Start the AI in a sandbox universe, like the "game of life". Give it a prior saying that universe is the only one that exists (no universal priors plz), and a utility function that tells it to spell out the answer to some formally specified question in some predefined spot within the universe. Run for many cycles, stop, inspect the answer.

I don’t think that a “lying case” analysis is correct. In both cases, lying or not, you are using all information you have to correctly estimate the “minimum acceptable price". In “lying case”, if you are rational, you know that you (probably) have given increased price so you decrease it back by your best estimation of increment. Now, the higher probability you lie for yourself, the less precise estimation you are left with. And, by being rational, you have to overestimate as likely as underestimate. So this strategy on gives you no consistent benefit.