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.
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.
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.