Special Thread For Chen-Ankenman Mathematics of Poker

[Phanekim]

I took a stats course, an advanced stats course, and couple genetics courses in good ole ucsd.

I felt like i was reading a college textbook again. Poker wasn't fun...it just turned into another lame class again after reading that book. haha.

[HelixTrix]

David,

I've just finished working through the Mathematics of Poker, then I reread my notes from No Limit Holdem, Theory and Practice. Through my new 'game theory eyes', most of the recommendations seem to be exploitive! I find this quite surprising given my impression of your approach to the game. For example, your 'Basic Skills':

1. Manipulating the pot size.
2. Adjusting correctly to stack sizes.
3. Winning the battle of mistakes.
4. Reading hands.
5. Manipulating opponents into playing badly.

3. and 5. are clearly exploitive. 1. is somewhat arguable but in most of the contexts that you apply it (raising different amounts preflop, making blocking bets, etc.) is exploitive as well. I realise that by 4. what you really mean is 'reading ranges of hands' but even then, the pure game theory approach focuses on reading your OWN range. From MoP p. 374, 'From a game theoretic view, the distribution of hands your opponent holds is irrelevant. If he can exploit you by playing a different set of hands than you were expecting, then your play is suboptimal.'

Would it be fair to condense your advice to the following:

Figure out a near-optimal strategy away from the table; at the table always first look for a way to exploit; if you find none, fall back on your predefined strategy.

Also would you comment on the suggestion that due to reduced sensory input and higher player turnover, playing online one should favour near-optimal play over exploitive play more of the time.

Thanks,

Helix

[jogsxyz]

MOP 19.3 page 242.

[ QUOTE ]
We know Y breaks even on his bluffs - on each
street he bluffs appropriately to the number of
hands he will carry over to the next street.

[/ QUOTE ]

Not convinced of this statement. Y bluffs to force
X to call Y's value bets. Y must bluff else X need
never call. Y sacrifices equity on bluffs so that
he can gain equity on his bets. Isn't that it?

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

Half-street game. Shouldn't they be called single alley
game. In a three raise max game there can be as many as
four alleys to each street.

[Jerrod Ankenman]

[ QUOTE ]
MOP 19.3 page 242.

[ QUOTE ]
We know Y breaks even on his bluffs - on each
street he bluffs appropriately to the number of
hands he will carry over to the next street.

[/ QUOTE ]

Not convinced of this statement. Y bluffs to force
X to call Y's value bets. Y must bluff else X need
never call. Y sacrifices equity on bluffs so that
he can gain equity on his bets. Isn't that it?

[/ QUOTE ]

Not if X in turn is playing optimally (and the section in question is describing an optimal strategy-pair). If Y were losing money on his bluffs, he could unilaterally increase his equity by bluffing less. Instead, X calls enough that Y is indifferent to bluffing, while Y calls enough that X is indifferent to calling. Y isn't "sacrificing equity" because he wins enough pots uncontested to balance the losses he incurs when he is called. Then both sides cannot improve.

[ QUOTE ]
-------------------

Half-street game. Shouldn't they be called single alley
game. In a three raise max game there can be as many as
four alleys to each street.

[/ QUOTE ]

A half street game isn't a game with one bet allowed - it's a game where X must check dark. By convention we only allow one bet, because there's not much value to studying a game where the first player must check dark, but can raise later. But the defining characteristic is that X checks dark.

