Shapley NTU Value Research Articles

On the values of Bayesian cooperative games with sidepayments

In this paper, we study the solution concept of value in transferable utility (TU) games with asymmetric information. In our model contingent contracts are required to be incentive compatible, and thus utility might not be not fully transferable. Our approach differs from the standard methodology of TU games with complete information, which summarizes the cooperative possibilities through the characteristic function. Instead, we consider a model in which monetary transfers are modeled as additional sidepayments in a non-transferable utility (NTU) game. Our main result states that [18] generalization of the Shapley NTU value and Salamanca (2020) extension of the Harsanyi NTU value are interim utility equivalent in our model with sidepayments. As a consequence of this result, we obtain a generalization of the Shapley TU value to games with incomplete information. Its formula, however, cannot be described by a simple closed form expression as in the case of complete information.

A generalization of the Harsanyi NTU value to games with incomplete information

In this paper, we introduce a solution concept generalizing the Harsanyi non-transferable utility (NTU) value to cooperative games with incomplete information. The so-defined H-solution is characterized by virtual utility scales that extend the Harsanyi-Shapley fictitious weighted-utility transfer procedure. We construct a three-player cooperative game in which Myerson's [Cooperative games with incomplete information. Int. J. Game Theory, 13, 1984, pp. 69-96] generalization of the Shapley NTU value does not capture some negative externality generated by the adverse selection. However, when we explicitly compute the H-solution in this game, it turns out that it prescribes a more intuitive outcome taking into account the informational externality mentioned above.

A Generalization of the Harsanyi NTU Value to Games with Incomplete Information

In this paper, we introduce a solution concept generalizing the Harsanyi non-transferable utility (NTU) value to cooperative games with incomplete information. The so-defined S-solution is characterized by virtual utility scales that extend the Harsanyi-Shapley fictitious weighted utility transfer procedure. We construct a three-player cooperative game in which Myerson’s [Cooperative games with incomplete information. Int. J. Game Theory, 13, 1984, pp. 69-96] generalization of the Shapley NTU value does not capture some “negative” externality generated by the adverse selection. However, when we explicitly compute the S-solution in this game, it turns out that it prescribes a more intuitive outcome which takes into account the above mentioned informational externality.

On the Values of Bayesian Cooperative Games with Sidepayments

In this paper we explore the relationship between several value-like solution concepts for cooperative games with incomplete information and utility transfers in the form of sidepayments. In our model, state-contingent contracts are required to be incentive compatible, and thus utility might not be not fully transferable (as it would be in the complete information case). When we restrict our attention to games with orthogonal coalitions (i.e., which do not involve strategic externalities), our first main result states that Myerson’s [Cooperative games with incomplete information. Int. J. Game Theory. (1984), 13, 69-96] generalization of the Shapley NTU value and Salamanca’s [A generalization of the Harsanyi NTU value to games with incomplete information. (2016), HAL 01579898] extension of the Harsanyi NTU value are interim utility equivalent. If we allow for arbitrary informational and strategic externalities, our second main result establishes that the ex-ante evaluation of Myerson’s solution equals Kalai and Kalai’s [Cooperation in strategic games revisited. Q. J. Econ. (2013) 128, 917-966] cooperative-competitive value in two-player games with verifiable types.

Nontransferable utility games with fuzzy coalition restrictions

A value for nontransferable utility (NTU) games with fuzzy coalition restrictions is introduced and characterized. In a similar way as the Shapley value for transferable utility (TU) games has been extended in the literature to study games with restricted cooperation, we extend the Shapley NTU value to deal with NTU games in situations in which there are fuzzy dependency relationships among the players.

Weighted values and the core in NTU games

Monderer et al. (Int J Game Theory 21(1):27–39, 1992) proved that the core is included in the set of the weighted Shapley values in TU games. The purpose of this paper is to extend this result to NTU games. We first show that the core is included in the closure of the positively weighted egalitarian solutions introduced by Kalai and Samet (Econometrica 53(2):307–327, 1985). Next, we show that the weighted version of the Shapley NTU value by Shapley (La Decision, aggregation et dynamique des ordres de preference, Editions du Centre National de la Recherche Scientifique, Paris, pp 251–263, 1969) does not always include the core. These results indicate that, in view of the relationship to the core, the egalitarian solution is a more desirable extension of the weighted Shapley value to NTU games. As a byproduct of our approach, we also clarify the relationship between the core and marginal contributions in NTU games. We show that, if the attainable payoff for the grand coalition is represented as a closed-half space, then any element of the core is attainable as the expected value of marginal contributions.

Consistency of the Shapley NTU value in G-hyperplane games

We study the Shapley NTU solution on the class of ssNTU games for which the feasible set of the grand coalition is given by a hyperplane (G-hyperplane games). It is shown that, by considering payoff configurations as solution outcomes, the Shapley NTU value is consistent according to a generalization of the reduced game proposed by Hart and Mas-Colell to the class of NTU games. Moreover, the Shapley NTU solution is characterized on this class of NTU games by means of this consistency property plus some plausible axioms, namely: maximality, covariance, symmetry, a null-player axiom, and an additional axiom requiring certain coherence in the payoffs of the intermediate coalitions.

