Social Choice: A Personal Theory of Power

Preferences, Choices, and Strategy


This essay, provided by George Mason University Computational Social Science PhD student Adam Elkus, is part of the Personal Theories of Power series, a joint Bridge-CIMSEC project which asked a group of national security professionals to provide their theory of power and its application. We hope this launches a long and insightful debate that may one day shape policy.

There are many sources of power, but this essay focuses on choice-theoretic research and social choice in particular. Social choice concerns the process by which individual preferences are aggregated into collective choices through group decision processes such as voting.

I hope to convince the reader that strategists ought to be thinking about social choice and its implications for the complexity of attaining stable agreement. It is by pondering the nature of social choice that we truly understand why the strategist is a “hero.”

Choose Wisely

Many strategic discussions default to simplistic explanations for why groups cannot achieve desired outcomes. It is easy to blame bureaucratic rivals, declare that salvation can be found in better strategy itself, make accusations of bad faith, or declare that America’s strategic culture in some way guarantees inconsistency and short-term thinking. However, there is a better, less value-laden approach to contemplating why strategy can sometimes seem like an illusion.

Colin Gray argues that his strategy bridge is a process of “dialogue and negotiation.” Carl von Clausewitz also famously noted that war is “politics by other means.” These formulations imply that both politics and strategy are products of group decision-making processes. This can be viewed on several levels — the group in the war room or the National Security Council, the “group” that routinely elects a President every four years, and everything in between.

I argue that one powerful method for analyzing the nature of group decision is the choice-theoretic perspective — and social choice in particular.

The choice-theoretic perspective models how decisions individuals make aggregate to group outcomes. While there are many different methods of analyzing choice, I talk about one in particular: social choice theory. Social choice concerns the problem of agreement by analyzing how individual preferences create collective decisions. Social choice is often used to analyze elections — or any other situation in which individuals rank the desirability of alternatives and must find some way to fashion a group preference out of individual choices.

From Condorcet to Arrow, social choice’s chief dilemma lies in the following problem:

Condorcet’s second insight, often called Condorcet’s paradox, is the observation that majority preferences can be ‘irrational’ (specifically, intransitive) even when individual preferences are ‘rational’ (specifically, transitive). Suppose, for example, that one third of a group prefers alternative x to y to z, a second third prefers y to z to x, and a final third prefers z to x to y. Then there are majorities (of two thirds) for x against y, for y against z, and for z against x: a ‘cycle’, which violates transitivity. Furthermore, no alternative is a Condorcet winner, an alternative that beats, or at least ties with, every other alternative in pairwise majority contests. Condorcet anticipated a key theme of modern social choice theory: majority rule is at once a plausible method of collective decision making and yet subject to some surprising problems.

How can social choice help us think about strategy? Consider the simplistic explanations previously mentioned. Rather than pine after the unicorn of an ideal strategy (usually the strategy the analyst is biased towards) or assume that collectively bad outcomes occur due to malicious or incompetent decisions, we could take a step back and make a humbler assessment.

Public policy problems are complex and there are multiple valid points of view about how to decide between alternative choices. Unfortunately, these points of view do not always aggregate into stable and sustained group agreement. Some problems of strategic formulation and execution boil down to the fact that even small groups of people tend to produce inconsistent group judgments out of even very well-specified individual preferences.

This is not to argue that there is necessarily a well-defined mapping between majority-rule assumptions to strategy formulation and execution. However, I invite the reader to ponder it as an intellectual exercise — for reasons I will make absolutely clear at the end.

Social Choice: Alcohol, Sports, and Politics

I draw the following example from Maki and Thompson’s Mathematical Modeling and Computer Simulation.

Let’s assume that each individual in a group has a preference ranking on a set of alternatives. Say, for example that you like wine better than beer and beer better than whiskey. Given a choice, you choose beer over whiskey. Second, if you prefer wine to beer, you also prefer wine to whiskey. These two assumptions — that you can make choices between alternatives and that your choices are also consistent — are very important, but the second assumption especially so.

Why is this latter assumption important? Consider the alcohol example again. There are 3! permutations of {wine, beer, whiskey}. Do you make consistent choices about alcohol preference, or do your preferences cycle (e.g. loop) due to an inconsistency in the ranking? If you prefer wine to beer to whiskey to wine, there’s a problem. Given that 3! = 6 unique preference combinations, this is all rather important information for any drinking buddies to know!

