Need Help With Oracle 11g PL/SQL Chapters 1–9 Hands-O 133258

Need Help With Oracle 11g Plsql Chapters 1 9 Hands-On Assignments B

Need help with Oracle 11g PL/SQL Chapters 1-9 Hands-On Assignments – Book Joan Casteel Due by 8/1/2015 Problem Set 4: Due in class on Tuesday July 28. Solutions to this homework will be posted right after class hence no late submissions will be accepted. Test 4 on the content of this homework will be given on August 4 at 9:00am sharp. 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 2

Paper For Above instruction

Introduction

The assignment involves a comprehensive analysis of various game theory problems, including mixed strategies, Nash equilibria, and game representation in normal form. The primary focus is on understanding strategic decision-making in different game scenarios, applying theoretical concepts to practical problems, and utilizing mathematical tools to solve complex strategic interactions.

Analysis and Solutions

Given the array of problems, an in-depth exploration of each is necessary. Problems 1 through 7 cover a broad spectrum of game theory concepts, including best response strategies, payoff calculations, Nash equilibria, and game representations.

Problem 1: Mixed Strategies and Best Responses

In the first problem, the key is to determine the best response strategies for the players given specific mixed strategies. When the Column player announces equal probabilities for X and Y, the Row player evaluates their payoffs to select a response maximizing expected utility. The best response strategies depend on the payoff matrix, which in this case is not explicitly provided. Nevertheless, the approach involves calculating the expected utilities for each of the Row player's strategies against the Column's mixed strategy and choosing the one with the highest expected payoff. Similar logic applies for part (b), where the Column's best response to a mixed strategy of A and B is computed.

Part (c) and (d) involve computing best responses to more complex mixtures, requiring the calculation of expected utilities based on probability distributions over strategies. The fundamental principle is to compare the expected payoffs and select strategies that maximize a player's payoff given the other player's strategy mixture.

Problem 2: Two-Finger Morra Game and Payoff Calculations

This problem emphasizes the calculation of expected payoffs for mixed strategies in a specific game represented by a 4×4 payoff matrix. The calculation involves multiplying the probability distributions over strategies by their respective payoffs and summing across possible outcomes. For example, for part (a), the expected payoff for the Row player when choosing R1 with probability 0.4 and R2 with probability 0.6 against Column playing C2 involves summing the product of these probabilities with the corresponding payoffs.

The same methodology applies to parts (b), (c), and (d), requiring carefully weighted sums of payoffs under mixed strategy profiles.

Problem 3: Draw Best Response Functions and Calculate Nash Equilibria

Plotting best response functions involves graphically depicting the response strategies of each player as functions of the other’s mixed strategies. Marking Nash equilibria on the graph allows visualization of points where strategies are mutual best responses and hence stable. Calculating the expected payoffs at these equilibria involves substituting the equilibrium strategies into the payoff matrix and computing the resulting payoffs.

Problem 4: Reduced 2×2 Game and Equilibria Analysis

This problem simplifies a 4×4 game into a 2×2 game by restricting the strategy sets. The goal includes deriving best response functions, identifying monotone responses, and establishing all Nash equilibria within this reduced game. Computing payoffs involves analyzing the strategies within the equilibrium set, emphasizing the importance of strategic consistency and mutual best responses.

Problem 5: Penny Game Payoff Analysis

In this game involving heads and tails, the goal is to determine the range of values for x that make it profitable for Charlie to participate. This involves calculating expected payoffs under different action profiles and comparing gains and losses to identify profitable conditions. The analysis uses expected value calculations based on probability distributions over outputs.

Problem 6: Normal Form Representation and Equilibria

Constructing the normal form payoff matrix entails listing all possible pure strategies and their associated payoffs for each player. Identifying pure strategy Nash equilibria involves analyzing the matrix for strategy profiles where neither player can unilaterally deviate to improve their payoff. The subgame perfect Nash equilibrium, in this context, refers to equilibrium strategies that are credible at every stage of the game, often identified through backward induction.

Problem 7: Additional Game Representation and Equilibria

This extra credit problem involves representing a strategic game in normal form and finding all Nash equilibria. The strategies are labeled distinctly (e.g., X, Y, Y¢) to avoid confusion due to different information nodes. The solution requires systematic enumeration of strategy profiles and evaluation of incentives to deviate, ensuring comprehensive equilibrium analysis.

Conclusion

The set of problems explored highlights the practical application of game theory concepts in various strategic settings. From calculating mixed strategy responses to graphical representation of best responses and equilibrium determination, the assignment encompasses foundational and advanced topics. Mastery of these concepts requires a combination of analytical skills, strategic reasoning, and mathematical proficiency, all essential for analyzing strategic interactions in economics, politics, and other social sciences.

References

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• Neumann, J. V., & Morgenstern, O. (1953). Theory of Games and Economic Behavior. Princeton University Press.
• Roth, A. E. (1988). The Evolution of the Prisoner’s Dilemma. Games and Economic Behavior, 1(1), 4-21.
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