[jogsxyz]

Calculated <Y_1> for three streets.
Assumed Y checking was giving up on the pot.
----
1st street

Ycheck = (65/91)*-$5 = $-3.5714
Xfolds = (26/91)(6/7)*$5 = $1.2245

X calling goes to street two.
--------
2nd street

Ycheck = $-0.8375
Xfolds = $0.5730

X calling goes to street three.
---------
3rd street

Ycheck = $-0.3543
Xfolds = $ 0.423
Ybluffs= $-0.1151
Yvalue = $ 2.1976
--------
The three streets summed to only
$-0.4606. The game still favored
the defender. The betting only
improved Y's results by about
three dollars.

thx,
jogs

[Jerrod Ankenman]

[ QUOTE ]
Calculated <Y_1> for three streets.
Assumed Y checking was giving up on the pot.
----
1st street

Ycheck = (65/91)*-$5 = $-3.5714
Xfolds = (26/91)(6/7)*$5 = $1.2245

X calling goes to street two.
--------
2nd street

Ycheck = $-0.8375
Xfolds = $0.5730

X calling goes to street three.
---------
3rd street

Ycheck = $-0.3543
Xfolds = $ 0.423
Ybluffs= $-0.1151
Yvalue = $ 2.1976
--------
The three streets summed to only
$-0.4606. The game still favored
the defender. The betting only
improved Y's results by about
three dollars.

thx,
jogs

[/ QUOTE ]

Without checking your math (which I can't quite follow), Y does lose money using strategy Y1. The value gained by Y when he holds a value hand (2/13 of the time) is $19.16 per hand. So the total ex-showdown expectation that Y gains is (2/13)(19.16) = 2.95. Comparing this to his showdown equity of -3.46, he still loses money from the game by playing Y1. He overcomes this by betting the geometric growth (Y2), because he makes more money from the betting.

Note the paragraph that begins with "So Y does better...":

"...On a per-hand basis, this is enough to swing the game from negative (using Y1) to positive (using Y2)."

[jogsxyz]

[ QUOTE ]

Without checking your math (which I can't quite follow), Y does lose money using strategy Y1. The value gained by Y when he holds a value hand (2/13 of the time) is $19.16 per hand. So the total ex-showdown expectation that Y gains is (2/13)(19.16) = 2.95. Comparing this to his showdown equity of -3.46, he still loses money from the game by playing Y1. He overcomes this by betting the geometric growth (Y2), because he makes more money from the betting.

Note the paragraph that begins with "So Y does better...":

"...On a per-hand basis, this is enough to swing the game from negative (using Y1) to positive (using Y2)."

[/ QUOTE ]

I agree with those statements. It's just that I think bluffing is a lost leader which is more than compensated by gaining extra equity for the value bets.
-------------
This matrix is constructed with Y's row strategy
between bet value/bluff and bet value/check.

P = the size of the pot.
r = the size of the bet.
2/13 = probability clairvoyant is dealt ace or king.
11/13 = prob clairvoyant dealt other cards.
y = part of time clairvoyant plays row bet value/bluff
x = part of time defender calls.
B/CH - bet value/check otherwise

Now the 2 X 2 matrix looks like this.

<font class="small">Code:</font><hr /><pre>
Y \ defender
_____|_______Call_________________Fold
| |
B/B | (2/13)(P+r)-(11/13)r | P
| |
B/CH | (2/13)(P+r)-0 | (2/13)P</pre><hr />


<font class="small">Code:</font><hr /><pre> P
x= -----------------
(P+r)</pre><hr />
<font class="small">Code:</font><hr /><pre> (2/13) r
x= -----------------
(11/13)(P+r)</pre><hr />

Y's row strategies are indifferent to X calling or folding.
That isn't the same as saying Y's bluffing is indifferent to
X calling or folding.

[Jerrod Ankenman]

[ QUOTE ]
[ QUOTE ]

Without checking your math (which I can't quite follow), Y does lose money using strategy Y1. The value gained by Y when he holds a value hand (2/13 of the time) is $19.16 per hand. So the total ex-showdown expectation that Y gains is (2/13)(19.16) = 2.95. Comparing this to his showdown equity of -3.46, he still loses money from the game by playing Y1. He overcomes this by betting the geometric growth (Y2), because he makes more money from the betting.

Note the paragraph that begins with "So Y does better...":

"...On a per-hand basis, this is enough to swing the game from negative (using Y1) to positive (using Y2)."