From the set of all individual preference rankings, we aim to construct a group preference schedule that details how the group ranks alternatives. A group preference schedule, like an individual preference schedule, ought to rank alternatives and exhibit consistency. However, this is not always possible.

Suppose we have a group of three people, labeled Joe, Anna, and Frank, and the alternatives are baseball, basketball, and soccer. They are trying to decide which sports game to watch. Let’s say that the rules of the vote are that we make pairwise comparisons and the most preferred alternative is the “winner.” Each group member has a preference ranking of the form {Sport 1, Sport 2, Sport 3} where 1 = highest preference and 3 = lowest preference.

Let Joe = A, Anna = B, and Frank = C . Let baseball = X, basketball = Y, and soccer = Z. A prefers X to Y to Z. B prefers Y to Z to X. C prefers Z to X to Y. Doing pairwise comparisons, we see that A and C both prefer X to Y. Additionally, A and B both prefer Y to Z. So, by majority vote, we get a preference schedule of {X, Y, Z}.

Problem solved? Not so fast. Both B and C prefer Z to X. Hence what we really have gotten is {X, Y, Z, X}. A majority vote system thus contains an inconsistency — the group prefers X to Y to Z but also simultaneously prefers Z to X. Unfortunately, as the number of individuals and possible alternatives for them to choose increases, the problem only gets worse.

To illustrate that social choice covers more than just cases of alcohol and ESPN remote control allocation, I relate something more relevant: elections. Kenneth Arrow famously developed a theorem that has some troubling implications for many systems of preferential voting. I relate a much more informal version of the theorem’s formal statement below.

Let us assume that the voting method is a function that enables voters to rank each candidate by order of preference, and the election re-sorts the candidate list in order of voter preference. Let us also assume, however, some following conditions:

No Dictators (ND): the outcome should not always be identical to the ranking of one particular person.
Pareto Efficiency (PE): if every voter prefers candidate A to candidate B, then the outcome should rank candidate A above candidate B.
Independence of Irrelevant Alternatives (IIA): the outcome’s relative ranking of candidates A and B should not change if voters change the ranking of other candidates but do not change their relative rankings of A and B.

Unfortunately, when we have three or more alternatives to choose from (see this page for a nice graphical visualization), the ability to produce non-cyclic group preference schedules given ND, PE, and IIA breaks down. Not all voting systems are described by Arrow’s theorem, the assumptions about the particular form that voter preferences take, and other factors are all controversial. Yet his theorem explains enough voting systems to make it one of the fundamental achievements of economics — and social science as a whole.

Conclusion: Social Choice and Strategy

I have only illustrated one small piece of a gigantic and complex literature. I ignore things like the median voter theorem, single-peaked preferences, mechanism design, stable matchings, and other desiderata. However, there was a reason why I chose social choice and the part of it that pertains to majority-vote elections.

Winston Churchill said that democracy is the worst form of government, except for all of the other forms of government tried. Like Churchill, it is the lot of the strategist to be unhappy with the nature of collective decisionmaking. Crossing the strategy bridge often entails bearing the burden of collective decisionmaking bodies and political entities that aggregate the desires of various stakeholders in a frustrating and sometimes counterproductive manner.

As I implied earlier, my simplified formulation of Arrow and related problems is only the start of a vast literature. The problems analyzed above are not destiny. But they underscore a fundamental problem: collective decision requires some means of gaining and sustaining agreement. At the root of many political and organizational problems is the simple and frustrating problem of doing so in a way that satisfies both normative and practical/instrumental expectations we have about the way collective decision ought to work.

The strategist could respond by denying all of this, placing their hopes in some unicorn of a sound strategy to gallop in and save the day or damning the systems and/or individuals they believe stand in their way. The strategist truly becomes Gray’s “hero” when they recognize that — for whatever reason — they likely must live with the imperfections of collective decision. The heroism of the strategist lies in his or her willingness to bear a heavy burden but nonetheless formulate and execute strategy in spite of it.

Perhaps one does not require social choice to come to this realization. But it definitely helps.