+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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