Monday, June 08, 2015

A Fixed Chess Match: Class and Case Probability

Keshav and I are going to play a chess match. Let us assume that when we have played in the past, we have been very evenly matched. There has been heavy betting on the match, and the Mob has gotten involved. They have persuaded the two of us to agree to fix the match, so that Keshav wins, 6 games to 4, and it is agreed as to which games he will throw, and which I will throw. There will also be a couple of games we play all out, just to make things look good, and a final fixed game, but for which it is unknown who will throw the game until the result of the two "real" games is known.

You happen to have gotten a hot tip about the fix, but your tipster only knows the final result, and not which games are which. You can place rational bets on the series or some number of games in the series, using the fact that there is a 60% chance Keshav will win any randomly chosen game in the series. In other words, you will accept any bet on Keshav that offers an expected return of over $0.66 on the dollar, and anything over $1.50 on the dollar for a bet on me.

However, for no game in the series is it objectively true that Keshav has a 60% chance of winning: two games are toss-ups, and in the rest it is 100% certain he will win, or 100% certain I will win. The 60% figure is a piece of knowledge about the whole class of games, and not knowledge about any game in the class. If you had knowledge about the particular games, your betting would be very different: you would take almost any odds and bet on Keshav in the games you knew he would win (almost any, because he might be struck by lightning during the game, etc.), and vice versa in the games you knew I would win.

Conclusion: the fact that we know that, for some class of events, there is probability p that X will occur in a group of such events cannot be automatically interpreted as a statement about some objective property of any particular event that is a member of that class.

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