Final Jeopardy wagering theory

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Vanya
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Re: Final Jeopardy wagering theory

Post by Vanya »

webster2000 wrote:This is my first post to this Board. I have been thinking about FJ Wagers for a couple of years and most intensely in the last two months. My point of departure has been my working knowledge of Game Theory. I have noticed that none of the FJ Wagering articles I have uncovered ever mention the need for mixed strategies, however the most usual combination of holdings at Final appears to require that the optimizing player avoid a rigid choice and resort to randomizing to insure their opponents do not anticipate his/her strategy. I finally found some basis for resolving my curiosity about a Game Theory solution to FJ and have obtained some basic results.My results are close to conventional wisdom except for the need for the top player to only bet high 2/3 of the time. If the players used this theoretically optimum strategy, the top player would win 1/2 the time while the second highest player would win 1/3, with the lowest winning 1/6 the time.

It is unlikely that Jeopardy players would ever or should ever or could ever use these heavily mathematical results in their heat of battle, but it seems instructive to know what theory implies. Given an expression of interest, I will post some of these results on Dropbox Public for your amusement and comments.
There isn't any advantage to randomizing as most players will only get one or two chances to wager in FJ.

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Re: Final Jeopardy wagering theory

Post by whoisalexjacob »

Judy5cents wrote: My advice is wager everything. It's not real money and you don't get to keep what's left over.
...unless you win.

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Re: Final Jeopardy wagering theory

Post by Judy5cents »

I'm curious. Among those of us who are champions, how many worked out a mathematical strategy for winning that takes into consideration all possible outcomes (good luck with that!) and how many just wrote down a number that looked good to them?

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Re: Final Jeopardy wagering theory

Post by Doug527 »

Judy5cents wrote:I'm curious. Among those of us who are champions, how many worked out a mathematical strategy for winning that takes into consideration all possible outcomes (good luck with that!) and how many just wrote down a number that looked good to them?
It's really easy to come up with such a strategy when you're in third place going into Final in a close game. Like, oh, say, this one. (To just randomly pick one out of thin air)

Only two ways for me to win there. Either I get it right and the other two get it wrong, or we all get it wrong. If either Anne or Bing gets it right, I'm obviously toast (assuming rational behavior). I expected Anne would have wagered $15,201 to cover me doubling (her higher wager surprised me as it sacrified her shot at 2nd place), and Bing would have (and did) wager $16,401 to cover Anne doubling. Under the circumstances, I had no reason to suspect any different behavior.

So, I knew I had to make a wager between $1,801 and $14,200, which would allow me to win in both of those two scenarios. I don't agree with the wagering calculator in this situation, because I don't think a $1,001-$1,800 bet is wise at all. What if it's a Triple Stumper and Bing irrationally wagers $0? So I guess I took that extra scenario into account.

But then I suppose I also wrote down a number that looked good to me, because I picked $3,800 out of that list so that, if I got it right, I'd have a nice round $20,000. Of course, I didn't get it right, and so I would up with a less round, but still quite nice, $12,400.

I have no doubt that someone who won from second place in a close game will have a much more interesting and complex answer than I do.
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Re: Final Jeopardy wagering theory

Post by slam »

Second place is always the hardest position to wager from. First place nearly always wagers for the simple lockout (Shoretegy notwithstanding). Third place usually has some straightforward reasoning available like Doug outlines above (though there are some wrinkles here and there). Second place often has to make a choice as to whether to bet big and lockout the 3rd place contestant or play for 3rd place to miss (since he has to assume that 1st place misses). And there are more wrinkles for second place to take into account.

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Re: Final Jeopardy wagering theory

Post by bpmod »

Doug527 wrote:I don't agree with the wagering calculator in this situation, because I don't think a $1,001-$1,800 bet is wise at all. What if it's a Triple Stumper and Bing irrationally wagers $0?
I think (although I may be mistaken here) that the wagering calculator assumes rational betting by the opponents.

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Re: Final Jeopardy wagering theory

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bpmod wrote:
Doug527 wrote:I don't agree with the wagering calculator in this situation, because I don't think a $1,001-$1,800 bet is wise at all. What if it's a Triple Stumper and Bing irrationally wagers $0?
I think (although I may be mistaken here) that the wagering calculator assumes rational betting by the opponents.

