Game Theory Basics for Poker

🎲 What Is Game Theory?

Game theory is the mathematical study of strategic decision-making between rational agents. It answers the question: "What is the best strategy when your outcome depends on what others do?"

Poker is a natural fit for game theory because:

• Multiple players whose payoffs are interdependent
• Sequential decisions with hidden information
• Zero-sum at the table (every dollar won comes from another player)
• Rational opponents who adapt to your strategy

Zero-Sum Games

In a zero-sum game, one player's gain exactly equals another's loss. The total payoff always sums to zero:

$$\sum_{i} u_i(\sigma) = 0 \quad \text{for all strategy profiles } \sigma$$

Cash game poker (ignoring rake) is zero-sum. This has profound implications: it means there is always an optimal unexploitable strategy — the Nash equilibrium.

⚖️ Nash Equilibrium

A Nash equilibrium is a strategy profile where no player can improve their expected payoff by unilaterally changing their strategy, given what all other players are doing.

Formally, strategy profile \(\sigma^* = (\sigma_1^*, \sigma_2^*, \ldots, \sigma_n^*)\) is a Nash equilibrium if for every player \(i\):

$$u_i(\sigma_i^*, \sigma_{-i}^*) \geq u_i(\sigma_i, \sigma_{-i}^*) \quad \forall \sigma_i$$

Poker Translation

In poker, a Nash equilibrium strategy means: if you play GTO and your opponent plays GTO, neither of you can exploit the other. You are both playing optimally given the other's strategy.

A Simple Example: The Bluffing Game

Suppose the pot is $10 and you can bluff or value bet for $10. Your opponent can call or fold.

If you always bluff, opponent always calls and you lose. If you never bluff, opponent never calls and you win less from value bets. The equilibrium requires mixing — bluffing some fraction of the time.

The equilibrium bluff frequency (from MDF analysis): with a pot-sized bet, bluff \(\frac{1}{3}\) of the time. The opponent is indifferent between calling and folding.

🛡️ Minimax Strategy

In a two-player zero-sum game, the Nash equilibrium is equivalent to the minimax strategy: minimize your maximum possible loss.

$$\sigma_1^* = \arg\max_{\sigma_1} \min_{\sigma_2} u_1(\sigma_1, \sigma_2)$$

Your opponent is trying to minimize your payoff, so you choose the strategy that maximizes your worst-case outcome.

The Minimax Theorem (Von Neumann, 1928)

For every two-player zero-sum game:

$$\max_{\sigma_1} \min_{\sigma_2} u_1(\sigma_1, \sigma_2) = \min_{\sigma_2} \max_{\sigma_1} u_1(\sigma_1, \sigma_2)$$

This theorem guarantees that every zero-sum game has a unique optimal value — even if both players use mixed strategies. This is the mathematical foundation of GTO poker.

Minimax in Practice: Defense vs Aggression

When you face a bet, the minimax perspective says: choose the calling frequency that makes the bettor indifferent between bluffing and value betting. This is exactly the MDF (Minimum Defense Frequency):

$$\text{MDF} = \frac{\text{Pot}}{\text{Pot} + \text{Bet}}$$

By defending exactly at MDF, you minimax — you make bluffs zero EV, preventing your opponent from exploiting you by over-bluffing.

🎰 Mixed Strategies — Why Randomize?

A pure strategy means always taking the same action in the same situation. A mixed strategy means randomizing between actions with specific probabilities.

Why Pure Strategies Fail

Suppose you always continuation bet the flop with top pair. Your opponent can then:

• Raise with two-pair+ to extract maximum value
• Float with draws knowing your range is narrow
• Fold bottom pair immediately instead of calling

Your predictability becomes exploitable. A mixed strategy prevents this.

The Indifference Principle

At a Nash equilibrium, whenever a player mixes between actions A and B, both actions must have equal expected value. If one had higher EV, the optimal strategy would be to play it 100% of the time.

$$EV(\text{bet}) = EV(\text{check}) \implies \text{mixing is optimal}$$

This is the fundamental principle behind solver outputs. When a solver says "bet 60%, check 40%," it means at those frequencies, both actions are equally profitable given opponent's response.

Practical Mixed Strategy Example

You hold top pair, second kicker on the river. The solver says: "Bet pot 30%, bet half-pot 50%, check 20%."

This mixing serves several purposes:

• Different sizings extract value from different opponent ranges
• Checking some top pairs balances your checking range
• Prevents opponents from knowing your hand strength from sizing alone

👑 Dominant Strategies and Dominated Actions

Strictly Dominant Strategy

An action is strictly dominant if it produces a better payoff than all other actions, regardless of what the opponent does. In poker, folding a dominated hand preflop (e.g., 72o from early position) is a near-dominant strategy — it is best regardless of opponent action.

Dominated Actions to Eliminate

Iterated elimination of dominated strategies simplifies game analysis. In poker, this means:

• Never bluff-raise the river if opponent never folds (dominated by calling or folding)
• Never slow-play the nuts if opponent never bets (dominated by betting yourself)
• Never call with zero equity and zero implied odds (dominated by folding)

The Prisoner's Dilemma Connection

The classic game theory "Prisoner's Dilemma" illustrates how individually rational decisions can lead to collectively worse outcomes. In multi-way poker pots, this analogy applies: two players both betting into a dry side pot may both lose EV compared to one betting and one folding.