ECO204Y5 Lecture Notes - Subgame, Takers, Westjet

Problem 1

In each of the three games shown below, let p be the probability that player 1 plays cooperates (and 1- p the probability that player 1 defects), and let q be the probability that Player 2 plays cooperates (and 1- q the probability that player 2 defects).

Prisoner’s Dilemma

Stag Hunt

Chicken

1. For each game, draw a graph with player 1’s best response function (choice of p as a function of q), and player 2’s best response function (choice of q as a function of p), with p on the horizontal axis and q on the vertical axis.

2. Using this graphs, find all the Nash equilibriums for the game, both pure and mixed strategy Nash equilibriums (if any). Label these equilibriums on the corresponding graph.

3. In those games that have multiple pure strategy Nash equilibriums, how do the expected payoffs from playing the mixed strategy Nash equilibrium compare with the payoffs from playing the pure strategy Nash equilibriums? Which type of strategy (mixed or pure) would players prefer to play in these games?

Problem 2

Two people are involved in a dispute. Player 1 does not know whether player 2 is strong or weak; she assigns probability α to player 2 being strong. Player 2 is fully informed. Each player can either fight or yield. Each player obtains a payoff of 0 is she yields (regardless of the other person’s action) and a payoff of 1 if she fights and her opponent yields. If both players fight, then their payoffs are (-1; 1) if player 2 is strong and (1;-1) if player 2 is weak. The Bayesian game is the following, depending on the type of player 2:

After writing all the strategies and payoffs in the same matrix, find the Bayesian Nash equilibriums, depending on the value of α (α ≤ 1/2 or α ≥1/2).

Consider the following game, which comes from James Andreoni and Hal Varian at the University of Michigan. A neutral referee runs the game.There are two players, Row and Column. The referee gives two cards to each:
2 and 7 to Row and 4 and 8 to Column. This is common knowledge. Then, playing simultaneously and independently, each player is · asked to hand over to the referee either his high card or his low card. The referee hands out payoffs- which come from a central kitty, not from the players' pockets- which come from a central kitty, not from the players' pockets-that are measured in dollars and depend on the cards that he collects. If row chooses his low card, 2, then row gets $2; if he choses his High card, 7 then Column gets $7. If column chooses his low card, 4, then column gets $4; if he chooses his high card, 8, then row gets $8.

(a) Show that the complete payoff table is as follows:

(b) What is the Nash equilibrium? Verify that this game is a prisoners' dilemma.

Now suppose the game has the following stages. The referee hands
out cards as before; who gets what cards is common knowledge. At stage
I, each player, out of his own pocket, can hand . over a sum of money,
which the referee is to hold in an escrow account. This amount can be
zero 'out cannot 'oe negative. When both have made then Stage l choices,
these are publicly disclosed. Then at stage II, the two make their choices
of cards, again simultaneously and independently. The referee hands
out payoffs from the central kitty in the same way as in the single-stage
game before. In addition, he disposes of the escrow account as follows.
If Column chooses his high card, the referee hands over to Column the
sum that Row put into the account; if Column chooses his low card,
Row's sum reverts back to him. The disposition of the sum that Column
deposited depends similarly on Row's card choice. All these rules are
common knowledge.
(c) Find the rollback (subgame-perfect) equilibrium of this two-stage game.
Does it resolve the prisoners' dilemma? What is the role of the escrow account?

Firms A and B are the only firms in the market for widgets. Each firm can choose between cooperating and fighting. If both firms choose to cooperate, each gets a profit of 10. If both firms choose to fight, each gets a profit of 5. If one chooses to cooperate and the other chooses to fight, the firm that chooses to cooperate gets a profit of 1 and the firm that chooses to fight gets a profit of 20. Each firm chooses the action that maximizes its profits.

(a) Suppose firms choose their actions simultaneously and they play this game once. Write the matrix of payoffs for this situation and find the pure strategy Nash equilibrium (or all equilibria) of the simultaneous-move game played by firms A and B.

(b) Suppose firms choose their actions simultaneously but the game is repeated for 10 years. So, when firms make their decisions in a specific year, each firm knows the choices of the other firm in the previous years. Find the equilibrium (or all equilibria) of this game.