Tilburg University

Are moral objections to free riding evolutionarily stable?

Güth, W.; Nitzan, S.

Publication date:1993

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Citation for published version (APA):Güth, W., & Nitzan, S. (1993). Are moral objections to free riding evolutionarily stable? (CentER DiscussionPaper; Vol. 1993-2). CentER.

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CBMR

84141993

2

~i i~ - DISCUSSIOri~;o~ paper~ ' Research

i!n~i Niii Nim ni n~ nm um ni~;u~ ~i

. ~.-~

No. 9302

ARE MORAL OHTECITONS TO FREERIDING EVOLiTITONARILY STABLE?

try Werner Guthand Shmuel Nitzan

January 1993

ISSN 0924-7815

ARE MORAL OBJECTIONS TO FREE

RIDING EVOLUTIONARILY STABLE?~

Werner Giith

Department of Economics

L'niversity of Frankfurtt

and

Shmuel Nitzan

Department of Economics

Bar Ilan Universityt

November 1992

'This paper waa written while both authore were viaiting the CentER for Economic Research at

Tilburg University. We are indebted to CentER Cor its aupport and excellent reaearch environment.

14'erner Guth gratefully acknowledgea the generous Alexander von Humboldt-award via the Dutch Sci-

ence Foundation (N14'O) which made his stay at CentER poesible.

tP.O. Box 11 13 32 (Fach 81), D-6000 Franlcfurt~M., Federal Republic of Germany

tRamat Gan, 52900, Israel

Abstract

In this study basic preferences, namely the utiGty impGcation of ftee riding, are

determined endogenously. By assuming individual rationality with respect to given

preferences, the motivational structure results from an evolutionary analysis which

compares the relative reproductive success of all possible preference types. The

tastes that emerge are usually different from those assumed in models resorting to

altruism, social sense of guilt, or moral obligations. In general, an effective social

conscience preventing (ree riding is not evolutionarily stable. Thus, voluntazy

conttibutions to the provision of public goods are not to be expected. In a more

general framework there are, however, two notable exceptions, namely unanimity

games and two person games where a contribution by a single player is sufficient

to produce the public good.

1

One of the most prominent social dilemmas arises when economic agents are asked to

voluntarily provide a public good. In such games the possibility of nobody contributing

(or contributing sufficiently) is a rather stable and viable one, e.g., in the sense of being

a strict equilibrium, that is, a combination of strategies where every player suffers from a

unilateral deviation. The dilemma is due to the existence of other strategy configurations

that can makeeverybody better off. Unlike in the so called prisoners' dilemma situations,

in binary games the players' equilibrium strategies need not be dominant strategies.

Typically, both contributing and not-contributing can be the unique best reply to the

others' behavior.

A rather unsatisfactory way to overcome the dilemma associated with private provi-

sion of public goods is to assume that people have moral commitment to social cooper-

ation or moral objections against free riding. In other words, to achieve better (more

efficient) allocation of public goods, one introduces a utility determinant which repre-

sents social conscience: utility is derived from acts of social cooperation or contribution,

or conversely, disutility is associated with non-cooperative behavior such as free riding.

James Andreoni (1989), for example, studies charity giving assuming impure altruism

where people derive some utility ("warm glow" ) from the act of giving. Thomas Palfrey

and Howard Rosenthal (1988) assume that the altruistic components of utility are drawn

from a distribution that is identical for all individuals and common knowledge. In their

game of incomplete information they analyze the effect of altruism on behavior. More

recently, Heinz Hollander (1990) offers an axiomatic model of social exchange in which

cooperative behavior is motivated by the expectation of emotionally prompted social

approval.

Although we do not deny that such motivational factors exist, there is a lot of evidence

that they are often insufficient to resolve the social dilemma. More importantly, such an

approach simply transforms the problem of resolving the social dilemma to the problem

of rationalizing or explaining the emergence of such preferences that alleviate or resolve

the social dilemma.

