The Irrationality of Committees and Courts: A Voting Paradox

Many things in life are decided by groups of people making judgments together—in courts, committees, board rooms, legislatures, etc. Collective decision-making has much to recommend it: groups of people can pool their wisdom, no one person can rule by fiat, organizations can benefit from the cohesion of people working together, etc. And, of course, there are drawbacks, as anyone who has suffered through interminable committee meetings knows. But suppose we had an ideal world, with all the best qualities of committee decision-making and none of the problems? Well, it turns out that that it’s a mathematical fact that even then committees can sometimes act irrationally.

In this post, I’m going to explain this paradox, called the discursive dilemma. The framework for the discussion is from a part of economics (or political science) called social choice theory, which is the study of voting procedures. Everyone ought to learn a bit about social choice theory, which is quite accessible (even though, at its core, it is math). I won’t take time to discuss the general theory here, since it isn’t really necessary to explain this specific paradox, so I’ll just jump right in.1

It’s easiest to explain by example. Suppose a committee of three experts commissioned by the UN must decide whether there really is global warming. Often, to decide a matter, a committee can just vote by majority rule. But to conclude that there is global warming, there are several intermediate logical steps, so the committee decides to vote on several logically interrelated statements (or "propositions", as they are called in logic) about global warming:

Proposition 1: If the amount of carbon dioxide in the atmosphere is above the level x, then there is global warming.

Proposition 2: The amount of carbon dioxide in the atmosphere is above level x.

Proposition 3: There is global warming.

We can represent this logically as follows. Let P be proposition "the amount of carbon dioxide in the atmosphere is above level x", and let Q be the proposition "there is global warming." What we’re asking the committee to decide on is:

Proposition 1: P implies Q

Proposition 2: P

Proposition 3: Q

There are a range of consistent—i.e., logically valid—judgments. You could decide that all three propositions are true. You could decide that P implies Q is true, but that P is false; in that case, you would be consistent if you chose Q to be true or false. And so on. But you couldn’t decide that P implies Q is true, that P is true, and yet that Q is false. (In fact, this is the only possible arrangement of judgments which is inconsistent.)

The three committee members, call them A, B, and C, vote on whether they think each proposition is true (T) or false (F), and then they use majority voting to determine the committee’s judgment of each proposition. Suppose the committee votes as follows:

\begin{array}{c||c|c|c} \text{Voter} &  P \text{ implies } Q & P & Q\\ \hline A & T & T & T\\ B &  F & T & F\\ C & T & F & F\\ \hline \text{Majority} & T & T & F \end{array}

Each member of the committee has made logically consistent choices. And yet, if we look at the majority decisions, the committee has decided that P implies Q and P are true, but Q is false. This is a contradiction!

Let me give two more examples. Suppose a committee at a university needs to decide whether to grant tenure to an assistant professor. The university’s standards for tenure are "excellence in teaching and in research." For simplicity’s sake, let’s assume there are only three people on the committee. Each member of the committee decides whether or not the candidate is excellent at teaching and whether or not the candidate is excellent at research. Then, if and only if they answer true for both, they vote yes on tenure. Again, let’s use some logical symbols to write down these propositions:

Proposition 1: The candidate is excellent at research (P)

Proposition 2: The candidate is excellent at teaching (Q)

Proposition 3: The candidate deserves tenure (both P and Q)

Here’s how the voting turns out:

\begin{array}{c||c|c|c} \text{Voter} &  P  & Q & \text{both } P \text{ and } Q \\ \hline A & T & F & F \\ B &  F & T & F \\ C & T & T & T \\ \hline \text{Majority} & T & T & F \end{array}

According to majority voting, the committee has decided not to award tenure, even though the majority thinks the candidate is excellent at research, and the majority thinks the candidate is excellent at teaching. Once again, the voting is inconsistent.

A final example comes from the legal system (where this paradox originated). Suppose an appellate court of three judges must decide whether or not to award damages for a potential breach of contract.2 The court must decide both whether or not there was a valid contract to begin with, and whether or not there was a breach of contract. If and only if both are true, the court awards damages. This problem is, in fact, identical to the problem about tenure above: let P be the proposition "the contract is valid" and let Q be the proposition "the contract was breached". If the judges vote as above, they will conclude that the contract is valid and was breached, and yet the majority of the court will vote against damages.

One can keep constructing examples along these lines. So the paradox is simply this: groups deciding logically related statements by majority voting can end up making decisions that are logically inconsistent.3

What is to be done? Well, it’s important to be aware of this inherent limitation that math has shown. Yes, groups can pool their collective wisdom ("the wisdom of crowds"), but you might have to sacrifice logical consistency. From there, the thinking to be done has to leave the domain of logic and mathematics, and get into the philosophical, legal, and practical implications.

  1. There are a whole range of voting paradoxes in social choice theory. The most famous of these, Arrow’s Impossibility Theorem, says that, when there are at least three candidates, there is no voting procedure—not plurality voting, nor instant runoff, nor anything else—that will satisfy a few seemingly innocuous and very desirable requirements. This discursive dilemma (also called the doctrinal paradox) is part of social choice theory called judgment aggregation. I’ve relied on this survey article of Christian List to refresh my memory, which was a bit hazy from when I learned it in a logic course back in college. (I could have sworn that I read in college that Oliver Wendell Holmes had noted this paradox, but my recent googling—the topic is rare enough that it took me a bit of time even to find out the name of the paradox—suggests that the first reference was a law paper in the 1980s by Kornhauser and Sager.) I also looked at an article by Philippe Mongin, which seems to draw a distinction between the doctrinal paradox and the discursive dilemma. (I didn’t go into any of the details.) The examples I use, though others could easily be constructed, are from these papers. List has a webpage with more resources. Caveat: I haven’t thought about social choice theory for a while; what I’m presenting here is pretty elementary, but it’s possible there might be some subtleties I’ve missed. Please let me know!

  2. This example is simpler than the legal reality, but it captures the essence of the problem. For one, it would be district courts which make such decisions; the appellate court might rule on an appeal. Moreover, contract law is often more complicated.

  3. One might wonder whether there’s an alternative to majority voting. There’s a theorem (usually stated in the more traditional framework without logical propositions) that say that majority voting is the only reasonably fair method. So this is, in a sense, a mathematically precise reflection of Churchill’s quip about democracy being the worst form of government except for all the others that have been tried.