Vsauce! Kevin here, and I have a simple coin-flipping

game, that requires no skill, has no catch or trick, and can lead to infinite wealth.

The thing is… nobody really wants to play it. Why? How is it possible that an incredibly easy

game with infinite upside causes virtually everyone to react with a massive yawn? Before we find out, a portion of this video

was sponsored by LastPass. They’ve been a great and repeated supporter of Vsauce2

and I genuinely use LastPass every single day. I signed up about two years ago now because

I was tired of getting locked out of accounts, forgetting passwords or dealing with websites

that have different rules for what a password can be. So I decided to just put everything

in LastPass and make my life that much easier. Let me tell you how it works and then we’ll

get back to our game. LastPass stores everything safely and securely for you and then autofills

your usernames and passwords everywhere. With unlimited password storage and cross-device

sync. So if you wanna eliminate your password frustrations forever just click the link down

in the description below. And thanks again to LastPass for sponsoring a portion of this

video. Now let’s get back to our game. To play this game, we’ll turn to the most

rational, calculating man in history: long-time friend of Vsauce2, Dwight Schrute. You walk

up to the table to flip a coin. Your prize starts at $2. If the coin flip results in

FALSE, the game is over and you win $2. If it lands on FACT, you play another round and

your prize doubles. Every time you get a FACT, you keep playing and the prize keeps doubling

— from $2 to $4 to $8 $16 $32 $64. $128, so on and so on… forever. But as soon as you get a FALSE, you are done

and you collect your winnings. So if you hit a FALSE in the third round, then your prize

is $8. If your first FALSE comes in Round 14, you’d walk away with $16,384. No matter

how unlucky you are, you’ll never win less than $2. If things go really well… then

things could go really well. Now that you know the potential payoffs, how

much would you be willing to pay to play this game? $3? What about $20, $100? The winnings

could be infinite so the question is: how much is a chance at infinite wealth worth

to you? We can determine the precise answer, but first

we need to know the game’s expected value, which is the sum of all its possible outcomes

relative to their probability. That determines the point at which we choose to play a game

— or, in the real world, the point at which we decide to take out insurance on our house

or a life insurance policy. If our risk is less than our likely reward, we should play.

If we’re paying too much relative to what we’re likely to get out of playing, then

we should not play. Here’s the expected value of Schrute. You’ve got a 50/50 chance of losing on your

first flip and heading back to the beet farm with $2. With a probability of ½ and a payoff

of $2, your expected value in the first round is $1. The probability of winning two rounds

is ½ * ½, or ¼, and your prize there would be $4. That’s another $1 in expected value.

For three successful flips, it’s ½ ^3 — or ⅛ — times $8. Another dollar. 1/16 * 16…

1/32 * 32… 1/64 * 64… For n rounds, the expected value is the probability

(½)^n * the payoff of 2^n — so no matter the value of n, the result will be 1. The

expected value of the game is 1 + 1 + 1 + 1 + 1… forever. Because each round adds

$1 of value no matter how rare the occurrence might be. The expected value is infinite. And there’s our paradox. Because, you’d

think a rational person would pay all the money they have to play this game. Mathematically,

it makes sense to pay any amount of money less than infinity to play. No matter what

amount of money you risk, you’re theoretically getting the deal of a lifetime: the reward

justifies the risk. But nobody wants to do that. Who would empty their bank account to

play a game where they know there’s a 75% chance they walk away with $4 or less? It’s confusing because expected value is,

mathematically, how you determine whether you’ll play a game. Look, if I offered you

a coin-flipping game where you won $5 on heads and lost $1 on tails, your expected value

of each round would be the sum of those possible outcomes: (50% chance * +$5) + (50% chance

* -$1). Half the time you’ll win $5, half the time you’ll lose $1. In the long run,

you’ll average +$2 for every round you play. So paying anything under $2 to play that game

would be a great deal. When the price to play is less than your expected value, it’s a

no-brainer. And since the expected value of the Schrute

game is infinite, paying anything less than infinite money to play it should also be a

no-brainer. But it’s not. Why? The thing that’s so interesting about this

game is how the math conflicts with…. actual humans. Enter: Prospect Theory. An element

of cognitive psychology in which people make choices based on the value of wins and losses

instead of just theoretical outcomes. The reason people don’t want to empty their

pockets to play this game despite its infinite gains is that the expected marginal utility

— its actual value to them — goes down as those mathematical gains increase forever.

