2.- ESTATUS CONSTITUCIONAL DE LOS PUEBLOS INDÍGENAS Argentina (1994)

First, as in PvBC river play, only the polar player will do any betting. Essen- tially, since any particular holding of his is the nuts or air, he knows whether or not he has the best hand. The bluff-catching player, on the other hand, is in the dark, and there is no reason for him to ever bet, since his opponent is able to play perfectly against it. If we check-call with a bluff-catcher, we always put in a bet against the nuts, but we have some chance of winning a bet from Villain’s air as well. However, if we bet our- selves, we ensure that we put a bet in whenever Villain has the nuts, and we give him the option to get away cheaply with his air.

Since the bluff-catching player will never bet at equilibrium, positions are essentially unimportant. The polar player will always get a chance to bet on each street. His opponent will check back if given the chance and will call or fold when facing a bet. This should agree with our experience of real play. For example, if the SB c-bets the flop and the BB check-calls, the BB will generally follow up with a check on the turn unless the turn card hap- pens to have a large effect on the relative hand strengths.

Turn Play: Polar Versus Bluff-catchers Redux

For the sake of the discussion, we will assume the SB is bluff-catching and the BB is polar, as if the SB has checked back the flop with a range of purely weak showdown hands. We will follow up with an example where this is the case. However, keep in mind that positions do not really matter, at least in the ideal case, due to the hand distributions. Our results will apply just as well when the SB is polar.

At the beginning of turn play, the pot size is P and both players have S be- hind. The polar BB will either check or bet B. If he checks, then so does the SB, and we get to the river with the same pot and stack sizes. If he bets, the SB can fold or call, and if he calls, we get to the river with a pot of P+2B and remaining stacks of S−B. In general, we will need to consider all of the pos- sible river cards separately, but for now, they are all effectively the same due to our assumption of static hand values. So, the decision tree is shown in Figure 10.1. It shows the turn action and the two river situations that arise after a bet does and does not go in on the turn.

Figure 10.1: Decision tree for the PvBC situation on the turn with static hand values.

As on the river, we have something of a special case if the polar player’s range is especially nut-heavy. In this situation, he can bet with all of his

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hands, all-in or even smaller, and the SB will still prefer folding over calling with his bluff-catchers. If this is possible, then this will be an equilibrium strategy for the BB, since it wins him the whole pot all the time, and he can expect no better than that at equilibrium.

Practically speaking, it will rarely be the case on the turn that the polar player’s range is much, much stronger than his opponent’s. For example, consider the two situations that we saw lead to a turn PvBC spot. In one, the SB checked back a flop with a range of weak made hands, and in the other, the BB check-called with a fairly similar range. In both cases, the bluff-catching player at least has something, and the majority of the polar player’s range likely falls into the air category by comparison. As they say, it’s hard to make a hand heads up. Perhaps we can imagine turn spots in 3-bet pots or after flop check-raises where ranges are narrower in general, and the strong portion of the polar player’s range is fairly large – possibly large enough that he can get away with betting all of his weak hands as bluffs. We will consider these possibilities in the context of some examples a bit later.

The criterion for this special case is similar to that on the river, and we will not go through it here. Once we solve the normal case, it will be easy to visualize the conditions under which our solutions break down. For now, we will proceed with the assumption that the BB’s range is not strong en- ough to just bet and take down the pot all the time on the turn. In this case, when facing a turn bet, the SB will sometimes call and see a river. If he did not, then the BB would bluff all his air on the turn in order to win the whole pot all the time, but then the SB would certainly want to start calling since we assumed that the BB did not have a strong enough range to make the SB always prefer folding.

Now, how do we find the equilibrium? We’ll start by identifying some in- difference relationships. They might seem obvious by analogy to the sin- gle-street case, but several subtleties arise because of the multi-street na- ture of the game, and these ideas won’t all hold in more complicated situations, so it’ll be helpful later if we understand exactly what’s going on. Once we have the indifferences, finding the strategies will be easy. Let us start with something we should already understand: play in the two

