How should we vote? Game theory and the optimal voting system

September 6, 2018
posted in
written by Florian Dendorfer

The search for alternative electoral systems has been a hot topic for quite some time.

In many political venues around the world election winners are determined by a relative majority, often leading to unsatisfactory results. Take the infamous US presidential election of 2000 as an example: back then the re-election of George W. Bush hinged on a 537-votes-lead in the state of Florida.1 However, apart from Al Gore, the Democratic presidential candidate at the time, Ralph Nader ran for office as an independent and won exactly 97,488 votes in Florida. With many of Nader’s voters leaning towards the Democrats, Al Gore would likely have won the state of Florida and, by extension, the presidency, if Nader had dropped out of the race. In the end, some Nader voters may have come to regret their vote. If they had voted for their second favourite choice, they might have prevented their least favourite candidate from becoming president.


Well, through the eyes of a game theorist, a voting system represents a mechanism, i.e. a game form, to be ‘played’ by voters. The outcome of this game results in the desired allocation, which in this case is the optimal winning candidate. What constitutes “optimal”, however, is open for discussion. Arguably, an election outcome, where the majority of voters (however slim) disapprove of the election winner, as was the case in 2000 in Florida, is not optimal.

Usually, in order to prevent third candidates from splitting the vote of a particular political party, i.e. a group of voters who cannot agree on their first and second choice, but share their disapproval of a certain candidate, primary elections take place. Primary elections allow voters of the same political leaning to take a coordinated approach and choose a particular candidate for the runoff election. Then again, if there are more than two candidates in any primary election, this election is itself prone to vote splitting. If candidates cannot agree to renounce their candidacy in favour of each other in the runoff election, there is no use in having primary elections.

Still, voters have the option to vote strategically, i.e. they do not state their preferences truthfully and do not always vote for their first choice but go for their best choice with the highest probability of winning. In 2000, voters who voted strategically should have marked Al Gore’s name on the ballot even though they preferred Ralph Nader.

Alas, voting strategically suffers from a very common coordination dilemma (for two players, you may also know the underlying game as the ‘Battle of the Sexes’), which is easily illustrated by a simple example as follows:

Suppose there are 100 voters in total and three candidates X, Y, Z. To win the election, a relative majority of cast votes is required. 49 voters prefer candidate X. 26 abhor X and prefer both Y and Z. Importantly, when push comes to shove, this group prefers candidate Y over Z. The remaining 25 voters want to prevent candidate X from winning as well, but prefer candidate Z over Y.

In conclusion, with 51 voters a majority want either Y or Z to win and X to lose the election. To translate their nominal majority into concrete votes, all of them must vote for Y. Alternatively they have to unanimously vote for Z. Even if all of 51 voters are willing to accept both their first and second choice as the winner of the election, every single one of them needs to match their vote with the votes of all residual 50 voters. As a result, assuming a secret ballot, voters will have a difficult time agreeing on a single candidate. Chances are that at least two voters deviate from the majority vote (for either Y or Z) and candidate X wins the election with a relative majority. Once again, voting produces a winning candidate who the majority of voters disapprove of.

A coordination dilemma of this kind is not a theoretical experiment: under California’s non-partisan blanket primary law, the two gubernatorial candidates with the most votes in the primary election qualify for the runoff election, independent of their party affiliation. With as many as 12 Democrats, but only 6 Republican candidates running in this year’s gubernatorial primary, California’s Democrats had to achieve a minimum amount of coordination in order to avert two Republican candidates being in the runoff election.2


Richie et al.3 endorse the following mechanism: All voters rank the candidates in order of preference. If no candidate is stated as the first preference on the majority of ballots cast right from the start, the candidate with the least first-preference-votes drops out (in the example above, candidate Z is eliminated with 25 first-preference-votes). Votes for the eliminated candidate are reassigned to the candidate stated as a second preference (in the example, the 25 votes indicate candidate Y as second best and are thus added to Y‘s vote count). This process is repeated until a candidate achieves a majority (in the example, after Z‘s elimination candidate Y is elected with 51 votes). This voting system is called ranked-choice or instant runoff voting (IRV).

So far, IRV has been implemented in some form or fashion in Australia, Malta, Ireland, and many cities in the US.4 IRV is credited with tackling many problems in today’s political environment, including raising voter participation (only one round of voting is required) and bridging the partisan divide (centrist candidates have the best chance of winning second-preference-votes). Notably, IRV allows voters to cast their vote for their first-choice candidate without fear that the candidate will act as a spoiler for their second favorite candidate: if their first choice flops, their vote will automatically go to their next choice. Thus, IRV reliably prevents electoral outcomes, where a majority of voters would like to change their vote afterwards to align their vote with those of like-minded voters. In short, IRV successfully resolves the coordination dilemma.

Despite all of this, it is hotly debated whether IRV or any one of the various alternative voting systems achieve the optimal electoral outcome. Naturally, the attractiveness of IRV and its alternatives depends on the objective. For example, if the objective is to obtain a truthful representation of voter preferences, IRV does not necessarily deliver. It leaves plenty of room for strategic voting, i.e. voters may not always state their true preferences on the ballot to increase the probability of their favorite candidate winning the election. To some degree, strategic voters adjust their voting behavior according to what they believe about their fellow voters (e.g. their preference structure). As beliefs sometimes turn out to be wrong, IRV may lead to electoral outcomes where voters would like to change their vote after the fact. When it comes to gubernatorial elections in the US state of Maine, however, IRV seems to be a solid choice. Both in 2010 and 2014, the Republican candidate was elected governor with a vote share of less than 50 percent because the Democratic and the independent candidate split the vote.5 With the introduction of IRV in this year’s election, this is bound to change.


1 Roberts, J. (2004, July 27). Nader to Crash Dems’ Party? Retrieved July 6, 2018, from

2 The New York Times (2018, June 11). California Primary Election Results. Retrieved July 6, 2018, from

3 Richie, R., Bouricius, T., & Macklin, P. (2001). Candidate number 1: instant runoff voting. Science, 294(5541), 303-306.

4 FairVote. Ranked Choice Voting / Instant Runoff. Retrieved July 6, 2018, from #where_is_ranked_choice_voting_used

5 Bureau of Corporations, Elections & Commissions (State of Maine). Tabulations. Retrieved July 6, 2018, from