by
University of Michigan
May 15, 1994
On the Evolution of Altruistic Ethical Rules for Siblings
Theodore C. Bergstrom
‘‘If you have remarked errors in me, your superior wisdom must pardon them. Who
errs not while perambulating the domain of nature? Who can observe everything
with accuracy? Correct me as a friend and I as a friend will requite with kindness.’’
--Linnaeus
ABSTRACT--This paper explores the evolutionary foundations of altruism among siblings and
extends the biologists’ kin-selection theory to a richer class of games between relatives. We
show that a population will resist invasion by dominant mutant genes if individuals maximize a
‘‘semi-Kantian’’ utility function in games with their siblings. It is shown that a population that
resists invasion by dominant mutants may be invaded by recessive mutants. Conditions are found
under which a population resists invasion by dominant and also by recessive mutants. (JEL C70,
D10, D63)
Most economic models are inhabited by selfish consumers, although in the ‘‘economics of the
family’’ it is permissable for parents to care about their offspring and perhaps also their spouses.
Typically these models treat preferences for selfishness or altruism as axiomatic rather than deduced
from a more fundamental model of human nature. In recent years there have been interesting
efforts to find deeper roots for consumer theory, drawing conclusions about the nature of human
preferences from the evolutionary history of our species. Gary Becker’s (1976) investigation of the
evolutionary foundations of the family and Jack Hirshleifer’s engaging manifesto (1978) on behalf
of an evolutionary basis for preferences were pioneering efforts. Robert Frank (1988) discussed
the evolutionary origins of emotions, Ingemar Hansson and Charles Stuart (1990) and Alan Rogers
(1994) discussed the evolution of time preference and attitudes toward saving and work effort,
Arthur Robson (1992a,1992b) suggested a theory of the evolution of preferences toward risk, and
H. Peyton Young (1993) developed an evolutionary theory of social conventions.
Evolutionary biologists such as William Hamilton (1964a, 1964b), Richard Dawkins (1976),
John Maynard Smith (1978), and Robert Trivers (1985) have investigated the theory of kinselection--the evolution of altruistic behavior between close relatives.
Dawkins’ evocative
expression of this view in The Selfish Gene, is that in evolution, the replicating agent is the gene
rather than the animal. Since a gene carried by one animal is likely to appear in its relatives, it
follows that a gene that makes an animal help its relatives, at least when it is cheap to do so, will
1
prosper relative to genes for totally selfish behavior. These ideas have also received attention from
economists. Gordon Tullock (1978) disputed the kin-selection argument, while Ted Frech (1978)
and Paul Samuelson (1983) rebut Tullock’s argument.
This paper is intended to be a contribution both to economics and to evolutionary biology,
but it is written with economist readers in mind. At the most direct level this paper explores the
economics of the family , offering an evolutionary explanation for the degree of altruism to be
expected between siblings. Apart from its direct applications, the biological model of kin selection
is likely to interest economists for its own sake. This is an elegant logical structure, sufficiently
similar to models of economic equilibrium to strike chords of familiarity, yet sufficiently different
to inspire fresh ways of thinking about economic and social problems. The synthesis of Darwinian
evolution with the Mendelian model of diploid inheritance is a rich prototype for cultural evolution,
where norms and culture arise as people copy the actions of others to whom they are related
through social rather than biological structures.
Several recent papers by game theorists (e.g. Daniel Friedman (1991), Ken Binmore and
Larry Samuelson (1992), Michihiro Kandori, George Mailath, and Rafael Rob (1993) and Young
(1993)) have proposed evolutionary foundations for solution concepts in the theory of games.
In these models, individuals play strategies against randomly selected opponents. Equilibrium is
a rest point of a dynamic process in which successful strategies are replicated more frequently
than unsuccessful strategies. In the current paper, the objects of selection are genes rather than
strategies. Because the players in games between relatives are likely to have similar genetic
makeup, it becomes necessary to model the Mendelian genetics of a sexually reproducing species
in more detail than has been attempted in previous economics papers.
As a contribution to evolutionary theory, this paper extends Hamilton’s model of kin-selection
from the special case of linear costs and benefits to a much richer class of games allowing nonlinear
interaction of individual actions, costs and benefits. It also extends the work of John Maynard
Smith and G. R. Price (1973) and Maynard Smith (1974, 1982) on evolutionary games to the case
where the players are relatives. Unlike most previous efforts to apply the theory of evolutionary
games to interaction among siblings, this paper makes explicit use of the Mendelian combinatorics
of diploid sexual reproduction and addresses the possibility of invasion by recessive as well as
dominant mutations.
1. The Game, the Players, and the Method of Reproduction
2
Strategies and the Game Siblings Play
This paper considers interactions between pairs of siblings playing a symmetric, two-person game.
Individuals are assumed to be able to distinguish their siblings from others, so that the strategies
they play in games with their siblings can be determined independently of the strategies they play
with non-siblings. There is a set
sibling plays strategy
played
is given by
of possible strategies for each individual such that when one
and the other plays strategy
, the payoff to the sibling who
and the payoff to the sibling who played
is
.
Individuals do not consciously choose strategies. Instead, their actions are programmed by
their genetic structure. Natural selection acts on the distribution of strategies in the population. The
probability that an individual survives to reproduce is higher, the greater the average payoff that it
receives in the games it plays with its siblings. Offspring tend to be like their parents, according to
the rules of Mendelian inheritance for sexual reproduction.
This paper is concerned with large populations for which it is assumed that expected proportions
are always realized. This makes it possible to treat the dynamical system as a deterministic system
of difference equations. Mating is assumed to be monogamous, so that siblings have two parents
in common, and mate selection is assumed to be random with respect to the genes that control
behavior toward siblings. It is assumed that an individual either dies without having any offspring
or survives to mate and have exactly
offspring.
Sexual Diploid Reproduction and Stable Monomorphic Equilibrium
In sexual diploid species like our own, every individual carries two genes at each genetic locus.
One of these genes is inherited from each parent. This gene is drawn randomly from the two genes
that the parent carries at the corresponding locus. Following common practice in the biological
literature, we use a ‘‘single locus model’’ in which the trait of interest is determined by the two
gene copies in one genetic locus. An individual’s genotype is specified by the contents of this
locus. An individual with two identical gene copies is said to be a homozygote. An individual with
two different kinds of genes in this locus is said to be a heterozygote. Let
and
denote two
different genes that could appear in the locus that determines behavior toward siblings. Suppose
that type
homozygotes choose strategy
heterozygotes of genotype
the gene
and type
homozygotes choose strategy
choose strategy , the gene
is said to be recessive (to ).
3
. If
is said to be dominant (over ) and
A population consisting of homozygotes, all of the same genotype, is called a monomorphic
population. This paper will be mainly concerned with stable monomorphic equilibrium. A stable
monomorphic equilibrium is a monomorphic population in which any mutant genes that might
appear would reproduce less rapidly than the predominant type of genes.
2. ‘‘Kantian’’ Ethical Rules, and Resistance to Dominant and Recessive Mutations
There is a simple and appealing intuitive explanation for the fact that in equilibrium, siblings will
not be purely selfish with each other. Evolution is a blunt instrument that does not change the
genetic structure of organisms in isolation. The Mendelian laws of inheritance imply that with
probability 1/2, a gene that influences my behavior will have the same influence on my sibling’s
behavior. Those individuals who are programmed to treat their siblings kindly are more likely than
the average member of the population to receive kindness from their siblings.
A brief look at the simpler case of evolutionary kin selection in a species with asexual
reproduction will make it easier to grasp the logic of kin selection in sexually reproducing species.
