Problem 212 - An Introduction to Game Theory

Question

Assume that there are \(n \geq 3\) candidates and \(q\) citizens, where \(q \geq 2 n\) is odd (so that the median of the voters' favorite positions is well-defined) and divisible by \(n\). Show that the game has a subgame perfect equilibrium in which no citizen's strategy is weakly dominated and every candidate enters the race and chooses the median of the citizens' favorite positions. (You may use the fact that every voting subgame has a (pure) Nash equilibrium in which no citizen's action is weakly dominated.)

Step-by-Step Solution

Answer

Each candidate chooses the median position, achieving a subgame perfect equilibrium with no weakly dominated strategies.

Step 1 - Understand the Problem

The problem involves proving the existence of a subgame perfect equilibrium (SPE) in a voting game with specific parameters: there are at least 3 candidates ( n ≥ 3), an odd number of citizens ( q ≥ 2 n ), and q is divisible by n . The key point is to demonstrate that each candidate chooses the median of the citizens' favorite positions.

Step 2 - Median of Citizens' Favorite Positions

Because q is odd, the median voter is well-defined. Suppose the citizens rank their preferred positions from leftmost to rightmost. The median voter is the one who is exactly in the middle of this ranking.

Step 3 - Game Players

There are two types of players in this game: the candidates and the citizens. Candidates choose their positions first, and then citizens vote for their preferred candidate. Our goal is to find an equilibrium where all candidates enter the race and choose the median position.

Step 4 - Candidate's Strategy

Each candidate needs to select a position such that none of the citizens' strategies are weakly dominated. A candidate should choose a position that maximizes their chance of winning, and given that citizens vote based on their preferred positions, the best strategy for candidates is to choose the median position.

Step 5 - Nash Equilibrium in Voting Subgame

Given that the voting subgame has a pure Nash equilibrium where no citizen's strategy is weakly dominated, we can ensure that each citizen votes for a candidate whose position is closest to their favorite position.

Step 6 - Subgame Perfect Equilibrium

By subgame perfection, each candidate anticipates the voting outcome and therefore chooses a strategy that results in securing the majority of votes. Since choosing the median position attracts the most potential votes, it ensures that no citizen’s voting strategy is weakly dominated and encourages every candidate to enter the race and choose the median position.

Step 7 - Conclusion

In conclusion, because the median position is the optimal choice for all candidates, they all select the median position. This choice simultaneously ensures no citizen’s strategy is weakly dominated and all candidates choose to enter the race. Hence, a subgame perfect equilibrium is achieved where all candidates choose the median position.

Key Concepts

Median Voter Theorem

The median voter theorem is a fundamental concept in voting game theory. It suggests that in a simple majority rule voting system, the candidate whose position is closest to the median voter's preference will win. This theorem relies on the idea that voters' preferences are distributed along a continuum, such as a left-to-right political spectrum. In our problem, since the number of voters ( the odd, the median is precisely defined and is the voter whose preferences lie in the middle. Candidates position themselves at this median to maximize votes since this strategy is a strong attractor for the majority of voters.

This theorem provides insight into why candidates in a two-party system often appear to adopt moderate positions: they aim to capture the median voter's support to secure victory.

Nash Equilibrium

Nash equilibrium is a key concept in game theory. It describes a situation in which each player, aware of the other players' strategies, chooses a strategy that maximizes their own payoff, assuming the others' strategies remain constant. In our exercise, each candidate chooses their position in anticipation of the strategies of other candidates and the voting behavior of citizens.

Given that citizens vote for the candidate closest to their own preferences, each candidate must consider this when choosing their position. The equilibrium occurs when all candidates choose the median position since this choice attracts the most votes from the citizens, assuming the others are also targeting the median. This creates a stable state where no candidate can improve their outcome by unilaterally changing their position.

Voting Game Theory

Voting game theory explores the strategies that candidates and voters use in elections. It examines how various rules and behaviors impact the outcomes of elections. In our scenario, voting game theory helps us understand why candidates choose the median position and how citizens' votes influence this decision.

The candidates first choose their positions, followed by citizens voting for their preferred candidate. The candidates' primary goal is to win by attracting the most votes. Given that citizens prefer candidates whose positions are closest to their favorite positions, it becomes a strategic game where each candidate positions themselves at the median to gain the majority support.

This interaction between candidates' strategies and voters' preferences is at the heart of voting game theory and highlights the connection between strategic behavior and electoral outcomes.

Strategic Behavior

Strategic behavior in voting games refers to the actions candidates and voters take to maximize their respective interests. Candidates act strategically by choosing positions that will attract the most votes, while voters act strategically by casting their votes in a way that aligns with their preferences.

In this exercise, candidates each choose the median position because this is the strategy that maximizes their chance of winning, knowing that citizens will vote for the candidate closest to their preferred position. This ensures a subgame perfect equilibrium, where every candidate choosing the median position means no citizen's strategy is weakly dominated.

This concept illustrates how strategic behavior shapes the choices made by both candidates and voters, leading to equilibrium where all participants act in their best interest given the actions of others.