UBERAGGRO

This is from part 4 page 14 of Endgame by Ferguson.

We see that Player I gets a definite advantage if
he is allowed to split his betting over two rounds rather
than betting the entire amount in one round.

Ferguson uses round. Street may be the more appropriate
term. Most books speak of betting in terms of the size
of the pot. To apply maximum pressure on opponents think
of betting in terms of the size of the smaller stack as
multiples of the pot. Consider 3 to 5 multiples of the
pot. Don't go all in on the turn. Bet 1/4 to 1/3 of
the smaller stack. This will force opponent into making
two fold or call decisions.
Does toy games apply to real poker? The turn is a non
threat card. Will your bet be believable? Will it exert
pressure on opponent? The flop bet is often a continuation
bet. Opener doesn't know why he's been called. He does know
defender called for a reason. So maybe the turn card doesn't
need to be a threat. The flop call by defender was threat
enough. Any turn bet is a threat.
The turn reduces poker to a two street game. On the turn
Clare bets all her winners. When using pot size bets she
also bets 5 losers to every 4 winners. Using half pot size
bets she bets 7 losers to every 9 winners. Either way it's
correct to bet nearly one loser to every winner.
Play úberaggro. Bet the turn more often. Try to be
Clare, not Dean.
Add more streets to the games. Each street increases the
clairvoyant's advantage.

Let's review the three street game. On the first street
Clare bets 19 losers for every 8 winners when making pot
size bets. With half pot bets it's betting 37 losers to
every 27 winners.

Here's a situation which happens frequently.
Hero has K:h: K:s:, opens for 3XBB. All fold to big blind
who calls.
Flop: A:d: 9:s: 3:d:
BB makes a donk bet of a little over half the pot.
What should our hero do? How many hundreds of times has
this hand been misplayed on TV. How many billions of times
has this hand been misplayed online?

If P > P_0 , then
(i) the value is V = a
(ii) it is optimal for Player II to fold on the first round, and
(iii) it is optimal for Player I to bet on the first round,
and to bet w.p. (P/(1-P))(b2/(2a+2b1 + b2))
(or w.p. 1 if this is greater than 1) on the second round.

If P =< P0, then all strategies are active,
(i) the value is V = a(2P - P_0)/P_0
(ii) it is optimal for Player II to fold on the first round
w.p. b1/(2a + b1),
and to fold on the second round w.p. b2/(2a + 2b1 + b2), and
(iii) with a winning card, Player I always bets; with a
losing card, he bets on the first round
w.p. P/(1 - P)*(1 - P_0)/P_0,
and on the second round w.p.
b2(2a + b1)/b2(2a + b1) + 2b1(a + b1 + b2).

If P > P_0 hero should always fold. In this case P_0 is
about 27/64. Does BB have aces or better at least 43% of
the time? If it's yes, our hero should fold. Against an
random unknown opponent hero's correct default action is
to fold. Does anyone really think donk bettors are bluffing
over 43% of the time? There's no need to agonize.
Just fold those kings.

That is the conclusion of the toy game in Ferguson's paper.

Bettor's Advantage

The math shows that poker is a bettor's game. The passive
defender is at a huge disadvantage.
The showdown game. Clare is dealt one card. It says
winner or loser. Half the cards are marked winner and the
other half are marked loser. She is playing against Dean.
This is a fair game. Clare wins 50% of the time.
Add a street of betting to make this a clairvoyant game.
Only Clare has the opportunity to bet. Clare can bet or
give up. Dean wins if Clare gives up. If Clare bets, Dean
can fold or call. With pot size bets Clare needs 1/3 winners
for this to be an even game. With half pot size bets she
needs 3/8 winners. Either way the bet favors Clare, the
clairvoyant.

Review the entire game. The pot starts as one chip. It
is treated as strange money. Clare is dealt one card
which is marked winner or loser. The winner's frequency
is known to both players. There are three streets of
betting. On each street Clare may check or bet the pot.
Dean can fold or call. Dean cannot benefit from betting
as only Clare knows who holds the winner. Clare always
bets the winner and bets a portion of her losers. Dean
will call 50% on each street.

Clare is dealt 16% winners.

Street one.

Clare should bet 19 losers to 8 winners. Clare bets 16
winners and 38 losers. She gives up on the remaining 46.
Dean calls half the time.

0 * 46 no net change in EV for Clare.
1 * 27 Clare wins one chip
The remaining 27 cases go to street two. The pot is now
3 chips. Clare has 1 chip invested.

Street two.

Clare bets 5 losers to 4 winners. Clare bets her remaining
8 winners and 10 of her losers. She gives up on the
remaining 9. Dean calls half the time.

-1 * 9 Clare gives up on this street.
2 * 9 Dean folds, Clare wins 2 chips.
The remaining 9 cases go to street three. The pot is
now 9 chips. Clare has 4 chips invested.

Street three.

Clare bets 2 losers to 4 winners. She gives up on the
remaining 3. Dean calls half the time.

-4 * 3 Clare gives up on this street.
5 * 3 Dean folds, Clare wins 5 chips.
-13 * 1 Dean calls and wins.
14 * 2 Dean calls and Clare wins.

Summary

Clare was dealt 16% winners. Yet she won 41 pots out of 100.

0 * 46 = 0
1 * 27 = 27
-1 * 9 = -9
2 * 9 = 18
-4 * 3 = -12
5 * 3 = 15
-13 * 1 = -13
14 * 2 = 28

Clare won a gross of 54 chips on 100 plays. She contributed
50 chips in antes. Her net winnings was 4 chips. With
three streets of betting Clare was able to convert 16%
winners to 4% favorite.

Uniform [0,1] game.

Now examine the uniform [0,1] game. Clare and Dean each
ante one chip. Clare and Dean are each dealt one number
from uniform [0,1]. Only Clare is allowed to bet. She
may check or bet the pot, two chips. Dean can fold or call.
Best number wins on showdown. With no betting the game is
obviously equal. With betting Clare bets all her winners.
Clare should also bet bet/(pot + bet) losers to every
winner. With pot size bets Clare is a 1/9 chip favorite.
With half pot size bets Clare is a 1/10 chip favorite.

STOP CALLING

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.
Clare bluffs one for every two winners. When dealt 2/3
winners Clare bets every hand and Dean should always fold.
1/2 Clare wins 1/2 pot per play.
1/3 Fair and equal game.

2. Two streets.
4/9 Clare has 100% equity.
Clare bluffs 5 for every 4 winners.
1/3 Clare wins 1/2 pot per play.
2/9 Fair and equal game.

3. Three streets.
8/27 Clare has 100% equity.
On the flop Clare should be betting 19 losers for every
8 winners.
2/9 Clare wins 1/2 pot per play
4/27 Fair and equal game.
In an equal game Clare has 46 losers for every 8 winners.
On street one she bets 19 losers for every 8 winners.
She gives on 27 losers. On street two she bets 10 losers
to her 8 winners. On street three she bets 4 losers to
her 8 winners. This overbluffing on earlier streets
is the optimal game theory strategy.

Critical Points for half pot bets.

1. One street.
3/4 Clairvoyant has 100% equity.
9/16 Clare wins 1/2 pot per play.
3/8 Fair and equal game.

2. Two streets.
9/16 Clare has 100% equity.
27/64 Clare wins 1/2 pot per play.
9/32 Fair and equal game.

3. Three streets.
27/64 Clare has 100% equity.
81/256 Clare wins 1/2 pot per play.
27/128 Fair and equal game.

Critical points for quarter pot bets.

3. Three streets.
64/125 Clare has 100% equity.
48/125 Clare wins 1/2 pot per play.
32/125 Fair and equal game.

CONCLUSIONS

When faced with a bet, the defender has three options.

