- Email: [email protected]
Cooperation should not be assumed
Update
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TRENDS in Ecology and Evolution Vol.21 No.9
5 Felsenstein, J. (1978) Cases in which parsimony or compatibility methods will be positively misleading. Syst. Zool. 27, 401–410 6 Hendy, M.D. and Penny, D. (1989) A framework for the quantitative study of evolutionary trees. Syst. Zool. 38, 296–309 7 Wa¨gele, J-W. (2005) Foundations of Phylogenetic Systematics, Pfeil 8 Paland, S. and Lynch, M. (2006) Transitions to asexuality result in excess amino acid substitutions. Science 311, 990–992 9 Smith, R.J. et al. (2006) Living males of the ‘ancient asexual’ Darwinulidae (Ostracoda: Crustacea). Proc. R. Soc. B 273, 1569–1578 10 Halkett, F. et al. (2005) Tackling the population genetics of clonal and partially clonal organisms. Trends Ecol. Evol. 20, 194–201 11 Peck, J.R. and Waxman, D. (2000) What is wrong with little sex? J. Evol. Biol. 13, 63–69 12 Marin, I. and Baker, B.S. (1998) The evolutionary dynamics of sex determination. Science 281, 1990–1994
13 Arnqvist, G. and Rowe, L. (2002) Correlated evolution of male and female morphologies in water striders. Evolution 56, 936–947 14 Harris, M.P. et al. (2006) The development of archosaurian firstgeneration teeth in a chicken mutant. Curr. Biol. 16, 371–377 15 Kearney, T.M. (2005) Hybridization, glaciation and geographical parthenogenesis. Trends Ecol. Evol. 20, 495–502 16 Ho¨rnschmeyer, T. (2005) Archostemata Kolbe, 1908. In Coleoptera, Beetles. Volume 1: Morphology and Systematics (Handbook of Zoology, IV. Arthropoda: Insecta; Part 38) (Beutel, R.G. and Leschen, A.B., eds), pp. 29–42, de Gruyter 17 Butlin, R.K. et al. (1999) Origin, age and diversity of clones. J. Evol. Biol. 12, 1020–1022 0169-5347/$ – see front matter ß 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.tree.2006.06.017
Cooperation should not be assumed John M. McNamara1, Ken Binmore2 and Alasdair I. Houston3 1
Centre for Behavioural Biology, Department of Mathematics, University of Bristol, Bristol, UK, BS8 1TW Department of Economics, University College London, London, UK, WC1E 6BT 3 Centre for Behavioural Biology, School of Biological Sciences, University of Bristol, Bristol, UK, BS8 1UG 2
Evolutionary game theory provides a framework for explaining social interactions, including those between males and females. In a recent article, Roughgarden et al. discuss a new approach to sexual selection based on cooperative game theory and argue that cooperation rather than competition is fundamental in interactions between the sexes. However, compelling reasons for adopting this approach are not given and the authors do not adopt it consistently. We argue that noncooperative game theory provides an adequate basis for understanding sexual selection, but that further work is needed to produce realistic models. We agree with Roughgarden and colleagues that bargaining is an important aspect of social interactions, but this is not a novel claim. Bargaining does not require the assumption of cooperation and does not necessarily lead to it. Introduction The current approach to explaining sexual selection is based on the idea that individuals act in their own best interests. This often leads to a conflict between males and females [1]. In a recent paper that criticizes this idea [2], Roughgarden and colleagues argue for a new approach in which non-cooperative game theory (i.e. competition) is replaced by cooperative game theory (i.e. cooperation). Many authors have discussed various aspects of the claims of Roughgarden et al. in the Letters section of Science for May 5 2006 (Vol. 312, pp 689–697). Here, we concentrate on the challenge to evolutionary game theory. Evolutionary game theory Evolutionary game theory provides a framework for understanding how evolution shapes interactions between organisms (i.e. it forms the basis for explaining social Corresponding author: Houston, A.I. ([email protected]) Available online 13 July 2006 www.sciencedirect.com
interactions). The standard use of evolutionary game theory assumes that individuals act so as to maximise their own fitness. Each member of a population has a genetically determined strategy that specifies how it interacts with other population members. The strategy adopted by most population members at evolutionary stability is called an evolutionarily stable strategy (ESS) [3]. Under this strategy, each population member is using the best strategy given that others are also using this strategy. Thus, at an ESS, no individual can increase its fitness by adopting a different strategy. This is the definition of a Nash equilibrium (NE) (which Roughgarden et al. [2] call a ‘Nash competitive equilibrium’ or NCE). Cooperation is not assumed, but can emerge from each individual trying ‘selfishly’ to maximise its own fitness. Many biological interactions between two organisms are modelled as a game in which each player makes a single decision in the absence of information about the decision of the other player. We refer to this as a one-shot simultaneous game. In such games, it is usual to assume that the choice of action of an animal is not influenced by the behaviour of its opponent, but is fixed, being determined by the genotype of the animal alone. Roughgarden et al. [2] are right to point out that one-shot simultaneous games are often unrealistic and we should take account of interactions between the players during the course of the game; however, this point has been made before in the context of interactions between males and females [4], and various models include interactions between animals [4–11]. When bargaining occurs, it is the rules (i.e. strategies) that govern bargaining that are genetically determined (and hence subject to evolutionary processes). At an ESS, rules are best responses to each other and are hence in NE [4]. Biological models in which rules evolve sometimes predict more cooperation than does the NE based on one-shot games, sometimes less [4,7,8].
