An Introductory Course on Mathematical Game Theory

Abstract

Page 114, Figure 3.3.5 b): The entry in the position (2, 1) of the table, corresponding to the strategy profile (MH, MM), should be (65, 75) instead of (75, 65).

Cooperative games and cost allocation problems

“…For instance, the Shapley value of a cost allocation problem may lie outside its core, even when the later is non-empty. The next result (whose proof can be seen, for instance, in González-Díaz et al 2010) relates the Shapley value with the core.…”

“…Moreover, it can be easily proved that if Co(c) = ∅ then N (c) ∈ Co(c). The proof of these two features can be seen, for instance, in González-Díaz et al (2010). There are several procedures to compute the nucleolus of a cost allocation problem, but its computation can be quite hard.…”

“…In other words, when the cooperation mechanisms are too complex as to be fully described by a mathematical model, we enter the terrain of cooperative game theory. It does not deal with strategic analysis, but with coalitions and allocations and, as indicated in González-Díaz et al (2010), it "considers groups of players willing to allocate the joint benefits derived from their cooperation (however it takes place)". The most important models within cooperative game theory are the so-called transferable utility games.…”

Topologically robust zero-sum games and Pfaffian orientation: How network topology determines the long-time dynamics of the antisymmetric Lotka-Volterra equation

“…For symmetric zero-sum games, one is typically interested in so-called optimal sets of actions (or briefly an optimal strategy), in which "optimal" may have different meanings depending on the context [45,75]. For example, an optimal set of actions may denote a probability vector (p * ∈ ∆ S−1 , that is, p * ∈ R S , p * ≥ 0, and S i=1 p i = 1), whose ith entry denotes the probability to play the ith action, and that maximizes the player's minimum expected payoff against all other sets of actions (a so-called mixed Nash equilibrium of the symmetric game [76,77]). For a symmetric zero-sum game, it is straightforward to show that a normalized positive kernel vector p * ≥ 0 of the payoff matrix (Ap * = 0) is an optimal strategy.…”

Experimental Demonstration on Quantum Sensitivity to Available Information in Decision Making

“…Our DM study is presented in a framework of game theory [6]. Game theory deals with the strategies by which players (decision makers in this paper) maximize their own rewards.…”