Collusive communication schemes in a first-price auction

We study optimal bidder collusion at first-price auctions when the collusive mechanism only relies on signals about bidders' valuations. We build on Fang and Morris (2006) when two bidders have low or high private valuation of a single object and additionally each receives a private noisy signal from an incentiveless center about the opponent's valuation. We derive the unique symmetric equilibrium of the first price auction for any symmetric, possibly correlated, distribution of signals, when these can only take two values. Next, we find the distribution of 2-valued signals, which maximizes the joint payoffs of bidders. We prove that allowing signals to take more than two values will not increase bidders' payoffs if the signals are restricted to be public. We also investigate the case when the signals are chosen conditionally independently and identically out of n = 2 possible values. We demonstrate that bidders are strictly better off as signals can take on more and more possible values. Finally, we look at another special case of the correlated signals, namely, when these are independent of the bidders' valuations. We show that in any symmetric 2-valued strategy correlated equilibrium, the bidders bid as if there were no signals at all and, hence, are not able to collude.