We offer a game-theoretic proof of Hamilton's rule for the spread of altruism. For a simple case of siblings, we show that the rule can be derived as the outcome of a one-shot prisoner's dilemma game between siblings.Evolution of altruism, Hamilton's rule, One-shot prisoner's dilemma game

Stark, Oded

Wang, You Qiang

English

Research Papers in Economics

139 
Reihe Ökonomie 
Economics Series 
 
 
 
 
 
On the Evolutionary Edge of 
Altruism: A Game-Theoretic 
Proof of Hamilton’s Rule for a 
Simple Case of Siblings
Oded Stark, You Qiang Wang 139
Reihe Ökonomie 
Economics Series 
   
 
 
 
 
On the Evolutionary Edge of 
Altruism: A Game-Theoretic 
Proof of Hamilton’s Rule for a 
Simple Case of Siblings
Oded Stark, You Qiang Wang 
 
September 2003 
 
Institut für Höhere Studien (IHS), Wien 
Institute for Advanced Studies, Vienna  
Contact: 
 
Oded Stark 
University of Bonn 
Walter-Flex-Strasse 3 
D-53113 Bonn, GERMANY 
email: ostark@uni-bonn.de 
and 
University of Vienna 
 
You Qiang Wang 
Tsinghua University 
 
Founded in 1963 by two prominent Austrians living in exile – the sociologist Paul F. Lazarsfeld and the 
economist Oskar Morgenstern – with the financial support from the Ford Foundation, the Austrian
Federal Ministry of Education and the City of Vienna, the Institute for Advanced Studies (IHS) is the first
institution for postgraduate education and research in economics and the social sciences in Austria. 
The Economics Series presents research done at the Department of Economics and Finance and
aims to share “work in progress” in a timely way before formal publication. As usual, authors bear full
responsibility for the content of their contributions.  
 
 
Das Institut für Höhere Studien (IHS) wurde im Jahr 1963 von zwei prominenten Exilösterreichern –
dem Soziologen Paul F. Lazarsfeld und dem Ökonomen Oskar Morgenstern – mit Hilfe der Ford-
Stiftung, des Österreichischen Bundesministeriums für Unterricht und der Stadt Wien gegründet und ist 
somit die erste nachuniversitäre Lehr- und Forschungsstätte für die Sozial- und Wirtschafts-
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Verantwortung für die veröffentlichten Beiträge liegt bei den Autoren und Autorinnen. 
 Abstract 
We offer a game-theoretic proof of Hamilton’s rule for the spread of altruism. For a simple 
case of siblings, we show that the rule can be derived as the outcome of a one-shot 
prisoner’s dilemma game between siblings. 
 
Keywords 
Evolution of altruism; Hamilton’s rule; One-shot prisoner’s dilemma game 
JEL Classification 
A13; C70; D64  
Comments 
We are indebted to an anonymous referee and to Uwe Cantner for helpful comments and suggestions.
Partial financial support from the National Institute on Aging (grant RO1-AG13037) and from the 
Humboldt Foundation is gratefully acknowledged. Contents 
1. Introduction  1 
2.  The game and a general result  2 
3.  The rule of imitation, survival, and the outcome:  
a special case  4 
4. Conclusion  6 
Appendix 7 
References 8 
 1.  Introduction 
  
  Evolutionary biologists have developed a powerful theory of the evolutionary 
foundations of altruism between relatives. The theory is based on the idea that 
individuals who are related by blood share genes.  Consider a gene that governs a 
particular behavior.  The likelihood that the gene will be replicated is higher when 
the gene takes into account not only the extra reproductive opportunities that the 
behavior confers on the host who carries the gene but also the extra reproductive 
opportunities that the behavior confers on relatives of the host who also carry the 
gene. William Hamilton, the pioneer of this theory, describes it 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 neighbor’s fitness against his 
own according to the coefficients of relationship appropriate to that situation” 
(Hamilton 1964, p.19). The coefficient of relationship between two individuals is the 
probability that a randomly selected gene in one of these individuals will have an 
exact copy located in the other individual as a result of descent from a common 
ancestor. In the case of a haploid population in which each parent has a single gene 
for being altruistic or selfish and mating is monogamous, the coefficient of 
relationship between two siblings is ½.  
  
  “Hamilton’s rule” is that altruism will spread in a population if the benefit 
obtained from giving times the coefficient of relationship exceeds the cost of giving. 
If c is the cost to oneself of helping a sibling, and b is the benefit to a sibling from 
receiving help, altruism will spread if b c >   ½ ⋅ , that is, if the benefit obtained from 
help exceeds twice the cost of helping. 
 
