good men (final)

8/2/2019 Good Men (Final)

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A Few Righteous Men: Critical Mass in the Dynamics of Cooperation

Serge Moresi, Charles River Associates

Steven Salop, Georgetown University Law Center

I. Introduction

In the Wealth of Nations, Adam Smith (1776) wrote that it was self-interest

guided by the invisible hand of the market that leads the economy to a beneficial

outcome, not the benevolence or humanity of the butcher, the brewer or the baker.1

The development of models of strategic interaction and imperfect information by Joseph

Stiglitz and others have demonstrated the failure of the invisible hand and the ubiquity

of market failures. Rothschild (1973), Stiglitz (1975), Spence (1976), Dixit and Stiglitz

(1977). The failure of cooperation also is nicely illustrated in the one-shot Prisoners

Dilemma game, where defection is the dominant strategy.

Escape from the Prisoners Dilemma is possible with enforceable agreements,

fear of legal, religious or social sanctions, altruism and other ethical motivations.

Ullman-Margarlit (1977), Frank (1988). In a repeated play context, mutual cooperation

can be sustained through reputation formation or adopting conditional cooperation

strategies (such as TIT-FOR-TAT and GRIM).2 Shapiro (1982), Kreps and Wilson

(1982), Axelrod (1984), Taylor (1987). [Evol Equil Cite]

1 But see Smith (1790). See also Sen (2000).2A player using a conditional cooperation strategy starts cooperating in the first round of play, andcontinues to cooperate as long as his partner cooperated in the previous round. If his partner ever

defects, the player stops cooperating and punishes his partner by defecting in the next round. TheTIT-FOR-TAT and GRIM strategies differ with respect to the length of the punishment. The GRIMstrategy punishes forever by never cooperating again (even if the partner learns a lesson and starts

cooperating). The TIT-FOR-TAT strategy leaves it to the partner to decide the length of thepunishment. That is, if the partner starts cooperating while he is being punished, then TIT-FOR-TATwill resume to cooperate (so that the punishment could be as short as one round). For other kinds of

conditional cooperation strategies, see Shubik (1959) and Taylor (1987).


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If individuals interact in multiple business relationships over time, then conditional

cooperation strategies would require information about previous interactions with other

partners. In the absence of this information, incentives to cooperate would seem to be

reduced. Someone who defects in a current relationship can exit and enter a new

relationship tomorrow. His incentive to cooperate today might be reduced even further

if the new partner will not know that he defected in a previous relationship.

In the context of potentially long-term relationships, cooperators make desirable

partners because they do not defect. Suppose there is a mass of ethical or religious

people who never defect (and who leave their partner if he defects). These righteous

individuals might increase or decrease the incentives of others to defect. On the one

hand, they might be the proverbial suckers-born-every-minute. A defector could exploit

this current cooperating partner and then exit the current relationship and find a new

cooperating sucker to exploit tomorrow. On the other hand, they might spur cooperative

behavior if it is the self-interest of others to cooperate in new relationships in the hope of

achieving a long-term relationship with a righteous individual. Salop (1978). In this

case, ethics and religion would be public goods.

In this paper, we analyze this public goods issue. We show that under certain

conditions, the adoption of religious/ethical norms by a critical mass will lead to a

cooperation bandwagon and increase the equilibrium amount of cooperation. In other

words, a critical mass of committed moral people can generate a moral rebirth for

society at large.3

3For example, in Genesis, God informs Abraham that He is going to destroy Sodom and Gomorrahbecause of its evil behavior. Abraham asks whether God would save the city if there were as few asfifty righteous inhabitants. When God assents, Abraham then asks about the outcome if there were

only forty-five (or forty or fewer) righteous inhabitants, and God ultimately concludes that the city would


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The paper is organized as follows. In Sections II and III, we set up the basic

model without any righteous individuals and solve for the steady-state equilibria. In this

model, there is always an equilibrium with 100% defection. Unlike standard models

(where players cannot change partner), there is never an equilibrium with 100%

cooperation. The incentive for opportunism is too strong. Under certain conditions,

there are also two partial cooperation equilibria, with different incidences of cooperation.

In Section IV, we discuss the main properties of these partial cooperation equilibria.

