Peter Sudhölter

Airport problems and consistent allocation rules

Abstract A class of single valued rules for airport problems is considered. The common properties of these rules are efficiency, reasonableness and a weak form of consistency. These solutions are automatically members of the core for the associated airport game. Every weighted Shapley value, the nucleolus, and the modified nucleolus turn out to belong to this class of rules. The τ -value, however, does not to belong to this class. As a side result we prove that, for airport games, the modified nucleolus and the prenucleolus of the dual game coincide. Furthermore, we investigate monotonicity properties of the rules and axiomatize the Shapley value, nucleolus, and modified nucleolus on the class of airport games.

Axiomatizations of the core on the universal domain and other natural domains

Abstract. We prove that the core on the set of all transferable utility games with players contained in a universe of at least five members can be axiomatized by the zero inessential game property, covariance under strategic equivalence, anonymity, boundedness, the weak reduced game property, the converse reduced game property, and the reconfirmation property. These properties also characterize the core on certain subsets of games, e.g., on the set of totally balanced games, on the set of balanced games, and on the set of superadditive games. Suitable extensions of these properties yield an axiomatization of the core on sets of nontransferable utility games.

The modified nucleolus as canonical representation of weighted majority games

A new solution concept for cooperative transferable utility games is introduced, which is strongly related to the nucleolus and therefore called modified nucleolus. It has many properties in common with the prenucleolus and can be considered as the canonical restriction of the prenucleolus of a certain replicated game. For weighted majority games this solution concept induces a representation of the game. In the special case of weighted majority constant-sum games and homogeneous games respectively the nucleolus and the minimal integer representation respectively are adequate candidates for a canonical representation see Peleg [Peleg, B. 1968. On weights of constant-sum majority games. SIAM J. Appl. Math.16 527--532.] and Ostman [Ostmann, A. 1987a. On the numerical representation of homogeneous games. Int. J. Game Theory16 69--81.]. Fortunately the modified nucleolus coincides with the just mentioned solutions in these special cases and can, therefore, be seen as a canonical representation in the general weighted majority case.

The positive core of a cooperative game

The positive core is a nonempty extension of the core of transferable utility games. If the core is nonempty, then it coincides with the core. It shares many properties with the core. Six well-known axioms that are employed in some axiomatizations of the core, the prenucleolus, or the positive prekernel, and one new intuitive axiom, characterize the positive core for any infinite universe of players. This new axiom requires that the solution of a game, whenever it is nonempty, contains an element that is invariant under any symmetry of the game.

Directed and Weighted Majority Games

In this paper we deal with several classes of simple games; the first class is the one of ordered simple games (i.e. they admit of a complete desirability relation). The second class consists of all zero-sum games in the first one.First of all we introduce a “natural” partial order on both classes respectively and prove that this order relation admits a rank function. Also the first class turns out to be a rank symmetric lattice. These order relations induce fast algorithms to generate both classes of ordered games.Next we focus on the class of weighted majority games withn persons, which can be mapped onto the class of weighted majority zero-sum games withn+1 persons.To this end, we use in addition methods of linear programming, styling them for the special structure of ordered games. Thus, finally, we obtain algorithms, by combiningLP-methods and the partial order relation structure. These fast algorithms serve to test any ordered game for the weighted majority property. They provide a (frequently minimal) representation in case the answer to the test is affirmative.

The nucleolus of homogeneous games with steps

Abstract Homogeneous games were introduced by von Neumann and Morgenstern in the constant-sum case. Peleg studied the kernel and the nucleolus within this framework. However, for the general nonconstant-sum case Ostmann invented the unique minimal representation, Rosenmuller gave a second characterization and Sudholter discovered the “incidence vector”. Based on these results Peleg and Rosenmuller treated several solution concepts for “games without steps”. The present paper treats the case of games “with steps”. It is shown that with a suitable version of a “truncated game” the nucleolus of a game is essentially the one obtained by truncating behind the “largest step”. As the truncated version has “no steps”, the case “with steps” is reduced to the one “without steps”, which is treated in the paper by Peleg and Rosenmuller.

The Shapley value of exact assignment games

We prove that the Shapley value of every two-sided exact assignment game lies in the core of the game.

Sequential Legislative Lobbying

In this paper, we analyze the equilibrium of a sequential game-theoretical model of lobbying, due to Groseclose and Snyder (Am Polit Sci Rev 90:303–315, 1996), describing a legislature that vote over two alternatives, where two opposing lobbies 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 the lobby moving first 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.

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.

Implementing the modified LH algorithm

The modified Lemke-Howson algorithm is a constructive procedure which enables us to compute equilibrium points of a bimatrix game. The algorithm as described by one of the authors is based on the original version invented by Lemke and Howson. However, it differs from this version with respect to several features. It works directly with the matrices defining the bimatrix game A and B. It has an easy and very direct geometrical interpretation; hence for small games we can follow the development of the algorithm geometrically. Finally, instead of being bilinear, the algorithm behaves rather like a piecewise linear program. This presentation closes a gap: although the algorithm has been described geometrically (and with a flow diagram), there has been no constructive procedure that can be implemented on a computer. This is provided by the present paper. We give all necessary proofs and computations in order to establish the following facts: There are two tableaus accompanying the proceeding of the algorithm. As the algorithm changes, moving alternatingly in the simplices of mixed strategies, so does the computational procedure alternatingly dealing with the two different tableaus. Each tableau contains six regions, depending on the various sequences of transitions the procedure has to perform. While this all is in marked difference to linear programming, there is also consolation: The well-known rectangle rule of linear programming can be modified easily (that is, there is a family of rectangle rules) so that changing the tableau alternatingly amounts to applying the appropriate rectangle rule. Thus, there is also close similarity to the familiar LP procedure. Thus, a complete description of the modified LH algorithm is provided that can immediately be implemented on any computer. In particular, we supply an APL program that, e.g., can be run on an IBM^(R) PC.