GT mixed strategy nash eq

Name: GT mixed strategy nash eq
Uploaded: 2022-08-13T20:15:55.000Z
Duration: 40 min 19 s

Mixed Strategy Nash Equilibrium Lecture

Introduction to Mixed Strategy Nash Equilibrium

This lecture focuses on extending the definition of pure strategy Nash equilibrium to mixed strategies. The aim is to explore examples and applications of this concept.

Definition and Conditions

The condition for a mixed strategy Nash equilibrium states that u_i(sigma^) geq u_i(sigma_i', sigma_-i^) for all alternative strategies sigma_i' for player i . This mirrors the pure strategy definition, substituting strategies with their mixed counterparts.

A mixed strategy must ensure that if there exists a better alternative strategy sigma_i' , it would imply that some pure strategy within the mix performs better than the current one, reinforcing the need for indifference among chosen strategies.

Example Analysis

Player 1's Choices

In analyzing player 1's choices, dominated strategies (W and Z) are excluded from consideration as they do not contribute positively to any mixed strategy involving X and Y. Thus, player 1 will only consider mixing between X and Y based on player 2's actions.

Best Responses

For player 1 to assign positive weight to both X and Y, these must be best responses against player 2’s mixture of L and R; otherwise, player 1 would favor a unique best response instead. This highlights the importance of identifying best responses in more complex games.

Constructing Mixed Strategy Nash Equilibria

Indifference Condition

To find a mixed strategy Nash equilibrium, players must be indifferent between any strategies they assign positive probability weights to; thus, player two needs to mix their probabilities effectively so as to maintain this indifference for player one between X and Y.

Dominated Strategies Consideration

Dominated strategies should receive zero probability in any proposed equilibrium; if an undominated strategy is included in calculations, it must not serve as a unique best response compared to other mixtures being evaluated. This ensures robustness in strategic choices made by players involved.

Probability Assignments

Player Two's Mixing Strategy

Using variable p for the probability with which player two plays L allows us to analyze how this affects player one's decisions while ensuring total probabilities sum up correctly (to one). Understanding these dynamics is crucial when determining optimal mixes for each player's strategies based on payoffs received from others’ actions.

Payoff Dependencies

It’s essential that while determining how each player's mixing occurs based on their respective payoffs, we recognize that understanding one player's payoffs can inform another’s strategic decisions—this interdependence is key in game theory analysis despite seeming counterintuitive at first glance.

Understanding Mixed Strategy Nash Equilibria

Player One's Indifference Condition

Player one must be indifferent between strategies x and y, which involves analyzing player two's probabilities (p and 1-p). The payoff for player one when playing x is calculated as 1p + 4(1 - p).

When considering the payoffs for strategy y, the calculation yields 3p - (1 - p), where the terms arise from the probabilities associated with player one's choices.

Coordination Game Example

An example of a coordination game illustrates that there are two pure strategy Nash equilibria: one where A is played and another where B is played. To find a mixed strategy equilibrium, we need to ensure player two is indifferent between A and B.

The analysis can focus on either player; however, it’s simpler to analyze just player one’s mixed strategy of playing A with probability p and B with probability 1 - p.

Pure Strategies as Degenerate Mixed Strategies

It’s important to note that a pure strategy can be viewed as a degenerate mixed strategy by assigning probability 1 to that pure strategy while giving all other strategies a probability of 0.

Payoff Calculations for Player Two

For player two's payoffs, if they play A, their expected payoff becomes 2p, while if they play B, it results in an expected payoff of 0 when combined with player one's actions.

After collecting terms from both players' strategies, we find that p = 1/3, indicating symmetric outcomes in this scenario.

Battle of the Sexes Game Analysis

In the battle of the sexes game, there exists a mixed strategy Nash equilibrium requiring positive weights on both m and o due to differing preferences between players.

Assuming player one mixes their strategies with probabilities p and 1 - p, we aim to make player two indifferent between their options.

Expected Payoff Simplifications

The expected payoff calculations show that when m is played by both players (with respective probabilities), it leads to different outcomes based on how often each player's choice aligns.

Simplifying these equations reveals that p = 2/3, establishing part of our Nash equilibrium framework.

Further Analysis on Player Two's Mixing Strategy

When examining how player two mixes their strategies using variables Q and 1 - Q, we derive new expected payoffs based on similar principles applied earlier.

Collecting terms leads us to conclude that Q = 1/3. This indicates an asymmetry in mixing compared to previous calculations.

Final Expected Payoff Evaluation

Both players’ expected payoffs can be verified through matrix analysis. For instance, if player one plays M, her expectation comes out as 2/3.

The final evaluation confirms consistency across various scenarios within the game structure leading up to established probabilities for each player's optimal strategies.

Understanding Mixed Strategy Nash Equilibrium

Payoff Calculations and Probabilities

The first term in the payoff calculation represents the probability of player 1 playing strategy M, which is two-thirds, multiplied by the probability of player 2 playing M, which is one-third.

The second term corresponds to the scenario where player 1 plays M.O., resulting in a payoff of zero for player 1 when this strategy is played.

Player 1's expected payoffs from both strategies (M and O) are calculated to be two-thirds each, indicating that she should be indifferent between these strategies in equilibrium.

Expected Payoff Simplification

In equilibrium, calculating expected payoffs can be simplified; players only need to consider their expected payoffs from either strategy rather than performing complex calculations with multiple terms.

Key concepts include ensuring that players randomize over undominated strategies and confirming that they are indifferent between their choices at equilibrium.

Intuition Behind Mixed Strategies

Players do not play randomly due to uncertainty but instead choose their strategies randomly while committing to them once chosen. This highlights a strategic element in decision-making.

Applications of Mixed Strategies

Mixed strategies are prevalent in various fields such as sports and national security, emphasizing unpredictability as a tactical advantage.