In Sections V and VI, we examine the impact of adding a mass of righteous

individuals into this model. Any mass of righteous individuals will prevent the total

defection equilibrium. In fact, if the mass of righteous individuals is sufficiently large,

then it will create a dynamic bandwagon effect that leads to a stable partial cooperation

equilibrium with the higher incidence of cooperation. In addition, we show that the

critical mass of righteous individuals (that is needed to induce other individuals to also

cooperate) tends to zero as the individuals discount rate tends to zero. Thus, it is true

that a moral commitment by a few righteous men can lead to a large increase in the

degree of morality in the long run. However, it is not powerful enough to lead to

universal cooperation.

II. Basic Model

At the beginning of the first period, a large number of ex ante identical individuals

are randomly paired. Each pair then plays the following (symmetric) Prisoners

Dilemma game: The players simultaneously decide whether to cooperate or defect. For

now, we assume that there are no commitments. Each obtains a payoffCif they both

be saved if there were at least ten moral inhabitants. Genesis (18;16-19:38). One possible


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cooperate or Dif they both defect. If one player cooperates and the other defects, the

cooperators payoff is S(for sucker payoff)and the defectors is T(for temptation

payoff). We assume T> C> D> S, so that the game is a Prisoners Dilemma.4

At the end of each period, after observing their payoffs, the two players

simultaneously decide whether to stay matched in the relationship and play together

again next period, or whether to exit the relationship and obtain a new partner from the

pool of unmatched individuals. The players exit and are re-paired with a different

partner unless both decide to continue with the relationship. If both decide to remain in

the match, they play the same game again in the next period. They again will choose

simultaneously whether to cooperate or defect. That is, we assume that individuals in a

relationship also cannot make binding commitments about their current or future

behavior. Similarly, if both exit and return to the pool of unmatched individuals, each

plays again with a different partner.

We assume that a new generation of individuals is born and arrives i n the market

at the beginning of each period. The new generation enters the pool of unmatched

individuals and joins the pairing process. This assumption ensures that there will

always be a large number of unmatched individuals. At the same time, a fraction of the

population may die and exit from their relationships or the unmatched pool. The impact

of birth and death rates on the equilibria is discussed in Section IV. Individuals

interpretation of this story is that the ten moral inhabitants might be enough to create a moral rebirth.4

Axelrod (1984) makes the additional assumption that 2C > T+S. As shown in Kuhn and Moresi (1995),this additional assumption ensures that mutual cooperation is Pareto efficient relative to the set of

mixed strategies. In Axelrods model, it also ensures that reverse TIT-FOR-TAT is nota bestresponse to TIT-FOR-TAT. In our model, however, this additional assumption is not necessarybecause cooperation can be sustained by threatening to leave a defecting partner (as opposed to

retaliate with TIT-FOR-TAT).


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evaluate future payoffs using a common discount factor, ? .5 For most of the analysis,

we assume 0 < ? < 1.

We focus our attention on steady-state equilibria in which there is a constant

fraction ? of unmatched individuals in each period who choose to cooperate with their

(new) partner. Similarly, among the individuals in relationships (i.e., the pairs who

decided to stay together), a constant fraction ? of them cooperate with their (long-term)

partner in the equilibria. In what follows, we derive the conditions under which ? > 0

and? = 1.

For an individual, a strategy is a complete plan of action, i.e., a set of decisions

rules that specify the individuals behavior in every contingency in which the individual

might be called on to act. In other words, an individuals strategy must specify what the

individual will do in all possible circumstances. In particular, it must provide an answer

to the following questions: (a) When the individual arrives in the market and is paired

up with his first partner, will he cooperate or defect? (b) At the end of that first period,

will he offer to remain with his partner for another period, or will he exit and re-enter the

pool of unmatched individuals? (c) If he decides to exit the relationship and obtain a

new partner, will he defect or cooperate with the new partner? (And at the end of that

period, will he offer his new partner to stay together or will he leave him too?) (d) If both

decide to remain matched for another period, will he cooperate or defect in that next

period? (And at the end of that period, will he exit or will he offer to remain matched?)

We assume that information is imperfect. Decisions in previous relationships

are private information.6 As a result, the individual cannot condition his decision on his

5The discount factor ?reflects both the rate of time preference and the mortality rate.


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partners history in previous relationships.7 This assumption is key to our results. It

rules out a conditional cooperation strategy of punishing (or rewarding) new partners

based on their conduct in past relationships with others. This assumption is central to

our analysis because it implies that universal cooperation is never an equilibrium

outcome, unlike the situation in standard models. In our model, a defector can exit his

current relationship and start fresh by reentering the pool of unmatched individuals

incognito.