Update
TRENDS in Ecology and Evolution
The approach of Roughgarden and colleagues In their recent paper, Roughgarden et al. propose that cooperation rather than competition is fundamental in social interactions (‘The distinction between our proposition and previous work is apparent in the use of the word ‘cooperative,’ which means only a mutually beneficial outcome in previous work but describes a process of perceiving and playing the game in our work.’ [2]). However the authors do not give adequate justification for this approach and do not adopt it consistently throughout the article. We consider Roughgarden et al. to also have an unusual view of ‘competitive’ and cooperative games: ‘In competitive games, the players do not communicate. In cooperative games, players make threats, promises, and side payments to each other; play together as teams; and form and dissolve coalitions’ [2]. Communication and threats are biologically important, but the standard view in both biology and economics is that they are included in what Roughgarden et al. call ‘competitive’ games [3,12]. To clarify terminology, we note that what Roughgarden et al. call a ‘competitive’ game is usually referred to as a non-cooperative game in both economics and biology. The use of the term ‘non-cooperative game’ assumes that agents act in their own best interests but does not preclude the emergence of cooperation. The contributions of Nash John Nash invented both the Nash bargaining solution (NBS, Box 1) [13] and the idea of a NE [14]. The NBS is part of a body of work usually described as cooperative game theory, whereas the NE is the basic concept of non-cooperative game theory. Roughgarden et al. offer cooperative game theory as an alternative to non-cooperative game theory, emphasising the role of the NBS [2]. Nash did not regard cooperative and non-cooperative game theory as rival approaches. On the contrary, he regarded them as complementary. In cooperative game theory, strategic issues are confined to a black box. Axioms are usually formulated that relate only to the payoff structure of a game. Ideally, the set of axioms will characterise a unique ‘cooperative solution concept’. The NBS is the prototypical example. Such a cooperative solution concept provides a simple prediction of the outcome of bargaining among the players that can be applied without getting into any messy analysis of who can do what and when. But when should we proceed as though the axioms apply? Nash’s answer to this is called the ‘Nash program’ [15–17]. It calls for the construction of a (usually much simplified) model of what the players can and cannot do while bargaining. The result is a non-cooperative bargaining game. One computes the NE of this bargaining game if one can, and hopes to be able to solve the problem of how to choose among the NE if there is more than one. If successful in the latter enterprise, one then compares the selected NE outcome with the prediction generated by the cooperative solution concept. If the two are the same, one has some support for the cooperative solution concept. If not, then the cooperative solution concept does not apply in the context under study. This makes cooperative game theory an auxiliary discipline to non-cooperative theory. It does not make it an alternative, as Roughgarden et al. appear to believe [2]. www.sciencedirect.com
Vol.21 No.9
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Box 1. The Nash bargaining solution Nash [13,14] introduced the Nash bargaining solution (NBS) and Nash equilibrium (NE) in successive years. NE arise more naturally in biology, because a system is always at a NE when EESs are played. However, is there also a role for the different idea of the Nash bargaining solution? The Nash bargaining problem consists of a feasible set (the set of pairs of possible payoffs to the two players) and a disagreement point (the payoffs that each player will get in the event of their failing to agree). Nash gave a set of rationality axioms that assign a unique solution to each such pair. The most important axioms are Symmetry and Independence of Irrelevant Alternatives. The latter says that if players sometimes choose x when y is available, then they will still choose x when y is unavailable. Using these axioms, Nash showed that there is usually a unique pair of payoffs to the players that are the Nash bargaining solution. Let P1 and P2 denote possible payoffs to Player 1 and Player 2, respectively. Let P1 