Friday, November 11, 2005


Explain the Gibbard-Satterthwaite (manipulability) theorems

Gibbard and Satterthwaite independently proved the susceptibility of almost all voting procedures to manipulation. As McLean summarises it, “A voting procedure would be strategy-proof if it satisfied the following conditions. For all individual preference profiles it would ensure that whenever an option became more popular its chances of success would at least get no less; and it would ensure that the result could not be manipulated by adding or withdrawing options. But this turns out to be the same as saying that such a procedure must satisfy conditions U, P, and I in Arrow’s Theorem. Therefore if there are more than two options any strategy-proof voting procedure might throw up a dictator”

In British General Elections, strategic-voting is quite common, for example if a voter ranks the parties (Labour, Lib Dem, Conservative), but thinks that the Lib Dems and Conservatives are the two most likely winners, he can vote Lib Dem – effectively reporting (Lib Dem, Labour, Conservative). This is by no means unique to FPTP plurality elections. If anything, the Borda count is even more susceptible, as here voters give a complete ordering, so there is more scope for shifting an option up our down one’s ranking. E.g. one might report (Lib Dem, Labour, Green, UKIP, BNP, Conservative) – even though one actually preferred the Conservatives to UKIP or the BNP, if one thought they were a bigger threat it might make more sense to minimise their points score.

The consequences of strategic voting are unclear. Sometimes co-ordinated strategic voting (e.g. log-rolling) can produce better outcomes for all involved, and maybe everyone. Alternatively winners can gain at the expense of losers, and if what they gain is less than others lose the overall consequences can be bad . Aside from the overall consequence, the purpose of strategic voting is for one person to get what they want, when they otherwise wouldn’t have. This distorts the winners and losers, and some have worried it is unfair. Theoretically perfectly informed strategic voters on either side can cancel out, but in practice the worry is those who are better informed are more likely to get their way if they are better able to manipulate the system. Some hold that this violates voter equality; however, it isn’t entirely obvious that this is so – everyone has a vote, the fact that they do in fact make differential use of this doesn’t show that it isn’t of equal potential value .

In any case, it is fairly easy to avoid strategic-voting, at least if we are willing to give up other possible requirements. Mackie quotes Hinich and Munger’s summary of the Gibbard-Satterthwaite theorem, but then goes on to point to how this conclusion can be avoided by a system like lottery-voting . Here voters express only first preferences, and every vote for A increases A’s chances, there is no possible way a voter can be better off by voting for B if he sincerely prefers A. Riker too notes the possibility of probabilistic solutions , but claims they violate the independence axiom. (It is not clear to me why he thinks this, or that it is necessarily a greater problem than the manipulability it overcomes. It is, however, true that such procedures could be said to violate Riker’s further ‘citizen sovereignty’ axiom – since the outcome depends not only on citizens’ votes but also a randomising device)

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