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