You’re talking about the best chance to win. I’m talking about the highest expected value in dollars, which is the only thing I would care about. So you’re right, it was discussed at length in OOT and no one actually proved what they claimed to have proved. The most vocal poster there often states emphatically that $0 and $1 are the only wagers that make sense, but then if you dig up his calculations he only shows that they give the highest chance of winning the game with no consideration of prize money.
If I’m understanding the argument correctly then this doesn’t really make sense and is actually the opposite. Isn’t the argument that 2nd place makes an “idiotic” wager pretty often, and so in order to exploit that the best response is to wager $1? If 2nd place is actually playing a bad strategy and we adapt to take advantage of that, then by definition it isn’t part of an unexploitable or equilibrium strategy aka “GTO.” I assume that’s what you meant by GTO which is a poker forum term that is never used by game theorists.
Single game EV (including factor in future equity from a win) is one calculation.
If you factor everything in (which is heavily biased toward your own estimation of getting FJ correct) and you come up with a +EV calc and then you come up on the - variance end (for n=1, if you lose there is no next hand), then live with it. You could be introduced for the rest of your life as J! Champion. Or not.
It’s a value statement including non-monetary value. It’s a valid way to play. I expect you’d find yourself in a significant minority as the preferred way.
Otherwise we are going to have to agree to disagree. In my world no amount of dollars in a J game is worth a single percent of win equity. Max win equity first, then max dollars.
But you didn’t calculate it, and ntnBO didn’t calculate it. That’s all I’m saying. The claim was that $0 and $1 are the only reasonable wagers you can make in a lock-tie situation, and there’s no actual proof provided for that claim.
So to be clear, the reason $0 and $1 are the only reasonable wagers in a lock-tie game is mostly due to the unquantifiable utility you get from being referred to as a Jeopardy champion? I’m trying to evaluate the strategy for this seriously and objectively because it seems complicated and not obvious. There may be important strategy implications beyond locking up the priceless title of Jeopardy champion.
The largest non optimal wager from 2nd is everything but a dollar. So that makes $1 the best win equity wager assuming we are more than 50% to get FJ correct. I don’t have the data on the frequency of that wager.
I don’t doubt that we can get a higher single game EV depending on:
1 our current total
2 assumed EV for winning in strictly cash terms
3 our relative estimations of us getting FJ correct or 2nd getting it wrong.
For example is we lead 8,000 to 4,000 the win equity term is going to be relatively large. If we lead 30,000 to 15,000 it would be relatively smaller than the first example.
Strictly speaking, it’s not “priceless”, as you’re guaranteed $1,000 for coming in 3rd place the next day. And that’s not even counting the EV of having a shot to win multiple games after that.
Again I’m not talking about single game EV. It’s the total present value of the strategy with all future games included. I’m using the numbers from the jboard lock-tie thread. Let’s go back to the hypothetical FJ scenario from that thread:
1st: 20,000
2nd: 10,000
3rd: -
Summary
Standard wagers of $0 and $1 from the lead show winning chances of 87%
Passive non-standard wagers from the lead show winning chances of 82%
Aggressive non-standard wagers from the lead show winning chances of 75%
where passive wager = $9,999 and aggressive wager = $19,999.
Obviously non-standard wagers greatly reduce winning chances. Some will argue that the extra money is worth the risk. But since the more the leader wagers the less chance they have of winning, it’s almost certainly not worth losing the chance to return to continue earning money.
Emphasis mine given the claim of only 5% win probability difference between $1 and $9,999 wagers and the fact that the argument is based solely on coming back to win money. As a gambler, those numbers should cause alarm bells to go off. We need the joint probability estimates for 1st and 2nd place FJ response outcomes, which are given as:
RR: 0.31
RW: 0.26
WR: 0.18
WW: 0.25
Here are the dollar values I get for the current game including guaranteed 2nd place money. For simplicity I’m rounding the inputs to nearest $100.
Wager $0 or $1: (0.87 x $20k) + (0.13 x $2k) = $17,660
Wager $9,999: (0.57 x $30k) + (0.25 x $10k) + (0.18 x $2k) = $19,960
Wager $19,999: (0.57 x $40k) + (0.25 x $2k) = $23,300
That implies the following equity differentials in the current game:
$0/$1 vs $9,999: $2,300
$0/$1 vs. $19,999: $5,640
There are several ways to look at this. One is to backsolve for the $EV required in future prize money to make the $0/$1 strategy break even against passive and aggressive wagering.
Wager 0/1 vs. Passive
(0.87 x F$EV) - (0.82 x F$EV) = $2,300
F$EV = $2,300 / 0.05 = $46,000
Wager 0/1 vs. Aggressive
(0.87 x F$EV) - (0.75 x F$EV) = $5640
F$EV = $5,640 / 0.12 = $47,000
Those numbers are closer than I would have guessed, and that’s likely significant in terms of extrapolating to a general strategy since there’s usually a third player we want to cover. However, the main issue is whether we think future expected winnings of $46,000+ is reasonable for a context-neutral Jeopardy champion. There’s no way it’s anywhere near that.
So, the stats, from October 4, 2004, through to January 6, 2017:
- The average winning score: $20,137.
- A defending champion wins 46.26% of the time.
Using the formula 1 / (1 – r) to determine the sum of the infinite geometric series that begins 1, 0.4626, etc., we know that after a player’s first win, they are expected to win another 0.8608 games.
