Educational project: Axelrod's counting optimal alliance partners

For those familiar with it, this is a problem as tackled on page 104 of Robert Axelrod's book ``The complexity of cooperation''.

It all boils down to counting all possible partitions of a set.

Below is Latex code. See the enclosed pdf-file for more readable text.

There are $n$ firms in the set $F$. Example: $F$ = \{$A$, $B$, $C$\}, $n$ = 3

An ``alliance'' is a subset of the firms. Example: \{$A$, $B$\} is an alliance.

A partition is a collection of alliances so that every firm is in precisely one alliance.


\newline \{\{A, B\}, \{C\}\} is a partition,

\newline \{\{A\}, \{B\}, \{C\}\} is a partition,

\newline \{\{A\}, \{A, B\}, \{C\}\} is not a partition (A is in more than one alliance)

\newline \{\{A\}, \{C\}\} is not a partition (B is missing)

Assumption: the set of firms can be divided in two subsets $C$ and $D$, ($C \cap D =

\emptyset$, $C + D = F$). All firms within $C$ are each other’s close rivals, all firms

within $D$ are each other’s close rivals. Moreover, for all firms in $C$, firms in $D$

are distant rivals, and for all firms in $D$, firms in $C$ are distant rivals (more on

the implications of this difference below).

The value of an alliance $A$ to firm $i$ equals

$U_i(A) = \sum_{j \in A} s_j - \left[ \alpha \sum_{j \in D} s_j + (\alpha + \beta)

\sum_{j \in C} s_j \right].$

where $s_j$ is the size of firm $j$, $\alpha$ is a firm's disincentive to ally with any

kind of rival, $\beta$ measures the additional disincentive to ally with close rivals

($\beta > 0$). Usually $\alpha > 0$, but for firms that are not rivals it could be equal

to zero, and for firms that happen to have some incentive to go together, a could be

smaller than zero.

One could rewrite the previous equation as $U_i(A) = \sum_{j \in A} s_j p_ij$ with $p_ij$

the propensity of two firms to ally, which is equal to $1-\alpha$ when $i$ and $j$ are

distant rivals, and equal to $1-(\alpha+\beta)$ when $i$ and $j$ are close rivals.

We want to know, given values for $s_j$, $\alpha$, $\beta$, and given subsets $C$ and

$D$, which possible partitions of $F$ are in equilibrium. That is, which partitions of

$F$ fulfil the requirement that none of the firms has an incentive to unilaterally change

to either another existing alliance or start an own ``one firm alliance''.

... see enclosed pdf for rest of text!

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