Awesome Jeopardy Math!

I’m a huge fan of the Jeopardy! game show. My viewing habits have ranged from diligently watching every episode live with my college roommates to DVRing every episode and catching up on the weekends to (currently) tuning in when I can, and not really stressing too much if I miss it (though I probably catch at least a portion of 3/5 episodes a week).  That being said – I’ve watched a lot of Jeopardy!! And as a math nerd (who really enjoys game theory), I love to analyze how much people wager (on Daily Doubles, but, more interestingly, on Final Jeopardy).

If you’re unfamiliar to the concept of Final Jeopardy! wagering, the general run-down is this: throughout the game, you’ve accumulated as much money as possible (while your two opponents have done the same — hopefully your trigger finger has been better than theirs!) and you are provided a topic for which the final Question/Answer combo is related to. Before you see the Answer to which you give your question (yeah, Jeopardy! is a game of you asking Questions to the provided Answers), you must wager an amount of money that neither of your opponents will see or know. You then get the Answer, provide your question, and Alex reveals each player’s answer (from last place to first) and their wager (taking them up or down accordingly).  You can wager any part of your current score (from $0 to all of it).

So there’s the setup – I watch a lot of Jeopardy!, and the wagering for the final question has the potential of being fun to analyze.

Tonight (January 10, 2013) had the following scores going into final Jeopardy:

Player A) 9600
Player B) 13000
Player C) 16400

Now, this is where I’d analyze a couple interesting situations that could come up.  There isn’t necessarily an optimal strategy, but there are a few different strategies that sometime make more sense than others.  Depending upon your confidence in the category that is revealed, and your confidence in other players’ abilities, you can pick the wager that you think is best for you.  So a run-down for me here would say this:

Player C knows that B has been doing pretty well this game (that’s why it’s not a sure thing that C is going to win).  So C might think “If B bets everything, she’ll have 26000, and so I need at least 26001 to beat her!”  and therefore A’s wager should be 9601 (that’s my thinking at least).

And Player B might know that C is thinking that way, so they’ll say “Well crap.. I can’t beat C if she gets it right… I’ve got to hope she gets it wrong.  But if A gets it right.. and A doubles her score up, then she’ll have 19200 — I’ve got to beat THAT!” and so B’s wager should be 6201 (again.. this is one possibility.  B could think “Well, if C’s playing to beat me, and I can only beat her if she gets it wrong, then I should wager 0 and hope it’s a super tough question that they BOTH get wrong.”  Another valid option.).

Finally, Player A is in a tough situation.  She’s 3400 behind B, and 6800 behind C.  If B & C get it wrong, they’re probably both wagering a lot, so maybe wagering nothing and hoping for incorrect answers would be good.  But really, she’s got nothing to lose, so she should probably just be confident in herself and wager everything. But crap, what if she lucks out and knows the answer — and C gets it wrong, and B went with the C getting it wrong “I’ll wager 0” — 3401 would beat B’s score and get the win! (I don’t actually know here.  3rd place strategy is tough for me — I just cross my fingers that I’m not in that situation when I get on the show.)

Alright, so now that we’ve analyzed, let’s see the question:  Ah crap, it’s something French that’s a law term — I have no idea.  Let’s see if A knew it — NOPE. 3401.. hmm, she went with the “Let’s assume B is wagering 0” strategy.. oh well.  Did B know it? Nope. 6201!  That’s an awesome play there, beat C and hope A screws up!  Too bad you missed it too :-(  All right, C was smart, they got it right, right?  NO!  Oh man, how much did they wager? 9601!  They wanted to beat B by $1, but instead… what’s this?! THEY TIED!!

So let’s see how this broke down:

Scores going into Final:

Player A) 9600
Player B) 13000
Player C) 16400

Wagers for Final:

Player A) 3401
Player B) 6201
Player C) 9601

Scores after Final

Player A) 6199
Player B) 6799
Player C) 6799

I was floored.  That was some really awesome wagering strategy by each player — they just had a super tough question (that Alex tried to make them feel bad about not knowing “What is Veeerdict. Verdict!”) and if any one of the 3 actually thought “This is going to stump all of us” and wagered 0, they would’ve had it made!

If you’re curious about why B and C ended up with the same final score, it’s because the difference between A and B was 3400 and so was the difference between B & C — it’s like magic!  The only cooler situation I could imagine is if A was half of C — and B split the difference.  That’d be only slightly cooler though.

Thanks for reading!  Feel free to tear apart my analysis in the comments.  I’m off to take the online Jeopardy! test in exactly 1 minute 25 seconds!  Wish me luck!