Game theory simplified: Why fixed strategies fail

• Mike Caro   Exit

This article first appeared in Card Player magazine.

Good morning. I was supposed to write this column last night. It’s 8:38 a.m. now Christene will log onto her computer at Card Player and look for it via e-mail. But, alas, it just won’t be there. Steve and Linda will be wondering where it is. They will swear aloud; I know they will. They will say that I am irresponsible. They will be right.

Shirley will phone me in desperation. Janet will be waiting to type these words into her terminal to be printed out in high resolution, just like you see it now. She’ll be mad, too. Sometimes, it’s just not pleasant being me. You’re wondering why I’m telling you this. Well, actually, I’m just babbling while my mind starts to focus. My cerebrum is in shock. The only times it has experienced this early stage of the morning is when it has accompanied me through a poker session carried over from the night before.

Clearly, we’re going to have to keep things very simple today as I strive to become more alert. I know. Let’s start by talking about game theory.

Exactly what is game theory?

Sure, if you’re an astute student of poker, you probably know what game theory is. If you’re schooled in higher mathematical concepts, you also know. Sort of. But the plain fact is, most poker players don’t have a clue. Don’t worry. Game theory is just a term you hear bandied about by poker nerds. Frankly, the term frightens most normal poker players, because it suggests that a few opponents know something they don’t. Something powerful. Something sinister.

Say it ain’t so, Mike. OK, it ain’t so. I haven’t seen anyone show me a purely game-theory approach to any real-life poker game that could come anywhere near the profit potential of even common-sense poker. And nobody is likely to show me one for many years.

And when someone does come up with a complete strategy for a given game, based on game theory, nobody will be able to apply it very successfully while playing. You’d need a secret computer to quickly analyze which action to take when. The game that probably comes closest to being cracked by pure game-theory principles is ace-to-five lowball. But even there, a naturally skilled player armed with some basic research will fare better.

So, relax.

Anyway, I never did tell you what game theory is, did I?

Game theory explained, finally

The legendary mathematician John von Neumann is credited with pioneering game theory about 70 years ago. Also sometimes called the theory of games, it doesn’t just apply to poker. In fact, its most powerful application today is not to what you and I think of games at all. It is used for military decisions (or should be), for politics, and for everyday choices.

All you need to know right now about game theory is that it tries to provide the best decision at a given moment for a poker player by taking into consideration what your opponent’s best strategy is. For instance, should you bluff for $20 at this moment, attempting to win a $300 pot? Well, if you always bluff, your opponent should always call with his semi-weak hands. And if you never bluff, your opponent should never call with his semi-weak hand.

The truth is that to perfect your strategy, you shouldn’t bluff all of the time, but you must bluff some of the time. You need to consider your opponent’s options. For instance, if he has a weak hand does this mean your bluff will succeed? What if he raises? Should you just give him the pot then, or should you re-raise? Through game theory, you can determine – if all goes well – what frequency you should bluff and what frequency you should take any other poker actions.

What’s important is that once you know the frequently you should try to bluff in a situation, you need to randomize your actions. What does that mean? It means that if the answer you came up with through game theory analysis is that you must bluff one out of 10 times in a certain situation, this does not mean that you must choose one time in 10 to bluff. If you do that, your actions theoretically will be predictable by an all-knowing opponent. He would say, “Gee, no bluffs observed in the last eight tries. This means my opponent is going to bluff once out of the next two times, so I better call.”

Randomizing your choices

To get around this problem, all decisions should be randomly generated. Every time that the same situation occurs, you should act as if it has never occurred before and randomly decide whether to bluff. You might go 50 times in a row or more without bluffing, or you might bluff three times in a row. Just so you choose randomly, you’re safe.

How do you choose randomly? There are several ways that are almost random. One method I sometimes use is to remember combinations of suits that I saw on the previous hand, but that’s another topic for another day. Just trying to be random in your own head may be close enough, in practice.

You need to know this about game theory. Your objective is to choose an action that leaves your opponent with no way to take advantage of it. In the case of bluffing, a perfectly randomized strategy will mean that it doesn’t matter to your opponent whether he calls or not, as long as his decision is also randomly based and he calls the correct percentage of the time in the long run. A theoretically correct game theory solution cannot lose through eternity, but it cannot win through eternity, either, when pitted against a similar strategy. The only reason such strategies win is because opponents make mistakes. That’s the same way any other poker strategy wins, by the way.

It turns out that some hands are almost always playable, and some hands are almost never playable. but many hands fall in the middle and they should be played sometimes, but not others. This is why I cannot usually answer questions that players pose about whether they should play a hand. The situation is usually borderline, thus prompting the question, and the answer is often that they should play the hand sometimes. This does not turn out to be a very satisfying answer in the minds of most players, but it is the right answer.

