Humans don't always make rational decisions. Even smart gamblers will sometimes refuse a wager when the expected payoff could be huge and the odds are fair. The St. Petersburg Paradox gives an example of a perfectly fair gambling game that perfectly healthy statisticians probably wouldn't play, just because they happen to be human. The standard decision-making process for statistically savvy gamblers involves figuring the average payoff for a hypothetical wager and the cost to play, and then determining whether they are likely to break even or, better yet, make a boatload of money. Though one could produce dozens of statistical analyses of gambling all about when a person should and shouldn't play, the psychology of the human mind sometimes takes over, and people will refuse to take a wager because it just doesn't feel right. The Game of St. PetersburgThe game of St. Petersburg is about 300 years old. The parameters of the game were described by Daniel Bernoulli in 1738. Here are the rules:
So far, it sounds pretty good and more than fair for you. But it gets better. We keep flipping until heads comes up. When it eventually arrives, I pay you $2n, where n is the number of flips it took to get heads. Great game, at least from your perspective. But here's the killer question: how much would you pay to play?
Deciding how much you would pay to play is an interesting process. As a smart statistician, you would certainly pay anything less than $2. Even without all the bigger payoff possibilities, betting you will get heads on a coin flip and getting paid more than the cost of playing is clearly a great bet, and you'd go for it in a shot. You also probably would gladly pay a full $2. You will win the $2 back half the time, and the other half of the time you will get much more than that! This is a game you are guaranteed to win eventually, so it's not a question of winning. When you don't get heads the first time, you have guaranteed yourself at least $4 back, and possibly morepossibly much more. So, maybe you'd pay $4 to play. Of course, occasionally, your payoff would be really big money$8, $16, $32, $64...theoretically, the payoff could be close to infinite. But how much would you pay? That's the 64-dollar question. Statistical AnalysisSome social science researchers suggest that most people would play this game for something around four bucks, maybe a little more. Few would pay much more. What about statistically, though? What is the most you should pay? Well, this is where I consider turning in my Stats Fan Club membership card, because I am afraid to tell you the correct answer. The rules of probability as they relate to gambling suggest that people should play this game at any cost. Yes, a statistician would tell you to play this game for any price! As long as the cost is something short of infinity, this is, theoretically, a good wager. Let's figure this out. Here's the payoff for the first six coin flips:
If you play this game 64 times, you will get to the sixth coin flip just once, but you will win $64. 32 of those 64 times you will win just $2. The average payoff sounds lowjust a buck. Occasionally, though, heads won't come up for a very long time, and when it finally does, you have won yourself a lot of money. When you start the game, you have no idea how long it will go and it could be very long indeed (a lot like a Peter Jackson film). Notice a few things about this series of flips and how the chances drop at the same rate as the winnings go up:
The decision rule among us Stats Fan Club members for whether to play a gambling game is whether the expected value of the game is more than the cost of playing. Expected value is calculated by adding up the expected payoff for all possible outcomes. You'll recall that the expected payoff for each possible trial is $1. There are an infinite number of possible outcomes, because that coin could just keep flipping forever. To get the expected value, we sum this infinite series of $1 and get a huge total. The expected value for this game is infinite dollars. Since you should play any game where the cost of playing is less than the expected value, you should play this game for any amount of money less than infinity. Why It Doesn't WorkOf course, in real life, people won't pay much more than $2 for such a game, even if they knew all the statistics. No one really knows for sure why smart people turn their noses up at paying very much money for such a prospect, but here are some theories. Infinite is a lotEven if you accept in spirit that the game is fair over the long run and would occasionally pay off really big if you played it many, many times, that "long run" is infinitely long, which is an awfully long time. Few people have the patience or deep enough pockets to play a game that relies on so much patience and demands such a large fee. Decreasing marginal utilityThe originator of the problem, Bernoulli, believed that people perceive money as valuable, but the perception is not proportional to the amount of money. In other words, while having $16 is better than having $8, the relative value of one to the other is different than the relative value of having $128 compared to $64. So, at some point, the infinite doubling of money stops being equally meaningful as a prize. Bernoulli also believed that if you have a lot of money, a small wager is less meaningful than if you have very little money. (Kind of like those wealthy cartoon characters who light their cigars with hundred dollar bills.) Risk versus rewardHumans tend to be risk averse. That is, they will occasionally risk something in exchange for a reward, but they want that risk to be fairly close to the chances of success. It is true that the game of St. Petersburg has a chance for a massive reward, but the chance might be seen as too little compared to a risk of even $4. Infinity doesn't existSome philosophers would argue that people do not accept the concept of infinity as a concrete reality. Any sales pitch to encourage people to play this game by promoting the infinity aspects would be less than compelling. This might be why I don't buy lottery tickets. I don't play the lottery because my odds of winning are increased only slightly by actually playing. In my mind, the odds of me winning are infinitely small, or close enough to it that I don't treat the possibility of winning as real. See Also
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