Abstract
Rosenthal (J Econ Theory 5:88–101, 1972) points out that the coalitional function form may be insufficient to analyze some strategic interactions of the cooperative normal form. His solution consists in representing games in effectiveness form, which explicitly describes the set of possible outcomes that each coalition can enforce by a unilateral deviation from each proposed outcome. The present paper detects some non-appropriateness of the effectiveness representation with respect to the stability of outcomes against coalitional deviations. In order to correct such inadequacies, we propose a new model, called deviation function form. The novelty is that, besides providing a detailed description of an outcome, the model captures new kinds of coalitional interactions that support the agreements proposed by deviating coalitions and that cannot be identified via the effectiveness form. Furthermore, it can be used to model the existent matching markets. Its formulation allows defining a new solution concept, which characterizes the cooperative equilibria and extends the stability concept of the existent matching models. The connection between the core and the cooperative equilibrium concepts is established.
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Notes
The concept of strong equilibrium point is due to Aumann (1967): a profile of strategies such that no coalition can gain by deviating from it while the others retain the same strategies.
These games were called coalitional games by Shapley.
These terms are negotiated and may be, for example, the activity to be developed by the members of the coalition, the time contributed to the coalition by each of its members and some times the monetary transfers, the partnerships that should be formed, the division of the resources, the prices, the discount rate over the negotiated price, etc.
In a labor market of firms and workers, for example, the rules might require that a firm hires a set of workers in block or through individual and independent trades. In a buyer–seller market, the rules might require that a buyer is not allowed to acquire more than one object from the same seller and is not obliged to acquire any of the objects of a seller. The restrictions might be, for example, the maximum number of coalitions a player can enter or the time each player owns to contribute to the coalitions he enters, the budget constraints, the initial endowments, etc.
For simplicity of exposition, along this paper we will refer to a player as “he”.
This concept contrasts with that of core: An outcome x is in the core if there is no coalition whose members can profitably deviate from x by interacting only among themselves. Therefore, every cooperative equilibrium is a core outcome.
In the matching markets the intuitive idea of cooperative equilibrium is captured by the concept of stability, which has been defined locally, for every matching model that has been studied, since Gale and Shapley (1962).
Originally (d) requires, for each coalition \(S\subseteq N\), an effectiveness function which maps every point \(x\in X\) into a collection of subsets of X.
In Example 2, suppose that the discount rate is not part of the rules of the market, but instead it is part of the agreement between the buyer and seller \(q_{{1}}\). In this case, we have to add the discount rate agreed between the buyer and seller \(q_{{1}}\) to the representation of the outcome, when these agents form a partnership. Since the feasible outcomes capture all relevant trades inside the coalitions that are formed, they are enough for the definition of the set of the feasible deviation outcomes, and so they are enough for the identification of the cooperative equilibria. This means that, if the effectiveness form uses that detailed description of the feasible outcomes, then it can be used as vehicle for any cooperative equilibrium analysis for that market.
Some times, as in the cooperative games derived from the normal form, it might be useful to include the set of agreements for each coalition as one of the primitives of the model. We decided not to do it because, according to Definition 1, the set of feasible outcomes already specifies such set. The set of feasible and minimal coalition structures is necessary to emphasize that an outcome is supported by a coalition structure.
It should be emphasized that the rules of the game are not negotiated among the agents, so they are not part of the agreements reached by them.
In some situations, as those represented by a matching market, the value \(U_{pB}((\partial ;B))\) only depends on the agreements made by the players in B. In some other situations, as those that can be represented in the cooperative normal form, such value may also depend on the agreements reached in the minimal coalitions that do not contain p.
This contrasts with the characteristic function form and the effectiveness form, in which the payoffs are one-dimensional.
