Replicating a Series-Winner Bet with Sequential Game Wagers

Summary

The question frames a best-of-seven series bet as a binary payoff and asks how to reproduce it using wagers placed on individual games. It proposes estimating the value of the position after the first game from the remaining-game win probabilities, then asks how to discount those conditional values to determine the initial wager. The document presents the problem but does not provide a solution to the replication or pricing question.

Its probability calculations assume each game is equally likely to be won by either team and use that assumption to estimate the chance of winning the series from an intermediate state. The discussion is therefore a prompt for reasoning about contingent claims and sequential betting, rather than a complete strategy. It does not address how to construct the hedge across later games, and the proposed intermediate values are not validated in the text.

Key ideas

• A series-winner payoff can be viewed as a contingent claim whose value changes as game outcomes arrive.
• The question proposes using remaining-game win probabilities to value the claim after the first game.
• The wager needed before the series depends on how to replicate the later conditional payoffs.
• The document leaves the replication method and initial stake unresolved.

Tags

# interview question : replication strategy of a betting game


# interview question : replication strategy of a betting game












Here is a question I found in a book I am not able to finish. Your help will be much appreciated! I also included where I have been so far.

Q: Team A plays team B in a series of 7 games, whoever wins 4 games first is the winner. You want to bet 100 that your team wins in which case you will receive 200 or 0 otherwise. However you can bet only on individual games (so game by game sequentially). You can bet X the day before the match and receive 2X if win otherwise 0 if lose. - Q1: How do you achieve the desired payout? - Q2: What amount do you bet on the 1st game?

For me this is like replicating a binary option paying 100 in case of success or losing 100 in case of default. The aim is to price this option (find this X). It seems to be a binomial tree of 7 states. We consider the first part of the tree (X, X_up, X_down). Probability of going up is a just the binomial probability of winning at least 3 games out of of 6 (we are already at the the state 1). So Proba_up = 1 - (1/2^6)*(1+15+6) = 21/32 Thus we can get the value of X_up = 100*21/32 - 100*(1-21/32) = 40.625 We then have Proba_down = 1 - Proba_up = 11/32 and then X_down = 100*11/32 - 100*(21/32) = -40.625

And now I am stuck: I cannot figure out the proba I need to use to discount these values. Many thanks for your help.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.