Calling all-in first hand with AK

In my recent post about my online SNG results, I included my self-assessments of the games in which I went out unusually early. One of them was having called an open-shove with AK on the very first hand of a tournament. I assessed the loss this way: "Guy went all in 1st hand with Q-9, I called with AK. He caught a 9. Definitely right move."

Commenter "Terry in Victoria" questioned this: "I just disagree on one thing where you called an all in on the first hand with AK. At best its a race for all your chips. Your a much better player than that and why risk them all when you can get your money in a much better times with less risk and more reward."

I'll admit that I had made my assessment on just a general sense that it was the right move, but this comment made me wonder. I decided to explore the question, since this is the kind of thing that math and modern poker software tools allow us to settle with a fair degree of confidence. It's a worthwhile question, because these low-stakes games have a small but steady number of players who shove all-in on the first hand no matter what they have and no matter what the action before them has been. I see it in maybe 15% of the games I play, so it's worth knowing how to respond to it.

First I went to an independent chip model calculator, here. If I lose, my equity obviously drops from 10% of the prize pool to zero. If I win, I go from 1500 chips to 3030 (doubling up plus the 10/20 blinds), which increases my prize-pool equity to 18.6%.

Let's assume that I call every time, and then set x to be the fraction of the time that I have to win in order to make the call profitable over a large number of trials. To simplify the math, I'll assume the prize pool to be $100. Then 18.6x (the amount I win) has to equal 10 (my original equity) to be a break-even call. X then must be 53.8%. I.e., if I win more than 53.8% of the time, it's a +EV move.

Next we have to figure out how A-K offsuit fares against the range of the shover. For that, the oddly named but wonderful PokerStove is the tool of choice. From it I learn that A-Ko is 65.3% to win against a random hand. That is plenty to make this a +EV call.

Now, one can still argue that it's not the best move, not on the basis that the call is -EV per se, but because one will likely be able to find spots to get it all in with even higher equity later. That may be so. I don't know, and I don't think it's a question that can be easily answered objectively. My gut sense, though, is that the call is indeed worthwhile.

Note that this calculation assumes everybody else will fold. A wrinkle is that that may not be so, and the hands with which one is likely to draw a second caller will be heavily populated with aces and kings, making it harder for my A-K to win. But I don't know how to account for that mathematically, so I'm conveniently disregarding it.

After finding the solution (or at least partial solution) to the case where I have A-K and the shover has any two cards, I wondered how much narrower his shoving range could be and the call still be a correct one mathematically. The answer surprised me. If an opponent is open-shoving not with every starting hand, but only with his top 50%, the answer changes only negligibly; I'm still 64.9% to win. If he tightens up and shoves with only the top 33% of his starting hands, I'm still 64% to win. Top 20% of his starting hands still leaves me with 62% winner, and even his top 10% of hands gives me 57%, just barely enough to be +EV.

The break-even point is if we somehow knew that he was only shoving with the top 7.4% of his possible starting-hand range, which means pairs 88 or better, suited aces with 10 or better kicker, unsuited aces with queen or better kicker, suited kings with jack or better kicker, and Q-Js. Maybe there exist players who adopt such a strategy (shoving with that range only), but I kind of doubt it. My impression is that those who open-shove do it with pretty much anything, hoping to get lucky and double up early or lose and start another game where they try the same gambit. These are not the world's most mathematical, analytical players.

An interesting related question is what range of hands I should call with if I know that my crazy opponent is, in fact, shoving on the first hand of the tournament with any two cards. Again my win percentage has to be about 54% to be mathematically correct. (I'm assuming the ideal case, where I'm either last to act and everybody else has already folded, or I somehow know that everybody else will fold behind me.) Top 50% of my starting hand range gives me 57.8% equity against a random hand, which is enough to do the trick. That's a huge swath of hands. I tried tightening it up somewhat: Any pocket pair, any suited or unsuited ace, and any two Broadway cards. That's about 28% of starting hands, and it yields 61.8% win equity, which is plenty.

I'm not sure I'll actually change my game to call that loosely--largely because of the fear of somebody else behind me overcalling with a much narrower range that has me crushed--but it's interesting and worthwhile knowing that the math says it works.