Final Jeopardy wagering theory

[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).

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.

Well, that's just because I don't respond incorrectly that often!

[creedofhubris]

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.

Well, that's just because I don't respond incorrectly that often!

I'm still working on that part.

[Vanya]

Am I missing something?

Yes. Analyzing statistics gathered over several games is meaningless. Each game is an independent event. The outcomes of previous games have no effect on the outcome of the current game.

[creedofhubris]

Am I missing something?

Yes. Analyzing statistics gathered over several games is meaningless. Each game is an independent event. The outcomes of previous games have no effect on the outcome of the current game.

It is prudent to know the true likelihood of events before you make a bet of any kind, whether on Jeopardy! or in a casino. Otherwise, how can you know whether you are making a good bet?

The outcomes of previous games reveal that likelihood.

[Vanya]

Am I missing something?

Yes. Analyzing statistics gathered over several games is meaningless. Each game is an independent event. The outcomes of previous games have no effect on the outcome of the current game.

It is prudent to know the true likelihood of events before you make a bet of any kind, whether on Jeopardy! or in a casino. Otherwise, how can you know whether you are making a good bet?

The outcomes of previous games reveal that likelihood.

No, they do not. Stop spreading this misinformation.

[Bob78164]

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?

A few things. First, sometimes the Shore's Conjecture wager wins the game even when the pattern is WRW. Second, sometimes (in either the Strong or Intermediate Form) the Shore's Conjecture wager wins the game when the second-place player bet "big," because the third-place player would beat the traditional shut-out bet. Third, simply getting the opportunity to play another game has a significant expectation value, because you might win, not just that second game, but one or two others after that. Thus, simply winning the game should get a big "bonus."

When I was measuring starting after Season 20, Shore's Conjecture jumped out to a lead in a ratio of about 2:1. That lead diminished in recent years -- shortly after Jeopardy! switched to on-line testing, the second-place player's propensity to wager "big" seemed to skyrocket. --Bob

[Bob78164]

Yes. Analyzing statistics gathered over several games is meaningless. Each game is an independent event. The outcomes of previous games have no effect on the outcome of the current game.

It is prudent to know the true likelihood of events before you make a bet of any kind, whether on Jeopardy! or in a casino. Otherwise, how can you know whether you are making a good bet?

The outcomes of previous games reveal that likelihood.

No, they do not. Stop spreading this misinformation.

Vanya -- On this subject it has become apparent that you don't know what you're talking about. I used to teach game theory. Creedofhubris is analyzing the issue correctly.

It's true that different games are independent events. Nevertheless, since the same writers are crafting clues and are endeavoring for clues of roughly equal difficulty, one's frequency of correct responses is in the past is predictive of one's frequency of correct responses in the future.

That's testable, by the way. Use the Archive to check out your variance of correct response percentages in consecutive years. (It's best to use consecutive years to avoid too much drift in the identity of clue writers.) That's probably a large enough sample size that the percentages won't differ by more than about 15%. --Bob

[creedofhubris]

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?

A few things. First, sometimes the Shore's Conjecture wager wins the game even when the pattern is WRW. Second, sometimes (in either the Strong or Intermediate Form) the Shore's Conjecture wager wins the game when the second-place player bet "big," because the third-place player would beat the traditional shut-out bet. Third, simply getting the opportunity to play another game has a significant expectation value, because you might win, not just that second game, but one or two others after that. Thus, simply winning the game should get a big "bonus."

When I was measuring starting after Season 20, Shore's Conjecture jumped out to a lead in a ratio of about 2:1. That lead diminished in recent years -- shortly after Jeopardy! switched to on-line testing, the second-place player's propensity to wager "big" seemed to skyrocket. --Bob

Thanks for the comments. I''m working on a post looking at the last few hundred Shore's games (not that much else on my plate right now). I'll trust the Archive to spot Shore wins in unusual spots.

