The Fight Is On: Game Theory Versus Pot Odds

What if I told you that sometimes pot odds don’t matter? You might say that it flies in the face of fundamental poker knowledge. Right. Well fortunately for my side of the argument, there is an area of mathematics that is specially dedicated to explaining this wizardry. It’s called game theory.

Game theory can get extremely complex when it is used to describe games like no-limit Texas hold’em. But, at the core of all the math, learning game theory optimal poker, or as close as we can get to it, is about learning how to create a strategy that can’t be exploited by your opponent. Now, if you have ever played poker, then you know that everyone makes mistakes constantly. Is your continuation bet (c-bet) the exact correct size? Should you bluff that river with that hand? Great players make solid decisions, but even when they try to be optimal, they fall short, because there are often just too many variables to optimize. The entire game of no-limit is too complex to say what is exactly right in all spots. And since game theory optimal poker is such a dense and mathematically complex topic, let’s create a bit of an intuitive example, without all the messy math.

We have two players, Sam and Shakira. Sam plays a fairly balanced play style, mostly betting his hands when their face value is greater than those of his opponents, and pairing those value bets with bluffs. He is a very solid player who is very careful about when he risks a bet. If he is checked to and has a weaker pair, he nearly always tries to show it down to see if he wins. On the other hand, Shakira is a bit less predictable. She can call on earlier streets lightly almost solely with the intention of bluffing later, and she isn’t afraid to bluff with a pair that she thinks isn’t going to win the pot very often at showdown. These are two very different, but also very common types of players.

Now, let’s look at one board run out, hand, stack size, and opponent. Let’s do this twice, so the first time we can pretend that we reach the river against only Sam, another time we are playing the river against Shakira. So that’s two different river scenarios, but the same board run out, cards, and stack sizes. Let’s get to it.

We are the out of position preflop raiser and we did some betting on the flop and turn of a K 7 5 10 board. The river is the A and we check our A J to whichever player we happen to be facing because we don’t think they will call with a worse hand if we bet. The remaining effective stack size is $100. The pot size is $200.

What if Sam bets all-in for $100? Well, we know that Sam is the type to check back with a hand like pocket nines and also the type to never have a bluff that was a missed draw on the turn. After all, most players as tight as Sam aren’t calling with hands like 8-6 suited preflop unless we have far deeper stacks than in this example, and even if he did, he’s going to fold it on the turn that’s the third heart, unless he made a flush. So if Sam bets $100, we know he simply is not bluffing enough of the time. Another important question to answer when facing players like Sam is that is he ever value betting a worse hand? No. What ace can he now have that isn’t otherwise connected with the board? A 10x beats us, so do A-K, A-7 suited and A-5 suited. Simply put, Sam is checking back all his losing hands in this spot, and betting with all his winners, which is of course, a pretty easy strategy to exploit.

What if Sam bet $5? That’s right, $5 into a $200 pot. We are getting 41-to-1 pot odds, which are probably about the most ridiculous odds you have seen offered on a bet. We only need to win the pot once in 42 times, or just over 2.3 percent of the time in order to make a profitable call; but, should we call Sam’s bet? Well, with our assumptions laid out above, Sam is not bluffing, nor is he value betting a worse hand. So unless the value of the information of the way Sam plays his hand is worth more to you than $5, and it rarely is since this is one hand in one spot from one player, then you have another easy fold. Another and perhaps more intuitive way to look at this is that Sam knows a few concepts as well. Remember, we said he’s fairly solid, so he is considering his options.

He knows that this river is a very scary one for you and a lot of your hands and that you checked to him, which indicates weakness as well; he knows that you could very well check/call with a pair of aces or a similar hand; and he knows that a very small bet is less scary than a larger one and that the small bet might induce a crying call, regardless of your holding. Well, now you know that Sam is choosing the bet size and that he’s never bluffing, so you simply fold to any bet he makes. The important concept here is that regardless of pot odds, sometimes a fold is simply a fold.

What if Shakira bets all-in for $100? Well, now you have a bit more complex of a decision. You are getting 3-to-1 odds on your money, the $300 in the pot and $100 to call and win it. So to break even, we need to win the pot 25 percent, or 1 out of 4 times. In other words, you need to call this bet with the top 25 percent of your river checking range; otherwise, Shakira can simply bet 100 percent of the time when checked to in order to profit much more than simply hoping that we have given up bluffing on the river always. Obviously, our river check isn’t always a busted semibluff; we have A-J here.

Shakira knows she probably shouldn’t check back with her hands like pocket nines. So she might bluff with nines in this spot, since they’re so unlikely to win the pot given the action, and that bluffs make for a more difficult river decision for you. Let’s assume that she has a balanced range of bluffs and value bets, a range made up of these weaker made hands with little to no showdown value, and of two pair hands and better. Your life is tough, but you decide that you want to avoid being exploited and that you are forced to call the top 25 percent of your river checking range. You think A J is definitely in the top 25 percent of your river checking range, so you call. You win some. You lose some. That’s poker people. Remember, your goal is to prevent Shakira from winning 100 percent of the time, so you have to call with some of your hands, even if they suck. This is different than facing Sam, who is not bluffing.

What if Shakira bets $5? Now this is an interesting spot. A lot of players who are as creative as Shakira with finding ways to bluff in spots where it looks like they can’t or shouldn’t have a bluff might also underbet the pot to deceive their opponent. But why is she trying to deceive you? Just like Sam, you know Shakira knows the dynamic of the river spot; but, what is interesting is that she also has a “balanced” value betting and bluffing plan to go with her $100 betting strategy. So why would she bet $5? It’s very likely that you’re facing an opponent using an exploitable strategy. What this means is that you can very likely always fold versus the $5 bet, but the implication of this size drastically affects the best way to play your other hands. For example, if you know that Shakira value bets all rivers for $5 and bluffs all rivers for $100, then you can always fold to the $5 bet and always call the $100 bet with hands that beat pocket nines. Pretty cool. Now you are making money on both ends, and you didn’t even need to see her cards to arrive at this conclusion, only extrapolate from her play and bet sizing. Nice. Be careful though, a world-class player will simply have balanced strategies that include both value bets and bluffs for multiple river bet sizes, and your attempts to outguess those types of players, rather than play based on avoiding exploitation, is not going to end well.

You can see how pot odds are very much a guideline when situations become borderline, and when your opponents are creative. Since opponents control their own betting frequencies, oftentimes the best you can do is choose a solid range of hands for reacting in a particular way. Boards run out in such a way that give one player the nuts much more often than the other player, and, especially if a lot of money and betting has already been done, one player is just along for the ride so long as the bettor is capable of bluffing in the spot. So make sure that you understand your opponent and his or her outlook on the game. Can he or she bluff when they bet into you? Should you ever call? Now you have a much better idea how to answer these questions for yourself. ♠

Reid Young is a successful cash game player and poker coach. He is the founder of TransformPoker.com.