A 300 year old probability paradox | St. Petersburg Paradox
Vložit
- čas přidán 27. 07. 2024
- The St. Petersburg game is a 300 year old probability paradox that has confounded people for centuries. It's counterintuitive in that the mathematical expectation is infinite, but no one in their right mind would actually pay huge amounts of money to play this game. Where is the disconnect?
There's still research done on this problem, although a lot of the discussion is no longer mathematical, but more economic/philosophical. The focus is now more on modeling how a rational agent will behave and how they should value this game. A good overview from the philosophy point of view is:
plato.stanford.edu/entries/pa...
From the math side, I think the best reference is "An Introduction to Probability Theory and Its Applications" by William Feller. Feller proved that the St. Petersburg game converges in probability to log_2 and explains this problem in detail in his book.
Timestamps:
0:00 Intro to Problem
1:55 Example
4:16 St. Petersburg Simulations
5:24 Estimating the Average Payout's Growth Rate
9:17 Other Considerations
I have major math anxiety.., for various reasons, but I found this easy to get invested in because you’re clearly passionate about it/interested in it. It’s much more beneficial to learn from someone enjoying it, so thank you!
I do love the how problems like the St. Petersburg paradox exist, hanging around to remind me that mathematics is crazy, and that I'm crazy for studying it. : )
There is no correct answer as its a matter of perspective and opinion due to the way the game works!
@@danquaylesitsspeltpotatoe8307you just have to know that value is subjective. Even when it comes to amount of money. Menger was right.
@@Georgggg Thanks for saying what i siad is correct while trying to say it wasnt! 🤦♂🤣
I see not many people are commenting and you really deserve it. Love the content. I am a math major who has now taken a side interest in optimisation problems and your videos are always great. I agree that the introduction of a utility function does not solve the underlying issue, and I don't think that it claims too. It was a way of reasoning that economists liked in order to do practical calculations (with the word practical used loosely). However, the paradox, which to me is simply the idea "the expected value is infinite, but when you play it it doesn't come out that way" is an entirely seperate issue
I appreciate the time you spent to explain this. It is unfortunate how so many people believe that happiness can be tied to a particular sum of money, but a study of the extremely wealthy will generally show an indication of overall dissatisfaction with life which no amount of money can solve. I believe philanthropists may be an exception to this rule since my belief is that true happiness and satisfaction in life are actually found (drumroll) ... by serving others.
And then there's those people who get 2^134 dollars first try, and no less any other. I know someone who legitimately rolled a twenty-sided die 10 times in a row. But it tends to balance out with the other person I know who does the same thing with 1s, to me this is indicative that odds are actually in a state of high entropy at any given time, where it always balances out, but in such a way that some are luckier than others.
Sounds a lot like Las Vegas or something similar.
Great video, thanks!
I think using an estimated average winning is not very useful here. The average gain of the last cell is actually infinite, so by capping it you didn't "solve" the paradox, you removed the key aspect of it and solved the non-paradox. If you add a couple of statements, you basically used the lower bound to prove that the average winnings series *does* tend to infinity.
Correctly estimating the huge chance to go broke is important and a big part of this paradox. But even with that chance, just trying to *maximize the expected money* is still in favor of playing, even if you only have $1k and the game costs $1k. We only say "but what if you run out of money" because the very-very low utility of having $0 is "common sense".
A small chance to win jillions is not worth the big chance to bust out exactly because of utility. Without it you have no reason to fear getting 0 and no reason to disregard a possible 2^999 outcome. If your utility actually was "number of dollars you own", you *would* play this game for any price.
And this is exactly why "average expected outcome" is not always a realistic thing to maximize in probabilistic scenarios.
Even if we accept the function average winnings = log(t) + 2 where t is the number of games you played, then playing 2^28 = 268,435,456 games would net you average winnings of $30 and approximate total winnings of (2^28)*30 = $8,053,063,680. Thus, if the game were to cost $30 to play, we need to be able to play more than 2^28 times to make a profit. (For example, if we played 269,000,000 times, then our estimated total winnings would be (log(269000000) + 2)×269000000 = $8,070,815,320.6811. Our total cost to play would be 269000000×30 = $8,070,000,000. And so our profit, on average, if we played 269,000,000 games, would be 8,070,815,320.6811 - 8,070,000,000 = $815,320.6811.)
So, we would need a little more than 8 billion dollars and time to play 269 million games just to make a little less than a million dollar profit given these circumstances. So, we can see the "paradox" is solved right here. It simply isn't worth the time and the money for the measly profits you would make. If you have 8 billion, $800,000 extra isn't gonna make a big difference. To make any money at all in this game, you need to have a high amount of money to begin with (to invest into each game), which means that if you're poor (less than 8 billion dollars to spend) then you shouldn't play, and if you're rich (more than 8 billion dollars to spend) the number of games you need to play/time it would take to play is probably not worth it for the amount of money you can make even if it can approach infinity the more you play, since your time at the point is probably more valuable to you than more money.
i think also its because the expected value calculation isn't taking into account the fact that you have to pay to play. its a lot less confusing that you're losing money despite an infinite EV when you remember that the infinite EV didnt factor in the $30 you paid
nevermind, i'm wrong. i think the adjusted EV that includes your initial payment still diverges
If we are just playing for numbers, then we are bound to win and to win big, so get the supercomputers humming, after all, it's just math.
But if we are playing for things, finite phisical things, then we are almost certain to lose, given our personal resources and our inability to collect on the imporbable winners, after all, it's just physics.
Mark Rober will be a trillionaire from this
Why would anyone pay more than $2?
$2. that's how much i'd pay.
i don't want to lose money half the playings.
I would probably sink like 100$ into this. 100$ isn't all that much and the chances of earning something substantial are still magnitudes higher than the lottery / ... I would still be sad that I lost 100$ but still. And before playing I'd run a few million simulations with different exit conditions and try to get out the best strategy.
The moral of the story: kids, don't do math.
So, basically to be able to win infinite amount of money, you need to already have infinite amount of money. Makes sense.
And playing makes zero sense.
you don't need to have an infinite amount of money. 2^28=268,435,456. so if you are a billionaire you can comfortably play this game
@@DendrocnideMoroides The amount of money that you can use per game is variable, so maybe even that is not enough.
The Utility function nonsense is nonsense, complete out of topic an irrelevant, tries to deny the fact that every human choice in based upon emotional grounds, complety tyrannical in the choice of an arbritary messure instead of the just natural interest analysis of the human mind with the use of the tools that one have.
I don't think that it denies emotional grounds, it is just that it is useful to make decisions objectively without taking into account emotions, this is what we do when we make logical decisions. I think it is just useful for making logical decisions.
@@prathampekamwar8751 It is impossible to make decisions without taking emotions, every decision will be at the end of the day determined by emotions, every use of logic in the decision process is no more than an analysis of how much that decision will suffice the desires of the emotions
@@friedrichhayek4862 But there can't be frameworks for decision making then because everyone has different emotional responses, then decision making would become nothing more than what u feel, am I understanding it right?
@@prathampekamwar8751 There can be recomendations, depending in your interests.
@@friedrichhayek4862 so isn't that what the utility function is? It asks the person how happy u will be with the money and that is used for recommendations on optimization as in the video.