  The purpose of this paper is to complement the large and important literature 
that followed Hamilton’s pioneering articles, both in evolutionary biology (notably 
Dawkins (1976), Grafen (1984), and Wilson (1987)) and beyond (Axelrod (1984), 
Hofbauer and Sigmund (1988), Binmore and Samuelson (1992), and Nowak and 
May (1992)), with a formal game-theoretic proof of Hamilton’s rule. Building on 
Bergstrom and Stark (1993), Bergstrom (1995), and Stark (1999), this paper provides 
a proof of the rule for a simple case of siblings. 
  1  In evolutionary economics, the study of altruism is motivated by two 
questions: where does altruism come from and what does it give rise to. The 
incentive to explore inclinations is not independent, however, from the density of 
implications. If the motivation to produce, the propensity to redistribute, and the 
tendency to accumulate and transfer – within families, societies, and across 
generations – matter both for individual well-being and for social welfare, and if 
these processes are governed or significantly affected by the incidence and intensity 
of altruism as a trait, we would like to find out how the trait evolves. The interest in 
economics, and beyond, in the evolution, survival, and extinction of institutions of 
various types cannot be orthogonal to the interest in the prevalence and intensity of 
altruism  if altruism gives rise to patterns and predispositions that completely or 
partially substitute for institutional mandates, impinge on the design of institutes, 
crowd out their roles, or render their mission superfluous. All the more so when there 
is a close correspondence between altruism and cooperation.
1  Since altruism is 
practiced and manifested socially, it is natural to start the search for its prevalence 
and origins in small social groupings such as the family. It is more likely that 
altruism will pervade large groupings such as the population at large if it evolves 
between siblings than if it fails to gain a foothold even within families. 
 
 
2.  The game and a general result 
 
  In each period there is an old generation and a young generation. A fraction 
of the old generation consists of altruists, a complementary fraction consists of 
nonaltruists. Members of the old generation are matched with uniform probabilities 
into pairs. Each pair breeds two children. The children constitute the young 
                                                 
1 The inquiry pursued in this paper relates to the study of institutions in yet another way. 
Schelling (1960, 1971, 1978) has shown how the interactions of individuals in environments 
characterized by bounded rationality and imperfect information coalesce over time into 
customs, norms, and institutions that govern economic and social life. Schelling’s pioneering 
work was recently supplemented significantly by Young’s study of economic and social 
institutions. To Young (1998) an institution is an established law, custom, usage, practice, 
organization. (Examples of institutions are aplenty: rules of the road, time of lunch, patterns 
of marriage, forms of economic contracts.) Young develops a theory that predicts how 
institutions evolve and characterizes their welfare properties. Viewing Hamilton’s rule as an 
institution places this paper’s inquiry in that research vein. 
  2generation. The two siblings play a one-shot prisoner’s dilemma game with each 
other. A sibling can help the other sibling at a cost to himself. Let c be the cost to a 
sibling of helping a sibling, and let b be the benefit to the sibling who receives the 
help, b>c>0.  We obtain the following payoff matrix: 
 
 
 
 
  Row  Sibling   
Column Sibling 
 
  C  D 
C  bc c − − ,b   −cb ,  
D   bc ,−    00 ,      
   
 
where playing C stands for providing help and playing D stands for not offering help. 
A sibling who plays C is altruistic, a sibling who plays D is nonaltruistic. To see this 
suppose the column sibling selects C. If the row sibling selects C rather than D, he 
give up b to receive the smaller b c − , whereas the column sibling gains since he 
receives b  which is larger than  c − c − . Suppose, alternatively, that the column sibling 
selects D. Again, if the row sibling selects C rather than D, his payoff declines (by c), 
while the column sibling’s payoff rises (by b). This defines altruism: giving up 
something for the sake of another. Thus, throughout the rest of this paper we identify 
altruism with playing cooperate in the one-shot prisoner’s dilemma game. 
 
 Let  (p, 1- p) denote the mixed strategy in which the row sibling plays C with 
probability p; and let (q, 1- q) denote the mixed strategy in which the column sibling 
plays C with probability q.  Then, for any given (p, q), the expected payoffs of the 
row and column siblings are qb – pc and pb – qc, respectively.  Let p1 and p0 be the 
probabilities that the row sibling plays C if the column sibling plays C  and  D, 
respectively.  We now provide a game-theoretic proof of Hamilton’s rule for a 
simple case of siblings. 
      
Proposition (Hamilton’s rule): If c is the cost to oneself of helping a sibling, and b is                    
the benefit to a sibling from receiving help, altruism will spread if       
() c > p p b 0 1 − ⋅ , that is, p 0 1 p −   is the equivalent of the coefficient of 
relationship. 
 