Putting this all together, we can determine the worth of a victory (not taking a ToC into account, we’ll do that later) with the formula:
(0.8608 * $20,137) + (0.5714 * $2,000) + (0.4286 * $1,000) = $18,907.
That is a context-neutral estimate for a champion’s expected winnings in future regular season play. We should adjust it by win% for each case and add to the current game $EVs to get the total value of each strategy:
$0/$1: $17,660 + (0.87)($18,900) = $34,103
Passive: $19,960 + (0.82)($18,900) = $35,458
Aggressive: $23,300 + (0.75)($18,900) = $37,475
The key point is that for this specific scenario, giving away $3,300 does not sound like an optimal strategy to me. Maybe someone could argue they’re kind of close, but what I certainly don’t see is a case for the assertion that $0 and $1 are the only sensible wagers. Remember this is based entirely on average 1st and 2nd place play over a large sample and without any situational context.
Of course, Jeopardy isn’t played context free so we can try to add some of the situational factors back and look for inflection points. The most obvious one with quantifiable value would be clinching a TOC spot. The analysis below also comes from the blog entry I linked above (edited it slightly to only show the value of successive wins).
If a player hasn’t qualified for the Tournament yet, we can use the percentages above to guess their chances of qualifying for the Tournament and add that percentage of [TOC prize money] accordingly.
1st win: $20,403.
2nd win: $22,142.
3rd win: $25,899.
4th win: $34,020.
5th win: $51,574.
6th+ win / TOC clinched / context-neutral: $18,907
We know that a 5th win at $51,574 crosses the threshold making $0/$1 wagering preferable. I’d argue that this is mostly intuitive, as someone sitting on four wins probably understands they have considerable TOC $ equity when wagering from the lead in FJ. However, this analysis doesn’t include the fact that total money can also factor into TOC entry which would counter this effect to some extent.
Right, so you understand that the evaluation can change significantly depending on the exact scenario. We still need to think about what to do with a 3rd place player as part of the general strategy. We need to know proper play under different assumptions about FJ category strength.
I think there’s an easy case to make that the span of sensible bets ranges from $0 up to nearly everything depending on the scenario. LOL at betting $1 if I have a lock-tie in SPORTS and can bet $20,000 getting the best of it. That’d be at least a $10,000 mistake and I’m snap firing there even if it’s my first appearance on the show.
2 Likes
Everyone understands what you are saying. They are just assigning value to something you are not.
If it’s not a tie scenario and the score is 30,000 to 14,000. You are a favorite to win. What do you wager? Probably not that hard to come up with math that says you should bet big from 1st.
Do you just take the lock game and wager 1,999?
Do you wager wager 15,999, which gets a win 82% of the time (only lose on WR)
Do you wager most of it because second is likely making a large wager and you still win say 75% of time (57% you win 18% you lose and you get a good split on the rest because 2nd goes close to all-in).
Do you risk a lock game for EV? Not even the most ruthless gambler that’s played the game has done that (James).
It seems like you are trying to explain something to people that don’t understand. We get it. The question is do you?
No they aren’t. I made sure to include that in the post and even bolded it.
To be fair, there is EV to gain at a small enhanced risk in the lock-tie scenario. Your concept is solid and the absolute minimum risk can decrease overall EV. Trading 10,000 of EV for 2% of risk would be a good risk to take except the desire to win is like one’s life.
10,000 bucks on the other side of the train tracks. You see someone a few hundred feet away that will reach the money first if you don’t make your move now. 98% you can make it without getting hit by a train.
98% of the time you got $10,000. 2% of the time you’re dead. Maybe not rational but losing is considered death.
Fell down a youtube rabbit hole and ended up on the first episode of the Twenty-One reboot from back in 2000.
Holy shit the strategy from the players is abysmal. Its a pretty simple game. Players in soundproof booths, each get the same set of questions and alternate their turns. They can choose any value in the set from 1 -11. (1 being easy etc.) They are blind and deaf so have no way of knowing where the opponent stands.
Standard strat would to me be three questions of 7 points to get to 21 in 3 turns. If you miss your first you absolutely HAVE to go for 10 or 11 on the second question to try to get to 21 in 3 turns and nobody fucking does. Its baffling
1 Like
Holy shit this dude just swept wheel of fortune. Took every single puzzle and solved the bonus round. 75k. They should give him a million for that
4 Likes
1 Like
Meh. The only WOF I am interested in is when the contestants crack under the pressure and fail to solve an obvious puzzle with one letter missing, something like:
ARNOLD SCHWAR_ENEGGER
“… J!”
2 Likes
Highly recommend watching Wednesday nights J! With no sharp objects or bricks within reach.
Lady in middle podium is ahead like 22k to 7k with a late DD, wagers 10k, misses and then misses again in FJ, this snatching defeat from the jaws of victory
1 Like
Personally, I thought it was hilarious.
1 Like
How much cash was left on the board when she hit the DD?
4 x $400
1 x $800
Could have bet $5, put the buzzer down, bet nothing on FJ and come away with over $20k and a spot in the next game.
1 Like
Thats infuriating. Especially after learning from someone not too long ago that the contestants essentially have an unlimited amount of time to make a wager. She could literally have added up the remaining clues and found the runaway amount and then bet to increase EV. Just awful
There have been a lot of really bad bets lately. I don’t hate betting $10k there if you’re an expert in the category. Hard to argue that anyone is specifically an expert in random people named Hans though and it felt like a trap all the way. Hans Holbein is a pretty tough name to come up with at $800, like definitely harder than Hans Zimmer at $1200 or Hans Krebs at $2000 imo.
2 Likes