(This post is a continuation of this previous post.)
Aumann and Myerson proposed that if political and ideological matters are put aside, 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. They expected that this idea would have some predictive value — even in reality, where political and ideological considerations are of importance. A few days ago Yair Tauman, another well-known Israeli game theorist, mentioned on TV this recipe as a normative game-theoretic recommendation in the context of the recent Israeli elections. (For Yair’s analysis see also this article. (I even sent a critical comment.))
Over the years, Aumann was quite fond of this suggestion and often claimed that in Israeli elections it gives good predictions in some (but not all) cases. The original paper mentions the Israeli 1977 elections and how delighted one of the authors was that four months after the elections a major “centrist” party joined the coalition, leading to a much better Shapley value for the party forming the coalition.
I was quite skeptical about the claim that the maximum-power-to-the-winning-party rule has any predictive value and in 1999 with the help of Sergiu Hart I decided to test this claim. I asked Aumann which Israeli coalition he regards as fitting his prediction the best. His answer was the 1988 election where Shamir’s party, the Likud, had a very large Shapley value in the coalition it formed. We checked how high the Shapley value was compared to a random coalition that the winning party could have formed.
As high as it was, the Shapley value of the coalition that was actually formed fell below the median in terms of the Shapley value of a random winning coalition. This observation suggests that the proposal of Aumann and Myerson does not have any predictive power when it comes to Israeli coalition formation.
This is a nice example of the difficulty of using game theory for real-life situations.