2b. Min-raise variation

SB min-raise or fold.

First bet.

SB post 1 chip, BB post 2 chips
SB can raise to 4 or fold
BB risk 3 to win 3
BB calls =< 0.5

{SB} payoff matrix
0*(1-s)*1 SB folds
3*s*(1-b) SB raises to 4 & BB folds
1*b² SB raises/BB calls and ties, blinds returned
-3*(s-b)b SB raises/BB calls; SB loses 3
BB calls 2 to win 6. b=3s/4. s=4b/3

3(1-b)s + b² - 3(s-b)b
3s-3bs + b² - 3bs + 3b²
3s-6bs + 4b²
4b-8b²+4b²
4b-4b²
/ds=4-8b
b=1/2
s=2/3

min-raise variation after SB's open.
{SB} = 4b-4b²= 2 - 1 = 1
net{SB} = 0
The min-raise was equal after only SB opening raise.
BB's raise will be EV- for SB.

net{SB}= -5/32
The 3XBB raise was -5/32 after SB's opening raise.
Which seems to indicate the larger SB's opening bet
the worst it is for SB's EV.

Second bet.

min-raise variation; BB can raise the pot after
SB min-raises.

6 chips in the pot after SB min-raises.
BB can pot raise with 2 to call plus 8 to raise.
8 chips for SB to call. SB should call 0.5
A simulation game suggested SB calling at 0.472
produce higher EV than calling at 0.5.

{BB} was +0.53125 or 17/32.

{SB} payoff matrix after BB raise option.
0*(1-s)*1 SB folds
3*s*(1-s) SB raises to 4 and BB folds
s² this area was solved by trial and error.

This is the graphic picture of the payoff matrix
for BB using net numbers.

The EV was 17/32 in favor of BB.
{SB} = 1-17/32 = 15/32
BB calls 2 to win 6. b=3s/4. s=4b/3

3s(1-s)+ 15s²/32
3s-3s²+ 15s²/32
3s - 81s²/32
/ds=3 - 81s/16 and set to zero
s = 48/81 = 16/27
b = 4/9

min-raise variation after BB's raise option.
{SB} = 3s-81s²/32= 3*16/27 - 81/32*16/27*16/27 =
16/9 - 8/9 = 8/9
net{SB} = -1/9

Line vector strategies are
SB raises =< 0.592 else folds
BB calls =< 4/9
BB reraises =< 1/9
BB bluff raises [0.571,0.592]
SB calls =< 0.21

The geometric expansion of the pot seem to favor BB.
SB can play fewer hands while min-raising than when
she limps. SB has a small negative EV by min-raising,
while limping was an even game with zero EV.

Game 2. 2-player 2bet game

Sara is the SB, posts 1 chip.
Bob is the BB, posts 2 chips.
Each is dealt a card from uniform [0,1].
Low values are best.

Sara may fold, limp, min-raise, or raise 3XBB.
Bob may fold, call, or raise the pot.

2a. Limp variation.

First bet.

Sara risks one chip to win three. Therefore she
should call 3/4 of the time.

Solve by calculus.

SB EV payoff matrix
Used braces for {SB} SB EV
"<" causes confusion in HTML.
0*(1-s)*1 SB folds
3*s*(1-s) SB limps & wins
1*s² SB limps & ties, return blind

3s(1-s)+s²
3s-3s²+s²
3s-2s²
/ds=3-4s and set to zero
s=3/4

{SB}=3s-2s²=3*3/4-2*3/4*3/4=9/4-9/8= 9/8
net{SB}= 1/8

Second bet.

Was only able to determine Bob's bet range by trial and
error. Bob bluff frequency was 1/2 of his value bet
frequency. Bob's value bet value frequency was 1/4
of Sara's limp frequency.
This game replaced the split pot area and part of the SB
limps/wins area. The new payoff for the split area and
for the overlap area is 1.