There have been several attempts to determine endogenously the players' motivational

structure or incentives to cooperate. For example, Robert Frank (1987) focuses on the

2

problem of identifying the kind of preferences that maximize the attainment of selfish

objectives. Guttman, Nitzan and Spiegel (1992) and Mark Gradstein (1992) study the

possibility of altruistic preferences determination by the individuals themselves in a

sequential strategic game with complete information. Preferences are determined in the

earlier stage of the game, e.g., by investment in education, and the public good provision

is determined in the later stage of the game. An alternative approach is pursued by

Ingmar Hansson and Charles Stuart (1990). Since social biology tells us (see, for example,

Richard Dawkins (1976) or Edward Wilson (1980)) that preferences and, in particular,

benevolence or altruism are mainly adoptive, they study natural selection of preferences

using a golden-age model with endogenous population in which in a long-run equilibrium

all agents have preferences with maximum biological fitness. Our study can be viewed

as another attempt of this latter approach. However, the endogenous determination of

the motivational structure in the current paper is based on arguments of evolutionary

stability in strategic settings.

Whereas in evolutionary biology, John Maynard Smith (1982), a genotype is assumed

to determine behavior, we rely on the modified evolutionary approach originally proposed

by ~~'erner Guth and Menachem Yaari (1992). According to this approach genetical

evolution determines preferences or motivational forces and players behave rationally

with respect to any given preferences. Equipped with this methodology, Guth and Yaari

(I992) are able to justify the existence of bilateral reciprocal incentives, i.e., a desire

to exchange in kind. Werner Guth (1990) generalizes this result to situations where

the other's type is not known. That is, one dces not know whether the other player is

eager to exchange in kind or not when interacting with him. Using a similar approach

Werner Guth and Klaus Ritzberger (1992) are able to rationalize resistance against mass

immigration and Werner Guth and Hartmut Kliemt (1992) explain cooperation based

on trust. Given these "positive" results, it is tempting to investigate also in the context

of voluntary provision of public goods whether a moral objection against free riding is

evolutionarily stable and whether its existence is sufficient to overcome the dilemma

inherent in such situations.

In the next section we present the basic model of voluntary contributions to the

3

provision of a discrete public good in the presence of two types of individual social

conscience. The unique solution of this multi-player basic game is derived in Section II.

Our evolutionary approach is then described in Section III. Our multi-agent adaptation

of the evolutionary approach (proposed by Guth and Yaari (1992) in the context of

two interacting agents) involves a number of problems that need to be addressed. In

particular, the definition of evolutionarily stable strategies must be modified. There

exist alternative specifications that we consider and compare to each other. Sections IV

and V contain, respectively, our results in the general case and in four special boundary

cases. It turns out that, in general, the assumption that individuals develop an effective

social conscience which is sufficient to prevent them from free riding is not justified. The

emergence of such a conscience is not evolutionarily stable.

The unique evolutionarily stable conscience is ineffective, namely with it the free-

riding problem cannot be resolved as in the resulting equilibrium no player contributes

to the provision of the public good. The exceptions to these conclusions are obtained in

two special boundary cases. In a unanimity game where the public good is produced only

when everybody contributes, or in two-player games where the public good is produced

if at least one player contributes. Section VI contains some brief concluding remarks.

I The basic model

Consider a group N-{ 1, ..., n} of individuals, e.g., a local community or tribe, who

may or may not voluntarily provide a discrete public good, e.g., a public library or

protection against invasion by foreign enemies. Each individual i has two choices, ó; - 1,

i.e., to contribute to the provision of the public good, or ê; - 0, i.e., not to contribute.

If the numbern

N(ó) - ~ b;i-1

exceeds the threshold m with n~ m~ 1, then the public good is provided. If the group

falls short of providing enough, that is, N(ó) c m, then the public good is not provided.

