The following items are culled from Chance News which is a database of current chance events as reported in the daily newspapers.
A poll released by the Program for International Policy Attitudes showed that 75% of Americans believe the United States spends too much on foreign aid, and 64% want foreign-aid spending cut. Respondents were also asked what they thought the share of the federal budget currently goes to foreign aid was.The median answer was 15% and the average 18%. The correct answer is less than 1%. The answers to the question of how much would be "too little" had a median of 3%.
Kinsley suggests that more interesting than what we learn about people's actual opinions from the poll is the observation that Americans are willing to have strong opinions about things they know very little about. He chides the pollsters for not asking whether the respondent knows anything about the topic of the poll and for assuming that people should have opinions on all subjects at all times and all their opinions should be weighted equally.
The game of "uniquely smallest"
Erich Friedmann has proposed the following interesting game.
1. Every player secretly writes down a positive integer (1,2,3, ...) and their name on a slip of paper.
2. The slips are collected, and the winner is the player who picked the lowest number that no one else picked.
3. If, by some coincidence, no such person exists, the game is a tie.
A dozen or so friends of Eric plan to play this game by e-mail and they have until midnight February 14 to figure out a good method for picking their number. A colleague of mine, Tom Sundquist, is one of the players and wants to make his choice by a probability distribution consistent with a Nash equilibrium point. Such a point can be described as follows. Assume there are three players and these players choose probability distributions p, q, and r to pick their numbers. Then the triple {p,q,r} is a Nash equilibrium point if no player can increase his probability of winning by changing his distribution while the choice of the other two players remain as specified by the equilibrium point.
If the possible numbers to submit is limited to (1,2,...,m), Tom has shown that there is a Nash equilibrium point in which each player uses the same probability distribution p to pick a number. This distribution p is characterized by the property: if all three players use the distribution p, then the probability that a player wins, given that he chose number x, is independent of x. Using this property, Tom computed the distribution p for 3 players as m varies. His results give:
m = 1 p = {1}
m = 2 p = {.5, .5}
m = 3 p = {.464, .268, .268}
m = 4 p = {.4578, .2517, .1453, .1453}
Further computations suggest these distributions converge rather rapidly to a limiting value as m tends to infinity. Knowing this limiting value for k players would help Tom make his choice and would be a fitting way to celebrate John Nash getting a Nobel Prize for his work on game theory. The first person to solve this problem will get special mention in the next Chance News.
Martin Baxter writes:
One distribution which does the job for the game is a geometric with parameter p where p is the unique solution (in [0,1]) of x^3 + x^2 + x - 1 = 0.
Numerically, p = 0.543689 and the first few terms are {{0.456311}, {0.248091}, {0.134884}, {0.0733352}, {0.0398716}}, says Mathematica.
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