Asexual Reproduction and the Kantian Golden Rule
Consider a population in which every individual that survives to reproductive age produces
two daughters. The daughters play a symmetric game with each other and the strategy that each
daughter plays is controlled by a single gene, inherited from her mother. Except in the case of rare
mutations, each of the two sisters will choose the same strategy as her mother did in the game she
played with her sister. The probability that an individual survives to produce daughters of her own
is higher, the higher the payoff she receives in the game played with her sister. Where
is the set
of possible strategies, suppose that there is a strategy
for all
in
such that
such that
. Then we claim that the only stable equilibrium population is one in which
every individual uses the strategy .
In a population where all individuals are programmed to take action , consider a mutant child
who takes action
. If
, then the mutant will get a higher payoff and
be more likely to survive than normal individuals. But although this genetic deviation increases
her own survival probability, it will be harmful to her progeny. Each of the mutant’s daughters
will, like their mother, take action , and therefore each will get a payoff of
.
Not only will the mutant’s daughters get lower payoffs and have lower survival probabilities than
4
the daughters of normal -strategists, but so will her granddaughters and of all their descendants.
Therefore the line of -strategists will reproduce less rapidly than the line of -strategists and will
ultimately become arbitrarily small as a fraction of the population. It follows that a monomorphic
population of -strategists cannot be invaded by a mutant using a different strategy.
Similar reasoning shows that in any population that does not consist entirely of -strategists,
the offspring of mutant individuals whose genes instruct them to take strategy
will replicate
more rapidly than the other types in the population.
It follows that in equilibrium, asexually reproducing sisters would act toward each other as if
they were guided by an operationalized version of Immanuel Kant’s Categorical Imperative.
The Kantian Golden Rule for Asexual Siblings.
Careful observers of human siblings will not be surprised to find that in sexually reproducing
species, equilibrium behavior is not so perfectly cooperative.
Sexual Reproduction, Dominant Mutants, and the Semi-Kantian Golden Rule
Consider a monomorphic population of individuals of genotype
strategy
with its siblings. Suppose that a mutant gene
(i.e., who carry one mutant
gene and one normal
mating is assumed to be random, so long as the
certainly mate with normal
, each of whom plays the same
appears and that individuals of genotype
gene) play the strategy
gene is rare, surviving
. Since
genotypes will almost
genotypes. Therefore the beachhead on which the mutant gene
will win or lose the battle for entry into the population is a family in which one parent is of type
and the other is of type
. The mutant gene will be eliminated from the population if the
expected payoff received by
genotypes born to such families is lower than that received by
normal offspring of normal parents. The mutant gene will ultimately achieve at least some positive
representation in the population if the expected payoff to
genotypes in such families exceeds
that received by normal offspring in normal families.
If one parent is of genotype
and the other is of genotype
will with probability 1/2 be of genotype
a family, if an individual is of genotype
probability 1/2, and of genotype
an
, then each of their offspring
and with probability 1/2 be of genotype
. In such
, then each of its siblings will be of genotype
with
with probability 1/2. In the game it plays with each sibling,
offspring will take action . With probability, 1/2, the sibling will also be of genotype
5
and take action
and with probability 1/2, the sibling will be of genotype
Therefore in each game it plays with a sibling, an
and take action .
genotype born to one
and one
parent
has an expected payoff of:
The normal population consists of pairs of siblings of genotype
expected payoff is
. It follows that when
, who take action . Their
genes are rare, the mutant
genotypes reproduce more rapidly than the normal population if
. Therefore a
necessary condition for a monomorphic population of -strategists to resist invasion by dominant
mutants is that
for all
defining property for the strategy
function
. This condition is familiar to game theorists as the
to be a symmetric Nash equilibrium for the game with payoff
. Similar reasoning shows that if
mutant gene
such that
for all
genotypes take action
, then a rare dominant
will be eliminated by natural selection.
Therefore a sufficient condition for a monomorphic population of -strategists to resist invasion
by mutant heterozygotes is that
for all
. A strategy
that satisfies this
condtion is said to be a strict symmetric Nash equilibrium for the game with payoff function
We will call the function
the semi-Kantian payoff function, since the payoff to an -strategist
in a population of -strategists is an average of the ‘‘Kantian ’’ payoff
payoff
.
and the ‘‘Nashian’’
. Stated in literary language, a necessary condition for a monomorphic population
to resist invasion by dominant mutants is that siblings abide by the precept:
The Semi-Kantian Golden Rule for Sexual Diploid Siblings.
In more formal dress, our first result is:
Proposition 1.
6
strict
Invasion by Recessive Mutants--When Mutants Mate with Mutants
Is it possible for a monomorphic population that resists invasion by dominant mutant genes to be
invaded by recessive mutant genes? In the special class of games studied by Hamilton (1964),
the answer is ‘‘No. ’’ But for games in which there are complementarities and substitutabilities
between the players’ actions, the answer is ‘‘Yes.’’ The reason that recessive mutants can
sometimes invade although dominant mutants can not turns out to quite instructive and interesting.
Let
be the normal gene and let
is a normal
be a recessive mutant gene. In a family where one parent
genotype and one parent is of genotype
genotypes or heterozygotes of genotype
behave in the same way as normal
genotype
. Since
, the offspring will either be normal
is recessive, individuals of genotype
genotypes. Therefore if at least one parent is of normal
, all offspring will take the same actions and receive the same payoffs as normal
offspring of two normal
include some
parents. Only if each parent has at least one
gene can their offspring
genotypes, and only then can payoffs to some offspring differ from those
received by normal offspring of normal parents.
In the Appendix we prove that when mating is random and when mutant
almost all of the
genes in the adult population are carried by
genes are rare,
genotypes rather than by
genotypes. Formally, this result is stated as follows:
Lemma 1.
It follows that when
genes are rare, almost all
both parents are of genotype
genotypes are born to families in which
. Therefore a necessary condition for a population of
genotypes
to be resistant to invasion by recessive mutants is that the expected rate of reproduction of
genes in families where both parents are of genotype
reproduction of normal genes in the population.
7
is no greater than the expected rate of
In a population of
double recessive
genotypes that use strategy , let
genotypes use strategy . In the Appendix, we calculate the expected payoff
gene found among the offspring of two parents
received by the carrier of a randomly selected
of genotype
payoff
be a recessive mutant gene, such that
. This expected payoff turns out to be a linear combination of the ‘‘Kantian’’
, the ‘‘Nashian’’ payoff
and an ‘‘altruistic’’ payoff,
which is the
payoff that a normal sibling of a mutant -strategist will receive. This expected payoff differs
from the semi-Kantian payoff function in the relative weights accorded to the Kantian payoff and
the Nashian payoff, but also differs because some weight is placed on the ‘‘altruistic’’ payoff. The
altruistic payoff gets some weight because where
in the offspring of two
normal action
is a recessive gene, some of the
parents are carried by individuals who are of genotype
, but whose siblings are of genotype
genes found
and take the
and take the mutant action .
Proposition 2.
strict
Second-Order Invasion by Dominant Mutants
If
is a symmetric Nash equilibrium, but not a strict symmetric Nash equilibrium, for the
semi-Kantian payoff function
, there remains one avenue by which dominant mutants can invade
a population of -strategists. Suppose that
genotypes take the Nash equilibrium action
carriers of the mutant gene
, but suppose that
of the mutant
take action
. Then carriers
gene will have the same survival probability as normals, so long as they have one
parent of genotype
and one of genotype
genes must depend on whether mutant
. Therefore the success or failure of an invasion by
genes reproduce faster or slower than normal genes in
families where the parents have a total of two or more
the parents have two or more
taking action
and
genes and
genes. As we show in the Appendix, if
, then dominant mutant genes for
will reproduce less rapidly than normal genes if
8
. This enables
us to state a ‘‘second-order sufficient condition’’ that is weaker than the assumption that
strict Nash equilibrium for
is a
.
