How will you react if you can make infinite wealth without any hard work just by flip a coin game? But still people don’t play this game.
If you don’t have the concept of infinite of infinity here is the basic information on it.
Yes you heard it right infinite wealth, the math of the coin game tells you that you have access to infinite wealth and unlimited expected value but the real life it tells you not to play the game.
Curious about what could that game be? Let’s discuss it today.
Flip a Coin Game
Suppose you walk up to the table to flip a coin and your price starts at $2. If the coin results tail, the game is over and you earn $2, if it results head you continue to play another round and your price double.
Every time you get head you keep playing and the price gets doubling. But as soon as you get tail you are done and you collect all your winning.
So if you get tail in the third round you get $8. If you got tail in 14 round you earn $16,384.
No matter how unlucky you are you get $2.
Now you got the idea of how much potential the game will pay, the question is How much are you willing to pay to play this game?
$2, 30$, 100$?
The winning could be infinite, the question is How much is a chance at infinite wealth worth to you?
We can get the answer of the question from the simple math. But before that we need to find the expected value of the coin game which is the sum of all its possible outcomes relative to their probability. Thats the point which determine the point where we should stop playing the game. In simple term we should play if risk is less than reward and if we are paying too much relative to what we can earn we should stop playing the game.
Here is the expected value of the game. First the probability of getting heads and tails is 50-50. So the probability of you loosing the game is 50-50 and make 2$.
With the probability of 1/2 and pay off of 2$, your expected value in the first round is 1$.
The probability of winning the 2nd round is 1/4 and pay off of 4$, thats another 1$ in expected value.
|n||P(n)||Prize||Expected Pay off|
Like that for the nth round the expected pay off is:
So no matter the value of n, the result will always be 1.
The expected gain of the game is 1+1+1+1+1+1+1 ……………………………
Because each round adds 1$ of value no matter how rare the occurrence is. So the expected value of the game is infinite.
So you think the rational person will pay infinite money to play the game. Mathematically it makes sense to pay any amount less than infinity to play the game because the risk is less than reward as you are theoretically getting the deal of you life.
But nobody wants to do this……….
Who would use all of their wealth to play the Flip a Coin game where they Flip a Coin three times, Flip a Coin 100 times, Flip a Coin 1000 times where they know there is 75% chance they walk away with $4 or less.
Its confusing because expected value is mathematically, how you determine whether you play a game.
For example: If I offer you a game where you earn +5$ for heads and loss -1$ on tail. Your expected value on each round would be the sum of those possible outcomes: (0.5*5$)+(0.5*(-1$)) = $2
In long run, in average you will make 2$ in every round you play. So paying under 2$ every round to play this game will be a great deal.
Since, the expected value of our initial coin game is infinite, paying anything less than infinite is a great deal.
But its not as it is seen.
The thing thats interesting about this game is how math conflicts with actual humans.
Humans make the choices based on the value of wins and losses instead of just theoretical outcomes of the mathematics according to Prospect Theory.
The reason people don’t want to empty their pocket to play this game despite they can make infinite amount of money is that the expected marginal utility (the actual value to them) goes on decreasing as those mathematical gains increases forever.
The expected utility of a game is moral expectation to differentiate it from mathematical expectation. How much a thing matter to you is relative to an individuals wealth and the each unit tends to be worth a little to you as you accumulate.
Winning $1,000 means a lot more to someone just broke than it would to, say, Elon Musk, but even winning $1,000,000 wouldn’t affect Elon Musk.
And there is also limit on players comfort with risk. John Maynard Keynes arguing that high risk is enough to keep a player from engaging in a game even with infinite expected value.
Elon Musk can afford to loose few million dollars even few billions. You probably can’t.
And value itself is subjective. If I win 1,000 chocolates I would be Thrilled but if someone allergic to chocolate win 1,000 chocolates they will be less thrilled.
Give all of these, how much can you afford to loose on the coin game? How badly you want to play?
Bernoulli comes with a logarithmic function to come 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, while someone with $1,000 would top out $10.95. Someone with 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 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 things is the game can’t even exist.
To earn the infinite wealth the other party should be capable of infinite loss.
So, if the important point is variable and the game doesn’t exist. Whats the point of it?
The game reminds of us, we all are more than maths. The raw numbers can convince the robot to play the coin flip game but deep down you know its the bad idea. Because you are not the expected value calculation. You are logarithmic function. The numbers are a part of you and help you live your life but at the end you are … you.
*** SOURCES ***
Original Bernoulli family correspondence
Play the St. Petersburg Paradox game
“Ending the Myth of the St. Petersburg Paradox,” by Vivian Robert William
“St. Petersburg Paradoxes: Defanged, Dissected, and Historically Described,” by Paul Samuelson