I ran across a passage in "The Mathematics of Poker" explaining bluffing strategy. Here is a paraphrased version.
Two players are playing heads-up ($1$ vs $1$) poker. They're on the final betting street (the river; so all $5$ community cards have been revealed). Player A has the winning hand $20\%$ of the time, and has the losing hand $80\%$ of the time. Both players are aware of these $20/80$ probabilities. For each given hand, Player A can see the cards of both players cards, and so knows who has the winning hand. The pot has $4$ dollars in it. Player A is first to act.
If A checks, B will never bet because A will know whether they have the winning hand or not in response and will raise / fold accordingly. So we know that A will not check the winning hands in order to try and milk more value out of them. But if A bets if and only if they have the winning hand, B will always fold. So, A will choose some $x$% of hands to bluff.
A's selection of $x$ is their strategy. We want to find the best $x$.
B loses $1$ dollar for calling when A has the winning hand, and B wins $(4 \text{ in the pot } + 1 \text{ bluff}) = 5$ dollars when A bluffs. B's calling strategy only applies when A bets, so the probability values below are conditional on A betting. The expectation of B's hand if he calls is:
$$ \begin{align} E(B ,\text{ call}) &= P(\text{A has winning hand})(-1) + P(\text{A bluffs})(5) \\ &= (0.2)(-1) + (x)(5)\\ &= 5x - 0.2 \end{align} $$
If B folds, his expectation is simply $0$, so $E(B ,\text{ fold}) = 0$
If A bluffs often B should call all the time. If A bluffs rarely, B should never call. Here is a graphical representation:
Based on this, the author claims that B should call when $x>0.04$, and fold when $x < 0.04$, which is obviously true.
But what about A? This question was initially about A's strategy, namely deciding $x$. I am confused about why should A bluff with a probability of $4\%$. Whether A bluffs $4\%$ of the time or $0\%$ of the time, B's EV is $0$. Within this range of $x$, B's strategy does not change and the EV does not exceed $0$. As a result, why would it matter how often A bluffs so long as $x \leq 0.04$?