Bidding and sequential coalition formation with externalities

We study coalition formation games with externalities where each agent’s payoff depends on the entire partition. As in Bloch (Games Econ Behav 14:90–123, 1996) and Ray and Vohra (Games Econ Behav 26:268–336, 1999), we assume that coalitions form sequentially and agreements are irreversible. Instead of a fixed order protocol, we use a “bidding mechanism” (see Pérez-Castrillo and Wettstein in Am Econ Rev 92(5):1577–1587, 2002) to determine proposals and transfers among the agents. We show that such a mechanism facilitates the attainment of efficiency; in particular, our game admits a Markov perfect equilibrium with the property of full dynamic efficiency. In addition, the aggregate equilibrium payoffs of our game exhibit monotonicity in time. Nevertheless, inefficient equilibria can also emerge.     

Multicriteria games and potentials

In this note we study how far the theory of strategic games with potentials, as reported by Monderer and Shapley (Games Econ Behav 14:124–143, 1996), can be extended to strategic games with vector payoffs, as reported by Shapley (Nav Res Logist Q 6:57–61, 1959). The problem of the existence of pure approximate Pareto equilibria for multicriteria potential games is also studied.      

The pseudo-average rule: bankruptcy,cost allocation and bargaining

A division rule for claims problems, also known as bankruptcy or rationing problems, based on the pseudo-average solution is studied (for 2-person problems). This solution was introduced in Moulin (Jpn Econ Rev 46:303–332, 1995) for discrete cost allocation problems. Using the asymptotic approach, we obtain a division rule for claims problems. We characterize the division rule axiomatically and show that it coincides with the rule associated to the equal area bargaining solution (this is not true for n = 3). Moreover, following Moulin and Shenker (J Econ Theor 64:178–201, 1994), we show that its associated solution for continuous homogeneous goods is precisely the continuous pseudo-average solution.     

The multichoice coalition value

In this paper we define a solution for multichoice games which is a generalization of the Owen coalition value (Lecture Notes in Economics and Mathematical Systems: Essays in Honor of Oskar Morgenstern, Springer, New York, pp. 76–88, 1977) for transferable utility cooperative games and the Egalitarian solution (Peters and Zanks, Ann. Oper. Res. 137, 399–409, 2005) for multichoice games. We also prove that this solution can be seen as a generalization of the configuration value and the dual configuration value (Albizuri et al., Games Econ. Behav. 57, 1–17, 2006) for transferable utility cooperative games.     

Equilibrium and revenue in a family of common-value first-price auctions with differential information

We study a discrete common-value auction environment with two asymmetrically informed bidders. Equilibrium of the first-price auction is in mixed strategies, which we characterize using a doubly recursive solution method. The distribution of bids for the ex post strong player stochastically dominates that for the ex post weak player. This result complements Maskin and Riley’s (Rev Econ Stud 67:413–438, 2000) similar result for asymmetric private-value auctions. Finally, comparison with the dominance-solvable equilibrium in a second-price auction shows the Milgrom–Weber (Econometrica 50:1089–1122, 1982a) finding that the second-price auction yields at least as much revenue as the first-price auction fails with asymmetry: in some cases the first-price auction provides greater expected revenue, in some cases less.     

Efficiency and equilibria in games of optimal derivative design

In this paper, optimal derivative design when multiple firms compete for heterogenous customers is studied. Ties in the agents’ best responses generate discontinuous payoffs. Efficient tie-breaking rules are considered: In a first step, the model presented by Carlier et al. (Math Financ Econ 1:57–80, 2007) is extended, and results of Page and Monteiro (J Math Econ 39:63–109, 2003, J Econ Theory 134:566–575, 2007, Econ Theory 34:503–524, 2008) are used to prove the existence of (mixed-strategies) Nash equilibria. In a second step, the case of risk minimizing firms is studied. Socially efficient allocations are introduced, and their existence is proved. In particular, the entropic risk measure is considered.     

Consistent cost sharing

A new concept of consistency for cost sharing solutions is discussed, analyzed, and related to the homonymous and natural property within the rationing context. Main result is that the isomorphism in Moulin and Shenker (J Econ Theory 64:178–201, 1994) pairs each additive and consistent single-valued mechanism with a corresponding monotonic and consistent rationing method. Then this answers the open question in Moulin (Econometrica 68:643–684, 2000; Handbook of social choice and welfare. Handbooks in economics, pp 289–357, 2002) whether such notion for cost sharing exists. The conclusion is that renown solutions like the average and serial cost sharing mechanisms are consistent, whereas the Shapley–Shubik mechanism is not. Average cost sharing is the only strongly consistent element in this class. The two subclasses of incremental and parametric cost sharing mechanisms are further analyzed as refinement of the main result.     

