Lecture 9: Mixed strategies in theory and tennis

What this lecture covers

This lecture makes mixed strategies rigorous after their informal introduction with rock-paper-scissors. It answers a practical question: if a player is willing to randomize between two or more strategies, what must be true about the payoffs those strategies give? The answer becomes the main tool for actually calculating mixed-strategy Nash equilibria, rather than just defining them.

The lecture sits early in the course's treatment of mixed strategies, right after they were first introduced, and before further applications to sports, dating, and tax compliance in the next lecture. After watching, you should be able to compute a two-player, two-strategy mixed-strategy Nash equilibrium by hand, and explain in words why each player's equilibrium mix depends on the other player's payoffs.

Key ideas

• Mixed strategy: a probability distribution P_i over a player's pure strategies; a pure strategy is the special case where one strategy gets probability 1.
• Expected payoff of a mix: the weighted average of the expected payoffs of the pure strategies in the mix, so it always lies between the best and worst of those payoffs.
• The indifference principle: if a mixed strategy is a best response, every pure strategy played with positive probability in that mix must itself be a best response, so all of them yield the same expected payoff.
• Finding an opponent's mix: to solve for player A's equilibrium mixing probabilities, set player B's expected payoffs from B's own two strategies equal to each other (not A's payoffs).
• No pure-strategy equilibrium: some games (like the Venus/Serena tennis example) have best responses that never coincide, so a mixed-strategy equilibrium is needed instead.
• Direct effect vs. strategic effect: improving one player's payoff from a strategy can push them to use it more (direct effect) while making the opponent target it less (strategic effect), and the two can pull the equilibrium mix in opposite directions.
• Comparative statics: re-solving an equilibrium after changing one payoff shows how the whole mix shifts, including for the player whose payoff didn't change.

Walkthrough

Defining mixed strategies (0:01)

The lecture opens by formalizing the mixed strategy notation P_i(s_i), the probability player i assigns to pure strategy s_i. It notes that a mixed strategy can leave some pure strategies at zero probability, and that assigning probability 1 to a single strategy just recovers a pure strategy — so pure strategies are a special case of mixed strategies.

Expected payoffs and the Battle of the Sexes example (4:14)

Using Battle of the Sexes, the lecture computes the expected payoff of a mixed strategy (1/5, 4/5) against an opponent mix of (1/2, 1/2). It first finds the payoff to each pure strategy against the opponent's mix, then averages them using the mixing weights. The result, 3/5, falls between the payoffs of the two pure strategies, illustrating that a weighted average always lies between its highest and lowest components.

Why every strategy in a best-response mix must be equally good (15:43)

Using examples like picking the tallest subgroup of TAs or students with the highest GPA, the lecture argues that if you're trying to maximize an average and any included item is worse than another available item, you should drop it. Applied to mixed strategies, this means: if a mix is a best response, every pure strategy used in it must itself be a best response, and therefore all must give the same expected payoff. This is formalized as the definition of a mixed-strategy Nash equilibrium, matching the earlier pure-strategy definition with P replacing S.

Setting up the tennis game (27:09)

The lecture introduces a game within tennis: Venus, at the baseline, chooses to hit a passing shot to Serena's left (backhand) or right (forehand); Serena, at the net, chooses to lean left or right. A payoff matrix (winning percentages for Venus) is given, calibrated so Serena volleys better to her right and Venus passes better to her left. Checking best responses shows they cycle without ever coinciding, so no pure-strategy Nash equilibrium exists.

Solving for the mixed-strategy equilibrium (40:40)

The core technique is demonstrated: to find Serena's equilibrium mix Q, use Venus's payoffs, since Venus is the one mixing between left and right and must therefore be indifferent between them. Setting Venus's expected payoff from "left" equal to her payoff from "right" gives one equation solved for Q = 0.6. The same trick run in reverse, using Serena's payoffs to find Venus's mix P, gives P = 0.7. The lecture translates the result into coaching advice: if the opponent's actual mix drifts from the equilibrium value, the best response snaps to a pure strategy in one direction or the other.

Comparative statics: improving Serena's backhand volley (55:33)

The payoff for Serena's backhand volley is raised (from 50/50 to 30/70 in Serena's favor), and the class predicts how equilibrium play should change. A direct effect suggests Serena should lean left more often since she now converts that volley more reliably. A strategic effect works the opposite way: Venus, anticipating the improved backhand, hits left less often, which should make Serena lean left less. Re-solving the equations gives the new Q = 0.5 (down from 0.6), showing the strategic effect dominates. Venus's new mix, found the same way, also shifts toward hitting right more often, confirming that a change to only one payoff moves both players' mixed strategies.

Before you watch

• Be comfortable with the definitions of dominant strategy and pure-strategy Nash equilibrium from earlier lectures in this course, since this lecture assumes and extends them.
• Recall how the class first introduced mixed strategies informally with rock-paper-scissors in the previous lecture.
• Basic comfort with expected value and solving a single linear equation is needed for the algebra in the tennis example.

Check your understanding

1. Why must every pure strategy played with positive probability in an equilibrium mix yield the same expected payoff?
2. In the tennis example, why does finding Serena's equilibrium mixing probability require looking at Venus's payoffs rather than Serena's own payoffs?
3. Starting from the original equilibrium, if Serena leans left more than 0.6 of the time, what should Venus do, and why?
4. Explain the difference between the direct effect and the strategic effect when Serena's backhand volley improves, and which one turned out to dominate.
5. Why does the improvement to only Serena's payoff also change Venus's equilibrium mixing probability?