Proposition 3.
3. Hamilton’s Inclusive Fitness, Additive Games, Complementarity and Substitutability
Evolutionary biologists have long been aware that natural selection can favor individual behavior
that reduces one’s own survival probability if there is sufficient increase in the survival probability
of one’s siblings and kin. William Hamilton (1964a,b) formalized this idea by proposing that
evolution would select for behavior that maximizes an individual’s inclusive fitness, where
inclusive fitness is a weighted average of one’s own survival probability and that of one’s kin,
with the weights applied to relatives being proportional to their degree of relationship. Hamilton
states this proposition, which has come to be known as ‘‘Hamilton’s Rule’’, as follows:
‘‘ The social behavior of a species evolves in such a way that in each distinct
behavior-evoking situation the individual will seem to value his neighbors’ fitness
against his own according to the coefficients of relationship appropriate to that
situation.’’ (W. Hamilton, 1964b, p 19.)
The coefficient of relationship between siblings is 1/2. In a symmetric game between two
siblings, if Sibling 1 takes action
of
and Sibling 2 takes action , then Sibling 1 will have a payoff
and Sibling 2 will have a payoff of
when one’s sibling takes strategy
. Therefore the inclusive fitness of strategy
is defined to be:
Stated more poetically, Hamilton proposed that in equilibrium, siblings will satisfy:
Hamilton’s Inclusive Fitness Rule for Siblings. ‘‘Love thy sibling half as well as thyself’’.
In a monomorphic population where all individuals are maximizing inclusive fitness, the action
taken by each individual must be a symmetric Nash equilibrium strategy for the game with
payoff function
. In general such a population need not be resistant to invasion by mutants,
but there is an interesting class of games with special additive structure for which a population of
inclusive fitness maximizers can not be invaded either by dominant or by recessive mutants.
9
Hamilton’s additive model
Hamilton’s 1964 papers do not claim that maximization of inclusive fitness characterizes evolutionary equilibrium for all games between relatives. Hamilton specifically restricts his attention to
the class of games in which the interaction between relatives is additive. For symmetric games
between pairs of siblings, Hamilton’s additive model takes the following form. Each sibling has
a set
action
of possible actions. Let
and
, it will confer a benefit of
sibling takes action
be functions defined so that if an individual takes
on its sibling at a cost of
and the other takes action
to itself. Then if one
, the payoff to the sibling who takes action
and the payoff to the sibling who takes action
is
is
.
In Hamilton’s additive model, the benefit one receives from a relative’s action is independent
of one’s own action, and the cost of helping a relative is independent of the amount of help one
receives. This is a strong restriction that is not satisfied in some of the most natural models of
economic interactions between individuals, including models of household production and familiar
examples of two-person games.
Hamilton’s additive model is of special interest because where it applies, the sets of symmetric
Nash equilibria for the three different payoffs,
,
, and
are identical. Therefore for additive
games a monomorphic population of -strategists is resistant to both dominant and recessive
invasion if the strategy
(strictly) maximizes Hamilton’s inclusive fitness for each individual.
Games that are not additive lack this happy coincidence of equilibria. Fortunately, even in the case
of non-additive games, there are clean, simple conditions that determine whether a monomorphic
population can resist invasion by dominant and by recessive mutants.
Symmetric Complementarity and Symmetric Substitutability
The relation between the set of equilibrium strategies under Hamilton’s Rule and the sets of
strategies that resist invasion by dominant and recessive mutants depends critically on whether
there is ‘‘complementarity’’ or ‘‘substitutability’’ between the actions of siblings. Stated loosely,
the presence of complementarity or substitutability depends on whether the total payoff is higher
when both players act similarly or when they act differently. More formally, a two-player
symmetric game will be said to exhibit symmetric complementarity if for any two strategies
and
, the expected total payoff to the two players is greater if one of the two strategies is selected at
random and played by both players, than if one player plays strategy
10
and the other plays strategy
. There is symmetric substitutability if this inequality is reversed.
Definition.
On the Nesting of Equilibrium Sets
It is useful to notice that the function
that represents fitness against recessive invasion is a
simple linear combination of Hamilton’s inclusive fitness function
function
. From the definitions of
,
, and
and the semi-Kantian payoff
, it follows that:
Lemma 2.
If the function
exhibits symmetric complementarity or substitutability, the sets of Nash
equilibria for the three payoff functions
,
, and
following proposition.
Proposition 4.
11
, are conveniently nested as described by the
In Hamilton’s additive model, the payoff function
in ,
has the property that for all strategies and
and therefore
In this special case,
.
exhibits both symmetric complementarity and symmetric substitutability.
It follows from Proposition 4 that in Hamilton’s additive model, a strategy is a symmetric Nash
equilibrium for the payoff function
each of the functions
and
if and only if it is also a symmetric Nash equilibrium for
.
Corollary 1---Hamilton’s Theorem.
Even without additivity, if there is either symmetric complementarity or symmetric substitutability, the equilibria that resist both dominant and recessive invasion can be quite neatly
characterized. If there is symmetric substitutability, then it follows from assertion (i) of Proposition
4 that a symmetric Nash equilibrium for the game with payoffs given by Hamilton’s inclusive
fitness function,
will resist both dominant and recessive invasion. (As we will show by example
in the next section, in the absence of symmetric substitutability, it can happen that even though
a symmetric Nash equilibrium for the game with payoff function
is
, a population of -strategists
can be invaded by dominant mutants.) If there is symmetric complementarity, then it follows from
assertion (ii) of Proposition 4 that if
semi-Kantian payoff function
is a symmetric Nash equilibrium for the game with the
, then a population of -strategists is resistant to invasion by both
dominant and recessive mutants. These results are collected in Corollary 2.
Corollary 2.
12
4. Examples of Games with Symmetric Complementarity and Substitutability
Example 1---Rousseau’s Stag Hunt
In order to demonstrate the game-theoretic approach to coordination problems, Drew Fudenberg
and Jean Tirole (1991) introduced a game called the Stag Hunt. The name of the game and the
story that goes with it are suggested by a passage from Jean Jacques Rousseau’s Discourse on the
Origin and Basis of Inequality among Men, written in 1754. Two hunters set out to kill a stag.
One of them has agreed to drive the stag through the forest, and the other to post at a place where
the stag must pass. If each performs his assigned task, they will surely kill the stag and each will
get a share of the prey worth
. During the course of the hunt, each hunter is tempted to
chase a hare that runs by him. If either hunter pursues his hare, he will catch it, but in so doing, he
will ensure that the stag escapes. An individual who catches a hare gets a payoff of 1. The payoff
matrix for this game is given below, where
where
denotes the strategy, persevere in the stag hunt and
denotes the strategy, desert the stag hunt to pursue the hare.
The Stag Hunt--Individual Payoffs
Sibling 2
S
H
Sibling 1
S
H
The Stag Hunt is an example of a game with symmetric complementarity, since the sum
of the diagonal payoffs exceeds the sum of the off-diagonal payoffs. Played between unrelated
13
individuals, this game presents a coordination problem; there are two Nash equilibria, one of which
is better for both players than the other. In one Nash equilibrium, both hunters faithfully play their
assigned roles and enjoy payoffs of
. In the other Nash equilibrium, each deserts the hunt
and each gets a payoff of 1. Although both players prefer the former equilibrium, if each believes
that the other will run off and pursue a hare, then both will choose to desert the stag hunt. As
Fudenberg and Tirole remark, ‘‘without more information about the context of the game and the
hunters’ expectations it is difficult to know which outcome to predict.’’