A. Always call.
B. Always fold.
C. Call pot/(pot+bet) part of the time where pot is
the original size of the pot.

For years while playing limit lowball (C) seemed like
the best default action. The pots were big and the
bets were small.
Multiple streets favor the aggressor. Even in fixed
limit hold'em calling for three streets is a huge
disadvantage. The aggressor can check and give up
on some of her bluffs. She will bet all her winners
every street.
Example. 10 small bets in the pot. The defender is
faced with a small bet on the flop. Figures it will
costs two more big bets to call down. The naive pot
odds approach says 5 bets to win 15 bets. A 25% chance
of winning should be sufficient for calling down.
This neglects that the aggressor bets all the winners
three times and will check many of the losers. The
clairvoyant three street method says the defender needs
31% equity to call with plus EV and 63% winners for call
down to be equal EV.
In nolimit when facing two or more streets of betting
the defender no longer has the luxury to call. It's
too expensive.
Here's the numbers for a two person pot with two streets
and half pot bets. Defender should call pot/(pot+bet)
part of the time. That's 2/3 in this case. Providing
the aggressor is bluffing frequently. She needs to
bluff seven times for every nine winners. Are very many
players bluffing that often? Probably not. If her bluffing
ratio is less, he should always fold.
Defender must have a good reason to call. Lacking a good
reason the best default action is (B), always fold.

Game 2. RECAP

Summary

Blinds are 1/2
SB folds or opens amount of column one.
BB folds, calls, or raises the pot.

__________________1st bet____________2nd bet
__________________s_________EV________ s________ EV
limp ................. 3/4 ......... 1/8 ......... 2/3 ......... 0
min-raise ........... 2/3 .......... 0 ......... 16/27 ....... -1/9
3XBB ................ 9/16 ....... -5/32 ........ 1/2 ........ -1/4
4XBB ............... 12/25 ....... -7/25 ...... 32/75 ....... -9/25
8XBB .........................................._ 0.2634 ..... -0.6049
16XBB ........................................_ 0.1476 ..... -0.7785


The EV in the chart is net EV. Any EV over -1 means
SB should open-raise whenever her card is equal to or
less than the s value on the chart. Low values are
stronger than high values.

It was surprising to be able to solve the net SB EV
for the two bet game without solving the one bet game.

As the opening raise size increases the percentage
of hands playable deceases. Also the expected EV
deceases. The EV approaches -1 but never reaches
-1, therefore any raise size is playable.
At 16XBB SB needs a card under .1476, AA is .14796
against a random hand. Therefore no starting two
hole cards is worth opening for 16BB according to
this chart.

These results seem to imply preflop when entering
a pot optimal strategy is being extremely selective
and enter very few pots.

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

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.

Set on the Flop

Sets on the flop are tricky to play. The correct amount of aggression
is board dependent.
On dry boards the set is very strong. A set against top pair is over
98% favorite. It's safe to bet small and attempt to string opp along.
You: 6s6h
Flop: 9s6c4d
Opp with As9h is way, way behind. Needs runner/runner to beat you.
As boards become more coordinated they become more dangerous.
Sets should fear flush and straight draws. Plan a betting
scheme which denies opp both pot odds and implied odds. On
these types of board sets want to be all in by the turn.
The river betting favors the chaser.

tiptoeing thru the turn

There are threads stating that many successful fixed limit
players have problems winning at no limit. No particular
reasons were given. The difference in turn play could be
the reason. In SS2 there's almost no space devoted to
turn play. The short turn section only discuss odds for
chasing. Nothing on way ahead/way behind situations.

Opening raiser misses the flop. She makes the continuation
bet. The defender calls. There's a void in the literature.
What should she do next? Why did he call? Occasionally
it's a big hand. Usually it's a pair or a draw. Should
she follow up with a turn bet? That's the mystery. Some
defenders call the flop with second pair and then fold on
the turn. Others call all three streets. Some call with
draws once and others twice. How do you know one from the
other?

In FL raiser has TPTK on the flop. She bets the flop and
is called by the defender. The turn is a blank. She bets
and is raised. In FL she can now be stubborn and call this
turn raise and a river bet. This will costs her two big
bets. In NL the defender can raise much more. He can bet
the pot or more on the river. In FL she is getting odds
to call. In NL she is giving odds to call. In a SnG the
stacks are usually small. Top pair makes one pot-committed.
Against aggression TPTK is harder to play in NL than FL
or a SnG.

Example A. She: As Qh
Flop: Qs Td 7c
She bets and he calls.
Turn: 5:c:
This card is very unlikely to improve defender's hand.
She bets. He raises. What can he have? In NL she has
to wonder why he called. In FL she doesn't care. She
knows it will only be two big bets to find out. In NL
it may costs the entire stack to find out.

The books say open raise first in. Then make a c-bet. But
what if the opponent calls? What does he have? What if
you have missed the flop and have nothing? What do you
do next?

That is the next phase of the game to investigate.

Cold calling

The current thinking on cold calling says you must hold
the top half of raiser's range. I believe they are all
wrong. They're focusing on the wrong statistic. The
critical statistic is raiser's continuation bet percentage.
When raiser rarely makes a continuation bet, he is playing
like an open book. It's possible to call with a vast
range of hands. Even greater than raiser's range. When
the raiser c-bets 100%, never call. It's too dangerous.
These pots always become larger pots than the other hands.
Also larger pots with you having at most a marginal EV
advantage.
All these pokerstove calculations are a waste of time.
The perceived ranges can't be trusted. The vector
doesn't include raiser's c-bet%. It doesn't measure
implied odds. It just gives the ex-showdown values.
There's no accounting for betting. None for future
chips at risk.
In the future a new vector in orthogonal space will be
developed. This vector include raiser's c-bet%, implied
odds and all other statistics which has measurable
effect on $EV.

MOP19.3. The pot size case.

The problem on 19.3 is resolved using geometric growth with
pot size bets.
Clare, the clairvoyant, will be dealt one card. If it's
an ace or king she wins, else she loses. Dean will be the
defender. There are three streets of betting. Clare may
bet or give up. Dean may call or fold. Both will ante
five units. Clare's bets will be pot size.

Question.
If both Clare and Dean are playing optimally, what's the
value of the game?

Answer.
Clare has a +EV of 0.19231 units.

The ex-showdown value of the game is -3.46. This is from
MOP 19.3. The game is solved by linking three matrices in
series.
First the three matrices are solved separately. On street
one Clare bluffs about 9% of her losers. If this were a
one street game, Dean would call 50% of the time. Since
this is a three street game, Dean can improve on his
-0.19231 EV by just folding to every bet. The game value
would be 5*(.76923)-5*(.23077)=2.6923. That's 2.6923
in Dean's favor.
To force Dean to call more often Clare must overbluff
street one. To produce +0.19231 EV Clare must bet so
that if Dean folds to every bet, she would still get
her +0.19231.

5*(x-(1-x))=0.19231
2x-1=0.19231/5
2x=1+.038462
x=.519231

Plug these numbers in and Clare is betting about 40% of
her losers instead of 9%.
On street two Clare had to overbluff again. She bets 35%
of her remaining losers instead of 21% suggesting by the
one street matrix.
By street three the game has been reduced to a one street
game. Clare bets losers at the suggested optimal strategy
frequency.
Clare overbluffs by over 300% on street one. This suggests
in a real game where Clare starts with 50-70% winners the
100% continuation bet may be viable. She only overbluffs
about 60% on street two.
Dean calls at the 50%(pot/{pot/bet}) frequency thruout all
three streets.

geogrowth overbluff
S1 Cgives 0.48077 __-5 ____-2.40385
S1 Dfolds 0.25962 __ 5 ____ 1.29808
S2 Cgives 0.11857 _-15 ____-1.77862
S2 Dfolds 0.07052 __15 _____1.05781
S3 matrix 0.07052 __28.63 __2.01888
____________________________0.19231
On street one Clare gives up 48% of the time. She loses
5 units on those. Dean folds about 26% of the time.
The street three matrix is played out as a one street
matrix game.

19.3 Game Implications.

On this static model the defender starts with nearly a 85%
probability of having the best hand and still was only
able to call three bets less than 10% of the time. With
barely over 15% winning cards the clairvoyant wins over
40% of the pots.
In dynamic models of real poker hands the defender usually
has less than 50% chance unless she hits the flop big.
I believe even in the dynamic case pot/(pot+bet) calling
frequency is still correct. Provided calling is still
a viable option. Often a new board card is so detrimental
to defender, his linear hand strength(LHS) becomes too low
to consider calling.

His best response is to call if and only
if F(y) > B/(2B + 2).

