This paradox hints that you shouldn't measure expected value like that in games like these, even if you want to objectify the subject and throw its values out the window to fit the theory. Nobody thinks that one will get 60$ when probability of winning 200$ is 30% when you play only one game. And in the coin flip game, there is a 1/128 probability of winning >100$, compared to 100% probability of winning 100$.
Thanks for covering this. It's worth noting that this game could never could never be offered because no party could ever handle the risk of the other side.
Diminishing marginal utility doesn't actually fix this problem, because that's only for money, which does have diminishing marginal returns, but lots of other forms of utility don't. For example, if you could choose between saving a million people from a slow painful death (or in econ speak, utility) or taking the bet with the coin flips and saving 2^n people, you'd always have to let the million people die and take the bet. Saving people doesn't have diminishing returns, so you're still stuck with something that seems obviously wrong. To be fair though, this was a great explanation of the concept and there's only so many topics you can cover in 7 minutes. Thanks for making this video! It was really helpful.
I don't agree with this logic, either. I totally see a "diminishing" marginal utility in saving lives. The probability of saving more than 1 million people is too low.
My statistics professor gave a different variation, where you pay 1,2,4... to flip the coin and once the coin lands on heads, you get 2^x as payout. The EV is also infinity, here's what I'm wondering. What if, we weigh the probability to the second order? 2*0.5*0.5 + 4*0.25*0.25, ... essentially summing the probabilities and getting 1. If you play the game an infinite number of times, you will still be infinitely rich but if you only play the game once, your winnings are much closer to 1 than infinity.
The important factor does not seem to be diminishing marginal value, but rather the perception of the value of the marginal change in probability, which is not clearly symmetrical with - or a counterpart to - marginal value although there does seem to be some similarity or connection; I suspect that in individual cases there are threshold points for each of those factors which would be sufficient to either get someone to play the lottery, not bothering to find out the odds of winning, or refusing to play the lottery on such long odds, no matter how much money is offered as a prize.
This helps a lot, i keep feeling stuck, the number of dice I'm rolling keeps getting smaller and smaller. Some I stop rolling all together. I'm hurting myself here waiting on coins I'm flipping. Ill never get where I'm going if I don't draw my guide lines somewhere. I'm sick of falling short. 💔
The problem could be reduced if we knew when the "house" would break. Surely they don't have infinite physical coins to give out. So the expected was never infinite. It's like always tripling your total losing bets on every coin flip. Garaunteed to win......unless someone demands a physical payoff at some time. Then......not a good strategy.
Couldn't you just, like, inlcude a sliding scale of rationality in this, depending on the numerical chances? Like, sure, the _possible_ benefit of playing this game is infinite, but the _probable_ benefit is dysmal. If you have a choice between just taking 100$, or a chance lower than 1% to get the same 100$, it's obviously more rational to take the 100 bucks.
I think that it would be more rational to play the game because if you decide to play you'll have a small probability of getting infinite money in return. So the value of infiniteXsmall probability is more than the value of just 100 dollars. The paradox is about the hypothetical value of a choice (more value=more rational) and not the highest probability.
This was a cool lesson. Thank you for explaining it so well. I wonder how much your insight at the end counts as evidence that rational choice theory (RCT) is flawed because it tries to quantify the projected happiness of choices. Economic theories which try to calculate gross national happiness based on diverse metrics of flourishing, taking into account both human and nonhuman well-being, seem to me far superior to the simplified metrics of RCT. Projects like the Capabilities Approach pioneered by Martha Nussbaum and Amartya Sen strike me as more promising.
I must be making some simple math error, because it looks like in the coin flip game, the probability of a given amount won decreases by the same factor which incrementally raises the reward. Hence the weighted probability calculation gives you a plus $2 straight across the board, for any number of coin flips. Why would someone not choose the $100 with a probability of one? Of course, from the perspective of one who is already playing the game, the probability of doubling my money on the next coin flip, with a 50/50 chance of heads, would lead to a tendency to continue playing. So it looks like the notion of marginal utility or marginal value enters for a person who has already made the choice, but does not come into play at the outset, when the subject must choose to take the $100 or to set off down that road of coin flips. In any event, I am not sure I see a paradox here.
