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On Sugden’s “mutually beneficial practice” and Berge equilibrium

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Abstract

Cooperative behavior is often observed in ordinary market transactions. To account for this observation, Robert Sugden proposes a team reasoning theory in which the common interest of team reasoners is defined by the notion of mutually beneficial practice. We study the relationships between mutually beneficial practices and Berge equilibria (a Berge equilibrium is a strategy profile such that a unilateral change of strategy by any one player cannot increase another player’s payoff). We propose two sufficient conditions under which a (strict) Berge equilibrium is a mutually beneficial practice.

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Notes

  1. Sugden (2015), p. 156.

  2. Ibid., p. 159. In addition to the notion of mutually beneficial practice, Sugden also presents a Schema of Cooperative Team reasoning. We shall not consider this schema in this paper.

  3. For a brief history of this game theoretical equilibrium concept, and the reasons why it has seldom been used so far, see Colman et al. (2011) and Courtois et al. (2015).

  4. Actually, in two-person two-action games a Berge equilibrium in which the payoff of each agent is higher than his maximin gain is always a strict Berge equilibrium.

  5. We shall always assume that the maximum and the minimum are realized.

  6. Definition 1 is slightly different from that proposed by Sugden. Sugden (2015, p. 17–18) defines the second condition of a mutually beneficial practice as follows: “...To formulate the second condition, let N be the set of players \(\lbrace 1, \dots , n\rbrace\), and consider any subgroup G, where G is a subset of N that contains at least one and fewer than n players. Let \(G'\) be the complement of G. For each player j in G, let \(v_{j}(G, \mathbf {s}^*)\) be the minimum payoff that j can receive, given that each member of G chooses his component of \(\mathbf {s}^*\). I will say that G benefits from the participation of \(G'\) in \(\mathbf {s}^*\) if and only if \(u_{j}(\mathbf {s}^*) \ge v_{j}(G, \mathbf {s}^*)\) for all j in G, with a strict inequality for at least one j. Condition 2 is that, for every subgroup G that contains at least one and fewer than n players, G benefits from the participation of \(G'\).” But the inequality \(u_{j}(\mathbf {s}^*) \ge v_{j}(G, \mathbf {s}^*)\) is always satisfied by definition of \(v_{j}\). What is required is that there is one agent in any subgroup for whom this inequality is strictly satisfied. I thank a referee for pointing out this fact.

  7. As noticed by a referee, this notion of equilibrium first appeared in Zhukovskiy (1985), who relied on the notion of P/K-equilibrium proposed by Berge (1957) (in French ‘point d’équilibre pour P relativement à un ensemble K, ou point d’équilibre pour P/K’, ibid, p. 88–89). A P/K-equilibrium is obtained when the players belonging to coalition K maximize the payoffs of the players belonging to coalition P. Since the notion proposed by Zhukovskiy is derived from that proposed by Berge, we call it a Berge equilibrium. In this, we follow Courtois et al. (2015). Larbani and Nessah (2008), Musy et al. (2012), Nessah and Larbani (2014), and Keskin and Saǧlam (2016) present existence results for a Berge equilibrium. Potier and Nessah (2014) study the relationships between Berge and Nash equilibria. Corley and Kwain (2015) propose an algorithm that computes all the Berge equilibria for an n-person game

  8. The notion of strict Berge equilibrium parallels that of a strict Nash equilibrium (see, e.g.,, Peters 2008, p. 118). I am grateful to a referee for suggesting the introduction of this definition.

  9. This does not seem to be too strong an assumption when the strategy sets are finite.

  10. Recall that a function \(f : S \rightarrow {\rm I}\!{\rm R}\) is injective if it never maps two different elements of S to the same element of \({\rm I}\!{\rm R}\).

  11. This notion of equilibrium was proposed by Vaisman (1994). Existence results for a Berge–Vaisman equilibrium can be found in Coleman et al. (2011), Courtois et al. (2015) and Pottier and Nessah (2014).

  12. There is not denying that the larger the strategy sets, the harder it is to check whether our sufficient condition is satisfied. I thank a referee for pointing out this limitation.

  13. The conditions given in Proposition 3 ensure that a profile of strategy is a strict Berge–Vaisman equilibrium. We do not know, however, under what conditions on the payoff functions and strategy sets the inequalities of this Proposition are satisfied. Moreover, these inequalities do not ensure the existence of a Berge equilibrium. I thank a referee for these remarks.

  14. This assertion can be proved by using the same argument as in the proof of Proposition 1.

  15. Two of these three games are extensive-form games. To compute the Berge equilibria of these games, we have converted each extensive-form game into its corresponding normal-form game. That is because, as noticed by a referee, a Berge equilibrium is only defined for normal-form games. The same remark applies for the computation of mutually beneficial practices.

  16. I thank a referee for stressing this difference between mutually beneficial practice and Berge equilibrium.

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Correspondence to Bertrand Crettez.

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I thank Gabrielle Smart for helpful comments on a previous version of this paper. I also thank two anonymous referees for stimulating remarks on the first submitted version of this work.

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Crettez, B. On Sugden’s “mutually beneficial practice” and Berge equilibrium. Int Rev Econ 64, 357–366 (2017). https://doi.org/10.1007/s12232-017-0278-3

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