This supplement provides a detailed special case of the proof of Proposition 3.1 in "The Expected Number of Nash Equilibria of a Normal Form Game."

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In Pinkse and Tan (2005) we show the existence of a new effect called the affiliation effect, which can cause equilibrium bids to be decreasing in the number of bidders in first-price auctions with conditionally independent private values. Here we analyze what happens when the number of bidders tends to infinity. We also derive sufficient conditions for the expected winning bid to be increasing in the number of bidders. We further generalize some of the results in Pinkse and Tan (2005) to general affiliated private-values models.

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