More power can be much less

Some people I’ve been talking with have been wondering how an extremely mediocre person such as Ulmert got to be the prime minister of Israel. Well, it is nigh time I start writing on subjects I fully comprehend. So, I present to you another lesson in election theory – when can more power mean less power?

We’ll take an example from voting theory, because it is a nice example with interesting lessons, and it demonstrates several aspects of game theory. Suppose we have three people on a committee: Alex, Bob, and Chris. They need to choose between options a, b, and c. A very reliable method of voting is the simple majority vote. In this method, each committee member gives his vote to one option. The option with the largest munger of votes wins.

The trouble is, what happens on a tie? Say Alex votes a, Bob votes b, and Chris votes c. Then each option got 1 vote, and it’s a tie. How do you break a tie?

There are two obvious (and common) tiebreaker rules. The first is to define some option as the default. In case of a tie, that specific option wins. Slightly more common is to define one of the committee members as chairman – giving him the right to decide in case of a tie.

So, suppose we have Alex as chairman, and the member’s preferences are as follows:

Alex rates option a first, option b second, and option c last.

Bob rates option b first, option c second, and option a last.

Chris rates option c first, option a second, and option b last.

This is the famous Condorcet’s paradox, named after the Marquis de Condorcet, who was the first to discuss it. But, given that we have a chairman, this paradox is resolved: we have a tie, and so Alex gets to choose, and, naturally, he chooses option a, so that would be the committee’s final decision. Right?
Wrong. What if Chris and Bob coordinate their steps? Both Bob and Chris prefer option c to option a. They can vote “c” together, which means Chris gets exactly what he wanted, Bob gets his second-best option, and poor old Alex loses everything. And that will, in all probability, be the outcome of this situation. Why so?

For Chris, of course, this is the best of outcomes. He will be very happy to coordinate with Bob, if Bob agrees to vote for c. Bob is better of, of course, and he will never agree to vote for a, which is the option he like the least. But why can’t Bob and Alex coordinate and vote b? Because Alex, unlike the other two committee members, cannot be trusted. When Bob and Chris are coordinating, Bob can rely on Chris for voting c, and Chris can rely on Bob, because Bob will not deviate from what they coordinated, otherwise a gets chosen, which is bad for Bob. If Bob tries to cut a deal with Alex, he has no guarantee that Alex votes b. In fact, he can be certain that Alex will deviate and vote for a, securing a victory. Alex is just is too powerful to be trusted. Whenever Bob and Chris are not coordinating, Alex can vote for option a, and win. He has no real incentive to compromise, and so he is bound to be left out, and lose.

So, sometimes having more power is a strategic disadvantage. Whenever a politician indulges in negotiations, he needs to be trusted by the politicians he negotiates with. And if he is too powerful, too charismatic, too smart, then the others might feel superfluous. If you don’t really need me, what are we negotiating, after all? So mediocre politicians, being less threatening to one another, are often “bundling together” to avert the greater risks posed by powerful men. You end up with having the mediocre types managing the government, and the mighty and powerful being left with little influence.

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