Lecture 20: Subgame Perfect Equilibrium: Wars of Attrition

What this lecture covers

This lecture uses a single extended example, the war of attrition, to show that costly, drawn-out conflicts can arise even among fully rational players with no pride, reputation, or irrationality involved. The class plays a live game where two players simultaneously choose to fight or quit each period; the first to quit gets nothing while the other wins a prize, and every period both fight costs both players. Real classroom rounds show fights persisting well past the point where losses exceed the prize, motivating the question the rest of the lecture answers formally.

Building on the previous lectures' method for subgame perfect equilibrium, the lecture solves a two-period version of the game, first for pure-strategy equilibria (which turn out to involve no real fighting) and then for a mixed-strategy equilibrium, which does produce fighting with positive probability. It then extends the same continuation-value technique to an infinite-horizon version of the game, showing wars of attrition can persist indefinitely, with a random, geometrically-shaped duration. This closes the course's run of subgame perfect equilibrium examples and previews the shift toward genuinely infinite games.

Key ideas

• War of attrition: two players choose fight or quit each period; whoever quits first gets nothing while the rival gets the prize V; every period both fight costs each player C; if both quit simultaneously, both get 0.
• Real-world wars of attrition: trench warfare in World War I, prolonged satellite-television competition between rival broadcasters, and bribery contests (all-pay auctions) all share this structure of accumulating losses in pursuit of an uncertain, possibly modest prize.
• Sunk costs are strategically irrelevant: costs already paid in earlier rounds cannot be recovered and do not affect the best choice going forward, so the decision problem looks the same in every period regardless of how much has already been spent.
• Pure-strategy subgame perfect equilibria involve no real fight: solving the two-period game by backward induction finds equilibria where one player always fights and the other always quits immediately, producing no costly conflict at all.
• Mixed-strategy equilibrium produces genuine fighting: each player fights with probability P* = V / (V + C) in every period, which is exactly the probability that makes the other player indifferent between fighting and quitting.
• Continuation values: the payoff from playing the equilibrium strategy in future periods, rolled back into the current period's payoff matrix in place of a stage-by-stage tree.
• Expected payoff in the mixed equilibrium is zero: because a player can always guarantee 0 by quitting, the mixed equilibrium's payoff from fighting must also equal 0 in expectation.
• Extending to infinite horizon: since a subgame beginning at any future stage looks identical to the original game once sunk costs are dropped and continuation values are computed, the same mixed strategy P* in every period is a subgame perfect equilibrium of the game with no fixed end.
• Comparative statics: the probability of fighting in equilibrium rises with the prize V and falls with the cost of fighting C, and the probability that the conflict has ended by any given period increases with time, meaning long, costly wars are possible but become progressively less likely the longer they persist.

Walkthrough

Playing the war of attrition live (0:01)

The professor sets up the fight-or-quit game with a small real cash prize and cost, then runs several rounds between student volunteers. Most pairs quit quickly, but one determined pair keeps fighting round after round, accumulating losses that exceed the prize. The class discusses possible explanations: pride, a desire to win for its own sake, reputation concerns, and the sunk-cost logic that losses already paid do not change the incentive to keep fighting.

Real-world examples and naming the phenomenon (around 17:40)

The lecture connects the classroom game to real wars of attrition: the prolonged trench warfare of World War I over small territorial gains, the costly competition between rival European satellite broadcasters, and bribery contests (all-pay auctions) where money paid is never recovered regardless of outcome. The goal stated here is to show that long, costly fights can occur even without irrationality or reputation effects.

Setting up and solving the two-period game (around 24:53)

A tree is built for a two-period version of the game, with the assumption V > C (as played in class). The lecture solves the final-period subgame first, finding two pure-strategy Nash equilibria (one player always fights, the other always quits), then rolls those continuation payoffs back into the first period. This produces two pure-strategy subgame perfect equilibria overall, but both predict a fight that ends immediately, since one player never fights at all.

Finding the mixed-strategy equilibrium and extending it across periods (around 50:41)

Since the pure-strategy equilibria fail to explain observed fighting, the lecture solves for the mixed-strategy equilibrium of the last subgame, finding each player fights with probability P* = V / (V + C), which yields an expected continuation payoff of zero. Rolling this zero continuation value back into the first period produces the same matrix and the same mixed-strategy equilibrium at the start of the game, so both players fight with probability P* in every period, giving a subgame perfect equilibrium in which real, costly fighting occurs with positive probability.

Extending to an infinite-horizon war of attrition (around 1:07:28)

The lecture argues that any arbitrary future stage of an infinite game looks structurally identical to the original one-period problem, since past costs are sunk and future play can be summarized by a continuation value. Because the mixed-strategy equilibrium's continuation value is always zero, the same probability P* of fighting each period remains an equilibrium indefinitely, meaning the game can in principle continue forever with no fixed endpoint.

Comparative statics and closing prediction (around 1:12:44)

The lecture closes by noting that the probability of a fight occurring rises with the prize V and falls with the cost C, matching intuition. It also derives that the probability of the war continuing to a given period shrinks geometrically over time, so very long, very costly wars of attrition remain possible in equilibrium but become increasingly unlikely the longer they run.

Before you watch

• Review the previous two lectures on subgame perfect equilibrium and the standard method of solving the last subgame first and rolling payoffs back.
• Be comfortable finding mixed-strategy Nash equilibria by setting one player's expected payoffs equal to make the other player indifferent, as covered earlier in the course (for example, in the Hawk-Dove game).
• Understanding sunk costs, and why they should not affect forward-looking decisions, will help make sense of why past fighting costs do not change equilibrium behavior.

Check your understanding

1. Why do the pure-strategy subgame perfect equilibria of the two-period war of attrition never actually involve both players fighting for a costly period?
2. How is the mixed-strategy probability P* = V / (V + C) derived, and why does it make the other player indifferent between fighting and quitting?
3. Why is the expected payoff from the mixed-strategy equilibrium equal to zero, and why does that matter for extending the analysis to more periods?
4. How does the concept of a continuation value let the lecture analyze a game that could in principle go on forever?
5. How does the equilibrium probability of a fight change as the prize V increases or the cost of fighting C increases, and why does that match intuition?