Brian
That would make sense, otherwise it wouldn't be the excellent tool that it is. If it took everything into account, it might as well just start making wild guesses, assuming people wouldn't behave rationally

I just wanted to explain what was going through my mind, and why I thought that $1,801 was a better minimum than $1,001.
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Re: Final Jeopardy wagering theory

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Judy5cents wrote:I'm curious. Among those of us who are champions, how many worked out a mathematical strategy for winning that takes into consideration all possible outcomes (good luck with that!) and how many just wrote down a number that looked good to them?
Mathematical strategy here. There are eight outcomes (assuming three players in FJ) as far as who is right and who is wrong, but you only really have to concern yourself with the outcomes in which your trailer(s) are right and/or your leader(s) are wrong, and they all make rational wagers. And, if you can, allow for the possibility of irrational or otherwise unexpected wagering. Granted, there are those who do this by betting on themselves to get it right and hoping those who lead them make a mistake or otherwise fail to cover. But there are more people who can say "I would have won if I had wagered less on FJ" than can say "I would have won if I had wagered more on FJ."

I entered FJ in the lead in five of my six regular games, so the strategy was generally pretty straightforward. Three of those were locks, so I just bet numbers close to the maximum I could wager while still guaranteeing myself victory.

One was a crush (I could bet less than my lead to cover a doubling second), so I shot for $28,801 (he had $14,400). I would have liked to go for $30,000, but such a wager would have meant I could lose a winnable game if second wagered irrationally small.

The other was a two-thirds game (I had to bet a couple thousand more than my lead); I thought about going for fancy numbers again, and simply saying "screw it, if he wagers right I'm guaranteed to lose on an incorrect response, and all my downsides are the same, so I might as well maximize my upside," but I knew how often players in second place can bet too much when they should stay put, and I didn't want to squander that opportunity if it happened. Sure enough, he bet almost everything, but he was the only one to be right, and it paid off handsomely.

The one game I wasn't in the lead, I was in a close second, with third place at exactly half my score. I didn't want to risk being overtaken by him, so I wagered $0 and hoped first place would fall behind me.
As it turned out, I was the only one to get FJ right, and while first place did fall behind me, she hadn't wagered to cover me doubling up.
So I got nothing extra for being right, and I live with the knowledge I would have regretted my wager if she had been right also. There's a part of me that wishes I had bet it all, but I don't question the logic of what I did.

As for my quarter-final game in the ToC, it's always a bit more haphazard with the wild cards. I went in with a plan to shoot for $12,225 if I entered FJ under that, and stay above $13,000 if I entered FJ above that. (If I was between, I'd aim for whatever was closer.) I had $14,800, so I wagered $1,225. I got FJ wrong, but I got the wild card.

My semifinal game is the main reason I make this post. I entered FJ with $13,200 to my opponents' $17,200 and $7,800. I realized I could go for $15,601 to cover third, and I'd win if everybody was wrong and first place wagered how I expected.
But I got scared that first might bet to stay above $16,000 or some such total, protecting him from a doubled third and dodging me if I only covered third.
I realized I could catch up to him and still have enough left over to win if everyone was wrong.
In fact, I could wager up to $5,199 and win on a triple stumper (I expected first to fall to $7,999 if he was wrong, so I could drop down to $8,001, leaving an extra dollar of padding), but I decided $4,001 was the simplest.

So I thought, "I don't have to be right to win. If third and first are both wrong, I'm set no matter what." I did what I could to answer FJ correctly, but third place's response was not the same as mine, and I heard Alex tell him he was correct. I closed my eyes and started taking stock of my final moments on the Jeopardy stage, when I heard Alex read his wager of $213 and I realized I might still be in it. My response was revealed, I lost money, but remained in second. First also got it wrong, dropped to third, and I became a finalist.

Of course, this seems to bolster your strategy, since third place would have won by wagering everything. However, he was in a specific situation where he should have been doubly motivated to do so: he had less than first's $7,999 he'd fall to if he covered me and was wrong (his downfall was fixating on clearing that total), and less than the $10,799 I'd fall to if I covered him and was wrong. Wagering everything would have won him the game, but not wagering everything won me the game; I ended up taking advantage of an outcome I never expected. That turned me from a $10,000 semifinalist to a $50,000 finalist (and very nearly a $250,000 one!). (I think I've gone on long enough, so I won't explain my Finals wagering strategy.)

