Question
Two rowing clubs compete in two races for a sponsor's prize. Each club has ordinary rowers and one specialist, who can be used in at most one of the two races. Before race 1, each coach secretly and simultaneously decides whether to field their specialist or save them. After the winner of race 1 is announced, a coach who saved their specialist may decide whether to field them in race 2. The clubs make those second decisions simultaneously, without learning whether the other coach used a specialist in race 1; each coach knows only their own choice and the public winner.
A race is won with certainty by the club fielding a specialist if the other club does not. If both field a specialist, or neither does, the race is won by an independent fair coin toss. Fielding a specialist in either race costs its club $28. The sponsor pays $100 to a club winning both races and $0 to the other club; if the races split 1–1, each club receives $30. Each club maximises its expected prize money less its own specialist costs. Randomised strategies are allowed.
Find a symmetric mixed-strategy equilibrium: the probability each coach uses the specialist in race 1, what a coach with a saved specialist does in race 2 after a win or a loss, and each club's expected net payoff. Explain how the information revealed by race 1 affects the relevant posterior beliefs; do not assume that coaches see their opponent's first-race lineup.
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Hint 1
At race 2, using a saved specialist always raises the probability of winning that race by 1/2, regardless of the other club’s unknown roster. But the monetary benefit of winning race 2 depends on who won race 1.
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Solution
1. Optimal race-2 decisions. Whether the opponent uses a specialist or not, adding your own specialist raises the probability that you win race 2 by exactly 1/2: the win probability changes either from 1/2 to 1 (if the opponent uses no specialist) or from 0 to 1/2 (if they do). This is true even when the opponent's first-race lineup is hidden.
If you won race 1, winning race 2 changes your prize from $30 (a split) to $100 (a sweep). The expected marginal prize from fielding a saved specialist is \(\frac{1}{2}(\$100-\$30)\)=$35, more than its $28 cost. You use it. If you lost race 1, winning race 2 changes your prize from $0 to $30. The marginal expected prize is \(\frac{1}{2}(\$30-\$0)\)=$15, less than $28. You bench it. These are strict choices, independent of posterior roster uncertainty.
2. Race-1 strategic payoff matrix. Now use those optimal continuation decisions. Entries below are A's expected net dollars, with A choosing the row and B the column:
| B uses specialist now | B saves specialist | |
|---|---|---|
| A uses specialist now | $12 | $37 |
| A saves specialist | $15 | $36 |
To verify each cell:
- Both use: race 1 is a coin toss; neither has a specialist left for race 2, also a coin toss. Each club has a 1/4 chance of a $100 sweep and a 1/2 chance of a $30 split prize. Gross expectation = \(\$25+\$15=\$40\); subtract the $28 first-race cost to obtain $12.
- A uses, B saves: A wins race 1 for certain. B, having lost, benches in race 2; A has no specialist, so race 2 is a coin toss. A's gross prize is \(\frac{1}{2}\cdot\$100+\frac{1}{2}\cdot\$30=\$65\), minus $28 = $37. B receives $30 on half the paths, so B's net is $15.
- A saves, B uses: symmetrically, A loses race 1 and benches in race 2, which is a coin toss. A gets a $30 split prize with probability 1/2, giving $15; B receives $37 net.
- Both save: race 1 is a fair coin toss. Its winner fields their saved specialist in race 2 and sweeps with certainty, paying $28 and collecting $100; the loser benches and collects nothing. Each club's expectation is \(\frac{1}{2}(\$100-\$28)\)=$36.
Suppose the opponent uses their specialist in race 1 with probability p. A earns \(12p+37(1-p)=37-25p\) by using now, versus \(15p+36(1-p)=36-21p\) by saving. Both choices are played with positive probability in a symmetric mixed equilibrium only if
\(37-25p=36-21p\) ⇒ \(p=\frac{1}{4}\).
The common expected payoff is \(37-\frac{25}{4}\)=\(\frac{\$123}{4}=\$30.75\). The off-diagonal first-race choices also form two asymmetric pure equilibria, but \(p=\frac{1}{4}\) is the symmetric mixed equilibrium requested.
3. Correct posterior beliefs after race 1. Suppose A saved their specialist. The equilibrium prior that B used theirs in race 1 is 1/4. A first-race win can occur only when B also saved and a fair coin favoured A: probability \(\frac{3}{4}\cdot\frac{1}{2}=\frac{3}{8}\). Hence
\(\Pr(B\text{ used in race 1}\mid A\text{ saved},\ A\text{ won})\)=0.
A loss occurs either because B used theirs (probability 1/4) or because B saved and won the coin toss (probability \(\frac{3}{4}\cdot\frac{1}{2}=\frac{3}{8}\)). Thus \(\Pr(A\text{ lost}\mid A\text{ saved})=\frac{5}{8}\), and
\(\Pr(B\text{ used in race 1}\mid A\text{ saved},\ A\text{ lost})=\frac{1/4}{5/8}\)=\(\frac{2}{5}\).
These posterior-weighted branches also verify A's expected payoff from saving. If A saved and won race 1 (probability 3/8), they field their specialist, beat the known specialist-free opponent and earn \(\$100-\$28=\$72\). If A saved and lost (probability 5/8), they bench. Conditional on that loss, B already used their specialist with probability 2/5, in which case race 2 is a fair coin and A earns $30 with probability 1/2; otherwise B saved, won race 1 and uses their specialist, so A loses race 2. The loss-branch expectation is \(\frac{2}{5}\cdot\frac{1}{2}\cdot\$30\)=$6. Total saved-strategy expectation = \(\frac{3}{8}\cdot\$72+\frac{5}{8}\cdot\$6=27+3.75\)=$30.75, matching the matrix. No coach has been allowed to condition on an unseen opposing lineup.
Assumptions
Each club has exactly one specialist and may field them in at most one race; choosing to never deploy costs nothing.
The two clubs choose first-race deployment secretly and simultaneously; after race 1 only the winner is announced, not either lineup.
Race-2 decisions by clubs with unused specialists are simultaneous and made without observing the opponent’s second choice.
Every fair coin used for a same-lineup race is independent; net utility is sponsor cash prize minus $28 per specialist deployment.
The question requests a symmetric mixed equilibrium; asymmetric pure equilibria may also exist.
Final Answer
In a symmetric mixed equilibrium each club fields its specialist in race 1 with probability 1/4, otherwise saves it. A coach who saved the specialist fields it in race 2 if they won race 1, and benches it if they lost. Each club's expected net payoff is \(\frac{\$123}{4}=\$30.75\). If a coach saved their specialist, their posterior that the opponent used theirs in race 1 is 0 after a first-race win and 2/5 after a first-race loss.