Module I. Centrality in graphs with applications to game theory and social choice (René van den Brink)
14, 15 y 16 de Noviembre de 11:00-13:20
This course begins with an introduction of different notions of centrality in graphs. For example, connectedness refers to the extent in which nodes are `well-connected’ to other nodes, betweenness refers to the possibility of nodes to connect other nodes, etc. Additionally, there is the notion of power or influence for graphs, which measures in the case of directed graphs to what extent asymmetric relations determine domination in graphs. Centrality or power measures are functions that assign to every graph on a set of n nodes ann-dimensional vector whose components are a measure of centrality or power of the corresponding node in that graph. We discuss several of such centrality measures for directed and undirected graphs. Next, the central ideas on cooperative games are presented. Cooperative games describe situations where players can earn certain payoffs by cooperating. A central question is how to allocate the worth that can be earned by cooperation as payoffs over the individual players. In other words, if there are n players and every subset of the player set is feasible, how to allocate the worths that can be earned by the nonempty subsets of players (called coalitions) over the n players. In applications of cooperative games, usually not all coalitions are feasible, but a certain structure among the players, often represented by a directed or undirected graph, determines the relations among the players and which coalitions are feasible. Centrality or power measures for graphs can be used to define solutions for such (directed and undirected) graph games that allocate the earnings over the individual players taking into account their position in the graph structure. We discuss how this can be done and compare solutions that are obtained by applying centrality or power measures with several known graph game solutions from the literature. Finally, ranking methods for digraphs are introduced, in fact, ranking methods turn every digraph into a complete and transitive digraph. Whereas not every digraph has a best element (i.e. a node that has an outgoing arc to every other node), in a transitive and complete digraph such a node always exists. Therefore, a complete and transitive digraph can be seen as a ranking of the nodes. Centrality or power measures can be used to define such ranking methods by ranking the nodes in any digraph according to their centrality or power. Applications of such ranking methods are, e.g. (i) ranking teams in a sports competition where each team plays against each other team once and there is an arc from team i to team j if team i won the match it played against team j, (ii) ranking web pages on the internet where the arcs are determined by how web pages have links to each other, (iii) turn any preference relation of an individual decision maker or a society into a complete and transitive relation having best elements, etc. We will mainly consider the last application to social choice theory where it is known that the (Condorcet) majority relation that is derived from a preference profile usually is not transitive (even when all individual preference relations are linear orders). Using power measures we can assign a complete and transitive social preference relation to every preference profile.
Module II. Game theory models and its applications to environmental management (Joaquín Sánchez Soriano)
30 de Noviembre, 1 y 2 de Diciembre de 11:00-13:20
In this course a (non-exhaustive) review of different models of game theory, both non-cooperative and cooperative, applied to environmental management will be presented. It is well known that the exploitation of natural resources and access to natural resources are sources of conflict, both between countries (international level) and between individuals (local level). For this reason, game theory can play an important role in resolving such conflicts and management of natural resources, including topics as relevant today as pollution or overexploitation. Different models of game theory to solve problems of environmental conflicts can be found in the literature, some examples are allocation of fishing quotas, water management or allocation of emission of greenhouse gases, among others. This course will introduce some basic concepts of the theory of games, both non-cooperative and cooperative, which will then be used to show some models of game theory applied to the management of natural resources, in particular, game theoretical models of pollution management, including greenhouse gas emission control.
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