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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 Presentation
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+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
𝐶+𝐷 𝑇[𝑁 ]
→
𝐶+2𝐷
𝐶+𝐷 𝑆[𝑁 ]
→
2𝐶+𝐷2𝐶 𝑅[𝑁 ]
→
3𝐶
2𝐷 𝑃[𝑁 ]
→
3𝐷
𝐶 𝑓 0→
2𝐶
𝐷 𝑓 0→
2𝐷
RC RR RS
RD RT RP
DC C+ +
Collisional population events
𝑑𝐷𝑑𝑡
= 𝜕𝐷𝜕𝑅𝐷
𝑑𝑅𝐷
𝑑𝑡+ 𝜕𝐷𝜕 𝑅𝑇
𝑑𝑅𝑇
𝑑𝑡+ 𝜕𝐷𝜕𝑅𝑃
𝑑𝑅𝑃
𝑑𝑡
𝑑𝐶𝑑𝑡
= 𝜕𝐶𝜕𝑅𝐶
𝑑𝑅𝐶
𝑑𝑡+ 𝜕𝐶𝜕𝑅𝑅
𝑑𝑅𝑅
𝑑𝑡+ 𝜕𝐶𝜕𝑅𝑆
𝑑𝑅𝑆
𝑑𝑡
4
Collisional population events
𝐶+𝐷 𝑇[𝑁 ]
→
𝐶+2𝐷
𝐶+𝐷 𝑆[𝑁 ]
→
2𝐶+𝐷2𝐶 𝑅[𝑁 ]
→
3𝐶
2𝐷 𝑃[𝑁 ]
→
3𝐷
𝐶 𝑓 0→
2𝐶
𝐷 𝑓 0→
2𝐷
RC RR RS
RD RT RP
𝑓 0𝐶+1𝑅
[𝑁 ] [𝐶 ]𝐶+1𝑆
[𝑁 ] [𝐷 ]𝐶+1
𝑓 0𝐷+1𝑇
[𝑁 ] [𝐶 ]𝐷+1𝑃
[𝑁 ] [𝐷 ]𝐷+1
𝑑𝐶𝑑𝑡
=( 𝑓 0+𝑅𝑝𝐶+𝑆𝑝𝐷 )𝐶 𝑑𝐷𝑑𝑡
=( 𝑓 0+𝑇 𝑝𝐶+𝑃𝑝𝐷 )𝐷
5
𝑑𝐶𝑑𝑡
=( 𝑓 0+𝑅𝑝𝐶+𝑆𝑝𝐷 )𝐶 𝑑𝐷𝑑𝑡
=( 𝑓 0+𝑇 𝑝𝐶+𝑃𝑝𝐷 )𝐷
𝑑𝑑𝑡
𝑝𝐷=𝑑𝑑𝑡 ( 𝐷
𝐶+𝐷 )=𝑑𝐷𝑑𝑡
(𝐶+𝐷 )−𝐷 𝑑𝑑𝑡
(𝐶+𝐷 )
(𝐶+𝐷 )2
𝑑𝑝𝐷
𝑑𝑡=𝑝𝐶𝑝𝐷 [ (𝑇 −𝑅 )𝑝𝐶+(𝑃−𝑆 )𝑝𝐷 ]
𝑑𝑝𝐶
𝑑𝑡+𝑑𝑝𝐷
𝑑𝑡=0STOP Check that total
probability is conserved
Evolutionary dynamics of demographics
𝑑𝐶𝑑𝑡
=( 𝑓 0+𝑅𝑝𝐶+𝑆𝑝𝐷 )𝐶 𝑑𝐷𝑑𝑡
=( 𝑓 0+𝑇 𝑝𝐶+𝑃𝑝𝐷 )𝐷
¿𝐶𝑑𝐷𝑑𝑡
+𝐷𝑑𝐷𝑑𝑡
−𝐷𝑑𝐶𝑑𝑡
−𝐷𝑑𝐷𝑑𝑡
(𝐶+𝐷 )2
¿𝐶 ( 𝑓 0+𝑇 𝑝𝐶+𝑃 𝑝𝐷 )𝐷−𝐷 ( 𝑓 0+𝑅𝑝𝐶+𝑆𝑝𝐷 )𝐶
(𝐶+𝐷 )2
6
Evolutionary dynamics of demographics
𝑑𝐶𝑑𝑡
=( 𝑓 0+𝑅𝑝𝐶+𝑆𝑝𝐷 )𝐶 𝑑𝐷𝑑𝑡
=( 𝑓 0+𝑇 𝑝𝐶+𝑃𝑝𝐷 )𝐷𝑑𝑝𝐷
𝑑𝑡=𝑝𝐶𝑝𝐷 [ (𝑇 −𝑅 )𝑝𝐶+(𝑃−𝑆 )𝑝𝐷 ]
Consider the example T > R > P > S
𝑑𝑝𝐷
𝑑𝑡=𝑝𝐷 (1−𝑝𝐷 ) [ (𝑇 −𝑅 ) (1−𝑝𝐷 )+(𝑃−𝑆 )𝑝𝐷 ]
> 0
> 0
> 0
> 0> 0
0
pD
1.0
0.5
t4321
7
Evolutionary dynamics of demographics
𝑑𝐶𝑑𝑡
=( 𝑓 0+𝑅𝑝𝐶+𝑆𝑝𝐷 )𝐶 𝑑𝐷𝑑𝑡
=( 𝑓 0+𝑇 𝑝𝐶+𝑃𝑝𝐷 )𝐷𝑑𝑝𝐷
𝑑𝑡=𝑝𝐶𝑝𝐷 [ (𝑇 −𝑅 )𝑝𝐶+(𝑃−𝑆 )𝑝𝐷 ]
Consider the example T > R > P > S
𝑑𝑝𝐷
𝑑𝑡=𝑝𝐷 (1−𝑝𝐷 ) [ (𝑇 −𝑅 ) (1−𝑝𝐷 )+(𝑃−𝑆 )𝑝𝐷 ]
> 0
> 0
> 0
> 0> 0
0
pD
1.0
0.5
t4321
Stable
Unstable
8
Evolutionary dynamics of demographics
𝑑𝐶𝑑𝑡
=( 𝑓 0+𝑅𝑝𝐶+𝑆𝑝𝐷 )𝐶 𝑑𝐷𝑑𝑡
=( 𝑓 0+𝑇 𝑝𝐶+𝑃𝑝𝐷 )𝐷
𝑑𝑝𝐷
𝑑𝑡=𝑝𝐶𝑝𝐷 [ (𝑇 −𝑅 )𝑝𝐶+(𝑃−𝑆 )𝑝𝐷 ]
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 individual’s 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 agent’s 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 → 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
Prisoner’s dilemma
Evolutionary game theory
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