and P2 denote the payoffs to Player 1 and Player 2, respectively, at the disagreement point. At the NBS, the payoffs P1 and P2 maximise the product (P 1 P1 )(P 2 P2 ). Under the Nash program, the NBS is justified in a particular bargaining situation if it arises as the NE of the bargaining game. This means that the fundamental biological assumption is that players maximise their own fitness; the resulting product maximisation is a consequence if the NE coincides with the NBS. Drawbacks to using the NBS as a predictor include its use of a symmetry axiom, its failure to recognise the potential importance to the players of their outside options, and its highly unrealistic assumption that the players have no secrets from each other. The NBS can sometimes be generalised to take account of the first and second of these problems, but attempts at an adequate generalisation in the face of incomplete information have yet to prove themselves [20]. The Nash program calls for the predictive power of cooperative solution concepts such as the NBS to be tested by modeling the negotiation process as a (non-cooperative) game by formalizing its implicit rules. If this negotiation game can be solved using the idea of a NE, we are in trouble unless this non-cooperative solution outcome coincides with the cooperative solution concept under study. A success for the Nash program came when Rubinstein [21] showed that the non-cooperative negotiation game in which two players exchange offers until someone says ‘yes’ has a unique (subgame-perfect) NE, provided that the players discount the unproductive passage of time at a positive rate. (A subgame-perfect strategy profile must include NE in all subgames, whether in equilibrium or not.) When the discount rates of the players are equal and the time interval between successive proposals is sufficiently small, the unique bargaining outcome in Rubinstein’s model approximates the NBS [19].
Commitments, threats and cheating The basic requirement for orthodox cooperative game theory is that the players are able to sign binding contracts that commit them to a course of action. But animals do not have a legal system. Any threats, promises, take-it-orleave-it demands and the like implicitly made by animals must therefore be credible if they are to be effective in modifying the behaviour of other animals. Cheaters and freeloaders will always be a problem when there are no binding contracts. As Roughgarden et al. point out [2], cheaters can be suppressed by penalties imposed by individuals that are prepared to enforce them. We agree, but there is usually a cost to policing. It is then necessary to show that policing is in the self-interest of the enforcer. Policing and enforcement have to be included as part of the process, and again one seeks the individually best strategies. In their examples [2], Roughgarden et al. talk in terms of a solution in which threats are effective because it is
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in the self-interest of the threatened individual to take note of credible threats. This is the standard approach of non-cooperative game theory, and is not consistent with the claim of Roughgarden et al. that cooperation is fundamental and hence can be taken as a starting point of the analysis [2]. The future Bargaining is part of non-cooperative game theory. An approach based solely on cooperation is neither justified nor required. Economic models show that the NBS can be the outcome of a negotiation process, but results are largely for the case of perfect information. When the players might have secrets from each other, economic models do not predict the simple version of the NBS considered by Roughgarden et al. Indeed, these models predict that cooperative behaviour will be unattainable much of the time, even with fully rational players using an optimal bargaining protocol [18]. The problem is that players have an incentive to lie about the least that they are willing to accept, and when both make such extravagant claims the result can sometimes be disastrous, and far from cooperative, even in ideally favourable circumstances. Neither is it true that adopting a bargaining framework that incorporates insight from cooperative game theory excludes the orthodox approach to sexual selection. It only appears so in the article by Roughgarden et al. because they seem unaware of later work that incorporates outside options into Nash bargaining theory. In the context of sexual selection, outside options are alternative partners with which an animal could interact. When the outside options of a player are sufficiently large, they trump the considerations that lead to the NBS [19]. But this is the case that is taken as the norm in the orthodox theory of sexual selection. Roughgarden et al. have correctly highlighted the need to consider interactions between animals. Unfortunately, the NBS is a black box that ignores this process. We need to understand when the outcome of the bargaining process can and cannot be predicted by the NBS. To do so, we need realistic models that do not assume symmetry, perfect information and no outside options. It is likely that the article by Roughgarden et al. should at least stimulate this research program.