It is assumed that the cost of contributing cannot be avoided even if N(ó) c m. The

4

utility of the public good is normalized to 1 so that the payoff U;(ó) of all players i E N

for all strategy vectors b- ( ó~, ..., 6„) with 6; E{0,1 } can be written as follows:

1-c if N(ó)~m and ó;-1

-c if N(ë) G m and À; - 1U;(ó) -

1 if N(ó) ~ m and ó; - 0

0 if N(ó) c m and 6; - 0

Here c, 0 G c C 1, is the cost of the individual contribution which can be avoided only

by choosing Á; - 0. This game was studied by Thomas Palfrey and Howard Rosenthal

(1984). A number of alternative extended versions of it have been more recently studied

by Anat Admati and Motty Perry (1991), Mark Bagnoli and Barton Lipman (1989) and

Shmuel Nitzan and Richard Romano (1990).

Observe that all strategy vectors ó' -(êi,...,ó;,) with N(ó') - m are pure strategy

equilibria of this game. (ê~,...,ê„) with N(ó) - 0 is also a strict equilibrium point in

the sense that every player loses by unilaterally deviating from his equilibrium strategy.

Here we are not interested in the model described above but in a slightly modified version

where the payoff function U;(ó) is substituted by U;(ë) which differs from U;(ó) only for

strategy vectors b with N(ó) ? m and ó; - 0 for which

U,`(ó) - 1- e instead of U;(b) - 1.

Palfrey and Rosenthal ( 1988) consider a related situation using a game of incomplete

information about the players' altruism.

The preference parameter e is not an exogenous parameter. As will become clear in

the next section, its endogenous determination is the distinctive and more interesting

characteristic of our approach. Assume, for instance, that e is positive. An individual i

facing a situation N(ó) ? m and 6; - 0 would thus have bad feelings when ( s)he free rides.

If e, however, is negative, such an individual would even enjoy free riding in addition to

getting the public good costlessly. Our main concern is with the determination of the

value of this parameter representing the individual social conscience.

5

Note that unlike in the otiginal model with e; - 0, for all i E N, now strategy

combinations ó with N(ó) 1 m can also be strict equilibria. Such a strategy vector 6 is

a strict equilibrium if

c~ e; for all i E N with 6; - 0

and

c G e; for all i E N with ó; - 1.

Henceforth, it is assumed that there are only two possible mutants e- and et which,

furthermore, constitute the option set for all n individuals. For the values e- and e} it

is only assumed that

e-GcGe}.

Thus the mutants e- and e} can be described as an e--type who prefers free riding if

he knows that the public good is provided even without his contribution and an et-type

who in such a situation would morally object free riding. For all individuals i E N the

preference parameter e; is therefore an element of the mutant set

E - {e-,et}.

Since individuals i and j can only differ with respect to their preference characteristics

e; and e„ group .V is completely described by the number .bI of individuals i E N with

e; - et where, of course, ~41 E N and 0 G Nf G n.

II The solution

One can easily see that there exists only one symmetry invariant - the solution does not

depend on the labelling of players and~or strategies nor on afíine utility transformations

- and strict equilibrium in all games with M G m, namely ó with N(b) - 0. For M 1 m

there exist two such solution candidates, namely 6 with N(ó) - 0 and b' with 6; - 1 for

all M et-type players i and ó; - 0 for all e--type players i. Now one has U;(b') - 1- c

6

for all et-type players i and U;(6') - l- e- for all e--type players i, wheteas U,`(6) - 0

for all players i in N. Since e- G c G 1,

U;(6') ~ U,`(6) for all i E N

which implies that in the range ;tif ~ m the strict, symmetry-invariant equilibrium 6'

payoff dominates 6 with N(b) - 0. To sum up,

Uniqueness: For e- G c G et, there exists a unique symmetry-invariant strict equilib-

rium which is not payoff dominated by another symmetry-invariant strict equilibrium,

namely b with N(6) - 0 for M G m, and 6' for M~ m.

