How to fix a broken bet – a puzzle

The point of a normal game is to win. But sometimes rules are written poorly and a player finds it beneficial to throw the contest.

The “winning by losing” strategy is a perverse incentive but it is not uncommon. Two examples come to mind. First, a while back I wrote about how Google won by losing in the FCC spectrum auction. Second, Mike Shor has a wonderful piece about soccer teams that scored own goals on purpose to improve their tournament draw.

How do you fix a broken game? This is an interesting question that keeps regulatory boards and sports agencies on guard.

I recently came across a puzzle on this topic I found interesting.

The puzzle – a broken bet

The reference is Charles Barry Townsend’s The World’s Best Puzzles, and the puzzle is called “the world’s best ‘racing’ puzzle.”

Two sporting gentlemen decided to stage a race where the buggy that crossed the finish line first would lose and the one that came in second would win.

Off they went, down a one-mile course, whipping their horses to a lather. As they neared the finish line, they both slowed down to a halt with only 100 yards to go. Realizing that they had made a dumb bet, the two drivers got down and went over to discuss the matter with a farmer who was watching them from his field.

When the farmer heard the story, he gave them a piece of advice that sent them leaping into the buggies and speeding down the course, as each one strained to be the first to cross the line.

The advice that the farmer gave in no way changed the terms of the original wager. Can you guess what it was?

[p. 96, extra paragraph breaks added by me]

A hint

The answer has to do with changing the game, and the trick is to think about the legal ways to alter the situation.

The answer is in the comments.



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  • http://www.mindyourdecisions.com/blog/ Presh Talwalkar

    The trick is seeing the loser is the first *buggy to cross the finish line. The farmer suggested that each rider get into the opponent’s buggy. The switch meant each rider was effectively racing to make the other person lose, just like in a normal race.

  • Scott

    It’s interesting to see how much lateral thinking plays in game theory.

    As a coincidence, “A Beautiful Mind” was on last night and this type of lateral thinking solution reminded me of the bar scene where Nash formulates a solution on how they can each get a girl by ignoring the blonde and going for her friends.

  • http://www.mindyourdecisions.com/blog/ Presh Talwalkar

    Here’s my post about that scene from A Beautiful Mind. (Ironically Nash’s solution is not a correct equilibrium)

  • JD

    Eh, puzzles like this are mildly amusing, but kind of cheap in my opinion.

    If the bet were for a standard race in which the buggy that crosses the finish line first wins, would one of the competitors be allowed to jump into the opponent’s buggy and sabotage it? Even if the rules don’t explicitly forbid that, such an act would be against the spirit of the bet and implicitly illegal.

    I don’t see how the bet in this puzzle is any different. The rules don’t specifically forbid taking control of the other person’s buggy, but that would be reasonably implied in any race. Maybe I am nit picking, but since the correct answer relies on a nitty technicality, I think it’s only fair.

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