Lecture 19: Subgame Perfect Equilibrium: Matchmaking and Strategic Investments

What this lecture covers

Following the previous lecture's formal introduction of subgame perfect equilibrium (SPE), this lecture works through examples to build intuition and a repeatable method: solve each subgame's Nash equilibrium first, then roll the payoffs back up the tree. It opens with a trust game the class plays live, where some Nash equilibria of the full game turn out to prescribe non-equilibrium behavior in a later subgame, and only one equilibrium survives as subgame perfect.

The second example, a matchmaking game with three players, shows that a game can have multiple subgame perfect equilibria, including one built on a mixed-strategy equilibrium in the final subgame. The lecture closes with an extended business-strategy case: whether a firm should rent a machine that lowers its costs. Comparing an accountant's answer, an economics answer, and a game-theory answer that accounts for a competitor's reaction shows why ignoring strategic effects, how other players change their behavior in response to your decisions, leads to the wrong conclusion. This lecture sits between the formal definition of SPE and the course's move into repeated games and infinite-horizon interactions.

Key ideas

• Solving by subgames: to find a subgame perfect equilibrium, solve the Nash equilibrium of the last subgame first, then roll that payoff back and repeat at each earlier subgame, working toward the front of the tree, just like backward induction.
• Non-credible equilibria persist in a matrix: converting a tree to a matrix can produce Nash equilibria that instruct a player to act against their own interest in a subgame that would never actually be reached under sensible play; these are eliminated by requiring Nash play in every subgame.
• A game can have several subgame perfect equilibria: whenever a subgame itself has multiple Nash equilibria (including mixed-strategy ones), each one can generate a distinct SPE of the whole game.
• Rolling back mixed-strategy payoffs: an earlier mover (like a matchmaker) can rationally choose not to enter a game if they expect the later players to coordinate on an inefficient mixed-strategy equilibrium, since its expected payoff can be worse than not playing at all.
• Cournot competition recap: firms simultaneously choose quantities; equilibrium quantity is (a - c) / 3b for a linear demand curve P = a - b(qA + qB) and identical marginal cost c.
• The accountant's mistake: evaluating a cost-cutting investment by assuming output stays fixed ignores that lower marginal cost lets the firm profitably produce more.
• The (incomplete) economics answer: accounting for the firm's own output increase (the extra "triangle" of profit from producing more) is still incomplete if it ignores the rival's reaction.
• Strategic substitutes: when one firm's best response shifts so that it wants to produce less in response to a rival producing more, an investment that lets a firm credibly commit to producing more induces the rival to cut back, softening competition and boosting the investor's profit further.
• General lesson on strategic effects: analyzing any policy, investment, or rule change by holding other players' behavior fixed (as in tax policy or curriculum design) misses the value or cost created by others' reactions.

Walkthrough

The "don't screw up" trust game (0:01)

The class plays a three-stage sequential game live, split into Player 1s and Player 2s. Backward induction predicts a specific outcome, but many Player 1s choose the "safe" early exit because they worry Player 2 might not trust them to follow through, and Player 2 might worry the same about Player 1. Converting the tree into a matrix reveals three pure-strategy Nash equilibria, only one of which matches backward induction.

Eliminating non-subgame-perfect equilibria (around 12:19)

Working from the last subgame backward, the lecture shows that two of the three Nash equilibria prescribe behavior that is not a Nash equilibrium within a smaller subgame, so they are eliminated. Only the equilibrium matching backward induction survives as subgame perfect, confirming that SPE generalizes backward induction to settings with imperfect information.

The matchmaker game: multiple SPEs including a mixed one (around 27:02)

A new three-player game has a matchmaker deciding whether to send two students on a date, after which the students simultaneously choose between two classes to meet in, a battle-of-the-sexes-style coordination problem. The lecture solves the final subgame first, finding two pure-strategy equilibria (both classes) and a mixed-strategy equilibrium, then computes what each is worth to the matchmaker. Because the mixed equilibrium yields the couple only a 4/9 chance of meeting, its expected value to the matchmaker is negative, producing a third subgame perfect equilibrium in which the matchmaker rationally chooses not to send them at all.

Reviewing the Cournot baseline (around 50:04)

Before the main application, the lecture recaps the Cournot duopoly model with a linear demand curve and constant marginal cost, computing baseline quantities, prices, and profits for two symmetric firms producing fertilizer, setting up the numbers used in the investment case that follows.

The strategic-investment case: three answers to "should you rent the machine?" (around 54:17)

A firm can rent a machine that lowers its marginal cost but carries a fixed rental cost. The "accountant's answer" assumes output stays fixed and concludes the savings don't cover the cost, so don't rent. The "economics answer" accounts for the firm producing more at the new, lower cost, adding extra profit from that increased output, but the total still falls short of the rental cost. Only the "game theory answer" also accounts for the fact that the rival's best response shifts down in this game of strategic substitutes: with the rival producing less in response, the investing firm's profit rises enough to justify renting the machine.

The general lesson on strategic effects (around 1:14:13)

The lecture closes by generalizing the mistake in the accounting and economics answers: any analysis that treats other players' behavior as fixed while your own decision changes is missing strategic effects. It connects this to real examples like tax policy design and curriculum reform, where ignoring how people change their behavior in response to new rules leads to systematically wrong conclusions.

Before you watch

• Watch the previous lecture on information sets, subgames, and the formal definition of subgame perfect equilibrium; this lecture assumes that framework.
• Be comfortable finding Nash equilibria, including mixed-strategy equilibria, in simple payoff matrices such as battle of the sexes.
• Recall the Cournot duopoly model (best responses, equilibrium quantity formula) from earlier in the course, since it is reused without full re-derivation.

Check your understanding

1. In the "don't screw up" game, why do two of the three Nash equilibria fail to be subgame perfect, and what specifically makes the surviving equilibrium different?
2. Why does the matchmaker choose not to send the couple on a date in one of the subgame perfect equilibria, even though sending them is profitable in the other two?
3. What is wrong with the accountant's method of evaluating the machine-rental decision, and what does the "economics answer" add that the accountant's answer misses?
4. Why does the rival firm reduce its output when the investing firm's costs fall, and why does this make the investment more attractive than the economics answer alone suggests?
5. Give an example, other than the ones in the lecture, of a policy or business decision where ignoring strategic effects (other players changing their behavior) would lead to a mistaken analysis.