EXAM 2 SPRING 2021 SOLUTIONS without solutions for #5

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OTM 732: Economics for Managers EXAM 2 1) (20 points) This is the “Rational Pigs Game.” Two pigs, one big and the other small, are placed in a large box. There is a lever at one end of the box which, when pressed, dispenses food at the other end. This means that the pig that presses the lever must run to the other end to get any food; and by the time it gets there, the other pig will have eaten some, and perhaps all, of the food. The big pig can prevent the little pig from getting any of the food when both are at the food. If the pigs can reason like game theorists, which pig will press the lever? For concreteness, let’s attach some hypothetical numbers to the game. Suppose that 12 kernels of corn are dispensed if the lever is pressed (by either pig). If the small pig presses the lever, then the big pig eats all 12 kernels before the small pig can run to the other side of the box where the food appeared. However, if it was the big pig who pressed the lever, then the small pig has time to eat 10 of the 12 kernels before the big pig pushes it away and eats the remaining two kernels. Suppose that in the unlikely event that they press the lever together, the small pig, because it can run faster, gets 4 kernels before the big pig pushes it away. And once that happens, the big pig will eat the remaining 8 kernels. Finally, suppose that pressing the lever and running to the other end requires effort equal to one kernel of corn. This leads to the following game: a) (5 points) Does the SMALL pig have a strictly dominant strategy? If so, identify it. YES, “DON’T PRESS” is a SDS for the small pig b) (5 points) Does the BIG pig have a strictly dominant strategy? If so, identify it. NO c) (5 points) What is the Nash equilibrium to this game? Which pig does better in the NE? NE is (DON’T PRESS by Small, PRESS by Big) for a payoff of (10,1). The Small Pig does better! d) (5 points) Is this game a Prisoner’s Dilemma game? Why or why not? No, this is not a PD because the Big Pig does not have a SDS….and there is not a non-equilibrium outcome that is strictly better for both of them.

***** This may seem like this is a silly game, but studies have shown that pigs can learn to behave in the ways described by this NE. 2) (20 points) You are playing a game with someone (not a friend and not someone you can communicate with). The game starts with both of you being awarded $3. The game’s rules are: • You announce whether you want to add $0, $1, $2 or $3 to your initial $3 award. Doing so reduces your opponent’s award by $0, $2, $4, or $6, respectively. (In words, they automatically lose twice what you gain.) • Your opponent simultaneously and independently has the same choices: add $0, $1, $2, or $3 to their initial $3 award, which automatically decreases your award by $0, $2, $4, or $6, respectively. Hi M So, for each of you, your final award equals: (your initial $3) + (your choice of $0, $1, $2, or $3) – 2 x (opponent’s choice of $0, $1, $2, or $3). If the game ends with a negative award for a player, that’s OK and you must pay that amount. a) (15 points) Create the game matrix for this game, showing players, actions, and payoffs. Identify the game’s Nash equilibrium strategies and payoffs for the players. 

 Each player (simultaneously and independently of the other) chooses to add to their award $0, $1, $2, $3, $4, ……$999,999, or $1,000,000. Doing so decreases the other player’s award by twice that amount. The game matrix for this expanded game has 1,000,001 rows and 1000,001 columns. DO NOT DRAW IT! But building on what you learned in part (a), answer the following two questions: i) What is the Nash equilibrium of this game? ii) What are the Nash equilibrium payoffs for the players in this game? 3)a) (10 points) A monopolist has the demand and MC (marginal cost) curves below. Assuming the monopolist must offer the same price to all potential buyers, draw the MR line, identify the profitmaximizing P* and Q*, shade the areas representing PS, CS, and any inefficiency. By calculating areas of triangles, report the numerical sizes of PS, CS, and any inefficiency

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