Disastrous Elections: Predicted by Game Theory

Recent elections have been disastrous. This statement is not a reflection of the results of those elections; rather, it is a conclusion which reflects that recent elections have been unrepresentative. In the United States, neither major presidential candidate received even 50 percent of the popular vote. This suggests that significantly more than half of the country would not have supported the winner regardless of the outcome.[1] Similar phenomena have occurred across the democratic world, such as in the recent U.K. parliamentary and Indian General Elections, and serve as reminders that these “democratic” elections are far from widespread democratic endorsements of governance.[2]

One of the causes behind these disastrous elections is gerrymandering, currently being reviewed by the Supreme Court in a potentially landmark case;[3],[4],[5] however, gerrymandering alone is not responsible for the failure of democratic elections to reflect democratic standards of representativeness and fairness. The primary agent undermining our elections is the voting system itself: first-past-the-post, a voting system in which everyone gets one vote and the winner is whoever receives a plurality of votes. Behind its deceptive simplicity, our voting system poses systemic problems which are predictable using game theory techniques. These same techniques can also be used to ultimately offer a solution to create truly democratic elections.

There are a few central game theory concepts required to understand how game theory can predict the failure of election systems. In game theory, “games” are merely situations in which at least two players must make rational decisions within a defined set of rules. Any set of decisions that players may make are referred to as a “strategy profile” or “strategy set.” A Nash Equilibrium—the concept behind game theory’s predictive power—is a strategy set which results from a situation where players cannot individually change their strategy and benefit. Finally, an optimal outcome is one which results in the best overall benefit as a summation of the benefits to each player individually. This is not necessarily a Nash Equilibrium. In fact, quite often game theory involves a discussion between how to design a game which leads to a Nash equilibrium which is also an optimal outcome.

One of the ways in which game theory attempts to predict complex outcomes is by dividing a game into easily solvable “toy games.” This principle can be applied to elections to demonstrate why first-past-the-post fails to produce truly democratic elections. Imagine there are five voters, from A to E.  Now imagine there are five candidates, from 1 to 5, such that candidate 1 is most like candidate 2, etc. As in first-past-the-post, the winner of the election is whichever candidate receives the most votes. To determine whether this system produces the optimal results, there needs to be some ‘accounting’ system for which to compare various outcomes. One such system is a utility function, which is often used in game theory problems. One way to model voter utility is through a quadratic utility function such as the one below, which reflects that voters prefer candidates most similar to their top preference:[6]

The question now is whether this Nash Equilibrium is optimal for the voting system. To determine this, consider an extreme situation where the results of the election are as shown in Figure 1. To find a Nash Equilibrium for the game, it is necessary to figure out how voters will vote. Consider a situation where all voters vote their preference. This strategy set of all votes is a Nash Equilibrium only if there is no individual incentive to deviate. Since the winner is determined by having the most votes, voting anything other than the preference decreases the likelihood of that preference winning—and thus the voter’s expected utility from the election. This is thus a Nash Equilibrium for our “toy election:” each voter will vote for their preferred candidate.

Voter Preference Winner Utility
A 1 1 16
B 1 1 16
C 1 1 16
D 5 1 0
E 5 1 0

 

The total utility for this election must be the sum of individual utilities, 48. It is easy to see that voters A, B, and C received high utilities for this election since their preferred candidate won, while voters D and E received no utility since they strongly dislike candidate 1. This is analogous to having two highly polarizing candidates in an election where voters are approximately evenly split between the two candidates. Can a better outcome than the Nash Equilibrium occur? Imagine that, somehow, it was determined that candidate 3 was the winner. The utilities of each voter would look like Figure 2.

Voter Preference Winner Utility
A 1 3 12
B 1 3 12
C 1 3 12
D 5 3 12
E 5 3 12

 

While not a single voter got their favorite candidate, the overall utility for this election is 60. This means that, when candidate 3 wins, the outcome is better overall even if some voters are not as happy as if their preferred candidate won! This demonstrates that, in an election with many candidates where voters cannot assume how other voters will vote, the outcome of the election will be non-optimal and voters will simply vote their preferences.

Now imagine a second toy game based on the first, and assume the outcome of the first toy game’s election was as shown in Figure 3.

