Short-chipped and all-in.

One of the simplest situations in poker is when you or an opponent is short-chipped or nearly forced to go all-in.  This comes up often in tournaments, but very rarely in cash games.  Most people play this very badly, but it's actually one of the simplest situations in poker, because there is not further action.  You don't have to worry about future bets and whether you can be confident with your hand, EV's and implied odds and such are all out the window.  You basically just have to know how your hand does in a race, that is a full board to a showdown.

Consider, for example, that the player in the big blind is nearly all-in.  He has put in a big blind of 400 chips and he has 100 chips left.  If he is a decent player, he won't fold any hand - even with a 23o he should call, since he'll be getting 9:1 odds and 23o wins about 32% of hands.  You're on the button, the question is, should you put him all in?  If you call, you should put him all-in, and in fact, if the small blind is still live, you should probably raise to try to isolate.  You want to play heads-up against the big blind.  So, you know he's on a random hand since he won't fold any hand.  Thus, in order to have a +EV move, you just need a hand that's better than 50% in a race against random hands.  We use computer simulations to compute this.  There's a full table for all hands in Appendix @@, but you can use some simple rules of thumb to know what to play.  The hands that are 50% to win are K2, Q5, J8, T9 (and of course any better hands).  So, you'll play any Ace (Ax), any King, and then QQ,QJ,QT...Q5, and so on.  Note that this is just in terms of pure EV for that call, so you usually don't actually want to just play hands that are 50%.  You have to consider your situation.  Assuming this is a tournament, what is your stack size?  Does it benefit you to knock him out?  Maybe you should wait to be even a bigger favorite.  This also depends on how he plays and how you play.  For example, if he is a much better player than you, you should be happy to have a 50% shot of knocking him out.  If you let him survive, he'll press his advantage against you and he'll have a better than 50% shot.  On the other hand, if you are also a short stack, it may not be worth taking the gamble.  

The correct thing to do is to consider both cases, and what your total EV would be.  You're considering taking a 50% shot, so you compare two cases : Case 1) you fold, you have C chips.  Case 2) you put the big-blind all in, 50% of the time you win, 50% you don't; compute a combined EV of 0.5 * EV if you win + 0.5 * EV if you lose.  Now you just look at which EV is better, and that tells you whether to go in or not.  The simplest case is at the end of a tournament.  You have A chips, and he has B chips.  Assuming you are equal players, your chance of winning first place is A/(A+B).  For simplicity, let's say first place pays 1 dollar and second place pays nothing (more realistic payouts have the same analysis).  Should you take a 50% shot?  In Case 1 your EV is just A/(A+B).  In Case 2, if you win your EV is 1 dollar; if you lose, your stack goes down to A-B, and your EV is now (A-B)/(A+B).  Your net EV for Case 2 is 0.5 * 1 + 0.5 * (A-B)/(A+B) = 0.5 * (A+B)/(A+B) + 0.5 * (A-B)/(A+B) = A/(A+B).  It's exactly a wash!  So, you need a slightly better than 50% shot to make it worth calling.  In practice, it depends more on how you think you play against each other, and exactly what your chip ratios are and how big the blinds are.  If the blinds are very big compared to your stack, then you need to push an advantage as soon as you have it.  If the blinds are very small compared to your stack (we assume your opponent's stack is roughly the size of the blinds), then you can fold a few times and wait for a slightly better advantage.  Note that your advantage is probably never going to be very big.  Even hands like KJ and A8 are only 60% favored against random hands.  That sort of 60:40 advantage is usually all you can hope for in Hold'Em - that small edge is how you make your money!  One big mistake that people make is to think that a 55:45 advantage is not very good - yes it is!  You're not going to get much more!  You can't sit around and wait to have pair-over-pair (which is an 80:20 favorite) or something like that.  Even AK against random hands is only about a 65% favorite.  Poker is not about having sure shots, it's always a gamble, you simply want to have the best of it, and maximize your EV and minimize your variance.

The opposite case is even easier.  When you're in the big-blind nearly all-in, you just call with just about any hand.  The worst hand heads up is 32o (not 72o as some incorrectly say), and it still has a 32% chance of winning against random hands.  Let's say half your stack is in as the big blind, so the big blind is 400 chips and you have 400 left.  It's heads up, and the button  (small blind) comes in for a raise, making it 800 to go.  You should call with any hand.  The pot odds are 3:1 since you need to put in 400 to win 1600.  I prefer to think in "pot percents", so the pot percent is 25% and your worst hand has a 32% shot at winning; that's an easy call.  If you fold in this scenario, you are making it way too easy for your opponent.  Now, heads up play is quite a special subject, so we'll address it in more detail later.

Consider this case.  You're playing short handed and you're short-chipped.  You raise with QT from the button.  The big blind has a bigger stack and puts you all-in.  Should you call?  Let's say there are now 30 chips in the pot after he sees your raise, and you have 30 more chips, so you are getting 2:1 odds, or 33% pot percent.  He would go all-in with any pocket pair, or any ace.  Against any under-pair you are about 50/50 ; against TT and QQ you are dominated.  Against JJ you are a 30% dog; against KK or AA you are a 20% dog; against Ax you are about a 40% dog, against AQ or AT, you are a 25% dog.  We don't even have to do the full math, since against most hands you are a 40% or 50% dog.  There are very few hands that give you worse odds, so calling a 2:1 shot is an obvious call.  You're quite sure you're beat here, but you still must call because you are getting very good odds.