We calculate the equilibrium fraction of cooperators in a population in which payoffs accrue from playing a single-shot prisoner’s dilemma game. Individuals who are hardwired as cooperators or defectors are randomly matched into pairs, and cooperators are able to perfectly find out the type of a partner to a game by incurring a recognition cost. We show that the equilibrium fraction of cooperators relates negatively to the population’s level of wealth.Equilibrium fraction of cooperators, Population's level of wealth, Single-shot prisoner's dilemma game

Stark, Oded

English

Research Papers in Economics

 
 
 
 
 
Cooperation and Wealth  
Oded Stark 
123 
Reihe Ökonomie 
Economics Series  
 
 
 
123 
Reihe Ökonomie 
Economics Series 
 
Cooperation and Wealth 
Oded Stark 
 
October 2002 
 
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 
 
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-
wissenschaften in Österreich. Die  Reihe Ökonomie bietet Einblick in die Forschungsarbeit der 
Abteilung für Ökonomie und Finanzwirtschaft und verfolgt das Ziel, abteilungsinterne 
Diskussionsbeiträge einer breiteren fachinternen Öffentlichkeit zugänglich zu machen. Die inhaltliche 
Verantwortung für die veröffentlichten Beiträge liegt bei den Autoren und Autorinnen. 
 Abstract 
We calculate the equilibrium fraction of cooperators in a population in which payoffs accrue 
from playing a single-shot prisoner’s dilemma game.  Individuals who are hardwired as 
cooperators or defectors are randomly matched into pairs, and cooperators are able to 
perfectly find out the type of a partner to a game by incurring a recognition cost.  We show 
that the equilibrium fraction of cooperators relates negatively to the population’s level of 
wealth. 
 
Keywords 
Equilibrium fraction of cooperators; Population’s level of wealth; Single-shot prisoner’s 
dilemma game 
JEL Classifications 
A13; C70  
Comments 
I am indebted to Theodore C. Bergstrom for stimulating correspondence and to Paul S. H. Lau for 
enlightening comments. Partial financial support from the Humboldt Foundation and the Government of 
Lower Austria are gratefully acknowledged. Contents 
1.  Introduction  1 
2.  The game and the payoffs  1 
3.  The types and their expected payoffs  2 
4.  Equilibrium with defectors and testing cooperators  
but without non-testing cooperators  4 
5.  The relationship between the equilibrium fraction of 
cooperators in a population and a population’s level  
of wealth  5 
6.  Robustness of the cooperation-wealth relationship  
when the testing cooperators are somewhat adventurous  7 
7.  Conclusion   9 
References  10 
   1
1.  Introduction 
 
  An example illustrates that the level of wealth of a population and the 
equilibrium fraction of cooperators in a population are inversely related.  It has been 
argued that the fraction of cooperators in a large society can be expected to be smaller 
than the fraction of cooperators in a small society (Binmore, 1998; Cook and Hardin, 
2001).  To the extent that a large society (say a city) is wealthier than a small society 
(say a town), the size effect may conceal a wealth effect. 
 
2.  The game and the payoffs 
 
  Consider the following two-player, two-strategy game in which a player who 
cooperates gets a payoff of R if his opponent cooperates, and S if the opponent defects.  
A player who defects gets T if his opponent cooperates, and P if the opponent defects.  
The game is a prisoner’s dilemma game:  . S P R T > > >   Hence defection is the 
dominant strategy for each player. 
 
  Let there be a large population of players consisting of individuals who are 
hardwired to be cooperators and individuals who are hardwired to be defectors.  
Individuals are randomly matched into pairs.  An individual does not know the type 
of the individual with whom he is matched, but he can obtain such information at a 
cost,  K K < < 0 , where  K  will be defined below.  The type-recognition test is 
perfect.  Thus, if an individual chooses to incur the cost and administer the test, the 
individual finds out whether he is matched with a cooperator or with a defector.  The 
individual can then decide to play or not to play.  If the individual decides not to play, 
he randomly picks another individual from the population and administers the   2
type-recognition test in the new match.  If individuals agree to play, they play their 
hardwired strategies, receive their respective payoffs, and leave the partner-seeking 
population to be replaced by new individuals.  In equilibrium (to be characterized 
below) the flow of individuals of each type who enter the population exactly replaces 
the flow of individuals of each type who exit the population. 
 
