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.