Where B is the bet and 2 is the pot.
This is from 5.1 pp13-14. Case 1 on page 14.
This is a paper by Thomas Ferguson.
www.math.ucla.edu/~tom/papers/poker1.pdf
The raiser's distribution is often stronger than
the defender's. Often it's the raiser always bets and
the defender always folds.
For the defender it becomes fold when LHS is too low.
Certain hands are 'clear' calls. There's an inbetween
region where defender should sometimes call and other
times fold. Call with pot/(pot+bet) frequency with these
hands. Unlike the [0,1] toy games defender shouldn't
always choose the hands with the highest LHS. LHS only
measures probability of finishing with the best hand at
showdown. He favors drawing hands with potential to be
betting or raising hands. There may be some other vector
which measures chip winning potential better. Call with
the highest pot/(pot+bet) on that chip vector.
The defender is at such a great disadvantage that it may
be wrong to ever cold call an opening raise. He will be
facing a c-bet by opener raiser over 50% of the time and
will get a piece of the flop at most 35% of the time.
There just doesn't seem to be any percentage in calling.

MOP chapter 19

jerrod said,

"Yes, that's right. Here's what happens. (Ignore checking
hands - assume that checking on any street is tantamount
to giving up)

Suppose we have street 3. We know what the bet size is
going to be on street 3, so Y ends up with a distribution
of all his value bets and exactly enough bluffs to make
his bluffing ratio on the river correct. Call his value
bets V and his bluffs B = (alpha)V for the appropriate
alpha = 1/(p+1). If the bets on the first two streets
are both $60 and the initial pot is $10, then we have
a pot size of $250 and a bet size of $60, so alpha is
60/(250+60) = 6/31.

So if we have V value bets, we need 6/31 bluffs. So the
total number of hands we need in order to play the river
correctly is (37/31)V - 31/31 of which are value bets
and 6/31 of which are bluffs.

Now let's look at the 2nd street. The trick here is to
use as your value bets the (37/31)V number and not V
itself. So here we have a pot of $130, a bet of $60,
and 37/31V value bets. alpha is 60/190 = 6/19. So now
we need (6/19)(37/31)V hands to bluff with on the 2nd
street (these hands will be given up on the river). So
we have V(222/589) hands that are 2nd street bluffs only.

Now our total number of hands to play the 2nd street
properly is 222V/589 + 37/31V, or 925V/589.

Now on the first street, we have an alpha of 60/(60+10)
= 6/7. So 6/7 of the 2nd street hands will be bluffs
on the first street. So that's (6/7)(925/589)V that are
bluffs on the first street.

I attached a little Excel file to this post - it shows
the relative frequencies of each strategy for X, and the
value of each of Y's strategies. You can see that using
this method, Y is indifferent between calling any number
of streets and folding immediately, and the value of any
strategy for him is -$19.17.

Y1=(14.29%)($60)+(9.77%)($60)+(7.88%)($60)=$19.16
After weeks still can't see the -$19.17. What's the
payoff for the rest of the space?
Y is guaranteed -$3.46. Y only needs to bet AceKing
and give up all others. Any strategy which produces
less than -$3.46 is dominated.
Didn't understand ($10). Y antes $5. If he loses his
ante, that's only ($5).
X is guaranteed -$5. Just fold every hand on street one.
Therefore the value of this game with both playing optimal
strategy is bounded by -$3.46 for Y and -$5 for X.

To solve this game this matrix was constructed.
Values are net payoffs for Y.

.....\....................X
__Y__\___fold____foldS2___foldS3___call
__V___|____5_____65____125____185
_BBB__|____5_____65____125___-185
_BBCh_|____5_____65___-125___-125
_BCh__|____5____-65____-65____-65
__Ch__|___-5_____-5_____-5_____-5

For each street the ratio of AKs/others(f) were known.
The value betting line was combined with the others.

_VBBB_|____5_____65____125___185(2f3-1)
_VBBCh|____5_____65____125(2f2-1)
_VBCh
__VCh_


fixed bet in series
S1 Y gives 0.71429 ___-5 ___-3.57143
S1 X folds 0.24490 ____5 ____1.22449
S2 X folds 0.04082 ___65 ____2.65306
Y expected value ____________0.30612

$0.30612.
Wasn't able to produce a plus value by over-bluffing method.
Decided to solve geometric growth game.

2. Geometric growth. Y bets 11.66, 38.86, and 129.48.
X calls 10/(10+11.66) of the time on each street.
This was the solution for three one street games in
series.



Finally settled on this tree structure as representative
of the game.



Fewer area values to solve. Fewer roundoff errors.


geometric growth in series
S1 Y gives 0.76333 ___-5 _____-3.81666
S1 X folds 0.12741 ____5 ______0.63704
S2 X folds 0.05882 ___16.66 ___0.97998
S3 matrix 0.05044 ____55.52 ___2.80036
Y expected value ______________0.60073

Y's win frequency for each street.
street one: 2/13
street two: 7/13
street three: 65%

X could improve his results by folding to all bets on
street one. Therefore Y needs to bet over 50% of
his hands on street one.
5y - (1-y)5 > .60073
y > .560073
56.0073% hands to be precise. Y has given up on
44% of his hands.
Y also needs to bet over 50% on street two.
16.66y - (1-y)16.66 > .60073 + 2.19964 - 1.50755
y > .5388
53.88%(30.174% of the original) the remaining hands
are bet on street two. Y gives up on another 25.8%
of his hands.
Only street three was treated as a one street
matrix game.


geometric growth over-bluffing
S1 Y gives 0.43993 ___-5 _____-2.19964
S1 Xfolds 0.30151 _____5 ______1.50755
S2 Ygives 0.11925 ___-16.66 __-1.98682
S2 Xfolds 0.07500 ____16.66 ___1.24954
S3 matrix 0.06432 ____31.56 ___2.03009
Y expected value ______________0.60073

On each street Y gives up a portion of his hands.
Y's win frequency and remaining hands for each street.
street one: 2/13, 100%
street two: 27.5%, 56%
street three: 51%, 30.2%

Just like the two street games, both the in series
method and the over-bluffing method produced the
same results. The $0.60 for Y is within the
bounds of -$3.46 and $5.
Y overbluffed both on streets one and two. Nearly three
times as many bluffs as value bets on street one. Nearly
as many bluffs as value bets on street two.
Y started with 2/13 winning frequency. X knows Y is
overbluffing on streets one and two. Still X is not
able to hold Y under parity.

<>Only Y, the clairvoyant, is allowed to bet in this
toy game. Dealt only about 15% winners, Y was able to
reverse -$3.46 equity into +$0.60 equity. This shows
the advantage of the bettor over the caller. This game
demonstrates the power of the continuation bet. Even
after being called, it is correct to frequently fire
more continuation bets. Only on the final street of
betting does Y revert to bluffing at the low frequency
of the one street game.

MOP Chap 19 simplified

X and Y start with stacks of 125 and each ante 5. Two streets of betting. Y is dealt a card marked winner or loser. 20% chance of the card is marked winner. Y wins the showdown if he holds the card marked winner.
Game A. Y bets 60 or gives up on each street. X may call or fold to a bet. On street one if Y gives up or X folds, game is over. If Y bets and X calls, street two is played.
Game B. Same as game A, accept Y bets 20 on street one and 100 on street two.

Matrix M
.... \ ...........X
....Y \_____call________fold
____VB___ f(S+b)-(1-f)b-S/2 ___S/2
__VCh_______f(S+b)-S/2______fS-S/2

V - Y value bets winner
B - Y bluffs loser
Ch - Y gives loser
f - 20% chance Y holds winner
(1-f) - chance Y holds loser
S - size of pot
b - size of bet
x - X calls
y - Y bluffs

Matrix N
----------Xcall----Xfold
Yvalue
Ybluff
Ycheck

Used matrix M to solve the strategy of the game and matrix N to calculate the value of the game.