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Re: Final Jeopardy wagering theory

Post by creedofhubris »

Vanya wrote:
webster2000 wrote:This is my first post to this Board. I have been thinking about FJ Wagers for a couple of years and most intensely in the last two months. My point of departure has been my working knowledge of Game Theory. I have noticed that none of the FJ Wagering articles I have uncovered ever mention the need for mixed strategies, however the most usual combination of holdings at Final appears to require that the optimizing player avoid a rigid choice and resort to randomizing to insure their opponents do not anticipate his/her strategy. I finally found some basis for resolving my curiosity about a Game Theory solution to FJ and have obtained some basic results.My results are close to conventional wisdom except for the need for the top player to only bet high 2/3 of the time. If the players used this theoretically optimum strategy, the top player would win 1/2 the time while the second highest player would win 1/3, with the lowest winning 1/6 the time.

It is unlikely that Jeopardy players would ever or should ever or could ever use these heavily mathematical results in their heat of battle, but it seems instructive to know what theory implies. Given an expression of interest, I will post some of these results on Dropbox Public for your amusement and comments.
There isn't any advantage to randomizing as most players will only get one or two chances to wager in FJ.
Irrelevant. The only thing that would keep you from applying the mixed strategy is whether your opponents are wagering in an exploitable fashion, in which case you abandon the randomization and exploit them. This happens to be generally true (first place almost always bets to cover, second place almost always bets big) so unless you're up against JBoarders (perhaps in the ToC) you'll be better off exploiting using the betting calculator.

Shoretegy represents "higher-level thinking" (2nd place thinks that I will bet big, and will bet small, so I will bet small) and is useful to exploit someone who thinks he is exploiting you, but that's probably not happening unless you're up against a very savvy 2nd place opponent who thinks you are naive.

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Re: Final Jeopardy wagering theory

Post by Vanya »

creedofhubris wrote:
Vanya wrote: There isn't any advantage to randomizing as most players will only get one or two chances to wager in FJ.
Irrelevant. The only thing that would keep you from applying the mixed strategy is whether your opponents are wagering in an exploitable fashion, in which case you abandon the randomization and exploit them. This happens to be generally true (first place almost always bets to cover, second place almost always bets big) so unless you're up against JBoarders (perhaps in the ToC) you'll be better off exploiting using the betting calculator.

Shoretegy represents "higher-level thinking" (2nd place thinks that I will bet big, and will bet small, so I will bet small) and is useful to exploit someone who thinks he is exploiting you, but that's probably not happening unless you're up against a very savvy 2nd place opponent who thinks you are naive.
Huh? That's not randomization; it's trying to anticipate what your opponents will do. Nothing random about that. You can call it "mixed strategy," but randomizing only applies when the same circumstances are repeated several times, which never happens in Jeopardy, as you are facing different opponents each time.

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Re: Final Jeopardy wagering theory

Post by creedofhubris »

Vanya wrote:
creedofhubris wrote:
Vanya wrote: There isn't any advantage to randomizing as most players will only get one or two chances to wager in FJ.
Irrelevant. The only thing that would keep you from applying the mixed strategy is whether your opponents are wagering in an exploitable fashion, in which case you abandon the randomization and exploit them. This happens to be generally true (first place almost always bets to cover, second place almost always bets big) so unless you're up against JBoarders (perhaps in the ToC) you'll be better off exploiting using the betting calculator.

Shoretegy represents "higher-level thinking" (2nd place thinks that I will bet big, and will bet small, so I will bet small) and is useful to exploit someone who thinks he is exploiting you, but that's probably not happening unless you're up against a very savvy 2nd place opponent who thinks you are naive.
Huh? That's not randomization; it's trying to anticipate what your opponents will do. Nothing random about that. You can call it "mixed strategy," but randomizing only applies when the same circumstances are repeated several times, which never happens in Jeopardy, as you are facing different opponents each time.
You can tell your opponent *beforehand* that you're betting to cover him 2/3 of the time, and if that's the optimal mixed strategy (it's something like that; higher % of cover bet if there are 3 players in contention) there's nothing he can do to improve his chances of beating you. Whether he bets high or low he'll still have the same chance of winning.

That's not true for a pure strategy -- if you tell him that you'll always be betting to cover him then he can choose a low bet and end up with an advantage over you.

If you can anticipate your opponent's bet then you don't need to bother with a mixed strategy. That's why I brought up the fact that most opponents are predictable.

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Re: Final Jeopardy wagering theory

Post by whoisalexjacob »

Judy5cents wrote:I'm curious. Among those of us who are champions, how many worked out a mathematical strategy for winning that takes into consideration all possible outcomes (good luck with that!) and how many just wrote down a number that looked good to them?
Now this post makes more sense to me, as I was wondering why you didn't bet it all in any of your games... Wagering strategy can get complicated, and I don't blame you for not caring to think about it (especially since you're already a champion and probably don't have to worry about doing it again). But, I think knowing how to wager improves one's chances of winning on the show.