We do assume that previous decisions within the relationship are known to both

matched players. Thus, both the decision to cooperate or defect next period and the

exit decision can be made conditional on the partners behavior in the current period.

III. Steady-State Equilibria

There is always an equilibrium with zero cooperation. There is never an

equilibrium with universal cooperation. They may be partial cooperation (i.e., less than

universal cooperation) equilibria.

Theorem 1. There is an equilibrium with zero cooperation (i.e., ? = ? = 0).

The proof of Theorem 1 is straightforward. Consider the following strategy: In

each period, I defect and then change partner. Clearly, if everybody is using that

strategy, nobody has an incentive to use a different strategy. Therefore, if everybody

uses that strategy then the market is in equilibrium and no cooperation ever arises.

Theorem 2. Universal cooperation (i.e., ? = ? = 1) is notan equilibrium.

6Nor does an individual know whether his partner is a new entrant or not.

7One could allow his decision rule to be conditional on his own history. However, this would not change

the set of equilibrium outcomes.


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The proof of Theorem 2 is also straightforward. If all the unmatched people

always cooperate, each individual has an incentive to defect and change partner in

every period. Thus, universal cooperation is not an equilibrium.

While universal cooperation is not possible, under certain conditions, there are

partial cooperation equilibria. By partial cooperation, we mean a situation in which

only a fraction of the unmatched individuals cooperate (i.e., 0 < ? < 1), and all the

matched individuals in long -term relationships cooperate (i.e., ? = 1). We now derive

the necessary and sufficient conditions for these equilibria to exist.

Let EW(d) be the expected present value of the current and future payoffs of an

unmatched individual who decides to defect on the partner he is paired with in the

current period. Similarly, let EW(c) be the expected present value of the current and

future payoffs of an unmatched individual who decides to cooperate with his current

partner. We then have:

EW(d) = ? T + (1-?)D + ? max{EW(d),EW(c)}, (1)

where ? T+(1-?)Dis the expected payo ff from defecting in the current period, given that

a fraction ? of unmatched individuals cooperates and a fraction 1-? defects. The term ?

max{EW(d),EW(c)} is the present value of future payoffs, given that the individual will

re-optimize his strategy at the beginning of next period. (We will check later that an

individual who defects will indeed be thrown back into the pool of unmatched

individuals.)

Similarly, if an unmatched individual cooperates, we have:

EW(c) = ? C/(1-?) + (1-?)[S +? max{EW(d),EW(c)}]. (2)


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Intuitively, his partner will also cooperate with probability ? , and the two players will

decide to enter a long-term relationship of mutual cooperation. (We will check later that

they indeed have an incentive to stay together and cooperate.) Thus, the expected total

payoff is equal to C/(1-? ) with probability ? . With probability (1-? ) his partner defects

and the two players then exit and re-enter the pool of unmatched individuals. (We will

check later that they indeed have an incentive to terminate their relationship.) The

expected total payoff is then equal to Sin the current period plus ? max{EW(d),EW(c)}

in future periods.

In any equilibrium with partial cooperation, an unmatched individual must be

indifferent between defecting and cooperating (since individuals are identical ex ante),

that is,

EW(d) = EW(c). (3)

One can show that Equation (3) is quadratic in ? , i.e., of the form:

A? 2- B? + F = 0, where (4)

A =? (T - D), F = (1-?)(D - S ) and B = A + F - (T - C). (5)

By the quadratic formula, Equation (4) has two solutions, i.e.:

? 1 = [B - (B2 4AF)1/2] / (2A), and (6)

? 2 = [B + (B2 4AF)1/2] / (2A). (7)

There are two real solutions (? 1and ? 2) that lie between 0 and 1, if and only if:8

B2> 4AF, B > 0, and 2A > B + (B2-4AF)1/2. (8)

8One can show that the two solutions are either both smaller than 1 or both greater than 1.


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These conditions are necessary for a steady-state equilibrium with partial

cooperation to exist. However, an additional condition is needed for sufficiency. In

particular, we need to show that ? =1, that is, that a matched individual prefers to

cooperate again with his (old) partner rather than defect on the second play and go

back to the pool of unmatched individuals. It is better to cooperate again if and only if:

C/(1-?) ? T + ? EWi, (9)

where EWi is the expected present value of the current and future payoffs of an

unmatched individual when ? =? i.