A.
..\.............. X
Y \_____call______fold__
VB______-39________5__
VCh_______9_______-3

x = 1/7, (1-x) = 6/7
y = .2143, (1-y) = .7857

Y gives up... 62.86%
X folds value 17.14%
X folds bluff 14.69%
Game values
Y checks -3.1429
X folds 1.5918
GV at end of str1 -1.2857
game continues to street two
value 0.02857
bluff 0.02449
total 0.05316
ratio winners/total 7/13

.........\............ X
__Y__\_____call________fold__
__VB______9.615_______65__
__VCh____38.308________5

x = 13/19, (1-x)= 6/19
y = .36848, (1-y) = .63152
matrix value 27.11 times .05316 = 1.4386

Added to Ychecks and Xfolds of str1
Game value -0.1124

Game B.
..\........... X
Y \_____call________fold__
VB_______-1.5________.5__
VCh________.1_______-.3

x = 1/3, (1-x) = 2/3
y = 1/6, (1-y) = 5/6
Y gives up... 2/3
X folds value 4/30
X folds bluff 8/90
Game values
Y checks -3 1/3
X folds 1 1/9
GV at end of str1 -1.6667

.....\.......... X
_Y_\_____call___fold__
VB______25______25__
VCh_____65_______5

x = 1/3, (1-x) = 2/3
y is 100%
matrix value 25 times 1/9 = 2 7/9
Added to Ychecks and Xfolds of str1
Game value 5/9

Whose Math?

2 player game. Each ante 10 chips. Each dealt a number
from uniform [0,1]. No betting. Just a showdown.
It is intuitively obvious the EV for each player is 10.

2 player game. Abe and Dan, each ante 10 chips. Each
dealt a number from uniform [0,1]. Abe may bet 10 chips
or check. Dan may call or fold. This is the TWC game.
Abe's EV is 11.

What if Dan greatly undercalls. Dan is known to call at .2.
Abe will maximize his value betting point. It's at .1.
That gives Abe a EV of 10.1.
Abe just bet all hands. Abe's EV is 14.8
Abe maximizes both his value betting and bluffing. Abe
should bet less than .1 and greater than .3.
Abe's EV is now 15.

The big EV gain was from betting all hands. It improved
Abe's EV from 10.1 to 14.8. Abe's optimizing only improved
his EV another 0.2. In poker the big EV gains are from
recognizing opp's style and exploiting it.

Same tests are run with Dan marginally undercalling. Dan
should call with .6 or better. Here Dan will call with .5
Abe bets all. Abe's EV is 10.
Abe maximizes. Abe's EV is 11.25

This Dan calls with .4
Abe bets all. Abe's EV is 11.2
Abe maximizes. Abe's EV is 12.

Players miss the flop 60-70% of the time. Only make top
pair of better 12-15% of the time. Unless a player is
willing to call with no pair, he will be calling 40% or
fewer. This means Abe should be overbetting. Since Dan
rates to undercall.
Recognizing opp's tendencies is the coarse adjustment.
Maximizing the exact betting and bluffing points is the
fine tuning. Coarse adjustment usually yields more than
fine tuning. The structure of NL forces players to play
a high risk game in order to be successful.

TWC 2 bet game.

TWC 2player 2bet game
Two players each ante one chip.
Bets are one chip each.
Two bets of one chip each.
Check/raise allowed.
Abe and Dan are each dealt card from [0,1].
Abe acts first.
Find optimal strategy for each and calculate
value of the game for Abe.

Abe is now allowed check/raise. Dan is no longer betting
with impunity from the last position. From alley 2b the
vector space near zero will now be checked by Abe. Abe
plans to raise if Dan bets. In alley 2b some of the areas
of the intersections with Abe calling will need to be
recomputed.



The figure on the left is the line strategies of Abe and
Dan. The figure on the right is Abe's C/R space rotated
90 degrees from the y-axis to the x-axis. The colored
payoffs are the EVs for Abe.

Most EV changes occurs by varying Abe's bet point during
alley 1 and Dan's bet point on alley 2a. The threat of
the C/R forced Dan to bet about 25-30% fewer hands.
After all the changes back and forth Abe was able to
increase his EV from -3.5% to nearly -2.7%.

This is fine tuning the strategy.

Small incremental gains

This is the chart of Abe's EV in the full two alley game.
To rid the chart of negative EV's the units were changed.
The pot starts with 20 chips. These chips are treated like
free money so both Abe and Dan will have positive EV.
In a fair game both Abe and Dan will have an EV of ten.
The bet and raises are fixed at ten chips.
The x-axis is the hands Abe bets originally. At x Abe
bets x and all hands better than x. The y-axis is Abe's
EV when both Abe and Dan plays optimally after the
first alley bet.
By playing perfectly Abe improves his EV only one
third of a chip. The standard deviation is 13 to 16 big
bets. Think about THIS. The sd is 130 to 160 chips per
100 hands. Optimal strategy improves Abe's results by
1/3 of a chip. This is really a small incremental gain
for optimal strategy.



This new chart shows calling strategies plotted against
optimal strategy. Notice that optimal strategy DOES NOT
dominate all other strategies.
On the river against a habitual bluffer always calling
dominates all other strategies, including optimal. Against
a passive non-bluffer always folding dominates all other
strategies, including optimal.
This chart also approximates EVs between streets. Abe faces
a turn raise or check/raise. With TPTK to trips Abe is way
ahead or way behind. He doesn't know which. If way behind
Abe should fold. If way ahead Abe should raise. In fixed
limit optimal strategy is to call pot/(pot+bet). Online
poker utilities hints at the correct response. Look at opp's
aggression factor. High AF(over 2.5) raise. Low AF(under 1)
fold. Otherwise calldown. A incorrect calldown, calling
a passive opp unlikely to bluff, costs two bbs. Making one
of these mistakes every 100 hands can be the difference
between winning and breaking even.

This chart graphically shows that optimal strategy should
be reserved for unknown opponents. Unknown could be a
pro who is unpredictable or just a new opponent of unknown
style. In the future a best default strategy may be found.

TWC Approach Naive

After adding extensions to TWC it was naive to assume the original bet, check, call and fold points would remain unchanged. Abe was able to improve his EV by betting less aggressively. Each of the two extensions were solved separately. This method proved invalid. The two extensions are elaborately interconnected.
The two extensions will be combined and solved simultaneously. This may cause a problem. It may be difficult to fit all the equations onto to a single page in Excel. As soon as the problem is solved, a new post with a solution to the combined extensions will be provided.
Abe's disadvantage should be less than 8%.

DUH! Didn't notice. Abe can check blind. That returns the game to the original TWC. In that game the player who could only call or fold was only minus 5%. Therefore Abe is less than minus 5% or less.
Combined the two extensions. Abe was negative 3.5%. Abe did best by betting only 24% of his hands. This is more conservative than expected. In hold'em it may be right to be aggressive on the flop and turn. But be conservative preflop and on the river. Of course this is only a toy game. Best toy game strategy may not translate well into best hold'em strategy.

TWC Ext 2.

Terms

*alley -- MOP referred to this as a half-street. Since there can be as many as five every street, it is more appropriate to call this an alley. An alley would consists of one player's action and opponent's response. If opp's response requires opp one to further respond, that would be another alley.
*within a street -- this is several alley strategies within a street.
*between streets -- the interaction of strategies from one street to the next street.

TWC Ext 2.

Ext 2 will focus on the area where Abe has bet in TWC. Dan is now given the option to raise.



On the first pass Abe called raises whenever his bet was for value. Folding only when the bet was a bluff. This method did not yield a good EV return for Abe. It was better to fold some of Abe's value bets.

Second pass. Find the best call point. Tough to resolve this. Dan should call S/(S+b) in TWC. That would be 2/3 of the time. The known solution for calling is 0.6. S/(S+b) is 4/5 for the raised pot. That would suggests .24 for calling. Will try that number first.
Ran TWC with pot at 40 and bet at 10. That would be similar to raising the pot. Attacker should bet 37.5% of the space and bluff 20% of the bet space. 3/8 of .30 is .90/8 or 0.1125. Bluff area is 0.0225. Ran four test runs for Dan bluffing. Dan did best bluffing with run 2. It was the high end of the call space from alley 1. Dan's call space was tested from 0.0-0.3 to 0.0-0.9. Abe's EV remained constant throughout the entire range. As long as the high end of Dan's call space is within Abe's check space Dan's EV remained max. On alley 2 Dan is allowed to raise. When Dan bluffed, his EV was max whenever his bluff space laid within the intersection of Abe's check space and his call space from alley 1. Following test run 2 place this bluff space into the high end(weakest hands) of the call space. This assures maximization in case Abe uses an inferior strategy. Dan's EV improved by +5.2%. Add this to ext 1. Dan's EV is +8% when both extensions are included.

