The reroll advantage in a die duel

HardProbability Market Making & Betting Games~10m

Alice rolls a fair $30$ -sided die; Bob rolls a fair $20$ -sided die. Each wants the larger value; ties go to Bob. Bob may (once) choose to reroll his die - without seeing Alice's - and must keep the new value. Under Bob's optimal strategy, what is the probability that Bob wins?

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Keep $b$ iff $\tfrac{b}{30}\ge$ reroll value $\mathbb{E}[b'/30]=0.35$ , i.e. $b\ge 11$ .
Combine: $\tfrac12\cdot\tfrac{15.5}{30}+\tfrac12\cdot0.35=\tfrac{13}{30}$ .

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0.4333 (± 0.005)

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Asked at: Optiver, SIG

The reroll advantage in a die duel

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