I am trying to understand Mathew Janda's example on pages 106-109. Can someone resolve the inconsistency I am encountering in calculating the optimal bluffing frequency on the turn? It may be a embarrassing arithmetic error despite many checks, but my calculation agrees with Janda when I assume the opponent folds the river.

The assumption is that "we" have the nuts on the flop 20% of the time and air the rest; our opponent has a bluff catcher.

Janda finds that we should bet 45% or our hands on the turn and 30% on the river. The 45% calculated for the turn assumed that our opponent folds on the river if we bet. Unless, I have made a mistake, our opponent will have a positive EV of if she calls on both streets. Here is my calculation, where our opponent is B:

With B calling on both streets,

1) 33.3% bet turn; check river (B wins 2 pots)
2) 22.2% (.1/.45) bet turn; bluff river (B wins 6 pots)
3) 44.4% (.2/.45) bet turn; bet river with nuts (B bets 1 on turn, 3 on river, and loses 1+3= 4 pots)

Thus B's EV = 0.33 (2) + 0.22 (6) - 0.44 (4) = 0.22

The numbers work for the cases where B folds either the river or the turn. For B loses -1 pot calling turn and folding river

EV = 0.33 (2) - (0.222 + 0.444) (1) = 0

I found our optimum river bluff frequency assuming B calls both streets to be 0.4, in which case,

EV = 0.25 (2) + 0.25 (6) - 0.5 (4) = 0

Anyone see a mistake or see a way to resolve this?

Thanks,

I dont recall this specific example but I was in a similar situation as you. Just google the 2p2 thread for the book and he has any mistakes corrected on the thread. Check if you can find the mistake you found in the thread for the book.

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Thanks. I found my way to the thread, and did not find an error listed for these pages. I had done a general google about optimum turn bets with his name in it before posting my question and did not find anything relevant, but I could still have missed something.

I don't think this is a mistake in the normal sense. Assuming no errors in my calculation, which seems to me to be likely since I duplicate Janda's number when I use his assumption of B folding on the river, it raises a question about how precisely we can work backward from the river to the flop. It looks like the optimum bluff rate depends on the decisions make on future streets. I have not chased this back to the flop to see what kind of a spread in rates results from different paths.

The book has been great. Working back to the flop from the river is really clever. However, my attempt to understand the calculation seems to lead to the conclusion that the optimum bluff rate is at least slightly less precise (e.g. the difference between 0.4 and 0.45 may be irrelevant in a game). I expect this is related to correctly handling the implied odds of earlier streets. Implied odds involve multiple streets so the streets cannot be considered "independently." At least that is what appears to be the case to me at this point.

B only wins 5 pots here. Original pot (1) + our turn bet (1) + our river bet (3). You can't count their turn call as money won.

B's EV = 0.33 (2) + 0.22 (5) - 0.44 (4) = 0

Thanks. Oops!

Janda assumes that on the flop, hero has 20% value (eq=1.0) and 80% bluff hands (eq=0). He then determines how many bluff hands he should bet on flop, turn, river so that villain, with only bluffcatchers, is indifferent to calling and folding on the river. With only pot size bets, the portion of value bets on each street is calculated and he shows hero needs 2 value bets to 1 bluff bet on the river. I don't think Janda discusses overall EV for this situation.

An extension of what OP asks is this:

What is villain's EV if he always calls playing against Janda's theoretic value-to-bluff ratio strategy for three streets.

Janda states that all checks that hero makees are losses to hero. This is so because he correctly reasons that hero will not check value hands. If I did the math right, hero will get to the river 45% of the time and then fold 15% bluff hands to maintain a required ratio of 2 value bets to 1 bluff bet on the river (20% vs 10%). This means he sustains losses on the 3 streets 70% of the time with losing checks .

I calculated the overall villain and hero EV assuming the decision point is the start of flop betting (was not easy). Villain's EV is 0.325 FlopPots if on each street hero pot bets and villain calls. I got hero EV =0.675 FP. The fact the EV sum is 1, the postflop pot, is somewhat comforting for this tricky analysis.

If I didn't screw up, if a player knows hero is playing this theoretical game and the assumptions are somewhat reasonable, always calling hero bets is +EV for the 80/20 case.

I'm still a bit dubious, though. The whole idea was to make villain indifferent on the river to betting or folding (EV=0), yet the result I got shows overall, he is +EV.

I thought 80/20 for flop sounds high to arrive balanced on the river. Maybe Janda is demonstrating how to calculate which percentage of bluffs to give up with on the river, if a range discrepancy gives hero an 80/20 mix given a certain flop texture?

Thanks for responding. Sorry to be late in seeing this. I stopped watching after someone pointed out my mistake. It is a sample problem to illustrate how to calculate bluff frequencies. "We" (I guess we go by Hero) have nuts 20% and air 80%. Opponent (Villain?) has a bluff catcher.

The technique is mine. Janda originally did it differently. It cannot be wrong in the same way that it cannot be right that a dice betting system has an edge. If your calculations say otherwise you must have made an error. It might be useful to try to find that error but there is no doubt there is one.

The general technique I speak of is assuming that a player folds when he is indifferent to calling or folding when you are trying to figure out multi street GTO strategies. It is logically impossible that this technique, used in an effort to simplify things, could fail. Do you see why?

I did make an embarrassing mistake which nolispeifaflaatoi pointed out to me. Trying to find it, if it was there as I originally suspected, was the point of my post so maybe I should not be quite so embarrassed. The technique is really slick. I found my way to Janda's book after hearing you mention it in a web interview a short while ago.