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Game theory in the popular press.

Machiavellian tactics modelled

Voting for your enemies can do you good.

Philip Ball
October 10, 2001
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Groucho Marx said he wouldn't want to join any club that saw fit to elect him. A new study suggests he might have been in more danger of being elected by a club that didn't want him.

Joining an elite is not just a matter of who you know, but of who knows who you know, and who they know, say Salvador Barbera of the Universitat Autonoma in Barcelona, Spain, and his co-workers.

The team has built a mathematical model of clubs and societies whose new membership is determined by the votes of its current members. Fellows of the Royal Society in the United Kingdom are elected in this way. The model is an example of a mathematical game, so Barbera's group have probed it using the tools of game theory, first devised in the 1930s.

Members seeking to advance their own cause will be keen to elect new members who are likely to support them, even if the grounds for their election are otherwise flimsy. They may also wish to exclude potential opponents even if they have shining credentials for membership. So members might vote for potential friends and vote against potential enemies, come what may.

But it may also be advantageous to vote for one's enemies, Barbera's team shows. If electing a new member relies on a consensus or a quota of votes, this is not too surprising - members might calculatedly elect an enemy to win the support of other members for their own choices. More surprising is that voting for enemies can be a good idea even when new members require only a single vote on their behalf.

Single-vote election to a club or society is rare, but the researchers have assumed this in their model because it is the simplest rule imaginable - so counter-intuitive results arising under these conditions are likely to persist when the voting rules are more complex.

The model consists of a club that grows steadily by selecting new members openly and repeatedly from a pool of candidates. Each member's decisions are determined by their 'utility function' - the value to him or her of letting a particular individual join.

Each member is assumed to regard others as either friend or foe. Unfortunately, one cannot dictate the company one's friends keep. As politicians know well, a friend can also be a friend to one's enemies.

Barbera and his colleagues searched for Nash equilibria. Familiar to economists, these are steady states of a model in which no individual can better themselves by changing their behaviour. For example, if the affinities between individuals are such that the club can include only mutual friends and exclude only members' enemies, that is an equilibrium state - albeit one requiring a rather special set of affinities.

The researchers find a disconcerting variety of possible equilibria, which depend on the detailed distributions of friends and enemies. For instance, voting in one undesirable person may stop other members electing many undesirables.

Nature News Service / Macmillan Magazines Ltd 2001