Extending the Nash solution to choice problems with reference points

In 1985 Aumann axiomatized the Shapley NTU value by non-emptiness, efficiency, unanimity, scale covariance, conditional additivity, and independence of irrelevant alternatives. We show that, when replacing unanimity by “unanimity for the grand coalition” and translation covariance, these axioms characterize the Nash solution on the class of n-person choice problems with reference points. A classical bargaining problem consists of a convex feasible set that contains the disagreement point here called reference point. The feasible set of a choice problem does not necessarily contain the reference point and may not be convex. However, we assume that it satisfies some standard properties. Our result is robust so that the characterization is still valid for many subclasses of choice problems, among those is the class of classical bargaining problems. Moreover, we show that each of the employed axioms – including independence of irrelevant alternatives – may be logically independent of the remaining axioms.

On the impact of independence of irrelevant alternatives: the case of two-person NTU games

On several classes of n-person NTU games that have at least one Shapley NTU value, Aumann characterized this solution by six axioms: Non-emptiness, efficiency, unanimity, scale covariance, conditional additivity, and independence of irrelevant alternatives (IIA). Each of the first five axioms is logically independent of the remaining axioms, and the logical independence of IIA is an open problem. We show that for n = 2 the first five axioms already characterize the Shapley NTU value, provided that the class of games is not further restricted. Moreover, we present an example of a solution that satisfies the first five axioms and violates IIA for two-person NTU games (N, V) with uniformly p-smooth V(N).

A VALUE FOR CEPHOIDAL NTU-GAMES

A Cephoid is an algebraic ("Minkowski") sum of finitely many prisms in ℝn. A cephoidal game is an NTU game the feasible sets of which are cephoids. We provide a version of the Shapley NTU value for such games based on the bargaining solution of Maschler–Perles. The value is characterized by a suitable set of axioms

Forming coalitions and the Shapley NTU value

A simple protocol for coalition formation is presented. First, an order of the players is randomly chosen. Then, a coalition grows by sequentially incorporating new members in this order. The protocol is studied in the context of non-transferable utility (NTU) games in characteristic function form. If (weighted) utility transfers are feasible when everybody cooperates, then the expected subgame perfect equilibrium payoff allocation anticipated before any implemented game is the Shapley NTU value.

Random marginal and random removal values

We propose two variations of the non-cooperative bargaining model for games in coalitional form, introduced by Hart and Mas-Colell (Econometrica 64:357–380, 1996a). These strategic games implement, in the limit, two new NTU-values: the random marginal and the random removal values. Their main characteristic is that they always select a unique payoff allocation in NTU-games. The random marginal value coincides with the Consistent NTU-value (Maschler and Owen in Int J Game Theory 18:389–407, 1989) for hyperplane games, and with the Shapley value for TU games (Shapley in In: Contributions to the theory of Games II. Princeton University Press, Princeton, pp 307–317, 1953). The random removal value coincides with the solidarity value (Nowak and Radzik in Int J Game Theory 23:43–48, 1994) in TU-games. In large games we show that, in the special class of market games, the random marginal value coincides with the Shapley NTU-value (Shapley in In: La Décision. Editions du CNRS, Paris, 1969), and that the random removal value coincides with the equal split value.

A comparison of non-transferable utility values

Three values for non-transferable utility games -- the Harsanyi NTU-value, the Shapley NTU-value, and the Maschler--Owen consistent NTU-value -- are compared in a simple example.

A Walrasian approach to bargaining games

The paper presents and discusses an alternative approach to bargaining games. N-person bargaining games with complete information are shown to induce in a canonical way an Arrow-Debreu economy with production and private ownership. The unique Walras stable competitive equilibrium of this economy is shown to coincide with an asymmetric Nash bargaining solution of the underlying game with weights corresponding to the shares in production. In the case of an economy with equal shares in production, the unique competitive equilibrium coincides with the symmetric Nash bargaining solution. As this in turn represents the unique Shapley nontransferable utility (NTU) value our paper solves a problem posed by Shubik, namely to find a model in which the Shapley NTU value is a Walrasian equilibrium. “There has been some controversy about the interpretation of the λ-transfer-value … no consensus has yet emerged on the significance of these concerns, which had been addressed… in numerous explorations of the λ-transfer-value as a tool for analysing games and markets … ” (Roth, 1985).

Cooperative games with imcomplete information

A bargaining solution concept which generalizes the Nash bargaining solution and the Shapley NTU value is defined for cooperative games with incomplete information. These bargaining solutions are efficient and equitable when interpersonal comparisons are made in terms of certainvirtual utility scales. A player's virtual utility differs from his real utility by exaggerating the difference from the preferences of false types that jeopardize his true type. In any incentive-efficient mechanism, the players always maximize their total virtual utility ex post. Conditionally-transferable virtual utility is the strongest possible transferability assumption for games with incomplete information.