3.  The types and their expected payoffs 
 
  Following Stark (1999, chapter 5), we study a population that consists of three 
types: defectors who play without incurring a recognition cost, cooperators who play 
after incurring the recognition cost, and cooperators who play without incurring the 
recognition cost.  While there can be an equilibrium with all three types present and 
an equilibrium with defectors only, i. there cannot be an equilibrium without defectors; 
and ii. there cannot be an equilibrium with only defectors and non-testing cooperators.  
The rationale for i . is that there cannot be an equilibrium with only non-testing 
cooperators because defectors will do better than cooperators; there cannot be an 
equilibrium with only testing cooperators because non-testing cooperators will do 
better; and there cannot be an equilibrium with only both types of cooperators because 
the non-testing cooperators will do better than the testing cooperators.  The rationale 
for  ii. is that there cannot be an equilibrium with only defectors and non-testing 
cooperators because defectors will do better than the non-testing cooperators. 
 
  Let the steady-state fractions of testing cooperators, non-testing cooperators, and 
defectors be  t p ,  nt p , and  d p , respectively,  . 1 = + + d nt t p p p   Given the manner 
in which a testing cooperator acts and plays, his expected payoff is 
   3
 
d
t
K
R V
p -
- =
1
.  (1) 
 
The proof is as f ollows:  The expected net payoff from administering the cost K 
(exactly once) and encountering a cooperator in the first match is 
); 1 ( ) 1 ( d d K R p p - - -  from failing to encounter a cooperator in the first match but 
encountering one in the second match is  ); 1 ( 2 ) 1 ( d d d d K R p p p p - - -  from failing 
to encounter a cooperator in the first two matches but succeeding in encountering one 
in the third match is  ); 1 ( 3 ) 1 (
2 2
d d d d K R p p p p - - -  and so on. Thus, 
L + - - - + - - - + - - - = ) 1 ( 3 ) 1 ( ) 1 ( 2 ) 1 ( ) 1 ( ) 1 (
2 2
d d d d d d d d d d t K R K R K R V p p p p p p p p p p
  ) 3 2 1 )( 1 (
1
) 1 ( 2 L + + + - -
-
-
= d d d
d
d K
R
p p p
p
p
 
  ] ) ( ) ( ) 1 )[( 1 (
2 2 2 L L L L + + + + + + + + + - - = d d d d d d K R p p p p p p  
  )
1 1 1
1
)( 1 (
2
L +
-
+
-
+
-
- - =
d
d
d
d
d
d K R
p
p
p
p
p
p  
  .
1 1
1
1
) 1 (
d d
d
d
K
R K R
p p
p
p
-
- =
-
-
- - =       
 
  The expected payoff of a non-testing cooperator who plays the game with 
whoever he is paired with in the first match is 
 
  . ) 1 ( S R V d d nt p p + - =   (2) 
 
  Since a defector always plays, that is, he plays when matched either with a 
non-testing cooperator or with a defector, his expected payoff is 
 
  ). (
1 1 1
1
P T T P T V
t
d
t
d
t
d t
d -
-
- =
-
+
-
- -
=
p
p
p
p
p
p p
  (3)   4
4.  Equilibrium with defectors and testing cooperators but without 
non-testing cooperators 
 
  From the discussion in the preceding section it follows that an equilibrium with 
defectors and testing cooperators but without non-testing cooperators is feasible.  If 
there are no non-testing cooperators,  ; 1 = + d t p p  the expected payoff of testing 
cooperators is  ;
t
t
K
R V
p
- =  and the expected payoff of defectors (who can play only 
with defectors) is  . P Vd =   In equilibrium, testing cooperators receive the same 
expected payoff as defectors.  Thus,  P
K
R
t
= -
p
 or 
 
  ,
P R
K
t -
= p   (4) 
assuming that  . K P R K ” - <  
 
  To help unravel the nature of the equilibrium, consider alternative values of K.  
Suppose that K were equal to  P R- .   t p  would then be equal to one.  But having 
a population with only testing cooperators cannot be an equilibrium because in that 
case the non-testing cooperators will do better.  Thus, we have a contradiction.  
Suppose that  . 0 ﬁ K   It follows that  0 ﬁ t p .  Yet suppose the opposite, that is, 
that  . 1 ﬁ t p   If such were the case, the population would consist of only testing 
cooperators which, from  i. in section 3, cannot hold.  As K assumes values that 
increasingly move it away from being close to  P R-  toward close to zero, the 
associated values of  t p  must become smaller.  To see the reason for this result, 
suppose that an equilibrium holds at  P
K
R
t
= -
0
0
p
 and consider the opposite, that is, 
as  K declines from  0 K  to  1 K ,  t p   increases from 
0 t p  to 
1 t p .  But then   5
0 1
0 1
t t
K K
p p
< , rendering it impossible to restore equilibrium at  .
0
0 P
K
R
t
= -
p
  As long as 
R and P are given, observing the equilibrium requires that  t p  and  K move in 
tandem. 
 