Conclusions. Bluff the bottom of the space. In extension 1 this was the bottom of the entire space. In extension 2 just give up with hands not worth calling. It's not worth two bets to bluff. Bluff with the bottom of the call space. This makes it a one bet bluff. This may give insight to playing between streets. Only make semi-bluff raises with hands worth calling.

Tom Weideman Challenge Extension

This challenge is for the specific case where each of two players ante one unit. Two players, called Abe and Dan, are each dealt a card from the line interval [0,1]. Abe is allowed to bet one unit or check. If Abe bets, Dan may call or fold. If Abe checks, Dan may bet or check. If Dan bets, Abe may call or fold. This is a two partial street game. In hold'em with check/raises there can be as many as six parts to a full street.

The two line strategy vectors are posted above. The one on the left is the TWC vector. The one on the right is the extension.

The solution for TWC was Abe should bet less than 0.3. Abe bluffs greater than 0.9. Greater than 0.3 and less than 0.9 Abe checks. Abe's EV is +0.1 units.
Dan's betting strategy in the next partial street is in the same ratios as Abe's in TWC. Dan bets 30% of Abe's 0.6 space. That's 18%. Add to the 0.3 where Dan has a known winner. Dan bets less than 0.48. Dan bluffs greater than 0.84. Abe calls 60% of his space. Abe calls less than 0.66 Abe's EV is -0.56.

Run 2. Dan's bluff space is shifted from 0.84 to 1 to 0.74 to 0.9. Dan's bet space and Abe's call space remains unchanged from run 1. Abe's new EV is -0.32.

Conclusions. Dan does better by bluffing his worst 16% of his hands. In this game Dan's positional advantage is 2.8%.

Simulation of A Three Player Game

Two player game
The attacker will be called Abe.
The defender is Dan.
Abe is dealt card from [0,1].
Dan is dealt 2 cards from [0,1].
Dan plays his better card.
Abe may bet or check.
Abe checks, showdown.
Dan may call or fold to a bet.
Dan is playing his best card from an unit square.

Dan is like two players. Abe is one player.
With no betting Abe should win 1/3 of the time.

The pot is S and the bet is b.



The above is the image of the Abe's payoff cube.
His strategy options are cut into three sections.
A is Abe betting for value. B is Abe checking.
C is Abe bluffing. Abe's payoffs are color
coded. The legion is in the image box.

On the first play of the game, the pot is 30 and the
bet is 10. Abe's EV is 10 with no betting.
Abe optimizes at .231 bet for value.
His EV is 11.622.
On the second play, the pot is 90 and the bet is 10.
This is similiar to a fixed limit game ratio. Abe's
EV is 30 with no betting.
Abe optimizes at .28 bet for value.
His EV is 32.304.

Part 5. GT: Attacker's Inequalities

	p	b
x = --	------	-----
	(1-p)	(S+b)

p = prob attacker has the best hand
S = size of pot
b = size of bet
This fraction is attacker's frequency for
choosing the bet/bet strategy line.

This fraction may be the secret to poker.

attacker's inequalities

5.1. pb/(1-p)(S+b) > 1
or
5.2. p/(1-p) > (S+b)/b

When attacker's inequality is true, the defender
should not call. The attacker is either
not bluffing or isn't bluffing often enough.

On the river p is either the probability the
attacker has made his hand or in many cases
the probability the attacker has made his
hand considering that he has bet.
Attacker's tendencies are unknown to defender. Defender
should call S/(S+b) part of the time. Attacker
rarely bluffs. Defender should not call. Attacker
overbluffs. Calling 100% of the time dominates
all mixed strategies. This confirms the correctness
of the crying call in fixed limit.

On earlier streets p is the probability the
attacker is in the lead. The p variable
is a function of the attacker's
aggressiveness. p is very high for passive
attackers. Passive players betting are very likely
to have top pair or better.
For aggressive players where a continuation bet on
a flop is nearly automatic after an opening raise p is very
low. Against these attackers bottom pair
may easily be in the lead. Defender must
call and raise much more liberally.
p is not an absolute constant. It's a relative
term. On any street defender has a fixed linear
hand strength against a random hand. But he has
a variable relative linear hand strength against
aggressor's hand range. Defender must still use
judgment to determine where he stands on a hand.
It will be dependent on the aggressiveness of the attacker.

Part 4. Game Theory Notes.

Chart and formulas are reposted for convenience.

S = the size of the pot.
b = the size of the bet.
p = probability attacker is dealt a higher card.
(1-p) = prob attacker misses.
x = part of time attacker plays row B/B.
y = part of time defender calls.

Now the 2 X 2 matrix looks like this.

______\_____defender
attacker	Call	Fold
Bet/Bet	p(S+b)-(1-p)b	S
Bet/Chk	p(S+b)-0	pS

x = pb/(1-p)(S+b)
y = S/(S+b)

Note 1. Optimal strategy is optimal like Pepsi Free is free.

Note 2. y=S/(S+b) suggests defender should call with high frequency. Look thru the algebra(in part 2)to calculate y. There was division by (1-p) on both sides of the equation.When p=1, (1-p)=0. Division by zero is undefined. If attacker never bluffs, defender should never call.
Note 3. x = pb/(1-p)(S+b) If this value of x>1, defender should never call.
p/(1-p) > (S+b)/b.
When attacker bluffs with too low a frequency, defender should never call. In fixed limit this would be close to rarely bluffing. In nl with pot size bets, defender should not call if attacker bluffs less than 1/3 of the time.
Note 4. In poker optimal strategy does not always dominate exploitive strategies. It's rarely right to call players who underbluff. It's usually right to call players who overbluff. Don't bluff calling stations.
You must know the math to play well. But the math is a guideline, not something to be followed blindly. That's the art part of the game; knowing which math applies.

Two street game.

The two players are X and Y. Each start with
equal size stacks S. Each ante A > 0. On the
first street no cards are dealt. On the 2nd
street each are dealt a card from [0,1]. The
player with the lower card wins. A/2 goes to
the house.
1st street. Y may bet any amount S-A. X may
call or fold.
2nd street. No betting. Best hand is awarded
the pot.
Optimal strategy on the 1st street must be
for X to call any bet. X is 50/50 to win
the game. X loses A if he folds and gets
better than one to one to call and play.

This demonstrates that optimal strategy is
not necessarily a winning strategy. It's
just a lose minimum strategy. As we learned
from the movie War Games not playing is
often the best strategy.

In hold'em the streets are not independent.
But they are not linearly dependent either.
There is only a positive correlation among
the streets. Being ahead on one street
increases one chances of being ahead on the
next street. But there are no guarantees.

That is another reason why optimal strategy
on the flop and turn may not always be the
desirable play option. While optimal strategy
exists on the flop and turn, it may not be
the most desirable +EV action.

Game Theory Limitations

Still not convinced. Game theory is only
clearly useful on the river and the final
half-street of betting. On earlier streets
strategy is influenced by game theory. On
the river there's a showdown winner. Only
changes if winner folds. Half-street is
when someone goes all in. Then it's a
jam or fold type chart.

Two player game.
The pot was P after the flop action.

Matrix A.

You---fold---call----RR----fold---call
Opp
C/R____-1_____M1_____xx_____xx_____xx
fold___xx_____xx____P+2_____xx_____xx
call___xx_____xx_____M2_____xx_____xx
cap____xx_____xx_____xx_____-3_____M3

On this matrix you have been check/raised
on the turn. Only when one player folds
is there a value in the cell. There's
another matrix when both players are still
competing. Most of the cells in the matrix
is empty.
Any time you are willing to fold the pot is
strange money. The C/R-fold cell is -1 because
you lose one chip on the turn by folding. The
C/R-call leads to the river matrix with the
new pot at P+2. The fold-RR you win the pot
plus two chips from opp.
This M1 has appeared in poker books in another
form. You are risking one chip to try to win
pot plus one chip. Ye is your pot
equity. Or your perceived pot equity.

