I watch the show often and I am amazed at how often contestants make the foolish mistake by trying to outbid each other by one dollar or some insignificantly small amount just to get clear first. This happens in the case of at least one contestant in at least 95% of episodes, based on my recollection, I am sure this can be checked. The reason it is a mistake is that according to the rules, if the game ends in a tie for first, BOTH contestants get their final money amount and return to the next show. So winning by $1 is worth $1, but losing by $1 is worth thousands of dollars in real money and thousands more in equity (since you don't get to return next episode).

For example, a hypothetical: A has 14,000, B has 10,000, C has 6000 (or less). Regardless of category confidence (except maybe in super extreme cases), the perfect wager for B is 2000. Notice I did not say the correct wager, but the perfect wager, taking into account the expected (exploitable) wagers of others and not the correct (unexploitable) wagers of others. The expected wager of A is 6001, in order to guarantee clear 1st as long as A is correct. So if everyone But if everyone misses, A loses by $1 and costs himself $8000 plus his equity on future shows. A further repercussion is that if A's expected wager was 6000, then B may be able to justify wagering his full 10,000 (depending on category confidence) because he would get his 20,000 whenever he guessed correctly.

Recently, there was actually a show where the contestants had 15,400, 10,200, and 5400. 3rd place risked 5399 and was correct, giving himself 10,799. 2nd place was a woman who foolishly bet 5201 (with the idea of winning clear first by $1 in case the leader risked nothing, which btw almost never happens. She was wrong, leaving her with 4999. Why was she foolish? She had to be correct in order to win (because 1st place had to risk no more than 5000 to guarantee clear first, and a wrong guess would have left her with 10,400. So a correct wager would have been 10,200 leaving her with 0, but again she needed to guess correctly to win in any case. But the best part is the leader who instead of wagering the 5000 needed to guarantee 1st, risked only 4600. Why? Because a wrong guess left her with - that's right - 10,800. Guess who got the shaft? The moron who wouldn't risk his last dollar. GG sir. You could argue this is being results oriented but I could argue back that 1st place used metagame to determine that 2nd place was too conservative to risk all her money, therefore making 4600 the perfect wager.

This hypothetical and others like it clearly show that it is godawful strategy for contestants to try and outbid each other by $1. And while Jeopardy is not a game that requires a lot of raw intelligence but rather memorization and recall skills, there is generally a decent correlation between interest in academic trivia factoid and general intelligence. So I would think most contestants should be able to figure out (based on the reasonable assumption that they are apprised of the rules) that tying for first = way better strategy than clear first. This doesn't even require any kind of superrationality or collusion because the loss of $1 or $100 or whatever is so insignificant compared to the potential loss of losing by that amount.

Therefore I can only conclude that one of two things is true- 1)either the contestants, despite being academically inclined enough to succeed on the show and answer most clues correctly, are really unintelligent and can't figure out basic wagering concepts, or 2) the producers of the show incentivize irrational bidding of $1 increments in order to avoid ties so that they can pay out less money and churn out more new contestants, while preserving the charade of having "fair" rules for ties. Obviously this would be a huge bombshell but I just can't get my head around the idea that acutely knowledgeable and learned contestants can be so naive to wagering strategy as to cost themselves potentially thousands of dollars in equity, so the other option seems relatively the most likely possibility.

Now tell me how mad I am for this crazy theory because it is disturbing me.

A lot of time the guy in the lead will bet an amount that will bring him to 2x (2nd place) +1$. The justification for it is that it eliminates being tied with an above average player, in which case he can face 2 weaker players in the next round.

A bigger mistake is when its something like A 18000 B 14000 C 3000
A's expected wager is 10000/10001. B, anticipating this should choose to wager between 5999 and 4000. Since he will win automatically if A gets the question wrong. Betting more than 6000 is a massive mistake since B can only win if he gets it correct and A gets it wrong.

A sick move by A would be to anticipate this and bet 2001.

OP's mistake is that if the values of people's 'stack' going into final jeopardy are different, then people have to get the answer 'right' regardless. So OP's argument doesn't work.

Stacks of 14000, 10000, and 8000.

Leader should wager 6001. Why do you think that this is irrational?

That's kind of what his whole post is about. You need an argument like rmoriar suggests to explain why outright winning is better than a tie, since the winner gets the same money (minus 1 dollar, obviously) regardless.

That said, rmoriar's explanation that a tie will often suggest a stronger than average player is a good reason why you might not play to tie. I have seen it said that they also strongly discourage people from playing that way, so perhaps it's just respecting the wishes of the producers.

Chris Ferguson's dad has a paper somewhere about Jeopardy wagering.

EDIT: I think most leaders are much better off betting with the expectation they'll get it right, as there are substantial money gains to be made from it. The only scenario where A is better off doing this is a triple-stumper type thing. Planning for those as one of the trailing contestants makes sense because it's the only way you can win, usually, but I think it would be pretty wrong from the leader.

Hmm I think your post has merit... all I can think as an explanation is that Final Jeopardy is generally easier than the rest of the game, as there is no time pressure to buzz in first. So I think all contestants usually wager on the assumption that all 3 will get it right.

This is similar to double jeopardy... I don't have the stats but I'm sure Double jeopardy questions are answered correctly a large majority of the time, and there is no time pressure to be first on the buzzer which is a huge edge over a typical question. It seems that the most +EV thing would be to bet it all every time you land on a DD, as that is almost like getting pocket AA in holdem, you want to get all your money in when you have the edge.

