The most interesting part of this paradox is the fact that the expected value is correct. If you play this game an infinite amount of times, no matter how much money you pay per play, you will always ("in the end" so to say) win an infinite amount of money. You can test this by simulating the game being played x number of times, then divide by x. Even though this should always be equal to the expected value, it will increase logarithmicly with x.
The expectation is that one loses or gains what one puts (I) minus the last toss money quantity obtained: 2^n ( if one, say, gets tails), where n is the number of tosses. This is to be weighted by the a priori probability to get only heads (1/2^n) or getting at least once tails (1-1/2^n) at toss n. The expected value for the gain equation for N tosses then must be (note that I appears both at the right and at the left of the equation): I = Sum(n=1:N) [2^n/2^n-I*(1-1/2^n)] -> I = 1 unit of money.
I think the paradox breaks down because we can only play a finite number of games for instance you can play the game in R using a real random number generator 1000 or 1,000,000 times presumably as the number of games grows the expected value deverges but there are only a finite number of games. So that the expected value if you could play 1000 or 1,000,000 games changes.
If we say that gaining the same amount of money would be worthless based on what money we already had, that does not mean that doubling your money has less value if you have $100,000 compared to $100. In both cases, since you're doubling your money, you gain relatively the same amount. So I would say that this increase is equivalent if we want to take the idea of relative value to its extreme. So we'll measure utility in relative currency. basically, it'll be a logarithm of the prize. That way gaining relatively the same will be equal to adding some constant, by the property of log(ab) = log(a) + log(b) In this case I'll use log base 2, to make it easy. That way doubling will lead to adding 1. The expected value chart then becomes: n | Prize | Utility | probability 1 | $2 | 1 | 1/2 2 | $4 | 2 | 1/4 3 | $8 | 3 | 1/8 4 | $16 | 4 | 1/16 so the utility is simply equal to n. Of course, the probabi Thus the expected winnings: 1/2*(1) + 1/4(2) + 1/8(3) + 1/16(4) + ... = 1/2 + 2/2² + 3/2³ + 4/2^4 + 5/2^5 + 6/2^6 + ... = 1/2 (1 + 2/2 + 3/2² + 4/2³ + 5/2^4 + 6/2^5 + ... ) = 1/2(S) Let's just call what's in the parentheses S. You should know the power series of 1/(1-r) which is equal to 1 + r + r² + r³ + r^4 + ... || under the condition that r²
Sometime I feel my evaluation going the other way. I would choose the riskier option if you told me to choose either to get a dollar or a one in a million chance at getting million dollars. One dollar just doesn't change anything for me.
And in fact this is how lotteries work: the prize is considerably less than the sum of the price of the tickets. Otherwise how would the people running the lottery make money?
Good presentation. I think it is Solution 2 not so much based on utility, but on risk aversion. I think of it this way: what would I pay for a one toss game. Well, up to the EV of 1. What would I pay for a two toss game. Well, up to the EV of 2. What would I pay for three toss game. Mmmm, 1:8 chance of 3 tails in a row. My risk averse nature says 2.5!!! Well below EV, and so it would continue. I want to profit from the uncertainty, not just get EV. Or think of it this way, what would you pay for a one off game of getting 5 tails in a row. EV is 1 but I wouldn't even play! (But neither would I offer the game!) However, gamblers willing and continually roll the dice and bet on "snake eyes" (1:36 chance) for a mere 30 potential payout!!! Rational?
I don't think the paradox is that the expected value is infinite--there are plenty of distributions with infinite or undefined expected values. The paradox is that despite the infinite expected value, most rational people would still refuse to pay too high of a price for this game--for example, I might pay $3 or even $10 (that might be about at my limit), but I would not pay $60.
