## "Burr's dilemma" flaw in Approval voting system – Executive Summary

1. The so-called "Burr Dilemma" is a problem with approval voting pointed out (and named) by Professor Jack H. Nagel (Univ. of Pennsylvania) in 2007. (Actually it is probably misnamed since this problem actually did not happen in the Burr 1800 election – Burr & Jefferson still placed top.)
2. The problem arises when two similar candidates A,B appear to be the "frontrunners." Then voters are strategically motivated to approve one but not the other.
3. This generates a "vote-split" between A and B, which if the split is even-enough could be enough to permit the far-less-liked candidate C to win – disaster. (Both A and B would have been far ahead of C with "honest" votes, the problem is caused by strategic exaggerated votes. The usual thinking had been that with approval voting there is no such thing as vote-splitting, and Nagel's point was that this Burr-effect could cause it to happen.)
4. This problem is lessened, but not eliminated, with range voting since voters could then provide a vote such as A=99, B=80, C=0.
5. Problem also lessened with better pre-election polling: if the voters knew C was thus-likely to win, then A and B would not be the two "frontrunners" (since C would be) and hence {A,B}-voters would be strategically motivated to place their approval thresholds between C and {A,B}... and the problem would disappear or diminish.
6. Also, C-supporters if they felt A,B were frontrunners, would be motivated not to "waste their vote" by approving C only; they would instead approve C and one among {A,B}. This too would eliminate or diminish the problem.
It is unsymmetrical/illogical to presume only the {A,B}-supporters would strategically exaggerate but not the C-supporters. The problem arises from this presumption. If the problem is genuine it is because of unsymmetrical/illogical human psychology, not logical strategy. This kind of illogical thinking actually is logical under plurality voting, and one may criticize Nagel as being "trapped in old-fashioned plurality-based voting thinking" rather than genuinely analysing the sort of thinking that would happen with approval voting. Nagel's response would (I think?) be that A & B would trigger the asymmetry because of an infighting-driven "retaliatory spiral." (That again definitely did not happen in the Burr 1800 election, with Burr indeed stating he would be happy if Jefferson won. Indeed, the main attempt to savage Burr so that Jefferson could win was in fact organized by Burr & Jefferson's enemy A.Hamilton, thus Nagel's Burr-1800-example actually supports the notion the C-voters would strategize and not the {A,B}-supporters, exactly the opposite of Nagel's thinking!)
Because our Bayesian regret computer simulations employed thus-logical strategic voters (in those sims involving strategic voters) the BR measurements were unable to see this whole problem (or only saw a small effect from it).
7. The only fairly-clear example known to me of an election which looks like it would have been victimized by the Burr Dilemma (if approval voting had been employed) is Portugal 1986.

The Burr Dilemma presumably is a genuine problem for approval and range voting, but it seems not to cause a great deal of Bayesian Regret. In other words, it numerically seems not to be a big enough problem to prevent range from being superior to, e.g. Instant Runoff Voting (IRV). And IRV also can experience Burr-like pathologies. (And so can many other voting methods.)

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