How Much Would You (Should You) Pay For A Gamble?

1. St. Petersburg Paradox You pay a fixed fee to enter the game. Then a fair coin is tossed repeatedly until a tail appears, ending the game. The pot starts at 1 dollar and is doubled every time a head appears. You win whatever is in the pot after the game ends. The paradox is named from Daniel Bernoulli's presentation of the problem and his solution, published in 1738. How Much Would You (Should You) Pay For A Gamble?

2. St. Petersburg Paradox Thus you win: 1 dollar if a tail appears on the first toss, 2 dollars if a head appears on the first toss and a tail on the second, 4 dollars if a head appears on the first two tosses and a tail on the third, 8 dollars if a head appears on the first three tosses and a tail on the fourth, etc. 2k−1 dollars if the coin is tossed k times until the first tail appears. How Much Would You (Should You) Pay For A Gamble?

3. St. Petersburg Paradox What would be a fair price to pay for entering the game? To answer this we need to consider what would be the average payout: With probability 1/2, you win 1 dollar; with probability 1/4 you win 2 dollars; with probability 1/8 you win 4 dollars etc. Of course the probability of gaining a head on the kth. play is 1/2k. How Much Would You (Should You) Pay For A Gamble?

4. How Much Would You (Should You) Pay For A Gamble? But how much should you pay?

5. The expected value (E) is thus How Much Would You (Should You) Pay For A Gamble?

6. The expected value (E) is thus, or notationally How Much Would You (Should You) Pay For A Gamble? where k counts the number of tosses.

7. This sum diverges to infinity, and so the expected win for the player of this game, at least in its idealized form, in which the casino has unlimited resources, is an infinite amount of money. This means that the player should almost surely come out ahead in the long run, no matter how much he pays to enter. How Much Would You (Should You) Pay For A Gamble?

8. While a large payoff comes along very rarely, when it eventually does it will typically be far more than the amount of money that he has already paid to play. According to the usual treatment of deciding when it is advantageous and therefore rational to play, one should therefore play the game at any price if offered the opportunity. How Much Would You (Should You) Pay For A Gamble?

9. Yet, in published descriptions of the paradox, e.g., (Martin 2004), many people expressed disbelief in the result. Martin quotes Ian Hacking as saying “few of us would pay even $25 to enter such a game” and says most commentators would agree. This is the paradox! Martin, Robert (2004). “The St. Petersburg Paradox”. In Edward N. Zalta. The Stanford Encyclopedia of Philosophy (Fall 2004 ed.). The Paper How Much Would You (Should You) Pay For A Gamble?

10. You will quickly discover, however, that you will not win an infinite amount of money playing this game. We seem to have a paradox. The expectation value is infinite, but certainly no one is going to put up an infinite amount of money, or even a million dollars, for the opportunity to play this game once. What is the solution to this paradox? How Much Would You (Should You) Pay For A Gamble?

11. The answer is that an expectation value is defined to be an average over an infinite number of trials (or the limit towards an infinite number). But you are simply not going to play an infinite number of games. In other words, the calculated expectation value doesn’t agree with your experiment, because your experiment has nothing whatsoever to do with the precise definition of an expectation value. How Much Would You (Should You) Pay For A Gamble?

12. To be sure, if you did somehow play an infinite number of games, then you would indeed have an infinite average for your winnings. The whole paradox arises from trying to make “expectation value” mean something it doesn’t. How Much Would You (Should You) Pay For A Gamble?

13. Expected Utility Theory The classical resolution of the paradox involved the explicit introduction of a utility function, an expected utility hypothesis, and the presumption of diminishing marginal utility of money. How Much Would You (Should You) Pay For A Gamble?

14. Expected Utility Theory In Daniel Bernoulli's own words: The determination of the value of an item must not be based on the price, but rather on the utility it yields…. There is no doubt that a gain of one thousand ducats is more significant to the pauper than to a rich man though both gain the same amount. How Much Would You (Should You) Pay For A Gamble?

15. Expected Utility Theory A common utility model, suggested by Bernoulli himself, is the logarithmic function u(w) = ln(w) (known as “log utility”). It is a function of the gambler’s total wealth w, and the concept of diminishing marginal utility of money is built into it. How Much Would You (Should You) Pay For A Gamble?

16. Expected Utility Theory By the expected utility hypothesis, expected utilities (EU) can be calculated the same way expected values are. For each possible event, the change in utility ln(wealth after the event) - ln(wealth before the event) . How Much Would You (Should You) Pay For A Gamble?

17. Expected Utility Theory For each possible event, the change in utility will be weighted by the probability of that event occurring. Let c be the cost charged to enter the game. The expected utility of the lottery now converges to a finite value: How Much Would You (Should You) Pay For A Gamble?

