A theory of 2-party domination

"NESD" and "NESD*" properties obeyed by some (but disobeyed by other) single-winner voting systems

PRELIMINARY web page. By Warren D. Smith, Nov. 2009.


Which voting systems tend to cause a country to degenerate into a state of "2-party domination" (2PD – or even worse, 1-party domination)? Note, this degeneration may happen gradually over a long time, perhaps more than 100 years, during which many elections happen; but once it happens it tends to be permanent. We propose a tentative answer to that question: voting systems failing the "NESD property," or the closely related "NESD* property," are the ones that fall into 2PD. Further, the systems obeying NESD not only are the ones avoiding 2PD, they also are the ones in which the influence of money on elections, inherently is greatly diminished. It is pointed out that NESD is incompatible with the "majority-top property," and we claim that (contrary to many careless claims by many people) the latter actually is an undesirable voting system demand.


A single-winner voting system obeys the "NESD property" if, when every voter (all voters assumed initially honest) changes her ballot to "exaggerate about A & B," i.e. to now rank A top and B bottom (or B top and A bottom; which depends on the voter and is "honest" about that voter's preference between A and B alone), leaving it otherwise unaltered – these alterations constitute NES, naive exaggeration strategy – that exaggeration-behavior does not always (ignoring very rare "exact tie" situations) cause A or B to win.

We also define the "NESD*" property (note the star) to be the same as NESD except A and B are to be solely-top-rated or ranked by all voters; we forbid coequal top.


Voting systems failing NESD will tend to cause a country to become 2-party dominated (or at least, will make that fate likely).

Voting systems failing and obeying NESD and NESD*

Systems failing NESD: IRV (instant runoff voting), plurality, Bucklin, and all Condorcet systems. [By the latter I here mean, with pure-rank-order ballots – no rank-equalities permitted.] Also, for score-style ballots the "greatest median score wins" system fails NESD... unless a third-party candidate is given the maximum allowed score by a (larger) voter majority.

It also is possible to consider multiwinner voting systems and ask whether heavy use of NES (naive exaggeration strategy) voting will force them to deliver 2-party domination. Then STV-PR systems (using rank-order ballots) and PR systems based on plurality-style voting such as "party list" and Simmons' flavor of asset voting both fail NESD.

Systems obeying NESD: Borda, approval and range voting. The two-round genuine runoff system (but not any one-election-only "instant" approximation). Incidentally, one similarly would predict a priori that with top-2 runoff, three-party domination should develop, since votes for the 4th-leading candidate are likely to be "wasted" and hence will be artificially diminished with strategically-aware voters, whereas votes for the leading three candidates are much more likely to be able to have an effect.

However: Borda restricted to 3-candidate elections fails NESD.

If we modify IRV to permit rank equalities by counting a ballot with K candidates co-equal top as 1/K votes for each, then the resulting system passes NESD, although I still feel uncomfortable about it because being co-equal top is plainly a lot worse and more vulnerable than being sole-top (there is kind of a "discontinuity," unlike in range voting where it is "continuous" as you move across the top score), so strategic voters might not do the former.

This is made clearer via an analogy. Consider plurality voting with "equal votes" permitted (i.e. you can vote "half" for Jefferson and "half" for Adams). This would clearly be stupid, i.e. would clearly be essentially equivalent strategically to plain plurality. Why? Consider each of your half-votes one at a time. If for the first, your best move was to vote Jefferson; then for the second, the same reasoning should apply. Hence you'd vote 100% for Jefferson and essentially never use the equality feature (unless you were a strategic idiot). Plurality with equals permitted, does technically pass NESD, but this reasoning suggests that is a misleading perception. Really, to avoid being misled, one instead should consider the NESD* property.

If we consider Condorcet systems with rank-equalities permitted, these also technically pass NESD but again there is that worrying "discontinuity." So we now enquire which voting systems pass NESD*:

Fail NESD*: IRV, plurality, Condorcet (all with rank-equalities permitted or forbidden, both ways they fail NESD*).