Why game theory fails

Game theory fails, because poker is just too complicated. There are too many things to consider in a real game, and calculating a tactic for a given situation is seldom possible with any degree of precision. Even if you could calculate the exact answers for every situation, you wouldn’t have the time to do this while you were at the table, so you’d need to memorize all the answers. Good luck.

Another problem is that the game theory solution is apt to be aimed at opponents who respond perfectly. That’s the main challenge behind game theory. Play perfectly and let your opponent destroy himself, if he chooses. Although it may seem strange to you, if you have a perfect game-theory solution, you don’t need to adjust. No matter what your opponent does, he cannot win. The more he strays from the perfect strategy that you’re playing, the more you’ll profit from his exceptions.

But wait! What if your opponent bluffs all the time? Are you just going to let him do it? Again, you don’t need to adjust. If you stubbornly stick to your predetermined game plan (which means calling most of the time in a limit poker game), you win more when your opponent bluffs too much. However, you won’t win as much as you would if you simply said, “To hell with this game-theory stuff. I’m going to call all the time.”

Yes, you could adjust game theory to the circumstances, but that would mean going to the poker table armed with many sets of incredibly complex strategies, and I’ve already said that I don’t think anyone could handle even the basic one.

I inserted the paragraph below in 2014. The concept is covered in other Poker1 entries, but this discussion also benefits by its inclusion.

Added paragraph: Although you can adjust a game theory strategy (one that’s intended to battle another perfect strategy), hoping to make more money against imperfect opponents, doing so is risky. Whenever you adjust a game theory strategy that’s targeted to hold off an opponent making perfect decisions, you’re gambling that the perceived opposing mistakes are real and not illusory. It’s theoretically possible that an opponent who seems to call way too frequently is actually playing perfect strategy, but randomly calling more than expected. If so, your adjustment, in an attempt to take advantage, will just cost money. That’s because it’s suddenly you who has strayed from perfect strategy. That doesn’t mean you shouldn’t adjust. Often adjusting is worth the risk. Just realize that it might be a mistake. And, if you stubbornly stick to game theory and were right about the opponents’ mistakes, you’ll still have a profit expectation — just not as large of one as you would by correctly adapting.

Another problem with game theory is that, although it can more readily be applied to two-player competition, poker games among three or more players add a complicated twist, due to the possible benefits of implied cooperation among opponents. This does not need to be cheating. The complication arises, even in honest multi-way confrontations. I’m not saying this cannot be resolved through game theory. I’m just saying, show me.

Having said all that, you still should realize that game-theory investigations are going on right now in poker, and that they will yield some valuable information. We will tune strategies accordingly. But it is unlikely that you will ever personally need to understand anything other than the basic concepts of game theory, because you certainly are unlikely to use such a complex strategy in a real poker game.

Why fixed strategies fail

Actually, you can home in on a very decent strategy by using a computer to try out tactics. You let the first “opponent” adjust until it is maximizing its profit. Then you let the second “opponent” adjust, and on and on. I have seen situations in which this won’t work, because the strategies stubbornly whipsaw back and forth, rather than finding a focal point. But it’s sometimes a very powerful technique. At the end of this experimentation, you arrive at a fixed poker strategy.

The main point I want to make today is that fixed strategies fail, whether they’re game-theory strategies or not. By fixed strategies, I mean ones that don’t adapt to the conditions at the table right now.

Even a perfect game-theory strategy, targeted at perfect opponents, is not the best approach in real-life games. That’s because, most likely, your opponents are not playing anything near perfect strategy. If they are, you probably don’t want to be in that game, anyway. So, the best strategies are those geared for imperfect opponents.

That is why much published hold ’em advice, for instance, is far off the mark when speculating that hands like A-J are seldom profitable. When opponents play A-9, A-8, A-4 with regularity, A-10 is more likely to beat up on another ace than to fall victim to one. Whether you should play A-J or not depends on the exact circumstances and the nature of your opponents. An A-K often makes more money against A-J than A-J makes against A-9, so you need to consider that, too.

Here is what’s wrong with fixed strategies in poker. The problem is threefold:

1. If you use a perfect strategy geared at holding off perfect opponents, you probably won’t maximize your profits against weaker opponents;
2. If, instead, you use a perfect strategy geared at typical weaker opponents, you may have to wait for the game to adapt to you (the wrong set of opponents can sit down and cost you money while you hope for a better group to arrive);
3. Even an apparently excellent strategy can be overcome by two or more opponents colluding and playing inconsistently with the true strength of their hands.

Anyway, I’m going back to bed now. — MC