One could justify the unique solution described above by the theory of equilibrium se-

lection developed by John C. Harsanyi and Reinhard Selten (1988). This theory gives

priority to stcict over non-strict equilibria and also to payoff dominance relationships

among equilibria. Furthermore, it is based on invariance with respect to isomorphisms

which implies symmetry invariance. The solution for the special case M- m can be

easily justified by adjusting the solution for market entry games as determined by Rein-

hard Selten and Werner Giith (1982), who apply the Harsanyi and Selten-theory of

equilibrium selection.

We prefer not to apply the Harsanyi and Selten-theory of equilibrium selection. In-

stead we rely on the concept of symmetry-invariant strict equilibria which are not payoff

dominated by other such equilibria as an ad hoc-selection theory for the class of games

which we envisage with two mutants e' and et satisfying e- G c G et. Fortunately, this

intuitive ad hoc-criterion ensures the existence of a unique solution in our basic game.

7

III The evolutionary approach

In evolutionary game theory (see, for instance, John Maynard Smith (1982), Reinhard

Selten (1988), Eric van Damme (1991)) one usually considers strategic interactions of

two genotypes which determine, respectively, the behavior of two strategically interacting

players in symmetric situations. More specifically, an evolutionary game G-(S, R) is

determined by its strategy~mutant set S and its reproductive success function R(s, s)

which assigns player 1's reproductive success to every strategy combination (s, s) with s

and s being, respectively, player 1's and player 2's strategy or genotype.

In an evolutionary game strategies are genetical programs which compete against each

other. Clearly, two players of the same species must rely on the same mutant set S, which

justifies the symmetry with respect to the strategy set S. Reproductive success measures

the expected number of offsprings. Thus, R(s,s) 1 R(s,s) means that a mutant s will

spread more rapidly in a population than a mutant s if both are confronted with a

genotype s. Clearly, player 2's reproductive success for (s,s) must be R(s,s), which

explains why it suffices to define reproductive success only for player 1.

A strategy~mutant~genotype s' E S is evolutionarily stable if

(i) R(s',s') ~ R(s,s') for all s E S,

(ii) for all s E S with R(s',s') - R(s,s'),R(s',s) ~ R(s,s).

Condition (i) requires that an evolutionarily stable strategy (ESS) is optimal in an s'-

monomorphic population. If in such a population another mutant s would also be op-

timally adjusted, then condition (ii) prevents the possibility that s spreads out in the

population. Since R(s', s) ~ R(s, s), the ESS s' earns a higher reproductive success in

every population containing s-genotypes in addition to s'-ones.

This approach has been recently applied by Guth and Yaari (1992) to encouters

of human players whose basic motivational forces are genetically determined. Our basic

model, described in Section II, assumes more than two players. To apply the evolutionary

approach and, in particular, the one proposed by Guth and Yaari (1992) in our multi-

player setting, we therefore have to adjust it to strategic encounters of more than two

players.

8

One possibility would be to assume the mutant set S- E,

E-{e-, et} with e' C c G et.

This would essentially mean that we restrict ourselves to analyzing whether the two

monomorphic populations, namely the one with e--, respectively, et-individuals only,

are evolutionarily stable. More specifically, an e-monomorphic population would be

viewed as evolutionarily stable if R(e, e) ) R(ê, e) for all é E E with é~ e. The more

general approach that we adopt will allow us to assess the evolutionary stability of such

e-monomorphic populations.

Another more general possibility is to let S- E where

~ - [0,1].

Here a mutant x E~ would denote the probability of developing as an et-individual, i.e.,

genotypes are genetical stochastic programs specifying the probability for developing as

an et- or e--phenotype. Clearly, by identifying e' E E with x - 0 and et E E withx- 1, the former mutant set E can be embedded into E.