The multi-core,balancedness and axiomatizations for multi-choice games

This note extends the solution concept of the core for cooperative games to multi-choice games. We propose an extension of the theorem of Bondareva (Problemy Kybernetiki 10:119–139, 1963) and Shapley (Nav Res Logist Q 14:453–460, 1967) to multi-choice games. Also, we introduce a notion of reduced games for multi-choice games and provide an axiomatization of the core on multi-choice games by means of corresponding notion of consistency and its converse.     

Credible deviations from signaling equilibria

In games with costly signaling, some equilibria are vulnerable to deviations which could be “unambiguously” interpreted as coming from a unique set of Sender-types. This occurs when these types are precisely the ones who gain from deviating for any beliefs the Receiver could form over that set. We show that this idea characterizes a unique equilibrium outcome in two classes of games. First, in monotonic signaling games, only the Riley outcome is immune to this sort of deviation. Our result therefore provides a plausible story behind the selection made by Cho and Kreps’s (Q J Econ 102:179–221, 1987) D1 criterion on this class of games. Second, we examine a version of Crawford and Sobel (Econometrica 50:1431–1451, 1982) model with costly signaling, where standard refinements have no effect. We show that only a Riley-like separating equilibrium is immune to these deviations.     

Interactions between government and firms: a differential game approach

In this paper we study an extended version of the model described in Gradus (J. Econ. 81:1092–1109, 1989) in order to determine the optimal taxation policy of a government and its effects on the stock of capital goods growth as a result of the activity developed by firms. It is shown that, by introducing a wealth tax, there exists an optimal wealth tax rate for which the open-loop/feedback Nash/Stackelberg equilibria coincide, maximizing the payments for both agents (government and firms), so that the open-loop Stackelberg equilibrium becomes both time consistent and subgame perfect. Partially supported by MEC (Spain) Grants MTM2006-13468, BMF2002-03493 and project ‘Ingenio Mathematica (i-MATH)’ No. CSD2006-00032 (Consolider-Ingenio 2010).     

Slightly Altruistic Equilibria

We introduce a refinement concept for Nash equilibria (slightly altruistic equilibrium) defined by a limit process and which captures the idea of reciprocal altruism as presented in Binmore (Proceedings of the XV Italian Meeting on Game Theory and Applications, [2003]). Existence is guaranteed for every finite game and for a large class of games with a continuum of strategies. Results and examples emphasize the (lack of) connections with classical refinement concepts. Finally, it is shown that, under a pseudomonotonicity assumption on a particular operator associated to the game, it is possible, by selecting slightly altruistic equilibria, to eliminate those equilibria in which a player can switch to a strategy that is better for the others without leaving the set of equilibria. Part of the results in this paper have been presented at: First Spain, Italy, Netherlands Meeting on Game Theory, Maastricht, 2005; Fifth International ISDG Workshop, Segovia, 2005; GATE, Université Lumière Lyon 2, 2005; XXX AMASES Workshop, Trieste 2006; CSEF, Università di Salerno, 2006.     

A characterization of convex TU games by means of the Mas-Colell bargaining set (à la Shimomura)

Within the class of zero-monotonic games, we prove that a cooperative game with transferable utility is convex if and only if the core of the game coincides with the Mas-Colell bargaining set (à la Shimomura, in Int J Game Theory 26:283–302, 1997).     

Loss of skills in coordination games

This paper deals with 2-player coordination games with vanishing actions, which are repeated games where all diagonal payoffs are strictly positive and all non-diagonal payoffs are zero with the following additional property: At any stage beyond r, if a player has not played a certain action for the last r stages, then he unlearns this action and it disappears from his action set. Such a game is called an r-restricted game. To evaluate the stream of payoffs we use the average reward. For r = 1 the game strategically reduces to a one-shot game and for r ≥ 3 in Schoenmakers (Int Game Theory Rev 4:119–126, 2002) it is shown that all payoffs in the convex hull of the diagonal payoffs are equilibrium rewards. In this paper for the case r = 2 we provide a characterization of the set of equilibrium rewards for 2 × 2 games of this type and a technique to find the equilibrium rewards in m × m games. We also discuss subgame perfection.     