What do you think the expected value of continuing your run is? I was going to use $12,500, but I'm pulling that number mostly out of thin air and I'd certainly be willing to accept another number if it's been crunched already.

[creedofhubris]

OK. I took all the Shore's games on the archive for the last 10 seasons. Before I analyzed the data I threw out all tournament games and anything involving a champ's game 5 or higher. I ended up with 215 games. In (parenthesis) is what a strategy would win without including losers’ money and the value of future games. I valued a winning game as the money won +$12,500 to reflect future opportunities, and losing was worth a varying amount between 1000 and 2000 depending on how often you finished 2nd as opposed to third. Someone betting Shoretegically will always finish 2nd when losing, for instance. and someone always going allin will always finish 3rd when losing.

Actual results: 130-85 ($3,073,914) $4,826,414
Results if leading player had played ABC, Always Bet (just enough to) Cover: 130-85 ($3,074,514) $4,827,014 in winnings

I’m calling ABC the baseline. The Shoretegy calculator already assumes that’s the bet rather than looking at the actual bet.

Results if leader had applied Shoretegic bets in all games: 143-72 ($2,454,423) $4,385,923 -$2,051/game vs ABC
Results if leader had gone allin in all gamex: 106-109 ($3,477,202) $4,911,202 +$392/game vs ABC

Results if leader had applied Shoretegic bets in "Strong" games, and ABC in all others: 148-67 ($2,983,747) $4,934,247 +$499/game vs ABC
Results if leader had applied Shoretegic bets in "Strong" games, and allin in all others: 127-88 ($3,253,611) $4,973,111 +$680/game vs ABC

Results if leader had applied Shoretegic bets in "Strong" games, and when top 2 players have low sum (<24000), and ABC in all others: 152-63 ($2,961,846) $4,975,246 +$689/game vs ABC
Results if leader had applied Shoretegic bets in “Strong” games, and when top 2 players have low sum (<24000), and allin bets in all others: 140-75 ($3,275,000) $5,160,000 +$1,548/game vs ABC

So, here are the take-home points as I see them:

-”Strong” Shoretegic situations seem to be positive expectation spots for Shoretegic wagers, while “weak/intermediate” situations are clearly big money losers for Shoretegy..

-Shoretegy is great if people miss a lot. If you and your opponent don't have high scores, it is evidence that you are both below average players and will miss final jeopardy more often than usual, so that’s a good spot for a conservative bet. I suspect if you look at Coryats rather than actual totals you'll be better able to specify the range where Shoretegy shines.

-Similary, if you think you’re likely to miss FJ due to the category you should strongly consider a Shoretegic wager. A Shoretegic wager “won” 61/109 times the leader missed FJ, vs 24/109 for an ABC wager. On the other hand, Shoretegic wagers won only 82/106 times the leader was correct, vs 106/106 for an ABC bet (obviously)

-Why aren’t more people going allin?

[danspartan]

Bumping an old topic for JBoard post #1.

This was the closest discussion I can find to what I call "Answer Combination Frequency". Its easy to envision in the basic example of what 2nd should bet when >2/3 and <3/4 (and specifically avoiding the exact ratios which lead to the tie bet scenarios). I see it touched on here and some talk of "past results don't matter in this game"--I respectfully disagree, especially until such time that 1st routinely offers the tie in this situation.

1st: 9000
2nd: 6500
3rd: Lets keep it simple, 0

The various calculators work up 2nds "maximum safe bet". Since 1st has to bet $4000 to get the double score, then 2nd's "maximum safe bet" is $1500.

I believe that betting exactly $0 is the correct way to go. We generally cannot win on Right/Right (RR) since 1st makes the lockout bet almost 100% of the time (exception:Arthur). Therefore we win on Wrong Right (WR) and Wrong Wrong (WW). We can ignore RW obviously.

Why? Answer Combination Frequency. I accumulated 4 seasons (26-29) and looked at the frequency of the various answer combinations.

1st was right 481 times, 1st was wrong 424 times. Once we set 1st though, we have a leading indicator on if 2nd will answer Right or Wrong.