Intuitively, suppose that all the pairs of matched individuals cooperate and stay

together. The expected present value of the current and future payoffs of a individual

that remains matched in a long-term relationship equals C/(1-?). If a matched individual

instead unilaterally decided to defect on his old cooperating partner, his payoff would be

equal to Tin the current period (since his old partner is cooperating) plus ? EWi in

future periods (since his old partner would exit at the end of the current period).

Equation (9) thus says that a matched individual prefers to maintain mutual cooperation

with his old partner rather than defect and go back to the pool of unmatched individuals.

Using Equations (1) and (3), we have:

EWi = [? iT + (1-? i)D] / (1-?). (10)

This is the expected utility of an unmatched individual who defects and changes partner

in every period. Substituting Equation (10) into Equation (9), we have the condition for

a player to prefer to remain in a cooperative relationship,

T - C ? ? (1-? i) (T D). (11)


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This is an additional necessary condition for a partial cooperation equilibrium to exist.

This also leads to the following theorem and corollary.

Theorem 3. There are two symmetric partial cooperation mixed strategy

equilibria if and only if the conditions in Equations (8) and (11) are satisfied. In that

case, the common probability of cooperation among unmatched individuals in the two

equilibria (? 1 and ? 2) are given by Equations (6) and (7), respectively.

Corollary to Theorem 3. There are two partial cooperation pure strategy

equilibria in which a fraction ? 1 (or ? 2) of unmatched individuals cooperate and a

fraction 1-? 1 (or 1-? 2) defect with certainty.

To summarize, partial cooperation can be sustained as an equilibrium by

adopting the following three-pronged strategy: (1) When placed in the pool of

unmatched individuals with a new partner, the player cooperates with probability ? (and

defects with probability 1-? ). (2) If the current outcome is not mutual cooperation, then

the player exits the relationship and re-enters the pool of unmatched individuals. If the

current outcome is mutual cooperation, then the player offers to stay together for

another period. (Of course, if his partner decides to exit, then the player re-enters the

pool of unmatched individuals.) (3) If both players in a period each choose to stay

together, then the player cooperates with probability 1 in the next period. (After that

play, the player follows Rule 2 again to decide whether or not to exit the relationship.) 9

9The partial cooperation equilibria can also be supported by an alternative three-pronged strategy where

both Rule 2 and Rule 3 are different from those described above. The modified Rule 2 is: If his partnerdefected in the current period, the player terminates the relationship and re-enters the pool ofunmatched individuals. If instead his partner cooperated, the player offers to stay together one more

period. The modified Rule 3 is: If both players in a period each choose to stay together then, in thenext period, the player cooperates with probability 1 if both players always cooperated (and he defectswith probability 1 if they did not always cooperate). If all the individuals change both Rule 2 and Rule 3

in this way, the market remains in equilibrium and the outcome is not affected. One can prove this


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The proof of Theorem 3 (and the Corollary) is contained in the Appendix. In a

nutshell, the proof involves showing that deviating from the three-pronged equilibrium

strategy reduces a players returns, when the conditions in Equations (8) and (11) are

satisfied.10 For example, when T= 5, C= 3, D= 1 and S= 0, the conditions are

satisfied if ? = 0.9. The conditions in Theorem 3 are less likely to be satisfied for lower

discount factors.11

IV. Equilibrium Properties

We assume hereafter that the conditions of Theorem 3 are satisfied at both ? =? 1

and ? =? 2, so that there are two equilibria with partial cooperation (in addition to the

equilibrium with no cooperation).

The following comparative statics results are straightforward and are thus stated

without proof:12

i) An increase in the discount factor (? ) reduces ? 1 and increases ? 2.

ii) As?

tends to 1,?

1 tends to zero and?

2tends to (C-D)/(T-D).

iii ) An increase in the payoff from mutual cooperation (C) reduces ? 1 and increases

? 2.

iv) As Ctends to T, ? 1 tends to F/A and ? 2tends to 1.

These results are generally intuitive. As individuals become more patient, the

high cooperation equilibrium incidence of cooperation (? 2) increases. However, even

when individuals are infinitely patient (i.e., as ? ? 1), the high cooperation equilibrium

version of Theorem 3 by showing that the modified three-pronged strategy is also a symmetric

equilibrium strategy. The proof is available upon request from the authors.10

We ignore the possibility of a single partial cooperation equilibrium when B2

= 4AFin Equations (6) and(7). As discussed in the section VI, this equilibrium is unstable.


