evolutionary game theory i: well-mixed populations
DESCRIPTION
Evolutionary game theory I: Well-mixed populations. Collisional population dynamics. Traditional game theory. +T. +R. p D. 1. +R. +S. +. +S. +P. +T. +P. t. 0. Collisional population events. Collisional population events. R C. R R. R S. R D. R T. R P. C. +. +. C. D. C. - PowerPoint PPT PresentationTRANSCRIPT
+R
+R +S
+T
+T
+S
+P
+P
Evolutionary game theory I: Well-mixed populations
1
Collisional population dynamics Traditional game theory
0
pD
1
t
+
2
Collisional population events
3
C
D
πΆ+π· π[π ]
β
πΆ+2π·
πΆ+π· π[π ]
β
2πΆ+π·2πΆ π [π ]
β
3πΆ
2π· π[π ]
β
3π·
πΆ π 0β
2πΆ
π· π 0β
2π·
RC RR RS
RD RT RP
DC C+ +
Collisional population events
ππ·ππ‘
= ππ·ππ π·
ππ π·
ππ‘+ ππ·π π π
ππ π
ππ‘+ ππ·ππ π
ππ π
ππ‘
ππΆππ‘
= ππΆππ πΆ
ππ πΆ
ππ‘+ ππΆππ π
ππ π
ππ‘+ ππΆππ π
ππ π
ππ‘
4
Collisional population events
πΆ+π· π[π ]
β
πΆ+2π·
πΆ+π· π[π ]
β
2πΆ+π·2πΆ π [π ]
β
3πΆ
2π· π[π ]
β
3π·
πΆ π 0β
2πΆ
π· π 0β
2π·
RC RR RS
RD RT RP
π 0πΆ+1π
[π ] [πΆ ]πΆ+1π
[π ] [π· ]πΆ+1
π 0π·+1π
[π ] [πΆ ]π·+1π
[π ] [π· ]π·+1
ππΆππ‘
=( π 0+π ππΆ+πππ· )πΆ ππ·ππ‘
=( π 0+π ππΆ+πππ· )π·
5
ππΆππ‘
=( π 0+π ππΆ+πππ· )πΆ ππ·ππ‘
=( π 0+π ππΆ+πππ· )π·
πππ‘
ππ·=πππ‘ ( π·
πΆ+π· )=ππ·ππ‘
(πΆ+π· )βπ· πππ‘
(πΆ+π· )
(πΆ+π· )2
πππ·
ππ‘=ππΆππ· [ (π βπ )ππΆ+(πβπ )ππ· ]
πππΆ
ππ‘+πππ·
ππ‘=0STOP Check that total
probability is conserved
Evolutionary dynamics of demographics
ππΆππ‘
=( π 0+π ππΆ+πππ· )πΆ ππ·ππ‘
=( π 0+π ππΆ+πππ· )π·
ΒΏπΆππ·ππ‘
+π·ππ·ππ‘
βπ·ππΆππ‘
βπ·ππ·ππ‘
(πΆ+π· )2
ΒΏπΆ ( π 0+π ππΆ+π ππ· )π·βπ· ( π 0+π ππΆ+πππ· )πΆ
(πΆ+π· )2
6
Evolutionary dynamics of demographics
ππΆππ‘
=( π 0+π ππΆ+πππ· )πΆ ππ·ππ‘
=( π 0+π ππΆ+πππ· )π·πππ·
ππ‘=ππΆππ· [ (π βπ )ππΆ+(πβπ )ππ· ]
Consider the example T > R > P > S
πππ·
ππ‘=ππ· (1βππ· ) [ (π βπ ) (1βππ· )+(πβπ )ππ· ]
> 0
> 0
> 0
> 0> 0
0
pD
1.0
0.5
t4321
7
Evolutionary dynamics of demographics
ππΆππ‘
=( π 0+π ππΆ+πππ· )πΆ ππ·ππ‘
=( π 0+π ππΆ+πππ· )π·πππ·
ππ‘=ππΆππ· [ (π βπ )ππΆ+(πβπ )ππ· ]
Consider the example T > R > P > S
πππ·
ππ‘=ππ· (1βππ· ) [ (π βπ ) (1βππ· )+(πβπ )ππ· ]
> 0
> 0
> 0
> 0> 0
0
pD
1.0
0.5
t4321
Stable
Unstable
8
Evolutionary dynamics of demographics
ππΆππ‘
=( π 0+π ππΆ+πππ· )πΆ ππ·ππ‘
=( π 0+π ππΆ+πππ· )π·
πππ·
ππ‘=ππΆππ· [ (π βπ )ππΆ+(πβπ )ππ· ]
Consider the example T > R > P > S
0
pD
1.0
0.5
t4321
Stable
Unstable
1. Enrichment in D because D is more fit than C (T > R and P > S)2. Loss of fitness of D (and of C) owing to enrichment in D (T > P and R > S)3. The fittest cells prevail, reducing their own fitness
Fitness of C Fitness of D
+R
+R +S
+T
