MOP: Fixed Limit

Fixed limit game: big blind versus player in the
field.
The blinds are one chip and two chips. Rakes are
ignored. A player in the field open raises. All
fold to the big blind. The big blind defends.
On the flop, turn, and river only the original
raiser may bet. The big blind may call or fold.

Assumptions. Randi Mae, a rational maniac, is the
open raiser. She will open 50% or more of the
pots with a raise. Against one opponent she will
make a continuation bet 100% of the time. On the
turn and river she will play optimal game theory
strategy. Billy is the big blind.

Added assumption. To simplify the example Billy's
pot equity will remain constant thru the final
three streets. When Billy isn't drawing to any
nut like hands, his pot equity is likely to
decrease with each new board card.

Question: What is the optimal game theory
strategy for Billy?

Now to solve this game using three methods.

Naive solution: Billy should call on every
street when his utility or pot equity is
positive.
9 chips after the flop. 4 from Randi Mae, 4
from Billy and 1 from the small blind.
Randi Mae bets all three streets.
If Billy calls all three streets, it will cost
10 chips. Billy can win 9 + 10 or 19 chips and
he risks 10 chips.
10/29 = .345
Billy needs .345 pot equity to call on the flop.
On the turn it's 8 chips for the 29 chip pot.
8/29 = .275
On the river it's 4 chips.
4/29 = .138

.345, .275 and .138 assumes Randi Mae is betting
100% of the time.

2nd way. This time the flop action is separate
from the turn/river action. On the flop Randi
makes the 100% continuation bet. Billy calls
100% of the time on the flop. The turn/river
is played as a combined street. Randi is
clairvoyant knowing whether she holds the
winner. Billy is the defender. Both Billy and
Randi will use optimal strategy for the
turn/river game.

Question: What is the optimal game theory
strategy for both players? What pot equity
does Billy need for this game to be even?

Solution.

S=size of pot
b=size of bet
f=frequency clairvoyant's equity
payoffs for defender

Chart A.

Randi\.......____.....Billy
_____\ fold ________call
b/g __| (1-f)S _ | __ (1-f)S-fb

b/b __| _ 0 _._ | _ (1-f)(S+b)-fb

Billy needs a EV of +2 to break even for his
flop call. When Randi gives up, Billy is assumed
to have won the pot.
S=13, b=8, f ranges from 0 to 1.
Find f so that Billy's EV is above 2.

Randi Mae\....._______...Billy
_________\____fold____________call
bet/give__|____5.031____|______.127
bet/bet___|______0_____|_____3.223

Randi's win frequency was .613 for Billy's EV
to equal 2. Billy's pot equity needed to be
.387. With this minor change in assumptions
Billy needed a 0.04 increase in pot equity to
made the flop call equal.

3rd way. This time each street is a separate
game.

The game is solved by placing the turn and river
in series. Solve the turn as a one street 2X2
matrix game. Then solve the river as a one
street 2X2 matrix game.
Billy will play the turn game. Then if Randi
Mae bets the turn AND Billy calls the turn, there
will be a river game.
Define f for frequency for checking or giving
up by Randi. Billy wins 13 chips whenever
Randi gives on the turn. 13f
Billy is guaranteed -2 by folding on the turn.
Therefore Billy will not play any river game
which has an EV less than -2. -2(1-f)
To force Billy to call the turn and play the
river Randi must overbluff. Billy then calls
with pot/(pot+bet) frequency.
Billy must win 2 to recover his flop call.
Solve for f in equation (1).

(1)13f - 2(1-f) = 2
__ 13f - 2 +2f = 2
________ 15f = 4
__________ f = 4/15 ~ 0.26667

Randi can give up at most 4/15 of the time on the
turn. She must bet 11/15 of the time else Billy
will never need call the turn to play the river
game.
On the turn S(pot) is 13 and b(bet) is 4.
On the river S is 21 and b is 4.
Billy calls pot/(pot+bet) of the time.
On the turn Billy calls 13/17.
On the river Billy calls 21/25.

______________ freq _ payoff _ value
turn RMgives _ 0.26667 _ 13 __ 3.46667
turn B folds _._ 0.17255 _ -2 _ -0.34510
river matrix __ 0.56078 _ -2 _ -1.12589
__________________________1.99568

Value of game for Billy.

Billy needed .488 pot equity to call on the flop.

On flops where the defender is drawing to no
hands worth betting he needs approximately
14% greater equity than a naive solution would
suggest.

On these aggressor/defender games the aggressor's
advantage is greater than her initial showdown
equity. For it to be profitable for defender
to play he needs about 15% more than showdown
equity. Defender can play profitability with
less equity only when he can reverse roles.
Meaning he goes from defender to clairvoyant.
This isn't possible with Axx rainbow boards.
There are no turn scare cards to inhibit the
aggressor.