Lecture 10: Mixed strategies in baseball, dating and paying your taxes

What this lecture covers

This lecture answers a question left open after the tennis example: how do you actually confirm that a mixed strategy is a Nash equilibrium, rather than just constructing one? It then asks a deeper question: what does it really mean for a rational person to randomize, and are there other ways to make sense of a mixed-strategy equilibrium besides literal coin-flipping?

The lecture follows directly from the previous one on mixed strategies and Venus/Serena, and sets up the next lecture's move into evolutionary game theory. After watching, you should be able to verify a candidate mixed-strategy equilibrium by checking pure-strategy deviations, and explain the three interpretations of mixing: literal randomization, beliefs about an opponent's actions, and proportions within a population.

Key ideas

• Verifying equilibrium: to confirm a mix is a best response, check that every pure-strategy deviation earns no more than the mix; since any mixed deviation is a weighted average of pure-strategy payoffs, checking pure strategies is enough.
• Randomization in sports: play-calling in football and pitch selection in baseball behave like mixed strategies, because a predictable pattern would let the opponent exploit it.
• The base-stealing fallacy: if a fast runner's expected return from attempting a steal looks similar to staying put, that is exactly what equilibrium predicts, not evidence that speed doesn't matter; the value shows up in the pitcher's altered behavior, visible in the next batter's statistics.
• Interpretation 1, literal randomization: airport security concealing where checks happen, or a wartime commander who randomizes convoy protection, are conscious attempts to be unpredictable.
• Interpretation 2, beliefs: in the "hapless dating couple" version of Battle of the Sexes, a player's mixed strategy can represent the other player's belief about what they'll do, not literal randomizing.
• Interpretation 3, population proportions: in the tax-compliance game, the taxpayer's mixing probability is better read as the fraction of a large population who cheat, while the auditor's probability really is a literal audit rate.
• Comparative statics in the tax game: raising the penalty for cheating changes the audit rate but not the compliance rate, because a player's own payoffs determine the opponent's equilibrium mix, not their own.

Walkthrough

Verifying the tennis equilibrium (0:01)

The lecture returns to the Venus/Serena passing-shot game and checks, step by step, that Venus's mix (0.7, 0.3) is really a best response to Serena's mix (0.6, 0.4). It computes Venus's expected payoff from playing pure left, pure right, and her actual mix, and finds all three equal roughly 0.62. Since no pure-strategy deviation is strictly better, and any mixed deviation must average between the pure-strategy payoffs, there is no profitable deviation at all — confirming the equilibrium.

Mixed strategies in sports (12:49)

Students suggest run-versus-pass calling in football and pitch selection in baseball as real-world mixed strategies. The lecture uses the "should a fast player steal bases" debate to show a reasoning trap: analysts who find that attempting to steal yields about the same expected return as not attempting it wrongly conclude speed is overrated. In a mixed-strategy equilibrium, that near-equality is exactly what theory predicts — the real value of a fast runner shows up in the pitcher throwing more fastballs and pitching from the stretch, which raises the batting average of the next hitter.

Literal randomization: security and the military (23:41)

The lecture applies the same logic to airport security after 9/11: concealing which baggage-screening boxes are active, or randomly selecting passengers for checks, makes attacks harder to plan even without eliminating them. A historical anecdote about a British commander in an insurgency who used a coin-flip-like method to decide which convoys to protect illustrates that literal randomization is sometimes the only reliable way for people to be unpredictable.

Beliefs: the dating couple revisited (28:17)

Returning to a Battle of the Sexes-style dating game (apple picking versus a play), the lecture finds the mixed equilibrium: Nina mixes (2/3, 1/3) and David mixes (1/3, 2/3), each favoring their own preferred activity, and their expected payoff is only 2/3 — worse than either pure-strategy equilibrium, because they end up meeting only 4/9 of the time. Since it seems implausible the couple is literally flipping coins, the lecture proposes reading each player's mix as the other player's belief about what they'll do, formed so that both remain indifferent between their options.

Proportions in a population: tax compliance (47:04)

A taxpayer-versus-auditor game (pay honestly or cheat; audit or not) has no pure-strategy equilibrium. Solving it the same way as before gives an equilibrium where two-thirds of taxpayers pay honestly and the auditor audits with probability 2/7. Here a third interpretation fits best: the taxpayer's mix is not one person randomizing but the proportion of a large population who comply, while the auditor's probability really is a literal audit rate (auditors are in fact required to randomize).

Policy experiment: raising the penalty for cheating (59:52)

Doubling the fine for getting caught cheating (from -10 to -20) is re-solved in the same framework. Because the taxpayers' equilibrium compliance rate is pinned down by the auditor's payoffs, which haven't changed, compliance stays at 2/3 — but the audit rate the taxpayers' payoffs pin down falls from 2/7 to 1/6. The lecture generalizes this: to change compliance, you must change the auditor's incentives (for example, making audits less costly or more rewarded), not just the penalty on cheaters. It also notes that richer taxpayers, who gain more from undetected cheating, get audited more often in equilibrium without necessarily cheating more.

Before you watch

• Have the tennis passing-shot example and the technique of solving for a mix by equating an opponent's pure-strategy payoffs fresh from the previous lecture in this course.
• Recall the original Battle of the Sexes coordination game and its pure-strategy equilibria from earlier in the course.
• Basic comfort with solving simple linear equations is used throughout.

Check your understanding

1. Why is checking pure-strategy deviations enough to verify a mixed-strategy Nash equilibrium, without checking every possible mixed deviation?
2. What is the flaw in concluding that base-stealing speed is "overrated" because the expected return to attempting a steal roughly equals the return to not attempting one?
3. In the dating game, why do Nina and David only meet 4/9 of the time in the mixed equilibrium, and how does that compare to the pure-strategy equilibria?
4. What is the difference between reading a mixed strategy as literal randomization, as a belief, and as a population proportion? Give an example of each from the lecture.
5. Why does raising the fine for tax cheating lower the equilibrium audit rate instead of raising the compliance rate?