  To complete the characterization of the equilibrium we note that in order for 
there to be no non-testing cooperators in the population, it has to be the case that if a 
non-testing cooperator were to enter the population, he will receive a lower payoff 
than that received by the testing cooperators and the defectors, that is, 
. ) 1 ( P S R t t < - + p p   Substituting 
P R
K
t -
= p  and rearranging terms we get 
K K
S R
S P
S R
P R S P
K <
-
-
=
-
- -
<
) )( (
 since  . S R S P - < -   Hence, exclusion of 
non-testing cooperators requires that  . K K
S R
S P
K ”
-
-
<  
 
5.  The relationship between the equilibrium fraction of cooperators in a 
population and a population’s level of wealth 
 
  Suppose we compare two populations that are equal in all respects except that 
one population is uniformly wealthier than the other population.  By “uniformly” we 
mean that there are no distributional differences in the payoffs to strategies; the only 
difference between the two populations is that in one population the payoffs are 
uniformly higher than in the other population, say by a factor of  . 1 > m   Holding K 
constant, 
) ( P R
K w
t -
=
m
p  of the wealthier population is smaller than 
P R
K
t -
= p  of 
the less wealthy population: the equilibrium fraction of cooperators in a wealthy 
population is smaller than the equilibrium fraction of cooperators in a (uniformly) less   6
wealthy population.
1 
 
  To appreciate the nature of this outcome consider the case of 
) ( P R
K
t -
=
m
p  
where  . ¥ ﬁ m   It follows that  . 0 ﬁ t p   The implication of a rising  m  is that the 
absolute difference between the payoffs R and P becomes increasingly larger.  With 
K held constant, if  t p  were, alternatively, to  rise, the expected payoff of testing 
cooperators will increasingly distance itself from the expected payoff of defectors 
(who, it will be recalled, play only with defectors) and equilibrium will not be 
restored. 
 
  Two comments regarding recognition costs are in order.  First, for the 
equilibrium to hold,  K can take a wider range of values than before since the 
constraint pertaining to K, which is now  , K K m <  is less stringent.  Second, the 
inverse relationship between the equilibrium fraction of cooperators and the level of 
wealth holds even when K increases with wealth, provided that the increase is less 
than  m .  An increase i n wealth is due to and entails a first order increase in the 
payoffs from trade and exchange but, at most, a second order increase in the cost of 
conducting trade.  Indeed, in a population whose level of wealth is higher, the 
recognition cost could be lower (for example, a computerized credit inquiry could 
                                                 
1 To rule out the possibility that, in spite of the payoffs to every cooperator and to every defector being 
higher in the wealthier population, the payoff per capita (and, since population size is held constant, 
total wealth) will be lower in the wealthier population, the sufficient condition that 
w
t
t
p
p
m m ” >  
can be added.  (This condition arises from the requirement that the per capita payoff in the wealthier 
population will be higher than the per capita payoff in the less wealthy population: 
.) ) 1 ( ) ( ) 1 ( ) ( P
K
R P
K
R t
t
t
w
t w
t
w
t p
p
p m p
p
m p - + - > - + -    7
replace a lengthy interview).  If  ) (m K K =  and  , 0 ) ( < ¢ m K  then 
) (
) (
P R
K w
t -
=
m
m
p  
and   
 
  ;
) (
) (
) (
) (
) (
) (
2 2 P R
K
P R
K
P R
K
w
t
-
- <
-
¢
+
-
- =
¶
¶
m
m
m
m
m
m
m
p
 
 
the adverse effect of a higher level of wealth on the equilibrium fraction of 
cooperators is stronger. 
 
6.  Robustness of the cooperation-wealth relationship when the testing 
cooperators are somewhat adventurous 
 
  Suppose that a testing cooperator acts in the following manner: with probability 
1 0 £ < q  he administers the type-recognition test.  With probability  q - 1  he does 
not resort to the test and plays with whoever he happens to be paired with.  (We 
know that q cannot be equal to zero because then we will have only defectors and 
non-testing cooperators which, from ii. in section 3, cannot be the case in equilibrium.)  
We seek to find out whether the result of section 5 holds in this setting too. 
 