In the Multiple partners assignment game of Sotomayor (1992), for example, the partners must agree on the division of the income they can generate by working together and a player may contribute to more than one partnership. In this model, a feasible outcome is given by a feasible payoff configuration, given by a feasible many-to-many matching together with a multi-dimensional payoff for each player. The coalition structure is formed with pairs of players, one of each side, or with single players. For a given agent \(p, u_{p}(x)\) is the sum of the individual payoffs of p. In the many-to-one job market of firms and workers, due to Kelso and Crawford (1982), the coalition structure is a partition of N, formed by coalitions consisting of one firm and the set of workers hired by it, or by single agents. In this case the payoff of each agent is one-dimensional. Thus \(u_{p}(x)\) is the payoff of p.
If \(q_{{1}}\) makes a new agreement and \(q_{{1}}\in S\) then this agreement does not result from a reformulation of \(p_{{1}}\). Also, if \(S=\{q_{j}\}\) and \(q_{j}\) does not sell any of his units at x, for \(j=1,2\), then \(q_{j}\) cannot make a new agreement. Then (C) implies that the set of feasible deviations from x via S is empty.
The individual payoffs of a player plays a crucial role in the identification of the minimal coalitions. For example, let \(x=(S{{1}}, S{{2}}, S{{3}}, \partial _{S{{1}}}, \partial _{S{{2}}},\partial _{S{3}})\) be a feasible outcome, where \(S{{1}}=\{p_{{1}},q_{{1}}\}, S{{2}}=\{p_{{1}}, q_{{2}}\}, S{{3}}=\{p_{{2}}, q_{{3}}\}\), and the agreement \(\partial _\mathrm{S1}\) is that \(p_{{1}}\) pays \(\$\hbox {s}_{1}\) to \(q_{{1}}\); the agreement \(\partial _\mathrm{S2}\) is that \(p_{{1}}\) pays \(\$\hbox {s}_{2}\) to \(q_{{2}}\) and \(\partial _{{S3}}\) is that \(p_{{2}}\) pays \(\$s_{{3}}\) to \(q_{{3}}\). Let \(y=(S{{4}}, S{5}, S{6}, \partial {'}_\mathrm{S4}, \partial {'}_{S{{5}}}, \partial {'}_{S{{6}}})\) be such that \(S{ 4}=\{p_{{1}},q_{{1}},q_{{2}}\}, S{ 5}=\{p_{{1}},q_{\textit{3}}\}, S{6}= \{p_{{2}}\}; p_{{1}}\) pays \(\$\hbox {s}_{1}\) to \(q_{{1}}\) and \(\$\hbox {s}_{2}\) to \(q_{{2}}\) under agreement \(\partial _{S{{4}}}\), and pays \(\$\hbox {s}_{4}\) to \(q_{{3}}\) under \(\partial _{S{{5}}}\). Suppose \(u_{p{{1}}}(x) =u_{p{{1}}}(y)=5\). Suppose further that \(p_{{1}}{'}\hbox {s}\) individual payoffs in S1 and S2 are 2 and 3, respectively. If \(p_{{1}}{'}\hbox {s}\) individual payoffs in S4 and S5 are 5 and 0, respectively, then y could not represent a feasible outcome, since S4 would not be minimal. However, if, for example, \(p_{{1}}{'}\hbox {s}\) individual payoffs in S4 and S5 are 4 and 1, respectively, y is a feasible outcome. The individual payoff of \(p_{{1}}\) in \(S_{{4}}\) determines if \(S_{{4}}\) is or is not a minimal coalition at y.
Because of this, the previous models, which were created to identify the core outcomes, do not care about the representation of the outcomes.
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M. Sotomayor is grateful to an anonymous referee and the Associate Editor for useful comments. This paper was partially supported by CNPq-Brazil. A draft of it was presented in the conference “Roth and Sotomayor: 20 years after”, May, 2010, Duke University. It is a revised version of a lecture delivered at the International Conference on Game Theory and Economic Applications at SUNY, Stony Brook, July 2013 and at the International Workshop on Game Theory and Economic Applications at USP, São Paulo, July 2014.
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Sotomayor, M. Modeling cooperative decision situations: the deviation function form and the equilibrium concept. Int J Game Theory 45, 743–768 (2016). https://doi.org/10.1007/s00182-015-0486-6
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DOI: https://doi.org/10.1007/s00182-015-0486-6