Quiz 3 Simultaneous games II

1. A professor presents the following game to Egdar and his five class-mates. Each of them simultaneously and privately writes down a number between 0 and 37 on a piece of paper, and they all give their number to the professor. Then he computes the mean (average) of these numbers and defines X to be the mean of the student’s numbers. The student who submits the number closest to two-thirds of X wins. If multiple students tie, they split the prize equally

   1. Show that choosing number 27 is a dominated strategy.[1]

   2. Edgar knows that all other class-mates will choose number23. What number should Edgar choose, so that his number is exactly equal to 2/3 of the average? [3] Please round the answer to one decimal place.

   3. Edgar knows that all other class-mates will choose number 23. What is the lowest number Edgar should choose to win this game? [5] Please round the answer to one decimal place.

   4. Find Nash equilibrium of that game [1]

2. Two carts (stands) selling coconut milk (from the coconuts) are located at 0 and 1, 1 mile apart on the beach. (They are the only two coconut carts on the beach.) The carts – Cart 0 and Cart 1 – charge prices p₀ and p₁, respectively, for each coconut. One thousand beachgoers buy coconut milk, and these customers are uniformly distributed along the beach between carts 0 and 1. Each beachgoer will purchase one coconut milk in the course of her day at the beach and, in addition to the price, each will incur a transportation cost of 0.5 times d², where d is the distance (in miles) from her beach blanket (or armchair) to the coconut cart. In this system, Cart 0 sells to all of the beachgoers located between 0 and x, and Cart 1 sells to all of the beachgoers located between x and 1, where x is the location of the beachgoer who pays the same total price if she goes to 0 or 1. Location x is then defined by the expression: p₀ + 0.5x² = p₁ + 0.5 (1 – x)². The two carts will set their prices to maximize their profit figures, П; profits are determined by revenue (the cart’s price times its number of customers) and cost (the carts each incur a cost of $0.25 per coconut times the number of coconuts sold).

   1. For each cart, determine the expression for the number of customers served as a function of p₀ and p₁. (Recall that Cart 0 gets the customers between 0 and x, while Cart 1 gets the customers between x and 1, or 1-x. [2]

   2. Write the profit functions for the two carts. [2] Find best response function for each cart as a function of their rival’s price. [2]

   3. Graph the best response function, [2] and then calculate the Nash equilibrium price level.[2]

Answers

1

a) The average of all numbers must be between 0 and 37. Thus the target number must be between (2/3)*0 and (2/3)*37, or 0 and 24,67. Therefore choosing any number above 24,67 is a strictly dominated strategy.

b) (5*23 + n)*2/3=n, therefore n=14,375

c) 23 - (5*23 + n)*2/3n>n, therefore n>4,06

2.

(a) Cart 0 serves x customers and Cart 1 serves (1 – x), where x is defined by the equation 1p₀ + 0.5x² = p₁ + 0.5 (1 – x)². Expanding this equation yields p₀ + 0.5x² = p₁ + 0.5 – x +0.5x², and solving for x yields x = p₁ – p₀ + 0.5. Thus Cart 0 serves p₁ – p₀ + 0.5 customers, and Cart 1 serves 1 – x, or p₀ – p₁ + 0.5, customers.

(b) Profits for Cart 0 are (p₁ – p₀ + 0.5)(p₀ – 0.25). Profits for Cart 1 are symmetric: (p₀ – p₁ + 0.5)( p₁ – 0.25). Expanding the expression for Cart 0 profits yields (p₁ – p₀ + 0.75)p₀ – (0.25 p₁ + 0.125). Solving for the profit-maximizing value of p₀ by completing the square or differentiating with respect to p₀ yields p₀ = 0.5p₁ + 0.375. Cart 1’s best-response rule is symmetric: p₁ = 0.5p₀ + 0.375.

(c) The graph is shown below. The Nash equilibrium prices are the values of p₀ and p₁ that solve simultaneously the two best-response rules found in part (b). Substituting Cart 1’s best-response rule into that for Cart 0, we find: p₀ = 0.5(0.5p₀ + 0.375) + 0.375 = p₀ + 0.5625. Solving for p₀ yields p₀ = 0.75 (75¢); Cart 1’s price is p₁ = 0.75 (75¢) also.