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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

#111

Post by mananaysiempre »

bambax wrote: post #94
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 shor…

> 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.

There are standard arguments (e.g. the Von Neumann–Morgenstern utility theorem) that an agent with rational preferences, with remarkably weak definitions of the word “rational”, must have an utility function and a subjective probability function such that their behaviour is always governed by the EV of that utility with respect to that probability.

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

#112

Post by trescenzi »

bambax wrote: post #94
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 shor…

This is interesting to me in the context of a post[1] yesterday about teaching logical thinking to children. One of the top comments was about how, yes, teach logical thinking but also teach other types too. SBF and those who think with a heavy bias towards EV show an extreme weakness towards reality. Of course SBF himself is a perfect example of this. I won’t pretend to know if his math was always right but one of his defenses on trial was essentially “just run the simulations a few more times and we’ll get all the money back” which clearly shows a lack of understanding of reality.

I’m glad to now know there’s a common example of that weakness.

[1]: https://news.ycombinator.com/item?id=41456472

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

#113

Post by iainmerrick »

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.

Right! I'm not sure if that's actually what he had in mind, but if he did, it's funny because it makes all this mathematical analysis completely pointless.

The OP has a complex randomized strategy that guarantees to average at least $0.07 against any adversary; meanwhile, just by delaying his "pick" and stringing you along, Ballmer makes you take seven guesses and owe him a dollar each time.

If you were expecting to win $0.07 on average, how many rounds would you play before you realise you're being scammed?

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

#114

Post by jessriedel »

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.

I mean, who know what he’s thinking, but based on the game description that strategy isn’t “adversarial”, it’s lying. Maybe the lesson is “don’t play games for money with people who will cheat”, but it would be a boring one.

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

#115

Post by ordu »

nighthawk454 wrote: post #31
michaelt wrote: post #29

Earlier quoted context omitted.

> Ballmer's argument is essentially about tail risk. Expected value is absolutely not a good way to make bets if you value survival, If he was trying to make that point, why set the bet at $1 - a loss that wouldn't imperil anyone? The situation is entirely fictional, why not fictionally gamble with a five-figure sum?

You could? There's no reason you couldn't do $50,000 with bets of $10,000. The point is you get 5 guesses before you lose. The bet sizes don't really matter.

They matter. I could play a game with negative expected value just for fun, if this negative expected value is not too negative. Or not for a fun, but to let Ballmer win to make him feel himself winner, to make him feel superior, hopefully it will help me to get what I want from him. To lose a game gracefully is a manipulation device, especially if the winner doesn't suspect, that you lose on purpose. And, the stories I heard about MS suggest that Gates, Ballmer and all those gray beards of MS were very competitive, so they are probably much more susceptible for this particular bait. $1 is a very small price for such a tactic. But $10,000... it depends on what I'm hoping to win, and on my detailed understanding of my further steps and probabilities of their success.

Life is more difficult than math abstractions of life. No one yet managed to mathematically describe any social situation to such level of details, so you could blindly believe your equations, like you do with physics.

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

#116

Post by energy123 »

n4r9 wrote: post #85
nopinsight wrote: post #21

Earlier quoted context omitted.

Kelly Criterion Betting more than the Kelly fraction increases the risk of ruin, especially in the long run. https://en.m.wikipedia.org/wiki/Kelly_criterion Note: Not saying that this is applicable in the original post's situation. It's relevant to the parent comment though, and very useful in many situations, such as investing.

What if I wanted to maximise the bottom 5th or 10th percentile of wealth?

Fractional Kelly?

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

#117

Post by cool_dude85 »

nighthawk454 wrote: post #31
michaelt wrote: post #29

Earlier quoted context omitted.

> Ballmer's argument is essentially about tail risk. Expected value is absolutely not a good way to make bets if you value survival, If he was trying to make that point, why set the bet at $1 - a loss that wouldn't imperil anyone? The situation is entirely fictional, why not fictionally gamble with a five-figure sum?

You could? There's no reason you couldn't do $50,000 with bets of $10,000. The point is you get 5 guesses before you lose. The bet sizes don't really matter.

The bet sizes here do really matter, because they are selected to ensure that there is no risk of ruin. Ballmer asks about playing the game once at a bet amount that anyone doing the job interview can cover.

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

#118

Post by nneonneo »

bambax wrote: post #94
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 shor…

The St. Petersburg "paradox" is not a paradox if we consider any real-world implementation of it. The EV accumulates at the rate of $1 per flip. So, if we want to make the EV at least $1,000,000, we must find a counterparty that is willing to pay at least $2^1000000 (or at least 2^1000000 units of "utility" if we're trying to avoid the depreciating utility effect). That's plainly unrealistic. As soon as the counterparty has any fixed upper limit to its ability to pay (or provide utility), the EV becomes finite.

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

#119

Post by qarl »

OmarShehata wrote: post #6
qarl wrote: post #2

And this, friends, is the perfect example of why the modern tech interview process is pure insanity.

is it? If I was forced to ask this question as an interviewer, and the candidate said, "actually, you're wrong, here's why" that's a very good signal. Do most people not do this? (typically there's discussion with all the interviewers and it isn't just "did the candidate get the question right or not). I personally think a lot of big tech interview questions are dumb but I think the process isn't as broken as I thoug…

I think you're missing something.

Presumably Balmer did ask this question. At least a few times. And yet he never heard of the correct answer, and believed the incorrect answer to be correct.

That tells you that if anyone did say "actually, you're wrong" he never listened to them.

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

#120

Post by dooglius »

Nice! I tried to solve this the other day too, but came from the other angle--trying to find a probability vector for Ballmer that always won (finding the best response tree is n^3 complexity best I could find). I'm somewhat surprised since I figured for sure Ballmer had a big edge by picking numbers near the endpoints, making the player pay a large cost to check them.
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