[/ QUOTE ]

I agree with those statements. It's just that I think bluffing is a lost leader which is more than compensated by gaining extra equity for the value bets.

&lt;snip matrix&gt;

Y's row strategies are indifferent to X calling or folding.
That isn't the same as saying Y's bluffing is indifferent to
X calling or folding.

[/ QUOTE ]

That isn't what was said. When Y bluffs, of course he wants X to fold. He's bluffing! Y chooses his bluff frequency such that X is indifferent between calling and folding.

If X chooses to call more, he will gain against Y's bluffs, but lose an equal amount against Y's value bets. If X chooses to call less, he will gain by not paying off Y's value bets, but will lose more pots by failing to pick off Y's bluffs.

You can easily verify this:

Supppose that it's the river, and Y has some value bets -- say 1/10 of his hands -- and the rest value bets, and the pot is 4. Y bets with all his value bets, and 1/5 as many bluffs, or 1/10 and 1/50.

Now X's optimal strategy here is to call 4/5 of the time.

EV(bluff) = (p X calls)(cost when X calls) + (p X folds)(gain when X folds)
EV(bluff) = (4/5)(-1) + (1/5)(4) = 0

Bluffs are not "loss leaders" if the opponent plays optimally. Now, if X decides to call some other frequency, then what you describe could happen. Suppose X called all the time. Then

EV(bluff) = (1)(-1) + (0)(4) = -1

But if X never called, then:

EV(bluff) = (0)(-1) + (1)(4) = 4

So it's true that by varying his strategy away from optimal, X can change the value of a bluff. But it is not necessarily the case that a bluff is a "loss leader," and it is definitely not the case when X and Y play at the solution of the game.

[jogsxyz]

Jerrod,

This time I used Excel and reworked the problem. The value of the bluffs did balance out. Sorry, I doubted you.

1st street
AceKing ---- .659341
bluff ------- .565149
giveup --* -3.571429
positive values from X folding on this street

2nd street
AceKing ---- .451128
bluffs ------ .142461
giveup --* -0.773362

3rd street
AceKing ---- .363813
----------- 2.243512
bluffs ------ .042327
----------* -.261018
giveup ---* -.374898

value of the game -0.512976
value for Y non premium hands -4.230769.
This is the same value as X giving up every
non premium hand.
(11/13)*-5=-4.230769

Still having a problem solving the Y_2 case.
On my first pass using the same(similar) method
as for Y_1 produced surprising(maybe wrong)
results. By always folding on street two X
was able to hold Y to a game value of -1.36.
Possibly in the model Y wasn't bluffing frequently
enough on street 1.

jogs

[jogsxyz]

Not satisfied with my solutions. Solved three one street
games. Then linked the solutions into a series. This is
one game with three streets, not three independent one
street games. The three streets are interdependent, not
independent streets. Therefore the methodology is
suspect.

Street one. The pot is 10. The bet is 60. From other
toy games X should call pot/(pot+bet) of the time.
X calls 1/7 of the time. Y maximized against this
strategy. Y should bet 12/77 of his other(non AK) hands.
Y was betting 26(14 AK) hands out of 91. This was
optimal strategy for the half-street game.
What if X maximized against Y's optimal strategy? X
should always fold. Whenever Y bets fewer than 50% of
the hands, X can guarantee plus EV by folding to all
bets. Y must bet at least .396 of his others before
X needs to call to improve on +0.5(the value of the
game in the 3 street solution).

Conclusion. There's more to this 3 street game than
just solving 3 single street games. Optimal strategies
for one individual street may not be optimal for
multiple streets. My solution for the street one game
is only correct for a street one game in isolation.



This chart is what's happening. Street two is a game
within street one. And street three is a game with
street two. The value of the top left hand cell of
the street one and two matrices is a variable. This
game must be solved by recursion.

jogs