Acknowledgments We thank Laurent Keller and two anonymous referees for comments on a previous version of this article. The research of J.M.McN. and A.I.H. on parental care is supported by the BBSRC.
References 1 Andersson, M. (1994) Sexual Selection, Princeton University Press 2 Roughgarden, J. et al. (2006) Reproductive social behavior: cooperative games to replace sexual selection. Science 311, 965–969 3 Maynard Smith, J. (1982) Evolution and the Theory of Games, Cambridge University Press 4 McNamara, J.M. et al. (1999) Incorporating rules for responding into evolutionary games. Nature 401, 368–371 5 Enquist, M. and Leimar, O. (1983) Evolution of fighting behavior decision rules and assessment of relative strength. J. Theor. Biol. 102, 387–410 6 Johnstone, R.A. (1996) Begging signals and parent-offspring conflict: do parents always win? Proc. R. Soc. B 263, 1677–1681 7 Killingback, T. and Doebeli, M. (2002) The continuous prisoner’s dilemma and the evolution of cooperation through reciprocal altruism with variable investment. Am. Nat. 160, 421–438 8 Sherratt, T.N. and Roberts, G. (2002) The stability of cooperation involving variable investment. J. Theor. Biol. 215, 47–56 9 McNamara, J.M. et al. (2003) Should young ever be better off with one parent than with two? Behav. Ecol. 14, 301–310 10 Taylor, P.D. and Day, T. (2004) Stability in negotiation games and the emergence of cooperation. Proc. R. Soc. B 271, 669–674 11 Pen, I. and Taylor, P.D. (2005) Modelling information exchange in worker-queen conflict over sex allocation. Proc. R. Soc. B 272, 2403– 2408 12 Luce, R.D. and Raiffa, H. (1957) Games and Decisions, John Wiley & Sons 13 Nash, J. (1950) The bargaining problem. Econometrica 18, 155–162 14 Nash, J. (1951) Non-cooperative games. Ann. Math. 54, 286–295 15 Binmore, K. (1987) Nash bargaining theory I. In The Economics of Bargaining (Binmore, K. and Dasgupta, P., eds), pp. 27–46, Blackwell 16 Binmore, K. (1987) Nash bargaining theory II. In The Economics of Bargaining (Binmore, K. and Dasgupta, P., eds), pp. 61–76, Blackwell 17 Binmore, K. (1987) Nash bargaining theory III. In The Economics of Bargaining (Binmore, K. and Dasgupta, P., eds), pp. 239–256, Blackwell 18 Myerson, R.B. (1991) Game Theory: Analysis of Conflict, Harvard University Press 19 Binmore, K. et al. (1982) The Nash bargaining solution in economic modelling. Rand J. Econ. 17, 176–188 20 Muthoo, A. (1999) Bargaining Theory with Applications, Cambridge University Press 21 Rubinstein, A. (1982) Perfect equilibrium in a bargaining model. Econometrica 50, 97–109
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476
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5 Felsenstein, J. (1978) Cases in which parsimony or compatibility methods will be positively misleading. Syst. Zool. 27, 401–410 6 Hendy, M.D. and Penny, D. (1989) A framework for the quantitative study of evolutionary trees. Syst. Zool. 38, 296–309 7 Wa¨gele, J-W. (2005) Foundations of Phylogenetic Systematics, Pfeil 8 Paland, S. and Lynch, M. (2006) Transitions to asexuality result in excess amino acid substitutions. Science 311, 990–992 9 Smith, R.J. et al. (2006) Living males of the ‘ancient asexual’ Darwinulidae (Ostracoda: Crustacea). Proc. R. Soc. B 273, 1569–1578 10 Halkett, F. et al. (2005) Tackling the population genetics of clonal and partially clonal organisms. Trends Ecol. Evol. 20, 194–201 11 Peck, J.R. and Waxman, D. (2000) What is wrong with little sex? J. Evol. Biol. 13, 63–69 12 Marin, I. and Baker, B.S. (1998) The evolutionary dynamics of sex determination. Science 281, 1990–1994
13 Arnqvist, G. and Rowe, L. (2002) Correlated evolution of male and female morphologies in water striders. Evolution 56, 936–947 14 Harris, M.P. et al. (2006) The development of archosaurian firstgeneration teeth in a chicken mutant. Curr. Biol. 16, 371–377 15 Kearney, T.M. (2005) Hybridization, glaciation and geographical parthenogenesis. Trends Ecol. Evol. 20, 495–502 16 Ho¨rnschmeyer, T. (2005) Archostemata Kolbe, 1908. In Coleoptera, Beetles. Volume 1: Morphology and Systematics (Handbook of Zoology, IV. Arthropoda: Insecta; Part 38) (Beutel, R.G. and Leschen, A.B., eds), pp. 29–42, de Gruyter 17 Butlin, R.K. et al. (1999) Origin, age and diversity of clones. J. Evol. Biol. 12, 1020–1022 0169-5347/$ – see front matter ß 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.tree.2006.06.017
Cooperation should not be assumed John M. McNamara1, Ken Binmore2 and Alasdair I. Houston3 1
