2. An Evolutionary Paradox

In their 2014 paper, “An Evolutionary Paradox for Prosocial Behavior,” Forber and Smead analyze how increasing the benefits of prosocial behavior impacts the fitness of spiteful individuals, relative to the fitness of prosocial individuals. Assuming pairwise interactions in a finite population, Forber and Smead model a game-theoretic scenario similar to the one described in the previous section. However, rather than analyzing a prisoner’s dilemma–style game, Forber and Smead propose a scenario in which mutual prosocial behavior should, theoretically, be less difficult to achieve. In such a game, sometimes referred to as a prisoner’s delight, the greatest payoffs are earned when both individuals play the prosocial strategyFootnote ¹ (Binmore Reference Binmore2007; Skyrms Reference Skyrms2007).

In Forber and Smead’s model, individuals are assumed to be one of two types—either prosocial or spiteful—and it is assumed that both behavior strategies are played unconditionally. In this model, interacting with a prosocial individual confers a benefit on the recipient, but prosocial individuals cannot benefit from their own prosocial behavior. Spiteful individuals benefit from interacting with prosocial individuals, while incurring a cost due to their own spiteful behavior. Due to the inherent costs associated with spite, one might expect selection to always favor prosocial behavior. However, Forber and Smead’s analysis reveals a paradoxical result: Increasing the benefit of prosocial behavior can increase the fitness of spiteful individuals, so that selection favors spite over prosociality.

Forber and Smead describe two effects that drive this relative increase in the fitness of spiteful individuals. The first is the “spite effect,” or the tendency of spiteful individuals in a finite population to reap disproportionate rewards from others’ prosocial behavior, due to anticorrelated interactions. Because prosocial individuals are more likely to interact with spiteful individuals than they are to interact with other prosocial individuals, spiteful individuals are more likely (than their prosocial counterparts) to receive the fitness benefits of prosocial behavior. The second is the “nearly neutral effect,” or the tendency for spiteful behavior to evolve by chance when the benefits of prosocial behavior are large and the fitness difference between the two behavior strategies is small. In such a situation, selection is “nearly neutral” with respect to behavior strategies, and the question of which strategy evolves in a given population is largely dependent on drift. As I will explore later in this article, it is possible to isolate the nearly neutral effect (or, in other words, to remove the spite effect) by removing the anticorrelations from the fitness functionsFootnote ² (Forber and Smead Reference Forber and Smead2014).

In this article, I will analyze both the spite effect and the nearly neutral effect by considering two other game-theoretic scenarios. Both scenarios share many characteristics with the model developed by Forber and Smead (Reference Forber and Smead2014), while including additional variables to describe the effects of synergism. First, I will analyze a synergistic-cost model, in which spiteful individuals incur an additional cost whenever they interact with each other.Footnote ³ Then, I will analyze a synergistic-benefit model, in which mutual prosocial behavior earns an additional benefit.Footnote ⁴ I will show that each of these synergistic models provides a partial solution to Forber and Smead’s paradox, by increasing the fitness of prosocial individuals relative to their spiteful counterparts.

A article by Ventura (Reference Ventura2019) likewise analyzed how the addition of synergistic effects to Forber and Smead’s model can provide a partial solution to the evolutionary paradox. However, Ventura’s definition of synergism differs from mine, with the result that our analyses and proposed solutions to the paradox are quite different. In Ventura’s model, the benefit of prosocial behavior is synergistic in the sense that it is a nonlinear function of the number of prosocial individuals, with the benefit decreasing as the number of prosocial individuals approaches the total population size. In my analysis, the benefit of prosocial behavior and the cost of spiteful behavior are not dependent on the number of either type of individual in the population. Rather, in my models, the cost of spiteful behavior is synergistic in the sense that a spiteful individual incurs an additional cost whenever it plays its spiteful strategy against another spiteful individual. Similarly, the benefit of prosocial behavior is synergistic in the sense that a prosocial individual reaps an additional benefit when interacting with another prosocial individual. The overall result is that mutual prosocial behavior is incentivized, mutual spiteful behavior is disincentivized, and the paradoxical effects noted by Forber and Smead (Reference Forber and Smead2014) are dampened to an extent.

3. A Game-Theoretic Model for the Evolution of Spiteful Behavior in Finite Populations

Forber and Smead (Reference Forber and Smead2014) investigate the evolution of spite in finite populations. As discussed in the previous section, they assume a game-theoretic scenario referred to as a prisoner’s delight, in which the greatest net benefits are gained through mutual prosocial behavior. Their analysis assumes pairwise interactions, with the payoffs shown in Table 1.

