EC220 Problem Set 3
November 19, 2020
This is the third problem set. Let me just take this space to re-iterate that it is nigh impossible to do well in this class if you don’t use the textbook.
Book Problems [will not be covered in Seminar]
Tadelis problems 10.4, 10.6 (in (c), consider only Nash Reversion strategies), 10.8, 12.3, 12.4, 12.5, 12.7, 12.9, 13.5, 13.10
Relevant readings: Tadelis chapters 10,12,13
1 Repeated Games: Cournot [will be covered in Seminar, time permit-ting]
Two irms I = 1, 2 compete repeatedly in a market. They choose their production quan- tities q1 , q2 every period from [0, 2/1]. Production is costless. The market then sets a price P (q1 , q2 ) = 1 - q1 - q2 at which each unit of the good is sold. Firms wish to maximize discounted proits, and they discount at a rate δ. As our Folk Theorems apply only to repeated games with inite stage games, they do not necessarily work here.
1.1 Find the set of Nash Equilibria of the stage-game. Find their payof vectors.
1.2 Find the feasible set F . (Hint: frst look for the largest sum the players’ payofs can be in the stage game, and the individual payofs they can attain with actions that maximize the sum of their payofs)
1.3 Suppose v 2 F. Show there is apure action profle that achieves v in the stage-game. 1.4 Suppose v 2 F and v is strictly above the Nash Reversion point wNE . Construct Nash Reversion strategies for the repeated game with payofs v. Find a sufficiently high discount factor such that at any greater discount factor, those strategies form an SPNE. (You can assume a pure action profle q(v) solves the previous question.)
1.5 Consider the payof vector (1/6, 1/12). Is it feasible?
1.6 Can the payof vector be achieved in a Nash-Reversion SPNE of the repeated game?
1.7 Find the minmax vector.
1.8 Consider strategies that revert to minmaxing the last deviator forever, or until an- other deviation occurs. Construct such strategies with payof vector (1/6, 1/12). Show that if the players are patient enough (provide a sufficient condition on δ) then these strategies form an SPNE.
Hints:
. The pure action proile (1/3, 1/6) gives stage-game payofs (1/6, 1/12).
. To check that a strategy proile is an SPNE, it’s su伍cient to check the unprof- itability of all‘one-shot’deviations - that is, strategies that difer from the original strategy at only a single history. More complicated deviations need not be checked, because in an SPNE, continuation play at any history (even one after a deviation) is itself an SPNE, and therefore players are already doing the best they can! (This always holds for repeated games with inite action sets, and just happens to hold in this setting as well despite the ininite action sets.)
2 Cournot with Incomplete Information about Costs [will be covered in Seminar]
Two irms I = 1, 2 compete once in a market. They choose their production quantities q1 , q2 from [0, 2/1]. Production is costly. The market then sets a price 1 − q1 − q2 at which each unit of the good is sold. Firms wish to maximize proits, that is, their sales revenue minus their cost of production. (All NE/BNE in this exercise are pure, and you can assume this)
Production costs for irm i is denoted ci, and it is a per unit cost, so that irm i pays q ici to produce qi units. First we’ll consider the case where the production costs of each irm are commonly known.
2.1 Assume production costs are commonly known to be (c1 , c2) = (0, 0) . Solve for the pure Nash equilibrium.
2.2 Assume production costs are commonly known to be (c1 , c2) = (4/1,4/1). Solve for the pure Nash equilibrium.
2.3 Assume production costs are commonly known to be (c1 , c2) = (4/1,0). Solve for the pure Nash equilibrium.
Now we assume that each irm only knows its own production cost. There is, however, a common prior over the production costs. These production costs are independent.
2.4 Assume production costs are identically and independently distributed with P (c1 = 0) = P (c2 = 0) = p and P (c1 = 0) = P(c2 = 0) = p and P(c1 = 4/1) = P(c2 = 4/1) = 1 − p. Solve for the pure Bayes-Nash equilibrium.
Now, we assume production costs are correlated, that is, each irm’s production costs contain statistical information about the other’s.
2.5 Assume production costs are distributed according to P (c1 = 0, c2 = 0) = P (c1 = 4/1, c2 = 4/1) = p and P(c1 = 0, c2 = 4/1) = P(c1 = 4/1, c2 = 0) = 2/1 − p. Solve for the pure Bayes-Nash equilibrium as a function of p.
2.6 How does the correlation parameter (p) afect production choices? Why? How do your answers here compare to your answers under commonly known production costs?