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Web Topic 13.1
Direct Benefit Models of Cooperation


Why humans should or do cooperate has been of concern to philosophers and economists for millennia. Humans often engage in extensive cooperation with unrelated individuals showing that kin selection and greenbeard biases are not necessary to compensate for the costs of being a cooperator. This immediately raises the question of whether the same mechanisms might explain the many cooperative interactions seen in other species; at minimum, kin selection cannot explain cooperative mutualisms between different species. In this Web Topic, we introduce the evolutionary game models advanced to explain cooperation without genetic compensations, and note particular cases that may be relevant to animal communication. Because the literature on this topic, especially that dealing with human cooperation, is so extensive, we can barely do justice to it here. However, we provide some further reading sources at the end of the module that provide gateways to this larger literature. Readers of this module should be familiar with the classification of evolutionary game models and the logic of “take” and “give” games reviewed in Web Topic 10.5.

Defining and classifying cooperation

A necessary condition for cooperation is that at least one individual undertakes an action at some cost to itself and other individuals benefit from the first individual’s investment. The first individual’s action could have been prompted by likely compensating benefits to itself, or its efforts might have been “purloined” as the result of theft, deceit, or coercion (Connor 1995). Setting coercive manipulations aside for the moment, there are a number of ways that a cooperative interaction could generate net benefits for both parties. In this module, we will also ignore genetic benefits as a result of kin selection or greenbeard biases: our focus here is on cooperation among unrelated animals, or at least, on contexts in which relatedness makes no significant contribution to the cooperation economics. These are usually called the “direct benefits of cooperation” (Bergmüller et al. 2007; Leimar and Hammerstein 2010).

Consider an animal B that is pursuing some direct benefit to itself. There are three ways that animal A might interact with B with net positive benefits to both parties and without either party invoking coercion or deceit:

Figure 1. Criterion tree for classifying types of cooperation among unrelated animals. Answering four basic questions can assign most biological examples to a specific category. Using the individuals mentioned in the text, the first question concerns whether animal A did or did not invest in animal B (which will pursue its own interests in any case). If animal A did invest, the second question concerns whether this investment was somehow forced or tricked out of animal A. If not, the third question is whether animal B responded by also investing in the interaction (above and beyond its ongoing self-investment). Whether this leads to pseudo-reciprocity or reciprocity, the final question is whether animal A’s compensation comes from animal B (direct exchange) or some third party (indirect exchange). (Modified from Bergmüller et al. 2007; Connor 2007.)

Combining these three alternatives with possible manipulation, several authors have proposed criterion trees for assigning real examples to categories. One example is shown in Figure 1. Where the party invested in by animal A is the same one who delivers the consequences to animal A, the exchange is said to be direct; if the feedback on animal A comes from a third party, the exchange is said to be indirect. Manipulation, pseudo-reciprocity, and reciprocity can all lead to either direct or indirect accountings. This schema, and others like it, have proved very effective at classifying known examples of animal and human cooperation: all of the classical examples such as cooperative hunting, joint predator surveillance, cooperative breeding, and symbiosis can be assigned to their own spot in such a tree (Connor 1995; Leimar and Connor 2003; Bergmüller et al. 2007; Connor 2007; Bshary and Bergmüller 2008; Connor 2010). This partitioning has greatly clarified economic and game theoretic modeling of cooperation as different parts of the classification tree require different assumptions to generate ESSs and exhibit quite different evolutionary dynamics. One comment on terminology: some authors have lobbied for the restriction of the word “mutualism” to interspecific relationships and “cooperation” to intraspecific ones (Bronstein 2001; Bergmüller et al. 2007). In this module, we use the two terms as synonyms because such use is already widespread in the literature and because convergent inter- and intraspecific associations can occur at the same point in classification trees such as the one in Figure 1. However, readers should be alert to publications where the authors specifically restrict the term mutualism to interspecific associations.

An overview of the basic problem

Before we examine specific cases, it is useful to stand back and identify the challenges that the evolution of direct benefit cooperation faces from the perspective of evolutionary game theory. There are two basic hurdles that usually hinder the evolution of cooperation. The first arises because many situations in which mutual cooperation could be beneficial are vulnerable to cheats and defectors. When modeled as a single-bout 2x2 symmetric contest, the result is invariably a take game with the Cheat or Defector strategy as the pure ESS (Figure 2A). The first challenge is thus to identify conditions under which take games can be replaced by a give games (Figure 2B). Once achieved, the second problem arises. As we saw in Web Topic 10.5, give games in the absence of genetic compensations such as greenbeard or kin selection invariably have two possible ESSs: one in which everyone cooperates, and one in which everyone defects. A few games (e.g., the Snowdrift Game discussed below) can lead to a mixed ESS in which each party cooperates some of the time, and defects the rest of the time. However, it is extremely difficult to find any realistic model in which pure cooperation is the only ESS. The second challenge is thus to find conditions that might get a population to the all-cooperation ESS and hold it there despite drift or other factors outside the basic game. While some of the literature on cooperation mingles both hurdles, an equally diverse amount focuses on only one of the two steps. We shall try to keep straight which hurdle we are considering in the following discussion.

