  The Minority Game (MG) is a basic multi-agent model representing a simplified
and binary form of the bar attendance model of Arthur. The model has an
informationally efficient phase in which the agents lack the capability of
exploiting any information in the winning action time series. We develop a
theory based on the ranking patterns of the strategies and the number of agents
using a particular rank of strategies as the game proceeds. The theory is
applied to calculate the distribution or probability density function in the
number of agents making a particular decision. From the distribution, the
standard deviation in the number of agents making a particular choice (e.g.,
the bar attendance) can be calculated in the efficient phase as a function of
the parameter m specifying the agent's memory size. Since situations with tied
cumulative performance of the strategies often occur in the efficient phase and
they are critical in the decision making dynamics, the present theory is
developed to take into account the effects of tied strategies. The analytic
results are found to be in better agreement with numerical results, when
compared with the simplest forms of the crowd-anticrowd theory in which cases
of tied strategies are ignored. Hence the present theory represents a
generalization of the crowd-anticrowd theory of multi-agent models, by focusing
on the evolution of strategy ranking patterns instead of the performance of
individual strategies.
  Paper to be presented in the 10th Annual Workshop on Economic Heterogeneous
Interacting Agents (WEHIA 2005), 13-15 June 2005, University of Essex, UK.