Suppose now that x E E is confronted with i E E' as assumed in evolutionary game

theory. The difficulty of S- P is that we cannot really judge the probability of certain

numbers M, M E N and 0 C M G n, of e}-type in society without specifying howmany individuals in society are of the et- and e--type. The obvious adaptation ofthe evolutionary approach would therefore be to define x' E~ as evolutionarily stable if

R(x', x') ~ R(i, x') for all i E E with i~ x'. Here R(i, x') would measure reproductive

success if only individual 1 is of the i-type whereas all other n- 1 individuals in society

are of type x'. We will analyze evolutionary stability also from the viewpoint of S- E.

The mutant set S- S allows only for probability distributions P(A1) on

~1-{MEN:OGMcn}

which implies either P(0) - 1, or P(n) - 1 or P(tL1) ~ 0 for all M E rl~l. Thus, there

9

is no possibility for a number M with 0 C M C n to describe an evolutionarily stable

constellation of a society with n interacting individuals. In our view, this, however, is an

interesting constellation which we do not want to neglect. An M-society with 0 G M C n

can be described as one which can be partitioned into a group of 11~f individuals who are

strictly against free riding and a group consisting of the remaining individuals who would

take advantage of the opportunity to free ride. Of course, also S-~ can imply such

a partitioning. But it necessarily implies a positive probability for all M E~l. Thus,

there is no way to have with certainty a constant M with 0 G M G n when relying on

S- E. This explains why we are also interested in the mutant set S- ~N.

To define evolutionary stability of numbers M E ~:~1, we go back to the basic idea of

evolutionary stability which is the stability of dynamic evolutionary processes. The basic

justification of an ESS is that in many circumstances (but not always, see, for instance,

Weissing, 1991) it defines dynamically stable situations of evolutionary processes, e.g.,

the so-called replicator equation (see Weissing, 1991). For the situation at hand with

S- ~t the dynamics are obvious: If for a given M E M an e}-type earns a higher

reproductive success than an e'-type, i.e., if

R~,(M) ~ R~-(,1f),

then M should increase if this is still possible. Conversely,

R~t(M) ~ R~-(~Lf)

should imply a decrease of itf if this is still possible. Thus a nmutant" bf' E Nl is

evolutionarily stable if for all ~4f E.til where M G M',

R~.(M) ~ R~-(M)

and for all M E~1 with M 1 M',

R~f(.tif) ~ R~-(~f).

lo

As in Guth and Yaari (1992), we assume for simplicity that reproductive success is a

linear function of the resources earned by the player. There remains the question whether

and how the preference parameter e is related to reproductive success. Here we do not

wish to impose any restrictive assumption. We therefore allow for close relationship or

for complete independence between e and reproductive success (the latter assumption is

made in Guth and Yaari (1992)). Precisely, let s, s E[0,1], be the share of e which relates

to reproductive success. That is, when the public good is provided a non-contributing

individual of type e has a reproductive success which is equal to 1- se. In all other cases

the reproductive success is equal to the payoff U; .

Due to our equilibrium selection criterion, introduced in Section II, the reproductive

success of an e-type in general depends on M as follows:

1- c for M~ m and e- et

R~(M) - 1- se for M? m and e- e-

0 otherwise

The mutant set ~1~t together with the reproductive success function R~(M) define

the evolutionary model for which one can study the existence of evolutionarily stabfe

mutants M' E ~1.

In the following section we will analyze evolutionary stability for S-~l as well as

for S- E and thereby also for S- E. Only for the special case of n- 2, which will be

discussed in Section VI, we will also apply the original ESS-concept for S- E since for

n- 2 and m- 1 this provides new possibilities of evolutionarily stable constellations.

Of course, an even more general mutant set than M would be the set

P- {P-(P1,...,P )E [0,1)": OC~P, c 1};-i

of probability distributions P over M. Here P, is the probability for M- i, i.e., Ibf - 0

results with probability 1- Pl -...- P,,. By identifying z E E with P E ~ where

nP; - z'(1 - z)"-' for i- 1,...,n,

i

11

the mutant set E' can easily be embedded into P. Of course, also JN can be embedded

into P by identifying M E A~1 with P-(P~,...,P„)where P~y - 1. The basic flaw of

S- P is that, in general, a mutant P E P can only be justified as a population strategy,

i.e., a societal genotype and not as an individual genetical program. For this reason we

prefer to neglect the possibility of S- P.

