4.4 The Indirect Evolution of Conditional
4.4.2 Evolutionary Adaptation to a Heterogeneous
Consider a society which faces one public good game which is repeated (finitely) many times within each generation. We focus on the case where there is scope for two possible equilibria,30 n∗
a and n∗b. Without loss of generality, we assume that n∗a < n∗b. Sometimes cooperation works rather well, sometimes it breaks down (e.g. due to exogenous shocks). Let the frequency that society coordinates on the equilibrium state
n∗
j be given by πj, forj ∈ {a, b}.31 The fitness of a strategy
¡ xa i, xbi ¢ is then given by w¡xai, xbi¢ = X j=a,b πj ¡ y(xji) +λ z(xji, n∗j)¢, (21)
wherexji denotes the action of agent iin equilibrium j. From n∗
a< n∗b and (6) follows ˆ
θ(n∗
a) < θˆ(n∗b). Hence, we will observe three different strategies: On the one hand, agents withθi ≤θˆ(n∗a) will free-ride in both equilibrium states. Agents withθi >θˆ(n∗b) on the other hand, will cooperate in both states. A third group of subjects, those with ˆθ(n∗
a) < θi ≤ θˆ(n∗b), behaves conditionally cooperative; such individuals would cooperate in equilibrium state a but defect in b. Making use of (1), (2), (5) and
πb = 1−πa, we can express the success of a type θ in the following way:
w(θ,θ, π¯ a) = −c for θ >θˆ(n∗ b) −πac−(1−πa)λ s(n∗b) for ˆθ(n∗a)< θ≤θˆ(n∗b) −πaλs(n∗a)−(1−πa)λ s(n∗b) for θ≤θˆ(n∗a) (22) withn∗
j = Φ(ˆθ(n∗j),θ, σ¯ 2). The crucial difference to the fitness function from (17) is the fact that intermediate θ-types obtain a fitness-payoff from two different actions. The success of the conditional cooperative strategy consists of the cooperation payoff for equilibrium state a plus the payoff from free-riding once the society has coordinated on stateb.
30Remember that there are (at least) two possible stable equilibrium levels of free-riding, as long
as assumptions A2 is fulfilled. Compare Lemma 1.
31The frequencyπ
a could be derived endogenously, according to the basin of attraction of equilib-
riumn∗
From (22) we can compute the mean fitness of the population for a given πa, ¯ w=−c+πa(c−λs(n∗a)) ˆ θ(n∗ a) Z −∞ dΦ(θ) + (1−πa) (c−λ s(n∗b)) ˆ θ(n∗ b) Z −∞ dΦ(θ). (23)
According to (14), the evolution of ¯θ is then determined by ∆¯θ= 1 ¯ wΨ with32 Ψ≡πa(λs(n∗a)−c) ¡¯ θn∗ a−θ¯∗a ¢ + (1−πa) (λ s(n∗b)−c) ¡¯ θn∗ b −θ¯∗b ¢ , (24) and ¯θ∗
j captures the mean level of θ among the free-riders in equilibrium n∗j. The evolutionary dynamics on ¯θ are then given by
∆¯θ >0 if Ψ>0 ∆¯θ= 0 if Ψ = 0 ∆¯θ <0 if Ψ<0
(25)
We can set up the following proposition:
Proposition 4 Consider a heterogeneous environment with 0 < πa < 1. (i) An
evolutionary equilibrium is characterized by Ψ = 0 with ne
a = Φ(ˆθ(nea),θ¯e, σ2) and
ne
b = Φ(ˆθ(neb),θ¯e, σ2)<1, where nea andneb are supported by a normal distribution with
mean θ¯e. (ii) In equilibrium there holds λs(ne
a)> c > λs(neb).
Proof. Part (i) follows immediately from (25). Part (ii) derives from (24): Note
that ¯θn∗
j > θ¯∗j as long as n∗j < 1. Hence, the first term in (24) would be negative if
c > λs(ne
a). Since nea < neb, (6) implies that the second term would be negative as well. We would get Ψ< 0. Therefore c > λs(ne
a) can not hold in an equilibrium. Iff
λs(ne
a) > c, the first term in (24) is positive. In order to get Ψ = 0, the second term in (24) must be negative, which holds for c > λs(ne
b). As long as ne
b <1,33 the distribution in an evolutionary equilibrium supports two equilibria such that λs(ne
a) > c > λs(neb). In terms of fitness, cooperation dominates free-riding in equilibrium statea, since −c >−λs(n∗
a). For state b, however, the oppo- site holds: As free-riding becomes more widespread, the (fitness) costs from sanctions
32The derivation of ∆¯θrespectively Ψ is analogous to the one of (19). Compare Appendix B. 33There could also exist an equilibrium with ne
b = 1 and λs(nea) = c. This case represents a
heterogeneous environment with a positive share of cooperation in equilibrium a and a complete break-down of cooperation in equilibrium state b. As the properties of this case are similar to the scenarios discussed in the previous subsection, we focus on equilibria withne
are lower than the costs of cooperation. From this follows that conditional cooperators have more success than ‘unconditional’ cooperators respectively free-riders.
