Democracy for Realists, Part 4 of 19

This is a mathematical fact: for a politician to win, they must advocate for some unpopular positions

From the book:

Democracy for Realists, 2016

Why Elections Do Not Produce Responsive Government

By Christopher H. Achen & Larry M. Bartels

Page 26-27

One of the most striking virtues of the canonical one-dimensional spatial model is that it identified a unique, normatively attractive and seeming feasible solution to the problem of aggregating individual preferences into a “democratic” policy choice - the policy located at the “ideal point” of the median voter. However, in their influential “expository development” of the spatial theory, Davis, Hinich, and Ordeshook (1970, 427, 428) noted “an important distinction between the unidimensional and multi-dimensional cases”: positions preferred by a majority of voters to every alternative position, “in general, do not exist for a multi-dimensional world.” As they observed, “The possibility that such a paradox exists poses a problem for majority decision-making.” Thus, even if voters' preferences in every issue domain are single-peaked, as the spatial model assumes, there may be no policy platform with a logical claim to represent “the will of the majority,” much less “the will of the people.”

Davis, Hinich, and Ordeshook (170, 438) described “conditions that guarantee the dominance of a single position for anynumber of dimensions” - the symmetry and unimodality of the electorate’s preference density in multi-dimensional space (Plott 1967). However they carefully noted “the eminent restrictiveness of these conditions” and concluded that “one should not presume the existence of dominant positions” in multidimensional models. Subsequent analyses along similar lines (Kramer 1973; McKelvey 1976; Schofield 1983) richly justified that caution by producing a series of so-called chaos theorems demonstrating with increasing mathematical generality that the sufficient conditions are exceedingly fragile: once the distribution of voters’ ideal points deviates from multidimensional symmetry, it is very likely that anyfeasible policy will be beatable by some other feasible policy in a straight majority vote.

This “problem for majority decision-making,” as David, Hinich, and Ordeshook called it, is a manifestation of the “paradox of voting” explored by a long line of social choice theorists, including the Marquis de Condorcet in the 18th century and Charles Dodgson (Lewis Carroll) in the 19th century. Kenneth Arrow’s (1951) much broader general possibility theorem demonstrated that any collective decision-making process satisfying certain reasonable-sounding conditions - not just majority rule - must be subject to similar difficulties. Arrow’s theorem demonstrated with mathematical rigor that what many people seemed to want - a reliable “democratic” procedure for aggregating coherent individual preferences to arrive at a coherent collective choice - was simply, logically, unattainable.

One commentator, Charles Plott (1976, 511-512), referred rather melodramatically to Arrow’s theorem and related theoretical work as “a gigantic cavern into which fall almost all of our ideas about social actions. Almost anything...anyone has ever said about what society wants or should get is threatened with internal inconsistency. It is as though people have been talking for years about a thing that cannot, in principle, exist, and a major effort is needed to see what objectively remains from the conservations.” Of course, Arrow’s theorem and the multi-dimensional spatial model are specific formulations of the problem of collective choice within the narrow framework of the folk theory of democracy, not the sum total of what “anyone has ever said about what society wants or should get.” Nonetheless, the remarkable fact that the populist ideal turned out to be logically incoherent within this simple, seemingly congenial theoretical framework did spur “a major effort” to reassess processes of preference aggregation in democratic politics.

Imagine a nation where there is only important issue: spending on education.

51% of the public wants more spending on education.

49% of the public wants less spending on education.

If you are a politician, it is easy to put together a 51% coalition that also supports 51% of all of the issues people care about, because there is only one issue that people care about.

What about a nation with 3 issues: education, environment, the military.

These are the factions that exist:

1. more education spending, more environment spending, more military spending.

2. more education spending, more environment spending, less military spending.

3. more education spending, less environment spending, less military spending.

4. less education spending, less environment spending, less military spending.

5. less education spending, less environment spending, more military spending.

6. less education spending, more environment spending, more military spending.

7. less education spending, more environment spending, less military spending.

8. more education spending, less environment spending, more military spending.

Now can you put together a 51% coalition that also holds the 51% popular position on all 3 issues? It is difficult. You’ll almost certainly have to get either some people who disagree with you on one issue, or you will have to advocate for an unpopular position, as part of putting together a larger coalition.

Now imagine a country that has 100 issues that the public regards as important. It is improbable, to the point of being close to an impossibility, that the 51% winning coalition also holds the 51% position on each of those 100 issues. In other words, the 51% winning coalition will advocate for the unpopular 49% side of at least some issues.

Put differently, for a politician to win, they must advocate for some unpopular positions. It is crucial to understand this, as it is one of the central reasons “Why elections do not lead to responsive government.”

Whoever wins an election, they come into power pushing for some unpopular policies. And that’s the average case. We can imagine extreme circumstances sometimes giving rise to the pathological case where the winning politician holds the unpopular position on the majority of issues (with a small handful of policies being popular enough that it still gives them 51% of the total vote).

There is no cure for this, it is purely a mathematical fact. Therefore, we must move away from any conception of democracy that has it representing “the will of the people.” While the benefits of democracy are well-documented, representing the will of the people is not one of those benefits. There is no aggregated will of the people that can be represented through elections. The benefits of democracy are caused by other factors (as we wrote about in the previous essay).