Jeopardy Wagering Strategies

This may be one of the more "trivial" blog entries I've ever written, but I felt some inner need to just get it all out...

The other night, while watching the popular trivia quiz show Jeopardy, they were on the final question, for which each of the three contestants could wager as much money as they had individually amassed throughout the game (before knowing what the question was, of course).

[Wait, switch "answer" for "question", for in Jeopardy, the answers ARE the questions (and vice versa).]

So anyway, this time around, when it was time for the final Jeopardy "answer", Contestant #1 had $21,000, Contestant #2 had $5,000, and Contestant #3 had $25,000 (those figures are approximate as my memory goes, as do the following...but the principle will be the same).  The correct "question" to the "answer", which all three should have gotten, was only written down by the leader, Contestant #3.  But he only had wagered about $10,000, giving him a final total of $35,000.  Had Contestant #1 gotten it right and wagered more than $14,000, he would have beaten Contestant #3 regardless whether the latter got it right or not.  This is important in Jeopardy because only the highest money winner gets to keep his or her earnings.  The other two get a nominal amount of money plus, appropriately, a supply of Alleve.  But Contestant #1 not only missed the final "answer", but only wagered about $4,000!  Both wagered stupidly, in my opinion.  But how should one wager, anyway, at the end of Jeopardy?

If the leader is has more than twice as much money as the second place contestant, than they should only wager enough that, should they lose, they would end up one dollar ahead of twice their nearest opponent's amount.  For example, if Ann has $20,000 and Dave, in second place, has $8,000, then she should wager no more than $3,999, giving her a dollar more than Dave at $16,001 should she miss at the end and he get it right.  And if a contestant is NOT in the lead when the final answer comes around, they should wager everything they have, since they won't be able to get their total winnings anyway should they finish second or third.  If the first place contestant isn't more than twice as "wealthy" as the second place contestant when it is time to wager, they have to wager enough for them to have at least one dollar more than twice their nearest opponent's amount should both write down the correct "question". For the third place contestant going into the final round, if his/her total so far is greater than the difference between the first two, than he or she can win it all, assuming of course that the others wager wisely (and miss the question as well), by getting the question right and by betting whatever money is theirs to wager. 

Looking at the specific game I witnessed, Contestants #1 and #2 should have wagered everything they had since the consolation prizes were insignificant compared to having the highest money total.  So Contestant #1 should have wagered all $21,000 and Contestant #2 should have wagered all $5,000 (but if memory serves me, she bet much less).  Contestant #3, aware that Contestant #1 would end up with $42,000 had he gotten the right "question", should have wagered #17,001 to ensure that he would still end up with the highest total. 

Of course, this all would be vastly different were the Jeopardy producers not such abject cheapskates and instead allowed all of the contestants to keep their hard-fought earnings regardless of their finish...