Tuesday, 12 July 2005

A little sample from my current paper

(The context is a discussion of cues and heuristics in voting - ie how voters can make decisions without being hugely knowledgeable about politics. But I think this is interesting anyway. It is partly inspired by the economics literature on "informational cascades". This is just a very simple example.)

If I trust that you know better than me how to vote, I may vote correctly. But I am certainly not helping make the democratic majority decision any more accurate. Under certain conditions, I may even make it less accurate.

Here's a simple formal example. Suppose that there are three voters on a simple Yes-No issue. One of these outcomes is unequivocally the right one. The voters each receive some information (a signal) about whether Yes or No is the right outcome. The accuracy of those signals varies: the voters have probabilities 0.7, 0.75 and 0.8 respectively of getting accurate information which suggests they should vote in the correct way. Also, these probabilities are known to all the voters in common, although the signals themselves are private. Suppose that the voters vote independently, and try to vote the right way. They all then follow their signals, which are more likely to be right than wrong. The chance of a correct decision is the chance of all three voting right, plus the probability of three different majorities of two voting correctly: 0.7*0.75*0.8 + 0.3*0.75*0.8 + 0.7*0.25*0.8 + 0.7*0.75*0.2 = 0.845.

Now suppose that the best informed voter, Mrs Point Eight, announces what her signal is (which way her information points) before the vote. The other two voters are then each in one of two situations. Either their signal agrees with Mrs Point Eight: if so they vote the same way anyway. Or their signal disagrees. If so, they now have two conflicting signals. But Mrs Point Eight's signal is more accurate than theirs, so they sensibly choose to ignore their own signal. In either case, they vote with Mrs Point Eight. Unfortunately, this means that the chance of reaching a correct decision has gone down to 0.8.1 By trusting a better-informed person, the individual voters have become more accurate (their chance of being right is 0.8 instead of 0.7 or 0.6) but have made the collective outcome less accurate.