No-Limit Holdem
The Mathematics of Short-stack Play
Johannes M.V.A. Koelman
Introduction
Many poker players believe that in cash games, it is best to have a stack at least as deep as your opponents. In their book "No Limit Hold'Em, Theory and Practice", Sklansky and Miller dispell this myth and stress the intrinsic advantages of playing short stack. Key advantage of being short-stacked is the fold equity provided to you 'free of charge' by the opponents betting after your all-in. The short-stack player is the parasite profiting from deep-stack hosts that battle amongst each other.
In the following I will demonstrate this by working out the maths for the extreme case of an ultra-short stack (a nano stack) in a three-player cash game. It will be demonstrated that buying-in ultra short stack is an unexploitable strategy with a positive expected payout as soon as the deep stacks start battling amongst each other following your all-in.
A simple three-handed game
Consider a No Limit Holdem cash game between three players. To simplify the analysis, we assume that the small blind and the big blind both have to post an equal amount of money ($1 each) prior to the cards being dealt. One player, the nano stack, has an ultra short stack of $1, and is therefore automatically all-in when seated at one of the blind positions. When on the button, the nano stack can decide to fold, or to make an all-in call. For the nano stack, this push/fold decision on the button is the only decision required during the game. The other two players are assumed to be deep-stacked, with stacks many times the blinds.
We first consider the case in which the deep stacks have a 'gentleman's agreement' of soft-playing each other and not attempting to push each other out of the pot. Hence, both deep stacks effectively don't use the full depth of their stacks and don't bet beyond the nano stack's all-in by limiting their actions to checking and calling.
Given this gentleman's agreement, the optimal play for the nano stack is, when on the button, to make an all-in call with those hands that have a winning chance against two random hands of at least 1/3, and fold with all other hands. According to PokerStove, this means going all-in on about 43% of the hands (see picture), which together have a pot equity of some 41%. Hence, the expectation value when on the button is EV = 0.43*(0.41*$3 - $1) = +$0.10.
Obviously, with the gentleman's agreement in place, both other players effectively leave their stacks beyond $1 untouched, and will check when on the blinds, and call with the same 48% of the hands when on the button. The result is that the EV on the blinds is EV = 0.57*(0.50*$2 - $1) + 0.43*(0.295*$3 - $1) = -$0.05 (here, 0.295 represents the pot equity of a random hand against another random hand and the 43% call-range hand). Obviously, the total EV for each player is neutral when summed over the three seats ($0.10 - $0.05 - $0.05 = $0.00), as it should be.
Now, when not hampered by a softplay agreement, the big stacks can individually do better whenever they receive a good hand, as they can improve their expectation value by raising the stakes. Let's try and see how well the deep stacks would do against the short stack, if they would start using their stacks and bet beyond the size of the short stack.
We take a straightforward case. Suppose one of the deep stacks is on the button and has a good hand - say 88 - whilst the other deep stack holds a very poor hand such as 32o. Under the gentleman's agreement, and in the absence of any raises, PokerStove yields a pot equity for the pocket eights of 60%, whilst the pot equity of the 32o is only 12%, and the remainder (28%) goes to the short stack (assumed to be a random hand).
The hand with the pocket eights can make a pre-flop raise to which the hand with 32o would fold. This is a profitable move for the raiser, as this increases his pot equity from 60% to 69%. However, the remainder of the pot equity (31%) now goes to the short stack, so that his pot equity increases from 28% to 31%. Such a 'free of charge' post-all-in equity increase for the short stack is not particular to this example, but happen in general. Whatever post-all-in folding takes place, the short stack will profit from it. As these short-stack equity increases happen against the EV-neutral base case characterised by the absence of post-all-in folding (i.e. with the gentleman's agreement in effect), the EV of the short stack is strictly positive (and therefore the EV of the deep stacks strictly negative) whenever the deep stacks attempt to optimise their own EV and make the other deep stack fold.
Conclusions
In the above three-player game, if one player buys in ultra-short stacked, she puts the other two deep-stacked players into a prisoner's-dilemma type of game. The only way whereby both deep stacks can avoid the expectation value of the nano stack to grow positive, and hence their own growing negative, is to cooperate and refrain from making any raises when the nano stack is all-in, even when making a raise would be EV-positive for their individual stacks. I.e. they have to leave the depth of their stacks beyond that of the nano stack untouched, and effectively have to play equally short stacked. Hence, playing short-stacked constitutes a dominant strategy.
Would nano-stack buy-ins be allowed, buying in for more than the big blind would be foolish. Fortunately, in all brick-'n-mortar and all on-line pokerrooms, a minimum buy-in (generally 10 or 20 times the big blind) is prescribed. But even then: given that nano stacks can not be exploited (no matter how strong the opponents!), buying in for more than the minimum is likely to render your play more exploitable, and should only be considered when all your opponents are deep-stacked as well, and you know you play considerably better than your opponents.
Keywords: Poker, Holdem, Math, Game Theory
