Monday, April 21, 2008

Jeopardy!


I'm a big fan of strategy, and economics satiates a good deal of my desire for it. Poker does too. Nonetheless, I've yet to encounter a diminishing marginal return to more competition in my life, so I enjoy as much of it as I can get.

Betting in final Jeopardy fascinates me. Some ideas I've been tossing around (and I'm sure this has been written about somewhere else many times; feel free to link it up):

- If you're in the lead, do you worry at all about the person in third place? I'm leaning now towards no, but I could be convinced otherwise. I think it may involve all three people being pretty close.

- If you're in third, obviously you want to two people ahead of you to be close, as presumably they both bet large amounts trying to beat each other and, if they both miss, the first person can slide in for the win (which happened tonight...on a final Jeopardy question category of "Military men" and having an enlisted serviceman in third place. Rigged? You decide.). But couldn't the top two people recognize this and mutually lower the stakes to effectively eliminate the lower player regardless of outcome? I know, there's no communication, it's a one time game, and even if both of those weren't true, it's not a stable equilibrium...nonetheless, I'm surprised this hasn't caught on in small bits here and there.

- I would think that if you're confident in your ability to answer one question-- and given the selection bias in people that can actually get on this show, I'd sense this is probably more true than not-- you'd find the top person always betting such that if they get it right and the second person bets it all and gets it right, they win by a dollar. Of course, knowing this, the second place person has an optimal betting strategy themselves, and then the third person too. We don't witness these respective strategies every single night, so where does this break down?

- I could be wrong, but in the past, if contestants tied, they brought them both back the next night. Why wouldn't people go for this? It's a chance to build trust with another contestant and make Jeopardy a repeated game. I suppose there's the fear of having someone approximately your equal back in the next game with you, but if you build the trust, they're not trying to get rid of you-- they could also go for the tie. Shirking is always a possibility...I'm surprised people haven't at least attempted the tie and move on in attempt to make friendly with another contestant. All you lose is one dollar for the chance to be playing the game with someone else. Maybe they don't let ties move on together anymore...in which case, strike all of this.

- Has the pattern of betting in final Jeopardy evolved over the years? There's totally a paper there. I'd suspect people learn and refine betting strategies; we may not be at the optimal betting solution, but given the path we took, I'd bet we're pretty near a locally optimal solution. Along the same lines, did Ken Jennings change his betting strategy throughout his run?

37 comments:

Andy Saunders said...

Hey:

Ken never substantively changed his wagering strategy over his run; granted, the wagering strategy was pretty trivial for most of his games due to his utter dominance of the main game.

The discussion over at Jeopardy!'s message board sometimes gravitates towards wagering strategies; some posters there have proposed strategies for the leader that technically would take 3rd place into account; however, they are considered "risky" due to the possibility of losing the game despite leading going into Final and getting the final question correct.

I personally have been meaning to do a write-up of wagering strategy for my own site, but haven't gotten around to it yet.

Andy Saunders
http://www.andysthoughts.com/

Andy Saunders said...

Also, regarding "forced ties"; while they would be possible in theory, if it continued for any length of time, I believe that the show's Standards and Practices auditors might get involved, worried that the show would appear to be "rigged".

Andy

Lisa Ross said...

I don't think a "planned tie" would work very well. First of all, the contestants are presumed to be strangers. Would you really trust a stranger to make that sort of deal with you right before a competition? It would be in the stranger's best interest to cheat on that deal. Second of all, it would only work if both players answered the final question correctly and they ended Double Jeopardy within a certain price range of one another.

I think it's more interesting to consider the betting strategies of second place. It seems to me they have three basic options:

1. Bet it all and hope that first place miscalculated their "one dollar more than double" wager.

2. Apply that same "one dollar more than double" strategy toward beating the third place contestant. Hopefully first place would wager too much and answer incorrectly. Thus allowing second place to beat out both contestants with a correct answer.

3. Calculate the amount first place is planning to wager (based on the "one dollar more than double the second contestant's score" strategy). The second place contestant can then plan to wager just enough to beat the first place contestant by one dollar if she gets the answer wrong. Of course, this strategy still needs to ensure that it is at least double the third place contestant's score.

I'm sure there are more possible strategies, but these are the ones that seem most obvious to me.

As far as first place goes, I would be sure to wager to win by at least one dollar more than double the second player's score. This is, in my opinion, the only strategy if you have any knowledge at all of the final Jeopardy category. Playing the video game version (with nothing to lose), I tend to wager allowing at least a $100 win over the second place contestant (just in case I miscalculated somehow). I guess it just depends how much risk you're wiling to take.

Bryce said...

This is a great topic. I get pissed (seriously) when people bet illogically in final jeopardy. I've seen the 2nd place person lost SO many times cause they wager it all for no reason. In my opinion, final wagering should follow a pretty basic pattern (in most cases):

First place wagers enough to guarantee a win if everyone answers correctly (Controlling your own destiny type thinking).

Second place should wager enough to equal the 1st place money total prior to final jeopary and enough to beat the 3rd place person if they get it right. This way, you win if you get final jeopardy correct and the 1st place person misses.

I actually think 3rd place wagering is most interesting in many situations. You realistically need both people ahead of you to miss, so in some cases, you should wager nothing and assume if the question stumps the two ahead of you, you're going to miss it too. Unless this total will put you under the probable totals after the other two miss, or you kick ass in the final category, in which case you should wager enough to pass their probable totals if they both miss and you get it.

Example:

1st: $1000 - bet $601
2nd: $800 - bet $201
3rd: $500 - bet $0 (or $100)

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