[Jerrod Ankenman]

Two things:

Zero-sum two-player games are a lot different than multiplayer games. Many of the terms and concepts that we define/explain in our book apply ONLY to ZSTPG, and not to other situations, such as multiplayer, in a technical sense, although they are still useful for understanding various situations.

So this bluffing into a dry side pot thing isn't really a very good example, because it's not an example of an "optimal" or "equilibrium" solution - it's an example of perturbing the (three-player Nash) equilibrium solution by threatening to transfer equity from one guy to the other in order to make the threatened guy pay you. This is obvious because you would prefer the guy to know that you bluff into dry side pots (with negative EV) because then he will call you and not otherwise. But the essential feature is the third player to whom you can transfer the first guy's equity.

Secondly, in a ZSTPG, it's obvious on reflection that IF strategy X is optimal (that is, is part of an optimal strategy-pair), and strategy X contains at least one mixed strategy where the number of non-zero strategic option weights is 2 or more, then each of those strategic options must have equal value against the other member of the strategy-pair. If this weren't true then the player playing X could improve his equity unilaterally by changing his mixed strategy to be pure in the higher-valued strategic option, which violates the condition that X is part of an optimal strategy-pair.