Duration of courtship effort with memory

1. Duration of courtship effort with memory Robert M Seymour Department of Mathematics & Department of Genetics, Evolution and Environment UCL

2. Acknowledgement to Peter Sozou LSE

3. Courtship as extended bargaining • Courtship between a male and a female is an asymmetric bargaining game extended over time • Time delay is costly • Participation involves costs to both male and female energy, predation risk, opportunity cost of time • Why do they pay these costs? • Why don’t they mate immediately?

4. Courtship over time Blue bird of paradise displays to a female by hanging upside down and vocalising for a prolonged period of time (Frith and Beehler 1998) A male signal, e.g. ornamentation, may be costly and can act as an honest signal of the male’s quality (Zahavi 1975, Grafen 1990)

5. Great Grey Shrike (Lanius excubitor) • A raptor-like passerine bird • Males give prey to females immediately before copulation • Prey are rodents, birds, lizards or large insects • Females select a mate according to the size of the prey offered Tryjanowski, P. & Hromada, M. (2005) Animal Behaviour 69, 529-533

6. Arthropods : Hanging fly (Bittacus apicalis) Thornhill, R.(1976) Am. Nat 110, no. 974, 529-548

7. And … Human courtship can involve a long sequence of outings, gifts….

8. The model : male types There are two types of male: Good males: high quality - a female wants to mate - she gets a positive fitness payoff Bad males: low quality - a female does not want to mate - she gets a negative fitness payoff Either type of male wants to mate with a female - he gets a positive fitness payoff A female does not have complete information about a male’s type A priori probability that a random male is good: P

10. The model : game tree per round game ends M begin next round quit courtship signal F F t accept reject and quit mate solicit new signal One game round - repeated until mate or quit

11. The model : costs and benefits Male Male’s cost per unit time of participating in courtship: x Payoff to good male from mating: Am Payoff to bad male from mating: Dm Female Female’s cost per unit time of participating in courtship: Payoff to female from mating with a good male: Af > 0 Payoff to female from mating with a bad male: -Cf < 0

12. Mating immediately The female’s expected payoff from mating immediately is Assume P is sufficiently large so that The female gets a positive payoff from mating immediately

13. The female doesn’t quit first t female quits Female gets positive expected payoff from mating Either the male will quit first Or the female will mate while she can still get a positive expected payoff Either way she doesn’t quit first Can assume that the female never quits

14. Pure strategies bad male best response bad male quits t female best response tG>tB >0 There are no non-trivial equilibria in pure strategies

15. The equilibrium mating strategy mate not mate quit not quit Expected payoff from not quitting = = 0 At equilibrium a bad male is indifferent between his pure strategies: quitting or not quitting Suppose the female mates with probability p = t At equilibrium the female’s mating rate is constant

16. A good male never quits mate not mate quit not quit At equilibrium Expected payoff from not quitting = when > 0 since Am > Dm A good male always gets a positive expected payoff from not quitting

17. With and without memory With memory Players have an internal clock They know how much the game has cost them at any time All rounds are distinguished Without memory Players cannot track objective time No information is acquired over time All rounds look the same to players Seymour R.M. & Sozou P.D (2009) Duration of courtship effort as a costly signal. J. Theor Biol 256, 1 - 13

18. Bad male quitting strategies quitting rate q(t) survival probability s(t) time t time t A bad male’s quitting rate q(t) is assumed to be conditioned on time (or equivalently, cost) Associated probability of survival function is

19. The female’s expected payoff Probability that male is Good Probability that male is Bad Probability that female mates at time t payoff payoff Probability that female mates at time t, before bad male has quit payoff Probability that bad male quits at time t, before female has mated

20. Scaling transformation: Laplace transform of s(t)

21. The female’s best response Solution * of: Solution * of: For a given bad male quitting rate function q(t), the female’s best response mating strategy  maximizes her payoff EF() which defines a maximum of EF() Equivalently which defines a minimum of F()

22. a constant Example 1: no memory F() Seymour R.M. & Sozou P.D (2009) Duration of courtship effort as a costly signal. J. Theor Biol 256, 1 - 13

23. = equilibrium mating rate Female’s best response curve

24.  = 0.8 q = 1 Example 2: increasing impatience F()

25.  = 0.8 q = 1 Example 3: fading memory F()

26.  for all > 0  Example 4: ‘perfect’ memory Suppose the female is indifferent between all her pure strategies (mating times tm) in response to a bad male quitting rate q(t) This is equivalent the female being indifferent between all her constant mating strategies 

27. Maximum endurance time for bad male Solution with initial condition s(0) = 1 has K = 1 A bad male will definitely have quit when s(t) = 0 This gives a maximum endurance time for a bad male

28. P = 0.2 = 0.2 time t ‘Perfect’ quitting rate

29. Maximum length of memory female can safely mate t bad male has definitely quit There are no viable equilibria with Viable equilibria require Length of memory = T For equilibrium to be possible the memory cannot be too long

30. q(t) s(t) T Tmax F() ‘Completing’ a perfect memory with f() a positive function defined for  0 F() is monotonically decreasing and is minimized at =  Female’s best response is to mate immediately

31. where is the Laplace transform of

32. lower bound upper bound mating rate  Bounds for F()

33. equilibrium mating frequency mating rate  memory length T Minimum of F0() This occurs at

34. probability that bad male quits during the perfect memory phase > * Bad male wants to decrease his quitting rate best response curve *(T) < * Bad male wants to increase his quitting rate

35. probability that bad male quits during the perfect memory phase > * Bad male wants to decrease his quitting rate < * Bad male wants to increase his quitting rate best response curve *(T)

36. Conclusions • There are extended courtship equilibria in which participants can condition their behaviour on time • There are no equilibria in pure strategies • In any such equilibrium neither the female nor a good male quits, and the game ends in mating • The female’s equilibrium strategy is a constant mating rate • There is a ‘perfect’ memory equilibrium in which the female is indifferent between her (pure) mating strategies (constant mating rates) • In this equilibrium a bad male will quit for sure in a finite time • There is a stable equilibrium in which a bad male follows the perfect memory quitting strategy for a finite time, and then adopts some other (possibly memoryless) strategy • There is a high probability that a bad male will quit before the female mates during the perfect memory phase

38. Expected payoff to female from the pure strategy: mate at time t Expected payoff (at time t = 0) to female from the mixed strategy  a constant (independent of t) Female indifference between pure strategies If the female is indifferent between all her pure strategies (mating times) then Hence is constant, independent of . That is, the female is indifferent between all her mixed strategies .

39. where is the Laplace transform of Conversely Hence, if EF() =, a constant (independent of ), then Therefore, taking inverse Laplace transforms is constant, independent of t. That is, the female is indifferent between all her pure strategies

40. Q is an increasing function of T Q is a decreasing function of  Probability that bad male quits during the perfect memory phase

41. Pure strategies in the memory game no male has quit good male has quit any male has quit t Male pure strategy: quitting time tG or tB Female pure strategy: mating time tm 0 < tGtB In all cases the female does better to mate immediately