Stick suggested I should create a new thread instead of posting into the dead one, so here goes:
A headsup NLHE poker match under the following conditions was discussed:
- Starting stack sizes are 200 units a piece
- Blinds are alternating, sb=1, bb=2, and never go up
- There are no antes
- The pushmonkey calls if the pro moves all in, if the pro limps or raises the pushmonkey pushes all in. In other words, the pushmonkey is all in every hand regardless
Questions: What is the optimal strategy for the pro? What is the pro's winning expectation?
This game is easy enough to be solved computationally, and complex enough to be interesting (to me at least, but then I have been pathologically attracted to optimal game solutions since seeing a chess endgame database for the first time), so I gave it a try. As I already mentioned in the dead thread, I was too lazy to incorporate ties properly, so my numbers refer to a holdem variant where there's no split pot, but the whole pot is awarded to the winner of a fair coin toss in case of a tie. That's obviously a bad variant for the pro, but I think it shouldn't matter much.
Results:
The winning expectation of the pro is 68.35% if pro is big blind in the first hand, 68.44% if pro is small blind in the first hand.
The winning expectation depending on pro's stack size (and only with pro being big blind at that point, as the differences are marginal) is plotted here:
Given this "match equity table", it is trivial to compute the optimal strategy (That is, if you know the preflop value of every hand vs random hand. That information is given here.)
The optimal strategy is summarized in the following plots:
The horizontal axis shows the pro's current stack size, the vertical axis the threshold value for hands worth calling/pushing/jamming (I don't know much about poker terminology) with. Two curves are plotted, where the upper one (tighter play) is valid when the pro is small blind. For better orientation, I put in horizontal lines that correspond to the values of AKs (upper line), 77 and AJs respectively in the third image, and to AT, JT and Q5 respectively in the second and fourth images. So at the beginning pro only calls with 88-AA, but soon AKs, 77 and AQs (which is very close to 77) are added to the mix. If one of the players is very short-stacked, the optimal strategy becomes similar to cash game strategy (i.e. threshold hand values are close to straightforward pot odds).
What I find notable about these graphs is
- the sharp peak at stack=200
- asymmetry: pro tends to play more loosely when leading than when trailing by the same amount. This is especially true close to stack=200
Changing the perspective:
When I first encountered a pushmonkey-vs.-pro thread in another forum, where very tight play was advocated, my initial reaction (silent, luckily) was: Why tight? Pro's play is governed by pot odds, he should call whenever he's the favorite in the hand, at least. Obviously my initial reaction was way wrong, because in this setup, the goal is not to maximise the (cash)-EV in the hand, but in the match. Pro is willing to make suboptimal cash decisions in a hand, if they increase the expected amount of outplaying (let's call it EAO) that occurs in the rest of the match accordingly. This is similar to backgammon with asymmetric METs, but much more significant.
The following plot shows the EAO for the pro playing the optimal match strategy depending on the stack size. Below the horizontal axis, the maximum amount of cash EV the pro is ready to sacrifice in one hand is shown.
With the optimal match strategy, pro has positive cash EV in every hand despite playing suboptimally in that sense. The following plot shows how much it is depending on stack size (crosses stand for pro being in the small blind) :
What happens if we increase the total number of chips?
With 300 units initial stack sizes each, pro is at 72.6%
With 500 units, pro is at 77.2%
With infinitely many units, pro is at 85.2% (i.e. the value of AA vs. random hand).