After 10 successful iterations you would get $1,024, after 20 you would get more than $1 million and after 30 you would get more than $1 billion. Finally, after 40 successful iterations you would get $1000 billions. Almost certainly the counterpart would not be able to pay more than $1 billion. In that case, it would be irrational to bet more than $30. But with so few total iterations, the entire exercise would be unlikely to take more than half an hour. The money will run out far far sooner than the time.
I can imagine an infinitely large economy, (e.g. an infinite number of inhabited planets connected by stargates, all using the same currency), which would presumably have an infinite capacity to absorb money, without causing inflation. It still seems irrational to pay everything you have to play this game in such a world, though.
It's like the issue of bitcoin. How many bitcoins should you have bought when they were 5¢ each? What about when they were 50¢? Was that the end? And when they were $5 each? $50? $500 each? And going forward, do you expect it to reach $5,000 each, $50,000? $500,000 each? Etc. Will it stop somewhere and reverse, or will it keep going? Many knew about bitcoins very early on, but they thought of mining them, not buying them. I remember, because people used to discuss whether the electricity spent in mining them was worth the return of actually mining some and selling them. Of course, they were calculating those returns based on the price at the day, which used to be low, or relatively little above that, so many decided they wouldn't be worth mining. Mining, and investing in mining gear, as well as buying, required faith. Some had this faith (or fate), some didn't. And in the same way, some will continue to have faith and keep invested, while some will call it quits. And what about cryptocurrencies in general? Should we invest in new cryptocurrencies? How much? How would we select them? And if our projections for a given cryptocurrency after some time prove wrong, in excess or otherwise, will we update our estimates and either withdraw or redouble our investment, or be on autopilot? And should we in the end invest in the whole lot of them as assets, and dedicate a part of our portfolio to them, or just for now? What about inventing new cryptocurrencies? Because of course programmers who invent a cryptocurrency get to mine in advance of others as many coins as they like, before it may be well known and the cost of mining increases. So arguably even a great effort such as giving up everything in your life and studying computing science and then cryptocurrencies and becoming so good at it so as to forge new ones, could eventually be a good investment and use of your time, even if it took you twenty years. Or are we past the cryptocurrencies "golden age" and we're now in the third or forth generation (of this cryptocurrencies' universe big bang, this 'inflation') and it is no longer worth it? How can we know? And what about inventing something other than a cryptocurrency that would give you unlimited wealth and power? Such as inventing the AI? Or is there something between cryptocurrencies and the AI that's still an 'object' and not a 'subject'? What would it be? Vast knowledge? A perfect predicting algorithm? A perfect way of manipulating public opinion / people / elections? What would the Gulag's present day forced labor be investigating? And do scientists and researchers see themselves as forced labor? Or only if the Soviets force them, but not when Google makes them do it? But what if Google made them do it because building the AI through using volunteer scientists is faster for Google - and even possible - than through using slaves -- so North Korea really doesn't stand a chance. Would you work for Kim Jong Un for $1,000,000, even if you could only fail? What about Trump? And yet you work for your boss, at which (i.e., at becoming the boss) you can only fail. How to solve the paradox? And should you plan to fix it, or should you start fixing it right now? [...] BTW While we think of this, Global Warming keeps advancing and Antarctic ice melting, so we'll all be fried. Unless all your infinite wealth can buy you the best refuges and hideouts to withstand any catastrophe, natural or man-made, and in so doing, why not? Buy you immortality, and the power to reach the stars. Who knows? Maybe that's how God got started. Musk doesn't want to be God, maybe only his father. 'Explore space, go and multiply', he tells Earthlings. Or maybe just so while he can sell us cars and shares in his companies, to elevate him as a high priest - then we'll die, and we won't know what happened. Who to follow then? And should we keep going and looking for one to follow, or give up while we still can? And if we give up and each go our own way, how will we keep track? (we won't). Should we stop watching or reading (anything), and talking and writing. What are we to do. Amen.