Corollary 1 In an evolutionary equilibrium within a heterogeneous environment with
ne
a < neb <1 and0< πa<1, conditional cooperators have a strictly higher fitness than
both, free-riders and cooperators.
Proof. From Proposition 4(ii) we know that λs(ne
a)> c > λs(neb). Using this in (22) proofs the Corollary.
Figure 4.3 (on the next page) graphically represents such an equilibrium. The graph on the left hand side captures a system with a distribution Φ(θ) and a function ˆθ(n), supporting two stable equilibrium states n∗
a < n∗b < 1. The graph on the right hand side depicts the fitness difference between the strategies for the two equilibria. As in this example the advantage of cooperators compared to free-riders in equilibrium state
a,λs(ne
a)−c >0, is smaller than the disadvantage in b, λs(neb)−c <0, the weight on state a – expressed byπa – must be sufficiently high in order that Ψ = 0 holds.
From figure 4.3 as well as from the analysis above (compare Proposition 2) it is clear that λs(0) > c is a necessary condition for such an evolutionary equilibrium to exist. In addition, assumption A2(ii) must be fulfilled, such that there are (at least) two equilibrium states.
Analogously to before, the necessary conditions for local stability is ∂∆¯θ ∂na +
∂∆¯θ ∂nb >0.
The formal analysis yields the following result:
Proposition 5 It is sufficient for an evolutionary equilibrium with ne
a < neb <1 to be
stable, if −s0(ne
a)<Γa and −s0(neb)>Γb holds, with
Γj ≡θ¯ Ã λ¡θn¯ e j−θ¯je ¢ λ s(ne j)−c +θˆ(n e j)2 s(ne j) φ(ˆθ(nej)) !−1
Proof. See Appendix B
As we know from Proposition 4, there holdsc > λ s(ne
b). Hence the denominator of the first term in the round brackets of Γb is negative and we could get Γb <0. In this case −s0(ne
b) >Γb would always be fulfilled. From Proposition 4 also follows Γa > 0. Therefores0(ne
Figure 4.3: Evolutionary Equilibrium in a Heterogenous Environment
Since the stability of an evolutionary equilibrium as characterized by Proposition 4 is in general ambiguous, we conducted a series of numerical simulations. We could not find even one single parameter combination, where the stability condition did not hold. Even in the case where one of the two (sufficient) conditions in Proposition 5 were violated, the equilibrium proved to be stable. We are therefore confident, that evolutionary equilibria within a heterogeneous environment are stable for a wide range of parameters.
The intuition for this finding is straight forward: Small shocks would not change the result from Corollary 1. Conditional cooperation would still perform more successfully than the two unconditional strategies. As conditional cooperators have intermediate values ofθ, preferences in the ‘middle’ of theθ-range are more successful and dominate more extreme – either low or high –θ-values. In contrast to the evolutionary adaptation to a homogenous environment, there is no scope for disruptive evolution. Hence, in
contrast to the analysis of the previous subsection, we find (potentially) stable equilibria where cooperators and defectors coexist.
The evolutionary dominance of conditional cooperators is the main result of our analysis. Holding extreme ‘anti-social’ (low θ values) or extreme ‘pro-social’ prefer- ences (high θ values) induces agents to play one particular strategy, irrespectively of the other agents behavior. In a stable evolutionary equilibrium within a homogenous environment, one of these two strategies will dominate the other. In a heterogeneous environment, there is scope for a third type of strategy, that of conditional cooperation. If individuals adapt to such a heterogeneous habitat, where they face either a ‘good’ or a ‘bad’ outcomes, the unconditional strategies prove less successful that the conditional strategy. Agents who free-ride in ‘bad’ equilibrium states but cooperate in a ‘good’ states, dominate the (unconditional) cooperators in the former and the free-riders in the latter environment. As compared to conventional one-game-one-equilibrium sce- narios, such heterogeneous environments appear as a more realistic description of social interactions. Therefore, the evolutionary pressure to adapt to several possible outcomes provides a simple explanation for the evolutionary success of conditionally cooperative behavior.