Behavioral Ecology Vol. 12 No. 6: 659-665
© 2001 International Society for Behavioral Ecology
Conflict of interest between sexes over cooperation: a supergame on egg carrying and mating in a coreid bug
a Department of Theoretical Ecology, Ecology Building, University of Lund, S-223 62 Lund, Sweden b Department of Biology, University of Oulu, Box 3000, Fin-90 014 Oulu, Finland
Address correspondence to R. Härdling, who is now at Graham Kerr Building, University of Glasgow, Glasgow G12 8QQ, UK. E-mail: rfh1n{at}udcf.gla.ac.uk .
Received 7 July 2000; revised 15 August 2000; accepted 15 August 2000.
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ABSTRACT |
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In the golden egg bug (Phyllomorpha laciniata Vill. Heteroptera: Coreidae) females lay eggs on the backs of conspecifics, often on courting males. Although the bugs do not provide care to the eggs, this decreases the risk of egg predation. As an effect males carry many eggs which are not their own. The male and female interests are in conflict; females need to find an oviposition site, and male fitness depends on the obtained number of matings. By using a very rare modeling approach, a supergame where the individuals actions change payoffs over time, we show that combinations of reciprocating strategies where males obtain a mating in return for a carried egg can be stable. The value of the mating, to males, is more important than the relatedness to the eggs in gaining their cooperation in carrying eggs. Females may also take advantage of the males without reciprocating. This is especially likely if the probability of future meeting is high and the value of a mating is high for the male. We relate our results to our own data from empirical studies and experiments on the species. In the light of the results we discuss the behavior of the bugs in relation to nuptial gifts. We also discuss the general applicability of the supergame approach.
Key words: cooperation, egg-laying, mating, Phyllomorpha, sexual conflict, supergame.
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONRESULTSDISCUSSIONAPPENDIXREFERENCES |
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Individuals are not expected to help unrelated conspecifics if the costs of helping are larger than the benefits. If the receiver is a relative then the costs of helping should be smaller than the benefits to the receiver times the relatedness (Hamilton, 1963
).
An individual's inclusive fitness depends not only on his own reproductive
output but also on that of his helped relatives. Occurrence of helping among
nonrelatives is often explained as reciprocity if the recipient later repays
the altruism so that both individuals get net benefit
(Trivers, 1971). Here we
theoretically analyze a case where individuals often behave altruistically
without reciprocity. We find that the conflict of interest between individuals
of different sexes may be resolved in such a way that males are
"forced" to help in order to increase their chance to
reproduce.
The species we consider is the golden egg bug (Phyllomorpha
laciniata Vill. Heteroptera: Coreidae). In this species, females lay eggs
almost exclusively on the backs of other individuals
(Jeannel, 1909;
Kaitala, 1996;
Reuter, 1909). In most natural
habitats, the eggs do not survive unless attached to the backs of conspecifics
due to predation (Kaitala,
1996; Kaitala and
Axén, 2000;
Kaitala et al., 2000). The
bugs do not, however, provide any direct care for the eggs. Although the
females do not prefer to lay eggs on males, males carry more than twice as
many eggs as females (Kaitala,
1996). Part of the reason for this is probably that males are
close to females in order to mate, thus they often receive eggs when courting
a female (Katvala and Kaitala, in preparation). Females as well as males
receive eggs when they are in copula and immobile. Then it is impossible to
avoid the eggs and females probably receive most of their egg load in these
situations.
A male's paternity does not seem to influence his willingness to accept
eggs (Kaitala and Miettinen,
1997). In fact, eggs carried by males are often not fertilized by
the carrier, but by unrelated individuals
(Kaitala and Miettinen, 1997;
Miettinen and Kaitala, 2000).
Egg carrying is at the same time costly due to increased predation risk
(Kaitala and
Axén, 2000;
Kaitala et al., 2000; Reguera
and Gomiendo, 1999). Thus male acceptance of eggs and egg-carrying behavior
cannot be easily explained as paternal care
(Kaitala and Miettinen,
1997).
Here we consider the case when males receive eggs during courtship. We
treat the acceptance of eggs as a behavior that the male can control, that is,
it is possible to avoid receiving such eggs while courting the female
(Kaitala and Miettinen, 1997;
Miettinen and Kaitala, 2000).
