Lecture 5: Nash Equilibrium, Bad Fashion, and Bank Runs

What this lecture covers

The lecture gives a formal definition of Nash equilibrium and answers a practical question: how do you actually find one in a game? It builds on the informal treatment from the previous class and works through several small two-player games by comparing best responses row by row and column by column, using "guess and check" as the general method.

The second half turns to a live class experiment, the Investment Game, which has many players but only two strategies each. It uses this to show that a game can have multiple Nash equilibria of very different quality, and that everyday coordination problems - bank runs, fashion trends, choice of meeting spots, technology standards - share this same structure. By the end, you can identify a coordination game, explain why communication (unlike in a Prisoner's Dilemma) can resolve it, and connect Nash equilibrium to the earlier ideas of dominance and best response.

Key ideas

• Formal definition: a strategy profile is a Nash equilibrium if every player's strategy is a best response to everyone else's strategies simultaneously.
• No regret: at a Nash equilibrium, no player would want to change their action after learning what everyone else did.
• Self-fulfilling beliefs: if everyone believes everyone else will play their part of a Nash equilibrium, then playing that equilibrium is indeed each person's best response.
• Guess and check: with many players but few strategies, the practical way to find equilibria is to propose a candidate outcome and verify no one wants to deviate.
• Strict dominance and Nash equilibrium: a strictly dominated strategy can never be played in a Nash equilibrium, because it is never a best response to anything.
• Weak dominance is fragile: unlike strict dominance, a weakly dominated strategy can still appear in a Nash equilibrium, which can produce equilibria that feel implausible.
• Coordination game: a game with multiple equilibria where players want to match each other's choice, and can get stuck at a bad equilibrium even though a better one exists.
• Pareto dominance between equilibria: one Nash equilibrium can be strictly better for everyone than another, yet play can still converge to the worse one.

Walkthrough

Defining Nash equilibrium and why it matters (0:01)

The lecture opens with the formal statement: a profile is a Nash equilibrium when each player's strategy is a best response to the others' strategies. It stresses that people do not always play Nash equilibrium - the earlier "guess a number" game converged toward it over repeated play but the one-shot outcome was not an equilibrium. Three motivations are given for studying it anyway: no regret (holding others' actions fixed, no one wishes they had done something else), self-fulfilling beliefs (if everyone expects the equilibrium, everyone's best response is to play it), and, later, convergence through repeated play. These ideas are tied back to last class's partnership game, where the equilibrium sits where the two best-response lines cross.

Finding equilibria in simple games (9:17)

Working through two 3x3 payoff-matrix games, the class finds each player's best response to each of the opponent's choices, marking them with circles and squares, and identifies the equilibrium as the cell where both markers coincide. In the first game, every strategy turns out to be a best response to something, so the dominance and best-response-elimination tools from earlier weeks would not have narrowed the game down at all - Nash equilibrium gives a sharper prediction. A brief digression shows that a "rational" player could still choose a non-equilibrium strategy by chaining beliefs about beliefs indefinitely, which is why equilibrium requires more than just rationality.

Nash equilibrium versus dominance (23:32)

Returning to the Prisoner's Dilemma, the lecture confirms that the Nash equilibrium coincides with the outcome from deleting strictly dominated strategies. It then proves generally that a strictly dominated strategy can never be part of a Nash equilibrium, since it is never anyone's best response. A simple 2x2 game then shows the limits of this logic for weak dominance: it has two Nash equilibria, one clearly sensible and one that relies only on ties, illustrating that weakly dominated strategies can survive inside an equilibrium.

The Investment Game: multiple equilibria in practice (32:47)

Students choose, without communicating, whether to invest $10 for a $5 net profit that only pays off if at least 90% of the class invests; otherwise investors lose their $10. The game has two Nash equilibria - everyone invests, or no one invests - found by guessing an outcome and checking that no one wants to deviate. Played repeatedly, the class's choices converge toward the "no one invests" equilibrium even though the "everyone invests" equilibrium is Pareto superior, because the first round starts below the 90% threshold and each subsequent round drifts further down. This demonstrates that starting conditions can determine which equilibrium a coordination game settles into, and that a bad but stable outcome is not the same as a Prisoner's Dilemma, since investing is not dominated - it becomes profitable once enough others invest.

Coordination problems across society (53:11)

The class generates real-world examples with the same structure: choosing which party or bar to go to, network goods like Microsoft software or a common HDTV standard, political bandwagons in primaries, listing on a single stock exchange, fashion trends, and bank runs. Each has multiple equilibria where people want to match others' choices, and coordination can fail even when everyone would prefer the better outcome. The bank-run discussion connects to a real 2007 run on the UK bank Northern Rock and to the film It's a Wonderful Life, where a persuasive speech shifts depositors from the bad equilibrium to the good one.

Communication can fix coordination games (1:04:26)

The Investment Game is played a fourth time, but first a student gives a short speech urging classmates to invest. Nearly everyone switches to investing, showing that mere communication - without contracts, payments, or enforcement - can move a group to a better Nash equilibrium. This would not work in a Prisoner's Dilemma, where the same appeal cannot overcome a strictly dominant strategy. The lecture closes by naming this as a further reason to care about Nash equilibrium: it can function as a self-enforcing agreement once people coordinate their expectations.

Before you watch

• Know the definition of best response and how to compute it from a payoff matrix, covered in the prior lecture.
• Be comfortable with the ideas of strictly and weakly dominated strategies from earlier in the course.
• Recall the Prisoner's Dilemma payoff structure, since it is used repeatedly for contrast.

Check your understanding

1. Why can a strictly dominated strategy never appear in a Nash equilibrium, while a weakly dominated one sometimes can?
2. In the Investment Game, explain why both "everyone invests" and "no one invests" are Nash equilibria even though one is clearly better for the group.
3. Why does a persuasive speech change behavior in the Investment Game but would not change behavior in a Prisoner's Dilemma?
4. What does it mean for play to be a "self-fulfilling belief," and how does that relate to the "no regret" property of equilibrium?
5. Give an example of a coordination problem not mentioned in the lecture, and identify its two possible equilibria.