Rigorous computation of smooth branches of equilibria for the three dimensional Cahn–Hilliard equation

In this paper, we propose a new general method to compute rigorously global smooth branches of equilibria of higher-dimensional partial differential equations. The theoretical framework is based on a combination of the theory introduced in Global smooth solution curves using rigorous branch following (van den Berg et al., Math. Comput. 79(271):1565–1584, 2010) and in Analytic estimates and rigorous continuation for equilibria of higher-dimensional PDEs (Gameiro and Lessard, J. Diff. Equ. 249(9):2237–2268, 2010). Using this method, one can obtain proofs of existence of global smooth solution curves of equilibria for large (continuous) parameter ranges and about local uniqueness of the solutions on the curve. As an application, we compute several smooth branches of equilibria for the three-dimensional Cahn–Hilliard equation.     

Compensations in the Shapley value and the compensation solutions for graph games

We consider an alternative expression of the Shapley value that reveals a system of compensations: each player receives an equal share of the worth of each coalition he belongs to, and has to compensate an equal share of the worth of any coalition he does not belong to. We give a representation in terms of formation of the grand coalition according to an ordering of the players and define the corresponding compensation vector. Then, we generalize this idea to cooperative games with a communication graph in order to construct new allocation rules called the compensation solutions. Firstly, we consider cooperative games with arbitrary graphs and construct rooted spanning trees (see Demange, J Political Econ 112:754–778, 2004) instead of orderings of the players by using the classical algorithms DFS and BFS. If the graph is complete, we show that the compensation solutions associated with DFS and BFS coincide with the Shapley value and the equal surplus division respectively. Secondly, we consider cooperative games with a forest (cycle-free graph) and all its rooted spanning trees. The compensation solution is characterized by component efficiency and relative fairness. The latter axiom takes into account the relative position of a player with respect to his component in the communication graph.     

Aubin cores and bargaining sets for convex cooperative fuzzy games

In this paper, we deal with Aubin cores and bargaining sets in convex cooperative fuzzy games. We first give a simple and direct proof to the well-known result (proved by Branzei et al. (Fuzzy Sets Syst 139:267–281, 2003)) that for a convex cooperative fuzzy game v, its Aubin core C(v) coincides with its crisp core C ^cr (v). We then introduce the concept of bargaining sets for cooperative fuzzy games and prove that for a continuous convex cooperative fuzzy game v, its bargaining set coincides with its Aubin core, which extends a well-known result by Maschler et al. for classical cooperative games to cooperative fuzzy games. We also show that some results proved by Shapley (Int J Game Theory 1:11–26, 1971) for classical decomposable convex cooperative games can be extended to convex cooperative fuzzy games.     

Generalized Eisenstein series and several modular transformation formulae

B.C. Berndt (J. Reine Angew. Math. 272:182–193, 1975; 304:332–365, 1978) has derived a number of new transformation formulas, in particular, the transformation formulae of the logarithms of the classical theta functions, by using a transformation formula for a more general class of Eisenstein series. In this paper, we continue his study. By using a transformation formula for a class of twisted generalized Eisenstein series, we generalize a transformation formula given by J. Lehner (Duke Math. J. 8:631–655, 1941) and give a new proof for transformation formulas proved by Y. Yang (Bull. Lond. Math. Soc. 36:671–682, 2004). This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2006-214-C00003). This work also partially supported by BK21-Postech CoDiMaRo.     

Probabilities of Pure Nash Equilibria in Matrix Games when the Payoff Entries of One Player Are Randomly Selected

The Nash equilibrium in pure strategies represents an important solution concept in nonzero sum matrix games. Existence of Nash equilibria in games with known and with randomly selected payoff entries have been studied extensively. In many real games, however, a player may know his own payoff entries but not the payoff entries of the other player. In this paper, we consider nonzero sum matrix games where the payoff entries of one player are known, but the payoff entries of the other player are assumed to be randomly selected. We are interested in determining the probabilities of existence of pure Nash equilibria in such games. We characterize these probabilities by first determining the finite space of ordinal matrix games that corresponds to the infinite space of matrix games with random entries for only one player. We then partition this space into mutually exclusive spaces that correspond to games with no Nash equilibria and with r Nash equilibria. In order to effectively compute the sizes of these spaces, we introduce the concept of top-rated preferences minimal ordinal games. We then present a theorem which provides a mechanism for computing the number of games in each of these mutually exclusive spaces, which then can be used to determine the probabilities. Finally, we summarize the results by deriving the probabilities of existence of unique, nonunique, and no Nash equilibria, and we present an illustrative example.