Turn Play: Polar Versus Bluff-catchers Redux

river situations. We know the BB will always bet the river with his nuts, and using all-in as his sizing is at least co-optimal. The indifferences that hold, however, are not immediately obvious, since we don’t know how much nuts and air he brings to each river spot. First, we can see that in both river spots, if the BB has a betting range at all, then his air must be indifferent to bluffing. He can’t have a non-empty betting range and still strictly prefer to check with his air at equilibrium, since that would mean he’s betting only with the nuts, and the SB’s response will clearly motivate him to begin bluffing. On the other hand, his air can’t strictly prefer to bluff either. Why not? We’ll consider the two river spots separately. First, consider river play after the turn goes bet/call. After the turn bet, re- maining stacks are S−B. If the BB strictly preferred to bluff his air here, then that would mean he bets his entire range. It would also mean that the SB sometimes folds to a bet, since otherwise there is no way the BB could want to bet his bluffs. But if the SB is folding to bets, then his EV when fac- ing a bet is S−B, and since the BB is always betting the river, the SB’s EV with a bluff-catcher on the river is always S−B. If this were so, then the SB’s strategy must involve always folding to the turn bet – why would he call and guarantee himself an EV of S−B when he can just fold the turn and achieve an EV of S? This cannot be the case, since we have already assumed that the BB’s range is in fact not strong enough to always win the pot on the turn. Thus, if the BB’s range is not strong enough to bet and always take it down on the turn, he also cannot strictly prefer to bet his bluffs on the river after the turn goes bet/call.

We now know a couple of things. First, the SB sometimes calls when facing a turn bet, and he sometimes calls the subsequent all-in on the river as well. Second, in order to make the SB want to call the turn, the BB must sometimes be getting to this river spot with air that gives up. Now, notice that the BB’s EV with air hands at the beginning of river play here is S−B: that is the EV of giving up on the pot, and his EV of bluffing cannot be any greater or else he would always do so. Since his EV with air is S+P if he bets the turn and gets a fold but is S−B if he bets the turn and gets a call, some- thing which happens with non-zero frequency, his EV of betting the turn with air is necessarily less than S+P. We will use this fact shortly.

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Now, the BB cannot strictly prefer to bet his bluffs after the turn checks through either. If he did, at equilibrium, that would mean that betting all his bluffs was still not enough to make the SB want to call with bluff- catchers. In other words, the SB always folds, and the BB’s bluffs here have an EV of S+P. We know that this is better than his EV of betting the turn, so the conclusion here would be that the BB saves all his air to bet the river and is still able to make the SB always fold to the bet. However, this can’t be true. An all-in on the river that included all the air in his turn starting range could only be a more attractive call for the SB than the same bet on the turn. Thus, if he could always bluff in this river spot, he could also al- ways bluff the turn, and we know that is not the case.

In neither river spot can the BB bet all of his air. At this point, we know that in both river spots, the BB must be indifferent to betting with all of the bluffs with which he gets there, at least if he has a betting range at all (i.e., if he has nuts that need balancing). Great, we can use that. We also saw that the SB is not always folding to a bet in either river situation, but he is clearly not always calling either. (Why not?) Since he sometimes calls and some- times folds when facing a bet, he must be indifferent between the two op- tions. So, whenever we get to the river, we know how things will go. The BB will bet all-in with all of the nuts he gets there with, and as many bluffs as it takes to make the SB indifferent to bluff-catching, and the SB will call enough to keep the BB indifferent to bluffing. Now, if the BB does not get to one of the river spots with any nuts, he won’t have a betting range there, since he can’t bet only bluffs. However, in river spots where he has a betting range, all the indifference relationships we might hope for turn out to be true. Now what about turn play? The SB cannot always fold to a bet, but can he always call? We need to suppose that he does, figure out the BB’s maxi- mally exploitative response, and then check if it incentivizes the SB to be- gin folding instead. This should seem reasonable, but things are a bit com- plicated. If this were the river, then the BB’s best response would be to simply stop bluffing since he is always getting called, and this would cer- tainly make the SB want to stop calling with bluff-catchers. On the turn, however, it is possible that the BB might still find it best to bluff some in order to continue on the river. A turn bluff certainly is not profitable in and of itself, but perhaps he can make up for that by betting again on the river.

Turn Play: Polar Versus Bluff-catchers Redux

What the BB will certainly not do, however, is just bluff the turn and then give up on the river – if the SB is always calling the turn, he is just throw- ing money away. So whenever the BB bets the turn here, it must be with the intention of continuing on the river. And whenever he decides to play bet-bet with a bluff, that line must have an EV of at least S, or else he would have just given up on the turn.

