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What continually surprises me is that more contestants don't bet to tie. In your example, why not bet $1000 instead of $1001? If they tie, they both come back.Let's talk FJ betting strategy, AKA second place person is an idiot. Let's call them One, Two, Three.
One & Two are close in money, both $20,000+. You know One is going to bet all/most of his money. One has to make sure he wins if he gets FJ correct. Therefore Two only needs to bet enough to pass One if One misses and Two is correct. So if One has $21,000 and Two has $20,000, Two should bet $1,001. This betting strategy has the additional benefit that if you BOTH miss, Two will only drop a little while One loses his rear-end.
This is a universally accepted strategy, and has been proven to be effective over the years. There are four outcomes for FJ and One/Two--(a) both right, (b) both wrong, (c) One right, (d) Two right. It is a given that a & c will lead to Two losing. But if Two follows this betting strategy, he can win under b & d. Whereas if he doesn't he will only win under d.
So whaddaya know--"b" happened and Two lost with a stupid all-in bet. And Three ended up winning!
I'm continually amazed how little Jeopardy contestants know about betting strategy. It can make the difference between winning & losing.