Lecture 3: Iterative Deletion and the Median-Voter Theorem

What this lecture covers

This lecture applies the iterative deletion technique from Lecture 2 to a concrete political model: two candidates choosing positions on a left-right spectrum to maximize vote share. It answers the question of what iterative deletion predicts for elections, why that prediction is called the Median Voter Theorem, and where the underlying model breaks down against real-world politics.

The second half of the lecture shifts to a new tool. When a game has no dominated strategies at all, the class needs another way to reason about it, and Polak introduces best response: the strategy that maximizes your payoff given a belief about what the opponent will do. By the end you can compute expected payoffs against a mixed belief about an opponent's play and read off best responses from a graph of expected payoff against belief.

Key ideas

• Spatial voting model: two candidates pick a position among ten evenly spaced points on a political spectrum; voters back the closest candidate, and each candidate wants to maximize vote share.
• Median Voter Theorem: iterative deletion of dominated strategies in this model eliminates every position except the two closest to the center, predicting that competing candidates converge toward the middle.
• Model realism check: uneven voter distribution barely changes the prediction, but multiple candidates, abstention, multiple election rounds, more than one policy dimension, and the inability of candidates to credibly commit to a position all can change the outcome.
• Best response: the strategy that gives the highest payoff given a specific belief (or probability distribution) about what the other player will do.
• Expected payoff: the payoff a strategy yields on average, weighted by the probabilities you assign to each of the opponent's possible choices.
• Rationalizing a choice with beliefs: even a strategy that is not always best, and does not dominate, can be the correct choice under some belief about the opponent, which is why "no dominated strategy" does not mean "no way to decide."

Walkthrough

Recap: iterative deletion as a method (0:01)

Polak recaps iterative deletion of dominated strategies from the previous lecture: identify all dominated strategies for all players, delete them, then repeat on the reduced game. He advises identifying every dominated strategy in a round before deleting, to avoid mistakes.

Setting up the two-candidate spectrum game (2:03)

Two candidates choose one of ten positions on a political spectrum, with voters evenly spread across the positions and voting for whichever candidate is closest (splitting evenly on ties). Payoffs are each candidate's share of the vote.

Showing position 1 is dominated (7:05)

Working through vote shares against every possible opponent position, the class confirms that position 2 strictly dominates position 1 by a consistent 5% margin, and by symmetry position 9 dominates position 10.

Iterating the deletion down to the center (15:26)

Once positions 1 and 10 are deleted, the same argument shows position 3 dominates position 2 (and 8 dominates 9) in the reduced game. Repeating this process eliminates every position except 5 and 6, which tie against each other, giving the Median Voter Theorem's prediction that candidates crowd toward the center.

Testing the model against reality (24:36)

Polak checks the prediction against historical U.S. and U.K. elections, then leads a discussion of what the model omits: uneven voter distributions (which turns out not to matter much), multiple candidates and abstention, primaries, multiple policy dimensions, and candidates' difficulty credibly committing to a stated position. He frames model-building as a cycle of abstracting, testing intuitions, then enriching the model to see what changes.

Introducing best response with a new example (40:00)

A new three-by-two game has no dominated strategies for either player, showing the limits of dominance reasoning. Polak asks what a rational player should do, and shows that each of the three available strategies for Player I can be justified as a best response to some belief about which strategy Player II will choose.

Expected payoffs and the best-response graph (46:08)

The lecture computes expected payoffs for each of Player I's strategies under different beliefs about the probability the opponent chooses Right, then plots these as straight lines against that probability. The upper envelope of the three lines shows exactly which strategy is the best response for any given belief, splitting the probability axis into regions favoring Up, Down, or Middle.

Before you watch

• Watch Lectures 1 and 2 first, since this lecture builds directly on strict dominance and iterative deletion of dominated strategies without re-explaining them.
• Basic comfort with percentages and solving one linear equation for one unknown will help with the vote-share calculations and the best-response graph.

Check your understanding

1. Why does position 2 strictly dominate position 1 in the spatial voting model, and why doesn't that same argument immediately rule out position 3?
2. What is the Median Voter Theorem's prediction, and through how many rounds of deletion is it reached in the ten-position model?
3. Which assumption behind the model turned out not to matter much when relaxed, and which ones did?
4. In the best-response example, why can three different strategies each be a justified choice depending on the player's belief about the opponent?
5. How does plotting expected payoff against the probability of the opponent's choice let you find the best response for any belief?