Numbers Contests

These are contests that I ran on the rec.puzzles and rec.games.trivia newsgroups many years ago. In these, entrants just had to pick one or more numbers, and the scores or winning entries were determined by both the entrant's number, and the numbers submitted by all other entrants. The rules were designed to generate interesting game-theory-type problems. Some contests were particularly interesting in that the "correct" game theory answers were easily calculable, but these game theory answers were often badly wrong. Here are some questions that I found particularly interesting from a game theory perspective:

Choose a positive integer. Your goal is to choose the second-lowest number of all numbers sent in. That is, all ties for the lowest number lose, and all ties for the second-lowest number win. (Contest NUM1, question D.)
Choose a number in the range [0,100]. Your goal is to get as close as possible to 4/5th the arithmetic mean of all responses to the question. (Contest NUM5, question B.)
Choose a positive integer. Your goal is to choose the smallest unique number sent in---that is, the smallest number that no one else submits. (Contest NUM1, question A.)

It's fun to figure out the game theory answers to these problems, and then think about whether those answers are actually good.

Below I have the original contests, and the the results of the contests. So you can look at the results if you want to see how good your answers are.

These contests, unsurprisingly, have an abstract, mathematical feel. Somewhat more engaging were my Common Entries Contests.