Poor presentation-the narration will confuse non mathematicians e.g. in the intro the narrator erroneously says 'if it lands tails you win $8 etc. Furthermore to the uninitiated the statement 'heads you win $2 and the game ends...' and then 'the price to enter can be infinite' is confusing i.e. you pay $100 000 to play; the coin is tossed; its heads you win $2 and the game ends. YOU JUST LOST $99 998! IT’S THE TOSS session (event) not the game that ends! The game goes on with the next toss which you have paid for. You pay only once and the coin is tossed through infinite sessions i.e. it starts at $2 every time. I see so many of these presentations done so poorly that for the initiated it makes sense but I have students repeatedly coming to me in a state of confusion due to ambiguities. It is anathema for mathematicians to tolerate an ambiguity.
"IT’S THE TOSS session (event) not the game that ends! The game goes on with the next toss which you have paid for" No. You don't pay for any particular number of coin flips. You pay to enter the game, and game ends when the coin lands heads. Here, for example, is the description from Hacking's Intro to Probability and Inductive Logic: "Imagine a fair coin, tossed in a sturdy chance setup. The coin is tossed once. If it falls heads, you receive a prize of $2, and the game stops. If not, it is tossed again. If it falls heads on the second toss, you get $4, and the game stops. If not, it is tossed again. If it falls heads on the third toss, you get $8, and the game stops. And so on." Perhaps there are other versions of the paradox out there that you have encountered. The version I have encountered was presented correctly in this video.
Tony Nelson I'm no mathematician and the presentation was absolutely fine. And your description of the paradox is just incorrect so if anybody is causing confusion it's you!
![[ไฮไลต์] ฟุตบอลชาย อุซเบกิสถาน vs สเปน รอบแบ่งกลุ่ม | โอลิมปิก 2024](http://i.ytimg.com/vi/2ps5WKV1Mq0/mqdefault.jpg)
Finally someone who explains it rationally.
Dude, I’ve been watching your channel every day like it’s Sesame Street for grown ups. XD
The most interesting part of this paradox is the fact that the expected value is correct. If you play this game an infinite amount of times, no matter how much money you pay per play, you will always ("in the end" so to say) win an infinite amount of money.
You can test this by simulating the game being played x number of times, then divide by x. Even though this should always be equal to the expected value, it will increase logarithmicly with x.
The expectation is that one loses or gains what one puts (I) minus the last toss money quantity obtained: 2^n ( if one, say, gets tails), where n is the number of tosses. This is to be weighted by the a priori probability to get only heads (1/2^n) or getting at least once tails (1-1/2^n) at toss n. The expected value for the gain equation for N tosses then must be (note that I appears both at the right and at the left of the equation): I = Sum(n=1:N) [2^n/2^n-I*(1-1/2^n)] -> I = 1 unit of money.
I think the paradox breaks down because we can only play a finite number of games for instance you can play the game in R using a real random number generator 1000 or 1,000,000 times presumably as the number of games grows the expected value deverges but there are only a finite number of games. So that the expected value if you could play 1000 or 1,000,000 games changes.
If we say that gaining the same amount of money would be worthless based on what money we already had, that does not mean that doubling your money has less value if you have $100,000 compared to $100. In both cases, since you're doubling your money, you gain relatively the same amount. So I would say that this increase is equivalent if we want to take the idea of relative value to its extreme. So we'll measure utility in relative currency. basically, it'll be a logarithm of the prize. That way gaining relatively the same will be equal to adding some constant,
by the property of log(ab) = log(a) + log(b)
In this case I'll use log base 2, to make it easy. That way doubling will lead to adding 1.
The expected value chart then becomes:
n | Prize | Utility | probability
1 | $2 | 1 | 1/2
2 | $4 | 2 | 1/4
3 | $8 | 3 | 1/8
4 | $16 | 4 | 1/16
so the utility is simply equal to n. Of course, the probabi
Thus the expected winnings:
1/2*(1) + 1/4(2) + 1/8(3) + 1/16(4) + ...