We ask the question if carrying of unrelated eggs by males may be the result
of adaptive behavior.
During the reproductive period, the bugs are associated with the host plant
Paronychia, sp. (Caryophyllaceae), and often occur in small groups of three to
six individuals although the host plant may be evenly distributed (Kaitala,
unpublished data; Kaitala and
Axén, 2000). The female
reproductive period is 1-2 months, and they lay a clutch of (on average) 1-3
eggs at a time. In the model described below, we will take clutch size to be
always one egg. Mating often lasts more than 10 h, and is likely to be costly
due to increased predation risk (Kaitala
and Axén, 2000), and due to a risk
of receiving eggs dumped by alien females while mating
(Kaitala and Miettinen, 1997).
One mating provides the female with enough sperm to fertilize her eggs for at
least one month (Kaitala, unpublished data). Nevertheless, females mate
repeatedly during the breeding season, and may copulate with several males
between egg layings. Last male sperm precedence is found in other
heteropterans studied (Arnqvist,
1988; Carroll,
1993; Rubenstein,
1989;
Sillén-Tullberg,
1981; Smith, 1979)
and is relevant here because males that have recently mated with the
egg-laying female may accept the eggs because of high expected paternity. When
courting a female, the males often receive eggs they have not fertilized
(Katvala and Kaitala, in preparation).
In this article, we explore the possible strategies for females and males with respect to sexual interest and risk of receiving eggs. The sexual conflict originates from the fact that both sexes are dependent on the presence of each other, but for different reasons: females need to find an oviposition site (a back), and male fitness depends on eggs fertilized. In any such situation, there is a possible trade between the male and the female.
The conflict situation may change over time, as the costs and benefits involved change in response to actions made by the participating male and female. This dynamical aspect of the conflict is usually neglected in theoretical work on sexual conflict, perhaps mainly because of difficulties in introducing different situations into one single game. In the mating conflict in Phyllomorpha laciniata, the assumed last-male precedence of sperm means that the male's situation varies, depending on whether he has mated with the female earlier or not.
To deal with this problem we have here used a supergame approach, which is
a game theory method specially developed to take account of this inherent
force of change (Friedman,
1986). The method has to our knowledge not been used before in
evolutionary models, although there may frequently be reasons to do so. We
will analyze the situation as a repeated asymmetric supergame. A game-theory
analysis is appropriate, as the players should choose strategies in the light
of the opponent's behavior. The game is repeated, that is, in a small mating
group the same male and female will meet repeatedly and play the game, and we
are interested in the long-term viable strategies. Further, in our case the
payoff matrix of males and females are unequal; this is why we have to
consider an asymmetric game. A supergame is a set of static games developed to
represent the way in which a conflict changes over time
(Friedman, 1986). The key idea
is that the outcomes in one game may lead to a new game with different
payoffs.
The model
The supergame Z contains two different subgames
{Z1, Z2} corresponding respectively to
the situations when the egg offered by the female is not the male's own
offspring, and when it may be with a certain probability r. Assume
that a male and a female meet repeatedly and each time, or in each stage of
the supergame, play either Z1 or Z2
(first time Z1). Let S = {C, D}
be the set of pure strategies for the players in the supergame Z.
These pure strategies will be called actions: both players may in each subgame
choose the action to cooperate (C) or defect (D). If the
male cooperates, he accepts to carry the eggs the female offers him. If he
defects he refuses to carry them. If the female cooperates she mates with the
male; if she defects she does not.
Let Mc be the payoff matrix of the male if the female cooperated
in the last game and Md the matrix if the female defected last time,
or if they meet the first time. F is the female's payoff matrix,
which is the same in both Z1 and Z2.
Thus Z1 consists of the payoff matrices {Md, F}
and Z2 consists of {Mc, F} for males and females
respectively.
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In formulating these fitness payoffs, we have already assumed a specific
sequence of events in the mating situation. First the male approaches, and the
female then tries to lay eggs on his back. He may then accept or reject the
eggs. We do not consider the possibility that rejecting eggs is impossible for
the male, that is, pure parasitism by females. Parasitism is possible, but at
least in laboratory conditions males can avoid accepting the eggs
(Kaitala and Miettinen, 1997;
Miettinen and Kaitala,
2000).
