Now I wonder if we're talking about the same thing or if you understand the concept of negative utility. You realize 99% of leaders always bet enough to cover a double-up by second place? So not only are you accusing most jeopardy points-leaders of being morons, you're dismissing their position (along with mine) out of hand w/o even bothering to offer a real rebuttal. Do I have this right or are we mis-communicating?

You're looking at this like making a +$EV thin value bet on the river in poker. Random poker game != national TV in front of friends and family. You have to give precedence to not looking stupid and/or having to question yourself later for pulling out some over-thinking-it strategy on the final wager amount.

I would have said Sheen.

Yeah that was my only guess too.

P1's failure is something I should've seen coming, who wears a leather jacket indoors? Does he not own a suit? Go to Goodwill or something for God's sake you're going to be on national television.


Tyler Durden- The Czechslovakia guess wasn't terrible because he didn't know the details of the Munich conference, it was terrible because the guessing process for that sort of question should start with "What countries do I know Germany invaded in WW2?", and the operation's name provides very few context clues, so you just gotta go from there with general question difficulty guiding you.

Guessing Poland there is totally reasonable. It's wrong but it's not dumb. His guess betrays either a criminally woeful understanding of history(like if you didn't even know Germany invaded Poland, the guessing process starts with "what countries border Germany?") but more likely a the guy was just really bad at the educated guess process, which imo is the core skill of Jeopardy.

There is a wagering calculator available for Jeopardy! that I think is pretty good and some of you might be interested in it based on the discussion here.

http://www.j-archive.com/wageringcalculator.php

There is a whole lexicon of types of wagers and wagering strategies on that site.

Also when you're in first place in some situations it can definitely be "correct" in some sense to not bet enough to cover a doubled bet by second place. This can occur when the 1st-place player's score is equal to twice the difference between the scores of the 2nd- and 3rd-place players.

In that case, second place should bet exactly enough to tie 1st place's score going into final Jeopardy! which is also exactly enough to protect against an all in bet of 3rd place.

If you're in first you should bet zero since then you can't lose. However you are kind of relying on 2nd to realize how he should bet, and it is going to hurt if he bets enough to overtake you, you both get it right, and you lose.

This strategy is called the "Faith Love" after a 4 time champion who went 0 for 5 in Final Jeopardy and successfully executed this strategy in one of her games.

To elaborate on the guessing process stuff, take a look at the last question. Father son acting team, portrayed real US Presidents. A bunch of guys are guessing Sheen(as did P1), but that is focusing on the father/son stuff instead the better elimination tool: playing a real President. Movies with real Presidents as major characters are pretty rare. I got the question right and I had no idea who played Reagan in that TV thing.

Yeah Brolin is a better guess if you remember that James Brolin played one in a movie. But if you don't even get that far you're SOL. Douglas might even be good too, hoping Kirk Douglas played one on TV in the 50s or something.

I did not watch this episode but WTF at the $4000 DD bet???

Another thing that tilts me is when the leader has, for example, $20,000, and there are just a few clues left. Second place has 9,800 and does not even attempt to ring in on the last 2 clues, even after it is clear that the leader is not ringing in either. Even if I know nothing about Shakespeare or whatever, in this situation, if I am the guy in 2nd, I am taking at least a wild guess on both of the $1600 and $2000 clues (in that order). (Have to at least have a shot to win in FJ FFS!)

And the teen/college tournaments where the winners advance as well as the 4 top non-winners...almost every year when one player has locked up the win, someone else with like 10K will make a wager of 5-6K. Yeah, hold onto that other 4-5K just in case you are wrong...see where that gets you.

On an unrelated note, I wish Alex would not practically give away answers to other players sometimes--like when someone barely mispronounces a name or something, Alex will grimace and say "ooohhhh no", then one of the other contestants rings in (who probably wouldn't otherwise), with the correct pronunciation to get the $$$. This, however, would be acceptable to me if he occasionally did that for wrong answers as well...such as

Contestant 1: "Who is John *Stenbeck*?"

Alex: (grimace) "Oooohh, no"

Contestant 2 (smugly): "Who is John *Steinbeck*?"

Alex: "No, sorry. It was Hemingway."

Yes, I see no problem with this. My point is that you should never assume that another player is going to make a bet like that, even if you think they are "correct" to do so. You basically have to guard against losing if the player below you doubles up and not allow that to happen if you get the question right.

Pudge,

Do you think that P2 will bet over $10,200? If so, and assuming P3 wants to maximize her winning chances, why would she ever bet $2400 or more? Betting less GUARANTEES her victory if P2 gets it wrong and she will NEVER win if he gets it right anyway assuming he bets +$10,200. The only way her bet is acceptable at all is if she:

a) believes that P2 will bet less than $10,000 and she can win even when they both get it right.
b) values (her bet size-2399) * (probabilty she gets question right AND P2 gets it wrong) more than winning and coming back the next day. I also think any reasonable estimate of that probabilty is going to be VERY low barrring some absurd category that she is expert in and knows he is likely to be weak in.

