Mixed Strategy Payoff Calculation Quiz

Expected payoff in a mixed strategy game is the probability-weighted average of all possible outcomes. It is calculated by multiplying each potential payoff by the probability it occurs and summing those values. This measure represents the average return a player can anticipate from a strategy when the opponent is randomizing across their available strategies.

Player 1 calculates the expected payoff from Up by weighting each payoff in the Up row by the probability that Player 2 plays each corresponding strategy. The Up-Left payoff is multiplied by q and the Up-Right payoff is multiplied by 1-q and these two values are summed to give the full expected payoff from playing Up against a randomizing opponent.

Finding Player 1's equilibrium mixing probability requires setting Player 2's expected payoffs equal across all of Player 2's strategies. Player 1's mixing probabilities are what create indifference for Player 2. This means solving the equation where Player 2's expected payoff from Left equals Player 2's expected payoff from Right yields the equilibrium probability that Player 1 must use.

If Player 2 plays Heads with probability 0.5 and Tails with probability 0.5, Player 1 earns 1 when both play Heads and loses 1 when Player 2 plays Tails. The expected payoff equals 0.5 multiplied by 1 plus 0.5 multiplied by negative 1 which equals 0. This confirms that Player 1 is indifferent between Heads and Tails, satisfying the equilibrium indifference condition.

When expected payoffs are equalized across strategies the player is indifferent between them. This indifference is the necessary condition for the player to be willing to randomize across those strategies. If one strategy yielded a strictly higher expected payoff the player would deviate to playing it with certainty which would destroy the mixed strategy equilibrium.

This statement is false. In a mixed strategy Nash Equilibrium each player chooses mixing probabilities that make the opponent indifferent between the opponent's own strategies rather than directly maximizing their own expected payoff. This is one of the most counterintuitive features of mixed strategy equilibrium and distinguishes it from standard individual payoff maximization.

Calculating a mixed strategy Nash Equilibrium involves expressing expected payoffs as functions of the opponent's mixing probability, setting those expected payoffs equal to identify the equilibrium probability, and verifying no player can profitably deviate. Directly maximizing one's own expected payoff is not the method for finding equilibrium mixing probabilities in a mixed strategy game.

Setting Player 2's expected payoff from Left equal to their expected payoff from Right creates an equation in terms of p which is Player 1's mixing probability. Solving for p reveals exactly what probability Player 1 must assign to Up to make Player 2 indifferent, which is the definition of Player 1's equilibrium mixing strategy in a mixed strategy Nash Equilibrium.

In a two player mixed strategy game each outcome payoff is weighted by the joint probability that both players choose the strategies producing that outcome. Multiplying each outcome payoff by its joint probability and summing across all possible strategy combinations gives the full expected payoff for each player in the game.

To identify the best response the firm calculates the expected payoff from each of its own strategies by weighting outcomes against the rival's mixing probabilities of 0.6 and 0.4. This expected payoff calculation reveals which strategy or mix yields the highest average return given the rival's randomization, which is central to identifying best responses in mixed strategy settings.

This statement is false. At a mixed strategy Nash Equilibrium the expected payoff from mixing equals the expected payoff from each individual pure strategy in the support of the mix. The mixed strategy does not produce a strictly higher expected payoff than the pure strategies it contains. All strategies assigned positive probability yield exactly the same expected return at equilibrium.

Mixing probabilities must lie between zero and one to form a valid probability distribution. A negative probability or one greater than one has no meaningful strategic interpretation. Verifying that solved probabilities fall within this range confirms the mathematical solution corresponds to a genuinely mixed strategy rather than a pure strategy or an infeasible mathematical artifact.

Expected payoffs in mixed strategy games depend on both the player's own strategy and the opponent's mixing probabilities. At equilibrium all strategies in the support of the mix yield equal expected payoffs. Changes in the opponent's mixing probabilities directly alter the player's expected payoffs. A strategy assigned zero probability does not yield a higher expected payoff than those in the equilibrium mix.

In a symmetric game where both players face identical payoff structures the equilibrium mixing probabilities are the same for both players. The symmetry of the payoff matrix means the indifference conditions faced by each player yield identical solutions. Rock Paper Scissors is a standard example where all players mix with equal probability across all three strategies.

The final verification step is confirming that no player has an incentive to deviate from the calculated mixing probabilities. This means checking that no alternative strategy or mixing proportion would yield a strictly higher expected payoff. If this no-deviation condition holds for all players the solution is confirmed as a valid mixed strategy Nash Equilibrium.