Ye(P+1) > 1
Ye > 1/(P+1)

You can call when Ye is greater than
1/(P+1). Some players think they can outplay
opp on the river and call with less. That's
impied odds.
In a FL calldown situation it's really

Ye(P+1+1) > 1+1
Ye > 2/(P+2)

Which means you need greater pot equity.

With TPTK and a C/R it's a wa/wb situation.
Usually you're wb. There's rarely enough
pot equity to call when you're wb. On a
static board with no flush possibilities
you will even be drawing dead to a hidden
set.

Properties of Variance

All the tight players like to brag about their low
variance while playing cash ring games. In SnGs
it's different.
When I figure out how it's done there will be a
chart. The losing SnGs players have a lower
variance than the winning ones. A player who
never finishes ITM has a variance of zero. Can't
get lower than that.
Average player will win one, one second, one third
and seven out of the moneys. The standard
deviation is 1.67 buy-ins.
The break even player will be ITM 33% of the time.
That's like 100 plays. 11 1st, 11 2nd, 11 3rd and
the rest out of the money. The s.d. is 1.76.
Also the better one does the higher the variance.
In SnGs the better players have higher variance
than the weaker players. This is also true for
MTTs.

It's Subjective

We know the solutions to the [0,1] games. On the
river our opponent has a hand from the space R [1,990].
This would accurately represent opponent's space if
he were not allowed to fold. Even maniacs are not
likely to hold each of the 990 hole card pairs with
equal likelihood. Our opponent's space after betting
the river is in Rb. Also Rb is independent
on opponent's style. Whose who best estimate Rb
for this opponent will achieve the best EV.
Note. The individual elements in Rb are not
equally likely. Players nearly always bets the nut
hands, while they only sometimes bet the bluffs.
Rb is not uniform.

Let's review the betting matrix on September 17.

Adam chooses bet/bet or bet/check from the matrix.
Bill then sees bet or check. Bill needs to respond
only to the bet line vector. When Adam bets, Bill
should call when Bill's LS is above S/(S+b). But
Bill only knows his LS from the space R. Bill is
estimating his LS from the space Rb. The bill
from a set of Bills who best estimate Rb will
be the ones who best deciphers the other player's
style.

Part 3. Applying game theory.

The introduction to the 3-card deck is posted on
8 NOV 2005.

On the turn Bill has been the aggressor and clearly
in the lead. Adam is drawing, possibly to a 4flush.
The river card apparently has not helped Bill. Bill
checks. On the river Adam and Bill flipflop roles.
The aggressor becomes the defender. The chaser
becomes the attacker.
What is Adam's best strategies? Assuming Adam bets,
what's Bill's best response?

The formulas are reposted.

S = the size of the pot.
b = the size of the bet.
p = prob river made Adam's hand best.
(1-p) = probability Bill's hand holds up.
x = part of time Adam plays row bet/bet(B/B).
--bet/bet means Adam bets whether his hand improves
or not.
1-x=Adam chooses the B/CH row.
--bet/check(B/CH)Adam bets a made hand and checks
when he misses.
y = part of time Bill chooses the call column.

Now the 2 X 2 matrix looks like this.

______\_______BILL
ADAM	Call	Fold
B/B	p(S+b)-(1-p)b	S
B/CH	p(S+b)-0	pS

	p	b
x = --	------	-----
	(1-p)	(S+b)

y= S / (S+b)

Now how do we use this info?

Example.

Bill has AsKd. Board on the turn is

[Kh 8h 2d],[4s]

Bill has been the aggressor and Adam has been calling.
The river card is a heart. Bill checks. Adam always
bets whenever he has made the flush. How frequently
should Adam bluff? How often should Bill call? The
answers assume both are playing optimal strategy.

Limit.

The size of the pot is large relative to the size
of the bet. Let's say the pot is 10 bb.
Bill should call S/(S+b) part of the time. That's
over 90% of the time. Bill should almost always call.
Against an unknown opponent don't bother to randomize
just call 100% of the time.
Adam should choose the B/B row

	p	b
x = --	------	-----
	(1-p)	(S+b)

part of the time. b/(S+b) is less than one in ten.
46 cards unseen and 9 cards make a flush. Therefore
p=9/46 and (1-p)=37/46. p/(1-p)=9/37~1/4 Adam
should bluff less than one in forty times. The
math supports the CW. In limit Adam should almost
never bluff. For pots of 10bbs or more it's close
to never bluff.

Pot size bets.

S=b. S/(S+b) = 1/2.
Bill should call half the time. This is quite
different from limit where Bill should call nearly
all the time. The larger the bet, the less often
Bill should call.

b/(S+b) = 1/2.
(1-p)=37/46
p/(1-p) = 9/37. Same as in limit.
But now [p/(1-p)][S/(S+b)]
= 9/37 X 1/2 = 9/74 ~ 1/8

Adam should be bluffing just under one in eight times
when he missed. Remember Adam is betting the flush
(9/46) nearly 20% of the time. When playing optimal
strategy a bet by Adam is a bluff 38% of the time.
This is for pot size bets.

When Adam bets he should have the made hand
(S+b)/(S+2b) part of the time.
**explain-Adam bets 100% of made hands.
--[p/(1-p)][S/(S+b)] of missed hands.

-------------
Bluffing

	p	b
x = --	------	-----
	(1-p)	(S+b)

37 of the 46 times Adam misses his 4flush.
Adam missed the draw (1-p)/1 part of the time.
What part of the 46 times should Adam be bluffing?

bluff freq= x times (1-p)

The (1-p) cancels out therefore
bluff freq = pb/(S+b)
nuts freq= p

For S=b
bluffs freq = p/2

For S=2b
bluffs freq = p/2

Take the set of hands which are bet. The proper
ratio of bluffing hands to made hands is the
fraction b/(S+b). When Adam bets the proper
ratio of made hands to bluffing hands he is
indifferent to whether Bill calls or folds.

Real world hold'em.

Those numbers are fine in theory. It doesn't work
in hold'em. Adam's drawing to a 4flush. The
river produces a non flush card. It's not a threat
card. If Adam bets, Bill will call. The reality
is Adam should never bluff when he misses the flush.

When should Adam bluff? Sometimes Adam will still
be in the pot drawing to something other than the
flush. A flush card appears on the river. Adam
should bet/bluff as if he made the flush. Adam
must paint a consistent picture of the hand.

Optimal strategy is a fine defensive approach to
poker. The winningest players are whose who know
when and why to deviate from this strategy.

------------

In the previous examples Bill, the one with the
better hand, was always the aggressor. Sometimes
the weaker hand is the aggressor.
In these new examples Bill will defending his blind
from a steal. Adam open raised from button or CO.
By the river Bill may feel he needs to improve
before calling. Adam should bluff more often when
Adam thinks Bill is willing to fold.

Example: Adam opens from CO with 8c6c.
Button and SB fold. Bill defends blind with Qs7h.
Flop: KsJdTh
Bill checks. Adam bets. Bill calls.
Turn: 5c
Both check.
River: 4c
Bill checks. What should Adam do?

If Adam thinks Bill was drawing to a straight, he
should bet. If Bill had a jack or a ten, Bill may
have bet the river. Since the turn and river did
not improve Bill, he may be willing to fold.
Adam must attempt to profile Bill. With which hands
is Bill calling? If they are drawing hands which
have likely missed Adam should bluff when Adam can't
beat the drawing hand. With a no pair ace high Adam
should showdown and see whose hand is best. If Adam
thinks Bill is calling with a pair, Adam should only
bet when he can beat that pair. Adam should not
bluff when Adam suspects Bill has a pair and will
call.

