Economic and Game Theory
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Consider the double auction with one seller and one buyer (Chatterjee and
Samuelson (1983)). The valuations vs and vb of the seller and the buyer are
independently drawn from the uniform distribution on the interval [0; 1].
Trade takes place if the bid pb of the buyer exceeds the bid ps of the seller.
That is, when pb ļ ps. The price against which trade takes place is set to
be p = 1/2 (pb + ps).
Suppose both bidders use linear bid functions,
pb(vb) = abvb + cb
ps(vs) = asvs + cs;
where ab, cb, as and cs are parameters.
(a) Show that the unique linear BNE of the double auction is given by the
pb(vb) =2/3vb +1/12
ps(vs) =2/3vs +1/4;
(b) Argue that (ps; pb) constitutes a BNE precisely when pĪs = vĪs = pĪb=vĪb .
(c) Suppose that (ps; pb) is a BNE. Write pĪ for the (common) value of the
parameters. Picture the region of ineąciency of the BNE. For which
value of pĪ is the probability of ineąciency minimized?
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