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Evolutionarily Stable Strategies (ESS)

Evolutionarily Stable Strategies (ESS). Chia-Yi Hu. 2009/10/9. ESS (Evolutionarily Stable Strategies). A strategy which is adopted by most members of a population, it cannot be invaded by the spread of any rare alternative strategy (Smith 1972) 1. Pure strategy

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Evolutionarily Stable Strategies (ESS)

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  1. Evolutionarily Stable Strategies (ESS) Chia-Yi Hu 2009/10/9

  2. ESS (Evolutionarily Stable Strategies) • A strategy which is adopted by most members of a population, it cannot be invaded by the spread of any rare alternative strategy (Smith 1972) 1. Pure strategy In condition c, each individual only play strategy A 2. Mixed strategy In condition c ,each individual play strategy A with probability pA, B with PB, C with PC, etc. P. 31

  3. Two Basic Types of ESS 1. Pure ESS All individuals adopt the fixed pure strategy 2. Mixed ESS All the strategies have equal fitness payoffs a. Each individual would consistently play only one pure strategy b. Individuals playing strategies randomly within the ESS probabilities P. 33-34

  4. Types of Biological Games 1. Contests Direct aggressive interaction, 2-person 2. Scrambles Indirect competition, n players  One-option scrambles (Ex. Parent-offspring conflict)  Alternative-option scrambles (Ex. Producer-scrounger, Ideal free searching) P. 34-35

  5. The Logic of ESS Analysis I : fixed strategy in a population J : any rare alternative strategy to I E(J, I) : the expected payoff to an opponent that plays J against an opponent that plays I E(I, I) : the expected payoff of I played against itself p : the frequency of strategy I q : the frequency of strategy J W0 : a constant which ensures fitness cannot be negative Fitness of I : W(I) = pE(I, I) + qE(I, J) + W0 Fitness of J : W(J) = pE(J, I) + qE(J, J) + W0 P. 36-37

  6. 1. ESS for Contests (Maynard Smith) Assumption: Strategy I is an ESS ifW(I)>W(J) for each alternative J is rare (0<q<<1) ESS 1st condition: E(I, I)>E(J, I) What if E(I, I)=E(J, I)? ESS2nd condition: If E(I, I)=E(J, I) E(I, J)>E(J, J) P. 36-37

  7. 2. ESSfor Scrambles (Hammerstein) W(J, I) : the fitness of a single J strategist in a population of I strategists W(J, Pq, J, I) : the fitness of a J strategist in a population that contains qJ + (1-q)I individuals ESS 1st condition : W(I, I) > W(J, I) ESS 2nd condition : If W(I, I) = W(J, I) for small values of q W(I, Pq, J, I) > W(J, Pq, J, I) P. 37

  8. Animal Contests 1. War of attrition models Two contestants compete for a resource V by persisting while constantly accumulating costs over the time t 2. Hawks-doves models The winner is the individual who injures his opponent in escalated fighting 3. Information acquired during a contest 1. Signals of intent 2. Information about resource value V 3. Information acquired in a contest P. 37-38, 47-48

  9. Model 1. War of attrition (continuous symmetric) V (benefit) : the value of winning c (cost) : the rate at which costs are expended (at time t cumulative costs are ct) Pure strategy T : a population that all play specific waiting time Pure ESS E(T, T) = V/2 – cT If cT < V/2 mutant strategy playing t > T will spread IfcT ≥ V/2 mutant strategy playing t = 0 will spread NO PURE ESS P. 38-39

  10. Model 1. War of attrition (continuous symmetric) V (benefit) : the value of winning c (cost) : the rate at which costs are expended (at time t cumulative costs are ct) Pure strategy T : a population that all play specific waiting time Mixed strategy I : plays a probability distribution of times t, p(t) Mixed ESS p(t) = (c/V)e(-ct/V) P. 38-39

  11. Model 2. Hawks-doves game (discrete symmetric) H (Hawk): fight as escalated level, retreat only if injured D (Dove): retreat immediately if opponent escalates, otherwise settle quickly without escalation V(benefit):the value of winning C(cost):the cost of injury P. 39-40

