Graphs and Matrices pp 157-164 | Cite as

# Positive Definite Completion Problem

## Abstract

In this chapter we consider two-person zero-sum games, or matrix games, in which the pure strategies of the players are the vertices, or the edges of a graph, and the payoff is determined by the incidence structure. We identify some cases where the value and the optimal strategies can be explicitly determined. We begin with a brief overview of the theory of matrix games. The Minimax Theorem and related results are reviewed. In the next section we consider the vertex selection game, whose payoff matrix is the skew-symmetric version of the adjacency matrix. Tournament games, which generalize the well-known “scissors, paper and stone” game, is considered. We show that in a tournament game, both the players have a unique optimal strategy. We then consider incidence matrix games, where the payoff matrix is the incidence matrix of a directed graph. A graph-theoretic description of the value and the optimal strategies is provided.

## References and Further Reading

- Berman, A., Shaked-Monderer, N.: Completely Positive Matrices. World Scientific, Singapore (2003)CrossRefMATHGoogle Scholar
- Golumbic, M.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980)MATHGoogle Scholar
- Grone, R., Johnson, C.R., Sá, E.M., Wolkowitz, H.: Positive definite completions of partial hermitian matrices. Linear Algebra Appl.
**58**, 109–124 (1984)MathSciNetCrossRefMATHGoogle Scholar