Keith uNL Game Theory

The Beginning

This all began with a pokertracker self-check session. I came to a shocking realization that I was playing really horrendously with every type of holding but one, pocket pairs. I don't remember what the $$$ amounts were exactly, but I had won twice as much with pocket pairs as all the money I'd won total. What I mean is if I'd won $400 with pocket pairs, I was only up $200 overall. This meant I was losing with other holdings enough to cut my winnings in more than half. Turns out I suck with suited connectors. I just couldn't play them at all. Most likely due to my extreme agression. I have difficulty with middle gears, I'm generally either at 0 or 100, no middle ground.

So I begain playing nothing but pocket pairs, and threw in AK and AQ because they really are quite strong holdings. Admittedly I would still at time play things like AJ suited in late position with limpers, or raise it from late if folded to me. I did quite well for myself and at the .05/.10 game I was 12-tabling averaging about $10 an hour. I had the lifebeat though, and now I'm playing .01/.02 and things don't seem to work quite as well.

The first thing is people don't respect bets period. Preflop raises are taking down the pot preflop only about 1/3 of the time! Further, cbets on the flop seem to be even less respected with the same result of pulling down the pot about 1 in 3 times. I started crunching numbers to get an idea about the equity of playing this tight style at the lowest of low limits, and got, well here's what I got....

Assumptions

First we're going to make some assumptions that can significantly alter actual results. It is assumed that:
1) We play only {AA-22, AK-AQ}
2) We raise all these hands preflop
3) We will be 3bet 1/5 of the time we raise preflop
*This may be a higher % than it actually occurs.
4) If we flop a set when 3bet preflop, we will stack our opponent.
5) If our cbet is called, we cannot win the hand without turning a set.
*I feel this point needs the most improvement. There
*should be more equations regarding showing down.
6) When we cbet, we will be raised 1/5 of the time.
7) If we are raised on the flop and have a set, we will stack our opponent.
8) I have not included hands in which we have AA-KK and after being
3bet preflop, we may shove for value against bad players calling {AK-AJ, QQ-99}.
9) Our opponent has a full stack to win (250 big blinds)

That covers our assumptions. There's probably more, but I can't think of them right now. On to the equations.

First is the likelyhood to be dealt one of our hands. There are 6 ways to be dealt a pocket pair, and 13 pairs for 78 total combinations. AK and AQ can be dealt to a player 16 ways each for 32 more combinations, a total of 110 combinations which comprises 8.75% of all possible dealings. This equates to about 1/11.5 hands. Round this to an even 1/12 which works nicely for a 6max table, where we will be dealt a playable hand about once every 2 orbits. In those 2 orbits we will pay 3bb (bb in this essay stands for big blinds NOT big bets), so we must subtract 3bb from whatever our equity equation totals.

Preflop Equity

Our raise will fold the blinds to us 1/3 of all hands played

(1/3)(1.5bb) = 0.50bb

One-fifth of the remaining 2/3 hands will be 3bet preflop. Of these, we will hold AQ 14.6% of the time and fold. We will call the 3bet with the other 85.4% of hands, flopping a set and winning our opponent's stack 11% of those hands, and missing the flop and folding 89%.

(1/5)(2/3)(.146)(-4bb) = -0.08bb
(1/5)(2/3)(.854)(.11)(250bb) = 3.13bb
(1/5)(2/3)(.854)(.89)(-12bb) = -1.22bb

Postflop Equity

The remaining situations comprise 4/5 of 2/3 of all outcomes. First our cbet of 1/2 the pot will win the pot 1/3 of the time.

(1/3)(4/5)(2/3)(11bb) = 1.96bb

Our cbet will be raised 1/5 of the time. If we have flopped a set (11%) we will stack our opponnent, but if we have not (89%), we will fold even with overpairs.

(1/5)(2/3)(4/5)(2/3)(.11)(250bb) = 1.96bb
(1/5)(2/3)(4/5)(2/3)(.89)(-9bb) = -0.57bb

Our cbet will be called 4/5 of the time. I have just noticed an error leading to too small an equity in my equations. I have not satisfied the scenario in which our cbet is called, but we have flopped a set. In lieu of this I will continue with my original equations as completed, and work out an updated version correctly expressing this parameter. Therefore in this equity equation if our cbet is flat-called we CANNOT have a flopped set. This is obviously not true, and therefore our equity shall in truth be larger. Without a set we will complete one on the turn 4% of the time, and win 1/2 our opponnent's stack on average, and will miss the turn 96% of the time, at which point we cannot win the hand, again a fallacy as we will win with our pair in the hole at showdown some % of the time.

(4/5)(2/3)(4/5)(2/3)(.04)(125bb) = 1.42bb
(4/5)(2/3)(4/5)(2/3)(.96)(-9bb) = -2.46bb

Analysis

The summation of the equities yields a result of 4.64 big blinds. Again, this gain is after paying 3 big blinds, so our net gain is 1.64 big blinds/12 hands, or 13.7 big blinds/100 hands, which in regular poker terms comes to 6.8 Big Bets/100 hands before rake.

As stated in this essay, the mathematics are not completely accurate, and the data is skewed to a lower winrate than what should be actually attained through employing this strategy. I would like to gather additional data in order to more accurately express the likelyhoods of opponent's folding, calling, and raising in various situations. Despite these shortcomings, I feel I have sufficiently supported my hypothesis that playing only {AA-22, AK, AQ} at uNL tables with stacks of 250bb is a very profitable strategy.