Someone on the Facebook Jeopardy Fan Group posted that the champ's bet was a mistake. an absurd comment. The actual mistakes were made by the other two players. The secod-place guy could have safely bet up to $1,200, ensuring the win if he and the champ both got it wrong. The third-place player bet $5,000, but she should have bet $6,000, ensuring the win if she got it right and the other two missed it.
If the champ had figured that the other guy would make the sophisticated bet of no more than $1,200, then he could have ensured the win by betting no more than $5,999.
3 comments:
I agree that in such a situation the leading player should almost always wager enough that if his/her answer is correct, it will guarantee a win. The only case where another strategy would make sense is if the Final Jeopardy category is one for which the player sees little hope of getting a right answer, and analyzes that a win will only happen if the opponent(s) within striking distance are wrong; or will bet insufficiently to overtake him/her. If I believe I am going to be wrong, I think the best bet is zero,
I agree that in practice the leader usually wagers enough to guarantee the win if he/she is correct, but is that really the best strategy?
Assuming mathematical sophistication of the other players, leader should reason that the trailer will follow the best strategy. If the trailer is within 2/3 of the leader, then he/she has the strategy of making a smaller wager than all-in as I discussed in the post. This wins for the trailer in two situations: one, they both miss it, and two, trailer gets it and leader misses. By contrast, the all-in strategy only wins in one situation: where leader gets it wrong and trailer gets it right.
So, the trailer's best strategy clearly is the smaller bet. But if the leader reasons this through, then he/she can also make a smaller bet. Of course, if trailer has less than 2/3 of leader's total, then leader bets to guarantee the win if he/she gets it right.
Is it good strategy for the leader to go through this sophisticated reasoning? Playing against the average person, probably not. But these are not average people, they are folks who got through the arduous process required to become a Jeopardy contestant. So hard to come up with a hard and fast rule here.
In a recent Jeopardy game, the scores going into Final Jeopardy were 19,200, 15,400, and 10,800. The bets were 11,601, 10,000, and 8,500.
I claim that only the leader gave their bet any thought. He made the standard "cover bet", meaning he bet enough to assure the win if he and the others all got it right. A more intelligent bet for the second-place player would have been 7,800, attempting to assure the win by one dollar if they both got it wrong.
But what about the third-place player? She could then try a bet of 3,199, going for the win if they all got it wrong! The scores would be 7,599, 7,600, and 7,601!
And what do the stats tell us? Roughly 25–30% of games end with a “triple stumper” (all wrong), while only about 10–15% of games see all three contestants correct. What this tells us is that it makes considerably more sense to bet as if all are going to get it wrong, than if all are going to get it right.
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