The Situation: Limited Resources
There is only one ventilator — or liver, or hospital bed, or any other potentially life-saving resource — at your hospital, but many people need it and are eligible for it. You are part of a group that must decide who gets the ventilator. You all agree about what kinds of things should be taken into account — the chance of treatment success, the number of healthy years the patient would have if they are treated, etc. — but you don’t have a perfect patient who is best on all these criteria, and your group doesn’t agree on how much you should care about each of these criteria. How can you vote on how to allocate the ventilator to best taken into account the different concerns of the members of your group? We’ll look at how well four different voting rules help you select a patient to receive the resource in a way that best takes into account the concerns of you and your fellow voters.
The model, informally
We have some number of candidates for our ventilator (or other limited resource). These candidates differ along several dimensions that we think are relevant to whether we should choose to give them the ventilator, such as the likelihood of a successful treatment, the cost of treatment, or the expected number of healthy years our patient would have if successfully treated. Our voters all agree on what makes a good candidate along all these dimensions (being likely to have a successful treatment, not having other concerns that could drive up the cost of the procedure, etc.), but do not necessarily agree on how important each of these dimensions are for determining who should get the ventilator. Since our candidates differ along all of these dimensions, a perfect candidate, who is the best along all dimensions, is unlikely to exist. Our voters will vote for a candidate to receive the ventilator using a variety of voting rules — plurality, plurality runoff, ranked choice, and Borda count, which are explained below. We will measure how well the winning candidates using these voting rules takes into account the concerns of all our voters to result in the most agreeable candidate to receive the ventilator.
The model, formally
A few gory details of the inner workings of the mathematical model I used.
The relevant properties of candidates are represented by dimensions in a (Euclidean) space. Candidates are uniformly distributed along the interval [0,1] along each dimension. These constitute the information about candidates that our voters have access to.
Each voter prefers candidates closer to 0 along each dimension. The voter utility functions are given by
Ui(xi1, xi2, …, xin)=m1xi1+m2xi2+…+mnxin,
where Ui is the the utility the voter assigns to candidate i, there are n dimensions (relevant properties of candidates that voters are taking into account), and -1≤mj≤0. The mj are uniformly distributed and then weighted so that they are -.5 on average for each voter. The greater |mj| is, the more a voter penalizes candidates for scoring badly on dimension j, i.e. the more a voter cares about that dimension.
We will have our voters vote for candidates to receive resources using Plurality, Plurality Runoff, Ranked Choice, and Borda Count voting rules, and then we will measure the average utility voters assign to winning candidates from each voting rule to see which voting rule is best at balancing the differing concerns of our voters. Voters will all vote honestly and fully complete their ballots by listing all the candidates for the Ranked Choice and Borda Count elections.
The voting rules we’ll consider
Plurality: Voters each vote for one candidate. The candidate with the most votes wins.
Plurality Runoff: Voters each vote for one candidate. The two candidates with the most votes advance to a final round, where voters vote between them. Whichever candidate gets more votes in this one-on-one election wins.
Ranked Choice: Voters rank all candidates. The first place rankings of all candidates are tallied, and the last place candidate is eliminated. Votes for the eliminated candidate are then reassigned to the next active candidate on each voter’s ballot. The new first place rankings of all candidates are tallied, and the last place candidate among all candidates who have not been previously eliminated is now eliminated. Votes are reassigned as before. Repeat this process until one candidate remains. The final remaining candidate wins.
Borda Count: Voters rank all candidates. Candidates receive points from each voter based on where they are ranked. For instance, the last place candidate on my ballot receives zero points from me, the second-to-last place candidate on my ballot receives one point from me, the third-to-last candidate on my ballot receives two points from me, and so on. The candidate with the most total points from all ballots wins.
Results: Borda Count promotes agreement
To get a rough idea of how these voting rules performed compared to each other, I ran simulations with different numbers of candidates, dimensions, and voters; for each combination of candidates, dimensions, and voters, I ran 1000 simulations.
The broad takeaways from these results are:
- Borda Count consistently outperforms the other voting rules in terms of best consolidating the differing concerns of our voters, as measured by the average utility that our voters assign to the allocating resources to the patient who receives them.
- Plurality Runoff is consistently second best after Borda Count. This is an especially strong result, as Plurality Runoff does not ask voters to list all candidates, so would be a strong option for situations in which we should not expect voters to be able to comprehensively rank all candidates.
- Ranked Choice’s performance relative to the other voting rules fluctuates significantly depending on the circumstances. It can tie with Plurality Runoff or perform far worse than the other voting rules. This is a particularly bad performance as Ranked Choice, like Borda Count, asks voters to rank every candidate on their ballots.
- Plurality typically is the worst performing voting rule, though it performs well in circumstances where the number of voters is greater than the number of candidates.
- As one would hope, all voting rules perform far better than selecting a candidate at random.
- The number of dimensions does not affect the relative performance of the voting rules.
What this tells us about real situations
Each of these voting rules can be rather straightforwardly extended to cases where we want to select more than one winning candidate, meaning that these results are likely to carry over to cases where we can allocate resources to multiple patients.
While Ranked Choice and Borda Count ask voters to rank all candidates, both of these voting rules can still work when voters do not rank all candidates. Looking at how well these voting rules perform when voters’ ballots do not systematically rank all candidates can help us understand how well these rules would perform in situations where voters cannot be reasonably expected to rank all the candidates for a resource.
Knowing how many voters, candidates, and relevant features of candidates that voters care about (dimensions in the model) there would be in situations where resource allocation decisions are being made would allow me to get some more targeted results and also work toward adapting the model so that our voters don’t need to participate in elections every time we need to choose how to allocate resources.
I assumed that candidates for resources are uniformly distributed along all relevant dimensions, and that their position along any one dimension is independent from the positions along all other dimensions. In reality, these factors will not be uncorrelated. If they are positively correlated, the relative difference between the voting rules might be lessened as the differing concerns of the voters become less relevant.
I also assumed that how much each voter cares about a given dimension is completely independent from how much other voters care about that dimension. Real voters might be more uniform in which dimensions they prioritize, which could also lessen the relative difference between the voting rules.
This model tries to best take into account the concerns that our voters actually have and not the concerns that they should have. I’m not going to tell the voters what to think. I’m not that kind of philosopher. Instead, I’ll take their opinions as valid and then determine how best to consolidate them with the differing concerns of others.
Though, as a current denizen of the Covid Era, I phrased this investigation in terms of the allocation of medical resources, this model can be used in other circumstances where a discrete (not continuous) limited resource is to be allocated.