Problem Set 4 Due In Class On Tuesday, July 28 Solutions ✓ Solved

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., 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)? X Y Z A B 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

Sample Paper For Above instruction

The provided problem set encompasses a variety of game theory concepts including strategic responses to mixed strategies, calculation of expected payoffs, graphical representation of best response functions, analysis of subgames, and equilibrium identification in both normal and extensive forms. This paper aims to systematically address each problem, illustrating principles of strategic decision-making through detailed mathematical and graphical analyses.

Problem 1: Analyzing Best Responses to Mixed Strategies

In this problem, we analyze the strategic responses of players given specific mixed strategies. For part (a), suppose the Column player commits to playing X and Y with equal probabilities (0.5 each). The Row player’s best response depends on the payoffs associated with each of their strategies against these probabilities. Specifically, if the Row player’s payoffs for their strategies are given in a payoff matrix, they will calculate expected payoffs for each and choose the strategy(s) maximizing their expected value. The dominant strategy or mixed strategy response is identified accordingly.

Part (b) involves the Column player responding to Row playing a 50-50 mix of A and B. The COLUMN player evaluates their expected payoffs from C, B, and C strategies based on the probabilities assigned to Row's strategies. The best response is the strategy (or strategies) yielding the highest expected payoff given Row's play.

In parts (c) and (d), similar principles apply: each player's best response to specific mixed strategies is calculated by evaluating expected payoffs, emphasizing the importance of the payoff matrices and the probability distributions over strategies. The responses are derived via the maximization of expected utility, considering the opponent's Mix.

Problem 2: Payoff Calculations in Two-Finger Morra

The Two-Finger Morra game involves players choosing among four responses, with payoff matrices dictating outcomes. The problem provides mixed strategy probabilities for different responses and requires calculating expected payoffs.

In part (a), calculating uR involves multiplying the payoffs associated with R1 and R2 by their respective probabilities (0.4 and 0.6) and summing them, considering the opponent’s play—similar for the other parts involving Column’s strategies and combinations of mixed strategies. These calculations exemplify how expected payoffs are weighted averages based on strategy probabilities, fundamental to understanding mixed-strategy equilibria.

Problem 3: Drawing Best Response Functions and Calculating Nash Equilibrium Payoffs

This problem deals with visualizing strategies through best response functions in a coordinate system, helping to identify Nash equilibria as intersection points where strategies are mutual best responses. The process entails plotting the best responses for each player against varying strategies and marking points where these responses intersect, indicative of equilibrium points. Subsequently, the expected payoffs at these points are calculated, emphasizing the strategic stability of equilibrium solutions.

Problem 4: Reducing a 4×4 Game to a 2×2 Game and Identifying Equilibria

This problem simplifies a larger game by focusing on mixed strategies involving only a subset of actions (R1, R2, C1, C3). The analysis involves plotting best response functions for the reduced game and determining all Nash equilibria through strategic considerations. Calculating payoffs at these equilibria provides insights into optimal strategies within the simplified context.

Problem 5: Analyzing the Penny Game with Payoff Variations

Here, the game involves strategic choice of heads or tails with payoffs depending on the combination. The core question is identifying values of x for which Charlie finds it profitable to participate, which involves comparing expected payoffs for Charlie based on the probability distribution of Lucy’s moves and the value of x, ensuring their expected utility is positive.

Problem 6: Normal Form Representation and Nash Equilibria Identification

This problem asks for constructing the payoff matrix for a sequential game, recognizing the critical difference between actions at different nodes in extensive form. It involves deriving pure-strategy Nash equilibria and pinpointing the subgame perfect equilibrium, which requires backward induction analysis. The distinctions in action nodes are crucial for accurate modeling.

Problem 7: Extra Credit - Normal Form and Equilibrium Computation

The final problem is an extension involving the creation of the normal form of a game with multiple strategies and identifying all Nash equilibria. The focus is on understanding the strategic landscape and the stability conditions of various strategy combinations, illustrating the comprehensive application of equilibrium concepts.

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