Since in theory you could flip 1 billion heads in a row, and keep going, the math says that the value of playing is technically infinite. Those have a low probability but they are not $0. The point is since there is no limit on how heads you can flip in a row, there is no limit on your upside potential. The $100 is certainly more probable. But the unlimited chance is more valuable.
personally even if i had the choice to 1. have probability 1 to get 100% more happiness or have probability 0,000.000.01 to get 1.000.000.000.000 % more happiness , i would chose the first option even though 1 x 100 = 100 and 1.000.000.000.000 x 0,000.000.01 = 10000 and so the second option is technically 100 times better but i still would go for the first option cuz my intuition tells me there is great value in certainty !
you can't measure expected value like that in a game you only get to play once. if you got to choose between those options over and over again, then the coin flip game would be the right choice, since you will eventually get a streak that will far exceed the $100
Why not? It is still true in a one-off game that a person who maximises expected utility will do better on average than one who doesn't, even though it is not certain.
@@sethapex9670 Whether we should consider the degree of certainty in rational choices is unrelated to what I was trying to say, which is that it doesn't matter if a game is repeatable or not.
The explanation with the utility function is unsatisfactory since the game can be altered in such a way that the paradox will reappear. To truly resolve the problem one needs to appeal to ergodicity theory, or in layman's terms, distinct between playing a single game or a large series of independence games (and say take the average of the rewards).
It takes seven flips of the coin to get to $128. Even with the repeated 50/50 chance; the law of averages will catch up. Another point to be made is lack of a penalizing result. Both outcomes result in increased money with no threat of loss. I believe in gambling they would establish a spread to make it interesting.
Shouldn't this be $2 raised to the number of flips total, rather than the number of heads? In the current formulation, the probability of winning $2 would be 0.25 not 0.5.
Good question. No, it is the number of heads. If you flip the coin once and get a tails right away, you get $0. If you flip once and get heads, you get $2 regardless of what happens next. The chances of getting that first $2 are 50/50. If we counted the number of flips, you would be guaranteed to get $2, since playing the game automatically gets you one flip and 2^1 =2
Infinity often breaks things. You could also say one has 10^-(10^10^10) or one to the number of atoms in the universe chance of winning an infinte amount of money and 0 in any other case. The expectation value would still be infinite and one really dumb to play that game in the real world with finite boundaries. One has to be careful not to think too much about infinities to avoid going insane like Georg Cantor or Bobby Fisher did.
Even setting aside the diminishing marginal utility of money, it seems like something is wrong with just the way of calculating the expected payout of the game, because surely there is only an infinitesimal chance that you will actually win the infinite money that this naive expected payout calculation predicts. You're more likely to win smaller amounts of money, because it's more likely that the game will end sooner. I don't know off the top of my head what a better way to predict the most likely payout would be, but as usual when you get an infinity in your calculations you probably did something wrong.
I disagree with the statement "as usual when you get an infinity in your calculations you probably did something wrong" If the game is finite but very long, the same issue will arise. I do agree that the diminishing marginal utility of money is a weird explanation that doesn't get to the heart of the issue.
Mmmm, my inner mathematician (edit: full disclosure, I'm a a biochemist, so to me mathematics is a useful tool I just use but make no claims to fully understand. I know how the screw-driver works, not how to make one) has been appeased: If the distribution of probabilities isn't linear, that's the cause of the paradox. And let's be honest, few things, if any, are linear.
This thing is dumb, and it’s not a paradox People aren’t willing to spend money on something that is expected to have infinite payout over an infinite number of time....... people live like 85 years. Of course they’re not going to pay high dollar to play a game that may not yield significant amounts of money for thousands of years. That ain’t paradox. It’s common sense
The expected value is a number which we should expect to be our average of n number of outcomes. For example, if you roll a dice 1000 times, you can expect that the average outcome is 3.5. Now if we simulate St. Petersburg experiment for as many times as we like, I am pretty sure the average outcome will never be anything "close" to infinity. For example, we will never encounter a series of, let's say 100 tails or more. That's why I can not accept that the expected value of this game is infinity, it will never happen in real life. The real expected value should be what we can observe empirically by making n number of simulations, n being a big number.