Figure 2. Take versus give games of cooperation. (A) The normal form matrix for this example shows that it is a take game: even though everyone cooperating would produce a higher per capita payoff (pareto optimum) than everyone defecting, the advantages to defecting when others are cooperating have an even higher payoff. The result is that cooperation degenerates until everyone is defecting as the best reply. Defect is then a pure ESS. (B) In this game, pure cooperation can be an ESS, but it is only one of two possible ones, the other being pure defect—which is found depends on starting conditions. If the initial condition is for everyone to be selfish and not cooperate, the system will stay there barring drift or some other outside factor.

The prisoner’s dilemma

Most modern treatments of the economics and/or evolution of cooperation begin with the Prisoner’s Dilemma. This model was first introduced to the evolutionary biology literature by Trivers (1971). In this game, two suspects are taken prisoner by the police and held separately. Each is urged to provide incriminating evidence about the other. The relevant game is typically modeled as a dyadic single-bout 2 x 2 symmetric contest. If both keep silent, they are likely to gain only a light sentence given the lack of detailed evidence (a per capita payoff of R). If one provides incriminating evidence on the other, that suspect get to go free (T) whereas the other gets sent to jail for a very long time (S). If both provide evidence on each other, they both go to jail but for less time than if they had not provided evidence (P). In this game T > R > P > S. The normal form matrix for the game is shown in Figure 3.

Figure 3. Payoff matrix for Prisoner’s Dilemma game. Given that T > R and P > S, Defect is a pure ESS. See text for justification of relative payoff values.

As is clear in Figure 3, this is a classic take game in which pure Defect is the only ESS. Cooperation will never evolve even though the pareto optimum of joint cooperation yields a higher payoff than the ESS (e.g., R > P). One can imagine many other scenarios in both animals and humans that would also be characterized by this game. As we shall see below, the only way to convert the Prisoner’s Dilemma to a give game with cooperation as one of the ESSs is to alter the type of game that is considered.

An intermediate: the Snowdrift game

As we saw in Web Topic 10.5, a single-bout 2 x 2 symmetric contest has four possible outcomes: each of the two strategies can be the only pure ESS, both can be pure ESSs, or there may be a stable mixture of the two strategies as the ESS. The Prisoner’s Dilemma game has a single pure ESS: defect. All give games have two pure ESSs, which one is seen depends on initial conditions or if populations are finite, on relative bases of attraction. The Snowdrift Game describes a cooperation context in which the ESS is a mixed strategy (Hauert and Doebeli 2004; Nowak 2006a). The scenario begins with two commuters trapped in deep snow on the highway. If both get out and shovel snow, they can get home sooner and split the effort. However, if one lets the other do the digging, he still gets home but evades all the work. If neither shovel snow, they remain stuck. Assuming a common currency for costs and benefits, let R be the net payoff if both drivers dig and share the work equally, T be the payoff to the driver who lets the other do all the work, S be the payoff to the driver who does all the digging alone, and P be the payoff to both drivers if neither shovels. This results in the same normal form matrix as the Prisoner’s Dilemma except that this time T > R > S > P. This is shown in Figure 4:

Figure 4. Payoff matrix for Snowdrift game. The best response is always the opposite of your opponent’s response. The only equilibrium is a mixed ESS in which players cooperate a fraction (SP)/(S + T P – R) of the time and otherwise defect.

This model gets us part of the way between the take game of the Prisoner’s Dilemma and a cooperative give game, but only part of the way. In addition, it only characterizes a subset of the types of cooperation seen in nature: specifically, those in which there is a common task whose completion benefits both parties. It does not help us with situations in which the two parties do not face a common task.

Replacing the take game with a give game

As long as there are opportunities to cheat or defect, it appears impossible to get a give game involving cooperation from a single-bout contest like the Prisoner’s Dilemma. In some contexts, we might be able to find stable mixtures of the two strategies (Snowdrift game). However, we can achieve a situation in which cooperation is a pure ESS only if we change the type of game. One solution is to allow participants to interact repeatedly: if early bouts can provide information that allows players to adjust strategies, the resulting sequential game might have an ESS policy favoring cooperation that is unavailable to single-bout contexts. A second way to change the game is to replace dyadic contests with multiplayer scrambles: as we saw in Web Topic 10.5, adding more players can open up ESS possibilities not available to dyadic games. For both approaches, we begin with the assumptions that the relevant populations are infinite in size and strategies are “well-mixed” throughout the population.