Vanya, I'm just going to give my opinion quickly that you're wrong and creedofhubris is right. You can still use a mixed strategy in 1 game, you flip a coin or a mental coin, and theoretically you can't be exploited since your opponent can't predict what you're going to do. He said specifically, though, that it was probably better to abandon randomization in favor of exploiting your opponents.

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Re: Final Jeopardy wagering theory

Post by Vanya »

omgwheelhouse wrote:
Vanya, I'm just going to give my opinion quickly that you're wrong and creedofhubris is right. You can still use a mixed strategy in 1 game, you flip a coin or a mental coin, and theoretically you can't be exploited since your opponent can't predict what you're going to do. He said specifically, though, that it was probably better to abandon randomization in favor of exploiting your opponents.
I think we're getting hung up on semantics. You can't use a "mixed strategy" in one game, you have to choose one strategy or the other. You can choose your strategy on the basis of a coin flip or whatever, but there is no advantage in doing so.

Webster2000 said, "...the need for the top player to only bet high 2/3 of the time." Now since most players only get one or two chances to be the top player, this observation is irrelevant to the individual player, unless I am misunderstanding Webster2000's point, which is entirely possible.

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Re: Final Jeopardy wagering theory

Post by Bob78164 »

creedofhubris wrote:
Vanya wrote:
webster2000 wrote:This is my first post to this Board. I have been thinking about FJ Wagers for a couple of years and most intensely in the last two months. My point of departure has been my working knowledge of Game Theory. I have noticed that none of the FJ Wagering articles I have uncovered ever mention the need for mixed strategies, however the most usual combination of holdings at Final appears to require that the optimizing player avoid a rigid choice and resort to randomizing to insure their opponents do not anticipate his/her strategy. I finally found some basis for resolving my curiosity about a Game Theory solution to FJ and have obtained some basic results.My results are close to conventional wisdom except for the need for the top player to only bet high 2/3 of the time. If the players used this theoretically optimum strategy, the top player would win 1/2 the time while the second highest player would win 1/3, with the lowest winning 1/6 the time.

It is unlikely that Jeopardy players would ever or should ever or could ever use these heavily mathematical results in their heat of battle, but it seems instructive to know what theory implies. Given an expression of interest, I will post some of these results on Dropbox Public for your amusement and comments.
There isn't any advantage to randomizing as most players will only get one or two chances to wager in FJ.
Irrelevant. The only thing that would keep you from applying the mixed strategy is whether your opponents are wagering in an exploitable fashion, in which case you abandon the randomization and exploit them. This happens to be generally true (first place almost always bets to cover, second place almost always bets big) so unless you're up against JBoarders (perhaps in the ToC) you'll be better off exploiting using the betting calculator.

Shoretegy represents "higher-level thinking" (2nd place thinks that I will bet big, and will bet small, so I will bet small) and is useful to exploit someone who thinks he is exploiting you, but that's probably not happening unless you're up against a very savvy 2nd place opponent who thinks you are naive.
It's not true that "second place almost always bets big." If it were true, Shore's Conjecture would be false. In Shore's Conjecture situations when I was measuring them in the old Shore's Conjecture thread, the second-place player bet "small" a clear majority of the time. For quite a while the ratio was almost exactly two-thirds. --Bob

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Re: Final Jeopardy wagering theory

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Vanya wrote:
omgwheelhouse wrote:
Vanya, I'm just going to give my opinion quickly that you're wrong and creedofhubris is right. You can still use a mixed strategy in 1 game, you flip a coin or a mental coin, and theoretically you can't be exploited since your opponent can't predict what you're going to do. He said specifically, though, that it was probably better to abandon randomization in favor of exploiting your opponents.
I think we're getting hung up on semantics. You can't use a "mixed strategy" in one game, you have to choose one strategy or the other. You can choose your strategy on the basis of a coin flip or whatever, but there is no advantage in doing so.

Webster2000 said, "...the need for the top player to only bet high 2/3 of the time." Now since most players only get one or two chances to be the top player, this observation is irrelevant to the individual player, unless I am misunderstanding Webster2000's point, which is entirely possible.
Well, I disagree. The word mixed doesn't mean you're doing two strategies at the same time. You can used a mixed strategy in 1 game, and I explained how. The theoretical advantage to doing so is that there's nothing your opponent can do that's always the right play against you.

