U(vi,b)=(vi-b)Fj(φj(b))
for i ≠ j = S,W
(1)
Since it is not worth bidding more than the maximum possible bid of one’s competitor, there
must exist some common maximum bid, b .2 Therefore, in equilibrium, the inverse bid
functions must satisfy the following two boundary conditions φi (0) = 0 and Fi (φi ( b )) = 1 for
i = S, W.
Maskin and Riley (2000a) show that in the Risk Neutral Nash Equilibrium (RNNE), inverse
bid functions φS* (b) and φW* (b) are determined as the solution to the system of non-linear
differential equations generated from the first-order conditions of (1) with respect to b. If we
assume symmetric bidders, then FS(v)=FW(v)=F(v),φS(b)=φW(b)=φ(b) and
IS ≡ IW ≡ I = [v; v ] so that (1) has a single first order condition with a unique solution for the
boundary condition φ(0) = 0 .3 Vickrey (1961) established that the RNNE bidding strategy for
symmetric bidders is
AM v(F ( x Ъ
(2)
b(v) =v-L- dx
v F(v)
In the case of asymmetric bidders, the system of non-linear differential equations that solve
the first-order conditions of (1) usually has no analytical solution and has to be determined
numerically.However, if bidders’ distributions of values satisfy a few additional assumptions,
Maskin and Riley (2000a, Proposition 3.5) predict that in equilibrium, Strong bidders bid less
2 See Plum (1992), Marshall, Meurer, Richard and Stromquist (1995), Corns and Schotter (1998), Landsberger,
Rubinstein, Wolfstetter and Zamir (1999), Lebrun (1999) and Li and Riley (1999) for other asymmetric auction
models that require a common ceiling on bids.
3 Maskin and Riley (2000b) show that if bidders are symmetric, then the equilibrium bidding strategy is unique
and monotone increasing.
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