The bulk of the literature assumes quasi-linear preferences(although risk-aversion does play a role, and is analyzed in several papers). However, this is clearly not sufficient to determine payoffs. We still need to specify: (i) how much the object is worth to each bidder we shall assume quasi-linear utility); and (ii)the bidders'transfers to the seller. Moreover unless we assume that players'valuations are known, we need to specify (iii) the form of payoff uncertainty we are interested in Specifying (i),(ii) and (ii) leads to a taxonomy of bidding games With respect to (i), we distinguish between private and interdependent values: in the former setting, each player knows her valuation for the object(but may be uncertain about her opponents' valuation); in the latter, one or more players may be uncertain about her valuation, which will typically be related to other players'valuations and or realizations of underlying state variables. In particular, if the object is worth the same to all bidders, but no bidder is certain of the value, we are in a pure common values setting Payment rules(ii)pin down the auction mechanism, the most popular being first-price, second-price and (to a lesser degree) all-pay auctions Finally, with respect to(iii), whatever information bidders receive or hold(be it their valuation or a signal thereof) may be either independent or correlated A few examples will clarify (i) and (ii: the differences may appear to be subtle, but they are substantial Suppose that each bidder i E N observes a signal si E 0, 1, that F denotes the joint cdf of the signals, and that bidder i s valuation is a function vi: [0, 1 -R+. This is a very common specification of payoff uncertainty n this simplified(but very popular) framework, (i) and (ii) entail restrictions on th functions v; and F respectively. If u (si, S-i)=v(si, s'i) for all s; E 0, 1] and s-i E [0, 1]N-1 then we are in a setting with private values: bidder i's valuation is uniquely determined by her own signal si, which she observes. In the simplest case, ui(si, s-i)= si(which, note well, also entails a symmetry assumption) I Sellers may also set a reserve price: all bids below the reserve price are simply rejected. In this case the object may remain unsold if all bids are below the reserve price. Although Myerson shows that, in certain situations, setting a positive reserve price maximizes expected revenues, we shall simply ignore this 2In the most common auction formats, only the winning bidder is required to pay. However, in all-pay auctions, as the name suggests, all bidders pay their bid. Indeed, one can conjure up just about any kind of payment rule: as long as the winning bidder is a high bidder, you can still call the resulting mechanism anThe bulk of the literature assumes quasi-linear preferences (although risk-aversion does play a role, and is analyzed in several papers). However, this is clearly not sufficient to determine payoffs. We still need to specify: (i) how much the object is worth to each bidder (we shall assume quasi-linear utility); and (ii) the bidders’ transfers to the seller.2 Moreover, unless we assume that players’ valuations are known, we need to specify (iii) the form of payoff uncertainty we are interested in. Specifying (i), (ii) and (iii) leads to a taxonomy of bidding games: • With respect to (i), we distinguish between private and interdependent values: in the former setting, each player knows her valuation for the object (but may be uncertain about her opponents’ valuation); in the latter, one or more players may be uncertain about her valuation, which will typically be related to other players’ valuations and/or realizations of underlying state variables. In particular, if the object is worth the same to all bidders, but no bidder is certain of the value, we are in a pure common values setting. • Payment rules (ii) pin down the auction mechanism, the most popular being first-price, second-price and (to a lesser degree) all-pay auctions. • Finally, with respect to (iii), whatever information bidders receive or hold (be it their valuation or a signal thereof) may be either independent or correlated. A few examples will clarify (i) and (iii): the differences may appear to be subtle, but they are substantial. Suppose that each bidder i ∈ N observes a signal si ∈ [0, 1], that F denotes the joint cdf of the signals, and that bidder i’s valuation is a function vi : [0, 1]N → R+. This is a very common specification of payoff uncertainty. In this simplified (but very popular) framework, (i) and (iii) entail restrictions on the functions vi and F respectively. If vi(si , s−i) = vi(si , s0 −i ) for all si ∈ [0, 1] and s−i ∈ [0, 1]N−1 , then we are in a setting with private values: bidder i’s valuation is uniquely determined by her own signal si , which she observes. In the simplest case, vi(si , s−i) = si (which, note well, also entails a symmetry assumption). 1Sellers may also set a reserve price: all bids below the reserve price are simply rejected. In this case, the object may remain unsold if all bids are below the reserve price. Although Myerson shows that, in certain situations, setting a positive reserve price maximizes expected revenues, we shall simply ignore this possibility. 2 In the most common auction formats, only the winning bidder is required to pay. However, in all-pay auctions, as the name suggests, all bidders pay their bid. Indeed, one can conjure up just about any kind of payment rule: as long as the winning bidder is a high bidder, you can still call the resulting mechanism an “auction”. 2