This solution was discovered a few years ago. A few hundred years ago. In 1738, Daniel Bernoulli published his “Exposition

of a New Theory on the Measurement of Risk” in the Commentaries of the Imperial Academy

of Science of Saint Petersburg — and what we now call the St. Petersburg Paradox was

born. Bernoulli didn’t dispute the expected value of the St. Petersburg game; those are

cold, hard numbers. He just realized there was a lot more to it. Bernoulli introduced

the concept of the expected utility of a game — what was, until the 20th century, called

moral expectation to differentiate it from mathematical expectation. The main point of Bernoulli’s resolution

was that utility, or how much a thing matters to you, is relative to an individual’s wealth

and that each unit tends to be worth a little less to you as you accumulate it. So, as an example, not only would winning

$1,000 mean a lot more to someone who’s broke than it would to, say, Tony Stark, but

even winning $1 million wouldn’t affect the research and development at Stark Industries. And there’s also a limit on a player’s

comfort with risk, with John Maynard Keynes arguing that a high relative risk is enough

to keep a player from engaging in a game even with infinite expected value. Iron Man can

afford to lose a few billion. You probably can’t. And value itself is subjective. If I won 1,000

peanut butter and jelly sandwiches, I would be THRILLED. If someone allergic to peanuts

won them, they’d be… less thrilled. So. Okay, okay. Given all this, how much can

YOU afford to lose in the St. Petersburg game? How badly do you want to play? Bernoulli used

the logarithmic function to come up with price points that factored in not only the expected

value of the game, but also the wealth of the player and its expected utility. A millionaire

should be comfortable paying as much as $20.88 to flip Schrutes, while someone with only

$1,000 would top out at $10.95. Someone with a total of $2 of wealth should, according

to the logarithmic function, borrow $1.35 from a friend to pay $3.35. Ultimately, everyone has their own price that

factors in their wealth, their desires, their comfort with risk, their preferences, how

they want to spend their time, what else they could be doing with their money, their own

happiness… And the thing is… this game can’t even

exist. Economist Paul Samuelson points out that a game of potentially infinite gain requires

the other party to be comfortable with potentially infinite loss. And no one is cool with that. So if the important elements are variable

and the game can’t exist, what’s the point? The St. Petersburg Paradox reminds us that

we’re all more than math. The raw numbers might convince a robot that it’s a good

idea to wager its robohouse on a series of coin flips, but you know deep down that’s

a really bad idea. Because you aren’t an expected value calculation.

You aren’t a logarithmic function. The numbers are a part of you and help you live your life.

But in the end, you are… you. Fact. And as always, thanks for watching. It is a fact that you should subscribe to

Vsauce2 and hit that notification bell because it takes me a little while to make these videos

so it’s not like you’re gonna be bombarded with notifications. But as soon as I do make

one, I’d like you to know about it. So hit the notification bell. If you want to watch

my interview with Derek Muller aka Veritasium you can do that over here and other than that.

You have yourself a wonderful day, morning, night. Tomorrow? Afternoon, did I say afternoon

already? Afternoon. Middle of the night if you can’t sleep, thanks for spending your

time with me. Bye.

Hi, people in the future!

If I would be able to play this game infinite number of times with all I have (left), I would play (until I get really lucky).

id pay 2 dollars

I wonder if you let the player choose heads or tails, every time, instead of stating which one wins or loses would affect the player behavior as if a false sense of control would make them more willing to play

infinime loss?

I BET MY SOUL

Thank you, have a good evening too.

12:07 how did he know

Well technically, having infinite luck will ultimately lead to a loss. How can you claim your prize if the game does not end?

But I'm not satisfied with the solution of the paradox! I can't stop thinking about it! Goodbye sleep 😭😭

Hello person! I know that it's 2025 but that's okay. Was this in your recommendations?