So again, how might the BB counter a SB who always calls a turn bet, and is the SB motivated to begin folding on the turn? Well, the SB will always fold on the turn if the EV of calling and playing a river is less than S. So, what is the value of the SB’s bluff-catchers on the river after he calls a turn bet, given that the BB is playing maximally exploitatively? It depends on the SB’s river play, of course, but the best possible case for him is that he too is playing a best-response strategy in the river subgame. But if both players are playing best responses, then the players play an equilibrium in the river subgame, and we know what that looks like. The BB bets the right mix of value and bluffs to make the SB indifferent to calling and folding. Thus, the SB’s EV is S−B regardless of his play, whenever he faces a bet. And, he always faces a bet, since the BB never bets the turn and gives up on the river. The BB will bluff just enough on the turn that he can always continue on the river and exactly make the SB indifferent to calling and folding. Thus, S−B is the value of the SB’s hand whenever he calls the turn bet. So, if the SB never folds the turn with his bluff-catchers, and the BB re- sponds maximally exploitatively, the SB’s EV is no more than S−B when fac- ing a turn bet. He could clearly do better by just folding on the turn to end up with S. In other words, if the SB is always calling when facing a turn bet, then even if he plays as well as possible on the river, the BB’s counter- strategy will incentivize him to start folding on the turn. This is what we needed to show that the SB must play a mixed strategy with his bluff- catchers on the turn when facing a bet.

We now know that the SB is playing to keep BB indifferent to bluffing in each spot where the BB actually has a betting range. We can apply the bluffing indifference to find the SB’s calling frequencies in these cases. First, the SB calls the turn just enough to make the BB indifferent to bluff- ing. Thus, for the BB’s air:

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The right side of the last equation gives the BB’s expected stack size if he bets the turn and then gives up on the river with a bluff. He always ends up with at least S−B, and he gets an additional B+P (i.e., he gets his bet back and wins the pot) whenever the SB folds the turn. Solving, we find that the SB folds to a turn bet with frequency B/(B+P). If he calls any less, then stab- bing (at least) once is necessarily better than giving up on the turn for the BB’s air. If the SB is calling any more, then just giving up is better than bluffing once. Bluffing twice, of course, could still be better than either of those options.

This result should look familiar. This is the same calling frequency that the bluff-catching player used in the river PvBC game. This is surprising! Call- ing with a bluff-catcher is much less appealing on the turn because of the possibility of facing a river bet. However, the frequency simply arises from the need to make the BB indifferent to betting once to try to win the pot with hands that will otherwise go to showdown and lose, the same as on the river. If the SB calls any less than this, the BB will prefer to always bet the turn rather than just give up with his bluffs.

We now know the SB’s play on the turn, and a couple familiar applications of the bluffing indifference give us his frequencies in the river spots, again, assuming that the BB gets to them with a range containing some value hands. After the turn checks through, the BB’s river bet sizing will be all-in,

S, into a pot of P, so the SB will call P/(S+P) of the time. After the turn goes

bet/call, the BB will bet S−B into a pot of P+2B, so the SB’s calling frequency will be (P+2B)/(S−B+P+2B). Notice that these expressions are essentially the same except that the river stack and bet sizes are different because of the turn bet.

How about the BB’s play? We now have enough information about the SB’s strategy that we can figure out the most profitable way for the BB to play his nuts, and we can then find the amount of bluffs he plays similarly for balance. His two choices on the turn are bet and check – if he holds the nuts, which is better? First, if he checks the turn, he will end up in a river spot where he will jam and get called with a certain frequency. We have

Turn Play: Polar Versus Bluff-catchers Redux

since he ends up with S+P all the time and another S the P/(S+P) of the time that the SB calls his river jam. On the other hand, if the BB bets B on the turn and continues all-in on the river, he ends the hand with at least S+P. He wins an additional B when the SB calls the turn and folds the river, and an additional S when the SB calls both streets:

(10.1)

The fractions (S−B)/(S−B+P+2B) and (P+2B)/(S−B+P+2B) are the SB’s folding and calling frequencies, respectively, after he gets to the river and faces a shove. Now that we have the EVs of the BB’s two options with the nuts on the turn, which is bigger? To check, we subtract EVBB(bet turn with

nuts)−EVBB(check turn with nuts) and, after some simplification, find a re-

sult which is necessarily positive. That means EVBB(bet turn with

nuts)>EVBB(check turn with nuts), which is what we wanted to know. It