  The expected payoff of an adventurous testing cooperator is 
 
    .
1
) 1 ( ) 1 (
d
d d a
t q
qK S q R
V
p
p p
-
- - + -
=   (5) 
 
The proof is as follows: when testing occurs with probability q, a match will confer a 
payoff either when the test is applied (at a cost K) and the partner in the match is 
found to be a cooperator, a case in which the play yields  ], ) 1 ( [ R K q d p - + -  or when 
the test is not applied, a case in which the payoff received is  ]. ) 1 )[( 1 ( S R q d d p p + - -    8
In the event that the test is applied and the partner to the match is found not to be a 
cooperator, which occurs with probability  , d qp  no payoff is received and the entire 
procedure is repeated thereby yielding  .
a
t V   Thus, 
a
t d d d d
a
t V q S R q R K q V p p p p + + - - + - + - = ] ) 1 )[( 1 ( ] ) 1 ( [  
  .
1
) 1 ( ) 1 (
d
d d
q
qK S q R
p
p p
-
- - + -
=        
  Since the combined population share of testing cooperators who happen not to 
administer the test and of defectors is  t qp - 1 , the expected payoff of a defector is 
P
q
T
q
q
V
t
d
t
d t
d p
p
p
p p
-
+
-
- -
=
1 1
1
 or 
    ). (
1
P T
q
T V
t
d
d -
-
- =
p
p
  (6) 
 
In equilibrium, adventurous cooperators receive the same expected payoff as defectors. 
Thus, from (5) and (6), 
 
    ). (
1 1
) 1 ( ) 1 (
P T
q
T
q
qK S q R
t
d
d
d d -
-
- =
-
- - + -
p
p
p
p p
 
 
Of course,  t t t q q p p p = - + ) 1 (  and hence  . 1 = + d t p p   We therefore have that 
 
    ). (
1
1
) 1 ( 1
) 1 )( 1 (
P T
q
T
q
qK S q R
t
t
t
t t -
-
-
- =
- -
- - - +
p
p
p
p p
  (7) 
 
Evaluating this last equality at  1 = q  yields  ) ( P T T
K R
t
t - - =
-
p
p
 or 
 
.
P R
K
t -
= p  
 
By continuity this last equality holds for values of q in (7) that are in the small 
neighborhood of 1.  Hence, the cooperation-wealth relationship alluded to in section   9
5 holds also when testing cooperators apply the test with a probability that is less than, 
but close to, one. 
 
7.  Conclusion 
 
  We calculate the equilibrium fraction of cooperators in a population in which 
payoffs are received upon playing a two-person single-shot prisoner’s dilemma game; 
individuals who are hardwired as cooperators or as defectors are paired randomly; 
cooperators check, at a cost, the type of individual with whom they are paired prior to 
executing a game, and play only with cooperators; and defectors play with whomever 
they happen to be paired with.  Measuring the wealth of a population by the level of 
the payoffs in the prisoner’s dilemma game, we show that the wealthier the population 
the lower the equilibrium fraction of cooperators.   10 
References 
 
Binmore, Ken. 1998.  Game Theory and the Social Contract II: Just Playing.  
Cambridge, MA: MIT Press. 
 
Cook, Karen S. and Hardin, Russell. 2001.  “Norms of Cooperativeness and 
Networks of Trust,” in Hechter, Michael and Opp, Karl-Dieter (eds.)  Social 
Norms.  New York: Russell Sage Foundation, pp. 327-347. 
 
Stark, Oded. 1999.  Altruism and Beyond: An Economic Analysis of Transfers and 
Exchanges Within Families and Groups.  Cambridge: Cambridge University 
Press.  
 
Author: Oded Stark 
 
Title: Cooperation and Wealth 
 
Reihe Ökonomie / Economics Series 123 
 
Editor: Robert M. Kunst (Econometrics) 
Associate Editors: Walter Fisher (Macroeconomics), Klaus Ritzberger (Microeconomics)  
 
ISSN: 1605-7996 
© 2002 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 and Beyond: An Economic Analysis of Transfers and Exchanges Within Families and Groups. Cambridge:

Author: Oded Stark Title: Cooperation and Wealth Reihe Ökonomie / Economics Series 123 Editor: Robert M. Kunst (Econometrics) Associate Editors: Walter Fisher (Macroeconomics), Klaus Ritzberger (Microeconomics)

Game Theory and the Social Contract II: Just Playing.

Cooperation and Wealth

http://www.ihs.ac.at/publications/eco/es-123.pdf

Cooperation and Wealth

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