St Peters Paper-1

Finite Versions of the St. Petersburg Paradox

Michael H. Birnbaum

CaliforniaStateUniversity, Fullerton and

DecisionResearchCenter

File: St Peters paper-1aDate: 4-29-04

Word counts:

Address:Michael H. Birnbaum

Department of Psychology H-830M

CaliforniaStateUniversity, Fullerton

P. O. Box 6846

Fullerton, CA92834-6846

Phones: (714)-278-2102(714)-278-3514 (Psychology Dept.)

fax:(714) 278-7134

e-mail:

Running Head:St. Petersburg Paradox

Footnotes

Author’s address: Department of Psychology, C.S.U.F., P.O. Box 6846, Fullerton, CA92834-6846. Email address: . Web URL

Support was received from National Science Foundation Grant, SBR-9410572. Thanks are due Teresa Martin, Juan Navarrete, Jamie Patton, and Melissa Lott for assistance in collecting the data of the first two experiments.

The St. Petersburg paradox was reviewed by Bernoulli in 1738 as a demonstration that people do not obey expected value when judging the value of a gamble. The paradox can be illustrated as follows: Suppose a fair coin will be tossed, and if it is heads, you win $2, and if it is tails, then it will be tossed again. If the second toss is heads, you win $4, and if tails, it will be tossed again. This process repeats, doubling the prize each time that tails occurs, and paying off only when heads occurs. The expected value of this gamble is as follows:

(1)

where EV is the expected value of the gamble, is the product of the probability and prize of outcome i. Since each term is 1, the sum is infinite. If people valued gambles by their EV, they should prefer this gamble to any gamble with finite EV, and to any finite amount of cash.

Although the expected value of the St. Petersburg gamble is infinite, most people refuse to offer to pay large sums to play the gamble. Furthermore, when given a hypothetical choice between playing the St. Petersburg game once or receiving a finite cash prize of $100, almost everyone prefers the cash.

Many theories have been offered to explain the paradox (a nice review is given in Bottom, Bontempo, and Holtgrave, 1989). The original explanation, by Bernoulli (and by Cramer), is that the utility of wealth is a negatively accelerated function of objective monetary wealth. Bernoulli proposed a logarithmic function u(x) = logx, and he noted that Cramer had earlier proposed a power function. If we suppose that utility is a function of increments in wealth (or if initial wealth is zero), we can write expected utility (EU) theory as follows:

(2)

where EU is the expected utility of the gamble; is the utility of outcome i.

If u(x) = log(x) then the cash equivalent value of the St. Petersburg gamble is $4, which is the median bid offered by both experts and students for a chance to play the game once (Bottom, et al., 1989). If u(x) = x.5, then the certainty equivalent is $5.82. These predictions are made by computing the cash certainty equivalent as follows: , where is the inverse of y = .

[Bottom, et al. (1989) listed expected utilities, rather than the certainty equivalents of the gambles; thus, the values in their Exhibit 1 are not cash equivalents according to EU theory.]

Bottom, et al. (1989) explored four variations of St. Petersburg paradox. These four variations were termed B for the basic game, described above, T2 for a variation that paid twice the payoffs of the basic game, P5 was a game that paid an extra $5 over the basic game, and P10, which paid an extra $10. The median judgments by students were $4, $8, $10, and $14 for B, T2, P5, and P10, respectively. The medians for “experts” were the same except for P5, which was $11 instead of $10. According to EU, with a logarithmic function, the predicted cash equivalents are $4, $8, $10.0, and $15.5, respectively, all within the inter-quartile range of the observed data.

Bottom, et al. (1989) noted that their values were also close to those predicted by an expectation heuristic (EH) that is based on finding the expected number of coin tosses and using the outcome for that number. This heuristic predicts values of $4, $8, $9, and $14, which are also quite close to the predictions of EU. Thus, the experiment by Bottom, et al. does not really distinguish the EU theory from the EH model.

Another form of explanation of the paradox is that people discount probabilities of winning very large outcomes. This discounting might occur for any of several reasons. People may in general set very small probabilities to zero. However, in the case of St. Petersburg paradox, there is another reason to discount small probabilities. No person can honestly offer the huge payoffs that might occur with tiny probabilities; therefore, the game must be a fraud. Thus, any probability associated with an outcome that exceeds the wealth of the house must be zero because the house can not make the payment. Indeed, the “paradox” would exist for anyone who believes that the offer is real.

This study includes finite versions of the St. Petersburg paradox, in which the game terminates after a given number of coin tosses. These finite versions of the games do not have infinite expected value, and they are games that a gambling house could reasonably offer. Indeed, participants in Exp. 3 had a chance to play one for real consequences. These games have payoffs that are the same as those of the St. Petersburg paradox, except if, after n trials, the heads did not occur, then the prize is double the prize for heads on trial n, and the game terminates. These games have EV = n + 1. Another version will also be used in Experiment 3 in which if no “heads” occurs after n trials, the payoff is zero. This version has an expected value of n.

