There are d democrats and r republicans voting in the US Presidential election. Each person decides whether to vote or not vote. The voters get a payoff of 2, 1, 0 if their preferred candidate wins, ties, or loses (respectively). Voters that vote incur a cost of 0.
In analyzing the voting behavior of Democrats and Republicans in U.S. presidential elections, it is insightful to model the situation as a strategic game where individual voters choose whether to participate or abstain, considering potential payoffs based on election outcomes. This paper explores the structure and equilibria of such a game under varying assumptions about party sizes, specifically when the numbers of party supporters are equal and small (d = r = 1) and when they are large and equal (d = r > 1).
When each party has exactly one supporter, the game simplifies to a two-player, two-strategy game where each player—representing a supporter—can choose to vote or abstain. The players are the Democrat and Republican supporter, each with preferences aligned with their party. The possible strategies are Voting (V) or Not Voting (NV).
Payoff structure:
If a supporter votes and their preferred candidate wins, they receive a payoff of 2.
If there is a tie, supporters receive 1.
If their candidate loses, the payoff is 0.
Voting incurs no cost for supporters, so the only considerations are the potential benefits based on election outcomes.
Constructing the normal form game table, the strategies are:
Repub Exists: V
Repub Exists: NV
Democrat: V
(2,2) — both vote; candidate with majority wins
(2,1) — Democrat votes; Republican abstains, Democrat wins
Democrat: NV
(1,2) — Democrat abstains; Republican votes; Republican wins
(0,1) — both abstain; no one votes; tie
The table reflects the possible outcomes, assuming voters vote sincerely according to their party preferences. Because voters want their preferred candidate to win, they will have incentives to vote if their vote can influence the outcome favorably. The game resembles a coordination game where both voters prefer to vote if they believe their vote can impact the result.
This game can be modeled as a coordination game with strategic complementarities. Analyzing the best responses, the Nash equilibrium occurs where both voters vote, as each prefers to avoid a situation where the other votes and they do not—leading to the candidate winning by a narrow majority or tie. Specifically, the strategy profile where both vote (V, V) forms a Nash equilibrium, because neither voter would prefer to unilaterally abstain when their vote could decide the outcome.
Thus, the unique pure-strategy Nash equilibrium in this case is both voters voting. This outcome aligns with the classic theory of voting where every supporter votes when individual votes can influence the outcome, especially in small electoral contests.
When the number of supporters per party increases beyond one, the analysis must consider the strategic incentives of a large electorate. Here, the game becomes a larger, symmetric voting game where each supporter decides independently whether to vote, recognizing that aggregate voting behavior determines the outcome.
Formally, the game involves:
Two large, symmetric groups, each with d = r supporters
Each supporter chooses to vote or abstain, with the payoff depending on the election result and individual influence
Possible Nash equilibria depend on the size of the groups and the expected benefits of voting. The key considerations are:
Everyone votes equilibrium:
When both groups find it optimal for each member to vote, considering that turnout might sway the election, the outcome is a profile where every supporter votes. This is likely when the expected benefit of influencing the result outweighs any cost of voting (here, assumed zero).
Partial turnout equilibria:
When only a subset of supporters votes, especially in tight elections, some supporters abstain because their individual vote is less likely to influence the outcome nor worth the effort.
Tie equilibria:
When each side expects that only a small turnout can lead to a tie, supporters might abstain or vote with lower probability, leading to mixed-strategy equilibria.
Winner by one vote equilibria:
These may occur in scenarios where the strategic incentive to sway the election is strong enough that supporters participate only in close margins.
Large margins (win by 2 or more votes):
Such outcomes are less likely as equilibrium predictions because individual votes have minimal impact, discouraging turnout.
The equilibrium set depends on the the margin where voters are indifferent. When the expected vote benefit exceeds the cost (which is zero here), the strongest candidate equilibrium involves full turnout (everyone votes). This aligns with the classic logic of rational voting, where rational supporters vote if their individual vote can determine the result.
In the case of symmetric large populations, theorems from social choice theory and game theory suggest that:
Full turnout (everyone votes) is a Nash equilibrium, especially in close elections.
Mixed-strategy equilibria, where supporters vote with some probability less than one, also exist, particularly when voters are uncertain about others' turnout and question whether their vote will matter.
Equilibria where only one side votes or neither votes are unstable unless under constraints like high voting costs, which are not assumed here.
Conclusion
The analysis demonstrates that in small electorates (d = r = 1), the pure-strategy equilibrium is both supporters voting, aligning with strategic voting theory. For larger, symmetric populations (d = r > 1), equilibrium configurations include full turnout, partial turnout, and mixed strategies, with full turnout being most plausible when voters believe their vote can decisively impact the outcome. These findings underscore the importance of voter influence calculus and strategic considerations underlying turnout behavior across different electoral contexts.
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