After SB limps, BB may bet 4.
0*(1-s) SB folds
3*s(1-9s/8) SB limps and wins
1*s*s/8 + 1*s² Bob's bet region

3s - 27s²/8 + 9s²/8
3s - 18s²/8
1st dev set to zero
/ds=3 - 9s/2
s = 2/3

3s - 9s²/4
{SB}
2-1=1

The line vector strategies are
SB limps =< 2/3
BB bets =< 1/6
BB bluffs [2/3,3/4]
SB calls =< 1/3

Net {SB} = 0

Bet the turn flush threat

2 Street Toy Game

New 2 street clairvoyant game. Only one card is dealt to the
clairvoyant(Clare). A quarter of the time it says winner. The
rest of the time it says loser. Dean is the defender. Each player
antes one chip. Clare is dealt her card.

1st street. Clare may bet pot or check. If checked proceed to
the 2nd street A. If Clare bets, Dean may call or fold. Dean
folds, Clare wins one chip. Dean calls go to 2nd street B.

2nd street.
A. If 1st street is checked, Clare draws a new card. Now Clare may
bet pot(2 chips) or check. Check means Clare holds loser and has
conceded. If Clare bets, Dean may call or fold. Dean folds,
Clare wins one chip. Dean calls, there's a showdown.

B. If 1st street was bet/call, Clare retains her original card.

Again Clare may bet pot(6 chips) or check. Check means Clare holds
loser and has conceded. If Clare bets, Dean may call or fold.
Dean folds, Clare wins three chips. Dean calls, there's a showdown.

Questions
1. Should Clare ever bluff on street one?
2. If Q1 is yes, what is the value of the game and what is the
optimal strategy for Clare and Dean?

Solving the questions.

First solve 2A. This is a one street game. Clare bets all winners
and a portion of her losers. Dean calls or folds.

Clare bets, Dean folds. Clare wins 1*1/4 = 0.25
Clare checks. Clare loses 1*3/4 = -0.75
Clare bets, Dean calls. Clare wins 3*1/4 = 0.75
Clare bluffs, Dean calls. Clare loses 3*3/4 = -2.25

Clare\Dean___folds___|__calls
bet/check____-0.5____|____0
bet/bluff____+1______|___-1.5

Dean folds x part of the time and calls (1-x).
-.5x + 0 = x - 1.5(1-x)
0 = 1.5x - 1.5 + 1.5x
1.5 = 3x
.5 = x
Dean calls and folds 0.5

Clare checks losers y part of the time and bluffs (1-y)
-.5y + 1 - y = 0 - 1.5(1-y)
1 = 1.5y - 1.5 + 1.5y
2.5 = 3y
5/6 = y
Clare checks 5/6 losers and bluffs 1/6 losers.

Calculate the value of each cell for the value of the line.
-.5(5/6)(1/2) + 0 + (1/6).5 - 1.5(1/6)(.5) =
-5/24 + 0 + 1/12 - 3/24 =
-6/24 = -1/4
This game is -.25 for Clare.
**********
Clare bets all winners .25. Clare bluffs 1/6 of losers.
(1/6).75 or 1/8. Dean calls half the time when Clare bets.
Clare bets .375 and Dean calls .1875.

2B. Clare bets, Dean calls. Clare enters this game with
2/3 winners. Solve this matrix. Dean should fold 100%
of the time and Clare bets 100% of all hands. Clare's EV
is 3 for this game.

Play the game in series.
.1875 Clare bets/Dean folds. Clare's EV = 1
.1875 Clare bets/Dean calls. Clare's EV = 3
.625 Clare checks. Clare's EV = -.25

Clare's total EV
.1875 + 3(.1875) - .25(.625) =
.75 - .15625 = .59375

Only in this game Dean can do better by folding to every
bet. Clare's new EV
.375 - .15625 = .21875

Clare must overbluff to force Dean to call. Clare wants
the street one game to call an EV of 1. Then Dean would
be indifferent to calling or folding. Clare needs to bet
the correct ratio of losers to winners.
Clare is betting her winners which is 25% of her hands.
Clare is betting 1/6 of her losers which is 1/8 of her
hands. She must bet at least 1/3 of her losers before
Dean needs to ever call.