Figure 1 plots the predictions of two models of decision making. According to the equations and parameters of the model of cumulative prospect theory (Tversky & Kahneman, 1992), certainty equivalents of these gambles should be a positively accelerated function of n, exceeding EV. The reason for this prediction is that the model of CPT uses a utility function that does not depart much from linearity, and the model assumes that the weight for small probabilities exceeds the probability, producing risk-seeking for small probabilities to win large prizes. For comparison are shown the calculations of the TAX model with utility of money approximated as a power function of objective cash, = .7 and  = -1. For cash amounts less than $100, it has been found that one can approximate data well with u(x) = x.

Insert Figure 1 about here.

Method Exp. 1

115 Introductory Psychology students were asked to complete a questionnaire that described the St. Petersburg gamble. The gamble was described (in part) as follows:

First, you choose Heads or Tails… If you chose "Heads" then the game ends as soon as the coin shows Heads. You will be paid $2 if the first toss shows Heads. However, if the first coin is Tails, the coin is tossed again, and if the second coin is Heads, you win $4, and the game ends. However, if the first two are Tails, the coin is tossed again, and this time Heads pays $8. Each time Tails occurs (before the first Heads), the winning amount doubles. The game is OVER as soon as the first Heads occurs. You can play the game only once, and you will be paid only once, based on the first Heads. If you initially chose "Tails", then everything works the same way, except the game ends when Tails occurs, and each Heads before the first Tails doubles the prize. In principle, this game could go on for a long, long time, in which case your winnings could be huge.

The probabilities and prizes for the first 5 outcomes were presented in a table, showing how each probability is half as large, but each prize is twice the size of the previous one. It was explained that the expected value of the gamble is infinite.

The concepts of highest buying price and lowest selling price were explained, using instructions similar to those used in Birnbaum and Sutton (1992). Subjects were asked to judge the value of the St. Petersburg gamble from both of these viewpoints.

Subjects were also asked, supposing they had a choice between the St. Petersburg gamble and cash prizes with certainty, which they would prefer. The sure cash amounts presented were $8, $16, $32, $64, $128, $256, $512, and $1024. Subjects were to circle all cash values that they would prefer to the St. Petersburg gamble.

Finally, subjects were asked to judge a receipt cash indifference value. They were asked to select a value of cash such that they would be indifferent between playing the gamble or receiving that amount of cash for sure.

The total length of the questionnaire was 184 words, printed on two pages. Subjects worked at their own paces, and most completed the task in 10 min.

Results of Experiment 1

The median value in the buyer's viewpoint was $8; 81% of the judges reported highest buying prices of $16 or less. The median judgment of the lowest selling price was $10; 64% gave judgments of $16 or less, and 75% gave judgments of $30 or less.

For the choice task, 16.8% chose $8 over the gamble; 32.7% chose $16 over the gamble; 56.6% chose $32 over the gamble; and 83.2% chose $64 over the gamble. The median judgment of cash indifference value of the gamble was $16; 20.9% giving responses of $4 or less, and 80% giving values of $80 or less.

Clearly, the value of the gamble depends on the procedures used to assess it. Nevertheless, by any of these methods, the St. Petersburg gamble appears to be worth about $8 to $16 on average.

Experiment 2

The instructions and procedure for Experiment 2 were similar to those of the first experiment, except that this experiment used both finite and infinite versions of the gamble. A new sample of 165 undergraduates were tested. First, the infinite gamble was explained, using the same instructions as in Experiment 1. Participants were instructed to judge the highest buying price, as in Experiment 1. Then, the finite versions of the gamble were explained as similar to the originalSt. Petersburggamble, except "they are limited to a fixed number of tosses, which limits the biggest prize that can be won. If there is one toss, you will win $2 if you call Heads and Heads occurs, otherwise, you win $4. The game always ends when your call turns up; however, this game ends after a fixed number of tosses, even if your call does not turn up." It was explained that (if a person calls "Heads") the prize for all Tails in the fixed game is double the prize for Heads on the last toss. The game rules were illustrated by a table that worked out the probabilities and prizes for finite games of one to four tosses, and space was provided in the table for the judges to work out the games up to 9 tosses.

Instructions requested only highest buying prices, which were described as, "the most you would bid in a sealed auction to buy the right to play the game once." Subjects were asked to judge prices for the infinite gamble, and the 9 finite gambles with 1 to 9 tosses.

Results of Experiment 2

The median buying price for the St. Petersburg gamble in Experiment 2 was $4, and the mean was $9.81. The mean seems considerably smaller in Experiment 2 than Experiment 1, but the difference is not statistically significant, t(278) = –1.24. The percentage of judgments less than $16 was 86%, which is comparable to the 81% figure for Experiment 1 for the same judgment.

