Mixed Strategy Nash Equilibrium for Two Generals Game

Problem 3: Two law firms, LexCorp and LawSmith Partners, are vying to win a high- profile class action lawsuit. Each firm has 4 strategies, and their success depends on how they allocate their resources to various legal tactics: Strategies: (i) File Motions: Dedicate resources to filing motions to control the proceedings. (ii) Gather Evidence: Focus on discovering new evidence. (iii) Hire Expert Witnesses: Invest in credible expert testimony. (iv) Media Campaign: Run a PR campaign to sway public opinion.

Solve the following problems carefully. Provide clear justifications for all answers, espe- cially for strategies, Nash equilibria, and reasoning about game outcomes. Problem 1: Two neighboring farmers, Farmer A and Farmer B, share access to a common water resource. Each farmer can decide to Share Water or Hog Water. The outcomes depend on their choices: • If both farmers Share Water, they each get a moderate payoff of 4, as the resource is fairly distributed. • If both farmers Hog Water, they each get a low payoff of 2, as neither gets enough water for optimal crop yields. • If one farmer Hogs Water and the other Shares Water, the one hogging water gets a high payoff of 6, while the one sharing water gets nothing. (a) Show the normal form representation and extensive form representation. (b) What are the best responses for each player, given the opponent’s strategy? (c) Identify any Nash equilibria in the game. (d) Does this game encourage Sharing Water or Hogging Water? Why?

Problem 8: In this game, two generals can choose between three strategies: Pass, One, or Two. The payoffs are as follows: Pass One Two Pass (1, 1) (0, 0) (0, 0) One (0, 0) (2, 2) (−1, 3) Two (0, 0) (3, −1) (0, 0) Find the mixed strategy nash equilibrium and expected utility of each player.