March 17, 2010
Poker software has played a huge part in the increased level of understanding of the average player. PokerStove is the pocket calculator of choice for the NLHE player. I cannot imagine how difficult it must be for anyone to progress quickly at their game of choice without an equity calculator at their disposal. Iâ€™ve written, or attempted to write, similar programs for the other games in the 8-game mix. Thankfully I donâ€™t have to rely on my own hackneyed efforts because of the brilliant propokertools (omaha and stud) and troutulator (27TD, lowball, badugi, 5-card draw). Given all these great tools, why, then, do so many people still suck so badly at these games?
Formulating a winning poker strategy is incredibly difficult, even with the wide range of tools and readily available knowledge-base of the poker community. A program like pokerazor (not currently available) or StoxEV can manage a modestly sized game tree and greatly assist the enthusiast wanting to develop a strong strategy for NLHE. Alternatively, a game tree can be sketched on paper, in Excel, or any other way you like. However, for all but the simplest sub-games of poker the task of defining a good strategy is monumentally difficult. Almost all of the â€œsolvedâ€ cases in poker game involve an information set that assumes unrealistic levels of certainty about the strategy and/or hand range of the opponent player.
The Nash tables for short-stacked HU NLHE give an unexploitable strategy for games where chips correlate linearly with equity. Even this is only a partial strategy as the player acting first is constrained to either shove all-in or fold. For stack depths of 20bb or greater it seems very likely that other strategies may exist that force a higher expected value for the player acting first. As of yet, no one has publicly announced a single comprehensive strategy that does this. The very-short-stacked HU NLHE game stands alone as an example of a comprehensive poker strategy that is common knowledge. Only when such a situation exists can alternative strategies can easily be identified as mistaken.
What can we do with the software we have? Apart from the Nash HU NLHE tables, Polaris has been shown to be play in a way that could said to be close to solving HUHU LHE. All other HUHU games so far appear to be way out of reach. Even then, mimicking the bot would not guarantee a strategy absent of mistakes. Memorising that strategy would be even more difficult. Certainly, most mistakes made at the highest levels in all poker games are small and are not easily quantified. Against some players, making these mistakes may even be the reason we are winning.
I am going to conclude with a chess analogy to demonstrate how little value we should put in being able to define what it means to make a mistake. Most chess players agree that there are certain first moves and opening lines that are inferior to others. It is a mistake to play 1. a3 as white, although a large public database of games still shows that white wins 50% of the time when making this move (cf. >55% for 1. e4 or 1. d4). It is possible that a very strong player could make this move against a well matched opponent and easily win a majority of his games; the other opponent may be totally unprepared to counter this opening. In the same way, a winning NLHE player may consistently open a very wide range from UTG and still maintain a very high win rate. I could easily make some assumptions and show that folding a few extra hands is probably more profitable. Fortunately, this small mistake is not enough to collapse the entire game-tree â€“ in fact it changes it dramatically; the counter-strategy is not simple enough for the opposing player to formulate and fortunately there are no programs that can do it yet, either.