## The problem.

OK, we had an election and have a new parliament with 120 members. The president has asked the leader of one party to form a coalition. (This has not happened yet in the Israeli election but it will happen soon.) Such a coalition should include parties that together have more than 60 seats in the parliament.

Can game theory make some prediction as to which coalition will be formed or give some normative suggestions on which coalition to form?

Robert Aumann and Roger Myerson made  (in 1977) the following concrete suggestion.  (Update: A link to the full 1988 paper is now in place.)  The party forming the coalition would (or should) prefer to form the coalition in which its own power (according to the Shapley-Shubik power index) is maximal. Of course this suggestion puts political and ideological matters aside.  Since the coalition-forming party has substantial freedom in forming the coalition and other parties can usually better serve their interests by joining the coalition, Aumann and Myerson expected that this idea would have some predictive value — even in reality, where political and ideological considerations are of importance. (Their paper and an earlier paper by Myerson give a framework for dealing with cases where certain parties cannot “live together” in the same coalition.)

## Example:

Suppose that there are four parties: the Blue party B with 50 seats, the Red party R with 30 seats, the Green party G with 20 seats, and the White party W with 20 seats.

The Blue party was asked to form a coalition. The power of B in a coalition with one more party is 0.5; the power of B in a coalition with two more parties is 0.66 and the power of B in a coalition with all other parties is 0.5. Therefore, according to AM the Blue party should form a coalition with two other parties.

## A more detailed explanation

1) Simple games

We have a set of players \$\{1,2,\dots, n\}\$  and a collection of subsets of players called the collection of winning coalitions. In our case the players are the parties and a set of players is winning if together they have more than 60 seats in the parliament.

2) The Shapley value

The Shapley value (or Shapley-Shubik power index) can be defined for every monotone simple game as follows:

A player $k$ is pivotal with respect to a coalition $S$ of players, if $S$ is a losing coalition and $S \cup \{k\}$ is a winning coalition.

The two major power indices are both based on this notion of pivotality and are defined as the probability that $k$ is pivotal with respect to a certain probability distribution for choosing the subset $S$ of the remaining players.

The Shapley-Shubik power index is based on the following probability distribution (that can be called the equal-slices distribution): we first choose the size of $S$ to be $j$ with probability $1/(n-1)$ for  $j=0,1,2,\dots,n-1$ and then choose $S$ to be a random subset of $N \backslash \{k\}$ of size $j$  according to the uniform distribution. Another equivalent way to define the Shapley-Shubik power index is to look at a random ordering of the players and to ask what is the probability that player $k$ was pivotal with respect to the coalition of earlier players.

The Banzhaf power index is the probability that $k$ is pivotal if $S$ is chosen at random uniformly from all subsets of other players. See the related post about influences. (The relations between the uniform distribution and the equal-slice distribution plays a role in a recent project discussed on Gowers’s blog.)

3) The Shapley value of the winning party after a coalition is formed.

Suppose that player 1 represents the party that forms the coalition, and $S$ represents the coalition that was formed. We can define a new game whose players are the members of $S$ with the same winning coalitions as before — parties that together have more than 60 seats in the parliament.

4) Aumann-Myerson suggestion:

Maximize the Shapley value of the winning party in the coalition it forms.

## What do you think?

So what do you think about this suggestion — is it a good normative suggestion? How can we test if it gives good predictions?