I think the math might be wrong. Instead of ($2 × 0.5) + ($4 × 0.25) + ($8 × 0.125)..., wouldn't it be ($1 × 0.5) + ($2 × 0.25) + ($4 × 0.125)...? The same paradox arises either way, but still.
@@ciroguerra-lara6747 You don't win $2 with 50% probability, though. The series shown in the video neglects the possibility that no heads are flipped and the player gets only $2^0, or $1. The probability of that outcome occurring is 50%.
@@ciroguerra-lara6747 That formulation of the paradox makes more sense. The one in this video is a different formulation of the paradox in which "a coin is flipped until it comes up tails" and "you are then paid $2^n where n is the number of times the coin came up heads." The version of the paradox in the link you provided doesn't allow an outcome in which n = 0. The one in this video does.
This theory is... Dumb. More rational to go for what is most likely, not what the greatest prize is times the chance. 1 or 0.1 percent chance to win a milli Lon dollars cannot possibly be better option than 100 percent chance to win 100 dollars.⚾
It seems to me you're evaluating whether It's good or not to take a certain bet assuming It's a one-time bet,but that's not necessarily the only place where expected value(EV) can be used.Take the example of an online poker player,each hand you play you are faced with a decision:Do you want to maximize EV by making the highest EV play,or make a less good EV play but that is more likely to make you win money on the spot?Considering these players play millions of hands every year,the highest EV play is always the most rational play,no matter how unlikely this is,as it increases your outcome by the end of the year.Since you've played such a big sample,variance plays a very small role,so whatever your EV is by the end of a huge sample of hands like they play,It's likely to be equal to your outcome,so outcome~expected value(after a big sample)
@@tlk1432 Maybe it is just the pessimist in me or maybe I'm just being dense on this topic or whatever but I just don't see the value (pun intended?) in Expected Value as is... I think it better to always go for the more likely scenario, even considering repetition of the gamble... I guess I just don't see how the likelihood amount multiplied by the prize amount gives us a relevant concept to use.
@@hendrikd2113 Interesting elucidation... Furthermore, after running some practical simulations involving Notepad.exe and a lot of dice rolls and scribbling I have come to the conclusion that I was wrong and dumb - and I am sorry for my bullheadedness. I guess this and Monty Hall are the sorts of things that my brain just doesn't want to gasp, grok, and accept.
@@philp521 Hey, thanks for the link, I'll check it out later. Neuroeconomics is sort of an offshoot of behavioral economics that focuses on the neural basis of economic decision-making. It's an exciting field of research and right now I'm working on it for my undergrad project, so I mentioned it because hearing some neurophilosophical considerations about it would be very interesting.
This paradox hints that you shouldn't measure expected value like that in games like these, even if you want to objectify the subject and throw its values out the window to fit the theory.
Nobody thinks that one will get 60$ when probability of winning 200$ is 30% when you play only one game.
And in the coin flip game, there is a 1/128 probability of winning >100$, compared to 100% probability of winning 100$.
Thanks for covering this. It's worth noting that this game could never could never be offered because no party could ever handle the risk of the other side.
Ending up with no happiness and no money?
Diminishing marginal utility doesn't actually fix this problem, because that's only for money, which does have diminishing marginal returns, but lots of other forms of utility don't. For example, if you could choose between saving a million people from a slow painful death (or in econ speak, utility) or taking the bet with the coin flips and saving 2^n people, you'd always have to let the million people die and take the bet. Saving people doesn't have diminishing returns, so you're still stuck with something that seems obviously wrong.
To be fair though, this was a great explanation of the concept and there's only so many topics you can cover in 7 minutes. Thanks for making this video! It was really helpful.
money of course have deminishing marginal utility. Is really $2 quintillion better than $1 quintillion, since it's almost all total money supply?