Centre for Behavioural Biology, Department of Mathematics, University of Bristol, Bristol, UK, BS8 1TW Department of Economics, University College London, London, UK, WC1E 6BT 3 Centre for Behavioural Biology, School of Biological Sciences, University of Bristol, Bristol, UK, BS8 1UG 2
Evolutionary game theory provides a framework for explaining social interactions, including those between males and females. In a recent article, Roughgarden et al. discuss a new approach to sexual selection based on cooperative game theory and argue that cooperation rather than competition is fundamental in interactions between the sexes. However, compelling reasons for adopting this approach are not given and the authors do not adopt it consistently. We argue that noncooperative game theory provides an adequate basis for understanding sexual selection, but that further work is needed to produce realistic models. We agree with Roughgarden and colleagues that bargaining is an important aspect of social interactions, but this is not a novel claim. Bargaining does not require the assumption of cooperation and does not necessarily lead to it. Introduction The current approach to explaining sexual selection is based on the idea that individuals act in their own best interests. This often leads to a conflict between males and females [1]. In a recent paper that criticizes this idea [2], Roughgarden and colleagues argue for a new approach in which non-cooperative game theory (i.e. competition) is replaced by cooperative game theory (i.e. cooperation). Many authors have discussed various aspects of the claims of Roughgarden et al. in the Letters section of Science for May 5 2006 (Vol. 312, pp 689–697). Here, we concentrate on the challenge to evolutionary game theory. Evolutionary game theory Evolutionary game theory provides a framework for understanding how evolution shapes interactions between organisms (i.e. it forms the basis for explaining social Corresponding author: Houston, A.I. ([email protected]) Available online 13 July 2006 www.sciencedirect.com
interactions). The standard use of evolutionary game theory assumes that individuals act so as to maximise their own fitness. Each member of a population has a genetically determined strategy that specifies how it interacts with other population members. The strategy adopted by most population members at evolutionary stability is called an evolutionarily stable strategy (ESS) [3]. Under this strategy, each population member is using the best strategy given that others are also using this strategy. Thus, at an ESS, no individual can increase its fitness by adopting a different strategy. This is the definition of a Nash equilibrium (NE) (which Roughgarden et al. [2] call a ‘Nash competitive equilibrium’ or NCE). Cooperation is not assumed, but can emerge from each individual trying ‘selfishly’ to maximise its own fitness. Many biological interactions between two organisms are modelled as a game in which each player makes a single decision in the absence of information about the decision of the other player. We refer to this as a one-shot simultaneous game. In such games, it is usual to assume that the choice of action of an animal is not influenced by the behaviour of its opponent, but is fixed, being determined by the genotype of the animal alone. Roughgarden et al. [2] are right to point out that one-shot simultaneous games are often unrealistic and we should take account of interactions between the players during the course of the game; however, this point has been made before in the context of interactions between males and females [4], and various models include interactions between animals [4–11]. When bargaining occurs, it is the rules (i.e. strategies) that govern bargaining that are genetically determined (and hence subject to evolutionary processes). At an ESS, rules are best responses to each other and are hence in NE [4]. Biological models in which rules evolve sometimes predict more cooperation than does the NE based on one-shot games, sometimes less [4,7,8].
Update
TRENDS in Ecology and Evolution
