Federalist 10 and the Chaos Theorem, Part II

On Tuesday, I provided a short survey of the chaos theorem. The theorem holds that, except under very special conditions, when there are at least three voters and two dimensions of policy, with pairwise, sincere, majority voting, an agenda setter can lead the voting outcome anywhere in the policy space, even outside the voters’ Pareto set (which is the set of outcomes all of voters would prefer relative to where they ended up). In short form, majority rule, under these conditions, cannot guarantee the rationality of social choices of groups composed of individually rational voters. Per Arrow’s impossibility theorem, rational outcomes require moving away from majority rule and toward dictatorship. (It is important to remember the thin definition of rationality Arrow used, meaning only that preferences are transitive and complete. His theorem does not suggest that dictatorship is “rational” in the everyday sense of the word.)

Scholars often style the possibility of democratic “chaos” or instability as implied by the chaos theorem as something bad that institutional designers should attempt to avoid. Sometimes articulate this thought explicitly, sometimes it is more subtly communicated in the oral tradition surrounding discussion of the results in grad seminars and academic meetings. And, indeed, “chaos” and “instability” sound like they’re bad things.

Yet, interestingly, a well-known line of constitutional thought in the U.S. treats majoritarian stability as suspect, at least some of the time. Correlatively, majoritarian instability can produce good things.

Like the word “chaos” in the context of McKelvey’s theorem, we need carefully to define what we’re talking about when referring to majoritarian “stability” and “instability.”

By stability I mean there are no proposals for legislators (or voters more generally) that would overturn an extant policy or proposal in a majority vote. Instability means that one or more majorities can be formed from the same set of voters that would vote to overturn an extant policy or proposal.

For example, if a legislature has only two factions, one being a majority, the other being a minority, and members of each faction have homogeneous preferences within the respective faction (preferences differ across factions), then majoritarian outcomes will be stable. The majority faction adopts the laws its members desire; there is nothing the minority faction can do to overturn those policies or proposals.

Say, however, the legislature has three factions, each constituting a third of the legislators. Any proposal therefore requires the concurrence of at least two factions. Say the proposal under consideration is a tax. (Recall Madison’s comment in Federalist 10 that, compared with legislative decisions on how to apportion taxes, there is “no legislative act in which greater opportunity and temptation are given to a predominant party to trample on the rules of justice.”)

Start by considering a proposal imposing a tax. It’s easy to illustrate the instability of majority voting over this policy. We could start with any proposal whatsoever, but let’s start with an equal allocation of the tax burden: The proposal is that each faction (or the communities each faction represents) bears one-third of the tax burden. It’s easy to overturn the proposal by majority vote: Two factions get together, they propose each of the two factions in the coalition pays only one-sixth of the tax, imposing a full two-thirds of the burden on the other faction. The proposal wins a majority vote.

But that’s not the end of the story. This proposal can also be overturned. The losing faction can cut a deal with one of the two winning factions, giving one of those factions a better deal, and now imposing a disproportionate tax burden on the other originally winning, but now losing, faction. Repeat ad nauseam.

In the initial case with only two factions, a majority faction and the minority, an unequal allocation of the tax burden is a stable policy. There’s nothing the minority faction can do to get out from under the policy imposed by the majority faction. In the case of three or more minority factions, however, the losing minority faction can always a cut a deal with one of the two winning factions to create a new legislative majority, overturning earlier majority-supported policies or proposals.

In the case of the three factions, we meet the requirements for McKelvey’s chaos theorem, three voters and at least two policy dimensions. (In the apportion-the-tax case, there are three policy dimensions, the proportion of the tax assigned to each factions is itself a separate policy dimension. Hence, cycles.)

In Federalist 10, James Madison argues factions will have less influence at the national level than at the state level because states more often have a single majority faction relative to the nation as a whole. Factious stability at the state level will be less likely at the national level because when all the states are added together, state-level majority factions will likely be only minority factions at the national level. Hence, a greater possibility of national-level voting cycles and majority instability at the national level than at the state level. Because all recognize that factious outcomes enacted by majorities of minority factions can be easily overturned by different coalitions put together from the same set of representatives, sustaining factious outcomes will be more difficult at the national level relative to the state level.

