By Akira Takayama
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Additional info for Analytical methods in economics
This left-boundary condition simply requires any bidder who draws the lowest valuation possible to bid his valuation. 9 We shall need to 9 If a reserve price r existed, then σ (r ) = r would be the relevant condition: the marginal bidder would n 0 0 0 bid the reserve price r0 . 49 On the Numerical Solution of Equilibria in Auction Models use the boundary condition(s) with the system of differential equations to solve for the inverse-bid functions, as discussed above. Specifically, we shall be interested in solving for a monotone pure-strategy equilibrium (MPSE) in which each bidder adopts a bidding strategy that maximizes expected pay-offs given the strategies of the other players.
In particular, we contrast features of this problem with those of the symmetric case presented in the previous subsection. Suppose that bidder 1 gets an independent draw from urn 1, denoted F1 (v1 ), while bidder 2 gets an independent draw from urn 2, denoted F2 (v2 ). Assume that the two valuation distributions have the same support [v, v¯ ]. The largest of the two bids wins the auction, and the winner pays what he bid. Now, U1 (s1 ), the expected profit of bid s1 to player 1, can be written as U1 (s1 ) = (v1 − s1 ) Pr(win|s1 ), while U2 (s2 ), the expected profit of bid s2 to player 2, can be written as U2 (s2 ) = (v2 − s2 ) Pr(win|s2 ).
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Analytical methods in economics by Akira Takayama