4-tier limit

The goal of 4-tier limit is to increase the skill
compotents relative to the luck in hold'em. After
35 years the theorists have been able to analyse
the game. They have been able to reduce the both
fixed limit and no limit to two card poker. The
intent of 4-tier limit is to increase the number
of decisions per hand. 4-tier limit will also
try to make each street have nearly equal effect
on a player's winrate.
In 4-tier limit the betting units would increase
every street. One unit for prf, two units for
the flop, four units for the turn, and ten units
for the first river bet.
First neutralize the effectiveness of playing tight.
Have three blinds rather than two. All blinds will
be big blinds. The button would also post a blind.
Preflop the action will start with the player to
the left of last big blind. He may open for one
or two units. The rest of the betting is like
regular fixed limit.
Always have at least three blinds each deal. If a
blind bust out or leave, move the button to the next
active player on the left, and have the two players
left of the new button position post blinds. New
players to the table may be dealt a hand by posting
a blind and a dead half blind.
On the flop the betting unit is two blinds or two
units.
On the turn the betting unit is four blinds.
The rules change drastically on the river. The
action will now start with the last aggressor.
That's the last player to bet or raise during this
hand. Check/raise will no longer be allowed. Each
player in turn is allowed to bet ten units. Raises
on the river will be twice the size of the previous
bet. In 4-tier most of the big action can only
occur after all the cards are dealt and the winner
is determined assuming he doesn't fold. This big
bet would also eliminate the automatic crying call
of fixed limit.
These changes would create a game more favorable
to strong post flop players.

river raises 2

Players seem to make raises with all sorts of hands.
They include no pair, one pair, two pair, set, and
better hands. Many player either don't understand
coordinated boards or just have no fear.

Calling strategy. On a board with no pair, four
to a flush or four straight, two pair is a must call
against a river raise. Against frequent raisers call
with top pair. You must to very sure before you can
fold a set or straight.
Repeat. This strategy refers to fix limit, not no
limit.

river raises

Started a study of river raises and check/raises in fix limit.
Have inspected about 700 hands. 50 raises and check/raises
appeared in the sample. Only two were out and out bluffs.
Two were ill advised and a few were weaker than top pair top
kicker. On balanced the raises and c/r were solid. Four
of the raises lost and one split.
Have decided river LHS has little value for assisting on
decisions on whether to call or reraise. Need the nut hand
to reraise against most players. Calling with TPTK can only
beat a bluff.
Will repost on this subject when over 10,000 hands are
inspected.

blind games [0,1]

A study of blind play.

SB is one chip. BB is two chips. The three chips
will be considered the pot's chips. All fold to
the blinds. This study will restrict itself to
fixed limit. Each players' choices will be initially
limited. The choices will be expanded as the study
processes.

1. SB will raise, bet or check at his turn to act.
BB can call or fold.

1.1. Each player has a stack of four chips. SB
open raises any two, meaning SB raises 100% of the
time. This is the same as a forced raise. BB may
call or fold.
There are 6 chips in the pot on BB's turn to act.
BB is getting 6 to 2 for his call. BB should call
with a LHS of 25 or higher.















SB's expected value of the game.
In the red area of the chart SB wins the blinds.
In the blue area SB loses two additional chips.
In the white the blinds chop the pot.

red= 1/4 X 3 = .75
blue= 1/4 X 3/4 X -3 = -.5625
white= 3/4 X 3/4 X 1 = .5625
SB's EV = 0.75

BB's EV

red= 0
blue= 1/4 X 3/4 X 6 = 1.125
white= 3/4 X 3/4 X 2 = 1.125
BB's EV = 2.25

SB's net EV is 0.75 minus his one chip blind.
His net is minus 0.25
BB's net EV is 2.25 minus his two chips blind.
BB's net is plus 0.25.

Conclusion: SB going all in blind against a
thinking BB doesn't work.

1.2 Each player has a stack of six chips. SB
open raises any two, meaning SB opens 100% of the
time. BB always calls. On the flop SB bets his
remaining two chips. BB may call or fold.
This is the same as both blinds are four chips with
the action starting after the flop. SB has a forced
continuation bet. BB may call or fold.
There are 10 chips in the pot on BB's turn to act.
BB is getting 10 to 2 for his call. BB should call
with a LHS of 16.67 or higher.

SB's EV

red= 1/6 X 5 = 0.8333
blue= 1/6 X 5/6 X -5 = -0.69444
white= 5/6 X 5/6 X 1= 0.69444
SB's EV = 0.8333
SB's net EV is neg 0.16667.

1.3 Each player has a stack of six chips. SB
open raises any two, meaning SB opens 100% of the
time. BB may call or fold. On the flop SB bets
his remaining two chips. BB may call or fold.
There are 10 chips in the pot on BB's turn to act.
BB is getting 10 to 2 for his call. BB should call
with a LHS of 16.67 or higher.
In this game the LHS values remain static. When
BB call prf BB will call on the flop.

SB's EV

red= 1/4 X 3 = 0.75
blue= 1/4 X 3/4 X -5 = -15/16
white= 3/4 X 3/4 X 1 = 15/16
SB's EV = 0.75
SB's net EV is neg 0.25

Conclusion: BB does better by folding hands with
LHS of under 25 prf. *In holdem no starting two
has a LHS less than 32.

1.4 Each player has infinite chips. SB open raises
prf and bets every street. BB forced call prf. BB
may fold on flop and calldown all bets.
18 to 10 for the calldown. BB should call with a
LHS of 35.714 or higher.

SB's EV

red= .35714 X 3 = 1.0714
blue= .35714 X .64286 X -13 = -2.9847
white= .64286 X .64286 X 1 = 0.4133
SB's EV = -1.5
SB's net EV is neg 2.5

Conclusion. This is a very poor strategy for SB.
BB can easily exploit SB by playing passively.

1.5. Each player has infinite chips. SB open raises
prf and bets flop. BB forced call prf. BB may fold
on flop and calldown all bets. SB may bet or check
on the turn/river. The turn/river is an 8 chip bet.
SB bets over 35.714 LHS. BB callsdown with over
35.714 LHS.

SB's EV

red= .35714 X 3 = 1.0714
blue= .35714 X .64286 X -5 = -1.148
white= .64286 X .64286 X 1 = 0.4133
SB's EV = -0.3367
SB's net EV is neg 1.3367

This is a clear improvement over continuely betting
hopeless hands. Maybe micromanaging SB's betting
points will improve SB's EV.

1.6. Each player has infinite chips. SB open raises
prf and bets flop. BB forced call prf. BB may fold
on flop and calldown all bets. SB may bet or check
on the turn/river. The turn/river is an 8 chip bet.
SB bets over 73.193 LHS. BB callsdown with over
35.714 LHS.















SB's EV

red= .35714 X 3 = 1.0714
pink= .26573 X .28572 X 11 = 0.83517
blue= .35714 X .64286 X -5 = -1.148
blue= .28572 X .26573 X -5 = -0.3796
white= .26573 X .26573 X 1 = 0.0706
white= .28572 X .28572 X 1 = 0.0816
SB's EV = 0.5312
SB's net EV is neg 0.4688

Conclusion: SB's EV has gone from -2.5 to -0.4688.
The real solution must be somewhere inbetween.
Either way opening any two with a raise has negative
expectation against anyone who defends his blind.

Note: It's not absolutely clear that my logic is valid.
The LHS of a starting hand remains constant in the [0,1]
game. In holdem the LHS changes with the appearance of
each new board card. Also starting cards catch a piece of
the flop from 32 to 40% of the time. With two unpaired
starting cards missing the flop can make the LHS value
drop drastically.

character set test

0

LHS PrF Starting Hands

Column A is the two hole cards.
Column D is for the hand space of a 10% preflop raiser.
Column E is for the hand space of a 20% preflop raiser.

The entry of the cells is the LHS against that space.

"Golden Mean of Poker"

http://groups.google.com/group/rec.gambling.poker/search?
q=&start=10&scoring=d&enc_author=2IMEHhgAAABIfsVGFKjfS-
anNRwnEVTLMxB39KJNQ76SLnMRgR9a0A&filter=0&as_drrb=
b&as_mind=1&as_minm=1&as_miny=2003&as_maxd=
31&as_maxm=1&as_maxy=2003&


This is from the [0,1]game series by Chen/Jerrod on
RGP. r is from part 3.

r=.414

r is the "golden mean of poker".

Use the reciprocal of r. Bet if your hand has a
linear hand strength(LHS) of .586 against your
opponent's range of hands. Range of hands sounds
so awkward. Will use hand space instead of range
of hands.
Square r, cube r and quad r for .171, .071 and
.029. The corresponding LHS are .829, .929 and
.971. You should raise with a LHS of .829.
Reraise with .929 and cap with .971.
This assumes both you and your opp are rational.
Both are trying to win. Also assumes neither
will fold.

