Probability Mixed Strategy Game Quiz

In a mixed strategy game probability represents the weight or likelihood a player assigns to each available strategy when randomizing. Rather than choosing one strategy with certainty the player selects each option according to a probability distribution, with all probabilities summing to one across the full strategy set.

When a player assigns probability p to one strategy and 1-p to the other these two values must sum to one since they represent a complete probability distribution over the player's strategy set. The value of p lies between zero and one inclusive, and the equilibrium value of p is determined by the condition that makes the opponent indifferent.

The probabilities a player assigns to their strategies form a valid probability distribution, which by definition must sum to exactly one. If the probabilities summed to more or less than one the player would be assigning inconsistent likelihoods to their strategies, which would violate the mathematical requirements of a well defined mixed strategy.

The equilibrium mixing probability for Player 1 is found by solving for the value that makes Player 2 exactly indifferent between Player 2s strategies. Player 1 does not directly maximize their own payoff when finding the mixing probability. Instead Player 1 sets probabilities to remove any incentive for Player 2 to deviate.

Assigning a probability of zero to a strategy is mathematically equivalent to removing that strategy from consideration. The player will never choose that option since it receives no weight in their randomization. This is consistent with rational play where strategies that are strictly dominated receive zero probability in any equilibrium.

In the Battle of the Sexes each player assigns mixing probabilities specifically to make the other player indifferent between their two options. The equilibrium probabilities are not equal and are derived from each player's payoff structure. This process of mutual indifference creation through probability setting is the hallmark of mixed strategy equilibrium computation.

All four statements are correct properties of probabilities in mixed strategy games. Probabilities represent long run frequencies, must sum to one, and are set to induce opponent indifference. When a player has a dominant strategy they assign it probability one and all other strategies probability zero, which is a degenerate case of a mixed strategy.

Player 2's indifference between Left and Right means that given Player 1s chosen mixing probabilities the expected payoff from playing Left equals the expected payoff from playing Right. This equality is created deliberately by Player 1s choice of probabilities and is the defining condition that pins down Player 1s equilibrium mixing strategy.

Expected payoff is calculated by multiplying each potential payoff by the probability that it occurs and then summing all of these probability-weighted values. This weighted average accounts for all possible opponent strategies and their associated probabilities, giving the average payoff a player can expect from choosing a particular strategy against a randomizing opponent.

Assigning probability 0.4 to High and 0.6 to Low means the player randomizes so that on average High is played 40 percent of the time and Low is played 60 percent of the time. This is the long run frequency interpretation of mixed strategy probabilities, which ensures the opponent cannot predict which strategy will be used in any specific interaction.

This is a True/False question. The statement is true. For a player to be willing to randomize across strategies all strategies assigned positive probability must yield the same expected payoff. If any strategy in the mix yielded a higher expected payoff the player would have a strict incentive to shift probability toward that strategy, which would break the equilibrium.

The player's mixing probabilities must induce opponent indifference because it is precisely the opponent's indifference that removes any incentive to deviate. If the opponent were not indifferent they would shift entirely to their better strategy, which would destroy the equilibrium. Sustaining equilibrium therefore requires making the opponent indifferent rather than maximizing one's own payoff.

Expected payoffs in mixed strategy games are probability-weighted averages of all possible outcomes, they must be equal across all strategies in the mix at equilibrium, and they depend directly on the probabilities the opponent assigns to their strategies. A higher certain payoff does not always translate to higher expected payoff since expected payoff depends on all scenarios weighted by probability.

In Rock Paper Scissors the unique mixed strategy Nash Equilibrium assigns equal probability of one third to each strategy. This equal randomization makes the opponent indifferent between all three options since each yields the same expected payoff. Any deviation from equal probabilities would allow the opponent to exploit the predictability by shifting toward the strategy that beats the over-used one.

Equilibrium mixing probabilities are calculated directly from the payoff values in the payoff matrix. If any payoff changes the indifference conditions used to derive the equilibrium probabilities will yield different solutions. This means mixing probabilities are sensitive to payoff structure and must be recalculated whenever the underlying payoff values in the game change.