Individuals are assumed to be one of two types: prosocial or spiteful. Payoffs are given for the player on the left-hand side. Prosocial behavior confers a benefit b on the recipient, while spite is defined as the withholding of this benefit b, at a cost a to oneself. If two prosocial individuals interact, then both individuals receive the benefit b. If two spiteful individuals interact, they both receive the baseline fitness (1), minus the cost, a, of their own spiteful behavior. If two individuals of different types interact, the prosocial individual receives the baseline fitness (1), while the spiteful individual receives the benefit b (from interacting with a prosocial individual) minus the cost, a, of its own spiteful behavior.

Using the payoffs given in Table 1, Forber and Smead (Reference Forber and Smead2014) calculate the fitness functions of both types:

(1) $$F\left( {p,\;N} \right) = {{b\left( {{x_p} - 1} \right) + {x_s}} \over {N - 1}}$$

(2) $$F\left( {s,\;N} \right) = {{(b - a){x_p} + \left( {1 - a} \right)\left( {{x_s} - 1} \right)} \over {N - 1}}$$

N represents the finite population size, while the numbers of prosocial and spiteful individuals are represented by x _p and x _s , respectively. Prosocial behavior is favored when F(p,N) > F(s,N), while spite is favored when F(p,N) < F(s,N). As discussed in the previous section, when the fitnesses of the two types are approximately equal, selection is relatively neutral with respect to behavior strategies, and drift plays the more significant evolutionary role (Forber and Smead Reference Forber and Smead2014). Because all interactions occur within a finite population, there is a small degree of anticorrelation between behavior strategies. As discussed in the preceding text, this anticorrelation opens the door to the evolution of spite, by enabling spiteful individuals to benefit disproportionately from others’ prosocial behavior. As Forber and Smead’s analysis shows, increasing the benefit (b) of prosocial behavior can increase the fitness of spiteful individuals, relative to the fitness of prosocial individuals, so that F(p,N) < F(s,N). Substituting the fitness functions into the two inequalities given in the preceding text, we see that prosociality is favored in this model when b < a(N – 1) + 1, while spite is favored when b > a(N – 1) + 1. Forber and Smead (Reference Forber and Smead2014) conduct simulations to show that, as the benefit (b) increases, spiteful behavior evolves in an increasingly large proportion of populations, and the effect is strongest for small (e.g., N = 25) population sizes.

In the final portion of their article, Forber and Smead (Reference Forber and Smead2014) remove the anticorrelations from the fitness functions to analyze the nearly neutral effect in isolation. That is to say, they remove the spite effect from the simulations by getting rid of the slight degree of anticorrelation that results from the finite population size. To facilitate this analysis, Forber and Smead first introduce the concept of a selection coefficient, which is a commonly used method for comparing the fitnesses of two traits.Footnote ⁵ Using s _c to represent the selection coefficient, and the fitness of prosocial individuals as baseline, the relative fitnesses of the two behavior strategies are as follows:

(3) $${F_r}\left( {p,N} \right) = 1$$

(4) $${F_r}\left( {s,N} \right) = \;{{F\left( {s,N} \right)} \over {F\left( {p,N} \right)}} = 1 - {s_c}$$

The symbol F _r is used to denote relative fitness. Substituting equations (1) and (2) for the fitnesses of the two types, Forber and Smead rearrange equation (4) to yield the following:

(5) $${s_c} = {{a\left( {N - 1} \right) - b + 1} \over {b\left( {{x_p} - 1} \right) + {x_s}}}$$

Observe what happens when the benefit (b) of prosocial behavior is increased. Because b is subtracted from the numerator of equation (5), but is a product in the denominator, increasing b decreases the selection coefficient. Decreasing the selection coefficient brings the ratio of the fitnesses [F(s,N)/F(p,N)] close to one, meaning that there is only a slight fitness difference between the two behavior strategies. As discussed in the preceding text, when the fitness difference between the two strategies is small, selection is effectively neutral, and the direction of evolution in any given population is governed primarily by drift (Forber and Smead, Reference Forber and Smead2014).