Voter Preference Winner Utility
A 1 1 16
B 1 1 16
C 3 1 12
D 4 1 7
E 5 1 0

The total utility for this election was 51. Now a new election in this district is approaching, and the voters and candidates remain the same. This time, voter D can observe the results of the first election. Since voter D prefers candidate 5 to 4, it’s possible that voter D decides to vote for candidate 5 instead—hoping that perhaps candidate 5 will win due to outspoken supporters.[7] Once voter D declares this preference, voter C now has a decision: throw the election to candidate 1 or candidate 5. Indifferent between the two, imagine voter C votes for candidate 5, as shown in Figure 4.

Voter Preference Vote Winner Utility
A 1 1 5 0
B 1 1 5 0
C 3 5 5 12
D 4 5 5 15
E 5 5 5 16

 

The total utility for this system is 43, meaning the outcome was even worse than before. Notice an interesting trend with the candidates. At this point, there are effectively only two candidates that can win: candidate 1 and candidate 5. At any point in time, candidate C can deviate to vote for candidate 1, and the outcome is still a Nash Equilibrium. Based on how voter C is wooed, either candidate could win any given election, and the outcome will always be suboptimal.

These two very simple toy elections illustrate several profound results of a first-past-the-post system. The first is that the system will devolve into a two-party system with committed base voters and some swing voters, just like the U.S. today. Equally importantly, the election result is suboptimal: while voters A and E are sometimes very satisfied and sometimes extremely unsatisfied with the election results, the “toy district” would be overall better when candidate 3 wins. This demonstrates that the first-past-the-post voting system fails to deliver optimal, truly democratic results.

Game theory methods also offer several solutions, one of which is the Schulze method. [8] The Schulze method always selects the optimal winner, which is often the Condorcet candidate—the candidate who would win in any one-on-one election against all other candidates. When such a candidate does not exist, the Schulze method is still capable of selecting the most optimal candidate.[9] Its application can be used to maximize utility for voters in an election system. If the United States is to maintain a democratic voting system it must switch to a smarter election system. Such a change would encourage better governance and real policy discussions rather than pandering to bases and swing voters. If we fail to alter our voting system, game theory predicts many more disastrous, undemocratic elections to come.

[1] “Presidential Election Results: Donald J. Trump Wins.” New York Times, accessed October 3, 2017, https://www.nytimes.com/elections/results/president

[2] “Electoral Systems.” ACE: the Electoral Knowledge Network, accessed October 12, 2017, http://aceproject.org/main/english/es/esy_in.htm

[3] Kendall, Brent. “Supreme Court to consider limits on partisan drawing of election maps.” Wall Street Journal, accessed October 2, 2017, https://www.wsj.com/articles/supreme-court-to-consider-limits-on-partisan-drawing-of-election-maps-1497881169

[4] Petry, Eric. “How the Efficiency Gap Works.” Brennan Center for Justice, accessed October 2, 2017, https://www.brennancenter.org/sites/default/files/legal-work/How_the_Efficiency_Gap_Standard_Works.pdf

[5] Reynolds, Molly E. “Vital Statistics on Congress.” Brookings Institute, accessed October 2, 2017, https://www.brookings.edu/multi-chapter-report/vital-statistics-on-congress/

[6] In this function, W is the number of the winning candidate and P is the number of the preferred candidate.

[7] Note that several other deviations are possible, such as {1, 1, 3, 3, 3} or {1, 1, 1, 4, 4} which are also Nash Equilibria. The true outcome will depend on the sequence of events. The chosen equilibrium was selected because it is realistic for fringe candidates to maintain steadfast supporters. In this toy election, it’s not unreasonable to assume that voter E will never change votes, forcing the chosen NE (if voting preferences are made known).

[8] Schulze, Markus. “A new monotonic, clone-independent, reversal symmetric, and Condorcet-consistent single-winner election method.” Social Choice and Welfare 36, iss. 2 (2011): 267-303, accessed October 3, 2017, https://link.springer.com/article/10.1007%2Fs00355-010-0475-4?LI=true

[9] The Schulze method is not a utility-derived model, but is instead based on satisfying several voting criteria. Nevertheless, these criteria closely align with the utility assumptions made in the “toy district” example.