Strategy x is ESS if

γ(x , x) ≥ γ(y , x) ∀ Stationary Strategies yIf y 6= x and γ(y , x) = γ(x , x), then γ(x , y) > γ(y , y)

Population Development by Replicator Dynamic˙̄xk = x̄k (γ(ek , x)− γ(x , x))

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Assumptions Continued

… a ij (s ) …p (s , i , j )

… a ij (s ), b ij (s ) …p (s , i , j )

j p j

i a ij p i a ij , b ij with b ij = a ji

state z

x j (s )

x i (s )

state 1 state s state z

j

i

state 1 state s

Fitness of Type k in Population x̄ = (x̄1, x̄2, . . . , x̄n)is Average Reward γ(ek , x)where x Stationary Strategy determined by x̄

Different Populations can give Same Stationary StrategyStationary Strategy x is ESS if

γ(x , x) ≥ γ(y , x) ∀ Stationary Strategies yIf y 6= x and γ(y , x) = γ(x , x), then γ(x , y) > γ(y , y)

Population Development by Replicator Dynamic˙̄xk = x̄k (γ(ek , x)− γ(x , x))

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Assumptions Continued

… a ij (s ) …p (s , i , j )

… a ij (s ), b ij (s ) …p (s , i , j )

j p j

i a ij p i a ij , b ij with b ij = a ji

state z

x j (s )

x i (s )

state 1 state s state z

j

i

state 1 state s

Fitness of Type k in Population x̄ = (x̄1, x̄2, . . . , x̄n)is Average Reward γ(ek , x)where x Stationary Strategy determined by x̄Different Populations can give Same Stationary Strategy

Stationary Strategy x is ESS ifγ(x , x) ≥ γ(y , x) ∀ Stationary Strategies yIf y 6= x and γ(y , x) = γ(x , x), then γ(x , y) > γ(y , y)

Population Development by Replicator Dynamic˙̄xk = x̄k (γ(ek , x)− γ(x , x))

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Assumptions Continued

… a ij (s ) …p (s , i , j )

… a ij (s ), b ij (s ) …p (s , i , j )

j p j

i a ij p i a ij , b ij with b ij = a ji

state z

x j (s )

x i (s )

state 1 state s state z

j

i

state 1 state s

Fitness of Type k in Population x̄ = (x̄1, x̄2, . . . , x̄n)is Average Reward γ(ek , x)where x Stationary Strategy determined by x̄Different Populations can give Same Stationary StrategyStationary Strategy x is ESS if

γ(x , x) ≥ γ(y , x) ∀ Stationary Strategies yIf y 6= x and γ(y , x) = γ(x , x), then γ(x , y) > γ(y , y)

Population Development by Replicator Dynamic˙̄xk = x̄k (γ(ek , x)− γ(x , x))

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Assumptions Continued

… a ij (s ) …p (s , i , j )

… a ij (s ), b ij (s ) …p (s , i , j )

j p j

i a ij p i a ij , b ij with b ij = a ji

state z

x j (s )

x i (s )

state 1 state s state z

j

i

state 1 state s

Fitness of Type k in Population x̄ = (x̄1, x̄2, . . . , x̄n)is Average Reward γ(ek , x)where x Stationary Strategy determined by x̄Different Populations can give Same Stationary StrategyStationary Strategy x is ESS if

γ(x , x) ≥ γ(y , x) ∀ Stationary Strategies yIf y 6= x and γ(y , x) = γ(x , x), then γ(x , y) > γ(y , y)

Population Development by Replicator Dynamic˙̄xk = x̄k (γ(ek , x)− γ(x , x))

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Some Remarks … a ij (s ) …p (s , i , j )

… a ij (s ), b ij (s ) …p (s , i , j )

j p j

i a ij p i a ij , b ij with b ij = a ji

state z

x j (s )

x i (s )

state 1 state s state z

j

i

state 1 state s

ESS Not Always ExistsReplicator Dynamic Not Always ConvergesLimit Points of Dynamic Not Always give ESS

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Some Remarks … a ij (s ) …p (s , i , j )

… a ij (s ), b ij (s ) …p (s , i , j )

j p j

i a ij p i a ij , b ij with b ij = a ji

state z

x j (s )

x i (s )

state 1 state s state z

j

i

state 1 state s

ESS Not Always Exists

Replicator Dynamic Not Always ConvergesLimit Points of Dynamic Not Always give ESS