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Re: Final Jeopardy wagering theory

Post by creedofhubris »

Bob78164 wrote:It's not true that "second place almost always bets big." If it were true, Shore's Conjecture would be false. In Shore's Conjecture situations when I was measuring them in the old Shore's Conjecture thread, the second-place player bet "small" a clear majority of the time. For quite a while the ratio was almost exactly two-thirds. --Bob
This is my mistake. I was relying on the Jeopardy! archive's "Shore Conjecture" page, and comparing wins vs losses over the last 4 years. Shoretegy went 9-17.

(a) that's a teeny sample, as I should've realized, and it did better before that and
(b) I didn't realize that most of the "draw" games represented a win for Shoretegy; results-oriented thinking on the archive makes it less useful than it could be

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Re: Final Jeopardy wagering theory

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Vanya wrote:
omgwheelhouse wrote:
Vanya, I'm just going to give my opinion quickly that you're wrong and creedofhubris is right. You can still use a mixed strategy in 1 game, you flip a coin or a mental coin, and theoretically you can't be exploited since your opponent can't predict what you're going to do. He said specifically, though, that it was probably better to abandon randomization in favor of exploiting your opponents.
I think we're getting hung up on semantics. You can't use a "mixed strategy" in one game, you have to choose one strategy or the other. You can choose your strategy on the basis of a coin flip or whatever, but there is no advantage in doing so.

Webster2000 said, "...the need for the top player to only bet high 2/3 of the time." Now since most players only get one or two chances to be the top player, this observation is irrelevant to the individual player, unless I am misunderstanding Webster2000's point, which is entirely possible.
"Mixed strategy" is a game theory term. It means a distribution of probabilities over your strategies/actions. For example, if you are in the lead going into FJ and it's a runaway so that your score is over double that of the next highest score (call this score b), playing the pure strategy of betting so that if you got the FJ wrong your score didn't go below 2b+1 is simply playing a mixed strategy and placing a 100-percent probability on that strategy.

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Re: Final Jeopardy wagering theory

Post by creedofhubris »

creedofhubris wrote:
Bob78164 wrote:It's not true that "second place almost always bets big." If it were true, Shore's Conjecture would be false. In Shore's Conjecture situations when I was measuring them in the old Shore's Conjecture thread, the second-place player bet "small" a clear majority of the time. For quite a while the ratio was almost exactly two-thirds. --Bob
This is my mistake. I was relying on the Jeopardy! archive's "Shore Conjecture" page, and comparing wins vs losses over the last 4 years. Shoretegy went 9-17.

(a) that's a teeny sample, as I should've realized, and it did better before that and
(b) I didn't realize that most of the "draw" games represented a win for Shoretegy; results-oriented thinking on the archive makes it less useful than it could be
OK, I looked some more at Shore's Conjecture. I obviously wasn't reading the (unarchived?) old board so I can't tell if I'm rehashing old (possibly rebutted) arguments, but:

(1) If the ratio of (RRR, RRW) to (WWR, WWW) responses is around 2:1, then second place player betting small 2/3 of the time makes Shoretegy a wash in terms of wins. You win the same number of games either way.
(1a) The games you win due to Shoretegy, you win less than you would have had you wagered to shutout on the games you lose due to Shoretegy. You'll win small amounts and lose big amounts.

(2) Of the games you "draw" -- 2/3 of the games where you both answer correctly will be shoretegic draws (opponent bets small), while 1/3 of the games where you both answer incorrectly will be shoretegic draws (opponent bets big). Given that it's a ~2:1 ratio already of both correct:both wrong, that's now a ~4:1 ratio of "drawn" games where you answered correctly to "drawn" games where you answer incorrectly. So most of the time when you have a shoretegic draw you cost yourself money with the shoretegy. You lose 3/5 of the difference between a shoretegic bet and a shutout bet on average in drawn games.

If you think you're at least 50% likely to get FJ correct (average) it seems like a mistake. If you think your chances are significantly <50% then it may become interesting.

Am I missing something?

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Re: Final Jeopardy wagering theory

Post by seaborgium »

If you win a game while wagering Shoretegically and responding incorrectly, you win more than if you had made the standard wager (particularly when the standard wager would have cost you the game).

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Re: Final Jeopardy wagering theory

Post by creedofhubris »

seaborgium wrote:If you win a game while wagering Shoretegically and responding incorrectly, you win more than if you had made the standard wager (particularly when the standard wager would have cost you the game).
Sure, but I predict you'll have about 4x as many games where you respond correctly and win as games when you respond incorrectly and win.

And about an equal number of games where Shoretegic bets give you the win as when they cost you the win. And the games where they cost you the win are higher cash totals (because you answered FJ correctly).

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