[jldugger]

I feel like you could probably 'fix' it by the following protocol:

1. Ask your participant to write down a random number. 2. After they've written that number down, inform them you will guess at their number, and give them a dollar if you guess wrong. 3. Ask if they'd like to change their random number. 4. Allow them to write down their new number.

The before/after would probably change the distribution, and pretty much demonstrate that people can be more random when motivated.

[gmac]

I'm sure the distribution would change, but not sure it would be closer to uniform. It's now a completely different problem: choose a number most resistant to guessing. For example, if the participant hypothesises that your guess will be a "random" number chosen by you, she should always pick 10.