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Some Remarks … a ij (s ) …p (s , i , j )

… a ij (s ), b ij (s ) …p (s , i , j )

j p j

i a ij p i a ij , b ij with b ij = a ji

state z

x j (s )

x i (s )

state 1 state s state z

j

i

state 1 state s

ESS Not Always ExistsReplicator Dynamic Not Always Converges

Limit Points of Dynamic Not Always give ESS

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Some Remarks … a ij (s ) …p (s , i , j )

… a ij (s ), b ij (s ) …p (s , i , j )

j p j

i a ij p i a ij , b ij with b ij = a ji

state z

x j (s )

x i (s )

state 1 state s state z

j

i

state 1 state s

ESS Not Always ExistsReplicator Dynamic Not Always ConvergesLimit Points of Dynamic Not Always give ESS

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Why Unichain?

1, 1 4, 3 3, 3 5, 4(0, 1) (.5, .5) (1, 0) (.5, .5)

3, 4 2, 2 4, 5 2, 2(.5, .5) (0, 1) (.5, .5) (1, 0)

1, 1 4, 3 3, 3 5, 4 4, 4 6, 7(.5, 0, .5) (.5, .5, 0) (1, 0, 0) (.5, 0, .5) (0, 1, 0) (0, .5, .5)

3, 4 2, 2 4, 5 2, 2 7, 6 5, 5(.5, .5, 0) (0, .5, .5) (.5, 0, .5) (0, 0, 1) (0, .5, .5) (1, 0, 0)

1, 1 4, 3 3, 3 5, 4(0, 1, 0) (.5, .5, 0) (1, 0, 0) (.5, .5, 0)

3, 4 2, 2 4, 5 2, 2 2, 2(.5, .5, 0) (0, 1, 0) (.5, .5, 0) (0, 0, 1) (0, 0, 1)

state 3

3, 3 5, 4(1, 0) (1, 0)

4, 5 2, 2 2, 2(1, 0) (0, 1) (0, 1)

state 2

state 1 state 2

state 1 state 2

state 1

state 3

state 1 state 2

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

A 2 State Example with Replicator Dynamics

1, 1 4, 3 3, 3 5, 4(0, 1) (.5, .5) (1, 0) (.5, .5)

3, 4 2, 2 4, 5 2, 2(.5, .5) (0, 1) (.5, .5) (1, 0)

1, 1 4, 3 3, 3 5, 4 4, 4 6, 7(.5, 0, .5) (.5, .5, 0) (1, 0, 0) (.5, 0, .5) (1, 0, 0) (0, .5, .5)

3, 4 2, 2 4, 5 2, 2 7, 6 5, 5(.5, .5, 0) (0, .5, .5) (.5, 0, .5) (0, 0, 1) (0, .5, .5) (0, 1, 0)

1, 1 4, 3 3, 3 5, 4(0, 1, 0) (.5, .5, 0) (1, 0, 0) (.5, .5, 0)

3, 4 2, 2 4, 5 2, 2 2, 2(.5, .5, 0) (0, 1, 0) (.5, .5, 0) (0, 0, 1) (0, 0, 1)

state 3

state 3

state 1 state 2

state 1 state 2

state 1 state 2

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

A 2 State Example with Replicator Dynamics

1, 1 4, 3 3, 3 5, 4(0, 1) (.5, .5) (1, 0) (.5, .5)

3, 4 2, 2 4, 5 2, 2(.5, .5) (0, 1) (.5, .5) (1, 0)

1, 1 4, 3 3, 3 5, 4 4, 4 6, 7(.5, 0, .5) (.5, .5, 0) (1, 0, 0) (.5, 0, .5) (1, 0, 0) (0, .5, .5)

3, 4 2, 2 4, 5 2, 2 7, 6 5, 5(.5, .5, 0) (0, .5, .5) (.5, 0, .5) (0, 0, 1) (0, .5, .5) (0, 1, 0)

1, 1 4, 3 3, 3 5, 4(0, 1, 0) (.5, .5, 0) (1, 0, 0) (.5, .5, 0)

3, 4 2, 2 4, 5 2, 2 2, 2(.5, .5, 0) (0, 1, 0) (.5, .5, 0) (0, 0, 1) (0, 0, 1)

state 3

state 3

state 1 state 2

state 1 state 2

state 1 state 2

(Trajectory)

