Optimal strategies for node selection games on oriented graphs: Skew matrices and symmetric games

In the node selection game ΓD each of the two players simultaneously selects a node from the oriented graph D. If there is an arc between the selected nodes, then there is a payoff from the “dominated” player to the “dominating” player. We investigate the set of optimal strategies for the players in the node selection game ΓD. We point out that a classical theorem from game theory relates the dimension of the polytope of optimal strategies for ΓD to the nullity of certain skew submatrix of the payoff matrix for ΓD. We show that if D is bipartite (with at least two nodes in each partite set), then an optimal strategy for the node selection game ΓD is never unique. Our work also implies that if D is a tournament, then there is a unique optimal strategy for each player, a result obtained by Fisher and Ryan [Optimal strategies for a generalized “scissors, paper, and stone” game, Amer. Math. Monthly 99 (1992) 935–942] and independently by Laffond, Laslier, and Le Breton [The bipartisan set of a tournament game, Games Econom. Behav. 5 (1993) 182–201].