Nash-solvable two-person symmetric cycle game forms

Abstract

A two-person positional game form g (with perfect information and without moves of chance) is modeled by a finite directed graph (digraph) whose vertices and arcs are interpreted as positions and moves, respectively. All simple directed cycles of this digraph together with its terminal positions form the set A of the outcomes. Each non-terminal position j is controlled by one of two players i ∈ I = { 1 , 2 } . A strategy x i of a player i ∈ I involves selecting a move ( j , j ′ ) in each position j controlled by i . We restrict both players to their pure positional strategies; in other words, a move ( j , j ′ ) in a position j is deterministic (not random) and it can depend only on j (not on preceding positions or moves or on their numbers). For every pair of strategies ( x 1 , x 2 ) , the selected moves uniquely define a play, that is, a directed path form a given initial position j 0 to an outcome (a directed cycle or terminal vertex). This outcome a ∈ A is the result of the game corresponding to the chosen strategies, a = a ( x 1 , x 2 ) . Furthermore, each player i ∈ I = { 1 , 2 } has a real-valued utility function u i over A . Standardly, a game form g is called Nash-solvable if for every u = ( u 1 , u 2 ) the obtained game ( g , u ) has a Nash equilibrium (in pure positional strategies). A digraph (and the corresponding game form) is called symmetric if ( j , j ′ ) is its arc whenever ( j ′ , j ) is. In this paper we obtain necessary and sufficient conditions for Nash-solvability of symmetric cycle two-person game forms and show that these conditions can be verified in linear time in the size of the digraph.