Optimal Betting to Replicate WS Series

This question considers a best-of-seven sports series, where the candidate can only bet on individual games but wants to replicate the payoff of a binary series bet on one team. The twist is that they must always stake a positive amount, and their final wealth must depend only on who wins the series, not on the exact game path. The setup forces the candidate to think in terms of wealth states after each game and the requirement that, no matter how the wins and losses arrive, the terminal wealth matches the series outcome payoff.

It leans heavily on dynamic replication along a finite tree, self-financing strategies, and backward induction. The candidate must express wealth after each possible scoreline and identify the bet sizes that make future wealth consistent across branches. An interviewer is watching for comfort with state-contingent claims, translating a path-dependent contract into per-period trading rules, and recognizing that model probabilities are irrelevant to replication. Precision with conditional reasoning on game counts, and the ability to organize the tree cleanly without getting lost in cases, are key signals of strength.

What it tests

This problem class is governed by the principle of replicating a contingent payoff using dynamic hedging or backward induction on a finite tree of possible outcomes. The key is that, for any path-dependent payoff that is determined by a sequence of binary events, you can construct a self-financing strategy by working backward from the terminal payoffs. At each node, you solve for the bet that ensures, regardless of the next outcome, your wealth will match the required value at the subsequent node. This approach is agnostic to the probabilities of each outcome: it is a purely mechanical replication, ensuring that the final wealth matches the desired contingent payoff structure. The process is analogous to constructing a replicating portfolio in option pricing, where the payoff is path-dependent and must be matched at every possible state.