  12. Model 2. Hawks-doves game (discrete symmetric) Pure ESS When C < V E(H, H) > E(D, H)  Pure Hawk Mixed ESS When C > V Payoff to a hawk must equal to a dove ∵ p[(V-C)/2] + (1-p)V = p*0 + (1-p)V/2 ∴p = V/C P. 40

  13. Symmetric v.s. Asymmetric (Parker 1981) • Payoff-relevant asymmetries 1. Resource value 2. Resource-holding potential (RHP) • Payoff-irrelevant asymmetries 3. Uncorrelated asymmetries (Ex. Owner-interloper) P. 42

  14. Model 3. War of attrition (continuous asymmetric) pA(t) : probability distribution when in role A pB(t) : probability distribution when in role B (t values in pB(t) are smaller than pA(t) ) V : the value of winning c : rate of fitness expending Mixed ESS pA(t) = (c/V)e(-ct/V) P. 44-46

  15. Model 3. War of attrition (continuous asymmetric) V : the value of winning c : rate of fitness expending The winning role A is having the high value for V/c, and role B with lower value for V/c Va/ca > Vb/cb When RHP(size) are the same ca = cb Va > Vb When resource value are the same Va = Vb ca < cb P. 44-46

  16. Model 4. Hawks-doves game (discrete asymmetric) Bourgeois strategy Fixed strategy I : play hawk in A (owner), play dove in B (interloper) Alternative strategy J : play dove in A with pA, play hawk in B with pB E(J, I) = ? P. 42-43

  17. Model 4. Hawks-doves game (discrete asymmetric) ∴ E(J, I) = (V/4)(1+pA) + pB[(V-C)/4] When C > V pB[(V-C)/4] is negative ∵ (V/4)(1+pA) < V/2 E(I, I) = V/2 ∴E(J, I) < E(I, I)  I strategy is an ESS P. 42-43

  18. Types of Biological Games 1. Contests Direct aggressive interaction, 2-person 2. Scrambles Indirect competition, n players  One-option scrambles (Ex. Parent-offspring conflict)  Alternative-option scrambles (Ex. Producer-scrounger, Ideal free searching) P. 34-35

  19. Model 6. Parent-offspring conflict One-option scramble Parents are in a scramble against other parents, and offspring scramble against offspring m : the value that the parent invest in each offspring x : the level of solicitation that the offspring can choose S(x) : survival rate in offspring P. 51-54

  20. Model 6. Parent-offspring conflict Optimal investment for mother : f′(mp) = f(mp)/mp ESS for the offspring : f′(mo) = f(mo)/2mo mo is always greater than mp P. 51-54

  21. Model 6. Parent-offspring conflict m* / x*: fixed strategy to parent / offspring m / x : alternative strategy to parent / offspring Parent alter m* m Offspring will alter from x*  x* (m/m*) • If m↑, then x*(m/m*)↑ • If the parent pays out more, expenditure on the solicitation is reduced Offspring alter x*  x Parent will alter from m*  m*(x/x*) • If x↑, then m*(x/x*)↑ • If the offspring soliciting more, the parent pays more ? P. 51-54

  22. Model 7. Ideal free searching Alternative-option scramble 1. The animal has alternative options (different parts of the environment) to search for some fitness-related commodity 2. The value of each option declines as competitors increases Payoff : resource items are captured within a patch 1. No resource renewal 2. Continuous resource renewal P. 55-56

  23. Model 8. Producer and scroungers Producer : invest time and energy in guarding or creating some resource Scrounger : parasitize Frequency-dependent P. 56-57

  24. Model 9. Phenotype-limited ESS P (Producer) : invest time and energy in guarding or creating some resource S (Scrounger) : parasitize s : size p(s): frequency distribution with s Payoffs always increase with size P. 56-59

  25. Model 9. Phenotype-limited ESS P (Producer) : invest time and energy in guarding or creating some resource S (Scrounger) : parasitize s : size P. 56-59

  26. Thank You for Paying Attention

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