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Assume we have a first-price auction with discrete and independent private values and identical distributions of all bidders. It is known that there exists a Bayes-Nash equilibrium where bidders bid on disjunct bid intervals. For example, we have valuations 0,1,2. Then a bidder with valuation 0 bids zero, a bidder with valuation 1 plays a mixed strategy on some interval $[0,\overline{b}_1]$ and a bidder with valuation 2 plays a mixed strategy on some interval $[\overline{b}_1,\overline{b}_2]$.

Is it true that this Bayes-Nash equilibrium is unique? If this is true, how one can show that overlapping bidding intervals are impossible? Or is it possible to show at least that this is the unique symmetric equilibrium, i.e. an equilibrium where all bidders employ the same strategy?

I tried to find the answer in the literature but I only found papers which derive the symmetric Bayes-Nash equilibrium with disjunct bidding intervals without discussing other possible equilibria. Thank you!

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Ruling out (strictly) overlapping intervals should be possible using the fact that players must be indifferent among all actions in the support of the mixed strategy, or that these type of auctions is efficient.

Uniqueness is tricky, simply because there are many possible interval structures like that one. But I guess you mean, equilibria are always of this form. In that case, uniqueness should be possible among symmetric equilibria, after ruling out the overlapping intervals, you need to show that the support of the mixed strategy is convex (which should be the case, otherwise money is left on the table), that the intervals are ordered (which comes from efficiency), and put restrictions on the bidders with the lowest valuation.

Without symmetry, you can have situations where players with the same valuation use different mixed strategies (even with different support), so the problem there seems untractable to me.

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