After the male has taken his action and shown whether he is cooperating she
decides to mate or not mate with him. With last-male precedence, any sequence
where the mating precedes egg laying will mean that his relatedness to the
subsequent eggs will be high, at least much higher than with the sequence
assumed above. The assumed sequence of events conforms to the normal pattern
(Kaitala and Miettinen,
1997).
Each outcome in one of the subgames provides a payoff to the players and a pointer to the next subgame to consider. Specifically, if the female chooses C in Z1, that is, mates with the male, this means that next time the male and the female will play the subgame Z2. This is because his expected relatedness to the eggs she tries to lay on his back is now higher. Had she chosen D, next game would again have been Z1. In the subgame Z2 if the female chooses D, that is does not mate with the male, then Z1 will be played next time because the males will then not be related to the eggs. We assume that the female will have mated with another male since two stages back in the game. Otherwise if she chooses C, Z2 will again be played next time. The individuals in the imagined group are assumed to meet and play the supergame Z repeatedly. After a game has been played, a new game will take place only with a certain probability q, so that the individuals do not keep on playing indefinitely.
The players' actions are assumed to depend on the history of the game, that
is, outcomes in earlier encounters. Either an actual memory of past meetings
or any ability to deduce earlier behaviors of the opponent could mediate this
dependency. One possibility is that females are able to identify their own
eggs on the back of the opponent for example, by chemical cues This would
reliably show the earlier cooperativeness of the male. Alternatively females
may simply test the cooperativeness by the attempt to oviposit (see the
"alternate choice" model below). Males have some means to
discriminate between individual females, as they court previous mating
partners significantly more often than other females
(Kaitala, 1998). In
male-biased sex ratios, males are more eager to accept eggs from their earlier
mating partners (Kaitala and Miettinen,
1997). If previous mating increases a male's willingness to accept
eggs, in effect his actions in a game are history-dependent. A pure strategy
for a repeated asymmetric supergame is a pair (t, f(
)): an
action t for the first stage of the game and then for each later
stage a function f() that makes the choice of action at that
stage contingent on the history of the game. Two different forms of
history functions will be considered separately. The first case we call
simultaneous choices, when a players action at stage s depends on all
players actions at the preceding stage (s - 1). One stage back there
are four possible histories of the game, 
{CC, CD, DC,
DD} where the player's own choice is listed before the opponent's choice.
In contrast, if choices are alternate a player's action is determined by the
opponent's last action. Males are assumed to always choose C or
D first in every stage, so with alternate choices the female is able
to respond to a male's action in the same stage. We will analyze a game
containing the pure strategies for males and females defined in
Table 1. We will separately
analyze the cases when the players' choices are simultaneous and
alternate.
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Simultaneous choices
To calculate the expected fitness gain in the repeated game between two
players using some of the pure strategies given in
Table 1, first envisage the
game as exemplified in Figure
1a. The players choose their actions simultaneously. Then sum the
payoffs in each stage of the game discounted by the probability of this stage
to be reached. This sum converges for each sex to an expected payoff
W(I, J) to the pure strategy I =
(tI, fI()) playing against the strategy
J = (tJ, fJ()). In general,
 | (1) |
),
[fI(), fJ()]) as the payoff
to the I - player in the game Z(), which is contingent
on history, given the players' actions fI() and
fJ(). In the Appendix, you find the resulting payoff
matrices for males and females for the pure strategies in
Table 1.
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Alternate choices
Each time the male and the female meet, females always have the option to
test the male's willingness to cooperate by trying to lay eggs on his back,
and respond according to the action then taken by the male. An example of the
contrasting outcomes with and without this ability to respond is shown in
Figure 1. The expected payoff
WM to a male I - player in the first game played
with a female J - player will be WM, I
(Z1, [tI,
fJ(tI)]). In later games, the history that
determines the action of the players will for males be the action of the
female in the preceding stage. For the female it is in contrast the action of
the male in the present stage (Figure
1b). We no longer have to assume the female action to depend on
earlier outcomes. Payoff matrices for males and females under the assumption
that choices are alternate are given in the Appendix. As the strategies
S and TFT have the same functions f(), we have
chosen to assume that their behaviors are a little different. A TFT
female simply respond according to the function
fTFT() to the initial action by the male. An
S female on the other hand always starts by responding D to
the first action, irrespective of what the male did. In later games she
follows the function fTFT(). Similarly, a
CTFT female always responds C in the first round
irrespective of the male action and in later games follow
fTFT().
