Mixed Strategies Nash Equilibrium: Intuition & Game Theory Logic Explained

For a mixed strategy to be evolutionarily stable, it must satisfy the ESS definition: (1) it is a symmetric Nash equilibrium, (2) it is not strict (since mixed equilibria make opponents indifferent), and (3) for any mutation, the original mix must do better against the mutation than the mutation does against itself. In the Chicken game, the 2/3-1/3 mix is evolutionarily stable because aggressive mutations do poorly against themselves, and passive mutations are exploited by the existing aggressive population.

Evolutionary game theory applies game-theoretic concepts to biological evolution. Symmetric games have identical strategy sets and symmetric payoffs (B = A^T). An Evolutionarily Stable Strategy (ESS) resists invasion by mutants: for any mutant x ≠ x*, there exists ε > 0 such that incumbents earn more than mutants in populations with proportion p mutants (0 < p < ε). The characterization theorem requires x* to be a symmetric Nash equilibrium and for any best response x to x*, π(x*,x) > π(x,x). In Hawk-Dove with V=2, C=6, the mixed equilibrium (1/3, 2/3) is an ESS.

An Evolutionarily Stable Strategy (ESS) is a strategy that, when adopted by a population, cannot be invaded by any mutant strategy; if a mutant strategy appears, it will be driven to extinction because it cannot achieve higher fitness than the resident strategy. In the Hawk-Dove game, the ESS is a mixed strategy where players choose Hawk with probability 2/3 and Dove with probability 1/3, meaning that when this strategy is widespread in a population, any alternative strategy introduced will not be able to spread and will eventually disappear.

This extensive section covers evolutionary game theory and its applications. A strategy is evolutionarily stable if, when it becomes dominant in a population, any small group of individuals using a different strategy (a mutant) will not be able to invade and replace the dominant strategy. The Prisoner's Dilemma demonstrates that neither Cooperate nor Defect is evolutionarily stable because each can be invaded by the other. The Hawk-Dove game illustrates a polymorphic evolutionarily stable strategy where 2/3 Hawks and 1/3 Doves coexist in stable proportions. The evolutionary stability criterion requires that a strategy be a symmetric Nash equilibrium and that mutants cannot invade by playing strategies that are also best responses to the equilibrium strategy. In games with three strategies, the mixed strategy equilibrium is found by setting the expected payoffs of all three strategies equal against the equilibrium strategy. The Rock-Paper-Scissors game demonstrates cyclical dynamics where the mixed strategy equilibrium (1/3 each) is not evolutionarily stable because all strategies can invade each other, creating perpetual cyclical dynamics. In Magic: The Gathering, the metagame can be predicted by constructing a payoff matrix based on win rates between decks and finding the Nash equilibrium, which predicts which decks will be most successful and which will be rare.

An evolutionarily stable strategy (ESS) is a symmetric mixed strategy Nash equilibrium that cannot be invaded by any small fraction of mutants. To check ESS, compare the mutant's expected payoff against the population distribution versus the average population payoff. If mutants cannot outperform the population even when meeting each other, the strategy is evolutionarily stable. The symmetric mixed strategy Nash equilibrium of the hawk-dove game (two-thirds hawk, one-third dove) is evolutionarily stable.

Behavioral game theory is an interdisciplinary research program that analyzes interactive strategic decisions using methods from game theory, experimental economics, and experimental psychology, focusing on how real human participants make choices that often deviate from the rational, utility-maximizing predictions of traditional game theory; unlike traditional game theory which is a normative theory prescribing what rational players should choose, behavioral game theory is a positive theory that explains actual decision-making through empirical experiments, revealing factors such as altruism, fairness, regret, social cooperation, and cognitive biases that influence real-world strategic interactions.

Behavioral game theory models explain human behavior through quantal response (stochastic choice based on utility differences) and cognitive hierarchy (iterative strategic reasoning). Level 0 agents are non-strategic (uniform randomization); Level 1 responds to Level 0; Level 2 responds to Level 1 and 0, and so on. Quantal cognitive hierarchy combines these: each level quantally best responds to levels below. This model dramatically improves prediction of unseen games compared to random guessing and mixed logit approaches.

