In Matthew Janda's book, he lays out a situation where we are OOP on the river, and our opponent has us beat 30% of the time. He goes on to explain that betting has a higher EV than checking, and this is because when we check, our opponent will be able to bet his 30% value hands, and also add 15% bluffs to make us indifferent. This results in him betting the river 45% of the time, and every time our opponent is able to bet the river with a balanced range, we have effectively lost the hand and the EV of our call will be zero. So the EV of shoving for one PSB will be .6, and the EV of checking will be .55.

My question is, ignoring optimal calling frequencies for a moment, does the fact that our opponent can add these bluffs and gain additional EV on the river affect our bluff catching decisions on the turn? I dunno if I'm misunderstanding something but is it more less similar to a reverse implied odds situation? For example, suppose we have a hand on the turn that we believe is good exactly one third of the time, and we are faced with a pot sized bet. Neither us or our opponent can improve on the river and there is still money behind. Do we have a break even call here, or does the same sort of concept apply to the river situation in Matthew Janda's book apply, and we would need to be good roughly 50% of the time to call, since he can add bluffs in a balanced fashion on the river? Thanks in advance for any help.

Steve

Ev for betting river assuming villain never calls with worse is -1*.3 + 1*.7 = 0.4

EV for betting river assuming villain calls 50% of the time is -1*.3 + 1*.5 + 2*.2 = 0.6.

I don't have MJ's book, so I don't know what the assumptions about the situation are supposed to be. Obviously, it makes a difference. Perhaps OP can clarify.

yes, money behind affects previous street decisions

I think this brings up an interesting point. If our range is composed with bluff catchers all with 70% equity and we shove, Villain will never want to call with any worse hands because it will only increase our EV. Therefore we choose the higher EV option of check/calling, check/folding. But if we add 50% complete air hands to our range (hands that are worse than Villains bluffs) We can then take the shove line and increase our EV to 0.6, In this case Villain will want to call 50% so that our bluffs are indifferent to betting/checking. Intuitively we are adding complete garbage to our range to increase our EV.

This is pretty easy to model in 1 card poker. Villain can hold A,K,Q, ,T, 9,8,7,6,5,4. (If your range is J, you have 70%.)

You'd rather be dealt from a deck that holds [Jacks, Deuces] that from a deck that holds Jacks only?

If so, think about whether this
still holds.

Bear with me. Ok so if we hold just the Jack in our range, what makes Villain want to call with worse? If we hold the 2 as well, Villain has incentive to call with cards worse than a Jack which makes shoving a reasonable option. If we shove with a balanced range so that Villains cards below a Jack are indifferent to calling/folding, it is reasonable for them to call with a combined frequency of
50% to keep us indifferent to bluffing/checking with our 2's. Correct?

Can you post the exact situation from the book? Modelling this with 1 card poker as was suggested, 4-T, Q-A vs J, checking is at least co-optimal for the OOP player with the range of J. The assumption that there is 1 100% pot bet left and we are only considering betting 100% of the pot.

I can post a Nash Equilibrium solution that proves this if desired (and a CREV file that can be used to verify the equilibrium numerically), but I want to make sure that I have the scenario right first.

The logic you are stating seems to ignore that we lose the amount we bet when we bet and he calls whereas we dont lose that amount when we check/fold. There is no reason to think he is going to call with worse right, or is assuming he calls with worse part of the situation for some reason.

If your range is a Jack or a Deuce, and you shove all of your range, what is villain's calling range? With a 4, he's winnning 50% of the time. The pot is not empty.

It is true that in this scenario (our range is half 2 half J) that always betting J and betting our 2 50% (not always!) is optimal and would induce our opponent to call 50%.

Anyway here are GTO strats with only J in our range and with J and 2 in our range. The program I used to solve this requires a valid board and hole cards. So I modeled it with a board of 22233. Then gave the IP player a range of 4h4c-ThTc, QhQc-AhAc. The two examples have the OOP player with just JcJh in the first and with JcJh,AsKs in the second.

With just J in our range:
http://gtorangebuilder.com/SolutionB...irtyNoAir.json


With J and 2 (modeled by AsKs):
http://gtorangebuilder.com/SolutionB...hirtyWAir.json

The example is on page 332. He says...

Suppose we are OOP and our opponent has us beat 30 percent of the time. If we go all in for a pot sized bet and he calls 50% of the time our EV is, 0.6= (0.50)(1)+(0.20)(2)-(0.30)(1). Then, paraphrasing a bit he says, in position we would never make this bet since we would lose over half the time when called, however when OOP, villian has the opportunity to bet with a balanced range of value bets and bluffs on the river. So when we check he will bet his 30% value, and add 15% bluffs. This results in him being able to bet 45% of the time, and whenever he able to face us w/ a bet from a balanced range our EV will be zero. Therefor the EV of us checking the river is .55, less than if we shoved.

