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The expected value of the game is positive regardless of Ballmer’s strategy

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Re: The expected value of the game is positive regardless of Ballmer’s strategy

#91

Post by fgna »

TheDong wrote: post #82

Edit: Oops, nope, this comment is wrong, ty fgna for pointing that out! I feel like there's an even simpler proof that you can beat adversarial-ballmer, with exactly the same expected positive outcome as binary search vs random ballmer. I call my algorithm "randomly offset binary search". It goes like this: 1. Pick a random number between 0-100, call this 'offset' 2. Perform the binary search algorithm, except at eac…

Unfortunately the numbers are not circular :( By offsetting the initial number, the binary search does not work optimally right? Imagine the number is below 50, and you start by guessing 60, now you have to search for 30 numbers instead of 25, and thus the binary search is not optimal. reply

Re: The expected value of the game is positive regardless of Ballmer’s strategy

#92

Post by TheDong »

fgna wrote: post #91
TheDong wrote: post #82

Edit: Oops, nope, this comment is wrong, ty fgna for pointing that out! I feel like there's an even simpler proof that you can beat adversarial-ballmer, with exactly the same expected positive outcome as binary search vs random ballmer. I call my algorithm "randomly offset binary search". It goes like this: 1. Pick a random number between 0-100, call this 'offset' 2. Perform the binary search algorithm, except at eac…

Unfortunately the numbers are not circular :( By offsetting the initial number, the binary search does not work optimally right? Imagine the number is below 50, and you start by guessing 60, now you have to search for 30 numbers instead of 25, and thus the binary search is not optimal. reply

Ah, yup, you're right. Ballmer's answer of "high or low" isn't in the offset number system, but the normal one, so my strategy doesn't work.

That's what I get for not thinking it through properly, thank you for pointing that out!

Re: The expected value of the game is positive regardless of Ballmer’s strategy

#93

Post by tromp »

lazyasciiart wrote: post #36
leni536 wrote: post #3

Can't wait for the paper that solves for the Nash equilibrium for this game.

Isn’t that what the Arthur O’Dwyer link gives?

Only for up to 5 numbers. Note that the Nash equilibrium for 100 numbers could require many pages to describe.

Re: The expected value of the game is positive regardless of Ballmer’s strategy

#94

Post by bambax »

nighthawk454 wrote: post #19

This sort of misses the forest for the trees, although neat application. Ballmer's argument is essentially about tail risk. Expected value is absolutely not a good way to make bets if you value survival, because you only get one shot. Same reason you wouldn't go all in every time you get a poker hand that's "expected" to win. Because you'll (very probably) be bankrupt in a few hands. Sure the mean is +$0.07 or whatev…

> Expected value is absolutely not a good way to make bets if you value survival

Yes! The St. Petersburg "Paradox" shows that we intuitively know that. I put "paradox" in quotes because I don't think it's a paradox, it's just a sane reaction.

(Sam Bankman-Fried was a big fan of EV and famously declared that he would toss a coin where heads would double the "value" (?) of the world but tails would destroy it.)

In short, the St. Petersburg paradox goes as follows: a fair coin is tossed until heads come up, and the player wins $2^n, where n is the number of times the coin was flipped. So for example if heads come up on the first flip the player gets $2, if it comes up on the second they get $4, on the third, $8, on the tenth $1024 (2^10), etc. It's easy to show that the expected value of the game is infinite (approaches infinity).

Therefore, someone perfectly rational (?) should be willing to pay virtually any amount of money to play the game, because any finite amount of money is less than an infinite amount of money, and therefore the expected gain is always positive.

Yet you will probably not find many people (except SBF?) willing to pay millions of dollars to play that game.

It's only a paradox if we think it shows that people are not "rational". But I think it simply shows EV is not a good measure of risk, and everyone knows it.

Very complete and fascinating article about the St. Petersburg Paradox here:

https://plato.stanford.edu/entries/paradox-stpetersburg/

Re: The expected value of the game is positive regardless of Ballmer’s strategy

#95

Post by fortydegrees »

rustybolt wrote: post #23
nighthawk454 wrote: post #19

This sort of misses the forest for the trees, although neat application. Ballmer's argument is essentially about tail risk. Expected value is absolutely not a good way to make bets if you value survival, because you only get one shot. Same reason you wouldn't go all in every time you get a poker hand that's "expected" to win. Because you'll (very probably) be bankrupt in a few hands. Sure the mean is +$0.07 or whatev…

I don't think this is true. Most people will not be bankrupt after losing a dollar. If this is true then Steve failed badly at communicating this context. To be honest, I think Steve just didn't grasp the mathematical deepness of the problem.

While it does seem that Ballmer doesn't have an understanding of the deepness of the problem, in his defence, he outscored BillG on the math SAT with a perfect score of 800, and graduated Harvard with a degree in applied mathematics.