This is also known as a Martingale strategy and was originally conceived of as a way to beat Roulette. The problem is that each individual spin or a roulette wheen like each flip of a coin is not determined based on the previous flip (just because the last flip came up heads does not mean the next is any more or less likely to come up tails.)

So as the odds do not change the utility of the return decreases after each subsequent flip.

The only time I have seen this actually work in any meaningful way is in trading as the next movement of a market has a direct relation to the previous movement and so the odds of up or down for example increase with every subsequent movement.

Well first of all, It is physically impossible for anything to complete an infinite amount of anything. And second of all even if it was possible, I'm pretty sure people would not want to risk just losing everything but 2$

Play with Zimbabweian Dollars 😆

It’s not infinite gain, because you can’t take the winnings until you get a loss.

Wait, wouldn't the pay off of the first game be $2 if you're guaranteed to get at least $2?

This is kinda like people betting on mortgage derivatives. You have to add in another factor they forgot, but turned out to be very important. What if you get really lucky and the house can't pay? Then they have insurance for such things…..but what if you are so lucky that the insurance company can't pay. And now we know how AIG went bankrupt.

I would never want to have infinite wealth, everyone in the world would be trying to kill me.

Much Simpler Example:

Flip a coin.

On Tails, you lose everything you own.

On Heads, you triple your total wealth.

Obviously good odds, but I wouldn't risk it.

(Curiously, although each individual game averages a net positive outcome, playing infinite times guarantees an irreversible failure)

A chance at infinite wealth is worth $1 to me

hehe its a coin with Dwight schrute on it and he says things like schrute and back to the beet farm which i am pretty sure is reffencing the show called "The Office"

Plenty of game theory in this video

I owe $1000 to the Mob tomorrow, and I have $1000 dollars. I'm not taking a bet, however good the odds are.

I owe $1000 to the Mob tomorrow and I have $5. I'm taking any bet, however long a shot – so long as it's the best chance I have amongst the alternatives.

Utility theory and opportunity cost in one small example.

h9 ZE

VSsuce "You're not a calculation"

Me: "News of the century"

It's not that hard guys. If you "lose" the first time you play you still get the 2 dollars. The maximum you should be paying to play is $1.99…

Vsauce..! Chipmunk here!

Mr beast should host the game

If I won a thousand peanut butter and jelly sandwiches I would be thrilled

If someone with a peanut allergy won them

They would be killed…

Wealth comes from family

Thank you for the video! All of you friends are super awesome! Oh, moments in this video are sad.

5.18 what game music it was?

You have a 100% chance of getting less than the expected value.

How else has only seen vesauce1

Karen takes the infinite kiddie.

yeah i'll be better off with just vsauce

7:12 I didn’t know bhad bhabie was interested in theory??

What if you play the game an infinite number of times, wagering $1 each time?

This is the famous st. Petersburg game ör paradox

1$

001$

1 dollar. Shut up now

I never understand what you are saying but whatever youre awesome

i thought this was called the martingale method?

I thought you were going to talk about having a dollar and going to the past and bringing it with you and then taking the dollar from the past so instead of 1 dollar you have 2 and repeat that

3:22 lottery players:

am i a joke to you?

everybody gangsta til kevin figures out the duplication glitch

The argument made in the video suggests that two infinite amounts cancel eachother out which would void the significance of playing the game at all, which is technically true that two infinites cancel eachother out but the video does not account for the fact that the infinites are not eaqual to eachother. And i can prove why…

for every round of the game you play the only circumstance which the infinites cancle eachother out is you spend the same amount as you win, this cannot be possible by only betting a singular ammout as there are an endless amount of values above any possible bet that you make, aswell as there being a finite amount of bets under the bet you make.

The obvious argument against this is what if you bet an infinite amount? And the answer to that is its not possible to do so therefore its not an option, why? Well because infinity as a concept meaning a number that is higher than every number before it does not make sense as a concept due to the way numbers work. Any given number is always eaqual to another number that is the same e.g. 1=1 and 26=26. This means that for any given number there is always a bigger one, because the "amount of amount" you have is consistent and follows an algorithm that is consistent itself an infinite bet makes no sense.

