Lecture 8: Location, Segregation, and Mixed Strategies

What this lecture covers

The lecture briefly finishes the candidate-voter model from last class, confirming that equilibria need not crowd candidates at the center, that entry on one wing can hand victory to the other wing, and that candidates can be no more than a bounded distance apart before a center candidate profitably enters. It then moves to a new topic: a live in-class location game (based on Thomas Schelling's work) where students choose between two towns, showing how individually reasonable preferences for a "mixed" neighborhood can still produce class-wide segregation.

The second half draws out the broader lessons - segregated equilibria are stable while an exactly integrated equilibrium is fragile, seemingly small modeling details can create equilibria that don't reflect reality, and centralized or individual randomization can restore integration. This sets up the final topic: since rock-paper-scissors has no Nash equilibrium in the strategies used so far, the lecture introduces mixed strategies, where players randomize over their choices, and shows that each player randomizing 1/3-1/3-1/3 is the game's Nash equilibrium.

Key ideas

• Candidates need not crowd the center: unlike the classic median-voter model, in the candidate-voter model equilibria can have two candidates positioned symmetrically away from the center.
• Entry can backfire: a candidate entering on one wing can split votes on that side and hand the election to the opposing wing, illustrated with real third-party election examples.
• Bounded spread: in the class's example, two candidates cannot be positioned beyond roughly the 1/6 and 5/6 marks of the spectrum without inviting a winning center entrant.
• Segregation from individual choice: even when everyone prefers a perfectly mixed neighborhood over being in the majority, uncoordinated individual choices can converge to full segregation.
• Strict versus weak equilibria and stability: the segregated outcomes are strict Nash equilibria and stable to small perturbations; the exactly integrated outcome is only weakly stable and unravels if disturbed even slightly.
• Tipping point: as in the earlier investment game, this location game has a threshold beyond which the system moves decisively toward one stable equilibrium or the other.
• Observed behavior doesn't reveal preferences: seeing segregation across many societies does not prove people prefer segregation - individually rational choices can aggregate into an outcome nobody actually wants.
• Mixed strategy: a randomization over a player's pure strategies; needed because some games, like rock-paper-scissors, have no Nash equilibrium among pure strategies alone.

Walkthrough

Wrapping up the candidate-voter model (0:01)

The lecture uses another row of students to re-demonstrate two equilibrium properties from the previous class: two symmetric off-center candidates can form an equilibrium, and a third candidate entering on one wing can cause the opposite wing to win outright. It then works out precisely how far apart two equilibrium candidates can be, showing with a sixths-based argument that candidates positioned just inside the 1/6 and 5/6 marks of the spectrum deter a winning center entrant, while candidates further apart invite one. The section closes by generalizing the lesson that guess-and-check, if done systematically (checking every type of possible deviation, including dropping out), is an effective method for finding equilibria.

Setting up the location and segregation game (14:22)

A new game is introduced with two towns (East and West) and two types of people (tall and short), each type preferring an evenly mixed town, but preferring to be in the majority over being in the minority if a town isn't mixed. Students are split into tall and short groups, given a deliberately slightly uneven starting distribution across the two towns, and repeatedly choose (by simultaneous show of hands) which town to move to.

Watching segregation emerge (roughly 22:28)

Despite the class's preference for mixed towns, repeated rounds of simultaneous choice quickly drive the room toward nearly complete segregation by height. The lecture attributes this to the deliberately imbalanced starting point: once a town skews even slightly toward one type, minority members in the other town increasingly prefer to switch, accelerating the drift toward full segregation. Students identify three equilibria: two segregated equilibria (each strict, since deviating strictly lowers payoff) and one exactly-integrated equilibrium (only weakly stable, since a party is indifferent about which town to live in).

Stability, tipping points, and irrelevant details (roughly 32:38)

The lecture explains why the segregated equilibria are stable - a small perturbation returns to segregation - while the integrated equilibrium is unstable, since even a tiny imbalance triggers a cascade back toward segregation, echoing the tipping-point idea introduced with the earlier investment game. It also surfaces a fourth, seemingly artificial equilibrium: everyone choosing the same town and being randomly reassigned by the town's capacity rule. This is used to illustrate that incidental modeling details, added only to make the game well-defined, can generate equilibria unrelated to any real-world mechanism.

Policy implications: bussing and randomized housing (roughly 52:00)

The lecture connects the model to real segregation debates, including school bussing policy in the 1960s and 70s and Harvard's move to randomize dormitory ("house") assignment after self-selection produced socially skewed houses. It stresses that observing segregation in many societies does not prove people prefer segregation; individually rational, majority-avoiding choices can still aggregate into a segregated outcome nobody actually wants, a version of Thomas Schelling's central insight.

Individual randomization and the road to mixed strategies (roughly 58:10)

The lecture distinguishes centralized randomization (a government randomly assigning residents) from individual randomization (each person independently flipping a coin to choose a town), showing the latter is also a Nash equilibrium once the town shares are equal, since a player is indifferent between the two options. This motivates a new class of strategy: a mixed strategy is a randomization over a player's existing ("pure") strategies.

Rock-paper-scissors and finding the mixed-strategy equilibrium (roughly 1:02:00)

Rock-paper-scissors is shown to have a payoff matrix with no pure-strategy Nash equilibrium, since best responses cycle indefinitely (paper beats rock, scissors beats paper, rock beats scissors). The lecture then verifies that both players randomizing equally, 1/3 each over rock, paper, and scissors, is a Nash equilibrium: against a 1/3-1/3-1/3 opponent, every pure strategy yields an expected payoff of exactly zero, so no deviation is strictly better, making the mixed strategy a (weak) best response for both players.

Before you watch

• Review the candidate-voter model setup from the previous lecture, since the opening section assumes familiarity with its rules and payoffs.
• Recall the investment game's two equilibria and its tipping point, referenced directly when discussing the segregation game.
• Be comfortable with expected value calculations, used to verify the rock-paper-scissors mixed-strategy equilibrium.

Check your understanding

1. Why can two candidates in the candidate-voter model sit apart from the center in equilibrium, yet not be positioned at the extremes?
2. Why do the segregated equilibria in the location game count as strict Nash equilibria while the integrated equilibrium is only weak?
3. What does it mean for a game to have a "tipping point," and how does the location game illustrate this?
4. Why can't you conclude from observing segregation across many societies that people prefer to be segregated?
5. Why does rock-paper-scissors have no Nash equilibrium in pure strategies, and how does the 1/3-1/3-1/3 mixed strategy solve this?