The prenucleolus for games with communication structures

It is well-known that the prenucleolus on the class of TU games is characterized by singlevaluedness, covariance under strategic equivalence, anonymity, and the reduced game property. We show that the prenucleolus on the class of TU games restricted to the connected coalitions with respect to communication structures may be characterized by the same axioms and a stronger version of independence of non-connected coalitions requiring that the solution does not depend on the worth of any non-connected coalition. Similarly as in the classical case, it turns out that each of the five axioms is logically independent of the remaining axioms and that an infinite universe of potential players is necessary. Moreover, a suitable definition and characterization of a prekernel for games with communication structures is presented

Nonsymmetric variants of the prekernel and the prenucleolus

A solution on a class of TU games that satisfies the axioms of the pre-nucleolus or -kernel except the equal treatment property and is single valued for two-person games, is a nonsymmetric pre-nucleolus (NSPN) or -kernel (NSPK). In this paper we investigate the NSPKs and NSPNs and their relations to the positive prekernel and to the positive core. It turns out that any NSPK is a subsolution of the positive prekernel. Moreover, it is shown that an arbitrary NSPK, when applied to a TU game, intersects the set of preimputations whose dissatisfactions coincide with the dissatisfactions of an arbitrary element of any other NSPK applied to this game. This result also provides a new proof of sufficiency of the characterizing condition for NSPKs due to the first author in his PhD thesis published in 1994 as a discussion paper. Any NSPN belongs to "its" NSPK. Several classes of NSPNs are presented, all of them are subsolutions of the positive core. It is shown that any NSPN is a subsolution of the positive core provided that it satisfies the equal treatment property on an infinite universe of potential players. Moreover, we prove that, for any game that has a nonempty anticore, any NSPN selects its prenucleolus as its unique element.TU game; Solution concept; Kernel; Nucleolus; Core; Equal treatment

The prenucleolus for games with communication structures

t is well-known that the prenucleolus on the class of TU games is characterized by singlevaluedness, covariance under strategic equivalence, anonymity, and the reduced game property. We show that the prenucleolus on the class of TU games restricted to the connected coalitions with respect to communication structures may be characterized by the same axioms and a stronger version of independence of non-connected coalitions requiring that the solution does not depend on the worth of any non-connected coalition. Similarly as in the classical case, it turns out that each of the five axioms is logically independent of the remaining axioms and that an infinite universe of potential players is necessary. Moreover, a suitable definition and characterization of a prekernel for games with communication structures is presented.TU game; solution concept; communication and conference structure; nucleolus

On extensions of the core and the anticore of transferable utility games

We consider several related set extensions of the core and the anticore of games with transferable utility. An efficient allocation is undominated if it cannot be improved, in a specific way, by sidepayments changing the allocation or the game. The set of all such allocations is called the undominated set, and we show that it consists of finitely many polytopes with a core-like structure. One of these polytopes is the L1-center, consisting of all efficient allocations that minimize the sum of the absolute values of the excesses. The excess Pareto optimal set contains the allocations that are Pareto optimal in the set obtained by ordering the sums of the absolute values of the excesses of coalitions and the absolute values of the excesses of their complements. The L1-center is contained in the excess Pareto optimal set, which in turn is contained in the undominated set. For three-person games all these sets coincide. These three sets also coincide with the core for balanced games and with the anticore for antibalanced games. We study properties of these sets and provide characterizations in terms of balanced collections of coalitions. We also propose a single-valued selection from the excess Pareto optimal set, the min-prenucleolus, which is defined as the prenucleolus of the minimum of a game and its dual.Transferable utility game; core; anticore; core extension; min-prenucleolus

The Canonical Extensive Form of a Game Form - Part II - Representation

This paper exhibits to any noncooperative game in strategic or normal form a 'canonical' game in extensive form that preserves all symmetries of the former one. The operation defined this way respects the restriction of games to subgames and yields a minimal total rank of the tree involved. Moreover, by the above requirements the 'canonical extensive game form' is uniquely defined.Games, Extensive Form, Normal Form, Strategic Form

On the impact of independence of irrelevant alternatives

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 5 axioms and violates IIA for 2-person NTU games (N,V) with uniformly p-smooth V(N).NTU game; Shapley NTU value; positive smoothness

Sequential legislative lobbying

In this paper, we analyze the equilibrium of a sequential game-theoretical model of lobbying, due to Groseclose and Snyder (1996), describing a legislature that vote over two alternatives, where two opposing lobbies, Lobby 0 and Lobby 1, compete by bidding for legislators’ votes. In this model, the lobbyist moving first suffers from a second mover advantage and will make an offer to a panel of legislators only if it deters any credible counter-reaction from his opponent, i.e., if he anticipates to win the battle. This paper departs from the existing literature in assuming that legislators care about the consequence of their votes rather than their votes per se. Our main focus is on the calculation of the smallest budget that he needs to win the game and on the distribution of this budget across the legislators. We study the impact of the key parameters of the game on these two variables and show the connection of this problem with the combinatorics of sets and notions from cooperative game theory.Lobbying; cooperative games; noncooperative games

The Bounded Core for Games with Precedence Constraints

An element of the possibly unbounded core of a cooperative game with precedence constraints belongs to its bounded core if any transfer to a player from any of her subordinates results in payoffs outside the core. The bounded core is the union of all bounded faces of the core, it is nonempty if the core is nonempty, and it is a continuous correspondence on games with coinciding precedence constraints. If the precedence constraints generate a connected hierarchy, then the core is always nonempty. It is shown that the bounded core is axiomatized similarly to the core for classical cooperative games, namely by boundedness (BOUND), nonemptiness for zero-inessential two-person games (ZIG), anonymity, covariance under strategic equivalence (COV), and certain variants of the reduced game property (RGP), the converse reduced game property (CRGP), and the reconfirmation property. The core is the maximum solution that satisfies a suitably weakened version of BOUND together with the remaining axioms. For games with connected hierarchies, the bounded core is axiomatized by BOUND, ZIG, COV, and some variants of RGP and CRGP, whereas the core is the maximum solution that satisfies the weakened version of BOUND, COV, and the variants of RGP and CRGP.TU game, core, restricted cooperation.

Generic finiteness of equilibrium distributions for bimatrix outcome game forms

We provide sufficient and necessary conditions for the generic finiteness of the number of distributions on outcomes, induced by the completely mixed Nash equilibria associated to a bimatrix outcome game form. These equivalent conditions are stated in terms of the ranks of two matrices constructed from the original game form.The first author acknowledges that this work was supported by a grant of the Romanian National Authority for Scientific Research and Innovation, CNCS-UEFISCDI, project number PN-II-RU-TE-2014-4-1827. The second author acknowledges financial support from the Ministerio Economía y Competitividad (Spain), grant ECO2016-75992-P, MDM 2014-0431, and Comunidad de Madrid, MadEco-CM (S2015/HUM-3444). The third author acknowledges support from the Spanish Ministerio de Economía y Competitividad under project ECO2015-66803-P and from the Danish Council for Independent Research|Social Sciences (Grant-id: DFFD 1327-00097)

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).The second and third author were supported by the Spanish Ministerio de Ciencia e Innovación under project ECO2009-11213, co-funded by the ERDF