Lasting Longer

by Douglas Zare
13 November 2013

Douglas Zare

At the end of my October column, I asked about last-longer bets in single-elimination backgammon tournaments, where two players bet on which of them will go farther. Unlike most situations in backgammon, this allows the possibility of a tie if both players are knocked out in the same round.

Years ago, two of the stronger regulars at the New England Backgammon Club were fond of making last-longer bets against each other. They would bet more than the tournament's entry on which of them would make it deeper into the tournament. If both were knocked out in the same round, then the bet was a wash. Byes counted, so a player could get a bye, lose his first match, and still win the bet.

Suppose it would be a fair bet if Player A offered Player B 11:10 odds on a heads-up match. What are the fair odds for Player A to offer Player B on a last-longer bet in a tournament?

The reader MSquonk argued that the 11:10 odds shouldn't change for the last-longer bet. Either the players meet each other, or they don't, and we can break up the wager into a weighted average of these cases. Player A should win 11/21 of the times they meet. If 11:10 odds are fair on the times that a player is knocked out without the two meeting, then 11:10 odds remain fair on the whole bet. Is that the case?

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