Since the Stag Hunt exhibits symmetric complementarity, it follows from Proposition 4 and
its Corollary 2 that a monomorphic equilibrium of
dominant and recessive mutants if
-strategists will resist invasion by both
is a strict symmetric Nash equilibrium for the game with
the semi-Kantian payoff function,
. The payoff matrix for
is
displayed below.
The Stag Hunt--Semi-Kantian Payoffs
Sibling 2
S
Sibling 1
If
H
S
H
, this game has only one Nash equilibrium; the efficient outcome in which both
faithfully pursue the stag. Therefore for
, kin selection ‘‘solves’’ the coordination problem
in the sense that the only population that is resistant to invasion by dominant and recessive mutants
is a population consisting entirely of individuals who pursue the efficient strategy .
Interestingly, this conclusion would have eluded us if we had tried to find the equilibria by
looking for the Nash equilibrium of the game with payoffs given by Hamilton’s inclusive fitness
function,
as displayed in the game matrix below.
14
The Stag Hunt--Inclusive Fitness Payoffs
Sibling 2
S
Sibling 1
For all
H
S
H
, the Stag Hunt with the inclusive fitness payoff function
has two Nash
equilibria--one in which both faithfully hunt the stag and one in which each chases the fleeting
hare. The Stag Hunt with semi-Kantian payoff
has only one Nash equilibrium--the outcome
where both hunt the stag. Since the Stag Hunt is a game with symmetric complementarity, we
know from Proposition 4 that every symmetric Nash equilibrium for the game with payoff
symmetric Nash equilibrium both for the game with payoff
and for the game with payoff
But as this example illustrates, not every symmetric Nash equilibrium for
is a
.
is a symmetric Nash
.
equilibrium for
Example 2---Prisoners’ Dilemma
Consider a single-shot prisoners’ dilemma game with two possible strategies, Cooperate and
Defect. Payoffs are given by the following game matrix, where the parameters satisfy the condition
.
Prisoners’ Dilemma
Sib 1
Sib 2
Cooperate Defect
RR
PP
Cooperate
Defect
As is well known, the only Nash equilibrium for this game is the outcome where both players
15
defect. But when prisoners’ dilemma is played between sexual diploid siblings, a much richer
variety of possibilities emerges. Depending on the parameter values, there may be zero, one, or two
equilibria that resist invasion by mutant genes. Moreover, some equilibria will resist invasion by
dominant mutants and not by recessive mutants. In the following discussion, we find the parameter
ranges for which each of the possible equilibrium configurations obtains.
The payoff matrix for the game with the semi-Kantian payoff function
is:
Prisoners’ Dilemma--Semi-Kantian Payoff Function, V
Sib 2
Cooperate
Defect
Cooperate
Sib 1
Defect
The set of Nash equilibria for this game depends on the parameters
there is a Nash equilibrium in which both players cooperate. If
and . If
,
, there is a Nash equilibrium
in which both players defect. As we show in Figure 1, parameter values can be found for which
one, both, or neither of these inequalities are satisfied. The region
above the line
and all points to the left of the line
of Figure 1 includes all points
. For parameter values in
this region, the unique symmetric Nash equilibrium is the outcome where both siblings cooperate.
For parameter values in region
of the figure, there are two symmetric Nash equilibria, one in
which both siblings cooperate and one in which both defect. For parameter values in region
,
the unique symmetric Nash equilibrium is the outcome where both siblings defect. For parameter
values in region
, there is no symmetric Nash equilibrium in pure strategies.
(Figure 1, about here.)
The prisoners’ dilemma game has symmetric complementarity for parameter values such that
. These parameter values are represented in Figure 1 by points on or above the
diagonal line
. For these parameter values, according to Corollary 2 of Proposition 4, a
monomorphic population that plays a strict symmetric Nash equilibrium for
will not be invaded
either by dominant or by recessive mutants. Therefore for parameter values in the part of Region
16
that lies above the line
, a monomorphic population of cooperators will resist invasion
both by dominant mutants and by recessive mutants, but a monomorphic population of defectors
would be invaded by dominant mutant genes for cooperation.
In Region
, since
and
equilibria for the game with payoff function
in the other both defect. Since Region
there are two distinct symmetric Nash
; in one equilibrium both players cooperator and
lies entirely above the line
, there is
strategic complementarity for parameter values in this region. It follows from Proposition 4 that
for parameter values in Region
, there are two stable monomorphic equilibria--a monomorphic
population of cooperators and a monomorphic population of defectors, each of which resists
invasion both by dominant and by recessive mutants.
In the part of Region
that lies above the line
, a monomorphic population of
defectors would resist invasion both by dominant and by recessive mutants, but a monomorphic
population of cooperators would be invaded by dominant mutant genes for defection.
In the region of Figure 1 that lies on or below the downward-sloping diagonal, there is
symmetric substitutability. According to Proposition 4, in the case of symmetric substitutability,
a monomorphic population thats resist recessive invasion will also resist invasion by dominant
mutants, but there are parameter values such that a monomorphic population which resists invasion
by dominant mutants will succumb to an invasion by recessive mutants. A monomorphic population
resists invasion by recessive mutants if the strategy used by its members is a symmetric Nash
.
equilibrium for the game with payoff function
Calculating these values for prisoners’ dilemma, one finds the payoff matrix:
Prisoners’ Dilemma with Payoff Function W
Sib 2
Cooperate
Sib 1
Defect
Cooperate
Defect
This game has a symmetric Nash equilibrium where both players cooperate if
and it has a symmetric Nash equilibrium where both players defect if
17
. In Figure
2, which displays the parameter values for which there is symmetric substitutability, Region
the set of parameter values for which
. For parameter values in
is
, a monomorphic
population of cooperators will resist invasion both by dominant and recessive mutants. Region
is the set of points for which
. For parameter values in this region, a monomorphic
population of defectors resists invasion both by recessive and by dominant mutants.
(Figure 2 about here.)
The lines making up the upper and lower boundaries of the region
and
have the equations
, respectively. Points above the line
represent
parameter values for which a monomorphic population of cooperators resists invasion by dominant
mutants. The set of all such points is the union of Regions
Region
and
. For parameter values in
, a population of cooperators resists invasion by dominant mutants, but can be invaded
by recessive mutants. Points below the line
represent parameter values for which a
monomorphic equilibrium population of defectors resists invation by dominant mutants. The set
of all such points is the union of the regions
and . For parameter values in Region
, a
monomorphic population of defectors can not be invaded by dominant mutants, but can be invaded
by recessive mutants. Finally, for parameter values in Region
, any monomorphic population
would be invaded both by dominant and by recessive mutants. For these parameter values, there is
no monomorphic equilibrium in pure strategies.
It is interesting to compare these results to predictions that would be obtained using the
symmetric Nash equilibria under Hamilton’s inclusive fitness function
. The inclusive-fitness
payoff matrix for siblings playing prisoners’ dilemma is the following:
Prisoners’ Dilemma--Inclusive Fitness Payoff Function, H
Sib 2
Cooperate
Sib 1
Cooperate
Defect
Defect
3P/2
For this game, there is a Nash equilibrium in which both siblings cooperate if and only if
, and there is a Nash equilibrium in which both defect if and only if
18
. In Figure 1,
the thin vertical line and the thin horizontal line represent the boundaries of these regions. Therefore
Hamilton’s rule partitions the parameter values in Figure 1 as follows: for parameter values in the
small square in the upper left, the only monomorphic equilibrium is a population of cooperators; in
the large square in the lower right, the only monomorphic equilibrium is a population of defectors;
in the rectangle in the upper right, there would be two distinct mononomorphic equilibria, one
with cooperators only and one with defectors only; in the rectangle in the lower left, there would
be no monomorphic equilibria. As we see, from earlier discussion, these predictions are not
entirely correct. Where
Nash equilibrium for
equilibrium for
, every symmetric Nash equlibrium for
and
, but not conversely. Where
is a symmetric Nash equilibrium for
is also a symmetric
, every symmetric Nash
, but not conversely.