So now that is clarified, we can now lay down a few rules to the way the game function.

For any bet made there will be a offset created by the (amount lost + amount won) unless they are equivalent. (This proves that the infinites cant be even unless under a singular circumstance i will talk about later.)

For any given bet the amount of prizes smaller than your bet is smaller than the endless amount of prizes bigger than your bet. (They are less likley yes but this proves there is going to be an offset made because they are in the sample space)

Now what if your wins were eaqual to your losses? You always every time for example got the 3rd option (win $8) and bet $8? Well this comes back to why the infinity argument is not consistent, if you assume any chance of this happening "a" a<1 then the accumilative chances of all the endless other options "x" then the chances will be.. a<x<1 so if run endlessly it is impossible to argue the chances of infinitely getting the 3rd option can ever outweigh the chances of not getting another option which is obvious enough.

Conclusion:

There is dialouge to be had about this game. But it is obvious that it is a game worth playing the real question now is how much do you bet per turn.

Obviously as close to nothing as you can, but how much should I bet to get eaqual spendings and winnings?

Let me know if any of you have any ideas on the subject and what you think of my opinion if its valid or overlooking something!

This better not be the milagro man arc all over again.

I mean it’s better than going to a casino.

Can you – in detail – describe which portion of the video was sponsored by LastPass? What made you choose those parts over others? What is the specific reason only a portion of the video is sponsored?

I kinda want that Dwight coin actually

Where do I play this game?

3:29 Actually you have a 57% chance of winning because the head side is heavier.

1738

…

Why did you yawn?! damn it.. it congaed me.

stonks

Basically, just use the Kelly Criterion

The winnings cannot be infinite. To get an infinitely large prize there must be an infinite number of turns but if that happens your actual winnings are $0 because you only get paid when the game ends and if there are infinite turns then it never ends.

"… In the end, you are you…" Then … what's the point…

Technically theres a 50% chance of 2 dollars, 25% chance of 4 dollars etc so the expected value of making it that far is 1 dollar not an additional dollar per round id pay a dollar to play but no one would make a game they automatically lose if they want to win 87.5% of the time charging between 8 and 16 dollars makes sense

1$

has iq of nearly 180Kevin: i KeEP FoRGe*TI*Ng

mYPa*Ss*WorDsI can't afford to lose a single dollar

I would pay $2 to play this…

as a long time farmhand at a major cattle livestock ranch, I find it insulting that I would go back there with venison, much less male venison.

Why are there 3 vsauses

Harvard wants to know your location

Ima play for 1 dollar

But doesn’t it even out if the winner looses and the the other guy wins ?

Now thear is 2 of them

I know this game, it's called gambling

Bro you got it all wrong from the begining. At the first round you have 50% to win 2$ and 50% to win 4$ or more. So, the expected value is 3$.

lastpass gas

Hey wassup future ok maybe no

Perhaps you subtract 1 from each prize.

There we go.

Or maybe you pay 1 dollar when you wanna continue after getting a heads.

I like how you made "less thrilled" appear as though you were gonna say "Dead"

this is what you call

stonksSomeone get MrBeast here immediately

Bruh I've already surprisingly figured this out last year

Leave risk assessment to the actuaries.

Can I borrow $1.35?

"If things go well, then…

things go well-Kevin

coin flipping? I thought you said this requires no skill…

If you say "b*t*h" a few times you'll look a lot like Jesse Pinkman

So LastPass is basically Dashlane's less popular little sibling

STONKS

yet nobody noticed that he was using his left hand during entire video

I would pay $3 to play it.

MEME?

Around 3:40 can anyone else hear the daredevil theme

Economy:

NoWhy should i not just spend 1$ on it? Do i need to pay every run?

What? Cool!

Love that classic opening:

“Vsause! Kevin here”

w a i tThis paradox

Everyone wants to play but no one wants to lose

Love your videos on paradoxes

This not the kind of infinite I wanted

That moment when you said to yourself that you would pay three dollars to pay and then he roasts you at the end saying I only have two dollars… He's right.

i like vsauce 2 more