Last week we looked at the dilemma faced by the contestant in third place at the start of Final Jeopardy. We assumed, for purposes of illustration, that three contestants were wagering with the following amounts:
2nd Place Contestant: $7,500
3rd Place Contestant: $6,000
We observed that in most cases, the 3rd Place Contestant should swallow the urge to “try to catch up” by making a sizable wager. Instead, a wiser bet would be zero (or close to zero), hoping for a Final Jeopardy answer that was so hard that none of the three would get it right, and that the first two contestants would lose enough money to leave the 3rd Place Contestant standing alone at the top. While that scenario might seem unlikely, it happens to be far more likely than the one most 3rd Place Contestants seem to favor when they bet the farm: the one whereby they (the 3rd Place Contestant) happen get the answer right, while the other two contestants both get it wrong.
Now let’s take a look at the winning strategy for the leader. Here, the math is much simpler, at least at first glance. We’ll discuss some complications in a moment.
A confident leader should wager as follows: (1) double the 2nd Place Contestant’s current dollar amount; (2) add one dollar, just to rule out a tie; and (3) bet the exact amount which will achieve that amount.
So, in the example given above, the leader will (1) double the $7,500 currently held by the 2nd Place Contestant, arriving at a total of $15,000; (2) add one dollar, which brings us to $15,001; (3) wager exactly $5,001, because that amount, when added to $10,000, will bring the leader to exactly $15,001 if the Final Jeopardy answer is correctly answered.
Note that by wagering this exact amount and no more, the leader is able to win under a wide variety of situations. If the leader uses this wager and gets the answer correct, there is no way either of the other two contestants can win, regardless of what they wager and regardless of whether they also get the answer correct. More importantly, by adopting this strategy, the leader is also likely to win when all three contestants get the answer wrong. In most games that I have watched, both the 2nd and 3rd Place Contestants tend to over-wager; i. e., they don’t follow the strategy I recommend in these blog posts. Thus, the leader’s final total of $4,999 (which is $10,000 minus the $5,001 wager) would be more than sufficient to win.
There are, of course, some scenarios in which the leader should vary from this calculation. One of those situations occurs when the leader is so far out in front that the 2nd Place Contestant cannot match the leader’s current score, even by betting everything. In that case, some leaders choose to wager zero, just to ensure they don’t lose any money. Other leaders bet a bit more, just to have fun, but they should never bet so much that the 2nd and 3rd Place Contestants would have a chance to win by betting all their money.
Another difficult wagering scenario occurs when the Final Jeopardy category seems so distasteful that the leader is loathe to bet anything at all. (In my case, that would be a category like “Great Romanian Operas.”) In such a case, the leader has little choice but to wager zero while praying neither of the other contestants–especially the one in 2nd place–will get the answer correct.
Next week we’ll look at the most complicated situation of all–the one faced by the 2nd Place Contestant.