  3Proof:  Cooperation will be globally stable if the expected payoff of a randomly 
selected cooperator child is larger than the expected payoff of a randomly 
selected defector child.  Without loss of generality, we randomly select the 
column sibling.  The expected payoff of a randomly selected cooperator 
column sibling is  c b p − 1 since ( ) ( ) 1 1, p q , p = , and the expected payoff of a 
randomly selected defector column sibling is p  since  .   
Hence, cooperation will be globally stable if p
b 0
c b 1
() (0 0, p q , p = )
b p0 > − , yielding the 
statement of the proposition.    
 
 
3.  The rule of imitation, survival, and the outcome: a special case 
 
  We assume that how a child plays, C or D, is determined through the 
imitation of his parents, and that each child imitates one of his parents with equal 
probabilities. The probability that a child survives to reproduce (to have his own 
children) is proportional to the payoff in the game. For example, consider a case in 
which the payoff positively influences the probability of reaching maturity and of 
being able to procreate.  
 
 Let  x be the proportion of cooperative parents, and let  x − 1  be the proportion 
of defector parents.  
 
Claim 1:  The probability that a randomly chosen child is a cooperator is x. 
 
Proof:   Let  N be the number of individuals in the old generation. Hence, the 
number of parent pairs is 
2
N . Cooperator children come either from 
cooperator-cooperator parent pairs or from cooperator-defector parent 
pairs. The number of cooperator-cooperator marriages is N x
2
2 . All 
N xN
2
2
2 ⋅=x
2  children of these marriages are cooperators. The number of 
cooperator-defector marriages is  () () [] ( x Nx x x x x
N
− = − + − 1 1 1
2 )  which is also 
  4the number of cooperator children produced by such marriages. Hence, the 
total number of cooperator children is  . In a population 
of  N  children, the probability then that a randomly chosen child is a 
cooperator is 
() Nx Nx x Nx
2 1 +− =
Nx
N
x = .   
2
1
2
x x +
=
−
x
2
 
Claim 2:   Given that a child is a cooperator, the conditional probability that its 
sibling is a defector is 
2
x 1− . 
 
Proof:   A cooperator-defector pair of children results from a mixed marriage. Half 
of these Nx  marriages produce mixed sibling pairs. The number of 
cooperating children in the mixed sibling pairs from these marriages is 
( x 1− )
( 1
2
1 Nx −
Nx
) x .  As already shown, the total number of cooperator children is 
.  Given that a child is a cooperator, the conditional probability that its 
sibling is a defector is 
()
2
1 1
2
1
x
Nx
x Nx −
=
−
.    
  
Given that a child is a cooperator, the conditional probability that its sibling is a 
cooperator is the complementary probability 
1
− 1 . 
 
  By replacing x  for 1   we get that given that a child is a defector, the 
conditional probability that its sibling is a cooperator is 
x −
. (Given that a child is a 
defector, the conditional probability that its sibling is a defector is the 
complementary probability 1 .)  
2
− x
 
Claim 3:  Cooperation will be globally stable if b>2c. 
 
Proof:  Since in this case 
2
1
1
x
p
+
=  and 
2
0
x
p = , the claim follows from the 
proposition. 
  5 
  Evolutionary biologists refer to “inclusive fitness” of an individual. In the 
present model the inclusive fitness of a cooperator parent is the expected 
payoff of a randomly selected cooperator child, and the inclusive fitness of 
a defector parent is the expected payoff of a randomly selected defector 
child. The inclusive fitness of a cooperator is larger than the inclusive 
fitness of a defector, and cooperation is globally stable if the benefit to a 
child from playing cooperate (helping) exceeds twice the child’s own cost 
of playing cooperate (helping).   
 
 
4.  Conclusion 
 
  We have shown that in a simple case of siblings, Hamilton’s rule can be 
derived as the outcome of a prisoner’s dilemma game between siblings. We 
employed several simplifying assumptions. These may be relaxed. For example, the 
formation of couples can be more selective than random. As shown in the Appendix, 
however, this change will only strengthen the case for cooperation. 
  6Appendix 
 
  To substantiate the claim that a non-random formation of couples will only 
strengthen the case for cooperation, note that if matching is purely (positively) 
assortative, the fractions of cooperator marriages and defector marriages are, 
respectively, x and 1-x.  To allow matching patterns that are intermediate between the 
polar cases of purely random matching and purely assortative matching, we define a 
parameter m where 0 , such that when matching is purely random m , and 
when matching is purely assortative 
1 ≤ ≤ m 0 =
1 = m .  The number of cooperator-cooperator 
marriages is then  )] 1
2
x
N
− ( [
2 mx x + , and the number of cooperator-defector marriages 
is  .  ) 1 ( ) 1 ( x x m N − −
 