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RESULTS |
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Simultaneous choices
The solution of the game is a pair (I, J) of male and female strategies which is a Nash equilibrium. This means that I must be an optimal choice of males if the females choose J: simultaneously, J must be an optimal choice for females if males choose I (Hofbauer and Sigmund, 1998
;
Maynard-Smith, 1982). The Nash
equilibria that can be found in the repeated game can be divided into three
classes. There is one set of uncooperative solutions, where the sequence of
actions is for both players a series of D. These solutions are
(AllD, AllD), (AllD, S), (S, AllD) and (S,
S). One solution is (TFT, TFT) and here the solution is for both
players a long series of C. The third set of solutions is the pair
(CTFT, S) and (S, CTFT) which means that one player starts
with the move D, but in subsequent games both players always
cooperate (see Appendix). Mathematical conditions for the equilibria to be
stable to invasion by other strategies are given in
Table 2. It should be noted
that the equilibria are not strict, for example, against any cooperative
strategy like TFT, the even more cooperative strategies CTFT
or C will give the same payoff. The same phenomenon occurs in other
repeated games (e.g., Axelrod and Hamilton,
1981). The choice of strategies in the game is to some degree
arbitrary, except that they should reflect a range of behaviors from very
cooperative to very uncooperative ones. Had we chosen to exclude the strategy
AllC, for example, the equilibria (S, CTFT) and (CTFT,
S) would have become strict Nash equilibria (see Appendix). The
biologically interesting question is under which conditions cooperation is a
probable outcome of the game. Our analysis will therefore concentrate on the
question under which biological conditions the cooperative solutions are
unstable to invasion by uncooperative strategies AllD and
S.
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One Nash equilibrium is (TFT, TFT), and females will cooperate as long as q > p/b, that is as long as the probability of a further game is high enough. If the probability is lower than this, then females would be better off if they chose AllD. Males should depart from the cooperative solution and choose AllD if qb(r + h) < c. Before this condition applies however, the more restrictive condition q2rb + qhb < c will be fulfilled, and then males will improve fitness by choosing S. The alternatives AllC and CTFT gives equal payoff against TFT so that this is not a strict Nash equilibrium.
(CTFT, S) can be bettered by males that always defect (choose AllD) if c > qhb + q2rb. With males playing CTFT, females should play S as long as q > p/b.
The solution (S, CTFT) is stable as long as qhb > c - rb. If this condition does not hold, then males could improve their situation by choosing AllD. If males are suspicious, then females should use the strategy CTFT as long as q > p/b. C and CTFT gives equal payoff against S.
Alternate choices
The equilibria are the same in this case: (AllD, AllD), (AllD,
S), (S, AllD), (S, S), (TFT, TFT), (CTFT,
S) and (S, CTFT). The mathematical stability conditions for the
equilibria are listed in Table
2. The solution (TFT, TFT) is now impossible to improve
on for males unless qrb + hb < c. In this case
males should always defect. This condition is more restrictive than in the
case of simultaneous choices. For females the condition for choosing
AllD is q < p/b as before. However, given a
choice TFT by the opponent the players have the alternatives
AllC and CTFT, which gives an equal payoff so that this is
not a strict Nash equilibrium.
The solution (CTFT, S) is now invadable by defecting males if c > q2rb + qhb. Males have the alternative strategy AllC giving equal payoff. Females can improve their situation by the choice AllD if q < p/b. Likewise, (S, CTFT) is an equilibrium given the conditions qhb > c - rb and q > p/b. Also, AllC is a choice that is equivalent for females, they get the same expected payoff with this choice.
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DISCUSSION |
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The solution (AllD, AllD) where both sexes defect is the simplest to explain. If males never carry any eggs then females should not agree to mate with them, and if females never mate with males, then males should not agree to carry any eggs. Remember that we only look at optimal behavior within a mating group, and assume that the female already has mated so that she can in principle fertilize her eggs. This solution corresponds to breakdown of the mating group system. The solutions (AllD, S), (S, AllD) and (S, S) also means that the mating group system has broken down. Their conditions (Table 2) are reversals of conditions for cooperation and can be understood in relation to these (see below). Generally, conditions favoring cooperation in mating groups were that the probability of repeated encounters was high, and that the costs of mating and egg carrying were low. The advantage of male egg carrying increased with relatedness to the eggs and male value of a mating. These results were stable for the case of simultaneous choices as well as alternate, and also if one sex could take a slight advantage of the other and choose not to cooperate in the first round.
