"Burr's dilemma" flaw in Approval voting system

See Executive Summary; By Tom Smith & Warren D. Smith (no relation)

Professor Jack H. Nagel (Univ. of Pennsylvania) in his paper The Burr Dilemma in Approval Voting, Journal of Politics 69,1 (February 2007) 43-58] pointed out the following problem with the approval voting system, which he called "Burr's dilemma":

Statement of the Burr Dilemma: When three or more candidates compete for an office that only one can win, and voters (V) may support two (or more) of them by casting equal (approval) votes, candidates (C1 and C2) seeking support from the same group (G) of voters will maximize their respective votes if all members of G vote for both C1 and C2. Both candidates thus have an incentive to appeal for shared support. However, if such appeals succeed completely and neither candidate receives votes from members of V-G, the outcome will be at best a tie in which neither C1 nor C2 is assured of victory. Each candidate therefore has an incentive to encourage some members of G to vote only for himself or herself. If both C1 and C2 successfully follow such a strategy, either or both may receive fewer votes than some other candidate C3 supported by members of V-G. The risk that both C1 and C2 will lose is exacerbated if a retaliatory spiral increases the number of single votes cast by members of G. At the limit, such retribution reduces approval voting to conventional single-vote balloting among the members of G or, if the problem is endemic, among all voters. The nearer that limit is approached, the lower the probability that advantages claimed for approval voting will be realized.

And according to Rob Richie:

Nagel went on to discuss how this played out in the 1796 and 1800 presidential election, where the presidential electors (who at that time had two equally weighted votes), made strategic mistakes with major consequences in both elections. (Nagel used to say approval voting was better, but now says IRV is better.)

Our response: Nagel has a valid point. You might have thought approval voting was immune to candidate cloning (and it is if voters vote the same on both clones). However, due to these irrational-or-strategic-voting "retaliatory spiral" effects that actually is not so in real life.

Quote from Nagel's abstract: It has not previously been recognized that the first four presidential elections (1788-1800) were conducted using a variant of approval voting. That experiment ended disastrously in 1800 with the infamous Electoral College tie between Jefferson and Burr. The tie, ..., resulted less from miscalculation than from a strategic tension built into approval voting, which forces two leaders appealing to the same voters to play a game of Chicken. Because the Burr Dilemma poses a significant difficulty for approval voting, this paper urges that researchers give more attention to "instant runoff" reform options, especially [IRV] and the Coombs rule.

But actually, Burr's dilemma did not result in any pathology in 1800, in the sense that the two "clones" Burr and Jefferson did not enter into a retaliatory spiral causing them both to lose – they actually both won. But Burr's dilemma plausibly would genuinely have resulted in a pathology in Portugal 1986. Therefore, "vote splitting" effects can still occur in approval voting – contrary to advertising.

However, in range voting, voters can still vote, say, C1=99, C2=97 thus expressing a slight preference without hurting the {C1,C2} candidate set too much. So, in practice we can expect that range voting would alleviate Nagel's Burr-dilemma problem with approval voting.

That seems to be another advantage of range voting versus approval voting.

As for Nagel's contention that this problem should perhaps cause us to prefer IRV or Coombs over Approval voting, that is dubious. The Coombs system is definitely a complete disaster for strategic voters in elections anything like contemporary US politics, since strategic voters will vote the "greater evil" artificially last, which should prevent any of the pre-election favorites from ever winning!

As for IRV, the question is which is more common – the "Burr dilemma" pathology for approval voting, or similar pathologies that happen for strategic voters in IRV elections. Nagel's paper erroneously suggests that no such pathology can happen under IRV. That is because Nagel completely ignores the "favorite betrayal" problem with IRV. Examination of our database suggests that such problems arise more commonly than the Burr dilemma pathology would arise under Approval voting – France 2007, Peru 2006, and Chile 1970 are examples of elections that would have yielded pathology under IRV, but the only election there that would seem to yield a Burr Dilemma pathology under approval is Portugal 1986. Of course, this is not enough examples to draw a statistically significant conclusion. But what evidence there is, goes the other way, indicating that Nagel's argument does not favor IRV over approval voting.

Return to main page