Axelrod and Bennett (1993) propose a model to analyse the formation of coalitions in different domains. The model has been successfully used to reproduce the alignment of European countries in the Second World War (Axelrod and Bennett 1993) and, with some refinements, to analyse the coalition formation of standard-setting alliances in UNIX operating systems (Axelrod et al. 1995). Axelrod and Bennett's model has been redefined and formally analysed by Galam (1996).

In Axelrod and Bennett's model each actor *i*
is a member of one of two competing coalitions. Actor *i*
decides to leave her current coalition or stay in it by comparing her
current frustration with the frustration she would have if she joined
the opposite coalition. Actor *i*'s
frustration depends on her affinity *a _{ij}*
with each of the other actors

where *d _{ij}(X)*
measures the distance between actors

To analyse this model within the Markov chain framework let us
define the state of the system as a certain configuration. The number
of possible states is the number of possible configurations, i.e. 2^{n}/2,
where *n* is the number of actors and the dividend 2
accounts for the symmetry in the system (any actor can be in one of the
sides, but which side is considered first is arbitrary).

The incremental assumption guarantees that any state where no
actor wants to change coalition unilaterally is absorbing. The
assumption *"no actor who wants to change coalition
is
permanently prevented from doing so"* guarantees
that any
state where there is at least one actor who wants to change coalition
is not absorbing. Thus, a state is absorbing if and only if no actor
wants to change coalition unilaterally. In game theory terms, a state
is absorbing if and only if it is a Nash equilibrium.

Since it is possible to reach (at least) one absorbing state
from any given state^{ [1]},
we can assert that there isn't any
closed communicating class with more than one state in the model.
Therefore, following Proposition 2 we can partition the state space as:
*S*
= {*abs*_{1}} ∪ {*abs*_{2}} ∪
… ∪ {*abs _{k}*} ∪

Axelrod and Bennett (1993) define a concept which is particularly
useful to identify absorbing states: the total energy of the system in
configuration *X*. The total energy is denoted by *E(X)*
and calculated as
follows:

This definition is convenient because, given that affinities
are symmetric, it is not difficult to prove that any movement under the
incremental assumption implies a reduction in the total energy of the
system (Axelrod and Bennett 1993; Axelrod et al. 1995), and absorbing states correspond to states where the energy has reached a local minimum^{ [2]}. In this way,
Axelrod and Bennett's (1993) formula nicely captures the idea
–inherited from physics– that systems tend to move
towards states of low energy.

In their study of the European coalitions in WWII, Axelrod and Bennett (1993) do not analyse the dynamics of the model; instead they calculate the total energy of the system for every possible configuration of the 17 considered countries. With their parameters, the system has only two local minima (i.e. two absorbing states) and one of them brilliantly approximates the Allies and the Axis coalitions in the conflict.

If we actually implemented and ran a simulation model satisfying the assumptions made here, we would see that the system would necessarily end up in one of these two absorbing states. The probability of finishing in one configuration or the other would depend on the initial conditions.

AXELROD R M and Bennett D S (1993) A Landscape Theory of
Aggregation. *British Journal of Political Science*,
23(2), pp. 211-233

AXELROD R M, Mitchell W, Thomas R E, Bennett D S, and Bruderer
E (1995) Coalition Formation in Standard-Setting Alliances. *Management
Science*, 41(9), pp. 1493-1508

GALAM S (1996) Fragmentation versus stability in bimodal
coalitions. *Physica A: Statistical Mechanics and its
Applications*,
230(1-2), pp. 174-188

^{1}
The validity of this statement can be easily appreciated using the
definition of the total energy of the system derived by Axelrod and
Bennett (1993).

^{2}
A configuration is a local minimum if and only if there is no
unilateral change of coalition that reduces the energy of the system.