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falls short of universal cooperation (i.e., ? 2< 1). This is not surprising in light of

Theorem 2. The result that the high cooperation equilibrium increases with the payoff

from mutual cooperation (C) is also very intuitive. Not surprisingly, as Ctends to T, the

temptation to defect vanishes and universal cooperation becomes possible in

equilibrium.

We now turn to the population dynamics. The steady-state equilibria involve a

constant fraction of cooperating individuals in the unmatched pool in every period. With

constant and exogenous birth rate band mortality rate m, the population growth rate is

equal to b-m.13

Given that a fraction ? of unmatched individuals cooperate, one can

show that the fraction Rof unmatched individuals in the entire population converges

to:14

11In the example, the equilibrium conditions are not satisfied if ? = 0.8.

12These results follow directly from differentiating Equations (7) and (8).

13If b


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R = [b + (1-m)m] / [b + (1-m)m + (1-m)2? 2)]. (13)

Thus, as long as there are positive births and deaths, the pool of unmatched individuals

does not disappear. The steady-state fraction of unmatched individuals, R, is higher if

the birth rate is higher, if the mortality rate is higher, or if the fraction of cooperative

unmatched individuals is lower (all else equal).

The mortality rate also affects the discount rate, i.e., the discount factor ? equals

(1-m)/(1+r), where ris the rate of time preference. A higher mortality rate thus implies a

lower discount factor, which in turn affects the equilibrium fraction of cooperative

unmatched individuals, increasing ? 1 and reducing ? 2. Comparing the high cooperation

equilibria, an increase in the mortality rate leads to a new steady state in which the

unmatched individuals cooperate less (? 2falls)and account for a greater fraction of the

population (Rrises). The fraction of defectors in society equals (1-? 2)R, and hence

increases with the mortality rate.15

V. Committed Cooperators

In this section, we assume that the market is initially in a steady-state equilibrium

with zero cooperation (i.e., ? = ? = 0) and the entire population is in the pool of

unmatched individuals (i.e., R=1). We then assume that a fraction ?of the population

become righteous individuals (i.e., committed cooperators). The behavior of righteous

individuals is exogenous; they cooperate in every period and change partner if and only

if their current partner defects. We assume that righteousness reproduces itself in

Ris a stable steady state since the derivative of Rt+1 with respect to Rt is less than 1.15

Comparing the low cooperation equilibria, an increase in the mortality rate leads to a new steady state

in which the unmatched individuals cooperate more (? 1 rises). In this case, the effect on the fraction R


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families; the offspring of righteous individuals also are righteous. The remaining fraction

1-? of the population are maximizers and their behavior is endogenous (as in the

previous sections).

We assume that an individuals type (i.e., whether he is righteous or a maximizer)

is private information. However, it is common knowledge that a fraction ?of the current

population are righteous individuals. In addition, it is also common knowledge that

maximizers and righteous individuals have the same mortality rate (m) and the same

birth rate (b), so that the fraction of righteous individuals in the entire population remains

constant over time.

Suppose for the moment that the maximizers do not change their behavior

despite the presence of righteous individuals. That is, each maximizer continues to

defect and change partner in every period. We then can derive the following

relationship between the fraction Rtof unmatched individuals in the entire population (in

period t) and the fraction ? tof cooperating individuals in the pool of unmatched

individuals:

1 - Rt + Rt? t = ?. (14)

Note that in the period in which the righteous individuals are first introduced into the

model (say, period T), we have ? T=? and RT=1. Then, in the following periods (i.e.,

periods T+1, T+2, etc.), both ? t and Rtfall over time as righteous individuals are

randomly matched together and enter long-term relationships.

of unmatched individuals is ambiguous. The ? 1 equilibrium, however, is unstable, as explained in

Section VI.


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One can show that, if the maximizers continue to defect and change partner in

every period, then Rt and ? twill converge to R(?) and ? (?), respectively, where R(?) and

? (?) are the solution of Equations (13) and (14). In particular, ? (?) is the steady-state

fraction of righteous individuals in the pool of unmatched individuals as a function of the

fraction ? of righteous individuals in the entire population. One can then show that ? >0

implies ? (?) >0. Thus, the market outcome will never converge to a steady state

equilibrium with zero cooperation. There will always be at least a fraction ? (?) of

unmatched individuals who cooperate (even if all the maximizers defect).

