reversibility

[hyunjimoon]

basic argument is explore and exploit states should co-exist - like markov chain states.

Paragraph 1 (Why the monotone curve reflects Detailed Balance).
In the sampling‐decision model above, $r$ (the ratio of action time to sampling time) effectively sets how "costly" it is to exploit versus keep exploring. As $r$ increases, additional data‐gathering is relatively cheaper than locking in a decision, causing the optimal $\smash{k^*}$ (number of samples) to rise—hence the upward, stepwise curve. This setup parallels detailed balance: when $r$ is large, "reversals" back to exploration are favored, so $\pi(\mathrm{A2E})$ grows. When $r$ is low, the chain tilts toward exploitation ($\mathrm{A2E}\to\mathrm{E2K}$) until new uncertainties force a transition ($\mathrm{E2K}\to\mathrm{A2E}$). In other words, the environment (via $r$) sets how readily a venture pivots, and the observed monotonic shift in sampling intensity captures the equilibrium between forward and backward transitions.

Paragraph 2 (Solving for the ratio $\pi(\mathrm{A2E})/\pi(\mathrm{E2K})$).
Once you know how costly actions are relative to sampling, you can "solve for" the fraction of resources in exploration vs.\ exploitation:

$$ \frac{\pi(\mathrm{A2E})}{\pi(\mathrm{E2K})} ;=; \frac{P(\mathrm{E2K}\to \mathrm{A2E})}{P(\mathrm{A2E}\to \mathrm{E2K})} $$

For instance, a battery startup with a high $r$ invests heavily in lab tests before committing, so $\mathrm{A2E}$ dominates; a small software venture (low $r$) exploits quickly unless major bugs force it to revert to R&D. This balancing act—akin to a reversible chain—ensures the firm's toggles between exploration and exploitation align with the environment's time‐cost ratio, preventing path‐dependent lock‐in.

Paragraph 3 (Reversible Jump MCMC and Detailed Balance).
The same principle appears in reversible jump Markov Chain Monte Carlo (RJMCMC), where we construct a Markov Chain whose stationary distribution is the desired posterior. Instead of enumerating all possibilities in a massive or even infinite state space, MCMC only needs relative probability ratios to move from one state to another. By satisfying detailed balance in each "jump" (proposing a new model parameterization and then either accepting or rejecting based on a posterior‐ratio test), RJMCMC maintains a steady‐state distribution that reflects the true posterior. Analogously in entrepreneurship, satisfying detailed balance among $\mathrm{A2E}$ and $\mathrm{E2K}$ "states" ensures resources flow back and forth in proportion to how beneficial it is to keep exploring versus scaling. Over time, just like MCMC converges on the posterior, the venture converges on a knowledge‐optimal mix of exploration and exploitation.