Which makes me wonder if it's related to another 'simple' game theory problem that came up in Matt Levine's money stuff:

"They made me do the math on 1000 coin flips. EV(heads) (easy), standard deviation (slightly harder), then they offered me a +EV bet on the outcome. I said “let’s go.”

They said “Wrong. If we’re offering it to you, you shouldn’t take it.”

I said “We just did the math.”

They said “We have a guy on the floor of the Amex who can flip 55% heads.”"

I like that anecdote and the takeaway, especially with regards to trading: if someone's offering you what seems obviously a +EV trade, why are they offering it to you and what are you missing? Whether that was Ballmer's intended lesson is another matter..

[0]https://www.bloomberg.com/opinion/articles/2024-05-14/amc-is...

Re: The expected value of the game is positive regardless of Ballmer’s strategy

#96

Post by GuB-42 »

rcxdude wrote: post #88

When Ballmer said 'adversarial', I considered this strategy: he's not actually required to pick a fixed number at the start at all. He can simply give the answer to each guess which leaves the largest number of possible numbers remaining, guaranteeing a loss regardless of strategy.

That's what I thought too, kind of like Absurdle, an adversarial variant of Wordle: https://qntm.org/files/absurdle/absurdle.html

It is by the author of HATERIS, a variant of Tetris that always gives you the worst piece.

Re: The expected value of the game is positive regardless of Ballmer’s strategy

#97

Post by wed239023 »

I watched the interview, and I see two problems:

- nowhere it says he has to choose whole number, he could choose fractions (55.25) or even irrational like PI. Number of questions can be infinitive.

- nowhere it says, he may not change his number while the game runs.

You pay upfront for each question, and you hope game is not somehow rigged. It is not just question of algorithms.

Also money you win is a taxable income, payments for hazard are not taxable expenses...

Re: The expected value of the game is positive regardless of Ballmer’s strategy

#99

Post by krisoft »

nighthawk454 wrote: post #41
qsort wrote: post #35

Earlier quoted context omitted.

I don't agree, I think he was just plain wrong. Unlike most people here I actually think questions like this are a decent way to see how people think. I would expect people with math/stats/cs background to be able to at least start the conversation about this problem. However when you hide hypotheses or add your own BS constraints as a gotcha without explicitly stating them is where you lose me. If the question is "w…

> If the question is "would you play this game" the reasonable mathematical translation is "determine if the expected value is greater than zero". Not really! And that may be the point of the question. It's not testing if you can pattern match to plausible CS concepts. If you get one play, and the goal is to win, do you take the chance? The whole question is about the difference in likelihood in the limit (expected v…

> It's not testing if you can pattern match to plausible CS concepts.

But thats a problem then, isn’t it?

Because if you abandon that framework there are many other possible frameworks to think in.

You say the question is about the difference in likelihood in the limit (expected value, infinite plays) and what is a likely outcome _of one round_. (Which may I say is just an other plausible CS concept you pattern matched.)

One might also say that you should play the game because even if you are playing like a chump and has absolutely the worst luck you are at max out of pocket of a 100 dollar. A real hustler can turn a personal meeting with Balmer into much more lucrative deal and earn back that hundred dollar million-fold that way.

Or you could say that the answer is no, because Balmer stinks and it would be ruinous to your personal reputation to be seen with him.

Or you could say “yes” because you know you don’t have cash at all. So even if he wins good luck trying to squeeze his reward out of you.

Or you could say “yes”, because while you distract Balmer with this inane game your associate will lift his valet and car keys.

Or you could say yes because what is the worst which can happen? You will spend a bit of a money, and have an awesome story to tell later.

Or you can answer no, because clearly a rich businessman does not have your best interest in his mind, so there must be some trap.

So this is what happens when you abandon the framework to answer the question as a plausible CS concept. You open the door to all these other alternative frameworks and more.

Re: The expected value of the game is positive regardless of Ballmer’s strategy

#100

Post by mattmanser »

wed239023 wrote: post #97

I watched the interview, and I see two problems: - nowhere it says he has to choose whole number, he could choose fractions (55.25) or even irrational like PI. Number of questions can be infinitive. - nowhere it says, he may not change his number while the game runs. You pay upfront for each question, and you hope game is not somehow rigged. It is not just question of algorithms. Also money you win is a taxable incom…

He also doesn't say he wants to play in this reality where the rules of maths hold. That his one and your one mean the same thing. That your accent doesn't have to match his. That you haven't got to be holding a certain pose when you say it. There are always implied rules, and Ballmer's implied rules are he'll use a whole number, not change his number and be fair. You could probably spend now until the end of time adding stipulations and he'd still be able to cheat.

I'd recommend never learning about philosophy as you'll disappear into nihilsm.

And lottery wins aren't taxable every where on the planet (e.g. the UK), so you made the same "mistake" as the author too!

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