IMO either of those assumptions are just about impossible to make.

PS- I'm sure my equation is technically wrong and it can be picked apart by people better at math than myself, but the principle I'm trying to illustrate is correct (I think) and I hope I was able to write it in a way where you can see what I'm getting at.

chalk this up as a thread I didn't think would take off, but did.

Ungo,

You realize that only the winner gets to keep the money right? The other players get "prizes" I believe that are on the order of recliners and what not, which you'd eventually get anyway after a win.

.

As another example of an interesting situation, imagine you are one of two players tied for first with $10,000 and the third place player has $4000. Remember if there's a tie you both come back as champs. I think I'd bet $0 and cough strongly in the direction of the other guy...if you both bet $0 you will both win no matter what obviously.

I'd probably then commit suicide after the other guy bets $1 and gets it right.

This is OOT trying to prove how much smarter we are than other people, of course it would take off.

This is an interesting question.

Say you are the 1001 guy and you know the other guy bets x. The correct move is to wager something between x+1 and x-1. If you're right then we will win. If you and your opponent are wrong then you will still win. The only way you lose is to be wrong and have him be right, which happens 25% of the time. Betting more causes you to lose even when you both are wrong while not gaining you any win equity, betting less is similar but you lose when you are both right. Since betting any amount is EV neutral ignoring winning, you can't gain from betting more or less, but it does cost you.

So your best response is to be in that (x-1,x+1) interval.

Say you bet y<999. If your opponent wagers x < y -1 then he will win whenever you are wrong and lose when you are right, making him win half the time. If he wagers x > y + 1 then he will win whenever he's right and lose whenever he's wrong, giving himself a 50/50 shot. If he wagers in between those, then he'll have only a 25% chance for the reasons I gave above when discussing what you should wager when he wagers x.

That makes his best response to be either greater than x+1 or less than x-1.

Because he can either go big or small given a fixed strategy for you, if he knows what number you are putting down then he can guarantee himself a 50% chance. Similarly, because an optimal wager for you if you know what he's putting down can be above or below it, you can always guarantee yourself a 3:1 edge if you cheat and know what he's putting down.

That means that there's no fixed equilibrium. The solution has to be something involving randomness.

I haven't actually studied FJ wagering, so I'm not sure if you can get an equilibrium by reducing the strategy space to just two possible wager sizes but it seems reasonable. That's a good place to start though.

Let's say the large bet is 1,000 and the small bet is 0. We'll say I'm the guy with 1,000 and you're the guy with 1,001. If I bet 1,000 with probability p and 0 with probability 1-p then you will win by betting 0 if either I'm wrong or I'm betting 0. This happens with probability .5p + 1 -p = 1-.5p. If you bet 1,000 you win if you are right, or if you're wrong but so am I and I bet 1,000 (you would win 1-0 in this case). The chances of that happening are .5 + .5*.5*p = .5 + .25p. As I showed above, it's not an equilibrium if either player is playing a fixed strategy. It only makes sense for you to randomize over the bet sizes if you are indifferent between playing either option, which only makes sense if the win% from betting 1,000 is the same as betting 0. That means that I must set p so that 1 - .5p = .5 + .25p or p = 2/3.

Say you bet 1,000 with probability q and 0 with probability 1-q. If I bet 0 then I win if you are wrong and bet 1,000. That probability is .5*q. If I bet 1,000 then I win if I am right and you are either wrong or bet 0. That probability is .5(1-q + .5q) = .5 - .25q. For the reasoning given above, we set these two equal and solve for q giving us q = 2/3.

So in this case, the unique equilibrium would be for each of us to wager 1,000 2/3 of the time and to wager 0 the other 1/3.

Actually, now that I think about it, this should be an equilibrium of the actual game. If you are doing this 2/3, 1/3 randomness then altering my wagers either doesn't matter or screws me. As I showed, I can't improve by wagering 0 or 1,000 all the time. I also can't improve by wagering anything in between. Any number less than 1 gives me a win in the same scenarios as betting 0 and anything 1 or greater gives me the same win scenarios as betting 1000. So I would have the same payoff across the whole spectrum of numbers, meaning that I can't do better by deviating whether that be playing a pure or mixed strategy.

For you, if I'm doing the 2/3, 1/3 mixing then betting less than 1 gives you the same win scenarios as betting 0. Putting anything between 1 and 999 is worse for you than betting 0 (and hence the mixed strategy) because you never win when you would have lost by putting 0 and you lose instead of win when you are wrong and I bet 0. Putting anything > 999 puts you in the same situation as wagering 1,000.

