4.4 The Indirect Evolution of Conditional
4.4.1 Evolutionary Adaptation to a Homogenous
Environment
Let θ be normally distributed according to θ ∼ φ(θ,θ, σ¯ 2), and the corresponding
cumulative distribution is given by Φ(θ,θ, σ¯ 2). Substituting for y(x
i), z(xi, n) and
xi =x(θi, n) from (1), (2) and (5), we can express individual fitness as a function of θ
22Alternatively one could also consider sanctions related to ostracism, e.g. in form of exclusion
from the public good consumption. Note that the impact of ostracism would also depend on the share of free-riders in society: Getting excluded from public good consumption in a cooperative society, with a high level of public goods provided, represents a more severe punishment, than exclusion in a society where cooperation fails.
23We do not include the public good payoff into the fitness function, since this would not alter our
results.
24Note that it is only the heterogeneity in actions – determined by different levels of θ – which
results in fitness differences. Within the group of cooperators and free-riders the heterogeneity inθi
does not result in different levels of fitness.
25In section 4.6 we discuss the crucial differences of this approach to an evolutionary process
and ¯θ, w(θ,θ¯) = −c for θ >θˆ(n∗) −λ s(n∗) for θ ≤θˆ(n∗) (17)
wheren∗ = Φ(ˆθ(n∗),θ, σ¯ 2) is an equilibrium, analogous to (7), for a normal distribution
with mean ¯θ (and σ2 is exogenously fixed). Note that individual fitness as described
by (17) is frequency dependent; as the θ-distribution changes, the share of free-riders
n∗ will change and thereby the fitness costs from the norm deviation. As the ap-
proach introduced in section 4.3 takes account of such payoff spillovers, the approach is applicable to this framework.
The mean fitness is defined by ¯w = R w(θ,θ¯)φ(θ). Using (17), we can express ¯w
as26 ¯ w=−c+ (c−λ s(n∗)) ˆ θ(n∗) Z −∞ dΦ(θ). (18)
Following (14), the evolution of ¯θ is determined by
∆¯θ= 1 ¯
w(λ s(n
∗)−c)¡θn¯ ∗−θ¯∗¢ (19)
where ¯θ∗ represents the mean level of θ among then∗ agents who defect in an equilib-
rium (compare Appendix B). As long as 0 < n∗ < 1, we can distinguish between the
following three cases:
∆¯θ <0 if λ s(n∗)< c
∆¯θ= 0 if λ s(n∗) =c
∆¯θ >0 if λ s(n∗)> c
(20)
From this the following results derive:
Proposition 1 (i) An evolutionary equilibrium where cooperators and free-riders co- exist is characterized by λ s(ne) = c and ne = Φ(ˆθ(ne),θ¯e, σ2), where 0 < ne < 1
constitutes an equilibrium share of free-riders which is supported by a normal distribu- tion with meanθ¯e. (ii) In such an equilibrium, w(θ,θ¯e)is the same for all θ and there
holdsλ = ˆθ(ne). (iii) An evolutionary equilibrium where cooperation fails, ne1 = 1, is
characterized byne1 = Φ(ˆθ(ne1),θ¯e1, σ2), supported by a normal distribution with mean
¯
θe1.
Proof. The proof of (i) follows immediately from (19). From (4) we know that
c = ˆθ(n∗)s(n∗) must hold for any equilibrium state. λ s(ne) = c then implies λ = ˆ
θ(ne). Using this in (17) and substituting for (4) proofs (ii). Part (iii) derives from
n∗ = 1 ⇒ θn¯ ∗ = ¯θ∗. Hence, for ne1 = 1 the term in the last brackets in (19) is zero
and ∆¯θ = 0.
The evolutionary equilibrium described in part (i) of the proposition is characterized by a positive share of cooperators such that there is no fitness differential between free- riders and cooperators. In equilibrium, the preferences of agents with θi = ˆθ(n), who are indifferent between defection and cooperation, coincide with the fitness function since ˆθ(n) = λ. In other words, these indifferent θ-types are ‘perfectly adapted’. In addition, there is also scope for an evolutionary equilibrium where cooperation breaks down; in such an equilibrium, the social norm has eroded and everybody free-rides. From Lemma 1 we know that n∗ = 1 constitutes a possible equilibrium state for
any distribution. Therefore, any level ¯θ could be the mean of the distribution in an evolutionary equilibrium with ne1 = 1. By the time the whole society free-rides, the
evolutionary pressure on ¯θ to decline vanishes and the system reaches a rest point.27
Let us now turn to the existence of these different types of equilibria.
Proposition 2 (i) Iff λ s(0) > c, there exists an evolutionary equilibrium with 0 < ne <1. (ii)There always exists an evolutionary equilibrium withne1 = 1. Ifc > λ s(0),
this is the only equilibrium.