Hamilton’s additive model applies only to parameters on the diagonal of Figure 1 where
. For these special parameter values, the Nash equilibria under inclusive fitness coincide
with the populations that resist invasion both by dominant and by recessive mutants. In the additive
model if
, the only monomorphic equilibrium is a population of cooperators, if
the only monomorphic equilibrium is a population of defectors, and if
,
, there will be two
monomorphic equilibria, one with defectors only and one with cooperators only.
5. On the Existence of Stable Monomorphic Equilibrium
Pure Strategy Equilibrium that Resists Both Dominant and Recessive Invasion
We have shown that a monomorphic population of -strategists will resist invasion by dominant
and by recessive mutants only if
payoff function
is a symmetric Nash equilibrium strategy for the game with
and also for the game with payoff function
. It is well known that there
are symmetric games that have no symmetric Nash equilibrium in pure strategies. One such a
game is Maynard Smith’s Hawk-Dove game, Maynard Smith (1982).
But there is a large, readily
identifiable class of symmetric games for which a symmetric, pure strategy Nash equilibrium
exists. The following result is found in many standard references on game theory. (e.g., Rasmusen
(1989, pp 123-127).
Lemma 3.
19
The results of Proposition 4 on the nesting of symmetric Nash equilibria for games with payoff
functions
,
, and
make the following conclusions immediate from Lemma 3.
Proposition 5.
Existence of Stable Monomorphic Equilibrium in Mixed Strategies
Since there are games for which no monomorphic equilibrium of pure strategists exists, it is of
interest to investigate whether there exists a monomorphic equilibrium in which all individuals are
of the same genotype and pursue the same mixed strategy. To study this question, let us confine
our attention to games in which there are a finite number
matrix such that
of possible strategies be the
dimensional simplex,
is a vector whose th component is the probability that the player uses the
th pure strategy. If an individual uses strategy
payoff to the
be an
is the payoff to an individual that chooses pure strategy while its sibling
chooses pure strategy . Let the set
where a strategy in
of pure strategies. Let
and its sibling uses strategy , then the expected
strategist is given by
Although the function
.
is continuous and linear in
for fixed , the semi-Kantian
is not linear, but quadratic in
payoff function
. Therefore,
simply adding the possibility of mixed strategies does not guarantee existence of a stable
monomorphic population of mixed strategists. On the other hand, the inclusive fitness function
is linear in
for fixed
(and linear in
for fixed
). It follows that there exists a symmetric Nash equilbrium in mixed strategies for the game
with payoff function
. Therefore, according to Proposition 4, if the payoff function
20
exhibits
symmetric substitutability, a symmetric Nash equilibrium for
equilibrium for
and for
.
will also be a symmetric Nash
Therefore we can claim the following:
Proposition 6.
In the previously discussed Prisoners’ Dilemma example, the only parameter values for which
there does not exist a monomorphic equilibrium of pure strategists that resists both dominant and
recessive invasions are in the region where there is symmetric substitutability. But according to
Proposition 6, in this case there is a monomorphic population of mixed strategists that resists
invasion by dominant and by recessive mutants. Therefore for all parameter values of the prisoners’
dilemma game, there exists a monomorphic equilibrium resistant to both dominant and recessive
mutants.
6. Differentiable Payoff Functions--A Partial Vindication for Hamilton’s Inclusive Fitness
If the payoff function
is differentiable, then the calculus first-order conditions for a symmetric
Nash equilibrium for the payoff functions
and
are the same as the first order conditions for
a symmetric Nash equilibrium in which the payoff is Hamilton’s inclusive fitness function
.
This means that in the differentiable case, a necessary condition for a monomorphic population
of -strategists to resist either dominant or recessive invasion is that
. However, as we will show, the second-order
equilibrium for the game with payoff function
conditions for Nash equilibrium for games with payoff functions
and accordingly, the condition that
be a symmetric Nash
,
, and
is a symmetric Nash equilibrium for
are not the same
is neither necessary
nor sufficient for a monomorphic population of -strategists to resist invasion by dominant and
recessive mutants.
Let the strategy set
lie in an -dimensional vector space and consider a symmetric game with
a twice continuously differentiable payoff function
to be an interior symmetric Nash equilibrium for is
of
. The first order necessary condition for
, where
is the gradient
with respect to its first argument, . The second order necessary condition is that the
Jacobean matrix
of second-order partials of
is negative semi-definite at the point
.
21
with respect to the first argument
Calculation shows that with a differentiable payoff function , the gradients
and
are proportional to each other ‘‘on the diagonal,’’ where
,
. This implies that
if the first-order necessary condition for symmetric Nash equilibrium is satisfied at
these payoff functions, then these conditions will also be satisfied at
for one of
for the other two. On the
other hand, the second-order conditions for symmetric Nash equilibria may be satisfied for one of
these payoff functions and not for the others. The second-order conditions for a symmetric Nash
equilibria for
and
are respectively that
and
be negative semi-definite
matrices. The relation between these matrices is as follows.
Proposition 7.
Since
, it follows that if
is negative semi-definite,
then the first and second order conditions for a symmetric Nash equilibrium for the game with
payoff function
will be satisfied whenever these conditions are satisfied for the game with
payoff function
. Similarly if
is positive semi-definite, then the first and second
order conditions for a symmetric Nash equilibrium in the game with payoff function
will be
satisfied, whenever these conditions are satisfied for the game with payoff function
. From
Proposition 4, we know that (whether or not
the game with payoff function
function
is differentiable) a symmetric Nash equilibrium for
will be a symmetric Nash equilibrium for the game with payoff
if there is symmetric substitutability and that the converse statement is true if there is
symmetric complementarity. The following calculus characterization of symmetric substitutability
and complementarity should therefore come as no surprise.
Lemma 4.
Proposition 7 and Lemma 4 suggest a practical four-step procedure for finding equilibrium
strategies for those games with differentiable payoff functions that display symmetric substitutability or complementarity. The steps are as follows: (i) Find a solution to the equation
Then it will also be true that
and
22
.
. (ii) Determine whether the game
exhibits symmetric substitutability or symmetric complementarity. This can be done by checking
whether the matrix
is negative semi-definite or positive semi-definite. (iii) If there is
is negative semi-definite. If so, then
symmetric substitutability, then check whether
not only satisfies the first and second order necessary conditions for a symmetric Nash equilibrium
for the game with payoff function
but also for games with payoff functions
If there is symmetric complementarity, then check whether
If so, then
. (iv)
is negative semi-definite.
not satisfies the first and second order necessary conditions for a symmetric Nash
equilibrium for the games with payoff functions
function
and for
and for
as well as for the game with payoff
.
The following example shows how this procedure can be applied to the economics of
partnerships.
