In Chapter 11 of MOP Chen analyzes some half street games. Half street games are poker games with one street of betting, in which only one player can do the betting. This situation comes up enough in practice to be useful to analyze. When at the river in position and the pot is big relative to stacks, the oop player often checks and his only choice is call or fold (effectively a half street game).

In these situations the in position player's optimal strategy is usually +EV. That is a powerful idea. It means that this player is +EV regardless (with the same EV) regardless of what the oop player does. Position is goot! Exploitive strategies need to be better than optimal to deviate. This is an obvious statement, but it illustrates that even players using exploitive strategies should understand some game theory in order to "know" when to deviate.

The key result of the chapter is the calculation of the optimal ratio of bluffs/valubets and the related optimal ratio of calls/folds for the ip and oop players. In the special case of NLHE with PSBs, this reduces to bluffing 1/3 often and calling 1/2 often.

There are a few ways to come up with the 1/3 and 1/2 frequencies. One way is to randomize your frequency with individual hands (call 1/2, fold 1/2 etc). That is the mixed strategy. Mixed strategies seem to occur when hand ranges are small. With large hand ranges, it's probably simpler to use absolute hand values to determine frequencies. For example, say you have 2 hands - tptk and two pair, both of which are the "same" against villain's range. The optimal strategy may have you call with two pair and fold tptk. Of course you could fold two pair and call tptk but that increases probability of bluffs with better hands. For most examples it appears that Chen models the range of hands with the interval [0,1] of real numbers. This makes calculations simpler.

Implementing optimal betting frequencies in practice requires an awareness of your own hand ranges. Obviously you need to know what your range is before choosing the 1/3 to bluff with. The final thought I want to mention is the inverse relation between payoffs and frequency. In optimal play, choices with high payoffs should be chosen with less frequently. The rowshambo example from chapter 10 is a good example. If you change the payoffs so that rocks has a higher payoff, the optimal solution requires that you choose rocks less frequently than in the equal payoff version. The intuitive reason is that you choose an optimal strategy vs the maximal exploitive strategy of villain. Since villain will also try to exploit the higher payoff, you have to do it less often to avoid exploitation. In real poker, you should bluff less often for $100 into $200 than you do when bluffing $200 into $200.