The median prices for games of one to nine tosses are as follows: $2, $2, $4, $4, $4.5, $5, $5, $5, and $5, respectively. Of the 165 judges, 63 assigned the same price to the infinite gamble as they did to the 9 toss gamble; 44 assigned a lower price to the nine toss gamble, and 58 assigned a lower price to the infinite gamble than the 9 toss gamble. Theoretically, the infinite gamble should be at least as valuable as any finite version; however, it is possible that judges do not believe that the infinite gamble is for real; it is also possible that the effect of judging the sequence of finite gambles makes successive gambles more attractive.

Method of Experiment 3

Experiment 3 was run via the WWW. Participants viewed the materials via browsers where they received the following instructions:

“This is a study in decision-making. On each trial below, you are asked to choose whether to play a gamble or take a "sure thing" of a given amount of cash. In this experiment, prizes will be awarded to three participants who will be selected at random from all participants. If you are selected, one trial will be selected at random, and you will receive either the amount of cash or the prize of the gamble you selected on that trial.

This study involved a famous type of gamble called the St. Petersburg Gamble. In this gamble, a fair coin is tossed and if it is "heads," you win $2, but if it comes up "tails," then the coin is tossed again. If this is heads, you win $4, but if tails, the coin is tossed again. If heads comes up on the third toss, you win $8, and if tails, the coin is tossed again. Whenever heads comes up, you are paid off and the gamble is over, but if tails keeps coming up, the coin will be tossed again and the prize doubles each time until heads comes up. In the original game, the game could go on and on forever, with the prize doubling each time. In this study, however, the game will end after a fixed number of tosses. If you have not won by that number of tosses, then you receive $0 (nothing).

For example, suppose the game has a limit of 4 tosses. On the first toss, if it is heads, you win $2; tails we toss again. On the second toss, if heads, you win $4, tails we toss again; on the third toss, if heads, you win $8; tails we toss again. On the fourth (and FINAL toss), if heads, you win $16, but for tails, you get $0 (nothing).

Here is a summary of the 4-toss game:

H ($2 and game ends)
TH ($4 and game ends)
TTH ($8 and game ends)
TTTH ($16 and game ends after 4 tosses)
TTTT ($0) game ends after 4 tosses.

In the 8-toss game, the coin will be tossed up to 8 times, until "heads" appears. If you get 8 "tails" in a row, you receive nothing; otherwise, you win $2, $4, $8, $16, $32, $64, $128, or $256, depending on when "heads" appears. In this game, the probability of winning $2 is 1/2; the probability to win $4 is 1/4;and so on, as follows:

8-toss game Summary:

H ($2 ) probability = 1/2
TH ($4) probability = 1/4
TTH ($8) probability = 1/8
TTTH ($16) probability = 1/16
TTTTH ($32) probability = 1/32
TTTTTH ($64) probability = 1/64
TTTTTTH ($128) probability = 1/128
TTTTTTTH ($256) probability = 1/256
TTTTTTTT ($0) probability = 1/256

On each trial, you decide whether you want the sure cash or one play of the n-toss game. Remember, the n-toss game always ends when "heads" appears, and it ends no matter what after n tosses.

For example, consider the first trial below, W1. This choice is a choice between the prize of a 2-toss game or $1 for sure. If you prefer the gamble, click the button beside "2-toss game", if you prefer the cash, click the button beside "Cash." If you chose the "game" and if you are one of the lucky winners, if this trial is the one randomly selected for you to play, you would receive $2 with probability of 1/2, $4 with probability 1/4, and $0 with probability 1/4. If you chose the cash, you would get $1 for sure. “

Following instructions, participants completed four warm-up trials and 40 experimental trials, composed of a 5 X 8, factorial design of Gamble X Cash Amounts, in which the 5 levels of the gambles were 2, 4, 6, 8, and 10-toss games, and the 8 levels of Cash Amounts were $1, $3, $5, $7, $9, $11, and $13. The complete materials can be viewed at the following URL:

Results of Experiment 3

Table 1 shows the proportion of participants who preferred the sure cash to each of the finite versions of the gamble. These data were used to calculate by interpolation the cash equivalent value of cash that would be preferred 50% of the time to the gamble. The values of these cash equivalents are ____, ____, ____, and _____, for the 2, 4, 6, 8, and 10 toss gambles.

Table 1. Proportions of participants who prefer the sure cash to each gamble.

Gamble / Value of Certain Cash
$1 / $3 / $5 / $7 / $9 / $11 / $13 / $15
2-toss
4-toss
6-toss
8-toss
10-toss

Figure 1. Predictions of Certainty Equivalents under the CPT model and TAX model for finite versions of St. Petersburg gambles.

Supplementary materials: Complete materials for Experiment 3 are presented here as a convenience to reviewers without Internet access. These materials are available from the following URL:

Instructions for St. Petersburg Gamble Decisions

This is a study in decision-making. On each trial below, you are asked to choose whether to play a gamble or take a "sure thing" of a given amount of cash. In this experiment, prizes will be awarded to three participants who will be selected at random from all participants. If you are selected, one trial will be selected at random, and you will receive either the amount of cash or the prize of the gamble you selected on that trial.