4. The frog and the scorpion: We begin with the fable of the frog and the scorpion. They were standing on the bank of the river, trying to figure out a way across. "I know," said the scorpion "I will climb on your back and you can swim across the river." The frog said, "But what if you sting me with your stinger?" The scorpion said, "Why would I do that? Then we would both die." The frog found this convincing, so the scorpion climbed on his back and they started across the river. Halfway across, at the deepest point, the scorpion stung the frog. Writhing in pain, the frog cried out, "Why did you do that? Now we are both doomed!" "Alas," said the scorpion, as he sank into the river, it is my nature." Let's look at the frog and the scorpion from the viewpoint of game theory. (a) Depict the sequential game with payoffs consistent with the story. Start at the bottom of the game tree. If the frog refuses the scorpion, both get nothing. Looking up one line, we see that if the frog carries the scorpion, he receives utility 10, for doing a good deed, and the scorpion receives a payoff of 6, for getting across the river. In line where the frog is stung, he receives a payoff of -20, and the scorpion gets a payoff of 10, representing the satisfaction from fulfilling his natural instincts. (7 points) (b) Solve the game by backwards induction. What is the SPNE of the game?. (4 points) (c) Define Pareto Efficiency. Is the subgame perfect equilibrium Pareto efficient? Explain. (4 points)