Since we’re in tournament mode, let’s turn our sights to next week’s twogame final match.
This scenario comes from the 2008 College Championship. The yellow scores are the totals from Game 1, which will be added to the totals from the present game. What should each player wager?
Thanks to wjmorris3 for pointing out that players keep whatever they win, subject to a minimum. The guarantees here were $100k for the champion, $50k for second, and $25k for third.
If you’re unsure how to approach this, fear not – I’ll explain it to you soon!
(Thanks to the J! Archive)
This situation comes from the third semifinal of the 2004 Tournament of Champions. What shall each player do? It’s trickier than it looks – and I’ll break my thinking into two sections.
(click to view as a slideshow)


(1/10) Arthur wagers 14,001 (remember the extra dollar!) to cover Tom.


(2/10) Tom can put up to 12,800 at risk to stay above Arthur.


(3/10) If Steve doubles up, he’ll have Tom’s score. But since this is a tournament, perhaps Tom tries to avoid the tie if he likes the category.


(4/10) But Tom will lose if he gets it wrong and Steve goes all in …


(5/10) … so a tiebreaker wouldn’t be such a bad thing if he doesn’t like the category.


(6/10) Steve can wager at most 5,200 to stay above Arthur should Arthur miss.


(7/10) Now, for Rule #3. Tom can wager at least 1,201 to beat out Arthur’s zero wager.


(8/10) Tom should also wager no more than 7,599 to stay above Steve’s preFinal total.


(9/10) No improvements for Arthur or Steve against one another.


(10/10) Here are our preliminary “best” wagers.
Now, we turn it up a bit to look at some advanced thinking – “poker” if you will.


(1/6) We start from where we finished in our initial goaround.


(2/6) If Steve wagers “small”, to stay above Arthur, Tom can wager up to 2,399.


(3/6) Now if Arthur does an “unsafe” wager …


(4/6) … and gets it right, Tom can beat him with a 2,400 wager.


(5/6) This is fine, but he’ll risk a tie with Steve in this case.


(6/6) In the end, all three players wagered poorly!
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A reminder that must be put in – on a two day final, if the contestants score higher than the stated prize, they keep the overage.
Okay, first off, I have Joey needing to bet 16,401 in order to hold off Danielle as Danielle would have 56000 if she doubled up.
He misses on this wager, and he’s down to $23,199 – which means Danielle could in theory bet everything and still finish ahead if they both got it wrong. But Danielle shouldn’t bet any more than $10,201. In fact, I would bet exactly $10,201 on it in an attempt to maximize winnings – that year, second place paid $50,000, so anything bet over $7,000 is money bet at no risk (!)
Andrew, unfortunately, is playing for second place here if bets are rational. But he needs to get it right in order to have any hope at second place, so he might as well bet it all. Again, maximizing winnings is the key here – third place paid $25,000 anyways so anything bet over $2,602 is potentially found money.
Good point! I’ve added that info to the post.
Your recommended wager for Danielle could run into some HUGE problems. I’ll explain next week, but if you’re chomping at the bit, check out the actual wagers in the link above.
I see 10201 as the maximum wager that still guarantees second place. Andrew gets to $32,798 at the most and Danielle would finish at $32,799 if wrong.
And now I see the huge problems. Maybe a bet of 0 is in order for Danielle, to force Joey to get it right to win.
What link?
@wjmorris3 – The situation you found is possible in multiday wagering but not in singleday matches because a smaller firstday total and a larger secondday one (Joey here) can give you the ability to cover and win if right despite not being cumulatively to that point in the lead, while a player with a large firstday total and a smaller secondday one (Danielle here) has the ability to wager to win on a doublestumper while still forcing the opponent to have to wager and get Final right to be able to win at all (in other words, unlike in a oneday match, a player in Joey’s position cannot make a small/unsafe wager and guarantee a win on a double stumper as an alternative to covering if he’s uncomfortable with the category).
Oops – should have been posted under the appropriate comment.
Yes, the score from game 1 can add some interesting kinks. For example: Joey is in the lead in this individual game, but if both he and Danielle wager zero, Danielle will win. And this situation is rather tame.
If my math is correct this kind of scenario will occur whenever the contestant with the higher cumulative total (in this game Danielle) and the highest maximum possible total (in this game Joey) at the end are not the same. In such a case the player in the latter position really has no option but to make the cover bet, while whoever is in the former position should limit his/her wager to stay above the opponent’s cumulative total and force him/her to be right to have any chance at winning (unless a larger bet is needed to cover the third player, which wasn’t the case in the example game).
Andrew’s situation is the simplest. I doubt that either Joey or Danielle will bet enough that Andrew can possibly get out of last place; thus he should ignore that possibility and just optimize his total payout. However, with third place paying a minimum of $25k, which is more than his present twoday total, Andrew literally has nothing to lose, and thus should bet everything — even if the category is much tougher than he would want to try!
Danielle has the highest carryover score and the highest present twoday total, but not the highest maximum possible result. In her shoes I would cap my wager at 10,202 (so that if she misses she’d still have 32,798, the same amount Andrew will have if he doubles up).
(This brings up an interesting question. What happens if there’s a tie? Unlike either a nontournament or the earlier rounds, there isn’t any subsequent game for you to get into by winning. Would a tie for second place mean they are each guaranteed $50k, or would the producers average the guarantees and give them each 37,500, or what? If it makes a difference to her own guaranteed payout I’d have Danielle bet $1 less, or 10,201.)
There is no reason for Danielle to bet less than this cap, since Joey can beat her maximum. Danielle cannot “control her own destiny” as you normally advise the leader to do.
Now for Joey. Joey is not in the lead but has the greatest maximum possible result, $64,000. Even that is much less than the guaranteed payout for first place. However, the marginal utility of money decreases with the amount you have, so in his shoes I would be much more concerned with avoiding third place than with getting first, nice though that would be. Therefore, in his shoes I cap my bet at 6,802 — thus giving him 32,798 (Andrew’s max) if he gets it wrong. As in Danielle’s case, reduce this by $1 if the guaranteed payout for being tied 2nd/3rd than for 2nd.
Last sentence should end “… is less than for 2nd.”