A theoretical game

[well]

So, if it's H the tth time, Mike pays 2^(t-2) for t>1.

Note that you switched H and T in your examples.

Let S be the stochastic variable denoting the amount of dollars Mike has to pay back.

Then ES (the Expectation value of S), is Sum{t=2..inf}: 2^-t*2^(t-2) = Sum{t=2..inf}: 1/4 = inf.

So, no price is good enoug.

Regards.