Rawlsian Voting Revisited December 22, 2025

Like all men in Babylon, I have been proconsul; like all, a slave.

Borges, “The Lottery in Babylon”

A few months ago, I wrote a short post on trying to justify random ballot elections using Rawls’s original position argument. In brief, I interpreted the original position argument as saying that a just society is one that maximizes your expected utility under the assumption that you inhabit a role selected uniformly at random. (Intuitively, a society is fair if you’d be happy entering as a randomly selected person.) I then applied this principle in the context of an election between two candidates $A$ and $B$.¹ If $n_A$ voters support candidate $A$, and $n_B$ support candidate $B$, what’s a “fair” value for the probability $p_A$ of $A$ winning?

Without loss of generality, let $n_A \ge n_B$. The result I presented was basically this: with a straightforward indicator utility function—in which you receive unit payoff if your favored candidate wins, and zero otherwise—it’s not hard to show that the optimal solution is $p_A = 1$, which is to say that we should just take a simple majority vote.² Some more exotic utility functions can give you more interesting results, such as $p_A = n_A / (n_A + n_B)$, which corresponds to a “random ballot” election, i.e. one in which we should draw a single ballot at random and let it dictate the result.

That’s the conclusion that I came to in the previous post, but I was unsatisfied with the argument, for two reasons. First, even if you grant that expected utility maximization has something interesting to say about this hypothetical election problem, the utility function that I proposed to justify random ballots seemed obviously contrived.³ Second, I did not actually engage with (because I did not know about) an important part of Rawls’s actual argument, which is that expected utility maximization is actually not the most reasonable framework to use when operating behind the veil of ignorance.⁴ Rawls’s famous book A Theory of Justice (1971), in which he first proposed the original position thought experiment, is actually a landmark rejection of utilitarianism, on the grounds of fairness.

I have spent some time recently mulling over both of these deficiencies in the original post. I think I have a satisfactory response to the first, although I’m still not too sure about the second. I think I will probably have more thoughts on the second later, but I thought that the argument for the first was independently interesting, so I’ve written it up below.

To present a less-contrived argument for random ballots, let’s modify the scenario from the previous post slightly. Instead of modeling each voter’s payoff as a random variable over the space of election outcomes, we instead model each voter’s payoff as a (presumably monotonically increasing, and for convenience differentiable) function $f : [0,1] \to \mathbb{R}$ of the probability of her favored candidate’s victory. That is, we are asserting that a voter is happier if her candidate is more likely to win, which seems like a reasonable assumption.⁵ Of course, a voter is happiest of all if her candidate wins with certainty, but our model allows for probabilistic outcomes.

In general, for some voter utility function $f$, the expected utility⁶ when entering society from behind the veil of ignorance is

$$E[U_f] = \sum_{i \in \mathcal{C}} \frac{n_i}{N} f(p_i),$$

where $\mathcal{C}$ is the set of candidates, $N$ is the total population, and the $p_i$’s form a probability distribution. What are some reasonable payoff functions $f$ that we might consider? Probably the most obvious is the identity function $f(x) = x$, which actually corresponds to the indicator utility in the setting of the previous post. In the simple case of $\mathcal{C} = \{A,B\}$, we have already shown in the previous post that $E[U_{\textsf{id}}]$ attains its maximum at $p_A = 1$, assuming $n_A > n_B$. That is, the identity payoff suggests we should just pick the candidate with the majority vote. (Or the plurality vote, if $|\mathcal{C}| > 2$.)

This accords well with how we tend to actually run elections, but there’s something very weird about using the identity function as a utility function, which is that an affine utility function is necessarily risk-neutral. That is, an $A$-voter with an identity payoff curve is indifferent between a world where $p_A = 0$ or $p_A = 1$ with equal chance, and a world with the certain outcome $p_A = 1/2$. Most people in real life are not risk-neutral! Getting a sure fifty dollars is preferable to a fair coin toss for one hundred dollars; we want to be paid a premium in order to take on risk. ⁷

Mathematically, we can represent this risk aversion by choosing a concave function $f$ for our voter payoffs: we are imposing a diminishing marginal utility for incremental gains in our favored candidate’s probability of victory.⁸ One intuitive way to do this is to say that the $A$-voter cares about percentage increases to $p_A$, not absolute increases as in the naively risk-neutral case. Put another way, the utility of the marginal increase in $p_A$ is inversely proportional to its current value (as opposed to constant), giving the differential equation $f'(x) = 1/x$. Of course, the solution is $f(x) = \ln x + C$, where we don’t really care about the constant.⁹ So we have:

$$E[U_{\textsf{log}}] = \sum_{i \in \mathcal{C}} \frac{n_i}{N} \ln p_i,$$

subject to the constraint that the $p_i$’s must sum to unity. We can discover the maximum with a Lagrange multiplier, setting $\nabla E[U_{\textsf{log}}] = \lambda\vec{1}$. This is just $\partial E[U_{\textsf{log}}] / \partial p_i = n_i / Np_i = \lambda$, which (normalizing to get a probability distribution) has solution $p_i = n_i / N$. But this is exactly a random ballot election: each candidate should have probability of victory equal to her proportion of the electorate!

I think this is an interesting argument for pure random ballot elections, starting from first principles. Now I don’t think it’s totally convincing, even if you allow the premise that expected utility maximization is reasonable here. In particular, although economists often use logarithmic utility functions because of their analytic convenience, it need not be the case that real people’s preferences follow such a rule. Indeed, taking a different function would yield a different result: consider the function $f(x) = 3x-x^2$, which is strictly concave and monotonically increasing over $[0,1]$. In the simple two-candidate election, $E[U_f]$ is maximized at $p_A = \max\{(4n_A-N)/2N,1\}$ (exercise for the reader!). Put plainly, this means that a society with such a utility function would prefer some change of $B$ winning if support for $A$ is below a three-quarters supermajority, but above that threshold it would prefer that $A$ win with certainty.

As I alluded to in the beginning of the post, there are also significant questions about whether maximizing expected utility (i.e. a naive classical utilitarianism) is even the right framework to apply here, with Rawls submitting significant objections himself. I have some half-baked thoughts, but no particularly coherent argument for either side, so I guess that will have to wait until a future post. At this rate, I’ll be impressed if I can get that post out before late 2026.