IV The evolutionarily stable social conscience and

number of contributors

Assume S-,M and the appropriate definition of evolutionary stability given in Section

III. Then,

Theorem 1 : There exists no M' E A'1 which is evofutionari(y staóle.

Proof: For M~ m since e- G c and s E~0,1], one obtains

R~.(.tif)-1-cGR~-(.4f)-1-se-

This proves that no .bf' ~ m is evolutionarily stable. For ~f G m the result is

R~i (M) - 0 - R~- (M)

which also proves that no M' E ~1~1 with M' G m is evolutionarily stable. Q.E.D.

The non-existence of an evolutionarily stable M in the range M G m causes no

surprise. By adopting the concept of a limit evolutionarily stable strategy (LESS) de-

veloped by Reinhard Selten (i 988), we can easily ensure existence of an evolutionarily

stable M' E~1. Let e be a positive minimum choice probability for ê, - 1 and all i E.N,

12

i.e., a player i E N can only use mixed strategies by which he chooses á; - 1 at least

with probability e. Let R~(M) denote the reproductive success of an e-type in a game

with e) 0. In the range M G m the possibility of the public good being provided can

no longer be disregarded. More specifically, in this case we obtain that for e- et and

e- e- and for all M G m,

~(M) - (1-E)(1-SC) ~ ~ n - I ~ ek(1-e)n-~-k~e(I-C) ~ ~ n - 1 ~ ek(1-e)n-~-k.k-m 1` 1C k-m-I 1C

Sincen-1

R~-(M) - R~t(M) - (1 - E)s(et - e-) ~n-1 lJ ek(1 - e)"-'-k ~ O ,

for e~ 0 and s) 0, one obtains that

~-(M) ~ ~.(M)

for all M with 0 C M G n. We have thus established that,

Theorem 2: Assume a positive minimum choice probability e Jor contributing and

s~ 0, i.e., e is not totally unrelated to reproductive success. Then the e-monomorphic

society, i.e., 1Lf - 0, is the only evolutionarily stable situation.

We call a mutant M' E Nl limit-evolutionarily staóle if one can find some arbitrarily

small minimum choice probability c for one of the possible moves in the game such that

M' E ~1~t is evolutionarily stable.

k

Theorem 3: For s ~ 0, i.e., if the preference parameter e is not totally unrelated to

reproductive success, there eaists a unique limit-evolutionarily staóle society, namely the

e-monomorphic society with M- 0.

13

Proof: If there is a positive minimum choice probability for contributing, it follows from

Theorem 2 that M - 0 is limit-evolutionarily stable and that M - 0 is the only such

mutant in J~l. Obviously, a positive minimum choice probability E for non-contributing

cannot change the result for M~ m if E(1 0) is small enough. For M C m it dces not

change the result since 6- 0 is chosen anyhow with maximal ptobability. If ó; - 1 and

ó; - 0 both have 0 minimum choice probabilities, no !1f E A'l is evolutionarily stable

since R~. - R~- and m~ 1. Q.E.D.

According to Theorem 2 there can be no society N with M 1 0, i.e., with a positive

number of et-type members who would voluntarily provide the public good given that

their total number M meets the threshold requirement m in the sense of M 1 m.

Assume now S- i- [0,1]. To rely on the adaptation of the LESS-concept, we first

have to assume a more restricted mutant set

S-ï'`-[E,1]

We conclude this section by showíng that,

Theorem 4: The only evolutionarily staóle r E~ is x' - E provided that E is small but

positive.

