Consider this simplified river spot. Hero is OOP with a condensed range against Villain’s polarized range. Hero’s hands can only beat Villain’s bluffs. Let’s also say that each player can only use an unmixed strategy for their entire range: Hero must always check-call or check-fold, Villain must always check or bet.

How does Villain’s EV change as a function of his range composition?

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Hero always calls

When Villain bets his entire range he begins to make money when his value bets outnumber his caught bluffs. Once we consider the money Villain loses from failed bluffs, we see that his EV is equivalent to his equity share.

Furthermore, we can integrate his EV equation from 0 to T and we get PT/2. This is equivalent to multiplying the mean value by total combos, T.

Villain checks his entire range and again his EV is equivalent to his equity.

Now we can compare Villain’s perfect...