Forber and Smead next apply this concept of a “nearly neutral model” to their original analysis to observe how increasing the benefit (b) of prosocial behavior can cause the evolutionary dynamics to become more neutral in character (Forber and Smead, Reference Forber and Smead2014). To do this, they isolate the nearly neutral effect by removing the anticorrelations from the original fitness functions. The revised fitness functions are then as follows:

(6) $$F\left( {p,\;N} \right) = {{b\left( {{x_p}} \right) + {x_s}} \over N}$$

(7) $$F\left( {s,\;N} \right) = {{(b\; - \;a){x_p} + \left( {1 - a} \right){x_s}} \over N}$$

Forber and Smead then repeat their simulations, using the revised fitness functions; the result is that, when the nearly neutral effect is isolated, spiteful behavior generally evolves less frequently for any given value of b. However, spite still becomes more and more likely to evolve in a given population as the benefits of prosocial behavior increase, with the likelihood approaching a limit of 0.5 as the fitness difference between the two strategies approaches zero. Plugging equations 6 and 7 into the inequality F(s,N) > F(p,N), we see that selection will never favor spite as long as the cost of spite is greater than zero (a > 0). Nevertheless, increasing the benefit (b) of prosocial behavior decreases the fitness difference between the two behavior types, making it more likely that spite will evolve in a given population due to drift alone.

4. The Synergistic-Cost Model

In the following analysis, I will consider two other game-theoretic scenarios. Both retain the general format used by Forber and Smead (Reference Forber and Smead2014), while introducing additional variables to describe synergistic effects. The first scenario involves a synergistic cost, c, which is incurred by two spiteful individuals whenever they interact with each other (Table 2).

The fitness functions for this scenario are as follows:

(1) $$F\left( {p,\;N} \right) = {{b\left( {{x_p} - 1} \right) + {x_s}} \over {N - 1}}$$

(8) $$F\left( {s,\;N} \right) = {{\left( {b - a} \right){x_p} + \left( {1 - a - c} \right)\left( {{x_s} - 1} \right)} \over {N - 1}}$$

Plugging these formulas into the inequality F(s,N) > F(p,N), we see that selection favors spite when b > ax _p + a(x _s – 1) + c(x _s – 1) + 1. Similarly, selection favors prosocial behavior when b < ax _p + a(x _s – 1) + c(x _s – 1) + 1. Consider a population size of N = 100, with a = 0.1 and c = 0.1 (Figure 1).Footnote ⁶ Observe that increasing the benefit (b) of prosocial behavior decreases the fitness of prosocial individuals; this is the paradox that Forber and Smead (Reference Forber and Smead2014) observed. At relatively low values of b, the selection coefficient remains above zero, meaning that the effect is not strong enough for selection to favor spite. However, as shown in Figure 1, at higher values of b (e.g., b = 26), the selection coefficient drops below zero, indicating that selection favors spite.

Figure 1. In all figures, a = 0.1 and N = 100. x _p on x-axis; s _c on y-axis.

Nevertheless, we see that the introduction of a synergistic cost provides a partial solution to the evolutionary paradox. If, as shown in Figure 2, we set c = 0, with N = 100 and a = 0.1, and increase the benefit (b) of prosocial behavior, the selection coefficient becomes negative at relatively low values of b (e.g., b = 14).Footnote ⁷ However, if we add a large synergistic cost by setting c = 1 (Figure 2), we see that the selection coefficient generally remains positive, unless the value of b is high (e.g., b = 26) and the number of prosocial individuals is large (e.g., x _p = 80). Therefore, if mutual spiteful behavior incurs synergistic costs, it is easier for prosocial behavior to evolve, and more difficult for spite to gain an evolutionary foothold.

Figure 2. In all figures, a = 0.1 and N = 100. x _p on x-axis; s _c on y-axis.

5. The Synergistic-Benefit Model

Having analyzed a synergistic-cost model, let us now turn to a synergistic-benefit model. Suppose that, when two prosocial individuals interact, they mutually confer a synergistic benefit, d, on each other. The payoffs for this scenario are given in Table 3.

The fitness functions are now as follows:

(9) $$F\left( {p,\;N} \right) = \;{{\left( {b + d} \right)\left( {{x_p} - 1} \right) + \;{x_s}} \over {N - 1}}$$

(2) $$F\left( {s,\;N} \right) = {{\left( {b - a} \right){x_p} + \left( {1 - a} \right)\left( {{x_s} - 1} \right)} \over {N - 1}}$$

Now, we see that selection favors spite when b > d(x _p – 1) + a(N – 1) + 1, and that selection favors prosociality when b < d(x _p – 1) + a(N – 1) + 1. Once again, the evolutionary paradox emerges. As the benefit (b) of prosocial behavior increases, the selection coefficient becomes smaller and smaller, and selection can favor spite at very large values of b. Nevertheless, we see that the inclusion of a synergistic benefit, like the inclusion of a synergistic cost, has the potential to significantly diminish these paradoxical effects (Figure 3). When there is no synergistic benefit (d = 0), and we increase the value of b, the paradox is noticeably present. At larger values of b, the selection coefficient falls just below zero, giving spite a slight fitness advantage (Figure 3). However, in the presence of a large synergistic benefit (Figure 3, d = 1), the selection coefficient generally remains above zero at larger values of b (e.g., b = 26). Spite is generally selected against, especially if x _p is large.

