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

2. Peter Sozou LSE Acknowledgement to

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. Blue bird of paradise displays to a female by hanging upside down and vocalising for a prolonged period of time (Frith and Beehler 1998) Courtship over time 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. Human courtship can involve a long sequence of outings, gifts…. And …

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

9. Good maleBad male Species with facultative paternal care Male finds female attractive and will stay and help after mating Male will desert after mating Species with universal paternal care Male is in good condition: likely to be a good provider Male is in poor condition: likely to be a poor provider Species with sexual selection and no paternal care Male is in good condition: likely to be of high genetic quality Male is in poor condition: likely to be of low genetic quality

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

11. The model : costs and benefits Male’s cost per unit time of participating in courtship: x Payoff to good male from mating: A m Payoff to bad male from mating: D m Male Female’s cost per unit time of participating in courtship: Payoff to female from mating with a good male: A f > 0 Payoff to female from mating with a bad male: - C f < 0 Female

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

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

16. A good male never quits quit not quit matenot mate At equilibrium Expected payoff from not quitting = when > 0 since A m > D m 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 A bad male’s quitting rate q(t) is assumed to be conditioned on time (or equivalently, cost) Associated probability of survival function is quitting rate q(t) survival probability s(t) time t

19. The female’s expected payoff Probability that female mates at time t payoff Probability that male is Good Probability that male is Bad 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 payoff

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

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

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

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

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

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

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

27. 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 Maximum endurance time for bad male

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

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

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

31. where is the Laplace transform of

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

33. Minimum of F 0 ( ) This occurs at mating rate memory length T equilibrium mating frequency

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

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

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

39. Conversely where is the Laplace transform of Hence, if E F ( ) = , 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 Male pure strategy: quitting time t G or t B Female pure strategy: mating time t m good male has quit any male has quit no male has quit t In all cases the female does better to mate immediately 0 < t G  t B