St. Peterburg paradox and human behavior

Coin flipping in the air. A guy can be seen behind it, blurry.

Reading through an amazing book about Math I bought a couple of years ago I found out about this thing: The St. Petersburg Paradox. Basically, it can be explained this way: to determine price for entering a game consisting on flipping a coin until getting tails, winning 2^(k−1) (2 elevated to the difference k minus one). Being k the number of times the coin was flipped until getting tails.

So, if the dealer/casino flips once and you get tails, you win just a dollar. If you get one heads before tails you get 2 dollars; with 2 heads you get 4, with 3 you get 8, with 4 you get 16… In essence it tells you that you can win a lot by just flipping a few times. It reminds me a little of the Wheat and chessboard problem, with which we could easily get quite a prize for anything (*blinks right eye*).

Back to the St. Petersburg paradox, we can see the real problem with what it’s needed: a price tag for entering the game. Knowing human behavior and “players”, many people who have studied the game have concluded that most people see the “possible earnings” as really high, and as so they would consider playing as long as the price is not too high. The “expected utility theory” calculates how much a person would be willing to pay considering how much money the person has, ranging from paying $2 if having the $2 to paying $10.94 if millionaire (that cheap are them. Although, considering they already have the money…).

Now, talking about human behavior, if you appreciate the bet, you can see a 50% chance to get a first heads, then 50% chance to get a second. In reality, it’s a 50% chance to get one heads, 25% chance to get two heads, 17.5% to get three and so on… The chances, as separated are 50% each, get 50% smaller with each coin. So, in reality, there’s a 1/2 chance to just get $1, as so 1/8 to get $4, 1/32 to get $16. That’s too little for a so small chance. And even then people buy lottery tickets.

It amaze me, because it’s true. We tend to see the winnings. And so, taking probabilities out, there is a chance you get infinite earnings from playing. Why not? some people are just that lucky. And so we start to believe in luck. The real thing is that in casino a lot of games are worse than this one. The difference is that this tells you the chances.

Photo by Pauli Antero.

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