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.