The approach of Roughgarden and colleagues In their recent paper, Roughgarden et al. propose that cooperation rather than competition is fundamental in social interactions (‘The distinction between our proposition and previous work is apparent in the use of the word ‘cooperative,’ which means only a mutually beneficial outcome in previous work but describes a process of perceiving and playing the game in our work.’ [2]). However the authors do not give adequate justification for this approach and do not adopt it consistently throughout the article. We consider Roughgarden et al. to also have an unusual view of ‘competitive’ and cooperative games: ‘In competitive games, the players do not communicate. In cooperative games, players make threats, promises, and side payments to each other; play together as teams; and form and dissolve coalitions’ [2]. Communication and threats are biologically important, but the standard view in both biology and economics is that they are included in what Roughgarden et al. call ‘competitive’ games [3,12]. To clarify terminology, we note that what Roughgarden et al. call a ‘competitive’ game is usually referred to as a non-cooperative game in both economics and biology. The use of the term ‘non-cooperative game’ assumes that agents act in their own best interests but does not preclude the emergence of cooperation. The contributions of Nash John Nash invented both the Nash bargaining solution (NBS, Box 1) [13] and the idea of a NE [14]. The NBS is part of a body of work usually described as cooperative game theory, whereas the NE is the basic concept of non-cooperative game theory. Roughgarden et al. offer cooperative game theory as an alternative to non-cooperative game theory, emphasising the role of the NBS [2]. Nash did not regard cooperative and non-cooperative game theory as rival approaches. On the contrary, he regarded them as complementary. In cooperative game theory, strategic issues are confined to a black box. Axioms are usually formulated that relate only to the payoff structure of a game. Ideally, the set of axioms will characterise a unique ‘cooperative solution concept’. The NBS is the prototypical example. Such a cooperative solution concept provides a simple prediction of the outcome of bargaining among the players that can be applied without getting into any messy analysis of who can do what and when. But when should we proceed as though the axioms apply? Nash’s answer to this is called the ‘Nash program’ [15–17]. It calls for the construction of a (usually much simplified) model of what the players can and cannot do while bargaining. The result is a non-cooperative bargaining game. One computes the NE of this bargaining game if one can, and hopes to be able to solve the problem of how to choose among the NE if there is more than one. If successful in the latter enterprise, one then compares the selected NE outcome with the prediction generated by the cooperative solution concept. If the two are the same, one has some support for the cooperative solution concept. If not, then the cooperative solution concept does not apply in the context under study. This makes cooperative game theory an auxiliary discipline to non-cooperative theory. It does not make it an alternative, as Roughgarden et al. appear to believe [2]. www.sciencedirect.com
Vol.21 No.9
477
Box 1. The Nash bargaining solution Nash [13,14] introduced the Nash bargaining solution (NBS) and Nash equilibrium (NE) in successive years. NE arise more naturally in biology, because a system is always at a NE when EESs are played. However, is there also a role for the different idea of the Nash bargaining solution? The Nash bargaining problem consists of a feasible set (the set of pairs of possible payoffs to the two players) and a disagreement point (the payoffs that each player will get in the event of their failing to agree). Nash gave a set of rationality axioms that assign a unique solution to each such pair. The most important axioms are Symmetry and Independence of Irrelevant Alternatives. The latter says that if players sometimes choose x when y is available, then they will still choose x when y is unavailable. Using these axioms, Nash showed that there is usually a unique pair of payoffs to the players that are the Nash bargaining solution. Let P1 and P2 denote possible payoffs to Player 1 and Player 2, respectively. Let P1 and P2 denote the payoffs to Player 1 and Player 2, respectively, at the disagreement point. At the NBS, the payoffs P1 and P2 maximise the product (P 1 P1 )(P 2 P2 ). Under the Nash program, the NBS is justified in a particular bargaining situation if it arises as the NE of the bargaining game. This means that the fundamental biological assumption is that players maximise their own fitness; the resulting product maximisation is a consequence if the NE coincides with the NBS. Drawbacks to using the NBS as a predictor include its use of a symmetry axiom, its failure to recognise the potential importance to the players of their outside options, and its highly unrealistic assumption that the players have no secrets from each other. The