In our last two posts, we discovered that the contestants who wound up in first and third place at the start of Final Jeopardy could maximize their chances of winning by using strategies that are fairly clear cut. We used a hypothetical example whereby the leader finished the Double Jeopardy round with $10,000, while the second-place contestant wound up with $8,000 and the third-place contestant had $6,000. The exact dollar amounts are not important. What’s important here is that any of the three contestants has a chance to win, depending on who gets the Final Jeopardy question right and how much each contestant decides to wager.

Recall that in Final Jeopardy, the contestants are required to make their wagers before seeing the answer (i. e., the question. This is Jeopardy, remember, where questions and answers are reversed.) And recall that in Jeopardy, only the winner gets to keep the actual dollar amount at the end of the game, and only the winner gets to come back for another game. The second and third place contestants take home consolation amounts ($2,000 and $1,000, respectively) that are often less than the dollar amounts of their ending totals.

Two weeks ago, we saw how the third place contestant should wager zero in most cases, hoping that the Final Jeopardy answer will be so difficult that none of three will get it right. That situation, though unlikely, gives the third place contestant better odds than the pipe dream alternative–namely, that he or she will magically get the answer right while the first two contestants (who outplayed the third contestant throughout the game) will suddenly get this one wrong.

One way to look at the third-place contestant’s situation is through the lens of mathematical probability. There are exactly eight possible finishes, where we consider who gets the Final Jeopardy question right or wrong, as follows:

(1) Leader right, 2nd place right, 3rd place right; (2) Leader right, 2nd place right, 3rd place wrong; (2) Leader right, 2nd place wrong, 3rd place right; (4) Leader right, 2nd place wrong, 3rd place wrong; (5) Leader wrong, 2nd place right, 3rd place right; (6) Leader wrong, 2nd place right, third place wrong; (7) Leader wrong, 2nd place wrong, 3rd place right; and (8) Leader wrong, 2nd place wrong, 3rd place wrong.

Having watched Jeopardy for many years, I can state with confidence that the first and last situations are fairly common. They tend to happen whenever the Final Jeopardy answer is either a bit too easy (so all three get it right) or a bit too hard (so all three get it wrong.) That’s an important piece of information to remember. Another, perhaps more important, piece of information is that the pipe dream situations described in situations (6) and (7)–where only one person gets the answer right, and that lucky person is not the leader–are relatively rare.

With proper wagering, the second and third place contestants cannot possibly win in situations (1), (2), (3), and (4), because the leader can shut out the other two contestants with proper wagering. And in situations (5) and (6), the third place contestant is still out of luck, because the second-place contestant got the answer right and started Final Jeopardy with more money to wager. Only in the case of (7) and (8) does the third contestant stand a realistic chance of winning. And in both situations, the third-place contestant can win by betting zero in the knowledge that contestants one and two would be forced to make sizable bets against each other. By betting zero, the third-place contestant is maximizing the chances of victory in the case of scenario (8), a far more likely scenario than (7), and still has a good chance of winning if (7) occurs. Most actual third-place contestants tend to ignore this strategy, placing all their hopes on scenario (7), thereby eliminating their chances of winning whenever the more likely (8) situation occurs.

The third-place contestant should make a sizable wager only in cases where the leader is so far ahead that he or she can close out the third-place contestant with the betting strategy discussed below, even in situation (8). In other words, the third-place contestant should bet on situation (7) only when that is the only winning scenario.

Now let’s review the leader’s strategy. The leader must guard against the possibility that the second place contestant will bet the farm. So, last week, we learned that the leader must look at the value of the second-place contestant’s wagering stash–in this case $8,000–and assume the second-place contestant will wager it all and will also get the answer right, thus ending up with $16,000. To ensure a victory with the least possible risk, the leader should therefore wager (in this hypothetical) exactly $6,001. So, in cases (1), (2), (3), and (4), there will be no way the other two contestants can top the leader’s final score of $16,001. This strategy will also give the leader a chance to win in situation (8), but only if the second-place and third-place contestants made sizable wagers, which is often the case.

Now, given all that, what should the second-place contestant do?

The answer may surprise you. The problem begins with the observation that most third-place contestants do not wager wisely. They tend to bet the farm, on the theory that “gee, I’m so far behind I need to bet everything to have any chance of winning.” So, the second-place contestant cannot assume that the third-place contestant has read these blog posts and will therefore wager zero. To cut to the chase, the correct strategy for the second-place contestant is to ignore the leader and wager only enough to close out the third-place contestant. The second-place contestant can maximize his or her chances of winning by adopting the very same strategy as the leader–but, whereas the leader should only look at the second-place contestant’s stash, the second-place contestant should not look back at the leader, but should only wager against the third-place contestant.

Here’s how the second-place contestant’s strategy would work in our hypothetical. The third-place contestant can wind up with a maximum of $12,000 by wagering his or her entire stash and getting the Final Jeopardy answer correct. Therefore, the second-place contestant should wager exactly $4,001, which will allow him or her to wind up with $12,001 by getting the question correct, while still allowing a final total of $3,999 if he/she gets it wrong. The total of $12,001 will ensure victory in situations (5) and (6), and the final total of $3,999 might be enough to win in situation (8). In fact, in this hypothetical case, the top two players will end up with tie scores if they both wager properly and scenario (8) happens to occur.

Notice that this more modest wagering strategy for the second-place contestant will only be incorrect in those cases where (a) both he/she and the leader get the Final Jeopardy answer correct, while also (b) the leader made a final wager that was rather modest but still managed to win; i. e., the leader conceded victory to the second-place contestant if the second-place contestant had bet the farm and got the answer right. Again, I can only observe that this type of clever wagering on the leader’s part almost never happens, except in cases where the Final Jeopardy category is so obscure that the leader is betting based on the likelihood of scenario (8).