= 1/2 + 2/2² + 3/2³ + 4/2^4 + 5/2^5 + 6/2^6 + ...
= 1/2 (1 + 2/2 + 3/2² + 4/2³ + 5/2^4 + 6/2^5 + ... )
= 1/2(S)
Let's just call what's in the parentheses S.
You should know the power series of 1/(1-r) which is equal to 1 + r + r² + r³ + r^4 + ... || under the condition that r²
Sometime I feel my evaluation going the other way. I would choose the riskier option if you told me to choose either to get a dollar or a one in a million chance at getting million dollars. One dollar just doesn't change anything for me.
And in fact this is how lotteries work: the prize is considerably less than the sum of the price of the tickets. Otherwise how would the people running the lottery make money?
Good presentation. I think it is Solution 2 not so much based on utility, but on risk aversion. I think of it this way: what would I pay for a one toss game. Well, up to the EV of 1. What would I pay for a two toss game. Well, up to the EV of 2. What would I pay for three toss game. Mmmm, 1:8 chance of 3 tails in a row. My risk averse nature says 2.5!!! Well below EV, and so it would continue. I want to profit from the uncertainty, not just get EV. Or think of it this way, what would you pay for a one off game of getting 5 tails in a row. EV is 1 but I wouldn't even play! (But neither would I offer the game!)
However, gamblers willing and continually roll the dice and bet on "snake eyes" (1:36 chance) for a mere 30 potential payout!!! Rational?
I don't think the paradox is that the expected value is infinite--there are plenty of distributions with infinite or undefined expected values. The paradox is that despite the infinite expected value, most rational people would still refuse to pay too high of a price for this game--for example, I might pay $3 or even $10 (that might be about at my limit), but I would not pay $60.
After 10 successful iterations you would get $1,024, after 20 you would get more than $1 million and after 30 you would get more than $1 billion. Finally, after 40 successful iterations you would get $1000 billions. Almost certainly the counterpart would not be able to pay more than $1 billion. In that case, it would be irrational to bet more than $30. But with so few total iterations, the entire exercise would be unlikely to take more than half an hour. The money will run out far far sooner than the time.
I can imagine an infinitely large economy, (e.g. an infinite number of inhabited planets connected by stargates, all using the same currency), which would presumably have an infinite capacity to absorb money, without causing inflation. It still seems irrational to pay everything you have to play this game in such a world, though.
It's like the issue of bitcoin. How many bitcoins should you have bought when they were 5¢ each? What about when they were 50¢? Was that the end? And when they were $5 each? $50? $500 each? And going forward, do you expect it to reach $5,000 each, $50,000? $500,000 each? Etc. Will it stop somewhere and reverse, or will it keep going?
Many knew about bitcoins very early on, but they thought of mining them, not buying them. I remember, because people used to discuss whether the electricity spent in mining them was worth the return of actually mining some and selling them. Of course, they were calculating those returns based on the price at the day, which used to be low, or relatively little above that, so many decided they wouldn't be worth mining. Mining, and investing in mining gear, as well as buying, required faith. Some had this faith (or fate), some didn't. And in the same way, some will continue to have faith and keep invested, while some will call it quits.
And what about cryptocurrencies in general? Should we invest in new cryptocurrencies? How much? How would we select them? And if our projections for a given cryptocurrency after some time prove wrong, in excess or otherwise, will we update our estimates and either withdraw or redouble our investment, or be on autopilot? And should we in the end invest in the whole lot of them as assets, and dedicate a part of our portfolio to them, or just for now?
What about inventing new cryptocurrencies? Because of course programmers who invent a cryptocurrency get to mine in advance of others as many coins as they like, before it may be well known and the cost of mining increases. So arguably even a great effort such as giving up everything in your life and studying computing science and then cryptocurrencies and becoming so good at it so as to forge new ones, could eventually be a good investment and use of your time, even if it took you twenty years.