18. Expected Utility Theory (Mathematical Detail) Which is convergent. Essentially the general term is k/2k and the ratio of the k+1th. term to the kth. converges to ½ < 1 (d'Alembert's 1768 ratio test). How Much Would You (Should You) Pay For A Gamble?

19. Expected Utility Theory This formula gives an implicit relationship between the gambler's wealth (w) and how much he should be willing to pay to play (specifically, any c that gives a positive expected utility). How Much Would You (Should You) Pay For A Gamble?

20. Expected Utility Theory For example, with log utility a millionaire (106) should be willing to pay up to $10.94, a person with $1000 should pay up to $5.97, a person with $2 should pay up to $2, a person with $0.60 should borrow $0.87 and pay up to $1.47. How Much Would You (Should You) Pay For A Gamble?

21. How Much Would You (Should You) Pay For A Gamble?

22. We should take into account that a casino would only offer lotteries with a finite expected value. Under this restriction, it has been proved that the St. Petersburg paradox disappears as long as the utility function is concave, which translates into the assumption that people are (at least for high stakes) risk averse. How Much Would You (Should You) Pay For A Gamble?

23. Probability Weighting Nicolas Bernoulli (he was older brother of Daniel Bernoulli, to whom he also taught mathematics) himself proposed an alternative idea for solving the paradox. He conjectured that people would neglect unlikely events. How Much Would You (Should You) Pay For A Gamble?

24. Probability Weighting Since in the St. Petersburg lottery only unlikely events yield the high prizes that lead to an infinite expected value, this could resolve the paradox. How Much Would You (Should You) Pay For A Gamble?

25. Probability Weighting The idea of probability weighting resurfaced much later in the work on prospect theory by Daniel Kahneman and Amos Tversky. However, their experiments indicated that, very much to the contrary, people tend to overweight small probability events. Therefore the proposed solution by Nicolas Bernoulli is nowadays not considered to be satisfactory. How Much Would You (Should You) Pay For A Gamble?

26. Rejection Of Mathematical Expectation Various authors, including Jean le Rond d'Alembert and John Maynard Keynes, have rejected maximization of expectation (even of utility) as a proper rule of conduct. Keynes, in particular, insisted that the relative risk of an alternative could be sufficiently high to reject it even were its expectation enormous. How Much Would You (Should You) Pay For A Gamble?

27. One Cannot Buy What Is Not Sold Some economists resolve the paradox by arguing that, even if an entity had infinite resources, the game would never be offered. If the lottery represents an infinite expected gain to the player, then it also represents an infinite expected loss to the host. No one could be observed paying to play the game because it would never be offered. How Much Would You (Should You) Pay For A Gamble?

28. Finite St. Petersburg Lotteries The classical St. Petersburg lottery assumes that the casino has infinite resources. This assumption is often criticized as unrealistic, particularly in connection with the paradox, which involves the reactions of ordinary people to the lottery. How Much Would You (Should You) Pay For A Gamble?

29. Finite St. Petersburg Lotteries Of course, the resources of an actual casino (or any other potential backer of the lottery) are finite. More importantly, the expected value of the lottery only grows logarithmically with the resources of the casino. As a result, the expected value of the lottery, even when played against a casino with the largest resources realistically conceivable, is quite modest. How Much Would You (Should You) Pay For A Gamble?

30. Finite St. Petersburg Lotteries If the total resources (or total maximum jackpot) of the casino are W dollars, then L = 1 + floor(log2(W)) (floor(x) is the largest integer less than or equal to x) is the maximum number of times the casino can play before it no longer covers the next bet. How Much Would You (Should You) Pay For A Gamble?

31. Finite St. Petersburg Lotteries The expected value E of the lottery then becomes: How Much Would You (Should You) Pay For A Gamble?

32. Finite St. Petersburg Lotteries Mathematical aside cut off at 2k-1 = W that is floor(log2(W)) = k-1 since an integer solution is required. How Much Would You (Should You) Pay For A Gamble?

33. Finite St. Petersburg Lotteries The following table shows the expected value E of the game with various potential bankers and their bankroll W (with the assumption that if you win more than the bankroll you will be paid what the bank has): How Much Would You (Should You) Pay For A Gamble?

34. Finite St. Petersburg Lotteries How Much Would You (Should You) Pay For A Gamble?

35. Finite St. Petersburg Lotteries A rational person might not find the lottery worth even the modest amounts in the above table, suggesting that the naïve decision model of the expected return causes essentially the same problems as for the infinite lottery. Even so, the possible discrepancy between theory and reality is far less dramatic. The assumption of infinite resources can produce other apparent paradoxes in economics. How Much Would You (Should You) Pay For A Gamble?

36. Simulation A typical graph of average winnings over one course of a St. Petersburg Paradox lottery shows how occasional large payoffs lead to an overall very slow rise in average winnings. After 20,000 game plays in this simulation the average winning per lottery was just under 8 dollars. How Much Would You (Should You) Pay For A Gamble?

