To make the supplier-buyer scenario described in the last post directly analogous to the Final Jeopardy! betting situation, one needs to convert the dichotomous choice between confident and timid proposals into a continuous choice that parallels the players' decisions to bet anything from nothing to all their money.

Additional specifications for the continuous variable case:

1) The advantage of supplier A is equal to x. X is a quantity known to both suppliers. (This condition parallels the known lead that player A has over player B going into the Final Jeopardy! round.)

2) A will win unless B's proposal exceeds A's in value by at least x. (This condition parallels the need of the trailing player in Final Jeopardy! to make up the difference with the leading player.)

3) A and B can both submit proposals on a range from 100% timid to 100% confident. A 100% timid proposal takes no risk and has a baseline value of 0; a 100% confident proposal takes maximum risk, and has value y if successful and -y if unsuccessful. All other proposals have an intermediate value; for example, a proposal that is 75% confident has a value of .75y if successful and -.75y if unsuccessful. (This parallels the ability of both players to bet any or all of their money.)

4) Y is greater than x. That is, there is no guaranteed win for A. (This is equivalent to a Final Jeopardy! round in which the second place player has more than 50% of the leader's total.)

The continuous variable case is not as straightforward to analyze in rational choice terms as the dichotomous variable case is. The mixed Nash equilibria are parallel to those described earlier for the dichotomous variable case: The leading player should tilt toward confidence, while the trailing player should tilt toward timidity.