NBS can sometimes be generalised to take account of the first and second of these problems, but attempts at an adequate generalisation in the face of incomplete information have yet to prove themselves [20]. The Nash program calls for the predictive power of cooperative solution concepts such as the NBS to be tested by modeling the negotiation process as a (non-cooperative) game by formalizing its implicit rules. If this negotiation game can be solved using the idea of a NE, we are in trouble unless this non-cooperative solution outcome coincides with the cooperative solution concept under study. A success for the Nash program came when Rubinstein [21] showed that the non-cooperative negotiation game in which two players exchange offers until someone says ‘yes’ has a unique (subgame-perfect) NE, provided that the players discount the unproductive passage of time at a positive rate. (A subgame-perfect strategy profile must include NE in all subgames, whether in equilibrium or not.) When the discount rates of the players are equal and the time interval between successive proposals is sufficiently small, the unique bargaining outcome in Rubinstein’s model approximates the NBS [19].
Commitments, threats and cheating The basic requirement for orthodox cooperative game theory is that the players are able to sign binding contracts that commit them to a course of action. But animals do not have a legal system. Any threats, promises, take-it-orleave-it demands and the like implicitly made by animals must therefore be credible if they are to be effective in modifying the behaviour of other animals. Cheaters and freeloaders will always be a problem when there are no binding contracts. As Roughgarden et al. point out [2], cheaters can be suppressed by penalties imposed by individuals that are prepared to enforce them. We agree, but there is usually a cost to policing. It is then necessary to show that policing is in the self-interest of the enforcer. Policing and enforcement have to be included as part of the process, and again one seeks the individually best strategies. In their examples [2], Roughgarden et al. talk in terms of a solution in which threats are effective because it is
Update
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TRENDS in Ecology and Evolution Vol.21 No.9
in the self-interest of the threatened individual to take note of credible threats. This is the standard approach of non-cooperative game theory, and is not consistent with the claim of Roughgarden et al. that cooperation is fundamental and hence can be taken as a starting point of the analysis [2]. The future Bargaining is part of non-cooperative game theory. An approach based solely on cooperation is neither justified nor required. Economic models show that the NBS can be the outcome of a negotiation process, but results are largely for the case of perfect information. When the players might have secrets from each other, economic models do not predict the simple version of the NBS considered by Roughgarden et al. Indeed, these models predict that cooperative behaviour will be unattainable much of the time, even with fully rational players using an optimal bargaining protocol [18]. The problem is that players have an incentive to lie about the least that they are willing to accept, and when both make such extravagant claims the result can sometimes be disastrous, and far from cooperative, even in ideally favourable circumstances. Neither is it true that adopting a bargaining framework that incorporates insight from cooperative game theory excludes the orthodox approach to sexual selection. It only appears so in the article by Roughgarden et al. because they seem unaware of later work that incorporates outside options into Nash bargaining theory. In the context of sexual selection, outside options are alternative partners with which an animal could interact. When the outside options of a player are sufficiently large, they trump the considerations that lead to the NBS [19]. But this is the case that is taken as the norm in the orthodox theory of sexual selection. Roughgarden et al. have correctly highlighted the need to consider interactions between animals. Unfortunately, the NBS is a black box that ignores this process. We need to understand when the outcome of the bargaining process can and cannot be predicted by the NBS. To do so, we need realistic models that do not assume symmetry, perfect information and no outside options. It is likely that the article by Roughgarden et al. should at least stimulate this research program.
