Game Theory in Cumulative Voting
Source: https://www.jstor.org/stable/2132065?seq=3#metadata_info_tab_contents
From 1902 to 1954, the state of Illinois had a very unique voting system for its state representative voting system. Voters would elect 3 state senators instead of only one. In addition, each voter received 3 votes instead of one. Voters could use all 3 votes on any number of candidates. For instance, they could give a single candidate 3 votes, give one 2 and another 1, or split all 3 votes among 3 candidates. The 3 candidates who received the most votes were elected into office. This is called Cumulative Voting. In The Accuracy of Game Theory for Political Behavior: Cumulative Voting in Illinois Revisited, the author, Goldberg, applies game theory analysis to determine how many candidates each political party should nominate in order to maximize the number of party members elected to the state senate.
For this game, there are 2 players, the Democratic Party and the Republican Party. Each player has 3 decisions to make; they can nominate 1, 2, or 3 candidates to run for election. The payoff of this game are the number of candidates who win the election.
In the paper, Goldberg states that voters practically never voted across party lines. If there was only one nominated candidate for the Democratic Party, all Democratic voters would give that candidate all 3 of their votes. If there were two Democratic nominees, Democratic voters would give one candidate 2 votes, and the other 1 vote randomly, giving each an average of 1.5 votes. Of course, if there were 3 Democratic candidates, Democratic voters gave 1 vote to each candidate. For his analysis, Goldberg assumes that all voters are either Democratic or Republican, and that they all display the same loyalty to their party described above. In addition, he assumes both parties know the percentage of voters who are members of their own party, and that both parties nominate their candidates simultaneously.
For this game, the Nash Equilibrium of decisions is dependent on the percentage party members for each of the two parties. The below picture from Table 1 of the article shows the payoff table for both parties if 62% of voters are members of the Democratic Party, and the other 38% of voters are members of the Republican Party.
In this example, we can find multiple states in Nash Equilibrium. The state of (2, 1), where the Democratic Party nominates 2 candidates and the Republican Party nominates 1, is in Nash Equilibrium. In this scenario, the Democrats will not gain anything by nominating 3 candidates, as they will still only win 2 seats, and if they nominate only 1 candidate they obviously only win 1 seat. The Republican Party will gain only one seat no matter how many candidates they nominate, meaning they have no incentive to switch as well. Similarly, (3, 1) is also in Nash Equilibrium. The Democratic Party will win the same number of seats if it nominates 2 candidates, and will win only 1 seat if it nominates 1. The Republican Party will not win any seats if they nominate more than one candidate.
While there is no dominant strategy for the Republican Party, since the best decision depends on the decision made by the Democrats, the Democratic Party certainly has a dominant strategy. If the Democratic Party nominates 3 candidates, they are guaranteed at least 2 members winning seats it the election. In addition to that, they have the possibility of winning 3 seats if the Republicans run more than 1 candidate.