Problem 3: Simultaneous vs Sequential Games (30 points) The Car Onwer and the Thief (10 points) Suppose the following conditions for a game between a thief and the owner of a vehicle. Both their benefits are measured for the car's valuation (higher is better) and the years of imprisonment (lower is better) . The car has a value of $50. . If the owner chooses to install an alarm, they must pay a cost of $5. . If the thief decides to steal the vehicle, they will face a penalty of -10 years when the car has an alarm. Otherwise, he can enjoy the full value of the car for himself. . If the thief decides not to commit any crime be will enjoy his freedom. 1. Define the strategies for the players players = {Oumer, Thief} 2. Represent the normal-form of the game when it is played simultaneously. 3. Define the Best response for each player. Is there a Nash Equilibrium? 4. Write the extensive form of the game, one when the owner moves first and the other when the thief moves first. Explain the process of backward induction to reach the SPE. Price competition (10 points) There is a duopoly in the market consisting of companies A and B. Both have a cost of $2 per unit produced (no fixed costs). The firms can choose between a high selling price ($10) and a low one ($5). When both arms choose a high price, the total demand is q = 10 units, which are divided equally. When both set a low price, the demand is q = 18. and again, it is divided equally. Finally, if one of the firms sets a low price and the other a high one, q = 15 and q = 2 are sold respectively. 1. Represent the normal-form of the game when it is played simultaneously. 2. Define the Best response for each play and find the Nash Equilibrium. 3. Does this game satisfy the two conditions of the prisoner's dilemma? Explain. 4. Write the extensive form of the game. (as the game is symmetric just do it with one of the firms). Explain the process of backward induction to reach the SPE. Colombia and Venezuela (10 points) In 2017, Colombia and Venezuela faced off in the meeting of defense ministers of Unasur, The discussion revolves around whether each country will or will not disclose to the other members of Unasur the conditions of their respective military agreements with major world powers. Each country had the option to reveal its agreement, not reveal it, or disclose only some of its details. Let's denote these options as R. N, and P (for "partially"), respectively. The utility each country receives from its choices depends on what the other does, as reflected in the following payoff matrix: Venezuela N R P N 4,4 9.0 6,0 Colombia R. 0.9 8.8 2,10 0.6 10,2 7,7 1. Use the Iterate Elimination of Dominated Strategies to either decrease the size of the matrix or find the Nash Equilibrium. Explain step-by-step the reduction. 2. Is there a unique Nash Equilibrium in this game?' Explain. 3. Suppose the countries play a sequential game where Venezuela decides first. Write the extensive form of the game and explain the process of backward induction to reach the SPE. 4. Is there any difference between the SPE and the Nash Equilibrium? Why