@@Acid31337 She wasn't doubting the veracity of diminishing marginal utility, but rather arguing that it does not sufficiently resolve the paradox.
I don't agree with this logic, either. I totally see a "diminishing" marginal utility in saving lives. The probability of saving more than 1 million people is too low.
It should be noted that while the rewards are infinite, you only get one shot at it. Only if you could play multiple games would it be worth it
My statistics professor gave a different variation, where you pay 1,2,4... to flip the coin and once the coin lands on heads, you get 2^x as payout. The EV is also infinity, here's what I'm wondering.
What if, we weigh the probability to the second order? 2*0.5*0.5 + 4*0.25*0.25, ... essentially summing the probabilities and getting 1. If you play the game an infinite number of times, you will still be infinitely rich but if you only play the game once, your winnings are much closer to 1 than infinity.
Philosophy of economics seems interesting. Keep it up!
The important factor does not seem to be diminishing marginal value, but rather the perception of the value of the marginal change in probability, which is not clearly symmetrical with - or a counterpart to - marginal value although there does seem to be some similarity or connection; I suspect that in individual cases there are threshold points for each of those factors which would be sufficient to either get someone to play the lottery, not bothering to find out the odds of winning, or refusing to play the lottery on such long odds, no matter how much money is offered as a prize.
This helps a lot, i keep feeling stuck, the number of dice I'm rolling keeps getting smaller and smaller. Some I stop rolling all together. I'm hurting myself here waiting on coins I'm flipping. Ill never get where I'm going if I don't draw my guide lines somewhere. I'm sick of falling short. 💔
The problem could be reduced if we knew when the "house" would break. Surely they don't have infinite physical coins to give out. So the expected was never infinite.
It's like always tripling your total losing bets on every coin flip. Garaunteed to win......unless someone demands a physical payoff at some time. Then......not a good strategy.
This is very useful! Thanks a lot
Glad to help! Thanks for watching.
Couldn't you just, like, inlcude a sliding scale of rationality in this, depending on the numerical chances? Like, sure, the _possible_ benefit of playing this game is infinite, but the _probable_ benefit is dysmal. If you have a choice between just taking 100$, or a chance lower than 1% to get the same 100$, it's obviously more rational to take the 100 bucks.
I think that it would be more rational to play the game because if you decide to play you'll have a small probability of getting infinite money in return. So the value of infiniteXsmall probability is more than the value of just 100 dollars. The paradox is about the hypothetical value of a choice (more value=more rational) and not the highest probability.
This was a cool lesson. Thank you for explaining it so well. I wonder how much your insight at the end counts as evidence that rational choice theory (RCT) is flawed because it tries to quantify the projected happiness of choices. Economic theories which try to calculate gross national happiness based on diverse metrics of flourishing, taking into account both human and nonhuman well-being, seem to me far superior to the simplified metrics of RCT. Projects like the Capabilities Approach pioneered by Martha Nussbaum and Amartya Sen strike me as more promising.
I must be making some simple math error, because it looks like in the coin flip game, the probability of a given amount won decreases by the same factor which incrementally raises the reward. Hence the weighted probability calculation gives you a plus $2 straight across the board, for any number of coin flips. Why would someone not choose the $100 with a probability of one? Of course, from the perspective of one who is already playing the game, the probability of doubling my money on the next coin flip, with a 50/50 chance of heads, would lead to a tendency to continue playing. So it looks like the notion of marginal utility or marginal value enters for a person who has already made the choice, but does not come into play at the outset, when the subject must choose to take the $100 or to set off down that road of coin flips. In any event, I am not sure I see a paradox here.
Since in theory you could flip 1 billion heads in a row, and keep going, the math says that the value of playing is technically infinite. Those have a low probability but they are not $0.
The point is since there is no limit on how heads you can flip in a row, there is no limit on your upside potential.
The $100 is certainly more probable. But the unlimited chance is more valuable.
