Arrow's social choice theory and democratic chaos
How inconsistency can arise from majority rule
The geneticist JBS Haldane conjectured in his Inequality of Man that voting was one of the four major civilizational advancements discovered between 3000 and 1000 BC. Three millennia later, in democracies, the majority rule is presented as an article of faith taught to schoolchildren as the best system for two centuries. In 1951, rigorous results show how inconsistency arises from this decision rule. For 70 years, these known limits results do not seem to have touched the democratic political discourse, which is rooted in symbolism and emotion.
After WWII, the economist Kenneth Arrow took over 18th-century work by Condorcet on votes and published in "Social Choice and Individual Values" the impossibility of finding a participatory decision-making system that led to coherent decisions.
Arrow is not only recognized for having defined a clear mathematical framework which has enabled the development of a new field of research (social choice theory) in the decades that followed. He also collaborated with Debreu in 1954 in the mathematization of market equilibrium, which he justified using a Kakutani fixed point theorem of 1941.
Condorcet: inconsistency of multiple decisions
In the 18th century, the Marquis de Condorcet's interest in voting methods was applied to the election of a new member of the Academy of Sciences.
He notices the non-transitivity of electoral choices. Even if a majority prefers C, it may be that B is preferable to A for a majority, and C is preferable to B for another majority. This property leads to an inconsistency of governance by majority vote.
Impossibility of Arrow
Besides Pareto's objection that voting for a social choice assumes an interpersonal comparison of individual utilities, Arrow shows that it is impossible to formulate an order of social preference that satisfies all the following conditions:
Non-dictature: the preferences of an individual should not become the classification of the group without taking into account the preferences of others.
Individual sovereignty: each individual must be able to order the choices in any way whatsoever: if each individual prefers one choice to another, then the group's classification should do the same
Lack of irrelevant alternatives: if a choice is deleted, the order of others should not change the uniqueness of the group rank: the method must give the same result whenever it is applied to a set of preferences.
The group classification must be transitive.
McKelvey-Schofield: Democratic chaos
In the rest of this article, we take up a result of McKelvey-Schofield which shows how a series of majority votes can lead to gradually moving away from what all voters want and lead them to a dystopian situation. This result is explained in a YouTube video in English. The video is here.
The visualization presents a quantitative choice (e.g. a decision to allocate spending to social programs) according to two axes (two separate programs). If the field of choice is restricted to exercise only on an axis, the choice of the median voter will be used, if the domain includes at least two axes, we see
how a series of votes can keep the choice of the centroid from preferences.
The initial state close to the average position is represented in blue, and the final state in red:
An indifference zone for each voter can be defined by the yellow circle.
The preference area of the majority is represented in green.
Policy 2 is the majority choice. It has made it possible to keep the state of the centroid system away and increase the possible area of possible majority votes.
Policy 3 can be chosen, which is now outside the convex envelope of what voters want.
Policy 4 is now likely to collect the majority of votes
The power to decide the agenda
This theorem shows how the power to pose the agenda allows a democracy to arrive at any decision. There is therefore a possibility of deliberate manipulation of political order in democracy: in the United States, for Noam Chomsky, the political agenda is manipulated by lobbies and the elite of the two parties. This would explain that the question of universal health insurance has never been posed directly in the USA and that the cost of health spending is now two to three times higher in Europe. In France, for Etienne Chouard (defender of the citizen’s initiated referendum), the choices presented in Parliament are the subject of preliminary discussions which are opaque. In general, any conspirator explains that the people are good, but that an invisible and secret group is responsible for social order. But is it enough to eliminate the influence groups to remedy the inadequacies of the decision by the vote?
Chaos potential inherent in the current order
In the visualization of the McKelvey-Schofield theorem, the closer the social order is to the centroid of preferences, the more the majority of decisions are acceptable by a majority. The more social order is distant, the more extreme changes can be voted by a majority.
The transition from policy 3 to policy 4 already corresponded to two social orders outside the convex envelope of desirable policies. It looks like an extreme discontinuity, as produced by a switch from communism to fascism. In fact, these transitions are so extreme that they would be classified as political discontinuity by Charles Tilly.
Thus, the switch from democracy to fascism resembles a situation where social order is already very unsatisfactory, and the next choices are made in a very wide space where moderates are stuck between communism and fascism. Likewise, China in the 1930s was in a chaotic situation of governance by warlords. Then everything becomes politically possible. Situations like the civil war in Congo and Sierra Leone show that warlords are capable of perversions that go far beyond disorder and instability. These are policies that 98% of a population would consider acceptable.
Even if there is no plot to decide the agenda, exogenous shocks impose an agenda, with different models defended by different factions. This agenda is in no way some to lead to a centroid policy.
Social capital, homogeneity, and polarization
When the Americans overthrew the Iraqi state in 2004, they dismissed all existing high-functionaries and launched a democratic policy based on an ethnic quota system. The result was a factious government, ethnically polarized, clientelist, unstable, and violent, creating crisis situations, but not showing interest in improving the fate of its citizens.
This contrasts with the resumption of Japan by the USA in 1945, where high-functionaries guilty of war crimes remained in place, and where democracy, albeit with a single ruling party for the next 70 years, was set up. The population was homogeneous.
The agenda is perhaps more about the political situation than deliberate choices by a secret clique. The democratic theories of Chomsky or Chouard are more suitable for a small homogeneous and already stable country like Liechtenstein or Guernsey than for a large country like the United States or France, where the useful vote and the protest vote have more potential to sway social choices away from median preferences.
Governance: Beyond the democratic symbol
In 1920, according to Walter Lippmann, democracy is legitimate because it is a symbol that brings together. But he insists that the median voter is not competent as an administrator, an economist, or a lawyer. He advised American presidents that a technocrat college is working to optimize the country's policy, while the country's social complexity has increased since its industrialization. He thought that a big country required a technocratic government.
Arrow's impossibility theorem applies not only to democracy but to any decision-making group. In fact, the governance problem is reproduced within the central committee as soon as a decision is put to the vote. One possibility is to modify the voting system so that each voter can report their level of utility rather than an ordinal level. In fact, such measures are already in place upstream of the vote: voters are allowed to invest and express their opinion on a subject that is important. They can train lobbies to report their financial interest in a cause that interests others less. However, we observe a capture of the decision by special interests and rent-seeking by groups operating in an asymmetrical situation, as explained in Olson's work on collective action.
One can imagine asking technocrats to calculate the usefulness of the constituency that they represent and thus aggregate votes of utility, but special interests can always come into account in this vote, insofar as it remains discretionary.
If the decision is only subject to a calculation, one decides by vote. The best solutions are (1) to reduce the size of the electoral college to a group that has homogeneous objectives, and (2) to reduce the scope of social choices to this homogeneous group so that it does not harm those in the out-group.
Amartya Sen will continue the work of Kenneth Arrow in his article "The impossibility of a paretian liberal", with a theorem of the impossibility of liberal governance if everyone is more concerned with having control over other people's lives than over their own.