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

replicator2.aviMedia File (video/avi)

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

A 2 State Example with Replicator Dynamics

1, 1 4, 3 3, 3 5, 4(0, 1) (.5, .5) (1, 0) (.5, .5)

3, 4 2, 2 4, 5 2, 2(.5, .5) (0, 1) (.5, .5) (1, 0)

1, 1 4, 3 3, 3 5, 4 4, 4 6, 7(.5, 0, .5) (.5, .5, 0) (1, 0, 0) (.5, 0, .5) (1, 0, 0) (0, .5, .5)

3, 4 2, 2 4, 5 2, 2 7, 6 5, 5(.5, .5, 0) (0, .5, .5) (.5, 0, .5) (0, 0, 1) (0, .5, .5) (0, 1, 0)

1, 1 4, 3 3, 3 5, 4(0, 1, 0) (.5, .5, 0) (1, 0, 0) (.5, .5, 0)

3, 4 2, 2 4, 5 2, 2 2, 2(.5, .5, 0) (0, 1, 0) (.5, .5, 0) (0, 0, 1) (0, 0, 1)

state 3

state 3

state 1 state 2

state 1 state 2

state 1 state 2

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

A 2 State Example with Replicator Dynamics

1, 1 4, 3 3, 3 5, 4(0, 1) (.5, .5) (1, 0) (.5, .5)

3, 4 2, 2 4, 5 2, 2(.5, .5) (0, 1) (.5, .5) (1, 0)

1, 1 4, 3 3, 3 5, 4 4, 4 6, 7(.5, 0, .5) (.5, .5, 0) (1, 0, 0) (.5, 0, .5) (1, 0, 0) (0, .5, .5)

3, 4 2, 2 4, 5 2, 2 7, 6 5, 5(.5, .5, 0) (0, .5, .5) (.5, 0, .5) (0, 0, 1) (0, .5, .5) (0, 1, 0)

1, 1 4, 3 3, 3 5, 4(0, 1, 0) (.5, .5, 0) (1, 0, 0) (.5, .5, 0)

3, 4 2, 2 4, 5 2, 2 2, 2(.5, .5, 0) (0, 1, 0) (.5, .5, 0) (0, 0, 1) (0, 0, 1)

state 3

state 3

state 1 state 2

state 1 state 2

state 1 state 2

(Trajectory)

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

replicator2random.aviMedia File (video/avi)

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

1951, George Brown / Julia Robinson

The Fictitious Play Process:Playing Best Replies against Observed Action Frequencies

For Matrix Games FP leads to Optimal StrategiesNo FP Convergence for Bimatrix Games (Shapley, 1964). . .

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

1951, George Brown / Julia Robinson

The Fictitious Play Process:Playing Best Replies against Observed Action Frequencies

For Matrix Games FP leads to Optimal Strategies

No FP Convergence for Bimatrix Games (Shapley, 1964). . .

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

1951, George Brown / Julia Robinson

The Fictitious Play Process:Playing Best Replies against Observed Action Frequencies

For Matrix Games FP leads to Optimal StrategiesNo FP Convergence for Bimatrix Games (Shapley, 1964)

. . .

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

1951, George Brown / Julia Robinson

The Fictitious Play Process:Playing Best Replies against Observed Action Frequencies

For Matrix Games FP leads to Optimal StrategiesNo FP Convergence for Bimatrix Games (Shapley, 1964). . .

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

A 2 State Example with Fictitious Play

1, 1 4, 3 3, 3 5, 4(0, 1) (.5, .5) (1, 0) (.5, .5)

3, 4 2, 2 4, 5 2, 2(.5, .5) (0, 1) (.5, .5) (1, 0)

1, 1 4, 3 3, 3 5, 4 4, 4 6, 7(.5, 0, .5) (.5, .5, 0) (1, 0, 0) (.5, 0, .5) (1, 0, 0) (0, .5, .5)

3, 4 2, 2 4, 5 2, 2 7, 6 5, 5(.5, .5, 0) (0, .5, .5) (.5, 0, .5) (0, 0, 1) (0, .5, .5) (0, 1, 0)

1, 1 4, 3 3, 3 5, 4(0, 1, 0) (.5, .5, 0) (1, 0, 0) (.5, .5, 0)