My question was, can we apply this same kind of logic to bluff catching the turn? This has been explained to me via PM so I understand the concept better now. The nuts air game with us calling X percent of our range on each street always hurt my brain. I had asked if we have a hand which we believe has 33% equity on the turn, neither hand or range will improve, do we have a break even call if he bets the pot on turn, laying us exactly the price we need, or does the fact that he can add these bluffs on the river make it a fold? Below was the answer I got which seems to make sense.

Turn: (Pot: $1) Villain bets $1, Hero calls $1.

River: (Pot: $3) Villain bets $3, Hero calls $3.
We're risking a total of $4 to win $5, so we need ~44.45% equity on the turn to call down profitably, and a hand with only 33.34% equity is losing $1 (the entire size of the pot on the turn--quite a large mistake).

This is why the 2:1 v-bet–to–bluff ratio is applicable only when there are no future streets of betting.

On the turn, it's 2:2.5, and on the flop, it's 2:4.75--assuming polarized ranges and pot-sized bets on all streets.

Villain is passing up EV by betting only a 2:1 ratio of v-bets to bluffs on the turn, assuming we use the proper counter-strategy. All of the combos that bet the river effectively "claim the pot" since they have an EV of +(pot), so to make our turn call indifferent, he can mix in additional combos that bluff the turn and give up on the river. By doing so, he forces us to call down some % of the time instead of allowing us to fold our entire range, and his value bets gain EV.

So 2 comments.

1. Unless I am misunderstanding, the reason to bet the J on the river is not just to stop him from bluffing, its also because he calls with worse. This is proved by the fact that the optimal strategy is to always check when our range is only a J, but to bet when our range contains a 2 proves this. If we were only scared of getting bluffed off of our J we would bet in both cases. The reason we only bet our J in the later is because that is whats required to make him call 50% and because he can bluff at us a lot more when our range is 50% air. We still should bet our J IP 40% when our range is has 2 in it (which we bet 20%), because we need a balanced betting range, and when we bet our 2 they will often fold better.

2. Your thinking about the turn is on the right track. You can use backwards induction to determine this. The basic idea is to solve for the EV of the river subgame when both people play GTO. Then assume that reaching that river pays the EV of that subgame. However, relying on equity to make this determination isn't very accurate, what matters is your range's ability to actually reach showdown, not your equity in particular.

Ok just ignore this, seems I was too many beers in yesterday to notice you're not shoving all of your range, which was the first impression I got somehow.

Anyway, you're OOP with a choise of range a) J or b) J,2. You're picking b) because you now bet with a J and get some calls from worse?

he doesnt gain anything by adding bluffs. We lose value allowing him (by checking) to play gto when we have bigger value hands-bluffs ratio in our range than necessary (7 to 3 when it should be 2 to 1).
In second part of your post we have only 30% of better hands so he definitely gains nothing
If he bluffs you must add additional 20% of worse hands (that beat his bluff of course) and just call turn-river. If he doesnt or bluff rarely your 30% is enough.

I think Janda is thinking about a wide betting range for the oop player in that spot: ie oop player also has nuts combos in his range, very strong combos, average value, total air.

In such a situation betting is basically I preemptive bet: looks like at current time there are at least 3 posts about this matter in the forum

Thanks for the responses. I think my confusion is stemming from something a lot more basic that I'm not quite wrapping my head around. In the example 30% of villains range is hands that beat us. Assuming for whatever reason we check the river, he can and should add 15% percent bluffs if he bets the pot. This makes his EV .45, since he's made us indifferent 45% of the time. So he is taking a situation where he's only good 30% of the time and winning the pot 45% of the time. But then I think about the fact that in theory, his 15% bluffs are breakeven, and basically function to prevent us from being able to fold our bluff catchers when he bets and not paying off his value hands.

So I guess in a way I'm still confused if the bluffs villain adding on the river, increase his EV, or just more less make it so we can't just exploit him by always folding.

You basically have it exactly right. He's still getting EV 0 with the part of his range he is actually bluffing with since we are calling exactly 50% such that his bluffs break even, but by bluffing with a proper frequency he is forcing us to call 50% against the top of his range as well, thus his over all EV with his entire range goes up from .3 (us always folding to his value bets) to .45 (him averaging 1.5x the pot when he bets the top of his range because he always wins 1x pot and 50% picks up another pot sized bet when we call).

In that link I posted for you (here it is again)

http://gtorangebuilder.com/SolutionB...irtyNoAir.json

If you click on the left most decision node, it will show you the EV for every hand in the hero's range. You can see his EV is 150% of the pot with the top 30% of his range, and 0 with the rest of his range, for an average of .45.

When your opponent is betting balanced, he's making you indifferent to calling or folding your bluff catcher. So you can safely assume you are going to always fold (only to compute your ev). If you always fold the ev of your bluff catcher when you are bet into is 0.
If you decided to always call your ev would still be 0, just a little more complex to compute.