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EC220 Problem Set 3

2020

1 Repeated Games: Cournot [will be covered in Seminar, time permit- ting]

Two ﬁrms 1 = 1, 2 compete repeatedly in a market. They choose their production quan- tities u1 , u2 every period from [0, ]. Production is costless. The market then sets a price P (u1 , u2 ) = 1 · u1 · u2 at which each unit of the good is sold. Firms wish to maximize discounted proﬁts, and they discount at a rate o. As our Folk Theorems apply only to repeated games with ﬁnite stage games, they do not necessarily work here.

1.1 Find the set of Nash Equilibria of the stage-game. Find their payoﬀ vectors.

1.2 Find the feasible set F . (Hint: ﬁrst look for the largest sum the players’ payoﬀs can be in the stage game, and the individual payoﬀs they can attain with actions that maximize the sum of their payoﬀs)

1.3 Suppose w e F . Show there is a pure action proﬁle that achieves w in the stage-game.

1.4 Suppose w e F and w is strictly above the Nash Reversion point -NE . Construct Nash Reversion strategies for the repeated game with payoﬀs w. Find a suﬃciently high discount factor such that at any greater discount factor, those strategies form an SPNE. (You can assume a pure action proﬁle u(w) solves the previous question.)

1.5 Consider the payoﬀ vector (1/6, 1/12). Is it feasible?

1.6 Can the payoﬀ 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 payoﬀ vector (1/6, 1/12). Show that if the players are patient enough (provide a suﬃcient condition on o) then these

strategies form an SPNE.

Hints:

● The pure action proﬁle (1/3, 1/6) gives stage-game payoﬀs (1/6, 1/12).

● To check that a strategy proﬁle is an SPNE, it’s suﬃcient to check the unprof- itability of all ‘one-shot’ deviations - that is, strategies that diﬀer 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 ﬁnite action sets, and just happens to hold in this setting as well despite the inﬁnite action sets.)

2 Cournot with Incomplete Information about Costs [will be covered in Seminar]

Two ﬁrms 1 = 1, 2 compete once in a market. They choose their production quantities u1 , u2 from [0, ]. Production is costly. The market then sets a price 1 · u1 · u2 at which each unit of the good is sold. Firms wish to maximize proﬁts, 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 ﬁrm p is denoted ci, and it is a per unit cost, so that ﬁrm p pays uici to produce ui units. First we’ll consider the case where the production costs of each ﬁrm 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 ) = ( , ) . Solve for the pure Nash equilibrium.

2.3 Assume production costs are commonly known to be (c1 , c2 ) = ( , 0) . Solve for the pure Nash equilibrium.

Now we assume that each ﬁrm 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) = g and P (c1 = ) = P (c2 = ) = 1 · g. Solve for the pure Bayes-Nash equilibrium.

Now, we assume production costs are correlated, that is, each ﬁrm’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 = , c2 = ) = g and P (c1 = 0, c2 = ) = P (c1 = , c2 = 0) = · g. Solve for the pure Bayes-Nash equilibrium as a function of g .

2.6 How does the correlation parameter (g) aﬀect production choices? Why? How do your answers here compare to your answers under commonly known production costs?