In our note “Kelly Can’t Fail” we shared the “Next Card Bet” game as described in Peter Winkler’s Mathematical Puz

Kelly Betting With Discrete Stakes – Win Vector LLC

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2024-12-21 16:30:04

In our note “Kelly Can’t Fail” we shared the “Next Card Bet” game as described in Peter Winkler’s Mathematical Puzzles. Our observation is the amazing zero variance strategy shown is in fact the Kelly criteria. Which means, this is a surprising situation where the Kelly strategy is zero variance. This may finally be a Kelly example that is safe for classroom demonstration.

As the Kelly strategy always returns about $9.08: when the strategy loses money partially through the game, the remaining deck is very valuable. If your stake is reduced to $0.1, then the value of the rest of the game is $90.8 per $1! Of course, the pricing of remaining games is central to history of probability theory, i.e “the problem of points.” This structure makes for a beautiful picture when we graph many betting trajectories.

The “Next Card Bet” game is essentially an instrument to examine the art of card counting (or exploiting uncertain information) without having put up with ugly details of a game such as blackjack (apologies to Edward O. Thorp!).

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