Mixed Strategy Nash Equilibrium Quiz

A mixed strategy Nash Equilibrium occurs when each player assigns probabilities to their available strategies and randomizes accordingly. The probabilities are chosen so that the opponent is made indifferent between their own strategies, meaning no player can gain a higher expected payoff by unilaterally changing their mixing proportions.

Every finite game with a finite number of players and strategies is guaranteed to have at least one Nash Equilibrium, which may be in mixed strategies. Games like Matching Pennies have no pure strategy Nash Equilibrium but always have a mixed strategy Nash Equilibrium where players randomize with specific probabilities.

In Matching Pennies the unique mixed strategy Nash Equilibrium requires each player to choose Heads or Tails each with probability one half. At these probabilities each player is exactly indifferent between both options since their expected payoff is identical regardless of which strategy they select, satisfying the equilibrium condition.

For a player to willingly randomize between two strategies neither can be strictly better than the other. Each strategy in the mix must yield exactly the same expected payoff given the opponents equilibrium mixing probabilities. If one strategy yielded a higher expected payoff the player would deviate to playing it with certainty, breaking the equilibrium.

This is a True/False question. The statement is false. In a mixed strategy Nash Equilibrium each player chooses mixing probabilities that make the opponent indifferent between the opponents own strategies, not to maximize their own payoff directly. This is a counterintuitive but fundamental feature of how mixed strategy equilibria are calculated and sustained.

Penalty kicks in soccer are a classic real world illustration of mixed strategy thinking. A goalkeeper who always dives the same direction is easily exploited by the kicker. By randomizing unpredictably the goalkeeper removes any systematic advantage the kicker could gain, which is exactly what a mixed strategy Nash Equilibrium prescribes for such zero sum interactions.

A mixed strategy Nash Equilibrium is defined by players randomizing over strategies, probabilities being set to make opponents indifferent, and no player being able to improve their expected payoff through unilateral deviation. It does not guarantee higher combined payoffs than pure strategy outcomes and may in fact produce lower collective welfare.

To find the correct mixing probability each player solves for the probability that equalizes the opponent's expected payoffs across the opponent's strategies. This indifference condition rather than direct own-payoff maximization is what pins down the equilibrium mixing probabilities and is central to how mixed strategy equilibria are computed.

A self enforcing equilibrium is one where each player independently chooses to maintain their strategy because deviation would not improve their outcome. In a mixed strategy Nash Equilibrium each player is indifferent between all strategies in the mix and has no incentive to deviate, so the equilibrium sustains itself without any external enforcement mechanism.

In a mixed strategy Nash Equilibrium all strategies assigned positive probability must yield the same expected payoff. This means the player's overall expected payoff from mixing equals the expected payoff of any individual strategy in the mix. This equality across strategies is precisely why the player is indifferent and willing to randomize.

The Nash Existence Theorem proved by John Nash in 1950 is one of the foundational results in game theory. It guarantees that every finite game with a finite number of players and strategies possesses at least one Nash Equilibrium when mixed strategies are permitted. This result assured economists that equilibrium analysis is always applicable.

In a two player zero sum game the mixed strategy Nash Equilibrium and the minimax solution coincide. This equivalence known as the Minimax Theorem proved by John von Neumann shows that each player's equilibrium strategy simultaneously maximizes their minimum guaranteed payoff and constitutes a Nash Equilibrium, linking these two foundational game theory concepts.

Matching Pennies, Rock Paper Scissors, and the penalty kick game all lack pure strategy Nash Equilibria and require mixed strategy analysis. In each game any predictable pure strategy choice can be exploited by the opponent. The Prisoners Dilemma has a pure strategy Nash Equilibrium where both players confess so it does not require mixed strategy analysis.

Randomization in a mixed strategy Nash Equilibrium is fully rational because predictable behavior can be systematically exploited by opponents. By randomizing in the correct proportions a player eliminates any advantage the opponent could gain from predicting their choice. This makes unpredictability itself a strategically valuable and equilibrium sustaining behavior.

In a mixed strategy Nash Equilibrium the opponent chooses their mixing probabilities specifically to equalize the player's expected payoffs across all strategies. This deliberate equalization by the opponent is exactly what creates the player's indifference. The indifference is an equilibrium outcome driven by the opponent's strategy, not by the player's own uncertainty or randomness.