37. Simulation The graph encapsulates the paradox of the lottery: The overall upward slope in the average winnings graph shows that average winnings tend upward to infinity, but the slowness of the rise in average winnings (a rise that becomes yet slower as game play progresses) indicates that a tremendously huge number of lottery plays will be required to reach average winnings of even modest size. How Much Would You (Should You) Pay For A Gamble?

38. Game / Simulate How Much Would You (Should You) Pay For A Gamble?

39. Iterated St. Petersburg Lottery The above might not be a very satisfying explanation, so let us get a better feeling for the problem by looking at a situation where someone plays N = 2n games. How much money would a “reasonable” person be willing to put up front for the opportunity to play these N games? How Much Would You (Should You) Pay For A Gamble?

40. Iterated St. Petersburg Lottery Well, in about half games (2n-1) he will win one dollar; in about one quarter (2n-2) he will win two dollars; in about one eighth (2n-3) games he will win four dollars; etc., until in about one game he will win 2n-1 dollars. In addition, there are the “fractional” numbers of games where he wins much larger quantities of money (for example, in half a game he will win 2n dollars, etc.), and this is indeed where the infinite expectation value comes from, in the calculation above. How Much Would You (Should You) Pay For A Gamble?

41. Iterated St. Petersburg Lottery But let us forget about these for the moment, in order to just get a lower bound on what a reasonable person should put on the table. Adding up the above cases gives the total winnings as 2n-1×1+2n-2×2+2n-3×4+…+1×2n-1) = 2n-1×n. The average value of these winnings in the N = 2n games is therefore 2n-1n/2n = n/2 = log2N/2. How Much Would You (Should You) Pay For A Gamble?

42. Iterated St. Petersburg Lottery A reasonable person should therefore expect to win at least log2N/2 dollars per game. By “expect”, we mean that if the player plays a very large number of sets of N games, and then takes an average over these sets, he will win at least 2n-1n dollars per set. This clearly increases with N, and goes to infinity as N goes to infinity. It is nice to see that we can obtain this infinite limit without having to worry about what happens in the infinite number of “fractional” games. Remember that this quantity, log2N/2, has nothing to do with a true expectation value, which is only defined for N → ∞. How Much Would You (Should You) Pay For A Gamble?

43. Feller’s Law Of Large Numbers More than 200 years after Daniel Bernoulli’s paper the case of the St. Petersburg problem was reopened by William Feller (1945). Feller insisted that the question for the fair price of a game of chance only made sense in connection with the law of large numbers. How Much Would You (Should You) Pay For A Gamble?

44. Feller’s Law Of Large Numbers According to this law, the average payoff in a long series of independent repetitions of the same game converges towards the expected value. Hence if the price charged per game is only slightly smaller than the expected value, the game in the long run becomes profitable for the player. How Much Would You (Should You) Pay For A Gamble?

45. Feller’s Law Of Large Numbers The traditional law of large numbers still holds for the St. Petersburg game and states in this case that the average payoff in a series of repeated games will converge to infinity: (E1+…+EN)/N → ∞ Ei being the expected payoff in the wth. game. How Much Would You (Should You) Pay For A Gamble?

46. Feller’s Law Of Large Numbers However, this convergence to infinity can be extremely slow. If you bet a fixed amount on every instance of the game, you are guaranteed to win in the long run. However you might have to wait until the end of your days before a large payoff washes away all your previous losses. How Much Would You (Should You) Pay For A Gamble?

47. Feller’s Law Of Large Numbers This gave Feller the idea that one should consider a variable price, dependent on the number of repetitions you are allowed to plan. Feller supported this idea by a Law of Large Numbers essentially stating that the average payoff per game in the long run is about log2(N) (E1+…+EN)/Nlog2(N) → 1 How Much Would You (Should You) Pay For A Gamble?

48. Feller’s Law Of Large Numbers Leading to the conclusion that the value of the St. Petersburg game ought to be log2(N) for someone who may play the game N consecutive times: How Much Would You (Should You) Pay For A Gamble?

49. Facebook and the St. Petersburg Paradox by Jason Zweig, The Wall Street Journal, February 4, 2012 A high-growth stock like Facebook is a lot like that St. Petersburg coin. The potential payoffs are enormous, although not infinite—and the game might peter out all too soon. Facebook - IPO Purchasers May Face A Paradox (IPO - Initial Public Offering)

50. At the end of 2010, 608 million people actively used Facebook every month; by this past Dec. 31, 845 million people did. If Facebook keeps growing that fast, more than 22 billion people will be using it 10 years from now, or three times the estimated population of the planet today. Facebook - IPO Purchasers May Face A Paradox