Of the 484 1st is Right, 2nd is Right 310 (64%) of the time.
Of the 424 1st in Wrong, 2nd is Wrong 251 (59%)... We can estimate that at about 3:2 if first is wrong, we will be wrong in 2nd as well.

Since we don't care about the RR scenarios (since in practice 1st bets to lockout at a very high rate, not tie).

So we are really betting on the WW and WR scenarios. I agree that our independent odds are a little better than 50/50 which would indicate a "max safe bet".

HOWEVER, we are not betting on our independent odds, we are betting based on the information that 1st was wrong. 1st being wrong is a leading indicator that the question is a little harder than normal, and shows a consistent frequency of 62% RR or WW, and in the specific case we worry about when 1st is WRONG, then we are only 41% to answer RIGHT.

So if we are a 3:2 dog to RIGHT in the only scenarios we care about, we should minimize our bet ($0).

To further highlight the trend, when looking at all three players, when first and 2nd are WRONG, 3rd is WRONG about 2/3 of the time.

Anyhow, this ended up longer than I wanted. Hopefully its coherent at least. Please point me to another thread if already discussed in detail. Interested to see the feedback.

[GoodStrategy]

@danspartan - What you described is the "correlation effect" - the odds of the eight (or four in a two-player scenario) outcomes aren't equal, but the ones where everyone gets it right or wrong are more likely than each individual "mixed" outcome where some solve FJ! and others don't. You're probably right in that if your outcome depends on someone else missing Final (as in betting small in a 2/3-to-3/4 scenario which doesn't require covering third, or another example being tied leaders and you're the trailer with less than half of their score) that a zero wager has the best EV, but in terms of "grading" a person's wager (like what's done here) any amount up to the maximum in the range is acceptable.

The correlation effect is also used in the argument for certain strategies, such as in Stratton's Dilemma betting for a TS win instead of covering third (which is most likely to be of benefit in the case of the leader being the sole one to miss, or makes a tie or non-cover bet), or if you're the leader in a lock-tie in a tournament (semifinal or other single-elimination) it's better to go ahead and bet a dollar to cover second instead of zero for the tie and a possible tiebreaker (since the only way you lose is if you miss and second gets it right, as opposed to second getting FJ! right and prevailing on the tiebreaker).

[danspartan]

Thanks, I should have included the Ev calc as well. Obv win% is much higher by betting small (about 47% vs 20-21%. I did the EV calcs on 0 and 1500 bets and using the RR RW WR and WW frequencies I have. 0 bet EV=3055; 1500 bet EV=2920 (ignoring equity for 3rd place). (I even did an EV on shoving and hoping for WR or RR with a 5% chance of being offered the tie--That was 2691. I just see lot of talk justifying 2nd betting the "max" in this scenario and it seems to be clearly a -EV play since the most likely winning scenario is a double miss. If we are "banking" on a miss, why risk anything if not necessary?

Related question--does a table exist out there where one could pull the % of time 2nd bets it all (or close) when clearly they shouldn't (most >2/3 situation). Gets interesting for 3rd. WWW happens at 19%, but WWR is 9%, that seems to be a big enough spread that while a game theory bet would be for 3rd to bet it all assuming the only win is WWR, when in actual practice, playing for WWW may be correct. Example 11,500, 11,000, 6000; clearly 2nd should bet 1000, and 3rd should bet it all, but often 2nd bets it all. It even happens when 2nd has a lockout over 3rd (Julia's last game).

If not, maybe one morning I will just crank through a few season in my own excel sheet, going throug J archive game by game.

[jeopardyhopeful]

So if we're leading going into FJ, we should use Shoretegy in "strong" situations, and ABC in the other situations? Now, remind me how to determine whether a situation is a "strong" Shoretegy situation. I couldn't really determine that using the j-archive glossary.

[nightreign]

It's strong if third place has more than twice the difference of first and second.