Favoring the cooperation ESS in a give game

The models reviewed above typically assume that focal populations are effectively infinite in size and well-mixed, so that different strategies are encountered in proportion to their global abundances. The payoff matrices for these games usually do not change if these assumptions are violated, and thus the basic outcomes (e.g., a give game) remain. What can change if one or both assumptions is violated is which of the alternative ESSs in the resulting give game is most likely to occur. Finite populations are subject to drift, and the smaller the population, the greater the role of drift relative to selection. The dynamics become stochastic, and this can significantly weaken the usual predictions.

Populations can also have heterogeneous distributions of alternative strategies; such populations are often called “structured” and modeled as networks (see Chapter 15). A well-mixed population is a network in which every individual is potentially linked to every other individual. Heterogeneously structured populations have a much more sparse network than that for a well-mixed population: only some individuals in structured populations are linked to any focal individual. There are two general classes of heterogeneous networks: a) those in which there is at least one path (however convoluted) linking any two individuals (a giant component); or b) not all individuals are linked by a path and thus the population is subdivided into multiple components.

All of the models below assume finite populations; however, they vary as to whether they focus on small population sizes where drift is significant, or sufficiently large ones where drift is not a major factor. They also vary depending on whether their structure is well-mixed, fully but sparsely linked, or subdivided into separate components. We cannot review all combinations here, but will provide a sampling of the more widely cited ones.

Matching field data to theory

Many examples of by-product mutualism and pseudo-reciprocity have been cited in the literature; we discuss some of these in a review of game theoretic models for environmental signaling in Web Topic 14.1. On the other hand, both direct and indirect reciprocity, policing, and market models have proved to be less easily assigned to specific biological examples. In part, this is because none of the alternative accountings listed above, this time including kin selection and greenbeard biases, need to be exclusive determinants of a given example of cooperative behavior. On the contrary, mixes of economics are likely the rule rather than the exception. For that reason, and because different authors cannot often agree on which accounting predominates for a given phenomenon, we do not provide a list of examples assigned to each accounting here. Instead, the reader is directed to several published reviews that attempt to make assignments including mixtures of accountings to examples of animal cooperation. General reviews include Dugatkin (1998, 2002), Sachs et al. (2004), Silk (2007), West et al. (2007a), Bshary and Bergmüller (2008), Clutton-Brock (2009), and Connor (2010). Melis et al. (2010) compare the mechanisms and accountings that justify cooperation in animals with those in humans.

Further reading

A number of authors have proposed taxonomies or classification trees for the economics of cooperation among unrelated animals. Stark (2010) combines all the classical 2 x 2 models into a common framework, with a particular focus on cases where partial cooperation may evolve in both single-shot and iterated contexts. More details on the tree shown in Figure 1 can be found in Connor (1995), Leimar and Connor (2003), Bergmüller et al. (2007), Connor (2007), Bshary and Bergmüller (2008), Connor (2010), and Bshary and Bronstein (2011). Nowak (2006b) adds spatial pattern effects to the list (but leaves out some of the other accountings). His introduction to spatial and finite game models (Nowak 2006a) is an excellent starting point. Nowak and Sigmund (2005) provide a thoughtful review of indirect reciprocity and its relationship to alternative accountings. Reviews of cooperation economics that compare direct benefit economics with kin selection are provided by Queller (1985), Sachs et al. (2004), Lehmann and Keller (2006), Queller and Strassmann (2006), West et al. (2007a), and Clutton-Brock (2009). Bowles and Hammerstein (2003) provide interesting contrasts in the application of market theory to human versus animal social contexts.

Given that cooperation is often at most a two-ESS give game, it should not be surprising to find that some mutualisms in the past have now degenerated into pure defect states. Sachs and Simms (2006) use molecular techniques to identify examples in which mutualism is now absent, but was likely present in the past. Sanfey et al. (2003) discuss which parts of human brains are involved in selected cooperation games, and Soares et al. (2010) provide similar perspectives on the neuroendocrine bases of cooperative behaviors.

Several recent books try to tie much of this together. The first chapter of Karl Sigmund’s The Calculus of Selfishness (2010) provides a good overview; the rest of the book relies very heavily on complex mathematics. Nowak and Highfield’s SuperCooperators (2011) is also readable, but somewhat polemical and not always even-handed.

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