  It follows that the probability that a randomly chosen child is a cooperator is x; 
given that a child is a cooperator, the conditional probability that its sibling is a 
defector is 
2
1
) 1
x
m
−
− ( ; given that a child is a cooperator, the conditional probability 
that its sibling is a cooperator is 
2
1
) 1 (
x
m
−
− − 1 ; and given that a child is a defector, 
the conditional probability that its sibling is a cooperator is 
2
) 1
x
m − ( .  (Given that a 
child is a defector, the conditional probability that its sibling is a defector is 
2
) 1 ( 1
x
m − − .)  Since in this case 
2
1
) 1 ( 1
1
x
m
−
− − = p  and 
2
) 1 ( 0
x
m p − = , it follows from 
the proposition that cooperation will be globally stable if  c
x
m
x
>  
 − −
−
2
) 1 (
2
m  
 − −
1
) 1 ( 1 b  
that is, if b .  c m 2 ) 1 ( > +
 
 
  7  8
References 
 
Axelrod, Robert. 1984. The Evolution of Cooperation. New York: Basic Books.  
 
Bergstrom, Theodore C. 1995. “On the Evolution of Altruistic Ethical Rules for 
Siblings.” American Economic Review 85, pp. 58-81. 
 
Bergstrom, Theodore C. and Stark, Oded. 1993. “How Altruism Can Prevail in an 
Evolutionary Environment.” American Economic Review 83, pp. 149-155. 
 
Binmore, Kenneth G. and Samuelson, Larry. 1992. “Evolutionary Stability in 
Repeated Games Played by Finite Automata.” Journal of Economic Theory 
57, pp. 278-305.   
 
Dawkins, Richard. 1976. The Selfish Gene. New York: Oxford University Press. 
 
Grafen, Alan. 1984. “Natural Selection, Group Selection and Kin-Selection.” In L.R. 
Kreps and N.B. Davis, eds. Behavioral Ecology: An Evolutionary Approach. 
Oxford: Blackwell Scientific. 
 
Hamilton, William D. 1964. “The Genetical Evolution of Social Behavior.” Parts I 
and II. Journal of Theoretical Biology 7, pp. 1-52. 
 
Hofbauer, Josef and Sigmund, Karl. 1988. The Theory of Evolution and Dynamical 
Systems. Cambridge: Cambridge University Press. 
 
Nowak, M. A. and May, R. M. 1992. “Evolutionary Games and Spatial Chaos.” 
Nature 359, pp. 826-829. 
 
Schelling, Thomas C. 1960. The Strategy of Conflict. Cambridge, MA: Harvard 
University Press. 
 
Schelling, Thomas C. 1971. “Dynamic Models of Segregation.” Journal of 
Mathematical Sociology 1, pp. 143-186. 
 
Schelling, Thomas C. 1978. Micromotives and Macrobehavior.  New York: Norton. 
 
Stark, Oded. 1999. “Siblings, Strangers, and the Surge of Altruism.” Economics 
Letters 65, pp. 135-142. 
 
Wilson, D. S. 1987. “Altruism in Mendelian Populations Derived from Sibling 
Groups: The Haystack Model Revised.” Evolution 41, pp. 1059-1070. 
 
Young, H. Peyton. 1998. Individual Strategy and Social Structure: An Evolutionary 
Theory of Institutions. Princeton: Princeton University Press. 
  
 
Authors: Oded Stark, You Qiang Wang 
 
Title:  On the Evolutionary Edge of Altruism: A Game-Theoretic Proof of Hamilton’s Rule for  
a Simple Case of Siblings 
 
Reihe Ökonomie / Economics Series 139 
 
Editor: Robert M. Kunst (Econometrics) 
Associate Editors: Walter Fisher (Macroeconomics), Klaus Ritzberger (Microeconomics)  
 
ISSN: 1605-7996 
© 2003 by the Department of Economics and Finance, Institute for Advanced Studies (IHS), 
Stumpergasse 56, A-1060 Vienna •  +43 1 59991-0 • Fax +43 1 59991-555 • http://www.ihs.ac.at  
  
ISSN: 1605-7996 
 

Altruism in Mendelian Populations Derived from Sibling Groups: The Haystack Model Revised.” Evolution 41,

Dynamic Models of Segregation.”

Evolutionary Stability in Repeated Games Played by Finite Automata.”

How Altruism Can Prevail in an

Individual Strategy and Social Structure: An Evolutionary Theory of Institutions.

Micromotives and Macrobehavior.

Natural Selection, Group Selection and Kin-Selection.”

The Evolution of Cooperation.

The Genetical Evolution of Social Behavior.” Parts I and II.

The Selfish Gene.

The Strategy of Conflict.

The Theory of Evolution and Dynamical Systems. Cambridge:

On the Evolutionary Edge of Altruism: A Game-Theoretic Proof of Hamilton's Rule for a Simple Case of Siblings

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On the Evolutionary Edge of Altruism: A Game-Theoretic Proof of Hamilton's Rule for a Simple Case of Siblings

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