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Plenary Sessions
The following plenary sessions are designed for the Meeting:
A survey of games on augmenting systems
Jesús Mario Bilbao Arrese (Department of Applied Mathematics II, University of Seville)
This contribution deals with cooperative games in which only certain coalitions are allowed to form. There have been previous models developed to confront the problem of unallowable coalitions. Games restricted by a communication graph were introduced by Myerson and Owen. In their model, the feasible coalitions are those that induce connected subgraphs. Another type of model is introduced in Gilles, Owen and van den Brink. In their model, the possibilities of coalition formation are determined by the positions of the players in a so-called permission structure. Faigle proposed a general model for cooperative games defined on lattice structures. We introduce a combinatorial structure called augmenting system which is a generalization of the antimatroid structure and the system of connected subgraphs of a graph. In this framework, the core and the Weber set of games on augmenting systems are introduced and it is proved that monotone convex games have a non-empty core. Moreover, we obtain a new characterization of the convexity of these games in terms of the core of the game and the Weber set of the extended game. In this framework, the Shapley value of games on augmenting systems is introduced and two axiomatizations of this value are showed.
How Democracy Resolves Conflict in Difficult Games
Steven J. Brams (New York University) and Marc Kilgour
Democracy resolves conflicts in difficult games like Prisoners' Dilemma and Chicken by stabilizing their cooperative outcomes. It does so by transforming these games into games in which voters are presented with a choice between a cooperative outcome and a Pareto-inferior noncooperative outcome. In the transformed game, it is always rational for voters to vote for the cooperative outcome, because cooperation is a weakly dominant strategy independent of the decision rule and the number of voters who choose it. Such games are illustrated by 2-person and n-person public-goods games, in which it is optimal to be a free rider, and a biblical story from the book of Exodus.
Some results and open problems on the theory of
power indices and its application to politics and finance
Some results and open problems on power indices, carried out in cooperation with several coauthors, are presented. These deal with studies at both a theoretical level and the relative applications to political and financial problems.
Cores and relatives
Stef Tijs (Department of Mathematics, Genoa and CentER, Tilburg)
The Nash equilibrium concept and the core concept are dominant solution concepts in game theory. In both concepts stability with respect to deviations plays a role. Also for both concepts there are studied refinements, coursenings and selections. Further available axiomatizations have similar axioms. Moreover, for both concepts interesting mathematical tools are used and also there are important applications available.
In this lecture the following topics around the core get special attention.
- The structure of the core (polytopes, extreme points).
- Additivity regions of the core correspondence.
- Refinements such as the aggregate monotonic core and the lexicore.
- Coursenings of the core such as the Weber set, the Milnor set and the core cover.
- Selections of the core, especially "toss and run" rules such as Alexia and also aggregate monotonic core selections.