Figure 3. In all figures, a = 0.1 and N = 100. x _p on x-axis; s _c on y-axis.

Because only prosocial individuals receive synergistic benefits, the addition of a synergistic benefit generally gives prosocial individuals a fitness advantage over their spiteful counterparts. For the previously mentioned reasons we see that, like the synergistic-cost model, the synergistic-benefit model provides a partial solution to Forber and Smead’s paradox.

6. Isolating the Nearly Neutral Effect

Recall that, in their analysis, Forber and Smead (Reference Forber and Smead2014) isolate the nearly neutral effect by removing the anticorrelations from the fitness functions. The result is that, as the benefit (b) of prosocial behavior increases, the fitness difference between the two behavior strategies approaches zero, and selection behaves in a “nearly neutral” manner. That is to say, when the fitness difference between the prosocial and spiteful strategies is very small, selection does not strongly favor one strategy over the other, and the question of which strategy evolves in a given population is largely dependent on drift.

I have already demonstrated that when mutual prosocial behavior earns synergistic benefits, or when mutual spiteful behavior incurs synergistic costs, selection is less likely to favor spite over prosociality, even when the benefit of prosocial behavior is rather large. Next, I will show that when the nearly neutral effect is isolated selection cannot favor spite if mutual spiteful behavior incurs synergistic costs or mutual prosocial behavior earns synergistic benefits. Still, as we will see, the nearly neutral effect poses a problem for the evolution of prosocial behavior: even if mutual spite entails synergistic costs, or mutual prosociality earns synergistic benefits, the influence of drift becomes stronger as the benefit (b) of prosocial behavior increases. When the benefits of prosocial behavior are very large, and spiteful individuals cannot reap the additional fitness benefits associated with anticorrelated interactions between behavior strategies, ^{Footnote 8} the direction of evolution depends primarily on drift—not selection—and so spite may still evolve in roughly half of all populations.

To observe this phenomenon, let us first isolate the nearly neutral effect in the synergistic-cost model. We remove the anticorrelations from the original fitness functions:

(6) $$F\left( {p,\;N} \right) = {{b\left( {{x_p}} \right) + {x_s}} \over N}$$

(10) $$F\left( {s,\;N} \right) = \;{{\left( {b - a} \right){x_p} + \left( {1 - a - c} \right){x_s}} \over N}$$

Spite is favored when c < –aN/x _s and prosociality is favored when c > –aN/x _s . In other words, if (1) the value of a is positive (meaning that spiteful individuals incur a cost from their own spiteful behavior, regardless of whom they interact with) and (2) mutual spiteful behavior incurs positive synergistic costs, selection will always favor prosociality and never favor spite. This result is shown in Figure 4; when c = 1, the selection coefficient remains positive, but when c = –1, selection generally favors spite.

Figure 4. In all figures, a = 0.1 and N = 100. x _p on x-axis; s _c on y-axis.

Next, consider the synergistic-benefit model. Again, we isolate the nearly neutral effect by removing the anticorrelations from the original fitness functions:

(11) $$F\left( {p,\;N} \right) = \;{{\left( {b + d} \right){x_p} + {x_s}} \over N}$$

(7) $$F\left( {s,\;N} \right) = {{(b\; - \;a){x_p} + \left( {1 - a} \right){x_s}} \over N}$$

Spite is favored when d < –aN/x _p and prosociality is favored when d > –aN/x _p . In other words, if (1) the value of a is positive and (2) mutual prosocial behavior earns positive synergistic benefits, selection will always favor prosocial behavior and will never favor spite. This result is shown in Figure 5. When d = 1, the selection coefficient remains positive, but when d = –1, selection generally favors spite unless x _p is small.

Figure 5. In all figures, a = 0.1 and N = 100. x _p on x-axis; s _c on y-axis.

However, as the benefit (b) of prosocial behavior increases in both models, the selection coefficient approaches zero. That is to say, as the benefit of prosocial behavior increases, the fitness difference between the two behavior strategies approaches zero, and at very high values of b, the question of which behavior strategy evolves in a given population is largely dependent on drift, not selection. It is true that, when the nearly neutral effect is isolated, selection cannot favor spite if mutual spite entails synergistic costs or mutual prosocial behavior earns synergistic benefits. Nevertheless, as the benefit of prosocial behavior increases in either model, it becomes more and more likely that spite will evolve through the effects of drift alone.