Pass NESD*: Range voting.

NESD* not applicable: Approval voting.

Motivation behind NESD

NESD stands for "Naive Exaggeration Strategy ⇒ Duopoly."

"NES" refers to the voter strategy of

  1. identify the "top two" candidates most likely to win;
  2. Exaggerate your (otherwise honest) vote to rank one top and the other bottom. (With NESD*, unique-top; with NESD, permitted to be co-equal top.) It appears that, in the real world, this is a pretty close approximation of what a very large percentage of voters in large well publicized+polled elections actually do (it does not necessarily always make complete sense that they do that, but the data indicates about 85% of Australian rank-order voters do it anyway CITE???, plus Swedish voters given score-voting-style ballots in a nationwide survey-study also did it).

The "D" part means: if all (or merely a sufficiently large percentage) of voters exhibit NES behavior, then one of the apparent-top-two frontrunners will always win (except in exceedingly unlikely "perfect-tie" scenarios), yielding 2-party domination.

And in fact, the same winner will arise as in strategic plurality voting, so any system failing NESD or NESD* can be accused (perhaps not with full justification, but certainly with some) of being "equivalent in the real world" to plain plurality voting. If so, it presumably over historical time will yield "duopoly," aka "2-party domination" where voters effectively only get one of two choices (or no choice) every election. This severely diminishes voter choice and "democracy" (as opposed to some system with more than 2 choices).

It's an interesting property (or two properties if we count NESD* also) and I think worth consideration.

You can now ask yourself other interesting questions, like "how can I design good voting systems passing NESD or NESD*?" etc.

Conflict with "Majority-Top" property

Definition: A voting system obeys "Majority-Top" if, whenever a voter-majority ranks X unique-top in their vote, then X must win the election.

Jonathan Lundell pointed out the (obvious, once you see it) fact that the (apparently desirable) NESD* property, conflicts with the (also apparently desirable) Majority-Top property: It is impossible for a voting system to obey both.

Lesson: the demise of Majority-Top

My personal lesson from this: Majority-Top is not an always-desirable property for voting systems, despite the common perception that it "obviously" is. Indeed, consider a range voting (0 to 9) election like this:

#votersTheir Vote
40A=9, B=0, C=7
51A=0, B=9, C=7
9A=0, B=0, C=9

A 51% voter-majority scores B unique-top. Therefore, in any voting system obeying Majority-Top, B must win. However, it seems pretty clear in this situation that C would generally be the best winner for society. And it is the range voting winner (totals: C=718, B=459, A=360) and also the one who would probably be elected by approval voting.

This situation often arises where there are two power-groups, the A-ites and the B-ites, each of whom wishes to destroy the other. If the A-ites gain a slight majority, game over. However, it could be there is some middle course ("C") which everybody, regardless of their power-group, finds reasonably attractive, although comparatively few regard C as the best option for them. Should it really be demanded, as a core principle of voting systems, that this middle course must, under all circumstances, always lose? I think not.

Another election example worth thinking about re this (devised by Balinski & Laraki) is here.

If you agree, even in a single election instance, then you must drop the Majority-Top property as a demand. (Remember, this logical property brooks no exceptions. You, by agreeing with even a single exception, disagree with the logical property.)

An analogous example-election in the rank-order-ballot world might be
#votersTheir Vote
Do you agree with me that declaring C, not B, winner would usually be better for society?


I do not see any feasible way to prove this whole theory. It would seem to require enacting lots of voting systems in lots of countries, then waiting 200 years to see what happens. That is not feasible. Nor probably ethical either. It looks to me that this theory is arguably supported by, or at least not refuted by, all evidence so far (from all countries using various voting systems so far). (The theory is falsifiable in principle.)

These same complaints unfortunately could be made about any theory of which voting systems engender 2-party domination. I'm not convinced my theory is correct, but do suspect it is going in the right direction, and largely approximately correct (all in some admittedly vague sense).

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