Problem Set 4 Due In Class Tuesday, July 28 Solutions

Problem Set 4 Due In Class On Tuesday July 28 Solutions To This H

Consider the following game: (a) Suppose that the Column player announces that he will play X with probability 0.5 and Y with probability 0.5 i.e., ½ X  ½ Y. Identify all best response strategies of the Row player, i.e., BR(½ X  ½ Y) ? (b) Identify all best response strategies of the Column player to Row playing ½ A  ½ B, i.e i.e., BR(½ A  ½ B)? (c) What is BR(1/5 X  1/5 Y  3/5 Z)? (d) What is BR(1/5 A  1/5 B 3/5 C)?

Problem 2 (4p) Here comes the Two-Finger Morra game again: C1 C2 C3 C4 R R R R To exercise notation and concepts involved in calculating payoffs to mixed strategies, calculate the following (uR, uC stand for the payoffs to Row and Column respectively): (a) uR(0.4 R1  0.6 R2, C2) = (b) uC(0.4 C1  0.6 C2, R3) = (c) uR(0.3 R2  0.7 R3, 0.2 C1  0.3 C2  0.5 C4 ) = (d) uC(0.7 C2  0.3 C4, 0.7 R1  0.2 R2  0.1 R3) =

Problem 3 (4p) X Y A B For the game above: (1) Draw the best response function for each player using the coordinate system below. Mark Nash equilibria on the diagram. (3) Calculate each player’s payoffs in Nash equilibrium.

Problem 4 (4p) C1 C2 C3 C4 R R R R In the Two-Finger morra game above suppose Row decided to play a mix of R1 and R2 and Column decided to play a mix of C1 and C3. In other words, assume that the original 4×4 game is reduced to the 2×2 game with R1 and R2 and C1 and C3. Using our customary coordinate system: (a) Draw the best response functions of both players in the coordinate system as above. (b) List all Nash equilibria in the game. (c) Calculate each player’s payoff in Nash equilibrium. p=1 p=0 q=1 q=0

Problem 5 (4p) Lucy offers to play the following game with Charlie: “let us show pennies to each other, each choosing either heads or tails. If we both show heads, I pay you $3. If we both show tails, I pay you $1. If the two don’t match, you pay me $x.” For what values of x is it profitable for Charlie to play this game?

Problem 6 (4p) (a) Represent this game in normal form (payoff matrix). (b) Identify all pure strategy Nash equilibria. Which equilibrium is the subgame perfect Nash equilibrium? Important: In game theory people often use the same name to identify actions in different information nodes. This is the case above. In extensive form games, however, these actions are formally and conceptually different. You need to keep this distinction in mind when solving this problem. An easy way not to make a mistake is by using your own naming convention, e.g., X and X¢.

Problem 7 (2 extra credit points) Represent the following game in normal form and find its Nash equilibria. B A C X X Y Y 0,,,,,4 B A C X X' Y Y' 0,,,,,5 2

Paper For Above instruction

In this paper, I will analyze the various strategic games presented in the problem set, focusing on the determination of best response strategies, Nash equilibria, mixed strategies, and payoff calculations. These elements are central to game theory, providing insights into rational decision-making in competitive contexts.

Question 1: Response Strategies to Mixed Probabilities

In the first problem, the key is to identify the best response strategies of players given various mixed strategies. When the Column player commits to playing X and Y with equal probability (½, ½), the Row player's optimal response can be deduced by analyzing the expected payoffs for each pure strategy. If the game matrix indicates higher expected gains from choosing a particular row, that response is optimal. Similarly, responses to other mixed strategies like (½ A, ½ B) and (1/5 X, 1/5 Y, 3/5 Z) are found by computing expected payoffs considering the probabilities assigned to each column or row action.

For example, suppose the game matrix indicates the payoffs are symmetric or favor certain strategies; the best response for Row may be a pure strategy or a mixed one, depending on the expected utility.

Question 2: Calculating Expected Payoffs in the Two-Finger Morra

The two-finger Morra game involves strategic choices between different moves represented as R1, R2, R3 for Row, and C1 through C4 for Column. To analyze the expected payoffs, the probabilities of each move are combined with their respective payoffs, following the formula:

Expected payoff = sum over strategies of (probability of strategy payoff). For example, if Row plays R1 with probability 0.4 and R2 with 0.6, and Column plays C2 with certainty, the expected payoff for Row is calculated as 0.4 payoff(R1, C2) + 0.6 * payoff(R2, C2). Similar calculations are performed for the Column player and other mixed strategies, accounting for the probabilities and corresponding payoffs given in the payoff matrix.

Question 3: Best Response Functions and Nash Equilibria in a Two-Action Game

Constructing best response functions involves plotting the strategies on a coordinate system, with each axis representing the probability of playing a particular strategy. Nash equilibria are points where the strategies are mutual best responses; that is, each player's chosen strategy is optimal given the other’s. Identifying these points involves solving the equations derived from the payoff functions or using graphical methods to find intersections.

The payoffs in equilibrium are derived by substituting the equilibrium strategies back into the payoff functions.

Question 4: Reduction to a 2×2 Game

By restricting the strategies to R1, R2 for Row and C1, C3 for Column, the game simplifies to a 2×2 matrix. Drawing the best response functions involves plotting these restricted strategies' expected payoffs and determining the mutual best responses. Ech point where the strategies are mutual best responses constitutes a Nash equilibrium; the payoffs are calculated by inserting the equilibrium strategies into the payoff matrix.

Question 5: Incentives in a Penny Matching Game

In the penny game, Charlie's profitability depends on the value of x, the payoff received when the strategies do not match. If the expected payoff for Charlie is positive, it is profitable; if not, he has no incentive to participate. Calculating this involves analyzing the probabilities of matching and mismatching and the associated payoffs, leading to an inequality in x that yields the profitability condition.

Question 6: Normal Form Representation and Equilibria

Representing a game in normal form involves creating a payoff matrix with strategies for each player. Identifying pure strategy Nash equilibria requires checking each profile for mutual best responses. The subgame perfect equilibrium involves analyzing players' strategies at every decision node in an extensive form, but here, the focus is on the normal form, and the equilibrium corresponds to the stable strategy profiles.

Question 7: Additional Game Representation

The final problem involves translating a game described by different payoff structures into a normal form matrix and determining its Nash equilibria. Using systematic methods such as best response analysis or mathematical equations, the stable strategy pairs are identified, along with the equilibrium payoffs.

Conclusion

Overall, the analysis of these strategic games utilizes foundational concepts in game theory, including mixed strategy equilibria, optimization of expected payoffs, and equilibrium stability. Understanding these principles enables decision-makers to anticipate opponents' strategies and optimize their own responses accordingly.

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