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.