What is the fairest way to divide the pot?

Suppose you and another player are playing a game in which you both have equal chances and the first player to earn a certain number of points wins. The game is interrupted with one player in the lead. 

In 1996 the Atlanta Braves beat the New York Yankees in the first 2 games of the baseball World Series, in which the first team to win 4 games is crowned champion. The fact that the Braves won the first 2 games didn’t necessarily mean they were the superior team. Still, it could be taken as a sign that they were indeed better. Nevertheless, for our current purposes we will stick to the assumption that either team was equally likely to win each game and that the first 2 games just happened to go to the Braves. Given that assumption, what would have been fair odds for a bet on the Yankees—that is, what was the chance of a Yankee comeback? To calculate it, we count all the ways in which the Yankees could have won and compare that to the number of ways in which they could have lost. Two games of the series had been played, so there were 5 possible games yet to play. And since each of those games had 2 possible outcomes—a Yankee win (Y) or a Braves win (B) –there were 25, or 32, possible outcomes. For instance, the Yankees could have won 3, then lost 2: YYYBB; or they could have alternated victories: YBYBY.The probability that the Yankees would come back to win the series was equal to the number of sequences in which they would win at least 4 games divided by the total number of sequences, 32; the chance that the Braves would win was equal to the number of sequences in which they would win at least 2 more games also divided by 32.

So in order to calculate the Yankees’ and the Braves’ chances of victory, we simply make an accounting of the possible 5-game sequences for the remainder of the series. First, the Yankees have been victorious if they had won 4 of the 5 possible remaining games. That could have happened in 1 of 5 ways: BYYYY, YBYYY, YYBYY, YYYBY, or YYYYB. Alternatively, the Yankees would have triumphed if they had won all 5 of the remaining games, which could have happened in only 1 way: YYYYY. Now for the Braves: they would have become champions if the Yankees had won only 3 games, which could have happened in 10 ways (BBYYY, BYBYY, and so on), or if the Yankees had won only 2 games (which again could have happened in 10 ways), or if the Yankees had won only 1 game (which could have happened in 5 ways), or if they had won none (which could happened in only 1 way). Adding these possible outcomes together, we find that the chance of a Yankees victory was 6 in 32, or about 19 percent, versus 26 in 32, or about 81 percent for the Braves. According to Pascal and Fermat, if the series had abruptly been terminated, that’s how they should have split the bonus pot, and those are the odds that should have been set if a bet was to be made after the first 2 games.

If the two teams have equal chances of winning each game, we will find, of course, that they have an equal chance of winning the series. But similar reasoning works if they don’t have an equal chance, except that the simple accounting we just employed would have to be altered slightly: each outcome would have to be weighted by a factor describing its relative probability. If we do that and analyze the situation at the start of the series, we will discover that in a 7-game series there is a sizable chance that the inferior team will be crowned champion.

Full Disclaimer: I am not a financial planner. The views expressed in this post are all mine and they may or may not suit your needs. Please do you own due diligence. I do not make money on any of the products suggested in this post. 


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