Because 2nd should risk 2000 or 0 and 3rd should risk 0. If 1st is wrong he will lose by 1 dollar. 6001 is literally the worst possible wager he could make, even 14,000 would be preferable. If he does the smart thing and bets 6000, he will win unless he is wrong AND second place is correct.

The argument for eliminating a strong opponent may be valid in theory but in practice you can almost never be certain enough that your opponent is in fact above average to justify the EV loss of risking losing by $1. The culprit is most often the leader who bids a dollar more than 2nd place can match. Unless 2nd was a champion of twice or more, the best case for knowing that 2nd place is well above average is if you and him both have really high scores (like ~17k+ without substantial help from double jeopardy) and even that could be because the categories were favorable or the third player just flat out sucked. The vast majority of the time there is insufficient evidence, due to sample size and variance issues, to assess how good your opponent is.

So it makes the most sense that the producers tell them to win by a dollar, which of course destroys the credibility of the show because it's not supposed to be, you know, rigged. Even if it was just tying that they didn't like, that still wouldn't explain the win-by-a-dollar strategy. Take rmoriar's example of 18k, 14k, 3k. If 1st is going to bet 10,001, he might as well bet 11,999, or at least 10,999, to beat 3rd place and undercut B's potential strategy to do the same (this is in the case that 3rd place is correct and the other two are wrong).

But of course nobody does that, so it must be that they are told to outbid by a dollar if they are in the lead. Doing it on their own would be abject stupidity. And Jeopardy is a difficult game that requires a lot of academic knowledge and good recall ability, which more often than not belong to intelligent people.

So 1st place wagers 6001. You argue that 2nd should wager 2000. But what if 3rd bets it all and wins? Now 2nd can't possibly win unless both 1^3 get it wrong.

I don't think that you've worked out the game theory properly. Game theoretically optimal strategies are those which allow the agent to do best regardless of what their opponent does.

This is assuming 2nd is going to bet exactly 2000. In which case A betting 0>>>>>>>>>>>>>>>>>>>>>>>>6000>>>>>>>>>>>6001

"Optimal" strategy isn't best if you have high enough confidence in what your opponents will actually do.

2nd can't win unless 1st is wrong in any case.

3rd place is almost never going to bet everything. If 3rd knew for certain that 2nd was going to bet 2k, even then he should only bet 8000 when he is at least roughly two thirds as likely to be right as his opponent is to be wrong. In other words, not if they are both likely to miss.

So since 3rd likely won't (and shouldn't) bet everything, 2000 is the correct wager for 2nd.

This chain of reasoning only applies because 3rd place can win without adding to his score. If he had 7800, then betting everything would be a no-brainer because he cannot win unless he is correct. But 2nd should still bet 2000 unless the category was quite easy.

Tonight's episode featured contestants entering final jeopardy with 8600, 9600, and 11,600 (the 1-time defending champ). Their respective wagers were 4599, 2001 (LOL), and 7601. More evidence that the producers are rigging the show. I'm just wondering how they incentivize people to wager that way, when the rules of the show state that you get all your money and return for the next show if you tie.

In what order do they bet?

all at the same time. they are first told the category, then make their wager during the "commercial break" (of course it's taped though so idk how long they actually have), then they get the "clue" and have 30 seconds to answer.

They give you as much time as you need and the staff can even "help" with the math.

According to j-archive, last season there was a 51.69% chance of supplying the correct Final response. Doesn't this mean that the leader should just double or nothing every time?

It seems to me that you should only maximize you chance for first when you have a legit shot at making it into the TOC, otherwise the best play is to just maximize your winnings.

I doubt there is any rigging. And I also doubt that people play to win and not tie, because they want weaker opponents the next time. For the most part all Jeopardy contestants are strong and there is no guarantee the next night's opponent won't be better than the one you (could have) tied with.

I really just think the mindset of the players is to win, and the thought process goes no further than that... I could be wrong

Here are a couple links:

From 2000: http://www.slate.com/id/86351/
From 1994: http://www.jstor.org/stable/2690846

Think about this one:

1st place has 12k
2nd has 9k
3rd has 6k

How do you bet as each player?

There's not a unique answer to that question.

So his use of the word "you" was fitting then.

No. There's not even a unique answer to how *I* bet that.

So if you were playing you would choose not to bet? flip a coin? read your opponents' body language for tells?

It depends on the (perceived) difficulty of the category.

How about for this category: Sides of a coin.

If I'm first, I best 6K
If I'm second, I bet 3K
If I'm third, I bet zero.

This seems obvious, but if it's not, let me know and I can clarify. The only assumption that I'm making is that when I put myself in each player's shoes, is that I assume the opponents are level one thinkers. I think this is a reasonable assumption since you don't normally know your opponents well enough to "out-level" them.

Yeah based on your assumption you are right. 1 can only lose if he is wrong and 2nd is right. However in this particular case, it would be a lot easier for 1 and 3 to think on level 2 or higher, because #1 has a high incentive to bet $0. The reason being that 2's wager seems obvious to be the only correct wager no matter what the others do, and if he makes it then 1 can guarantee a win (and a future appearance) by betting $0, even if he gets it wrong. This possibility may compel 3 to bet 6k as it will guarantee him a $12k win if he is right, compared to needing a wrong answer from both 1 and 2 (and an 6k wager by 1) to win if he bets $0. Of course #2 could throw a wrench in the works by making a foolish bet of say 3001, or 9000, for which your proposed wagers would work the best.