Proof: According to the definition of evolutionary stability for S- E, described in

Section IV, one has to show that

R(E, E) - E ~ R~1 ~ n- 1 ~ Ek(1 - E)"-1-k - CI }(1-E) I n~' ~ n- 1 ~ Ek(1 - E~"-1-k - 3E-

kvml-1 k J IIILkL`cm 1C ~

n-1 n - 1 1i R(2,E) - 2 ~ ~ ~ Ek(1 - E)"-1-k - CJk-m-1 1C

"~1 (n-1t(1 - x) ~ I` Ek(1 - E)"-1-k - se-k-m k

14

for all x E ï with x~ c. By writing x - e~- ~ with 0~ 0 one obtains

R(E, E) - R(x, E) - ~ ~k ~ 1 ~ n k 1~ Ek( 1- E)~-1-k - C~

"-1 n - 1-~

~~ ~

Ek(1 - E)~-1-k - SC-k-m k

- Q ~ Em-~(1 - E)~-m - ~(C - 3E-).

Since c~ e- and s E[0,1~,

R(c, c) - R(x, e) G 0

if e(~ 0) is sufficiently small. Q.E.D.

By Theorem 4, the only evolutionarily stable mutant x E ï in the sense of the LESS-

concept developed by Selten (1988), is therefore (im~.-,ox' - Iim~-,ac - 0. This justifies

the claim we made in Section III that our evolutionary analysis for S-~1 will imply

the same conclusion as S - E and thereby S- E.

V Border cases

Except for the assumption regarding the nature of the production function of the public

good and the obvious restrictions 1) c 1 0 and e- G c G et which define our stylized

setting, the main restriction of our model is the requirement

m-1

1GmGn.

15

In this section we consider the border cases

(i) m - n,

(ii) m - 1,

(iii) n ~ oo,

(iv) n - 2.

For m- n our basic game is a unanimity game. Notice that

R~. (n) - 1- c~ R~-(n) - 0.

andR~t(M) - 0- R~-(M) for all M C n.

Introducing a positive minimum choice probability c for b; - 1 as in Theorem 2 there-

fore implies that the only limit-evolutionarily stable M' E~1 is t~1' - n.

Remark 5 : For aff m - n~ 2 there is a unique limit-evolutionarily stabfe 111' E.1~1,

namely M' - n(the et-monomorphic populationJ.

Notice that the same result can also be obtained by the direct application of the LESS-

concept to the mutant set E' which justifies z' - 1 as the only LESS-stable genotype in

~ - ~o, l~.

For m- I one obtains

R~t(:b1)-l-cCR~-(:Lf)-1-e- for M11

and

R~.(0) - 0 - R~-(0)

This proves

16

Remark 8 : For all n~ m- 1, the only evolutionarily staóle M' E~l is M' - 0(the

e--monomorphic population~.

The results of Section IV apply to all cases where 1 G m G n. Thus the validity of

Theorem 3 is retained also for the limit case where n-. oo and m remains finite.

The remaining border case is n- 2. Since the case n- 2- m is covered by Remark

5, we only need to study the game with n- 2 and m- 1. For our original approach

based on S-~l, the general conclusion for all games with n~ m- 1, Remark 6,

applies. We therefore want to demonstrate that for the special case of n- 2 a direct

application of the ESS-concept can yield a different result.

For the case n- 2 and m- 1 the reproductive success function R(e~, ez) is described

in Table 1.

e~el et e-

et 1-c 1-c

e- 1 - se' 0

Table 1.

Since e' G c and s E[0, 1],

R(et, e}) - 1- c G R(e-, et) - 1- se-,

i.e., et E E is not evolutionarily stable. Similarly,

R(e-, e-) - 0 G R(et, e-) - 1- c

implies that e- E E is also not evolutionarily stable.

Remark 7: For n- 2 and m- 1 there is no evolutionarily staóle strategy e E E-

{e-,et}.