So yeah, that is an equilibrium for the game. There are definitely others, probably some where you could possibly bet any number, but that is the simplest IMO. A key to this is that each play is 50/50 to get it right meaning that the only thing that matters is winning or losing. If that weren't the case then my argument for it being an equilibrium would break down. For example, betting 100 might be better than betting 0 if you are a 3:1 favorite to be right because you gain $50 in equity which might be worth the hit you take in chances of being champion.

edit: I wasn't grunching but made such a long post that I missed several responses. Apparently second doesn't get to keep their money, which changes things a bit, though I suspect not much.

Dude. How long have you watched this show?

Baltazar(sp?) is a moron.

idiotic wagering on daily doubles drives me nuts, also final betting in Jeopardy. how can guys know all this stuff and have no clue about game theory. there is also skill in picking which squares you pick which i see a lot of players do poorly.

finally I had to stop watching millionaire because all these idiots get on the show and have zero clue on how to properly use their lifelines.

ugh its tilting me just to think about it.

It would have helped if I noticed the word "real" in the answer.

Isn't the whole question of strategic betting complicated by the fact that you win the amount you have at the end? If you're in first and you are confident in the category with a high probability of getting the question right, then you gain value by betting big. It's not purely a defensive decision.

This is absolutely true. However you can't really analyze it in a general way because it will be so tailored to the individual.

Even keeping it in the realm of playing solely to maximize your chance of winning the game, in many cases you have a range of wagers that don't affect your chance of winning. You can bet more if you're aggressive and less if you are passive within that range of wagers. On the j-archive site I linked earlier, they use the terminology "martian" for more aggressive wagers and "venusian" for less aggressive wagers.

In many instances if you are the trailing player, you have a range of bets such that if the leading player gets the question wrong, you will win regardless of if you get it wrong or right. This is making the assumption that the leading player will bet enough to cover an all in bet by you, which will generally be the case.

by the way someone wrote a phD thesis on final jeopardy wagering if you google around you can find it - it's of course unreadable, but it shows that it's more complicated than at least OP is giving it credit for being.

Great post Jared. I understood most of it, I think. Trying to teach myself basic game theory but it'll take several months before I can do that kind of calculation even though it isn't too difficult. Maybe it is simpler if the dolllar amounts for both players are only 3 and 2.

Of course, but I don't see how that's relevant to anything I said. I just think you're overrating the value of coming back for another day. What is the average amount won on a single day of Jeopardy? I looked but couldn't find any decent stats anywhere, but I can't imagine it's much higher than $15000. Keep in mind you have days like in the OP to balance the outliers where they win $30,0000+. If the returning champion wins again, let's say 50% of the time they come back, EV of future games would be:

0.5 * ($15,000 + 0.5 * ($15,000 + 0.5 * ($15,0000 + ... ))), which equals $30,000.

I think that's being very generous, especially with the 50% estimate. (If you make him an average player with a 33% chance of repeating, the EV plummets to $22,500.)

If P3 thinks he has a very good chance of answering the question correctly when P2 gets it wrong, let's say 80%, then it's very close to neutral EV for him to bet everything:

(0.8) * ($11,900 + $11,900 + $30,000) = $43,040
(0.8) * ($11,900 + $2,399 + $30,000) + (0.2) * ($11,900 - $2,399 + $30,000) = $43,339.4

Again, you can change the numbers however you want to get a different result, but it's hardly an idiotic bet. Winning more money on future shows is hard, and all I'm saying is that giving up nearly $10,000 in immediate EV just to guarantee a win isn't a no-brainer.

wat. I mean I guess if you have a ******edly skewed utility curve, where you care more about , what randoms think about you and randoms possibly busting your balls you are right. But I care much more about maximizing my expectation, than caring what my dad thinks I should have wagered. If someone wanted to reduce variance instead of pushing there 5% edge, I would have no problem with that, but as someone who is insecure, I find this hypothetical person to be cartoonishly insecure.


Yes. If we assume shorties bet 0 and you are more than 50/50 to get the question right. Shoving is +EV. There is no difference between 10,200 and 13,000 in terms of P(win), but you are passing up P(correct)*2800 in EV.



Because she doesn't want to guarantee victory, she wants to maximize her expectation.

Guaranteeing victory isn't the end game. Using your same logic why doesn't P3 bet 201? He will always win if he gets it right and P2 bets optimal 2400, and he will win if the other two get it right? ITT you are saying A is stupid, than B is stupid for adjusting to A's stupidity, C is stupid for adjusting to B an A's stupidity.

These aren't that unlikely outcomes if she feels P3 is less than 50/50 to get the question right, she would be better off shoving.