How is game theory used to study voting systems?
The normal-form table illustrates the concept of a saddlepoint, or entry, in a payoff matrix at which the expected gain of each participant (row or column) has the highest guaranteed payoff.
In the context of mathematics, game theory is used to analyze voting systems by examining how strategic interactions and institutional rules influence outcomes. It distinguishes between the formal resources a voter possesses—such as the number of votes or tie-breaking authority—and their actual power to influence results.12
Strategic Voting and Power
Game theory models how voters may act "sophisticatedly" rather than "sincerely." In some scenarios, a player’s apparent advantage, such as a tie-breaking vote, can backfire if other voters adjust their strategies to form implicit coalitions against them. By using the successive elimination of dominated strategies, analysts can identify Nash equilibria to predict likely outcomes in noncooperative voting games.12
Measuring Voting Power
To evaluate systems where voting weight does not equate to actual influence, researchers use power indices rooted in cooperative game theory:
- Banzhaf Value: This index calculates power based on the frequency with which a voter acts as a "critical voter"—someone whose support is necessary for a coalition to win.2
- Applications: These indices have been used to identify "dummy" voters (who have no influence on outcomes) and to analyze the fairness of systems like the U.S. Electoral College, the United Nations Security Council, and various legislative boards.2
These mathematical tools are essential for assessing whether voting systems uphold principles like "one person, one vote" and for designing more equitable decision-making structures.2