3, 4 2, 2 4, 5 2, 2 2, 2(.5, .5, 0) (0, 1, 0) (.5, .5, 0) (0, 0, 1) (0, 0, 1)

state 3

state 3

state 1 state 2

state 1 state 2

state 1 state 2

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

A 2 State Example with Fictitious Play

1, 1 4, 3 3, 3 5, 4(0, 1) (.5, .5) (1, 0) (.5, .5)

3, 4 2, 2 4, 5 2, 2(.5, .5) (0, 1) (.5, .5) (1, 0)

1, 1 4, 3 3, 3 5, 4 4, 4 6, 7(.5, 0, .5) (.5, .5, 0) (1, 0, 0) (.5, 0, .5) (1, 0, 0) (0, .5, .5)

3, 4 2, 2 4, 5 2, 2 7, 6 5, 5(.5, .5, 0) (0, .5, .5) (.5, 0, .5) (0, 0, 1) (0, .5, .5) (0, 1, 0)

1, 1 4, 3 3, 3 5, 4(0, 1, 0) (.5, .5, 0) (1, 0, 0) (.5, .5, 0)

3, 4 2, 2 4, 5 2, 2 2, 2(.5, .5, 0) (0, 1, 0) (.5, .5, 0) (0, 0, 1) (0, 0, 1)

state 3

state 3

state 1 state 2

state 1 state 2

state 1 state 2

(Trajectory)

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

2StatesFictitious250generations.aviMedia File (video/avi)

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

A 3 State Example with Replicator Dynamics

1, 1 4, 3 3, 3 5, 4(0, 1) (.5, .5) (1, 0) (.5, .5)

3, 4 2, 2 4, 5 2, 2(.5, .5) (0, 1) (.5, .5) (1, 0)

1, 1 4, 3 3, 3 5, 4 4, 4 6, 7(.5, 0, .5) (.5, .5, 0) (1, 0, 0) (.5, 0, .5) (0, 1, 0) (0, .5, .5)

3, 4 2, 2 4, 5 2, 2 7, 6 5, 5(.5, .5, 0) (0, .5, .5) (.5, 0, .5) (0, 0, 1) (0, .5, .5) (1, 0, 0)

1, 1 4, 3 3, 3 5, 4(0, 1, 0) (.5, .5, 0) (1, 0, 0) (.5, .5, 0)

3, 4 2, 2 4, 5 2, 2 2, 2(.5, .5, 0) (0, 1, 0) (.5, .5, 0) (0, 0, 1) (0, 0, 1)

state 3

3, 3 5, 4(1, 0,) (.5, .5)

4, 5 2, 2 2, 2(.5, .5) (0, 1) (0, 1)

state 2

state 1 state 2

state 1 state 2

state 1

state 3

state 1 state 2

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

A 3 State Example with Replicator Dynamics

1, 1 4, 3 3, 3 5, 4(0, 1) (.5, .5) (1, 0) (.5, .5)

3, 4 2, 2 4, 5 2, 2(.5, .5) (0, 1) (.5, .5) (1, 0)

1, 1 4, 3 3, 3 5, 4 4, 4 6, 7(.5, 0, .5) (.5, .5, 0) (1, 0, 0) (.5, 0, .5) (0, 1, 0) (0, .5, .5)

3, 4 2, 2 4, 5 2, 2 7, 6 5, 5(.5, .5, 0) (0, .5, .5) (.5, 0, .5) (0, 0, 1) (0, .5, .5) (1, 0, 0)

1, 1 4, 3 3, 3 5, 4(0, 1, 0) (.5, .5, 0) (1, 0, 0) (.5, .5, 0)

3, 4 2, 2 4, 5 2, 2 2, 2(.5, .5, 0) (0, 1, 0) (.5, .5, 0) (0, 0, 1) (0, 0, 1)

state 3

3, 3 5, 4(1, 0,) (.5, .5)

4, 5 2, 2 2, 2(.5, .5) (0, 1) (0, 1)

state 2

state 1 state 2

state 1 state 2

state 1

state 3

state 1 state 2(Trajectory)

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

3StatesReplicator.aviMedia File (video/avi)

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

A 3 State Example with Fictitious Play

1, 1 4, 3 3, 3 5, 4(0, 1) (.5, .5) (1, 0) (.5, .5)