Got here because someone referenced this in a Bitcoin debate 🤔
Very cool along with readings
personally even if i had the choice to 1. have probability 1 to get 100% more happiness or have probability 0,000.000.01 to get 1.000.000.000.000 % more happiness , i would chose the first option even though 1 x 100 = 100 and 1.000.000.000.000 x 0,000.000.01 = 10000 and so the second option is technically 100 times better but i still would go for the first option cuz my intuition tells me there is great value in certainty !
you can't measure expected value like that in a game you only get to play once. if you got to choose between those options over and over again, then the coin flip game would be the right choice, since you will eventually get a streak that will far exceed the $100
Why not? It is still true in a one-off game that a person who maximises expected utility will do better on average than one who doesn't, even though it is not certain.
@@edisonyi1188 yes, so it's better to at least get what IS certain.
@@sethapex9670 Whether we should consider the degree of certainty in rational choices is unrelated to what I was trying to say, which is that it doesn't matter if a game is repeatable or not.
The explanation with the utility function is unsatisfactory since the game can be altered in such a way that the paradox will reappear. To truly resolve the problem one needs to appeal to ergodicity theory, or in layman's terms, distinct between playing a single game or a large series of independence games (and say take the average of the rewards).
It takes seven flips of the coin to get to $128. Even with the repeated 50/50 chance; the law of averages will catch up.
Another point to be made is lack of a penalizing result. Both outcomes result in increased money with no threat of loss. I believe in gambling they would establish a spread to make it interesting.
I find it hilarious how a person can so unashamedly cite the "law of averages," on in this case the "gambler's fallacy"
Shouldn't this be $2 raised to the number of flips total, rather than the number of heads? In the current formulation, the probability of winning $2 would be 0.25 not 0.5.
Good question. No, it is the number of heads. If you flip the coin once and get a tails right away, you get $0. If you flip once and get heads, you get $2 regardless of what happens next. The chances of getting that first $2 are 50/50. If we counted the number of flips, you would be guaranteed to get $2, since playing the game automatically gets you one flip and 2^1 =2
The problem is that you will not be able to reach the end of a possibly infinite game
Splendid topic , I would love to see more videos on the same
Infinity often breaks things. You could also say one has 10^-(10^10^10) or one to the number of atoms in the universe chance of winning an infinte amount of money and 0 in any other case. The expectation value would still be infinite and one really dumb to play that game in the real world with finite boundaries. One has to be careful not to think too much about infinities to avoid going insane like Georg Cantor or Bobby Fisher did.
Yes, please!
Very distracting slides; too much info on the slides
I have prefered the explanation from Kane B: FOR ME (emphasized here), it was clearer.
Even setting aside the diminishing marginal utility of money, it seems like something is wrong with just the way of calculating the expected payout of the game, because surely there is only an infinitesimal chance that you will actually win the infinite money that this naive expected payout calculation predicts. You're more likely to win smaller amounts of money, because it's more likely that the game will end sooner. I don't know off the top of my head what a better way to predict the most likely payout would be, but as usual when you get an infinity in your calculations you probably did something wrong.
I disagree with the statement "as usual when you get an infinity in your calculations you probably did something wrong" If the game is finite but very long, the same issue will arise. I do agree that the diminishing marginal utility of money is a weird explanation that doesn't get to the heart of the issue.
this breaks down at a certain amount because who can spend infinite money?
Why can't someone just reformulate the paradox in terms of units of happiness instead of money?
Mmmm, my inner mathematician (edit: full disclosure, I'm a a biochemist, so to me mathematics is a useful tool I just use but make no claims to fully understand. I know how the screw-driver works, not how to make one) has been appeased: If the distribution of probabilities isn't linear, that's the cause of the paradox. And let's be honest, few things, if any, are linear.
This thing is dumb, and it’s not a paradox
People aren’t willing to spend money on something that is expected to have infinite payout over an infinite number of time....... people live like 85 years. Of course they’re not going to pay high dollar to play a game that may not yield significant amounts of money for thousands of years. That ain’t paradox. It’s common sense
Dunning-Kruger. Would you choose to play if the game is instantaneous?
