Unformatted text preview: minus its total cost,
ui (s1 , s2 ) = si p (s1 + s2 ) − csi
where p (q ) is the price of the good (as a function of the total amount
q ), and c is unit cost (same for both ﬁrms).
Assume for simplicity that c = 1 and p (q ) = max{0, 2 − q } Consider the best response correspondence for each of the ﬁrms, i.e.,
for each i , the mapping Bi (s−i ) : S−i � Si such that
Bi (si ) ∈ arg max ui (si , s−i ).
si ∈Si Why is this a “correspondence” not a function? When will it be a
function?
13 Game Theory: Lecture 2 Strategic Form Games Cournot Competition (continued)
By using the ﬁrst order optimality conditions, we have s2
1 B1(s2) Bi (s−i ) = arg max(si (2 − si − s−i ) − si )
si ≥ 0
� 1−s
−i
if s−i ≤ 1,
2
=
0
otherwise. 1/2 B2(s1)
1/2 s1 1 The ﬁgure illustrates the best response correspondences (which in this case
are functions).
Assuming that players are rational and fully knowledgeable about the
structure of the game and each other’s rationality, what should th...
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This document was uploaded on 03/19/2014 for the course EECS 6.254 at MIT.
 Spring '10
 AsuOzdaglar

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