Consider the case in which the fraction of righteous individuals is relatively small,

i.e., ?< ? 1. It can nonetheless have an effect on the equilbrium selected. The two

partial cooperation equilibria described in Theorem 3 are still equilibria. For example,

the market outcome could move to the high cooperation equilibrium if a fraction (? 2-

?)/(1-?) of maximizers decided to cooperate in period T(so that a fraction ? 2of

unmatched individuals would be cooperating in period T). In period T+1, and in each of

the following periods, the fraction of unmatched individuals who cooperate could then

remain equal to ? 2. Thus, the introduction of righteous individuals into the model could

result in the market moving immediately from the zero cooperation equilibrium to the

high cooperation equilibrium.16

In fact, such an immediate jump to the high cooperation equilibrium is possible

even without introducing any righteous individuals. This is because there is no inertia

in the pool of unmatched maximizers. That is, an unmatched maximizer is constantly

16In this case, the fraction of matched individuals in the entire population and the fraction of cooperatingindividuals among unmatched maximizers would gradually increase over time and converge to their

steady-state level.


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re-optimizing his behavior and nothing in the model prevents him from switching to

cooperation overnight (if that is what he wants to do). More realistically, however,

individuals do not constantly re-optimize their behavior, and hence the transition from

zero cooperation to high cooperation would not occur overnight. We return to this point

in Section VI.

Consider next the case in which the fraction of righteous individuals is relatively

large, i.e., ?> ? 1, and assume for simplicity that ?< ? 2. As explained above, the high

cooperation equilibrium (? =? 2) is still an equilibrium and can be reached immediately.

However, if ? (?)>? 1, the low cooperation equilibrium (? =? 1) will no longer be a steady-

state equilibrium. This also leads to the following theorem.

Theorem 4. For constant birth and mortality rates (b,m), if ? >?*then the only

steady state equilibrium is the high cooperation equilibrium, where:

2

1

2

1

2

1)1()1(

)1()1(*

?

???

mmmb

mmmb

????

????? . (15)

The proof of Theorem 4 is straightforward. As explained above, ? (?) is the

steady-state fraction of righteous individuals in the pool of unmatched people as a

function of the fraction ? of righteous individuals in the entire population, and assuming

that all the maximizers always defect and change partner. Thus, the inverse function

?(?) is the fraction of righteous individuals in the entire population that is needed to

ensure that the fraction of cooperating unmatched individuals will always be greater or

equal to ? .17 The right-hand side of Equation (15) is equal to ?(? 1), and thus the fraction

of cooperating unmatched individuals will always be greater than ? 1 if ? >?*(even if all


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the maximizers always defect). It follows that the only steady-state equilibrium is the ? 2

equilibrium.

Corollary of Theorem 4. As the discount factor ? tends to 1, the critical mass ?*

of righteous individuals tends to zero. In the limit, the only steady-state equilibrium is

the high-cooperation equilibrium regardless of how small the fraction of righteous

individuals.

This result supports the view that a few righteous men may well be enough to

move society from no cooperation to a high incidence of cooperation.18

VI. Stability and Dynamics

In this section, we consider the alternative assumption that maximizers do not

constantly re-optimize their behavior. This alternative assumption is a useful device to

assess the stability and dynamics of the equilibria described above. Initially, suppose

that each maximizer must choose a strategy only once in his lifetime. In particular,

suppose that when he first arrives into the market, he must decide once and for all for

his whole life whether to be either (i) meanspirited, that is, always defect and change

partner in every period, or (ii) gracious, that is, always cooperate in every period and

change partner if and only if his current partner defects. The offspring of the original

mass of righteous individuals are all righteous themselves. Therefore, in each period,

the only individuals who choose their behavior are the new generation of maximizers.

As above, we assume that an individuals type is private information.

Figure 1 shows the expected payoff of a maximizer who decides to become

gracious. His expected payoff is the line labeled EWGand is a function of the fraction ?

17The function ?(?) can be obtained by using Equation (13) to substitute for Rin Equation (14).


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of unmatched individuals who cooperate. Similarly, the line labeled EWM is the

expected payoff of a maximizer who decides to become meanspirited.19 One can check

that the results of the previous sections still hold. On Figure 1, the two points where the

two lines intersect correspond to the two partial cooperation equilibria (and the origin to

the zero cooperation equilibrium).