The expected value is a number which we should expect to be our average of n number of outcomes. For example, if you roll a dice 1000 times, you can expect that the average outcome is 3.5. Now if we simulate St. Petersburg experiment for as many times as we like, I am pretty sure the average outcome will never be anything "close" to infinity. For example, we will never encounter a series of, let's say 100 tails or more. That's why I can not accept that the expected value of this game is infinity, it will never happen in real life. The real expected value should be what we can observe empirically by making n number of simulations, n being a big number.
In 1000 flips, even 10tails in a row would be too optimistic to think of as 2¹⁰=1024>1000
50% chance of getting 1$*...
I think the math might be wrong. Instead of ($2 × 0.5) + ($4 × 0.25) + ($8 × 0.125)..., wouldn't it be ($1 × 0.5) + ($2 × 0.25) + ($4 × 0.125)...? The same paradox arises either way, but still.
Oh yea, it is possible to get 0 heads
n starts at 1 since getting 0 heads will give you no benefit
@@garbunka 2^0 = 1, so zero heads still gets you $1
@@ciroguerra-lara6747 You don't win $2 with 50% probability, though. The series shown in the video neglects the possibility that no heads are flipped and the player gets only $2^0, or $1. The probability of that outcome occurring is 50%.
@@ciroguerra-lara6747 That formulation of the paradox makes more sense. The one in this video is a different formulation of the paradox in which "a coin is flipped until it comes up tails" and "you are then paid $2^n where n is the number of times the coin came up heads." The version of the paradox in the link you provided doesn't allow an outcome in which n = 0. The one in this video does.
This theory is... Dumb. More rational to go for what is most likely, not what the greatest prize is times the chance. 1 or 0.1 percent chance to win a milli
Lon dollars cannot possibly be better option than 100 percent chance to win 100 dollars.⚾
I just don't see the value of this theory or paradox... Why does this exist...
It seems to me you're evaluating whether It's good or not to take a certain bet assuming It's a one-time bet,but that's not necessarily the only place where expected value(EV) can be used.Take the example of an online poker player,each hand you play you are faced with a decision:Do you want to maximize EV by making the highest EV play,or make a less good EV play but that is more likely to make you win money on the spot?Considering these players play millions of hands every year,the highest EV play is always the most rational play,no matter how unlikely this is,as it increases your outcome by the end of the year.Since you've played such a big sample,variance plays a very small role,so whatever your EV is by the end of a huge sample of hands like they play,It's likely to be equal to your outcome,so outcome~expected value(after a big sample)
@@tlk1432 Maybe it is just the pessimist in me or maybe I'm just being dense on this topic or whatever but I just don't see the value (pun intended?) in Expected Value as is... I think it better to always go for the more likely scenario, even considering repetition of the gamble...
I guess I just don't see how the likelihood amount multiplied by the prize amount gives us a relevant concept to use.
Would you take 1$ right now, or 100000$ in 1 hour? You might have a stroke within the next 60 minutes. What would be your choice?
@@hendrikd2113 Interesting elucidation... Furthermore, after running some practical simulations involving Notepad.exe and a lot of dice rolls and scribbling I have come to the conclusion that I was wrong and dumb - and I am sorry for my bullheadedness.
I guess this and Monty Hall are the sorts of things that my brain just doesn't want to gasp, grok, and accept.
I really like this topic. The philosophy of neuroeconomics would be very interesting; if it exists, of course.
By neuroeconomics, do you mean behavioral economics?
If so, this is a good read!
www.cmu.edu/dietrich/sds/docs/loewenstein/BehavioralEconomics.pdf
@@philp521 Hey, thanks for the link, I'll check it out later.
Neuroeconomics is sort of an offshoot of behavioral economics that focuses on the neural basis of economic decision-making. It's an exciting field of research and right now I'm working on it for my undergrad project, so I mentioned it because hearing some neurophilosophical considerations about it would be very interesting.