In practice this isn't necessarily true. Many
opps fail to adjust to new info. Against these
opps you can and should often rebet your same
values.

Tom Weideman Challenge

It's time to reexamine the Weideman Challenge. This
challenge is for the specific case where each of two
players ante one unit. Tom and Cal are each dealt a
card from the line interval [0,1]. Tom is allowed to
bet one unit or check. If Tom bets Cal may call or
fold. This is essentially the same as betting half
the pot.

Tom should bet any card .7 or higher. Tom should
bluff .1 or lower. Cal should call .4 or higher.
Tom has a +EV of 0.1 bets.

Let's example other cases. Betting the size of the
pot. Now Tom should bet less often. Cal should call
less often. For Tom, bet .78 and bluff .11. Cal
should call .53. Tom has a +EV of 0.056 bets.

In fixed limit the bets will be much smaller than the
pot. Instinctively I tend to value bet less. This
is wrong. Value bet more often. Cal will be forced
to call more often due to the size of the pot. Tom's
bluffing frequency relative to his betting frequency
will be lower.

Note: the uniform [0,1] games now use low values as
the best values. This is a flip-flop from the
previous high values as best.

Linear Hand Strength(LHS)

Linear Hand Strength: This is a measure which ranks
your hand in the range of possible hands.
Specifically, all possible hands are evaluated,
and ordered, and the measure of the hand is the
fraction of all hands which it beats or ties. By
convention, this measure is mapped to a value in
the range [0,1].

I wasn't aware that there was a term for ranking
hands. So from now on linear hand strength will
be used to rank hands. But zero to 100 will be
used instead of [0,1]. All the odds calculators
give results in percentages. I will also use the
term LHS for preflop, flop, turn and river.

For on the flop each hand type will be given a
general LHS value. The flop is three small cards.
Each possible two overs will be given a LHS value
against various range of hands. Normally it's
worth calling a continuation bet with 35% or
higher.

I have started to compile a file of these values
in Excel. When it gets organized in some readable
form, the table will be posted on this blog.

Shifting gears

When Doyle Brunson coined the term shifting gears it was a sophisticated play only known to experts. Today all advanced players know the term and many know when to use it. Tomorrow, in the future, this will be a intermediate concept.
Shifting gears is so basic. One changes speeds on a one-dimensional line. Hold'em is much too complex for best strategy to be one-dimensional. Shifting gears should be an integral part of an overall strategy.
Shifting gears only cover your range of preflop starting hands. This set of hands was always a function of many factors. These factors include number of players on the table, relative chip stack sizes, stack sizes in terms in big blinds, your starting position, have others all ready entered the pot, styles of the other players, etc. and etc. This would suggests a strong player is shifting gears from hand to hand. Sometimes during a hand.
Phil Ivey was asked of his strategy starting a new tourney.
Ivey replied that he had no preplanned strategy. He would
observe the table and adjust.
There isn't a fixed relationship between the weighting of
the importance of each variable affecting the opening of
hands. When your chip stack is large relative to the big
blind you may choose among many opening styles. If your
chip stack is very small you are restricted to jam or fold.
You should be shifting gears from hand to hand.

Top no pair rank within the 990

Will make a table to rank high no pair among
all possible two card starting hands after
the board is completed.
ABCDE means five cards of different rank,
meaning no pair on the board.
wxyz will represent the four suits.

*4th or 5th try. Hope this is the right answer.

1. The first board ranked will be ABCDE with
no straight or flush possible.

Top no pair is ranked 384-393.

2a. AxKyQz9w6z
Board didn't pair. No flush possible. Only
one way for a straight which requires both
hole cards. 16 combinations for each way for
a straight.

368-377.

2b. KxQyJz8w6z
Board didn't pair. No flush possible. Only
two ways for a straight which requires both
hole cards. 16 combinations for each way for
a straight. 32 combinations.

352-361.

2c. QxJyTz7w5z
Board didn't pair. No flush possible. Three
ways for a straight which requires both
hole cards. 16 combinations for each way for
a straight. 48 combinations.

334-343.

3a. A straight that needs only one hole
card. 280 combinations

280-289.

3b. A straight that needs only one hole
card two ways. 240 combinations

192-201.

Mapping Distributions

LOWBALL

In a raised pot you and a stranger have both
drawn one card. You've never seen him play.
He could be drawing to an eight or a wheel.
After the draw you don't know the shape of the
distribution of his stronger hands. Regardless
of his draw the distribution of the tails end,
the weaker hands is the same. He has the same
number of ways to catch each high card and pair.
The distribution of all one card draws are
similiar.

This distribution maps easily into the uniform
[0,1] distribution. When you see your card, you
know where your hand lies in the uniform [0,1]
distribution. The relative value of the hand
is known because opponent's draw is known.

Lowball distributions are static. They fit
well into Tom Weiderman's Challenge problem.
Therefore it is easy to apply this to a game
theory model.
The drawing hands are of a continuous range.
On a one card draw we know it's right to call
with a king or better, when the other player
draws one. We also know that if we bluff
when we pair tops, we are bluffing at the
right frequency.
Amazingly many longtime players don't know this.
They call and bluff by feel. I play by rote,
only deviating against predictable players.

Hold'em

In hold'em there are straights and flushes. This
makes the distribution disjointed.

Two-way pot.
From the board and your two hole cards, opp has
a discrete and countable distribution. There
are 990 possible ways for his two hole cards.
But when you look at your hole cards unless it's
the nuts it isn't readily clear where your hand
ranks among the 990 possible. At least it isn't
clear to me.

On the flop it is even more murky. There are
over 200,000 ways to choose opp's two, turn
and river cards. The distribution is dynamic.

On a raised pot between two players.
The top 65% of the hands of the distribution is
about the correct calling frequency for a flop
continuation bet. But what's the top 65%?

Example:
you in BB: AsJh
Button raises and you call.
All others have folded.
flop: 9s6c3d
You check and button bets. You know button is
betting nearly 100% of the time. Does you hand
beat 65% of the hands in his range? This is not
easy to determine even when you know his exact
range.

What if the flop were changed to 9s6d3d. Does
that change the ranking of your two hole cards?
If so, how much?

Bowling Analogy

Lowball is like spot bowling. There are static
points of reference in the distribution. Once
the drawn card arrives you know the relative
rank of your hand to opp's distribution.
In hold'em the distribution is board dependent.
The possibility of boats, flushes and straight
are dynamic and in constant flux. It's like
bowling with the spots removed from the lanes.

Lowball is a simple math game. Hold'em has no
points of reference. The relative rank of your
hand isn't obvious. There are only 990 possible
combinations after the river. Assuming you
don't hold the nuts, it's extremely difficult
to know where your hand ranks among the 990.
Most of the 990 possible hands are unlikely to
be part of opp's set of playable hands.

Hold'em is too complex for players to make
calculations in real time. In hold'em you
must trust your gut. Your gut is often
based on air.

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not maniacs

maniac

(n) A player who bets, raises, and reraises without regard to the quality of his hand; someone to whom getting in the last bet is a matter of pride. Such a player is most often found in flop games.

Been playing fixed limit hold'em for about 12 months. Think I have played much less than 1% of my hands against a pure maniac. Only a player with unlimited funds can play this way and remain in action.
Have seen many threads on poker forum by members stating that they have problems against maniacs. The major reason for their problems is that they misidentify players as maniacs.
A maniac plays wildly on every street. Are not detered when they encounter aggression. Very few players actually play that way.

flopsters

(n) A player who bets, raises, and reraises without regard to the quality of his hand; on the flop only.

One sees these players frequently. Many players mistake these flopsters with maniacs. Flopsters will apply pressure on other streets. They also will attempt to play well postflop. This means they try to avoid bad calls. Many players are giving the flopsters too much action postflop. On the flop the flopster's 3bet means nothing. Postflop flopsters are willing to laydown no hand no draw to aggression. 3bet on the turn means the flopster has a strong hand. Don't blindly call flopster's reraises with no pair. Postflop, don't overplay the 3bet and cap against a flopster.