In addition, Figure 1 shows that the partial cooperation equilibrium with a low

incidence of cooperation (? 1) is unstable. Indeed, if ? is slightly smaller (greater) than

? 1, then EWG is smaller (greater) than EWM. Thus, starting at ? =? 1, if ? falls (increases)

by a small amount, then the new generations of maximizers will choose to be

meanspirited (gracious) which will further reduce (increase) the fraction ? of

cooperating unmatched individuals. If the market is initially to the left of ? 1, the market

will converge to the zero cooperation equilibrium (assuming that there are no

exogenously righteous cooperators among the new generations of individuals). If

instead the market is initially to the right of ? 1, then it will converge to the high

cooperation equilibrium (? 2).20 In this model, the maximizers make a one-time decision

of whether to act cooperatively in all their relationships. In contrast, in the basic

equilibrium model with committed cooperators, the maximizers choose each period

18The proof follows directly from Equation (15) and the fact that ? 1 tends to 0 as ? tends to 1.

19 EWMand EWGare different from EW(d) and EW(c) (as defined by Equations (1) and (2)) because here

a maximizer must choose whether to defect or cooperate and must then stick to his decision for therest of his life. However, the equation EWM= EWGhas the same two solutions, ? 1 and ? 2, as theequation EW(d) = EW(c). Intuitively, if an unmatched individual is indifferent between cooperating and

defecting, then it does not matter whether the individual can or cannot re-optimize his decision infuture periods.

20It is standard practice in stability analysis to assume that individuals react to each other sequentially

and myopically. That is, each maximizer (among the new generation) sequentially chooses whether to

become gracious or meanspirited, assuming that the fraction ? of cooperating unmatched individuals

will remain constant at its current level. Under this assumption, the critical mass ?*of Theorem 4 can

be reduced to ?*=? 1.


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whether to act cooperatively. These two models can be viewed as extreme cases along

a continuum. That is, an intermediate model could assume that the maximizers make

an initial choice but then revisit their choice with some probability each period. That

model would behave similarly to the model set out in this section.


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References

Axelrod, Robert. 1984. The Evolution of Cooperation. New York: Basic Books.

Dixit, Avinash and Stiglitz, Joseph E. 1977. Monopolistic Competition and Optimum

Product Diversity. American Economic Review67, 297-308.

Frank, Robert. 1988. Passions Within Reason: The Strategic Role of the Emotions. NewYork: W.W. Norton.

Kreps, David L. and Wilson, Robert. 1982. Reputation and Imperfect Information.Journal of Economic Theory27, 253-79.

Kuhn, Steven and Moresi, Serge. 1995. Pure and Utilitarian Prisoners Dilemmas.Economics and Philosophy11, 333-43.

Rothschild, Michael. 1973. Models of Market Organization with Imperfect Information: ASurvey. Journal of Political Economy81, 1283-1308.

Salop, Steven C. 1978. Parables of Information Transmission in Markets. In The Effectof Information on Consumer and Market Behavior, ed. A. Mitchell: American MarketingAssociation.

Sen, Amartya. 2000. What Difference Can Ethics Make?Presented at the InternationalMeeting on Ethics and Development of the Inter-American Development Bank.http://www.iadb.org/etica/ingles/lis-doc1-i.cfm.

Shapiro, Carl. 1982. Consumer Information, Product Quality, and Seller Reputation. BellJournal of Economics13, 20-35.

Shubik, Martin. 1959. Strategy and Market Structure: Competition, Oligopoly, and theTheory of Games. New York: Wiley.

Smith, Adam. 1776 (republished 1910). An Inquiry into the Nature and Causes of theWealth of Nations.London: Dent & Sons.

Smith, Adam. 1790 (republished 1976). The Theory of Moral Sentiments. Oxford:Clarendon Press.

Spence, A. Michael. 1976. Product Selection, Fixed Costs, and Monopolistic Competition.Review of Economic Studies43, 217-35.

Stiglitz, Joseph E. 1975. Information and Economic Analysis. In Current EconomicProblems,eds. Parkin and Nobay, Cambridge: Cambridge University Press, 27-52.


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Taylor, Michael. 1987. The Possibility of Cooperation. Cambridge: Cambridge UniversityPress.

Ullmann-Margalit, Edna. 1977. The Emergence of Norms. Oxford: Clarendon Press.

< Add Cite to Evolutionary Equilibrium article>


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Appendix: Proof of Theorem 3

Suppose that every individual uses the three-pronged strategy with ? =? i (where

either ? i=? 1for every individual, or ? i=? 2for every individual). The proof of Theorem 3

consists in showing that no individual has an incentive to unilaterally change strategy if

the conditions in Equations (8) and (11) hold.