Or are we past the cryptocurrencies "golden age" and we're now in the third or forth generation (of this cryptocurrencies' universe big bang, this 'inflation') and it is no longer worth it? How can we know?
And what about inventing something other than a cryptocurrency that would give you unlimited wealth and power? Such as inventing the AI? Or is there something between cryptocurrencies and the AI that's still an 'object' and not a 'subject'? What would it be? Vast knowledge? A perfect predicting algorithm? A perfect way of manipulating public opinion / people / elections? What would the Gulag's present day forced labor be investigating? And do scientists and researchers see themselves as forced labor? Or only if the Soviets force them, but not when Google makes them do it? But what if Google made them do it because building the AI through using volunteer scientists is faster for Google - and even possible - than through using slaves -- so North Korea really doesn't stand a chance. Would you work for Kim Jong Un for $1,000,000, even if you could only fail? What about Trump? And yet you work for your boss, at which (i.e., at becoming the boss) you can only fail. How to solve the paradox? And should you plan to fix it, or should you start fixing it right now?
[...]
BTW While we think of this, Global Warming keeps advancing and Antarctic ice melting, so we'll all be fried. Unless all your infinite wealth can buy you the best refuges and hideouts to withstand any catastrophe, natural or man-made, and in so doing, why not? Buy you immortality, and the power to reach the stars. Who knows? Maybe that's how God got started.
Musk doesn't want to be God, maybe only his father. 'Explore space, go and multiply', he tells Earthlings. Or maybe just so while he can sell us cars and shares in his companies, to elevate him as a high priest - then we'll die, and we won't know what happened.
Who to follow then? And should we keep going and looking for one to follow, or give up while we still can? And if we give up and each go our own way, how will we keep track? (we won't). Should we stop watching or reading (anything), and talking and writing. What are we to do. Amen.
Poor presentation-the narration will confuse non mathematicians e.g. in the intro the narrator erroneously says 'if it lands tails you win $8 etc. Furthermore to the uninitiated the statement 'heads you win $2 and the game ends...' and then 'the price to enter can be infinite' is confusing i.e. you pay $100 000 to play; the coin is tossed; its heads you win $2 and the game ends. YOU JUST LOST $99 998! IT’S THE TOSS session (event) not the game that ends! The game goes on with the next toss which you have paid for. You pay only once and the coin is tossed through infinite sessions i.e. it starts at $2 every time. I see so many of these presentations done so poorly that for the initiated it makes sense but I have students repeatedly coming to me in a state of confusion due to ambiguities. It is anathema for mathematicians to tolerate an ambiguity.
"IT’S THE TOSS session (event) not the game that ends! The game goes on with the next toss which you have paid for"
No. You don't pay for any particular number of coin flips. You pay to enter the game, and game ends when the coin lands heads. Here, for example, is the description from Hacking's Intro to Probability and Inductive Logic:
"Imagine a fair coin, tossed in a sturdy chance setup. The coin is tossed once. If it falls heads, you receive a prize of $2, and the game stops. If not, it is tossed again. If it falls heads on the second toss, you get $4, and the game stops. If not, it is tossed again. If it falls heads on the third toss, you get $8, and the game stops. And so on."
Perhaps there are other versions of the paradox out there that you have encountered. The version I have encountered was presented correctly in this video.
Tony Nelson
I'm no mathematician and the presentation was absolutely fine. And your description of the paradox is just incorrect so if anybody is causing confusion it's you!
This is too drenched in facts... It needs a TINY bit of filter.
It seems that you are not preparing enough with your voice unless you just speak like that then I apologize.
Speak like what? Anyway, I don't bother "preparing" my voice. I just talk.
Kane B For some reason it just seems like you're reading off something in a sort of semi-monotone fashion.
Oh I see. That's because I am in fact reading off something in a semi-monotone fashion. (I write a script for all my videos.)