Acknowledgments We thank Laurent Keller and two anonymous referees for comments on a previous version of this article. The research of J.M.McN. and A.I.H. on parental care is supported by the BBSRC.
References 1 Andersson, M. (1994) Sexual Selection, Princeton University Press 2 Roughgarden, J. et al. (2006) Reproductive social behavior: cooperative games to replace sexual selection. Science 311, 965–969 3 Maynard Smith, J. (1982) Evolution and the Theory of Games, Cambridge University Press 4 McNamara, J.M. et al. (1999) Incorporating rules for responding into evolutionary games. Nature 401, 368–371 5 Enquist, M. and Leimar, O. (1983) Evolution of fighting behavior decision rules and assessment of relative strength. J. Theor. Biol. 102, 387–410 6 Johnstone, R.A. (1996) Begging signals and parent-offspring conflict: do parents always win? Proc. R. Soc. B 263, 1677–1681 7 Killingback, T. and Doebeli, M. (2002) The continuous prisoner’s dilemma and the evolution of cooperation through reciprocal altruism with variable investment. Am. Nat. 160, 421–438 8 Sherratt, T.N. and Roberts, G. (2002) The stability of cooperation involving variable investment. J. Theor. Biol. 215, 47–56 9 McNamara, J.M. et al. (2003) Should young ever be better off with one parent than with two? Behav. Ecol. 14, 301–310 10 Taylor, P.D. and Day, T. (2004) Stability in negotiation games and the emergence of cooperation. Proc. R. Soc. B 271, 669–674 11 Pen, I. and Taylor, P.D. (2005) Modelling information exchange in worker-queen conflict over sex allocation. Proc. R. Soc. B 272, 2403– 2408 12 Luce, R.D. and Raiffa, H. (1957) Games and Decisions, John Wiley & Sons 13 Nash, J. (1950) The bargaining problem. Econometrica 18, 155–162 14 Nash, J. (1951) Non-cooperative games. Ann. Math. 54, 286–295 15 Binmore, K. (1987) Nash bargaining theory I. In The Economics of Bargaining (Binmore, K. and Dasgupta, P., eds), pp. 27–46, Blackwell 16 Binmore, K. (1987) Nash bargaining theory II. In The Economics of Bargaining (Binmore, K. and Dasgupta, P., eds), pp. 61–76, Blackwell 17 Binmore, K. (1987) Nash bargaining theory III. In The Economics of Bargaining (Binmore, K. and Dasgupta, P., eds), pp. 239–256, Blackwell 18 Myerson, R.B. (1991) Game Theory: Analysis of Conflict, Harvard University Press 19 Binmore, K. et al. (1982) The Nash bargaining solution in economic modelling. Rand J. Econ. 17, 176–188 20 Muthoo, A. (1999) Bargaining Theory with Applications, Cambridge University Press 21 Rubinstein, A. (1982) Perfect equilibrium in a bargaining model. Econometrica 50, 97–109
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