Pure vs Mixed Strategy Quiz

A pure strategy is one in which a player chooses a single specific action with complete certainty, assigning probability one to that action and zero to all others. Unlike a mixed strategy there is no randomization involved. Pure strategies are the simplest form of strategic choice and form the foundation of game theory analysis.

A mixed strategy involves assigning probabilities to two or more available strategies and then randomizing among them according to those probabilities. This contrasts with a pure strategy where one action is chosen with certainty. Mixed strategies allow players to be deliberately unpredictable which is sometimes the only equilibrium solution available.

Any pure strategy Nash Equilibrium is a special case of a mixed strategy Nash Equilibrium where the chosen strategy receives probability one and all others receive probability zero. This means games with pure strategy equilibria always have mixed strategy equilibria as well since pure strategies are degenerate mixed strategies at the boundary of the probability simplex.

Matching Pennies has no pure strategy Nash Equilibrium because whenever one player settles on a pure strategy the other player always has an incentive to deviate. The only equilibrium of the game is in mixed strategies where each player randomizes with equal probability. This makes Matching Pennies the standard example of a game requiring a mixed strategy analysis.

This is a True/False question. The statement is false. Not every finite game has a pure strategy Nash Equilibrium. Matching Pennies and Rock Paper Scissors are well known examples of finite games with no pure strategy Nash Equilibrium. In such games the only equilibrium solution involves players randomizing according to a mixed strategy.

When an opponent can predict a player's pure strategy choice they can exploit it by always choosing the best response to that known strategy. By randomizing the player eliminates this predictability making it impossible for the opponent to systematically exploit any fixed pattern. This strategic value of unpredictability is the primary reason mixed strategies are used.

Pure strategies assign probability one to one action while mixed strategies spread positive probability across two or more actions. Pure strategies are correctly understood as a special case of mixed strategies at the probability boundary. Mixed strategies do not always produce Pareto superior outcomes and may lead to collectively worse results than some pure strategy outcomes.

In Rock Paper Scissors any pure strategy can be beaten by a specific counter strategy. If a player always plays Rock the opponent always plays Paper. This predictability means no pure strategy choice is a best response to the opponent also playing a pure strategy, which rules out any pure strategy Nash Equilibrium and necessitates a mixed strategy solution.

In games without pure strategy Nash Equilibria the mixed strategy Nash Equilibrium provides the only stable solution where no player has an incentive to deviate. The equilibrium mixing probabilities are chosen so that all players are indifferent across their strategies and no unilateral deviation can increase any player's expected payoff.

This is a defining property of any Nash Equilibrium including mixed strategy ones. At the equilibrium mixing probabilities each strategy in the mix yields the same expected payoff. Switching to a pure strategy means playing one of these strategies with certainty which yields the same expected payoff as the mixed strategy, meaning there is no gain from deviating.

When a player always chooses Left with certainty this is a pure strategy. It assigns probability one to Left and probability zero to all other strategies. A pure strategy is the simplest form of strategic choice in game theory and is classified as a degenerate mixed strategy at the extreme of the probability distribution over available actions.

A game can have both pure and mixed strategy Nash Equilibria simultaneously. The pure strategy equilibrium represents a stable outcome where players choose actions with certainty while a separate mixed strategy equilibrium may exist where players randomize. The Battle of the Sexes is a well known example with multiple pure strategy equilibria and one mixed strategy equilibrium.

Pure strategies are degenerate mixed strategies with probability one on one action. Mixed strategies can include probability zero for actions not in the support of the equilibrium. Games can and do have multiple equilibria in both pure and mixed strategies simultaneously. It is not true that mixed strategies are only needed when pure strategy equilibria are absent.

The Battle of the Sexes has two pure strategy Nash Equilibria where both players attend the same event and one mixed strategy Nash Equilibrium where each player randomizes. The two pure strategy equilibria favor different players while the mixed strategy equilibrium produces lower expected payoffs for both, making coordination on a pure strategy preferable if feasible.

At a mixed strategy Nash Equilibrium the player's expected payoff from randomizing equals the expected payoff from each individual pure strategy assigned positive probability in the mix. This equality is the defining indifference condition. If any pure strategy in the mix yielded a different expected payoff the player would not be willing to randomize over it.