Lecture 4: Best Responses in Soccer and Business Partnerships

What this lecture covers

This lecture answers two questions: how do you apply best-response reasoning when a game has no dominated strategies at all, and what happens when strategies are continuous rather than a short discrete list? Using a soccer penalty-kick game, Ben Polak shows that a strategy can be eliminated for never being a best response to any belief, even though nothing is strictly dominated. He then gives the formal definitions of best response to a strategy and best response to a belief.

The second half applies this to a profit-sharing business partnership with continuous effort levels, using calculus to find each player's best-response function, and iteratively deletes strategies that are never a best response until the two players' choices converge to a single point. That convergence point is introduced, by name only for now, as Nash equilibrium: a pair of strategies where each player is playing a best response to the other. This concept becomes the central tool for the rest of the course.

Key ideas

• A strategy can be eliminated without being dominated: in the penalty-kick game, shooting to the middle is never a best response to any belief about the goalkeeper, even though no strategy strictly dominates another.
• Formal best response definitions: a strategy is a best response to a specific opponent strategy if it maximizes your payoff against it; a strategy is a best response to a belief if it maximizes your expected payoff given a probability distribution over the opponent's choices.
• Model realism: the simple penalty-kick model ignores footedness, shot power versus placement, and the goalkeeper staying put; real data from professional leagues roughly confirms the qualitative conclusion while showing the numbers are not perfectly symmetric.
• Continuous strategy spaces: unlike earlier discrete games, the partnership game lets players choose any effort level in a range, so best responses are found with calculus rather than by checking a short list.
• Externality in profit sharing: when partners split profits but each bears the full marginal cost of their own effort, both contribute too little effort relative to the efficient level, because each captures only half the marginal benefit.
• Nash equilibrium (preview): the point where two players' best-response functions cross, where each player's strategy is a best response to the other's, so neither wants to unilaterally deviate.

Walkthrough

The penalty-kick game and dominance fails (1:01)

Polak sets up a stylized penalty kick: the shooter picks left, middle or right, and the goalkeeper dives left or right, with rough scoring probabilities as payoffs. Checking for dominated strategies finds none, showing that dominance reasoning alone cannot resolve this important, realistic game.

Best response and the middle-shot lesson (6:10)

Plotting expected payoff for each shot direction against the believed probability the goalkeeper dives right shows that shooting to the middle is never the highest line, for any belief. This gives the broader lesson: never choose a strategy that is not a best response to some belief, even when nothing is strictly dominated.

Testing the model with real data (15:25)

The class lists what the simple model misses: natural kicking side, whether the goalkeeper can stay in the middle, and the tradeoff between power and accuracy. Real scoring-probability data from a published study roughly matches the made-up numbers, though not perfectly symmetric, and Polak notes that power-versus-accuracy tradeoffs can make shooting to the middle rational for some players.

Formal definitions of best response (23:34)

Using the class's notation, Polak states two definitions: a strategy is a best response to a specific opponent strategy if it maximizes payoff against it, and a strategy is a best response to a belief (a probability distribution over opponent strategies) if it maximizes expected payoff given that belief.

The profit-sharing partnership game (30:56)

Two partners each choose a continuous effort level, splitting a profit function that includes a synergy term between their efforts, while individually bearing a cost proportional to the square of their own effort. Polak sets up the payoff functions for both players.

Solving for best responses with calculus (38:07)

Each player's best response is found by differentiating their payoff with respect to their own effort and setting the derivative to zero, yielding a linear best-response function for each player in terms of the other's effort. Plotting both lines shows they cross at a single point after iteratively deleting effort levels that are never a best response.

Nash equilibrium as mutual best response (1:07:15)

The point where the two best-response lines intersect is named as a Nash equilibrium: each player's strategy is simultaneously a best response to the other's, so neither wants to deviate. Polak connects this back to the earlier guessing game, where everyone choosing 1 is the Nash equilibrium, and notes that repeated play of that game did converge toward it.

Before you watch

• Watch Lecture 3 first, since it introduces best response and the expected-payoff graph technique reused here.
• Basic single-variable calculus (taking a derivative and setting it to zero to find a maximum) is used to solve the partnership game; a review is suggested for anyone rusty on it.

Check your understanding

1. Why can a strategy be ruled out for never being a best response even when no strategy strictly dominates it?
2. What real-world factors does the simple penalty-kick model leave out, and how does at least one of them change the analysis?
3. How do the two formal definitions of best response (to a strategy versus to a belief) differ?
4. In the partnership game, why do both partners end up contributing less effort than the efficient level, and what economic concept explains it?
5. What does it mean for a pair of strategies to be a Nash equilibrium, and how does that relate to where the two best-response lines cross?