In the game show Jeopardy, Bob with $10,000 and Dan with $6,000 are about to place their

Question:

In the game show Jeopardy, Bob with $10,000 and Dan with $6,000 are about to place their bets in Final Jeopardy. (The third player has so little money that he cannot possible win.) Each secretly places his bet and then answers a final question, winning his bet with a correct answer, losing it if he is wrong. Both know that either's chance of a correct answer is .5 (and these chances are independent). After answers are given and bets are added or subtracted, the person with the most total money wins (keeping his money and returning to play again the next day). The loser gets nothing.
This situation is equivalent to a zero-sum game, where Bob seeks to maximize his chance of winning (and Dan wants to minimize Bob's chance). As shown in Table A, Bob's strategic options are to make a shut-out bid, $2,001, giving him an unbeatable $12,001 if he answers correctly, or to bid nothing, $0. Dan's options are to bid his entire winnings, $6,000, or to bid nothing, $0.
a. In Table A, supply Bob's winning chances for the two missing entries. (For example, the lower-left entry shows that if Bob doesn't bet but Dan does, Bob's winning chance is .5 - i.e., when Dan answers incorrectly.) Then, determine both players' equilibrium strategies and the value of the game (i.e., Bob's winning chances). Does either player use a mixed strategy?

b. As before, Bob has $10,000, but now suppose that Dan has $8,000. Complete the missing entries in Table B, and find both players' equilibrium strategies. Does either player use a mixed strategy? Now, what is Bob's chance of winning?