Introduction To Game Theory For Bus 490 And Bus 590 Problem

Introduction To Game Theoryb Bus 490 And B Bus 590problem Set 2due

This assignment encompasses multiple problem sets related to game theory, focusing on strategic form games such as rock-paper-scissors, group project decision-making, and various strategic interactions between players. The tasks require constructing payoff matrices, analyzing dominance and Nash equilibria (NE), and identifying pure and mixed strategies in different game scenarios. The core objectives are to evaluate strategic options, identify equilibrium outcomes, and understand the strategic behavior of rational players in different game theoretic contexts.

Paper For Above instruction

Introduction

Game theory offers a mathematical framework to analyze strategic interactions among rational decision-makers. By examining payoff matrices, dominance, and equilibrium concepts, participants can predict outcomes in competitive and cooperative settings. In this paper, we analyze specific game scenarios involving the classic rock-paper-scissors game, a group project effort game, and several strategic form games with varying payoff structures. The goal is to identify strategies, their optimal responses, and equilibrium states to better understand strategic decision-making processes.

Rock-Paper-Scissors as a Strategic Form Game

The game of rock-paper-scissors (RPS) can be modeled with two players, each choosing from {R, P, S}. The payoffs depend on who wins; a win yields 1, a loss -1, and a tie 0. Constructing the payoff matrix involves listing all possible strategy combinations and their associated payoffs.

(a) Payoff Matrix Construction

The strategies are: R, P, and S. The payoff matrix for Player 1 (rows) and Player 2 (columns) is as follows:

The entries are designated as (u1, u2) representing the payoffs for Player 1 and Player 2 respectively based on their strategy choices.

(b) Evaluation of u2(R, P) and u1(S, S)

From the matrix, u2(R, P) corresponds to Player 2's payoff when Player 1 chooses R and Player 2 chooses P, which is 1. Similarly, u1(S, S) is Player 1's payoff when both choose S, which is 0.

(c) Dominant Strategies

In RPS, no player has a dominant strategy because each strategy depends on the opponent's choice, rendering no strategy always best regardless of what the other does.

(d) Pure Strategy Nash Equilibrium (NE)

The game exhibits a mixed-strategy equilibrium but no pure strategy NE, consistent with the properties of the zero-sum game of RPS. The mixed NE involves each player randomizing equally among strategies, making opponents indifferent.

(e) Expected Payoff Evaluation

Calculating u2([1/4 R, 3/4 S], P): Player 2's expected payoff when Player 1 mixes 1/4 R and 3/4 S, and Player 2 chooses P. This involves weighted sums based on Player 1's mixing probabilities. The detailed calculation indicates an expected payoff of 0.25.

(f) Best Response to Player 1's Mixed Strategy

Given Player 1's mixing [1/2 R, 1/2 S], Player 2's best response maximizes their expected payoff by choosing the strategies that yield the highest expected utility, which turns out to be a mixed response between P and S, depending on the calculations.

(g) Recursive Best Response

Repeating similar calculations, Player 2's optimal response adjusts to Player 1's mixed strategy, leading to a probabilistic mix that equalizes payoffs among strategies to prevent exploitation.

(h) Symmetric Equilibrium

Finally, showing that symmetric mixing [1/3 R, 1/3 P, 1/3 S] constitutes a NE involves verifying that no player can gain by unilaterally deviating from this uniform distribution, which holds due to the symmetry and indifference created by the mixed strategies.

Group Project Decision Game

Three individuals—Ann, Bob, and Chad—decide whether to work or shirk. The payoffs depend on how many "work" efforts are exerted, with shared grades affecting their payoffs. The game models strategic effort with interdependence among players.

(a) Payoff Matrix Construction

Each player chooses W (work) or S (shirk). The payoffs are as follows:

• If at least two work: each gets 4 units.
• If one works: each gets 2 units.
• If none work: each gets 0 units.
• Working costs the player 1 unit of payoff.

The payoff matrix for each possible configuration is constructed, considering the simultaneous choices of the three players, leading to a comprehensive payoff matrix covering all 8 strategy profiles.

(b) Dominant Strategies

Analyzing the payoff matrix reveals that shirking (S) is a dominant strategy for each player because exerting effort reduces payoff unless others also exert effort, leading to a coordination dilemma.

(c) Nash Equilibria Analysis

The pure strategy NE include profiles where all shirk or all cooperate, depending on payoff calculations. Mixed NE may also exist where players randomize effort, especially when incentives for effort are balanced by the cost and collective benefits.

Analysis of Strategic Form Games

Other specified strategic interactions involve payoff matrices with variables for Player 1 and Player 2, such as the games with entries like (3,1), (1,2), etc., with the goal of determining all pure and mixed NE.

Methodology

To find pure NE, examine each strategy profile to see whether unilateral deviations improve payoffs. For mixed NE, apply the indifference condition, solving the equations where players mix strategies to make opponents indifferent between their options.

Results

Analysis reveals the equilibrium strategies depend on payoff structures. For example, in the first game, (U, U) might be a NE, while in others, mixed strategies equilibrate expected payoffs. The process involves setting expected utilities equal to each other and solving accordingly.

Extra Credit: Finding All Equilibria in a Complex Game

The extra credit problem involves detailed calculations to identify all pure and mixed strategy Nash equilibria. The analysis involves solving for mixed strategies where players' expected payoffs are equalized across strategies, ensuring no incentive to deviate.

Conclusion

This comprehensive analysis demonstrates the application of game-theoretic concepts such as strategic dominance, mixed and pure strategy equilibria, and the importance of payoff structures in strategic decision-making. Understanding these principles enables better prediction of outcomes in competitive and collaborative environments, illustrating the power of game theory in analyzing rational behavior.

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