Humans consistently fail to generate truly random sequences. When asked to choose a random number between 1-10, people choose 7 most often (the 'seven effect'). When generating random digits 0-9, they avoid repeating the same digit twice in a row (3% vs. expected 10%), prefer adjacent digits, and avoid 1 and 9. Rock-Paper-Scissors demonstrates game theory principles: the optimal strategy is choosing each option with equal probability (1/3 each), preventing opponents from exploiting predictable patterns. This state is Nash Equilibrium—neither player can improve by changing strategy. Research shows humans don't play randomly: they choose rock 35.4% of the time, paper 34.1%, and scissors 30.5%, because they intuitively believe rock is strongest. Common knowledge (everyone knows what everyone knows) affects strategic decisions. Randomization is essential in sports (tennis serves, penalty kicks) and law enforcement to prevent exploitation. Research published in the American Economic Review found tennis players exhibit the same randomization bias as humans, making them predictable to observant opponents.

Game theory provides a mathematical framework for strategic interactions but systematically fails to predict human behavior. While it predicts unique dominant solvable Nash equilibria (like zero in the beauty contest game), empirical evidence from the New York Times Upshot experiment with over 60,000 participants showed most people chose numbers around 50, not zero. This motivates behavioral game theory, which models how humans actually reason in strategic contexts. The core challenge is learning functions that map arbitrary game descriptions to predicted action distributions, evaluated by generalization to unseen games. Maximum likelihood estimation fits model parameters to training data, then assesses prediction quality on test data. Traditional approaches like logistic regression and mixed logit fail to capture behavioral patterns effectively, motivating exploration of more sophisticated behavioral models that incorporate psychological insights about human decision-making.

Action Game Theory (or Behavioral Game Theory) is a field that combines game theory with experimental methods to study how people actually behave in strategic situations. It recognizes that people may not behave according to game-theoretic predictions and seeks to understand the psychological and social factors that influence decision-making. This field is considered a bridge between traditional game theory and behavioral economics.

A Bayesian Nash equilibrium is a solution concept for games with incomplete information where players have private types (information) and must choose strategies based on their beliefs about other players' types; to find it, one calculates expected payoffs by weighting outcomes according to the probability distribution over different game states and verifies that no player can improve their expected payoff by unilaterally changing their strategy.

Bayesian Nash Equilibrium is a game theory concept where players have incomplete information about each other's types, and it can be solved using two methods: (1) converting the game with chance nodes into a matrix form and finding Nash equilibrium, or (2) treating each type as a separate player and solving the game accordingly.

Bayesian Nash equilibrium is a solution concept for Bayesian games where each player chooses a strategy (a mapping from their type to actions) that maximizes their expected utility at the interim stage, accounting for both strategic uncertainty (expectations about other players' actions based on their types) and payoff uncertainty (how other players' types may affect one's own payoffs); this equilibrium concept, developed by John Harsanyi in the late 1960s, extends standard Nash equilibrium to settings with incomplete information by requiring that every player's strategy is a best response given their beliefs about others' types and strategies.

Bayes-Nash Equilibrium extends Nash Equilibrium to games with incomplete information where players have multiple types; it is found by proposing a strategy for one player, calculating best responses for each type of the other player, and verifying that the best response of the first player matches the proposed strategy. In the Battle of the Sexes example with Maria having two types and João not knowing her type, the equilibrium depends on the probability of Maria being type 1: when this probability is between 0.25 and 0.75, the equilibrium is (Futebol, Telenovela) for Maria and Futebol for João; when probability is less than 0.25, no equilibrium exists; when probability is greater than or equal to 0.75, the equilibrium is (Telenovela, Futebol) for Maria and Futebol for João.

In Bayesian games with incomplete information, players have private types with associated probabilities, and equilibrium strategies must account for beliefs about opponent types. The Bayesian Nash Equilibrium is found by determining optimal strategies for each player type: Player 1 plays Left when type is Left (payoff 1 > -2) and Right when type is Right (payoff 2 > 0), while Player 2 always chooses Y (payoff 0 > -2 and 2 > 0), regardless of Player 1's type.