3, 4 2, 2 4, 5 2, 2(.5, .5) (0, 1) (.5, .5) (1, 0)

1, 1 4, 3 3, 3 5, 4 4, 4 6, 7(.5, 0, .5) (.5, .5, 0) (1, 0, 0) (.5, 0, .5) (0, 1, 0) (0, .5, .5)

3, 4 2, 2 4, 5 2, 2 7, 6 5, 5(.5, .5, 0) (0, .5, .5) (.5, 0, .5) (0, 0, 1) (0, .5, .5) (1, 0, 0)

1, 1 4, 3 3, 3 5, 4(0, 1, 0) (.5, .5, 0) (1, 0, 0) (.5, .5, 0)

3, 4 2, 2 4, 5 2, 2 2, 2(.5, .5, 0) (0, 1, 0) (.5, .5, 0) (0, 0, 1) (0, 0, 1)

state 3

3, 3 5, 4(1, 0,) (.5, .5)

4, 5 2, 2 2, 2(.5, .5) (0, 1) (0, 1)

state 2

state 1 state 2

state 1 state 2

state 1

state 3

state 1 state 2

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

A 3 State Example with Fictitious Play

1, 1 4, 3 3, 3 5, 4(0, 1) (.5, .5) (1, 0) (.5, .5)

3, 4 2, 2 4, 5 2, 2(.5, .5) (0, 1) (.5, .5) (1, 0)

1, 1 4, 3 3, 3 5, 4 4, 4 6, 7(.5, 0, .5) (.5, .5, 0) (1, 0, 0) (.5, 0, .5) (0, 1, 0) (0, .5, .5)

3, 4 2, 2 4, 5 2, 2 7, 6 5, 5(.5, .5, 0) (0, .5, .5) (.5, 0, .5) (0, 0, 1) (0, .5, .5) (1, 0, 0)

1, 1 4, 3 3, 3 5, 4(0, 1, 0) (.5, .5, 0) (1, 0, 0) (.5, .5, 0)

3, 4 2, 2 4, 5 2, 2 2, 2(.5, .5, 0) (0, 1, 0) (.5, .5, 0) (0, 0, 1) (0, 0, 1)

state 3

3, 3 5, 4(1, 0,) (.5, .5)

4, 5 2, 2 2, 2(.5, .5) (0, 1) (0, 1)

state 2

state 1 state 2

state 1 state 2

state 1

state 3

state 1 state 2(Trajectory)

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

3StatesFictious250Generations.aviMedia File (video/avi)

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

About this Model

Existence of Symmetric Equilibria for SymmetricStochastic Games?Relation between Replicator Dynamic and Fictitious Playfor Symmetric Stochastic Games?Some Stability Issues on Population Dynamic

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

About this Model

Existence of Symmetric Equilibria for SymmetricStochastic Games?

Relation between Replicator Dynamic and Fictitious Playfor Symmetric Stochastic Games?Some Stability Issues on Population Dynamic

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

About this Model

Existence of Symmetric Equilibria for SymmetricStochastic Games?Relation between Replicator Dynamic and Fictitious Playfor Symmetric Stochastic Games?

Some Stability Issues on Population Dynamic

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

About this Model

Existence of Symmetric Equilibria for SymmetricStochastic Games?Relation between Replicator Dynamic and Fictitious Playfor Symmetric Stochastic Games?Some Stability Issues on Population Dynamic

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Other ‘Evolutionary’ Work in Maastricht

Examining the effects of periodic fitness in replicator dynamics

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Other ‘Evolutionary’ Work in Maastricht

Examining the effects of periodic fitness in replicator dynamics

(Trajectory)

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

CenterCycle.aviMedia File (video/avi)

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Other ‘Evolutionary’ Work in Maastricht

Examining the effects of local replicator dynamics

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Other ‘Evolutionary’ Work in Maastricht

Studying sex choice ovipositioning behavior of parasitoid wasps

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

Introduction Stochastic Games Evolutionary Games Evolutionary Stochastic Games Concluding Remarks

Thanks

Thank you for your attention!Any comment is welcome!

This presentation will be available atwww.personeel.unimaas.nl/F-Thuijsman

Frank Thuijsman, Maastricht University

Population Dynamics in Stochastic Games

IntroductionStochastic GamesEvolutionary GamesEvolutionary Stochastic GamesConcluding Remarks