Proof. (i) Since c > λs(1) = ε and s(.) is continuously decreasing in n, λ s(0) > c
assures that there exists a level ofn where λ s(n) = cholds. Moreover, we can always find a distribution φ(θ,θ, σ¯ 2), a function s(n) and a level c, which supports such an
equilibrium share of free-riders ne. (ii) From Lemma 1 we know that n∗ = 1 is
supported by any distribution as long as A1 and A2(i) hold. Proposition 1(iii) implies that any equilibrium with n∗ = 1 constitutes an evolutionary equilibrium ne1. From
A1 followsc > λ s(0) ⇒c > λ s(n) for alln ∈[0,1]. It therefore follows fromc > λ s(0) that there can not exist an equilibrium withne<1, as @ n with λ s(n) =c .
A graphical representation of this result is provided in figure 4.2, where we have plotted λs(n)-curves for two different levels of λ. In the case of the higher curve,
27Theoretically, we could also describe an evolutionary equilibrium with ne = 0. For this case,
n∗ = 0 ⇒θn¯ ∗ = ¯θ∗. Hence, the last bracket term in (19) would equal zero and ∆¯θ= 0. However,
an equilibrium state withn∗= 0 would only be supported by a distribution with ¯θ→ ∞. Since this
λ s(0) > c and there exists an equilibrium with ne < 1. In the case of the lower curve, there is no intersection with the c-line. The costs of cooperation are higher than the the fitness impact of sanctions, even for the state where n∗ = 0; free-riding
dominates cooperation in terms of fitness for any level of norm-violations. Starting from anyn∗ <1, the evolutionary process induces ¯θ to fall and society moves towards
an equilibrium withne1 = 1.
Figure 4.2: Fitness Payoffs
Finally, we address the stability of the system. An evolutionary equilibrium is locally stable if ∂∆¯θ
∂n > 0. If this condition holds, small mistakes in the adaptation process would not affect the evolutionary equilibrium. Consider for example that, starting from an equilibrium distribution with ¯θe, some agents would acquire a ‘too’ low level of θ. In this case, the share of free-riders would exceed ne and the stability condition would imply ∆¯θ > 0. An increase in the mean norm sensitivity would then provide a pressure to adapt ‘back’ towards the initial equilibrium. In our case, however, an evolutionary equilibrium where cooperators and free-riders coexist can never be stable.
Proposition 3 An evolutionary equilibrium with 0 < ne < 1 is never stable. In
contrast, an evolutionary equilibrium with ne1 = 1 is locally stable.
Proof. See Appendix B.
Due to assumption A1,s0(n)≤0. Hence, any small deviation fromne would tip the balance in fitness-payoffs between the two strategies. If the level of free-riding would slightly exceedne, the norm payoff would become less important and we getc > λ s(n). Free-riders, i.e. types with lowθ-values, earn a higher level of fitness and consequently
¯
θdecreases. The system moves to an equilibrium withne1 = 1.28 Note that the system
would return to such an equilibrium after small shocks – e.g. if some agents mistakenly cooperate – as in the neighborhood of ne1 = 1 there holds c > λ s(ne1) since s(1) =ε.
Hence, ¯θ would decline and thereby trigger an increase in free-riding which would push behavior back towards the equilibriumne1.
The analysis provided so far yields an unsatisfactory result. While there may exist an evolutionary equilibrium where free-riders and cooperators coexist, such an equilib- rium turns out to be instable. With the pattern of sanctionss(n) considered in A1, the evolutionary adaptation induces disruptive selection respectively disruptive evolution: Typically, either one or the other strategy dominates in terms of fitness. The system either evolves towards an equilibrium where the norm has eroded and everybody free- rides, or ¯θ → ∞ and society would evolve towards full cooperation (compare footnote 27). While this is in conflict with the coexistence of free-riders and cooperators ob- served in real life social outcomes, we are nevertheless convinced that the crucial model assumption from A1, s0(n) ≤ 0 – which is quite common in the literature29 – as well
as the fitness function introduced in (16) do make sense and can result in reasonable, stable, evolutionary outcomes. Here we have studied the adaptation to a homogenous environment. Agents encounter one particular situation, and evolution shapes their preferences such that they are fit for this particular environment. In reality, however, we typically face heterogeneous environments, as social interaction are repeated in different situations with quite diverse outcomes. The level of cooperation varies for different collective action problems, along time and along space. In the next section we show how we can capture such a heterogeneous environment within our model frame- work. In contrast to the case of a homogenous environment, the evolutionary process can result in stable equilibria where cooperators and free-riders coexist.
28If, on the one hand, the share of free-riding falls short of ne, we getλ s(n) > c. Cooperators
would be more successful than free-riders, ¯θwould increase andn∗would decline further. The system
would evolve towards ¯θ→ ∞. (Compare Footnote 27.)
29Compare e.g. Akerlof (1980), Corneo (1995), Lindbeck et al. (1999), Naylor (1989), Romer