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# Seminars on “Some remarkable polyhedra in cooperative game theory” and “Modeling anonymous influence with anti-conformist agents” were held

DeCAn lab organized a visit of Professors M. Grabisch and A. Rusinowska-Grabisch from Centre d’Economie de la Sorbonne of University Paris I (France). On November 22 a course of seminars (joint with the all-Russian seminar "Mathematical methods of decision analysis in economics, finance and politics") was held for the DeCAn faculty.

(1) “Some remarkable polyhedra in cooperative game theory” (M. Grabisch)

The characteristic function of a TU-game is a set function defined on a finite universe vanishing at the empty set. Set functions appear in many domains of Operations Research and decision theory (capacities, pseudo-Boolean functions, polymatroids, etc.) and induce interesting polyhedra. Remarkable families of set functions form polyhedra, e.g., the polytope of capacities (monotone TU-games), the polytope of \$p\$-additive capacities, the cone of supermodular games, etc. Also, the core of a set function, defined as the set of additive set functions dominating that set function, is a polyhedron which is of fundamental importance in game theory, decision making and combinatorial optimization. This survey gives an overview of these notions and studies all these polyhedra. We put an emphasis on the (still unsolved) problem of finding the vertices of the core.

(2) “Modeling anonymous influence with anti-conformist agents” (A. Rusinowska-Grabisch)

We study a stochastic model of anonymous influence with conformist and anti-conformist individuals. Each agent with a "yes" or "no" initial opinion on a certain issue can change his opinion due to social influence. We consider anonymous influence, which depends on the number of agents having a certain opinion, but not on their identity. An individual is conformist/anti-conformist if his probability of saying "yes" increases/decreases with the number of "yes"- agents. In order to consider a society in which both conformists and anti-conformists co-exist, we investigate a generalized aggregation mechanism based on ordered weighted averages. Additionally, every agent has a coeffi- cient of conformism which is a real number in [−1, 1], with negative/positive values corresponding to anti-conformists/conformists. The two extreme values −1 and 1 represent a pure anti-conformist and a pure conformist, respectively, and the remaining values – so called "mixed" agents. We consider two kinds of a society: without mixed agents, and with mixed agents who play randomly either as conformists or anti-conformists. For both cases of the model, we deliver a qualitative analysis of convergence, i.e., find all absorbing classes and conditions for their occurrence.