We begin with a few observations. First, the fraction of unmatched individuals who

cooperate is equal to ? i (since there is a large number of unmatched individuals).

Second, if an unmatched individual unilaterally decides to defect, then his expected

payoff is given by Equation (1) since his current and future partners are using Rule 1

and Rule 2. If instead he decides to cooperate, then his expected payoff is given by

Equation (2) if the individual himself is using Rule 2 and Rule 3. Therefore, if the

individual uses Rule 2 and Rule 3, then he has no incentive to (unilaterally) change Rule

1. Indeed, the individual is indifferent between defecting and cooperating with a new

partner. This follows from the conditions in Equation (8) and the definition of ? i. We are

thus left with proving that the individual has no incentive to change Rule 2 or Rule 3 (or

both).

Third, according to Rule 2, if the current relationship is not one of mutual

cooperation, then the individual terminates the relationship. If instead the individual

decides to offer his partner to stay together (despite the lack of mutual cooperation), the

individuals expected payoff does not change because his partner terminates the

relationship anyway (given the lack of mutual cooperation). Thus, the individual has no

incentive to change Rule 2 in this way (even if he can also change Rule 1 and Rule 3 at

the same time).


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Fourth, according to Rule 2, if the current relationship is one of mutual

cooperation, then the individual offers his partner to stay together. Suppose that the

individual instead decides to terminate the relationship, and let EWibe his expected

payoff evaluated at the beginning of the following period. Consider now the alternative

decision of staying with his old partner one more period and defect on him in that

period. This would give the individual a payoff equal to T +? EWi, which is necessarily

greater than EWi. (Indeed, EWi > T +? EWiis not possible since it implies EWi > T/(1-

?), and T/(1-?) is the maximum feasible payoff.) Therefore, it is not a rational decision to

terminate a relationship of mutual cooperation (given that the partner is using the three-

pronged strategy). It follows from this observation (and the previous one) that the

individual has no incentive to change Rule 2. We are left with checking the rationality of

Rule 3.

According to Rule 3, if an individual stays with his partner, then he cooperates.

Given that his partner is cooperating and willing to stay together, the individual will have

no incentive to terminate the relationship. Thus, if the individual follows Rule 3, the two

players will stay together and cooperate in every period. This gives the individual an

expected payoff equal to C/(1-?).

Suppose that the individual instead decides to defect on his old partner. Since

the latter is using Rule 3 and Rule 2, the individual obtains a payoff equal to Tin the

current period and then goes back to the pool of unmatched people. There are two

cases to consider:

a) Suppose it is optimal to defect on new partners. Then, the individuals

expected future payoff is equal to ? EWi, where EWi is given by Equation (10). Thus,


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his total payoff from defecting on his old partner is equal to T +? EWi, which is not

greater than C/(1-?) because of condition (11). Therefore, in this case, the individual

has no incentive to change Rule 3.

b) Suppose instead that it is optimal to cooperate with new partners. Then, the

individuals expected future payoff is equal to ? EWi, where:

EWi = (1-? i)(S +? EWi) +? i(C +? (T +? EWi)). (A1)

Intuitively, with probability 1-? i, his new partner will defect and thus his future payoff will

equal Sin that period plus ? EWiin the following periods. With probability ? i, his new

partner will cooperate and thus his future payoff will equal Cin that period plus ? (T +?

EWi) in the following periods. (If it is rational to defect on the current old partner, it will

also be rational to defect on future old partners.) Thus, his total payoff from defecting

on his old partner equals T +? EWi, where EWiis given by Equation (A1). One can

then check that T +? EWiis smaller than C/(1-?) because of condition (11). This

shows that the individual has no incentive to change Rule 3. This also completes the

proof of Theorem 3.

Finally, the Corollary of Theorem 3 follows from three observations. First, the

pure-strategy and mixed-strategy equilibria differ with respect to Rule 1 only. Second,

in a pure-strategy equilibrium, each individual faces the same decision problems as in

the corresponding mixed-strategy equilibrium (because there is a large number of

unmatched individuals). Third, each unmatched individual is indifferent between

cooperating and defecting. Thus, any strategy profile where Rule 1 results in a fraction

? iof cooperating unmatched individuals is an equilibrium strategy profile (holding Rule 2

and Rule 3 unchanged).


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Figure 1: Steady-State Equilibria(T=35,C=25, D=10, S=0, d=0.9)

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EWM

EWG

aa1 a2

good men (final)

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