7. Economic Partnership Between Siblings--An Application
Two partners work together on a farm. Expected total output depends on the total amount of labor
effort that they contribute. The partners are not able to monitor each others’ efforts and hence they
simply divide total output equally between them. The payoff to each depends positively on the
amount of output he gets and negatively on the amount of effort he expends. Specifically, where
is effort by partner , total output is
is
and the cost to partner of his own labor effort
. Since each partner gets half of the total output, the payoff to partner is
This is a standard incentive problem. Efficiency calls for each partner to work to the point
where the marginal cost of his effort equals its marginal product. But in Nash equilibrium, where
each chooses his effort level independently, a partner gets only half of the marginal product of
extra effort and therefore he works only enough so that the marginal cost of his effort equals half
of its marginal product. The first-order necessary condition
to be a Nash equilibrium effort
level for each partner is:
Now suppose that the partners are siblings and that their effort levels are determined genetically
rather than by selfish maximization. As we discovered in the previous section, with a differentiable
payoff function, a first-order necessary condition for a population of -strategists to resist invasion
23
both by dominant and recessive mutants is that be a symmetric Nash equilibrium strategy for the
game with Hamilton’s inclusive fitness payoffs. Hamilton’s inclusive fitness is given by
The first order necessary condition for in the interior of
for
to be a symmetric Nash equilibrium
is that
Thus, sibling-partners, while not achieving full efficiency would work harder than unrelated
partners.
As we found in the previous section, the second-order necessary conditions for a population
of -strategists to resist dominant and recessive invasion will be different depending on whether
there is symmetric complementarity or substitutability. Calculation shows that
. It follows from Lemma 4 that there is symmetric substitutability if
for all , which is the case of ‘‘non-increasing returns to labor,’’ and there will be symmetric
complementarity if
for all , which is the case of ‘‘non-decreasing returns to labor.’’
Consider the case of symmetric substitutability, where
for all
. According
to Corollary 2 of Proposition 4, when there is symmetric substitutability, if
is a symmetric
Nash equilibrium for
, then a monomorphic population of
and recessive invasion. The strategy
equilibrium with payoff
strategists will resist both dominant
satisfies the second order necessary condition for a Nash
if
(and the second order sufficiency
condition if the last inequality is strict). Since, by assumption
, the second order
sufficiency condition will be satisfied if there is increasing marginal cost of effort, in which case
.
Now consider the case of symmetric complementarity, where
for all
case, according to Corollary 2 of Proposition 4, if is a symmetric Nash equilibrium for
. In this
, then a
monomorphic population of -strategists will resist invasion by dominant and recessive mutants.
From the definition, we have
24
Calculation shows that
Then we find that the second order sufficiency condition is:
In this case there is symmetric complementarity so that
. Therefore the second order
condition will be satisfied only if there are sufficiently strongly increasing marginal costs of labor
effort so that
.
8. Discussion and Extensions
Cultural Evolution
Cultural evolution can be studied as a formal structure similar to that of genetic evolution. Suppose
that individual attitudes, beliefs, skills, and knowledge are copied from ‘‘cultural parents.’’ Of
course the possible patterns of cultural inheritance are much more varied than the rules for sexual
diploid genetic inheritance. For example, one could have any number of cultural parents who
might or might not be one’s genetic relatives. Despite the great number of possibilities, orderly
patterns of cultural inheritance can be found. Marcus Feldman and Luigi Cavelli-Sforza (1981)
and Robert Boyd and Peter Richerson (1985) have laid promising foundations for this endeavor.
Bergstrom and Oded Stark (1993) outline a simple spatial model of cultural inheritance in which
individuals copy their parents’ most prosperous neighbor. They also examine a model of cultural
inheritance in which children will with some probability copy the actions of one of their parents
and with some probability copy the actions of a randomly chosen member of the population.
Most existing formal models of cultural evolution are ‘‘sexual haploid’’ models, in which each
of one’s cultural parents possesses a single ‘‘gene’’ for the trait in question. Probabilistic rules
are proposed that determine from which cultural parent one inherits a cultural trait. The genetic
25
model of diploid inheritance suggests an interesting way of enriching these models. For example
in cases where there are two cultural parents, one can construct models of cultural inheritance that
are isomorphic to the genetic model of sexual diploid inheritance, with dominant and recessive
cultural traits. A ‘‘recessive’’ cultural trait might not be expressed in one’s actions unless more
than one of a person’s cultural parents carry this trait. It may then be the case that recessive mutant
cultural traits can invade a population that resists invasion by a ‘‘dominant’’ mutant cultural trait.
One of the common threads of models of cultural inheritance is that an individual who inherits
a particular behavior is more likely than a randomly selected member of the population to interact
with others whose behavior is copied from the same cultural parents. This phenomenon produces
a striking effect, quite similar to sibling altruism. If the actions of prosperous individuals are more
likely to be copied than the actions of poorer individuals, one might first think that in equilibrium,
everyone would take the action that maximizes his own wealth. But suppose that a person’s
prosperity depends not only on his own actions but also on the actions of those with whom he
interacts and that an individual is more likely to interact with ‘‘cultural relatives’’ than with a
random member of the population. Then those who are helpful to others will have learned to behave
in this way from cooperative role models and are likely to have the good fortune of interacting with
other cultural relatives who are copying the same cooperative behavior. The richest and hence the
most copied individuals will not necessarily be those who selfishly attempt to maximize their own
wealth. These individuals would have learned their behavior from selfish role models and would
be more likely than average to interact with others who are copying selfish models.
Altruism is sustained in this model not because people realize that being generous ‘‘pays’’
nor is it sustained because a generous person believes that his generosity will be observed and
reciprocated. The truth is that, given the actions of others, it would pay to be selfish. The reason
the equilibrium population need not be selfish is that instead of deliberately choosing actions so
as to maximize their own payoffs, individuals select strategies by copying successful role models.
The logic is the same as that in our model of sibling interaction. Those who have learned to treat
their cultural siblings generously will tend to receive kindness from cultural siblings who learned
from the same role models. Hence they are likely to be relatively successful and to be copied by
members of the next generation.
Economics of the Family
The evolution of behavior toward siblings is only one of the pieces that is needed to build an
26
evolutionarily-based theory of the economics of the family. Evolutionary arguments also have
much to tell us about the conflict and commonality of interests between mother and father, between
parents and offspring, and within kinship groups. Adding parents as players will greatly enrich
the model of the game played by families. In equilibrium for this game, it is to be expected that
siblings would act as if they valued each others’ probabilities of survival at approximately half
of their own, while parents would value the survival probability of their children approximately
equally. As Trivers (1985) remarked, this leads to perpetual conflict between parents and offspring,
with parents attempting to get their offspring to act less selfishly toward their siblings than is their
natural inclination.
David Haig (1992) proposes an interesting structure for a model of conflict between older and
younger siblings. To quote Haig, ‘‘A simple analogy may be helpful. Suppose that a mother buys
a milkshake to be shared among her children, but the milkshake has only a single straw. If the
first child takes a drink and passes the remainder on to the second, and so on down the line, then
the greater the consumption of each child, the fewer children receive a drink.’’ Haig’s milkshake
analogy differs from the symmetric two-person games between siblings that have been modeled
here because of the asymmetries of the game between younger and older siblings and because of
the participation of the mother (and possibly the father) whose interests in each child’s payoff is
approximately equal.
The theory of kin-selection extends to other relatives such as cousins, nephews and nieces.
These relationships are likely to be important when one considers patterns of nonrandom mating
and especially of inbreeding. It is likely that through most of history, our ancestors lived in small,
rather isolated populations with significant amounts of inbreeding. In such a population, most
of an individual’s neighbors are likely to be relatives. If our preferences was shaped in such
an environment, kin selection is likely to have favored preferences that reflect concern for the
well-being of neighbors as well as near kin. The strength of such concern for neighbors would
depend on the average degree of relationship between neighbors. It should be possible to construct
relatively simple models of inbreeding and outbreeding for which the Mendelian combinatorics of
relationship between neighbors could be calculated so as to allow predictions could be made about
the degree of altruism between neighbors.
