Evolutionary game theory I: Well-mixed populations

1. Evolutionary game theory I: Well-mixed populations Collisional population dynamics Traditional game theory +T +R pD 1 +R +S + +S +P +T +P t 0

2. Collisional population events

3. Collisional population events RC RR RS RD RT RP C + + C D C D

4. Collisional population events RC RR RS RD RT RP +1 +1 +1 +1 +1 +1

5. Evolutionary dynamics of demographics Check that total probability is conserved STOP

6. Evolutionary dynamics of demographics >0 >0 >0 >0 >0 pD Consider the exampleT > R> P > S 1.0 0.5 t 0 1 2 3 4

7. Evolutionary dynamics of demographics >0 >0 >0 >0 >0 pD Consider the exampleT > R> P > S Stable 1.0 0.5 Unstable t 0 1 2 3 4

8. Evolutionary dynamics of demographics Fitness of C Fitness of D Enrichment in D because D is more fit than C (T > R and P > S) Loss of fitness of D (and of C) owing to enrichment in D (T > P and R > S) The fittest cells prevail, reducingtheir own fitness pD Consider the exampleT > R> P > S Stable 1.0 0.5 Unstable t 0 1 2 3 4

9. Evolutionary game theory I: Well-mixed populations Collisional population dynamics Traditional game theory +T +R pD 1 +R +S + +S +P +T +P t 0

10. Self-consistent quantity maximization +? +? C D ? ?

11. Self-consistent quantity maximization +R +? +T +? +R +S C C C C C C D D D D D D ? ? ? ? ? ? ? ? ? ? +S +P +T +P

12. Self-consistent quantity maximization +T +R +R +S C C D D ? ? ? ? +S +P +T +P

13. Self-consistent quantity maximization Consider the exampleT > R> P > S Individuals attempt to maximize payoff by adjusting strategy +T +R +R +S +S +P D D C C +T +P

14. Self-consistent quantity maximization Consider the exampleT > R> P > S Individuals attempt to maximize payoff by adjusting strategy +T +R +R +S +S +P D D C C +T +P

15. Self-consistent quantity maximization Consider the exampleT > R> P > S Individuals attempt to maximize payoff by adjusting strategy +T +R +R +S +S +P D D C C +T +P

16. Self-consistent quantity maximization Consider the exampleT > R> P > S Individuals attempt to maximize payoff by adjusting strategy +T +R +R +S +S +P D D C C +T +P

17. Self-consistent quantity maximization Consider the exampleT > R> P > S Individuals attempt to maximize payoff by adjusting strategy +T +R D-vs.-D is a stable strategy pair in that neither agent can increase payoff through unilateral strategy change +R +S +S +P D D C C +T +P

18. Self-consistent quantity maximization Consider the exampleT > R> P > S Individuals attempt to maximize payoff by adjusting strategy +T +R D-vs.-D is a stable strategy pair in that neither agent can increase payoff through unilateral strategy change +R +S +S +P D D C C Guided to solution D-vs.-D because T > R and P > S +T +P Each individual obtains less-than-maximum payoff (P < T) owing to the other individual’s adoption of strategy D

19. Evolutionary game theory Evolutionary dynamics providing insight into a related game theory model Game theory +T +R +S +R +S +P pD +T +P 1 Prisoner’s dilemma Consider exampleT > R> P > S Consider exampleT > R> P > S T, R, P, and S are cell-replication coefficients associated with pairwise collisions Agents try to maximize payoff Rationality Replicators with fitness Solution := no agent can increase payoff through unilateral change of strategy. E.g., D-vs.-D (T> R and P > S). Stable homogeneous steady state, i.e. pD → 1 because T > R and P > S. ESS Nash equilibrium Each agent obtains less-than-maximum payoff (P < T) owing to other agent’s adoption of strategy D Fortune cookie Enriching in D reduces fitness of both cell types (because T > P and R > S) t 0