Anna has to choose whether to pursue a training course that costs her $2,000. Beppe, Anna's employer, has to decide whether to pay her a fixed wage of $420,000 or share with her the revenues of the enterprise on a 50/50 with Anna. The output is positively affected by both training and revenue sharing. Indeed, with no training and a fixed wage total output is $40,000, while if either training or revenue sharing is implemented, output rises to $42,000. If both training and revenue sharing are implemented the output raises further to $50,000.

a. Set up the corresponding normal form game G = {I, Si ,i }, identifying all the elements of this strategic situation.

b. Identify the payoff matrix for this game.

c. Draw the cooperative payoff region for this game. Can you identify the set of Pareto efficient outcomes?

d. Are there dominant strategies in this game?

e. Are there Nash equilibria?

f. If Anna and Beppe could be able to write down a contract to enforce a specific (efficent) behavior, would it make any difference for the outcome of this game?

EX. 2.

Anna and Beppe care both about their own payoff just about the payoff their mate.

Take the payoff matrix of a standard Prisoners' dilemma and redefine utilities in the game in this manner. Write down the payoff matrix, cooperative payoff region and the set of Nash equilibria in this game and compare with standard Prisoner's Dilemma.

EX. 3.

Anna and Beppe are two firms serving serve the same market. They have constant average costs of e2 per unit. The firms can choose either a high price ($10) or a low price ($5) for their output. When both firms set a high price, total demand = 10,000 units which is split evenly between the two firms. When both set a low price, total demand is 18,000, which is again split evenly. If one firm sets a low price and the second a high price, the low priced firm sells 15,000 units, the high priced firm only 2,000 units.

a. Analyse the pricing decisions of the two firms as a noncooperative game.

Figure 1: The Stag Hunt Game. (matrix)

stag, stag = 2,2

stag, hare = 0,1

hare, stag = 1,0

hare, hare = 1,1

In the normal form representation, construct the payoff matrix, where the elements of each cell of the matrix are Anna and Beppe's short run profits.

Derive the set of Nash equilibria for this game.

what this game is meant to represent and compare with other games.

EX. 4

Consider the familiar "Rock, Paper, Scissors" game. Two players indicate either "Rock", "Paper", or "Scissors" simultaneously. The winner is determined by:

1. Rock crushes scissors

2. Paper covers rock

3. Scissors cut paper

a. Write down the normal form for this game. b. Identify its equilibria. c. What is the strategic situation this game represents?

EX.5

Consider the normal form game of Figure 1:

Figure 1: The Stag Hunt Game Stag Hare Stag 2,2 0, 1 Hare 1,0 1, 1 1. In the normal form representation, construct the payoff matrix, where the elements of each cell of the matrix are Anna and Beppe's short run profits. 2. Derive the set of Nash equilibria for this game. 3. Explain in words what this game is meant to represent and compare with other games discussed in class. EX. 4 Consider the familiar "Rock, Paper, Scissors" game. Two players indicate either "Rock", "Paper", or "Scissors" simultaneously. The winner is determined by: 1. Rock crushes scissors 2. Paper covers rock 3. Scissors cut paper 1. Write down the normal form for this game. 2. Identify its equilibria. 3. What is the strategic situation this game represents? 4. What is your own way of playing this game? EX.5 Consider the normal form game of Figure 1: 1. Draw the cooperative payoff region and identify the set of Pareto efficient outcomes 2. Identify the set of Nash equilibria for this game. 3. This game has two Nash equilibria, each of which has its pros and cons. Identify them and comment (max = page)