It would be useful to relax the assumption of monogamous mating. If there is uncertainty
about paternity, then if an individual inherits a rare mutant gene from its father, the probability
that this gene is shared by the mother’s other children is less than 1/2.
27
The theory developed in
this paper can be extended to deal with uncertain paternity, using different probability weights for
calculating the payoff functions
and
. This extension allows for quite tractible comparative
statics, relating the amount of cooperation between siblings and the average payoff to siblings
to the probability of shared paternity. Such comparative statics results are likely to be helpful
for understanding the evolution of monogamy, polygyny and other institutions that determine the
distribution of paternity.
Free Choice Versus Determinism
A literal interpretation of our model of kin selection has individuals who do not choose their
strategies, but act according to the program written in their genes. Such rigidly programmed
strategies would work well in a static environment where the individual payoff functions
do not change substantially from year to year or from generation to generation. But in a more fluid
environment, a preprogrammed strategist may not be able to respond as effectively to variations in
its environment as would a creature that is endowed with ‘‘preferences’’ and the capacity to solve
problems.
It is interesting to reinterpret our model as one in which there is evolution of preference
orderings rather than of programmed strategies. In this interpretation, natural selection acts to
shape the preferences of a population of ‘‘utility-maximizers’’. A population of individuals with
utility functions that are not consistent with the semi-Kantian utility function,
, would be invaded
by dominant mutants with utility functions that recommend actions yielding higher payoffs as
measured by
. Hence equilibrium would tend to produce a population of individuals with
preferences that could be represented by the utility function
.
The conception of humans as ‘‘problem-solvers’’ whose preferences were shaped by evolution
seems to offer a useful perspective on the ancient problem of whether individuals are agents
exercising free will or are simply puppets, acting out their predestined roles. From this perspective,
we can see some glimmerings of how it can be that though we pride ourselves on our rationality,
we sometimes seem compelled to take actions that are manifestly not in our self-interest. Though
our preferences have been shaped by heredity, we have also evolved the ability to assess payoffs
and to solve problems as they arise. This capacity evolved because, given the variability of payoff
functions across time and space, natural selection has not found hard-wired strategies that serve us
as well as the ability to choose ‘‘best actions’’ according to preferences that have been molded by
evolution.
28
Appendix
Proof of Lemma 1:
If mating is random and the proportions of
and
genes and
, then the expected proportions of genotypes
of this generation are respectively,
,
genes in the adult population are
,
, and
, and
found in the offspring
. These proportions are known
to geneticists as the Hardy-Weinberg ratios. At birth, therefore, the ratio of the number of
genotypes to the number of
genotypes is
. Evidently this ratio approaches zero as
approaches zero. This does not quite finish the proof, since we need to show that among those
offspring that survive to reproduce, the ratio of the number of
genotypes also approaches zero as
individuals of genotype
genotypes to the number of
approaches zero. Since the mutant
gene is recessive,
choose the same actions as those of genotype
. Let
be the
survival probability of individuals in families where all siblings take the same actions as the
genotypes. As
approaches zero, the probability that all of the siblings of an
either of type
or of type
an
genotype are
approaches 1. Therefore as approaches zero, the probability that
genotype survives to reproduce approaches
. The probability that an
genotype
survives to reproduce can be no larger than 1. Therefore the limit as approaches zero of the ratio
of the survival probability of
the ratio of the number of
zero as
genotypes can be no larger than . Since
genotypes born to the number of
approaches zero and since in the limit as
probability of
limit as
genotypes to that of
genotypes to that of
approaches zero of the ratio of the survival
genotypes is bounded from above, it follows that in the
approaches zero, the ratio of the number of surviving
surviving
genotypes born approaches
genotypes to the number of
genotypes approaches zero.
Proof of Proposition 2:
When the
gene is recessive, an individual carrying mutant
from that received by normal
genes gets a payoff that differs
genotypes only if this individual or its sibling is of genotype
There are three critical types of sibling pairs in which at least one sibling is
Type I.
Both siblings are of genotype
Type II. One sibling is of genotype
Type III. One sibling is of genotype
.
. These are
.
and the other is of genotype
and the other is of genotype
The only types of parent pairs that can produce offspring of genotype
29
.
.
are matches between
two parents of genotype
either of genotype
or matches in which one parent is of genotype
or of genotype
to adults of type
and the other is
. According to Lemma 1, the ratio of adults of type
approaches zero as the fraction of mutant genes in the population approaches
zero. Therefore when the mutant gene is rare, almost all mated pairs that produce offspring of
genotype
will be two individuals of genotype
. It follows that a rare recessive
reproduce more or less rapidly than normal genes depending on whether the
among the offspring of two
parents reproduce more or less rapidly than
genes will
genes that appear
genes in the normal
population.
Where both parents are heterozygotes, the probability that any pair of their offspring is of type
I is 1/16, the probability that the pair is of type II is 1/8, and the probability that the pair is of
type III is 1/4. In each sibling pair of critical type I, a total of four
sibling pair of type II, two
are found. Of those
genes are found. In each
genes are found, and in each sibling pair of type III, three
genes
genes that are found in a critical type of sibling pair, the fraction located in
type I sibling pairs is
and the fraction located in type II sibling pairs is
The remaining 3/5 of critical type pairs will be of type III.
In type I sibling pairs, both siblings take action , so each
type II sibling pairs, each of these
sibling takes action . Hence each
genes is found in an
One third of the
are
genes are carried by
. In type
genotypes who take action ,
genotypes, take action . These
genes found in type III sibling pairs are carried by
genotypes. These
. In
genotype who takes action while its
gene in a type II sibling pair gets a payoff of
III sibling pairs, two thirds of the type
while their siblings, who are
gene gets a payoff of
genes get payoffs of
.
genotypes whose siblings
genes are carried by individuals who take action , while their siblings
take action . Accordingly they get payoffs of
in the offspring of two parents of genotype
. Putting these facts together, we find that
, the expected payoff experienced by an
conditional on it being found in a sibling pair of one of the critical types is:
30
gene
Since offspring who do not belong to a critical type of sibling pair get the same payoff as
normal offspring, it follows that among the offspring of two
parents, the expected payoff to
genes will be greater or smaller than the expected payoff to normal individuals depending on
whether
is greater or smaller than
.
Therefore a necessary condition for a population of -strategists to resist invasion by recessive
mutants is that
for all
and a sufficient condition is that
.
Proof of Proposition 3:
In each of the possible cases where the parents have a total of two or more
probability that any offspring has at least one
gene is at least 3/4. (If either parent has two
genes, then the offspring is sure to have at least one
then with probability
gene
of genotype
gene. If both parents are of genotype
, the offspring will have at least on
, individuals who carry at least one
genes, the
,
gene.) For a dominant mutant
gene take the deviant action , while individuals
take action . Therefore the expected payoff to a mutant offspring of parents
who carry two or more
population of
genes is
where
. It follows that a
genotypes will resist second-order invasion by dominant mutants if the following
condition is satisfied.
whenever
Condition A.
.
Using the equation in Condition A to eliminate the expression
from the inequality,
one finds Condition A is equivalent to the inequality
, this inequality in turn is equivalent to
. Since
.
Proof of Proposition 4:
Let us define
By definition, for all
and
in , if there is symmetric substitutability, then
there is symmetric substitutability, then
. Let us also define
31
and if
Then is a symmetric Nash equilibrium for the payoff function
for all
. Furthermore,
only if
for all
is a symmetric Nash equilibrium for the payoff function
if and only if
; and
for all
.
,
, and
According to Lemma 2,
, that for all
and
then it must be that for all
. If there is symmetric substitutability, it must also be that
in . It then follows from equation
that
, it follows from equation that
equation
it also follows that
complementarity, if
is a symmetric Nash
. This proves assertion (i).
