Wagering in the 2008 College Championship Finals
In my previous Wagering Practice post, I gave you the following situation, from the 2008 College Championship Finals.
When I picked this situation for my example, I didn’t realize how complicated it was. In fact, under one entirely feasible scenario, the three players might be separated by a total of just $2.
This underscores how important it is to know wagering theory backwards and forwards before you go into battle.
I’m going to break this up into two slideshows. The first shows the wagers you would calculate under “basic” game theory conditions. The second delves more deeply into the “mind game” aspect of wagering, which has a potential to rear its head every so often.
Let’s begin. (Click the upper-left square to enter the slideshow.)
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- (1/7) Let’s set this up in our familiar format: we’ll calculate the Game Two totals on the top, then add those to the Game One totals (highlighted).
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- (2/7) First, we see where each player stands by calculating his maximum score. Joey’s in the lead, with Danielle in second.
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- (3/7) Rule #1, first vs. second: Joey wagers 16,401 to lock out Danielle.
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- (4/7) Rule #2, first vs. second: if Joey responds incorrectly with that wager, he’ll have 23,199. That’s less than Danielle’s total from Game One, so she has to do nothing here.
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- (5/7) Rule #1, second vs. third: Andrew’s maximum score is only 2,798 more than Danielle’s first-day total. So she can wager up to 10,201 to stay above him.
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- (6/7) Rule #2, first vs. third: should Joey respond incorrectly with his lock-out wager, Andrew will need to wager at least 802 – and respond correctly.
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- (7/7) No player can improve under Rule #3, so our optimal wagers are fairly straightforward.
Ok, so this seems like an easy puzzle.
But riddle me this, Batman: how did it happen that Joey wagered much less than he was “supposed” to, yet still managed to win by just $400 – even though both he and Danielle responded correctly?
This is where things get tricky – the “mind games” aspect of wagering, which has the potential to take on subtle nuances in the multiple-game format. For example, Joey is in the lead overall, but if he gets it wrong and wagers 16,401, he’s going to lose to Danielle – even if Danielle wagers everything!
I’d also like to point out that had Danielle wagered 10,201 instead, and both she and Joey responded incorrectly, we’d have this situation:
Joey beats Danielle by a dollar – who beats Andrew by a dollar! What is going on here?
Let’s delve into the “second-order” calculations – how Joey and Danielle each might respond to suboptimal wagers by the other.
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- (1/8) We start where we left off in the first slideshow.
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- (2/8) If Joey wagers zero, Danielle can still lose 3,399 and win. So she might as well wager at least that much, because there appears to be very little downside.
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- (3/8) If she responds correctly with that wager, she’ll have 46,399. A wager of 6,800 covers that for Joey. (AHA!)
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- (4/8) Now notice that if Joey wagers 6,801, he will beat Andrew by a dollar if Andrew doubles up. So Joey has a very tight “alternative” range.
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- (5/8) So Danielle should restrict her maximum wager by two dollars, to stay above Joey should he miss and go big.
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- (6/8) If Danielle wants to take her destiny into her own hands, she can wager at least 3,402 to cover that maximum 6,801 “alternative” wager by Joey.
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- (7/8) If Joey wants to cover a zero wager by Danielle, he’ll need to put up at least 3,401. Nothing that benefits him …
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- (8/8) … but if he gets it wrong, Danielle can wager at most 6,800 to still win against him. So that becomes her new maximum wager.
Joey’s wager makes a little more sense in this light. Danielle should aim at the low end of that range only if she thinks Joey will wager zero. She must have been scared of that, which is why she stayed below 3,400.
But on Jeopardy!, you play to win – not to not lose. Danielle’s 3,000 was a fearful wager, one that cost her the trophy, an additional $50,000, and a trip to the Tournament of Champions. Joey’s was, too – but his bluff won this hand of poker.