Lots of things to talk about in your post. In your example, pot odds tell you that you should call the turn because you're being given 3-1 on the turn and have a hand than is only a 2-1 dog. Gto is a tricky thing and generally speaking when people refer to it in forums they're talking about defending enough to stop villains from auto-profiting by betting their entire ranges versus us. So if we're using that framework we're generally going to want to defend enough of our range to prevent that from happening. That being said, some boards/turns/rivers are better for our range than villain's (so we'll be able to call/raise more than the minimum defense frequency) and some are worse (so we can call/raise less often). Also, when you're thinking about defending enough of your range on flops and turns those hands don't necessarily need to be able to beat a bluff because they have equity to improve and can steal the pot with bets later on. Finally, if you get to the river and you can't defend enough of your range to stop villain from auto-profiting with a bet you might want to look at how you structured your range on earlier streets as that might be a problem OR the river might just suck for your range. Hope that helps a little.

The equity of a hand against a range is static at a particular street; however, the EV depends on the actions. Equities and Expected Value are not related.

GTO is the analysis of the correct frequencies of each action (betting, bluffing, bluff catching) to optimize your EV.

> Vilain bets 1/2 pot. GTO tells me I should call with 66% of my range which can beat a bluff.

Let's assume the situation is on the river, the bettor's range is perfectly polarized and you have a bluff catcher (for simplicity's sake, otherwize, other factors make things more complicated to analyze and understand).

To optimize his EV with a 1/2 pot bet, Villain must bluff at an Optimal Bluffing Frequency which is always equal to the pot odds he's offering, here 25 %.

The 66.66 % represents the Optimal Calling Frequency on the River to prevent the bettor to make a profit on his bluffs. This calling frequency optimizes your EV.

The spot you describe is more complex because it doesn't tell us anything about Villain's range : he might hold a NFD, that beats you if you hit, etc.

http://www.runitonce.com/nlhe/breaking-apart-misconceptions-about-gto/

http://www.runitonce.com/chatter/gto-confusion/

http://www.runitonce.com/chatter/gto-simplified/

Videos:

- Lefort's Advanced Theory Principles  and the 6-max concept series.

- Ben Sulsky : the Toy Gaming series

Robert, James,

I see I have more to learn than I imagined.

Thanks for your guidance.

huh, yeah this is a good discussion.  I think it may help also to use a river example where no more cards come:  villain bets 20 value combos and 10 bluffs for a pot size bet on the river and he's perfectly polarized.  Given his bet size, we have to defend 50% to keep him from being able to bluff any two profitably.  Also, given his bet size, we have to have 33% equity.  And...we have 33% percent equity effectively since we win 1/3 of the time (2 value for 1 bluff).  So both concepts are satisfied here.

If villain bets 1/2 pot, we have to defend 67%. 
He needs to have 75% value with this betsizing (according to my math...I can expand if you want).

We'd need 25% equity to make a breakeven call, and that's exactly what we have. 

So I think those two concepts aren't actually contradicting.  In these examples, our whole range has the breakeven required equity, but we still only want to defend the proper frequency, or we'd be vulnerable to villain exploiting us.

I'd go as far to venture that the situations where these concepts diverge (for example, we need to defend with, say, 50% in a spot, but half of our range doesn't have 33% equity) represent spots where either a) we've misplayed or misbalanced our ranges, b) the board developed in such a way that sets our opponent up for an inherently profitable situation, or c) villain had to take a risk or potentially -EV move earlier in the hand to get to this situation. 

Thoughts welcome, I kinda realized/worked through this as I was writing. 

Yeah agree it can get confusing, particularly on streets leading up the river.  I still tend to think in terms of equity which be leading to problems.  Consider a spot where we have three bet preflop with KK and get 4 bet.  We call a flop bet, and on the turn, we are faced with a pot sized bet, and we think his range is 8 combos of AA and 8 combos of AK.  From the nuts air game assuming nobody improves, we know he should be betting 5/9 bluffs 4/9 value on the turn to make us indifferent.  So here, ignoring our implied odds when we river a K and his A outs when he has AK, we still need to fold even though we have 50% equity VS his range and we're getting 2:1 on a call, assuming he plays the river correctly.  This leads me to start thinking about how much "extra" equity I need than the price I'm being offered to call a bet when I have a bluffcatcher on a street before the river.  Is this line of thinking wrong or not often applicable, since in real poker one range will almost never be able to bet all three streets with its value hands?