+T
+S
+P
+P
Evolutionary game theory I: Well-mixed populations
9
Collisional population dynamics Traditional game theory
0
pD
1
t
+
?CD
10
Self-consistent quantity maximization
?
+?
+?
?DCC
11
Self-consistent quantity maximization
C
D
?
? C D?
?
?
+?
+?
C+R
+R +S
+T
??
??
DC
+T
+S D D
+P
+P
12
Self-consistent quantity maximization
C
D
?
? C D?
?
+R
+R +S
+T
+T+S
+P+P
13
Self-consistent quantity maximization
C
D
+R+R +S
+T
+T+S
+P+P
C D Consider the example T > R > P > S
Individuals attempt to maximize payoff by adjusting strategy
14
Self-consistent quantity maximization
C
D
+R+R +S
+T
+T+S
+P+P
C D Consider the example T > R > P > S
Individuals attempt to maximize payoff by adjusting strategy
15
Self-consistent quantity maximization
C
D
+R+R +S
+T
+T+S
+P+P
C D Consider the example T > R > P > S
Individuals attempt to maximize payoff by adjusting strategy
16
Self-consistent quantity maximization
C
D
+R+R +S
+T
+T+S
+P+P
C D Consider the example T > R > P > S
Individuals attempt to maximize payoff by adjusting strategy
17
Self-consistent quantity maximization
C
D
+R+R +S
+T
+T+S
+P+P
C D Consider the example T > R > P > S
Individuals attempt to maximize payoff by adjusting strategy
D-vs.-D is a stable strategy pair in that neither agent can increase payoff through unilateral strategy change
18
Self-consistent quantity maximization
C
D
+R+R +S
+T
+T+S
+P+P
C D Consider the example T > R > P > S
Individuals attempt to maximize payoff by adjusting strategy
D-vs.-D is a stable strategy pair in that neither agent can increase payoff through unilateral strategy change
Guided to solution D-vs.-D because T > R and P > S
Each individual obtains less-than-maximum payoff (P < T)owing to the other individualβs adoption of strategy D
19
+R+R +S
+T
+T+S
+P+P
Consider example T > R > P > S
Agents try to maximize payoff
Solution := no agent can increase payoff through unilateral change of strategy. E.g., D-vs.-D (T > R and P > S).
Each agent obtains less-than-maximum payoff (P < T) owing to other agentβs adoption of strategy D
Rationality
Nash equilibrium
0
pD
1
t
Consider example T > R > P > S
T, R, P, and S are cell-replication coefficients associated with pairwise collisions
Stable homogeneous steady state, i.e. pD β 1 because T > R and P > S.
Enriching in D reduces fitness of both cell types (because T > P and R > S)
Replicators with fitness
ESS
Evolutionary dynamics providing insight into a related game theory model
Game theory
Prisonerβs dilemma
Evolutionary game theory
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