On Fri, 25 Mar 2005, Jobst Heitzig wrote:
Forest, you wrote:And:Sorry, I was thinking in terms of equilibria that are stable under Rob's ballot-by-ballot DSV procedure.In Rob's algorithm, once A is in the lead, the ABC voters stop approving B.
But why should they do so when A wins already?
They have no incentive whatsoever to change their behaviour and thus provoke reactions which can only lead to a worsening of their outcome, have they?
True, in this case greed doesn't pay.
But suppose that all of the votes are not in, and all you know is that the approval order (so far) is A>B>C, which happens to match your preference order. Where would you put your approval cutoff?
For me it would depend on whether B was closer to A or to C in my estimation.
If closer to A, then I would say to myself, "Approving A and B will keep A in the lead, and put a cushion between B and C."
If closer to B, then I would say to myself, "I better put the cushion between A and B."
At any rate, Rob's strategy A, which says, "Put the approval cutoff next to the current approval leader on the side of the current approval runner up" is the simplest DSV strategy (for ordinal ballots) that almost always converges to the CW when there is one.
As you have demonstrated, in general it is not always possible to determine from ordinal information alone where rational voters should put their cutoffs, even given accurate polling information or information about partial election results.
The reason?
Because there truly is information in approval ballots that cannot be derived even from a set of completed ordinal ballots.
It seems to me that this supports our idea that voter supplied approval cutoffs do indeed add valuable information.
Finally, here's another (half-baked) idea that I have been thinking about:
Suppose that L is a lottery with maximum support S(L) such that for each positive epsilon and each candidate C in S(L), there is a lottery L' within epsilon of L such that C wins (or ties for first) when voters approve according to L'.
In other words, by announcing winning probabilities close to L, we can make any candidate in S(L) win the approval contest, IF the approval voters believe our predictions (with enough faith to place their approval cutoffs so as to maximize their expected payoff under L).
It seems to me that each member of S(L) should have some probability of winning.
What should that probability be?
The lottery L is kind of like the basketball referee setting up the ball between the two centers to start play.
So perhaps L is a fair way of assigning probabilities.
On the other hand, perhaps, having used L to identify the set S(L), we can now throw away L, and pick the winner by random ballot from S(L), giving direct support a chance.
One reason I like this lottery L is that we can mimic a constructive (piecewise linear) path following proof of Sperner's Lemma to find such a lottery.
But, under what conditions is it unique?
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