Combinatorial Aspects of Parker’s model

Last modified: 2014-06-09

#### Abstract

Individuals are located at the vertices of the (simple) graph G = (V; E), and at time t play either 0 or 1. At time (t + 1) an

individual will play 0 if, and only if, more of their neighbours were playing 0 at timet than were playing 1 at time t. This is

the Majority Game, and we shall assume that all vertices have odd degree so that there is never equal numbers of 0s and 1s

amongst neighbours. This game is a special case of a threshold game and so the system must converge to ﬁxed point or to a

two cycle. We shall explore features of the dynamics on some particular regular graphs, most importantly the hypercube. The

Minority game is dual to this. At (t + 1) an individual will play 0 if, and only if, more of their neighbours were playing 1 at

time t than were playing 0 at time t. We shall brieﬂy examine some of the consequences of having a mixture of majority players

and minority players, when the dynamics is more complex than the homogeneous game.

individual will play 0 if, and only if, more of their neighbours were playing 0 at timet than were playing 1 at time t. This is

the Majority Game, and we shall assume that all vertices have odd degree so that there is never equal numbers of 0s and 1s

amongst neighbours. This game is a special case of a threshold game and so the system must converge to ﬁxed point or to a

two cycle. We shall explore features of the dynamics on some particular regular graphs, most importantly the hypercube. The

Minority game is dual to this. At (t + 1) an individual will play 0 if, and only if, more of their neighbours were playing 1 at

time t than were playing 0 at time t. We shall brieﬂy examine some of the consequences of having a mixture of majority players

and minority players, when the dynamics is more complex than the homogeneous game.

#### Keywords

majority/minority game, dynamics, thresholds, cycles.