The female condition for cooperation was for all equilibria q > p/b. This means that the probability of repeated encounters should be high, and as expected the benefit of laying one egg must always be higher than the cost of a mating. What follows concerns male conditions only. In the case of simultaneous choices, the solution (TFT, TFT) required c < q(rb + hb) for males not to defect. The benefit term is discounted by the probability of a repeated game q, showing the importance of frequent encounters. The average value of a carried egg is rb and the value of a mating is hb, and these two components influence the condition equally in this case. If males always accept to carry eggs, they will end up carrying a lot of eggs they are not related to. However, if the probability of repeated encounter is high, and accepting is the only chance they have to continue to mate they may be forced to keep the system up. The solution (CTFT, S) where females are suspicious and do not mate unless the male has already accepted eggs is stable as long as c < qhb + q2rb. This translates into the biological conditions that male carrying costs are low, the probability of further games is high, the male relatedness to carried eggs is high and that the fitness discount factor h takes a high value. It does when the female mating frequency is low, and male mating costs are low. The condition for males to "forgive" females in this way does not allow for as high carrying costs as (TFT, TFT). Further, for a given q, the condition is more dependent on the value of a mating than that of a carried egg. With suspicious females, the relatedness to carried eggs is relatively unimportant for the male decision whether to accept eggs or not. If the condition does not hold, males should always defect; this leads to the solution (AllD, S), as seen in Table 2.
It is interesting to note that there is a somewhat paradoxical solution
(S, CTFT) if qhb + rb < c. Males are suspicious,
and females are willing to mate in order to get places for the eggs. However,
a female has other options besides laying eggs on courting males so that the
value of a male back may be low for her. Other mating couples can also be
targets for egg-laying females, since mating individuals are unable to resist
receiving eggs laid by alien females
(Kaitala and Miettinen, 1997).
If females in this case choose the strategy AllD, the solution
(S, AllD) will be reached. Second, a male could in this situation try
to avoid egg carrying by leaving the mating group after having mated with all
the females there. Even in this case he will run a risk of receiving eggs
dumped by females while mating. If such roaming male behavior would spread, it
would break up the group structure. The model only captures the question of
stable behavior combinations within a mating group and implicitly assumes that
moving between groups is impossible. The question of when a male might benefit
from leaving the group is treated elsewhere (Kaitala and
Härdling, in preparation)
The case of alternate choices echoes the earlier solutions and has the same equilibria. The difference is that the stability conditions are different and generally make cooperation (TFT, TFT) a more probable outcome (Table 2). This is as expected, as males have less opportunity to get away with defection if the female can take countermeasures at once.
Our treatment of the situation as a sexual game implies that the actions
taken by males and females are voluntary. It is assumed possible for females
to refuse to mate and for males to refuse to accept eggs. Mating interactions
have been studied mainly in the lab, but egg-laying attempts may be
unsuccessful because of male resistance, although males often are still and
accept the eggs. Females may refuse to mate after oviposition
(Miettinen and Kaitala, 2000).
Copulating pairs are probably unable to avoid receiving eggs from alien
females.
Enquist et al. (1998) have
modeled a mating system involving a pair and the female's lover. They showed
that a female can obtain more assistance from her pair mate by being receptive
to other males. The pair male will then provide more such assistance that also
increases his confidence of paternity. The Phyllomorpha case is
similar in that here also female sexuality regulates social behavior. Neither
the model by Enquist et al. nor our model assumes that females prefer any
particular male. Both models further predict that a male that neglects his
female will suffer loss of paternity. This is supported by data that
Phyllomorpha females mate more than once and they may mate with
different males between oviposition bouts
(Kaitala, 1998;
Kaitala and Miettinen, 1997).
The present work focuses on a more specific question than the model by Enquist
et al.: when is reciprocal cooperation expected to be stable in
Phyllomorpha mating groups?