If there is symmetric complementarity, then
it then follows that if
is a symmetric
and
and hence that
equilibrium for the game with payoff function
equation
. Since
,
for all
and hence that
Nash equilibrium for the game with payoff function
for all
:
and therefore
is a symmetric Nash equilibrium for the payoff function
and
if and
is a symmetric Nash equilibrium for the payoff function
It follows directly form the definitions of
If
if and only if
for all
for all
for all
and
, then
. From
and then from
. Therefore in the case of symmetric
is a symmetric Nash equilibrium for the game with payoff function
is also a symmetric Nash equilibrium for the games with payoff functions
and
, it
. this proves
assertion (ii).
From Equations and it follows that
Using the same kind of reasoning we used in proving assertions (i) and (ii), we see that if there
is symmetric complementarity, then every symmetric Nash equilibrium for the game with payoff
function
is also a symmetric Nash equilibrium for the game with payoff function
and if there
is symmetric substitutability, then every symmetric Nash equilibrium for the game with payoff
32
function
is also a symmetric Nash equilibrium for the game with payoff function
. This
proves assertions (iii) and (iv).
Proof of Proposition 7:
Since,
, calculation shows that
and hence
. Since
, it must
be that
and therefore
. This proves the first claim of Proposition 7.
Differentiating once again, we find that
and that
Therefore
and
It is
then immediate that
.
33
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Footnotes
Department of Economics, University of Michigan, Ann Arbor, Mi., 48109. I am grateful to
Carl Bergstrom for teaching me some rudiments of population genetics and suggesting useful lines
of inquiry, and to Ken Binmore, Jack Hirshleifer, Alan Rogers, and Oded Stark for advice and
encouragement.
1. A very readable introduction to the theory of evolutionary games is found in Binmore (1992).
2. In contrast, Foster and Young (1990), Young (1993), and Kandori et.al. (1993) study limiting
probability distributions as time goes to infinity in models where evolution is described by
stochastic differential equations. It would be interesting and useful to extend the models discussed
here to a truly stochastic environment.
3. A polymorphic equilibrium is an equilibrium in which more than one type of gene is present.
In polymorphic equilibrium, the coexisting genes all reproduce at the same rate, while any mutant
genes reproduce less rapidly than the types already present in equilibrium.
4. Readers familiar with Maynard Smith’s definition of ESS may be surprised to see that this
condition differs from Maynard Smith’s ‘‘second-order condition’’ for ESS. Maynard Smith’s
condition is appropriate in the case where creatures encounter randomly chosen strangers, but not
for the case of games between siblings.
5. Hamilton studied the more general case of simultaneous interaction within an entire network of
relatives, and he also treats the case of haplodiploid species like ants and bees. Our attention here
will be confined to relations between pairs of sexual diploid siblings.
6. The coefficient of relationship between two individuals is defined by biologists to be the
probability that a randomly selected gene in one of these individuals will have an exact copy
located in the other individual by descent from a common ancestor. (Trivers (1985)) For parent
and child the coefficient or relationship is 1/2, for half-siblings it is 1/4, and for cousins it is 1/8.
7. The properties of symmetric complementarity and substitutability are weaker than the more
familiar concepts of strategic complementarity and it strategic substitutability (See Bulow,
37
Geanakopolis, and Klemperer (1985), Vives (1990) or Milgrom and Roberts (1990).) Strategic
complementarity requires the stronger property that for all , ,
, then (
)
(
)
(
)
(
, and
in
). If is differentiable and
strategic complementarity requires that the cross-partial derivatives
, while symmetric complementarity requires only that
(
)
, if
and
is a real interval, then
(
)
0 for all
0 for all
in
and
in
. Symmetric
substitutability is related to strategic substitutability in essentially the same way.
8. The original passage as quoted by Fudenberg and Tirole is ‘‘If a group of hunters set out to take
a stag, they are fully aware that they would all have to remain faithfully at their posts in order to
succeed; but if a hare happens to pass near one of them, there can be no doubt that he pursued it
without qualm, and that once he had caught his prey, he cared very little whether or not he had
made his companions miss theirs.’’
9. The payoffs
(
(
) 2 =
) 2+(
(
10.
)=(
) are calculated as follows:
) 2+(
The payoffs
(
2+0 =
2,
(
(
)=(
) = (
) 2+(
) 2+(
) 2=
,
(
)=
) 2 = 1 2 + 1 2 = 1,
) 2 = 1 2 + 1 2 = 1.
) are calculated as follows:
(
) = (
)+(
) 2 = 0 + 1 2 = 1 2, (
(
)=(
)+(
) 2 = 3 2.
(
) = (
) = (
)+(
)+(
) 2 = 3
2,
) 2 = 1 + 0 = 1,
11. Some definitions of prisoners’ dilemma add the additional restriction that the sum of the two
players’ payoffs is greater if both cooperate than if one cooperates and one defects, which means
that
1 2. As it happens, adding this condition does not affect the qualitative results relevant
to our discussion.
12. Another example, familiar to poultry lovers and game theorists, is the game of chicken.
13. A payoff function (
and
in , and for all
) is defined to be concave in a player’s own strategy if for all
[0 1], (
+ (1
)
)
(
) + (1
remains true if the assumption of concavity is weakened to quasi-concavity.
38
)(
,
,
). The lemma
14. An alternative way to find a sufficient condition for existence of a Nash equilibrium for
observe that the function
(
) will be concave on the simplex
is to
of own strategies if the matrix
is negative semi-definite on the simplex (which is equivalent to the standard bordered-Hessian
conditions on
. ) It can be demonstrated that the matrix
will be negative semi-definite on the
simplex if and only if exhibits symmetric complementarity.
15.
Proof of Lemma 4 is a straightforward exercise in multivariate calculus. A more general
version of this result is known as Topkis Characterization Theorem, Milgrom and Roberts (1990).
16. Haig (1992) observes that this conflict of interest begins even before birth. According to Haig,
there is medical evidence of a kind of ‘‘arms race’’ between human fetuses and their mothers,
in which the body of the fetus acts to extract more nutrition from the mother and to stay in the
womb longer, while the mother’s body acts to limit the resources devoted to the fetus and to hasten
birth. In the game played between mother and fetus, the biological interests of the mother are
not identical with those of the fetus. Her genetic interest in maintaining her strength to produce
additional children is stronger than the genetic interest of the fetus in having more siblings.
17. The genetic consequences of various patterns of inbreeding have been extensively studied
because of practical applications for the breeding of livestock. A useful reference to models of
nonrandom mating is Karlin (1969). Samuelson (1983) makes the interesting suggestion that in
models of kin selection, there may be genetic advantages to inbreeding which must be weighed
against the costs of increased probability of being homozygous for rare, harmful recessive genes.
18. Biologists (see, for example Haig (1992)) have established that some genes are able to act
differently, depending on whether they inherited from the mother or the father. This phenomenon,
known as genomic imprinting presents another interesting problem, since if mating is promiscuous
and genomic imprinting is present, there will be a conflict of interest between maternally inherited
and paternally inherited genes residing in the same individual, with genes inherited from the
mother being more altruistic than those inherited from the father.
39
Figure 1. Equilibrium Regimes With Semi-Kantian Payoffs
R
1
B
A
1)/2
(P+
=
R
2/3
C
D
=1
+P
R
R=
2P
1/2
1
1/3
40
P
Figure 2. Equilibrium Regimes Under Symmetric Substititutability
R
1
F
3/5
1/2
G
R+
P
D
=1
H
0
I
1
1/5
41
P