yup and i haven't solved it all the way through.  I imagine there will be some results along the lines of:

if it's optimal to publish at (a,b) then it's also optimal to publish at (a+1,b)

if it's optimal to secretly mine at (a,b), then it's also optimal to secretly mine at (a,b-1)

or something like that.  Haven't proved either result though.

That makes sense. I was implicitly assuming that if she didn't publish at b, then she still wouldn't at b+1, but maybe that's wrong. She probably cuts her losses at some point.

Undisciplined

This is almost correct, except Alice still has to decide whether to publish what she has when Bob finds the next block, or keep mining privately.

I'll give you the bounty since it was close enough.

, i.e. when it would be preferable to mine in secret versus publishing.

math

bitcoin

Bitcoin Math Puzzle #004: Selfish Mining Pt. 4

SimpleStacker

This is almost correct, except Alice still has to decide whether to publish what she has when Bob finds the next block, or keep mining privately.

So the correct equation is:

$$
\begin{align*}
r V(a, b) = & - h_A c \\
& + \lambda_A \left[ \max \left\{ P(a+1,b), V(a+1,b) \right\} - V(a,b) \right] \\
& + \lambda_B \left[ \max \left\{ P(a, b+1), V(a, b+1) \right\} - V(a,b) \right] 
\end{align*}
$$

I'll give you the bounty since it was close enough.

The trick is now figuring out when $$V(a,b) > P(a,b)$$, i.e. when it would be preferable to mine in secret versus publishing.