## Introduction

Gambling is at the foundation of poker; it doesn’t take too much skill to win a pot if you manage to hit a lucky hand, like a royal flush. Yet, there’s a reason certain names appear at the world finals each year, with many amassing total winnings in the millions. Along with incredible intuition and lots of experience, these people are able to exploit the probability of the game to win in the long run.

Winning in the *long run* is key. Game theory can’t guarantee you’ll win any given pot, but over a large number of hands, playing according to its rules will theoretically turn you a profit.

The following examples and strategies are for no limit Texas hold’em. The game begins with each player being dealt two cards which are hidden from the other players. A round of betting follows, where players can check, bet, call, or raise. A player can check or bet if no amount has yet been made in the current round of betting and a player can call (match the amount already bet) or raise (bet an additional amount) if the an opponent bets. After the initial round of betting (pre-flop), three community cards come out (flop). Another round of betting proceeds before the fourth card, and then the fifth and final card. After all cards are out, there is one last round of betting before the players’ hands are compared (showdown). Game theory optimal poker revolves around inferring probabilities through the many rounds of betting and making decisions that are profitable in the long run.

## Pot Odds

Understanding pot odds and calculating your hand’s expected value is key to game theory optimal poker. As an example, let’s say you’re heads up with \(8\clubsuit 9\clubsuit\). The turn comes \(2\clubsuit A\heartsuit 5\clubsuit J\spadesuit\), and your opponent bets $50 into a pot of $100. What do you do?

First, we need an estimate of how often we win. Since we have a flush draw, there are 9 club-suited cards left in the deck that can make our flush. This means that we have approximately a 18% chance of hitting our flush on the river. Now, we can calculate the expected value of calling with our hand as the probability of winning multiplied by the size of the pot after betting minus our contribution:

\(E(hand) = \frac{18}{100}(100 + 2(50)) - 50 = -14\)

Therefore, we will lose approximately $14 every time we call in this situation in the long run, so we should fold. In a real game, we don’t need to necessarily calculate the exact expected value. We know that we have about a 18% chance of winning, and our *pot odds*, or the ratio of the bet to the pot size, is 3:1 or 25%. Our chance of winning should be greater than the pot odds to call profitably.

## Balance

Although the expected value of your hand is an important tool, playing solely by it can make you easily exploitable. If you always bet proportionally to the profitability of your hand, your opponents can simply fold when you bet large and play aggressive when you check or bet small. After all, bluffing is as integral to poker as value betting. Therefore, we need to balance our ranges.

As an example, say the river comes, and we lead betting $100 into a pot of $100. Our opponent has 2:1 pot odds and needs to win 33% of the time to be profitable. Therefore, we should bluff 33% of the time. For simplicity, let’s say that at showdown, a value bet means you have the better hand and a bluff means you have the worse hand.

Let’s see how much money you can gain in the long run with different ranges:

Case 1: Value bet 100%, Bluff 0% If you always value bet, your opponent can always fold. You make $100 from the pot.

Case 2: Value bet 0%, Bluff 100% If you always bluff, your opponent can always call, and you lose your $100 bet plus the $100 pot.

Case 3: Value bet 50%, Bluff 50% If your opponent always calls, you win $200 50% of the time and lose $200 the other 50% of the time for a net profit of $0. If your opponent always folds, you make $100 from the pot.

Case 4: Value bet 66%, Bluff 33% If your opponent always calls, you win $200 66% of the time and lose $200 33% of the time for a net profit of $100. If your opponent always folds, you make $100 from the pot.

As we can see in Case 4, a perfectly balanced range is not exploitable. No matter what your opponent does, you will profit the same amount.

## Closing Thoughts

Understanding how to exploit game theory optimal poker is a necessary foundation to any great poker player. But being able to know when and how to adjust your ranges based on the gameplay at the table is what separates the great poker players from people who simply understand the basic concepts. At the end of the day, experience and talent for reading others, along with pure luck, are also important factors to consider.