Poker Math: Multistreet Bluffing

Scenario: Cbet on Flop
-Hero raise BTN to 2.5x, BB calls

FLOP: [Ah9d3d]

Hero Cbet half pot
-> What is our value to bluff ratio?

Villain needs to call 2.5BB to win 7.5BB
= 3:1 odds

OUR VALUE-TO-BLUFF-RATIO has to be 3:1
(25% bluff and 75% value range)
therefore, we bet bigger when we have many bluffs in our range

Scenario: Hero has 2:1 Value to bluff ratio:
Turn: Pot=$60, Hero bet $60 and villain calls
River: Pot= $180, Hero bet $180, villain calls

What is the EV of Villain calling down two streets?
EV(call turn/river) = EV(Villain calling and loses) + EV(Villain calling and win)
EV(total) = (2/3*-(60+180)) + (1/3*(60+60+180))
= -160+ 100 = -$60

Impact on Defense
if the call of our EV is 0 = we call/fold 50% on the river
as this is the optimal calling frequency...
if villain has 2:1 bluffing ratio and our pot odds is also 2:1


Scenario: Hero has 3 combos of AKs on flop of [QJT]
Hero bets $25 into $50 and continues to bet half pot until the river
-how many bluffs combo should hero add to their range when betting this size on the flop?
WORK BACKWARDS

On the river, hero should have 3:1 value-to-bluff-ratio therefore we can add 1 bluff on the river
On the turn, now we know that we have 4 combos to bet the river with, we now have to make our turn bet consists of 3:1 ratio as well; where we have 4 combos of value therefore: we can add 1.33 more combos to bet with on the turn: Total combos = 5.33 combos
On the flop, we know that on the turn we're betting 5.33 combos, therefore we have to add 1.77 more combos in our betting range in order to make our value-to-bluff ratio static at 3:1

So our range on the flop consists of 3 value combos and 4.11 bluff combos

Villain's EV of calling the flop
EV (calldown) and lose = (3/7.11*-175)
EV (call flop and hero give up) = (1.77/7.11*75)
EV (call turn and hero give up) = (1.33/7.11*125)
EV (calldown) and win = (1/7.11*225)
EV(total) = ~0

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