A group of people are asked to play the following game. Each person in the group picks a number between 0 and 100. There is a large reward for getting an answer that is closest to 2/3 of the average answer given by the other members of the group.
(a) Sam Smarty-pants thinks that the other players in the game are really stupid. He believes that they will choose numbers between 0 and 100 at random, with an expected value of 50. Sam will maximize his expected winnings by choosing a number close to ___________
(b) Oona Onestep realizes that the average cannot exceed 100 and so half the average cannot exceed 50. She thinks that the other people in the group are about as smart as Sam Smarty-pants. Given that these people will choose numbers close to ___________ Oona will choose a number close to __________
(c) Talullah Twostep thinks that nobody could be as stupid as Sam Smartypants, but that everyone else will reason just like Oona Onestep. Talullah notices however that if everybody else acts like Oona, then she should not do the same thing that Oona does, but should instead choose a number close to ___________
(d) Raquel Rational notes that Sam, Oona, and Talullah all think they are smarter than everybody else. She asks herself, “What number would I choose if I thought everyone reasons in the same way that I do?” For what value of X is it true that X = 2X/3? __________
(e) This game is known as the “Beauty Contest Game.” The Beauty Contest game has been played in many economics laboratory experiments. It has also been played by several thousand people who submitted reader entries in three separate contests conducted by business newspapers in the UK and Spain and by Spektrum der Wissenschaft, the German edition of Scientific American. Readers were asked to submit numbers between 0 and 100 with a prize awarded to the number closest to 2/3 of the average. The game has also been played in laboratory experiments by groups such as undergraduates at Cal Tech and UCLA, students in Germany and Singapore, high school students, the Cal Tech board of trustees, and portfolio managers. In these experiments, typically only a small proportion of participants respond with numbers of 50 or more. In graphs showing frequencies of the possible responses, there are spikes at 33.3, at 22.2, and a modest spike at zero. The mean number selected by readers of Spektrum der Wissenschaft was 22.08. Mean scores among the various laboratory groups varied considerably from a low of 21.9 to a high of 46.1. Which laboratory group do you think chose the lowest number? ____________ To find out more about this game, where it got its name, and how each of the groups did, look at “Progress in Behavioral Game Theory,” a paper in the Journal of Economic Perspectives, Autumn 1997, by Professor Colin Camerer of Cal Tech.