Duration of courtship effort with memory

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Duration of courtship effort with memory
Robert M Seymour Department of Mathematics Depar
tment of Genetics, Evolution and Environment UCL
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Acknowledgement to
Peter Sozou LSE
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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 dont they mate immediately?

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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 males
quality (Zahavi 1975, Grafen 1990)
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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
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Arthropods Hanging fly (Bittacus apicalis)
Thornhill, R.(1976) Am. Nat 110, no. 974, 529-548
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And
Human courtship can involve a long sequence of
outings, gifts.
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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 males type A priori probability that a random
male is good P
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Good male Bad 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

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The model game tree per round
One game round - repeated until mate or quit
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The model costs and benefits
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Mating immediately
The females expected payoff from mating
immediately is
Assume P is sufficiently large so that
The female gets a positive payoff from mating
immediately
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The female doesnt 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 doesnt quit first
Can assume that the female never quits
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Pure strategies
tG gt tB gt0
There are no non-trivial equilibria in pure
strategies
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The equilibrium mating strategy
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 females mating rate is
constant
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A good male never quits
A good male always gets a positive expected
payoff from not quitting
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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
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Bad male quitting strategies
A bad males quitting rate q(t) is assumed to be
conditioned on time (or equivalently, cost)
Associated probability of survival function is
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The females expected payoff
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The females best response
For a given bad male quitting rate function q(t),
the females best response mating strategy ?
maximizes her payoff EF(?)
which defines a maximum of EF(?)
Equivalently
which defines a minimum of F(?)
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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
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Example 2 increasing impatience
F(?)
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Example 3 fading memory
F(?)
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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 ?
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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
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Perfect quitting rate
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Maximum length of memory
Length of memory T
For equilibrium to be possible the memory cannot
be too long
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Completing a perfect memory
with f(?) a positive function defined for ? ? 0
F(?) is monotonically decreasing and is minimized
at ? ?
Females best response is to mate immediately
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Bounds for F(?)
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Minimum of F0(?)
This occurs at
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best response curve ?(T)
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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 females 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

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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 ?.
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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
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Probability that bad male quits during the
perfect memory phase
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Pure strategies in the memory game
Male pure strategy quitting time tG or tB
Female pure strategy mating time tm
0 lt tG ? tB
In all cases the female does better to mate
immediately