9.- DERECHO DE PARTICIPACIÓN EN LA VIDA POLÍTICA DEL ESTADO Bolivia

What is the equilibrium of the ideal turn PvBC model, given the stack sizes and ranges in this hand? How will it compare to the real computationally- generated equilibrium? First, the pot is P=5 BB at the beginning of turn play, and S=27.5 BB remain behind. Plugging into the GGOP bet sizing for- mula above, we find that we need to bet about 1.23 times the pot on the turn and river in order to get all-in. That is, we lead the turn for B=6.16BB with all of our betting range, and this leaves S−B=21.34BB to bet into a pot of P+2B=17.32BB on the river.

Now, any made hand a pair of sevens or better is ahead of all the SB’s range. About a quarter of the BB’s turn starting distribution falls into this category – let us call it exactly 25% for simplicity. The majority of the rest of his hands lose to A-high. So, as a first approximation, the BB is value-betting a quarter of his turn starting range on the turn and river, and the remainder of his range may be used as bluffs. Referencing the solutions above, we find that at the equilibrium, the BB is jamming enough on the river so that bluffs make up (S−B)/[2(S−B)+(P+2B)] =35.6% of his shoving range. (It would be 33.3% if he were making a pot-sized bet, but the bet here is a bit larger, so Hero can bluff a bit more frequently.) He bluffs enough additional air hands on the turn so that the fraction of turn betting hands that give up on the river is B/(2B+P) =35.6%. Again, the relationship of his single-barrel bluffs to his total river jamming range is the same as the relationship of his river bluff shoves to his river value shoves. So, overall, he bet-bets for value with 25% of his turn starting range, double barrel bluffs with 13.8% of it, and sin- gle barrels with 21.5% for a total turn betting frequency of a bit over 60%. His turn leading range contains more bluffs than value hands, and he gives up with a majority of them on the river.

How about the SB? Facing a bet on the turn, he folds B/(B+P)=55.2% of the time and calls the other 44.8%. He folds a bit more than half the time since he is facing a somewhat larger than pot-sized bet. Then, if he faces a sec- ond bet on the river, he folds an additional (S−B)/(S+P+B)=55.2% of the hands with which he gets to the river. That is, overall, if he faces a bet-bet, he gets to showdown with 20.0% of the bluff-catchers he started with on

Turn Play: Polar Versus Bluff-catchers Redux

the turn. Of course, he shows down more often overall, since Hero cannot always bet-bet, and he always wins the pot in those cases.

What do the EVs look like here? The players begin the turn with 27.5-BB stacks and a 5 BB pot to contest. The BB had the best hand a quarter of the time. Thus, if they had just skipped the turn and river action and showed down their hands, the BB would have ended up on average with

27.5+[(1/4)×5]=28.75 BB and the SB would have expected

27.5+[(3/4)×5]=31.25 BB. However, thanks to the betting, the SB’s EV is only around 29.5 BB, and the BB’s is over 30.5 BB. The BB expects 27.5 BB with his bluffs, of course. This is the same as if they had just checked down – they win none of the pot. His nuts, however, average a bit over 39.5 BB at the equilibrium! They win significantly more than the whole pot at the be- ginning of the situation. The turn and river betting gained Hero’s nuts tons of value above their raw showdown equity, and in fact he expects to make a tidy profit in this spot despite holding the worst hand three quar- ters of the time.

Here we see again a very important theme of NLHE play: the shape of a player’s distribution has a large effect on his ability to contest and win “his share” of the pot. If we only have a bit of equity, it is best that it be distrib- uted such that we have some very strong and some very weak hands rather than a bunch of mediocre ones. A middling hand is often only a lit- tle better than air, while a 100% equity holding is much more valuable than a middling equity one. That is, the relationship between EV and eq- uity is nonlinear.

This helps to explain why it is often better to play hands such as 5♥-4♥ pre- flop rather than less “playable” ones such as K♠-3♦, which may actually have more raw equity versus an opponent’s range. The suited connector is a lot more likely to make us a particularly good or particularly bad hand post-flop, and thus has a higher EV than the other, which is more likely to make a bluff-catcher. We can apply this idea post-flop as well. If we face a bet on the turn, and there is money left behind, we usually much prefer to continue with a draw than a made hand if the two have about the same amount of equity. The draw will sometimes turn into air on the river, in which case it is only a little less valuable than a mediocre made hand.

Expert Heads Up No-Limit Hold ’em, Volume 2

However, the draw can also become very strong, at which point it will be very valuable.

One lesson from this work that we can immediately take to the bank has to do with bet sizing. We have focused so far on the case where the BB leads the turn and river using GGOP sizings. These are over-bets (bets larger than the size of the pot). However, many players stick with a smaller sizing with most of their betting range on both streets and especially on the turn. How does the BB’s EV here compare to the case where he uses more stan- dard sizes? In other words, what is these hands’ EV at the equilibrium of the static PvBC game if we constrain the BB to use smaller sizings when- ever he bets? Suppose the BB makes 3/4-pot sized bets on the turn but still uses an over-bet all-in on the river. Plugging into Equation 10.1, we quickly find that the EV of the BB’s nuts here is about 39.3BB, down about 20 BB per 100 hands from the optimal choice. Try the case where the BB uses 3/4- pot sized bets on both the turn and the river yourself.

Suppose the players are playing the same idealized PvBC turn game except that the BB bets 3/4 of the pot on the turn and river with all of his betting range. How much does BB expect to end up with at the equilibrium? Hint: we cannot plug directly into Equation 10.1 since it assumes that we jam the river. How- ever, this situation is equivalent to one where the effective stack is shorter and the two small bets do get us all-in, since none of the additional money behind can ever come into play anyway. In fact, since we are betting the same fraction of the pot on both streets, 3/4-pot is the GGOP sizing for that alter- nate stack size. Find it, and then apply Equation 10.1. 1

1 You should have found that the BB’s nuts expect approximately 37.7 BB in the case that he

uses the more “standard” sizing on both streets. Although much better than checking down, this strategy loses his nuts over 180 BB/100 relative to the optimal sizing! Certainly there are cases where smaller turn and river leads are appropriate. However, over-bet lead- ing the later streets from the BB versus the flop check back should certainly be a common adjustment to players who check back primarily mediocre made hands. When we are in the SB, we must be capable of balancing our flop checking range when facing a BB who is ca- pable of making such an adjustment.

Turn Play: Polar Versus Bluff-catchers Redux