There is a non zero chance that betting 6001 (as opposed to to 6000) could result in you not winning (you guess wrong, P2 wagers >2k and guesses wrong, P3 wagers 0).

There is ZERO chance betting 6001 could result in you winning when 6000 would make you lose (since no one can reach 20001, the score required for someone to cause you to lose despite guessing correctly with 6000, and clearly wagering 6001 can only be better than 6000 in the cases where you guess correctly)

Wagering 6001 is dominated by wagering 6000, in terms of "chance of winning" (ie ignoring the physical wagering of $1)

You cannot solve this problem properly unless you know some details that unfortunately in real life can only be slightly approximated making a solution elusive typically.

First of all you need to know the win probabilities for the question for all 3 . You also need to know what is the utility of a victory in terms of expected future return. I mean if for example by winning your expected earnings are say 30k down the road and you now stand at 12 you have an incentive to go for it with maximum probability even if this reduces the current game average profit.

So lets establish a few details; what are the prizes the players get here? The first gets to keep the winnings and advance to a new game. The second and third do what?

You can then start a model with p1,p2,p3 correct answer probabilities for a given category and positions s1,s2,s3 each one connected with a given utility for each position finish and then try to maximize these utilities by selecting strategies..

You could simplify the problem posed with 12,9,6 by adding a future income prize for winning (by advancing to future games and winning more prizes there) of say 30k and an answer right probability of say 50% for each. Then you could try to find an answer in this simplified model.

Sorry to bump the thread, but I don't think anyone mentioned this (someone else explained this to me). Knowing how to buzz in fast is half the battle for Jeopardy. You're better off playing against people without as much buzzer experience, which is why it makes sense to bet to win.

something along the lines of what masque said, simply pulling number $xx,xxx out of your ass with one or two calculations is no way to declare an optimal strategy

thats because its final jeopardy(!)