Clare bets, Dean folds. Clare wins pot*(1-p)
Clare checks. Clare loses 3*p
Clare bets, Dean calls. Clare wins (pot+bet)*(1-p)
Clare bluffs, Dean calls. Clare loses (pot+bet)*p

Set Clare's bet/check line to 1
Dean folds 50% and calls 50%
pot = 6 and bet = 6
((pot)p-pot/2)/2 + ((pot+bet)p-pot/2) = 1
6p-3 + 12p-3 = 2
18p = 8
p = 4/9
Equilibria is at Clare betting 4/9 of losers.
Clare bluffs five losers for every four winners.
1/4 * 5/4 = 5/16
Clare is betting 1/4 + 5/16 or 9/16 of all hands.

Q2 answers.

*Line strategies.
Street one.
Clare bets 100% winners.
Clare bluffs 4/9 of losers
Dean calls/folds 50/50.
Street two.
When Dean calls st1.
Clare bets 100% winners
Clare bluffs 40% losers.
Dean calls/folds 50/50
Clare's EV=1
When Clare checks st1.
Clare bets 100% winners
Clare bluffs 1/6 of losers
Dean c/f 50/50
Clare's EV is -0.25

*Clare's EV for the game.
9/16(1) - 0.25(7/16)
9/16 - 7/64 = (36-7)/64 = 29/64
= 0.453125
*****
Golden ratio!?

The clairvoyant maximizes her EV at five bluffs for every four
winners. This ratio holds for clairvoyant dealt 1/6 to 4/9
winners. If the clairvoyant is dealt greater than 4/9 winners,
she should bet every hand and the defender should fold every
hand.
How can Clare benefit from this information? If Dean thinks
Clare has 4/9 winners or better, Dean will always fold. Can
Clare ever convince Dean she has 4/9 winners?
**
Two player game. Dean opens the pot. Clare calls.
The flop is two in one suit and the third card in another suit.
This flop occurs about 55% of the time. Then Clare checks and
calls after Dean makes a continuation bet. This creates the
illusion that Clare is likely to be drawing to a flush. The
turn card is the third card in the suit, about 22% or about
13% for the joint conditions. The passive player, Clare,
should bet the size of the pot on the turn. Clare must be
drawing to a flush at least 4/9 of the time else why would
she call the flop bet.

There are additional conditions. Dean must be a competent
player capable of recognizing Clare's flush threat and being
able to fold hands drawing dead to the flush. The bet must be
at least the size of the pot, not less. Both players must have
chip stacks at least four times the size of the current pot.
The math says Dean should always fold if and only if subject
to a second pot size bet or greater. On smaller bets Dean can
call pot/(pot+bet) part of the time. On the flop the bet/call
is a necessary condition. Absent the bet/call Clare would
only have a 5% chance of drawing to a flush.

When the conditions are perfect for the illusion the two players
flipflop roles. In a two player game where there is a bet on the
flop, the passive player should become aggressive and always bet
when the third flush card appears on the turn. The aggressor
drawing dead to a flush should always fold to a turn bet.
Clare's expected value for this play is the entire pot.

NEVER CALL

When faced with a pot sized bet on the turn defender
should never call unless one of the following
conditions are met.
The hand being represented by aggressor is worst than
defender's hand or defender is drawing live to that
hand. Don't bother drawing to 4 or 5 outs.
Or.
The aggressor is a known high frequency bluffer likely
to be bluffing more than 5/9 of the time.
In most cases just folding to large turn bets is
probably best strategy.

Critical Points.

1. One street.
2/3 Clairvoyant has 100% equity.
1/3 Clare has 50% equity.

2. Two streets.
4/9 Clare has 100% equity.
2/9 Clare has 50% equity.

3. Three streets.
8/27 Clare has 100% equity.
4/27 Clare has 50% equity.


jogs