17

Let us consider the more general mutant set ï-[0,1] for n- 2 and m- 1 where

x E (0, 1] is the probability for developing as an et-type. By identifying et with x- 1

and e- with x- 0 the former mutant set E-{e', et} can be embedded into ~-[0,1],

i.e., E C ï. Clearly, the only candidate x' E E for an evolutionarily stable x E ir is the

mixed strategy equilibrium x' of the game presented in Table 1 which is given by:

x' -1-c

1 - se- ~

e' G c and s E[0,1] ensure that 0 G x' G 1.

Obviously, every other mixed strategy x E E is an alternative best reply against

x'. Thus, we must check whether condition ( ii) of an evolutionarily stable strategy is

satisfied for all x E E with x~ x'. Nows

R(x',x) - (1 - c) t x(1 - se-),1 - se-

whereas

R(x, x) - x(1 - c) t(1 - x) . x(1 - se-).

Condition (ii) for x~ x' requires therefore that

z j

11-se)~xll-c-x(1-se-)].

Now both the left and right hand sides of the above inequality are positive.

To check whether this inequality is satisfied we can therefore investigate the maximum

of the function

f(x) - x[1 - c- x(I - se")]

From

f'(x) - 1- c- 2x(1 - se")

la

one obtains that

Since se- G 1

1-cx- -x'.

1 - se-

J"(x) - -2(1 - se-) G 0

Hence, J(x) achieves its maximum at x- x' and f(x) G J(x') for all x~ x'. Now

J(x') - 0 whicó proves

Theorem 8 There exists an evolutionarily stable strategy x' E E, namely the mixed

equilébrium strategy

x' - 1-c1 - se-

oj the symmetric game in Table 1.

In an environment with two-player strategic encounters, one may expect voluntary

cooperation in the provision of a public good which requires the contribution of at least

one player since in such an environment the positive probability x' of developing the

e}-type is evolutionarily stable. The most important example seems to be monogamy,

i.e., the long term relationship between just one female and one male. Observe, however,

that x' justifies all possible distributions of the burden of providing the public good,

i.e., there is a positive probability that it is not provided at all, that only one of the two

carries the burden, and that both provide it (although this is inefficient).

VI Concluding remarks

1n this study we have applied the stability concept of evolutionary biology to the problem

of voluntary provision of a discrete public good. The motivational structure, i.e., the

pattern of social conscience emerges endogenously in the evolutionary game where the

payoff (reproductive success) functions are closely related to the payoffs in the public

19

good provision game. Our analysis illustrates that the original notion of an E5S and its

adaptation by Guth and Yaari (1992) can also be applied in a multi-person setting. We

have carried out the analysis for a particular model of public good provision. It is our

hope that futher studies of alternative frameworks will be inspired by our analysis.

The results obtained in our model imply that, in general, the evolution of individual

social conscience enhancing the resolution of the free-riding problem is not to be expected.

Within our framework evolutionarily stable mutants either do not exist at all or generate

a monomorphic society characterized by ineffective social conscience which, in turn,

results in universal non-contributing. The well observed fact that the bulk of voluntary

cooperation occurs in small social units is also supported by the border case ofour model,

namely, the case of two players. Our analysis predicts that in bilateral encounters there

exists an evolutionarily stable pattern of social conscience that alleviates the free-rider

problem. The same conclusion is obtained for unanimity games in which the production

of the public good depends on universal cooperation.

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DLgcussion Paper Series, CentER, Tilburg Unlverslty, Tbe Netherlands:

(For previous papers please consult previous discussion papers.)

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9137 L. Mirman,L. Samuelson andE. Schlee

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9139 A.de Zeeuw

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9142 J.D. Angrist andG.W. Imbens

9143 A.K Bera andA. Ullah

9144 B. Melenberg andA. van Scest

9145 G. Imbens andT. Lancaster

9146 Th. van de Klundertand S. Smulders

9147 J. Greenberg

9148 S. van Wijnbergen

9149 S. van Wijnbergen

9150 G. Koop andM.FJ. Steel

9151 A.P. Barten

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~ V I M I I Í~ IÍÁÍIÍÍVIÍGÍI~ÍÍ~IÍnI N fl I Y R