Saturday, March 17, 2007

A Computer Scientist in Jeopardy

The long-running Jeopardy television game show had a first on Friday, when all three players ended up tied for the first time.
The three contestants on the venerable game show all finished with $16,000 after each answering the final question correctly in the category, "Women of the 1930s," on Friday's show. They identified Bonnie Parker, of the famed Bonnie and Clyde crime duo, as a woman who, as a waitress, once served one of the men who shot her…The show contacted a mathematician who calculated the odds of such a three-way tie happening — one in 25 million. In that final round contestants choose how much of their winnings to risk, so it is impossible to give a probability in such a setting. It's more an issue of simple game theory. Before the Final Jeopardy round the totals were$13,400, $8000 and$8000. Both of the $8000 decided to risk all of their money so they wouldn't be overtaken by the other one. The$13,400 belonged to Scott Weiss, a computer science professor at Mount St. Mary's University in Maryland. Since $13,400 is between 1.5 and 2 times$8000, the standard strategy is to bet enough so that if you win you have more than $16,000 and if you lose you have more than$8000, for example betting $3000. Had Scott done so, he would have taken home all his winnings and come back for the next show. Instead Scott bet$2600, leading to the \$16,000 tie. By doing this, Scott gets to take home all his winnings and comes back for the next show.

It takes a computer scientist to make the most conservative bet, knowing that the rules of the game give no particular advantage to winning over tying and leading to the first three-way tie ever.

1. Maybe there's even an advantage to tieing? That way you know that your opponents in the next game will be people you've already shown yourself able to beat.

2. Scott is a member of the first class of graduating undergrad computer science majors at Carnegie Mellon.

3. I'm more interested to know where the 1 in 25 million claim came from!

4. There might be an even bigger advantage. If the opponents of Scott Weiss learn something, they might copy his strategy. Then, Scott has better chances of going yet another time to Jeopardy.

5. It's more an issue of simple game theory.

Is this true? As you point out, there is no particular advantage to the outcome of a tie game. Each person knows this. Since a tie results in each of them keeping their money and coming back another day (for as long as they like), there is a strong motive for collusion.

Each contestant could agree to the following strategy. Let the person who has the least amount of money bet zero. Then the other two bet an amount equal to the difference of their amount and the 3rd place player. Then these two will intentionally get the question wrong resulting in a guaranteed tie.

You've just ensured (as long as no one ends up with 0 or negative money at the end of the game) that all three of you will be able to win an unbounded amount of money.

6. The advantage to winning over tying is this: when your opponents each have one game of experience rather than zero games of experience (because you would play two fresh contestants if you won outright), they will perform better on the signaling button.

Any buzzing advantage you had over them will be smaller the second time you all play, more likely leading to your defeat.

7. Building on anon 5's post, the players would probably want to follow a strategy somewhat like this during the game (to maximize the third place player's money going into final jeopardy): the first and second place players pause long enough for the third place player to answer if he can, then first place player pauses for second place player to answer, then first place player answers. But what if the first place player thinks he will not be able to correctly answer any questions in the other categories and will end up in third going into final jeopardy? Then he shouldn't pause and should answer the current question first. But players don't know the other players' assessments of their future performance ... and what about daily doubles? How should they play?? Maybe there is some interesting game theory here ...