Although b in our model denotes the value of a carried egg,
assumed to be equal for males and females, the benefits of cooperation are
probably not the same for both sexes. A female gets enough sperm from one male
to fertilize all of her eggs for more than a month (Kaitala, unpublished
data). Within the group there are also other options for laying eggs besides
courting males (Kaitala and Miettinen,
1997). There-fore one could suspect that females would not benefit
from additional matings. There are many reasons why multiple mating may be
adaptive for females, for example, bet-hedging or genetic benefits
(Jennions and Petrie, 2000).
However, this model suggests that one reason may be that frequent mating makes
it necessary for males to stay with the female, which is in her interest.
In this interpretation, one can compare the carrying of eggs by males to
nuptial gifts. Nuptial gifts such as feeding of the mate during courtship or
copulation are common in insects and are often held to be sexually selected
because they serve to attract mates or ensure fertilizations
(Choe and Crespi, 1997;
Thornhill and Alcock, 1983).
Egg-carrying in male Phyllomorpha laciniata shares several
characteristics of a nuptial gift: first, it increases the survival of
offspring by providing a high-quality oviposition site. This is the
interesting benefit for females who need to lay eggs. Also, egg carrying is
costly for males and the nurturing male should therefore have a high
confidence of paternity in the carried eggs. The high female mating rate
together with last male precedence of sperm may mean that males can not rely
on the value of one single mating. The only way to ensure offspring is then to
mate repeatedly within the mating group, and then it is necessary to cooperate
and accept to carry eggs. If the male does this, some of the eggs he carries
will be his own, and the proportion depends among other things on the size of
the group and whether females mate randomly. In the end, both sexes benefit
from staying in the mating group.
One solution of the game was that males should cooperate in the first
meeting, even if females never did so. This possibility for females to take
advantage of the male without reciprocating is especially interesting. Under
natural circumstances, males commonly get eggs without mating
(Kaitala, 1998) which is
paradoxical from a naive reciprocal altruism argument
(Trivers, 1971). Our model
shows that a solution where females "test" the cooperative
intentions of the male can be stable. The conditions for this solution (low
carrying costs, high probability of further games, high value of a mating for
the male and less importantly high male relatedness to carried eggs) are not
violated by what we know of the natural conditions. In conclusion, we find
that there is no need to invoke parental care arguments to explain the
egg-carrying behavior of male Phyllomorpha laciniata.
With this new supergame approach it is possible to take into account
structural time dependencies in a game
(Friedman, 1986), which are
present if the payoffs associated with a certain time period or stage depends
on actions in earlier games. This method can often be used as an alternative
to a dynamic programming approach and in that case gives more easily
understood results, and also retains the rigor of a game theoretical analysis
(Hofbauer and Sigmund, 1998;
Maynard-Smith, 1982). It
should be rewarding to use this modeling approach to study the evolution of
cooperation, especially evolutionary stability in games with
pseudo-reciprocity, synergistic effects, and by-product mutualism
(Connor, 1995). As far as we
know, this is the first instance where the supergame approach has been used in
an evolutionary context, although other authors have recently considered cases
where individuals' behaviors influence the probability of further games
(Webb et al., 1999).
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APPENDIX |
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TOPABSTRACTINTRODUCTIONRESULTSDISCUSSIONAPPENDIXREFERENCES |
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We assume that the players have the set of pure strategies (R1,..., R5) in Table 1 to choose from, and order them from one to five as follows: {AllC, AllD, S, TFT, CTFT}. We first assume that the players choose actions simultaneously. The expected payoffs calculated from equation (1) are given by the matrices A for males and B for females. Thus the male using strategy I against a female using strategy J obtains the payoff aij, and the female obtains bji. In our game, strategy I consists of playing the pure strategies (R1,..., R5) with probabilities given by the unit vector ei (i = 1... 5). A pair (I, J) of male and female strategies is said to be a Nash equilibrium in pure strategies if for the corresponding payoffs the following relations hold
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In Figure 1a, the expected
payoff to the female can be mathematically described by the series
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ACKNOWLEDGEMENTS |
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R.H. was supported by a mobility grant from NorFA, and A.K. was supported by the Academy of Finland (project # 42587). We thank Anders Brodin and Juha Tuomi for comments on the manuscript.
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REFERENCES |
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TOPABSTRACTINTRODUCTIONRESULTSDISCUSSIONAPPENDIXREFERENCES |
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