1. [30 points] A factory is suspected of hiring illegal immigrants as workers. The authority is deciding whether to conduct an inspection. If the factory has illegal workers and an inspection takes place, the workers will be discovered. The cost of an inspection to the authority is 100 units. The benefit from the inspection is 500 units if illegal workers are found, but 0 if none are found. The cost and benefit to the authority in case they choose not to conduct an investigation is 0. The authority's objective is to maximize their net benefit (the benefit minus the cost). For the factory, the benefit from having illegal workers is 200 units, from not using illegal workers is 0. The factory incurs a cost of 500 units if they are caught using illegal workers. Like the authority, the factory is also trying to maximize its net benefit (the benefit minus the cost). The factory must decide whether or not to use illegal workers, and the authority must decide whether or not to conduct an inspection. a. [5 points] Complete the payoff matrix below with the payoff numbers to represent this interaction. Use the net benefits as described above in the question as the payoff numbers. Authority Inspect Don't Inspect Hire Factory Not hire b. [5 points] Is there a Nash Equilibrium in pure strategies? If so, what is it? Show your best response analysis clearly. c. [10 points] Is there mixed strategy Nash Equilibrium? If so, what is it? Show each step of your analysis clearly. d. [10 points] Using your answer to part (c), plot the best-response functions of both players, and clearly mark the equilibria on the figure