unJon
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Minty
April 19th, 2019 at 12:15:56 PM permalink
Quote: Wizard

I think he mentioned he had a daughter and ended a Daily Double wager in her birthday.

However, for the sake of the question at hand, assume happiness is a function of the log of your wealth. In other words, a change in wealth affects happiness not by the dollar amount of the change, but the degree of change relative to the initial wealth.

I smell a Kelly.
The race is not always to the swift, nor the battle to the strong; but that is the way to bet.
GWAE
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Minty
April 19th, 2019 at 12:32:35 PM permalink
Quote: Wizard

I think he mentioned he had a daughter and ended a Daily Double wager in her birthday.

However, for the sake of the question at hand, assume happiness is a function of the log of your wealth. In other words, a change in wealth affects happiness not by the dollar amount of the change, but the degree of change relative to the initial wealth.



Imo, the difference between 0 and 1 mil is so much greater than 1 mil and 2 mil.

The difference between 400k and 1 mil is so much more than 1 mil to 1.6 mil.

What I am trying to say is I would risk the amount that I think won't affect my life for the worse. Risking is all and losing is going to be a huge negative whereas winning the additional 1 million is not a huge positive when it comes to lifestyle.

I think I would wager 300k and thank you for the free money.
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charliepatrick
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April 19th, 2019 at 12:33:33 PM permalink
While I can see the argument of a LOG I think it depends on whether, for instance, you could buy a nice (house/car/holiday) with a specified amount and hence had "fun money" to spare.
I'm not sure of the maths but a spreadhseet said your best EV was to gamble 80%. I'm not expected any brownie points for that!
CrystalMath
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April 19th, 2019 at 1:19:33 PM permalink

If you want to maximize your expected happiness, then you must maximize:
0.9(log(1,000,000 + w)-log(1,000,000)) + 0.1(log(1,000,000-w)-log(1,000,000)

Taking the derivative, we get:

0.9/(1,000,000+w)/ln(10) - 0.1/(1,000,000-w)/ln(10)

Set to zero and solve.

w = 800,000, and your average happiness will increase by about 16%.
I heart Crystal Math.
gordonm888
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April 19th, 2019 at 1:36:30 PM permalink
All these answers do ignore state and federal tax rates, including the federal Alternative Minimum Tax.

Given the taped delay in filming Jeopardy, I wonder if James Holzhauer won/wins all his money in the same tax year?
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FinsRule
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April 19th, 2019 at 1:55:41 PM permalink
Quote: gordonm888

All these answers do ignore state and federal tax rates, including the federal Alternative Minimum Tax.

Given the taped delay in filming Jeopardy, I wonder if James Holzhauer won/wins all his money in the same tax year?



I think there’s a 2 month delay? So yes.
Wizard
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April 19th, 2019 at 5:03:28 PM permalink
Quote: CrystalMath


If you want to maximize your expected happiness, then you must maximize:
0.9(log(1,000,000 + w)-log(1,000,000)) + 0.1(log(1,000,000-w)-log(1,000,000)

Taking the derivative, we get:

0.9/(1,000,000+w)/ln(10) - 0.1/(1,000,000-w)/ln(10)

Set to zero and solve.

w = 800,000, and your average happiness will increase by about 16%.



I agree!
It's not whether you win or lose; it's whether or not you had a good bet.
Wizard
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April 19th, 2019 at 5:30:31 PM permalink


Here is my analysis of episode 153, James' 5th appearance, which aired April 10 (the infamous day of the Burning Man main ticket sale). Click on any image for a larger version.

James got the first Daily Double. The category was Holidays and Observances. Here was the board:



Scores at the time:
James: $3,000
Jeff: -$1,000
Laura: $1,800

This was obviously very early in the game, so it pays to be aggressive. James correctly went all in. Given my alleged 90% success rate, it is obviously the right move.

This was a rare one that James got wrong, by the way.

Here is the second Daily Double, which James got. The category was the Civil War. Here was the board.



Scores at the time:
James: $9,200
Jeff: $200
Laura: $5,200

It was still early in the Double Jeopardy round and both competitors had low scores, so James correctly goes all in. No big surprise there. He got it right.


Here is the third Daily Double, which Laura got. The category was Mollusks. Here was the board.



Scores at the time:
James: $28,400
Jeff: $1,400
Laura: $12,000

Before making a wager, Alex even hinted she be aggressive with this comment, "At the moment, the game is a runaway for James." However she wagered $7,000 only. WTF?! What a terrible bet. If she went all in and got it correct, she would have had $24,000, within striking distance of James. Maybe she wouldn't have surpassed him before Final Jeopardy, but in Final Jeopardy she would have had hope to swing him with more than half. She got it wrong, by the way.

Here were the scores in Final Jeopardy

James: $33,200
Jeff: Negative (eliminated)
Laura: $5,800

In this "can't be caught" situation, the correct strategy is simply to bet your score less double your opponents score, less $1, or less. In this case, that maximum wager would have been $33,200 - 2*$5,800 - $1 = $21,599.*

James wanted his final score to end in 1122, to honor somebody's birthday, so he wagered $21,122. Note that if he bet $31,122, he would have risked being swung by Laura. Assuming the sentimental value of the final score ending in 1122 was worth $477 to James, he did the right thing. As usual, he got it right.

In conclusion, this wasn't the best episode to illustrate James' brilliance in wagering, as every decision was fairly obvious.

I welcome all comments, as always.

* Corrected
Last edited by: Wizard on Apr 20, 2019
It's not whether you win or lose; it's whether or not you had a good bet.
EdCollins
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April 19th, 2019 at 5:46:44 PM permalink
I did notice one "strategy" mistake he made the other day. (Well, I believe it's a strategy mistake.)

In the Double Jeopardy Round, there are, of course, two Daily Doubles to be found. They are NEVER in the same category. That's by design.

Given that it's a big advantage to find these Daily Doubles yourself, so that your opponents cannot take advantage of them, after finding the first one you should NOT select any further dollar values that are available in the same category. Instead, use your choice of selection (which you may never get back if don't answer any further questions correctly), to select ANY OTHER dollar value in ANY OTHER category.

I saw James select a dollar value from a category that contained the first Daily Double, when the final DD was still hidden.

Now, if one specifically selects a dollar value in that same category with the argument that they wish to build up their winnings as much as possible before finding that final Daily Double, I won't argue that fact. But usually that's NOT what they are doing.
EdCollins
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April 19th, 2019 at 6:15:24 PM permalink
Shoot, I have to stay away from any social media for the next 3+ hours. (I don't want to accidentally find out about tonight's outcome beforehand.)

Jeopardy normally airs here at 7:00 pm on the local ABC network, just prior to Wheel of Fortune. However, because of the Celtics/Pacers game tonight, I see that it's scheduled to air at 9:30 pm., after they show Wheel at 9:00 pm.

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