Madison’s argument is well known, nonetheless it’s worth quoting:

The smaller the society, the fewer probably will be the distinct parties and interests composing it; the fewer the distinct parties and interests, the more frequently will a majority be found of the same party; and the smaller the number of individuals composing a majority, and the smaller the compass within which they are placed, the more easily will they concert and execute their plans of oppression. Extend the sphere, and you take in a greater variety of parties and interests; you make it less probable that a majority of the whole will have a common motive to invade the rights of other citizens; or if such a common motive exists, it will be more difficult for all who feel it to discover their own strength, and to act in unison with each other.

Beyond Madison, so, too, the “discrete and insular minorities” language from footnote 4 of Carolene Products betokens judicial protection not simply for minorities, but minorities which do not have ordinary opportunities to form coalitions with other legislative minorities in order to overturn factiously oppressive legislation. Per the third paragraph of the footnote, democratic processes fail for these groups because of the absence of normal majoritarian instability. Therefore courts need to step in: “[P]rejudice against discrete and insular minorities may be a special condition, which tends seriously to curtail the operation of those political processes ordinarily to be relied upon to protect minorities, and . . . may call for correspondingly more searching judicial inquiry.” Think for example of the stability of race laws adopted in Georgia, circa 1930. Not a good thing.

Notwithstanding his argument in Federalist 10, Madison is not all in on the side of policy instability. In Federalist 62 Madison observes how policy instability can undermine reliance interests: “[G]reat injury results from an unstable government . . . . What prudent merchant will hazard his fortunes in any new branch of commerce when he knows not but that his plans may be rendered unlawful before they can be executed?”

It would seem the stability, or instability, of majoritarian outcomes is neither uniformly bad nor uniformly good. We cannot assess the good or bad of democratic stability, or instability, without also considering the content of policies that are stabilized or destabilized.

McKelvey and Arrow might shrug their shoulders at this conclusion, observing with equanimity their theorems hold no matter whether one assesses their implications as good or bad (or indifferent). I don’t think Madison could view the conclusion with the same equanimity. His argument in Federalist 10 for factions being more probably controlled at the national level relative to state levels relies on the more probable existence of voting cycles at the national level, with an associated increase policy instability. Yet the instability cheers in Federalist 10, if implicitly, he decries elsewhere in The Federalist. On Madisonian terms, whether a higher incidence of voting cycles at the national level is on balance good or bad for the country would depend on weighing the benefits of less factious activity at the national level with the cost of reliance interests overturned by the corollary policy instability. At the very least, it makes his conclusion in Federalist 10 regarding the benefit of nationalizing policies to avoid faction a closer run thing.

Reader Discussion

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on May 03, 2018 at 13:58:25 pm

I've always thought the solution to this is fairly simple. Require rotating selection of agenda, and allow a larger majority to overrule a previous decision issued by a smaller majority. So if you have a group of 100 people. One person is randomly selected to propose something to vote upon. Lets say that gets 40% of the vote. Another person who hasn't gotten a chance to vote is randomly given the second chance to set the agenda. Lets say his proposal changes the first proposal and gets 50% of the vote. A third person is randomly selected and his proposal only gets 45% of the vote to change it once again and fails because it is less than 50%. Then a fourth person is selected and proposes to change it once again and gets 60% of the vote. This locks the new change in at 60% of the vote and requires that changes to it need more than 60% of the vote to change it again.

This releases the precondition of majority vote in favor of a supermajority system. Where the larger the supermajority the more it gets its way. You can then "lock in" broad principles with very high levels of agreement, and not let simple majorities override those. It doesn't defeat Arrow’s impossibility theorem, as it rejects the premises, but shows that Arrow’s theorm isn't really that important.

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Devin Watkins
on May 03, 2018 at 14:04:09 pm

This doesn't totally solve the three person in-transitivity (in that each of the people's vote is respected in the end result), but it does produce a solution that is transitive. One person will randomly lose, and not get their preferred outcome. One person will get their preferred outcome, and one person will get their second best outcome. The end result can still be transitive though.

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Devin Watkins

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