Otherwise, valuations are interdependent; for instance, a pedagogically useful and con- venient specification is vi (s)= ieN S;. This is known as the "wallet game (note again the implicit symmetry assumption The latter is a specification with pure common values. The expression"interdependent values"allows for intermediate cases such as vi (s)=a;:+2itiS, for some ai>1 either case, the signals si, i E N, may be independent(s F(s)=leN Fi(si) for some collection of one-dimensional cdf's Fi, i E N or correlated; for instance, suppose that there exist N+l i.i.d. random variables o, 11,. N, uniform on 0,1, such that Si=ro+ai for every iE N The key intuition is that, since bids will be functions of the signals, when signals are correlated, bidder i can make inferences about her opponents equilibrium bids based on her own signal. This is of course true regardless of whether values are private or interdependent biste inally, payment rules are relatively simple to specify. In a first-price auction, the winning m2(a) ani∈H(a) 0 i H(a) In a second-price auction, she pays the highest non-winning bid: that m2(a) 0xa,i∈H(a) i g H(a Auctions are modelled as (infinite)Bayesian games. In the simplified setting we are looking at, signals contain all payoff-relevant information(although other formulations could considere ed ). It is typical to let g2= s and t(s)=s1×[0,1]-1 reflecting the assumption that players know their signal, and nothing else. Indeed, in this setting, the distinction between the signal si and the corresponding type si x 0, 1]N-Iis blurred, and I will follow conventional usage by adopting whatever term is most convenient The cdf F is then used to define a common prior on Q. As I have already noted, this has several implications for equilibrium analysis; one could entertain different hypotheses, but this is not the point of today's lecture Also note that the resulting game G=(N, Q, (Ai, ui, tien)is infinite both because action spaces are(Ai=R+) and because the state space is. It should be noted that bidding An 3The idea is as follows: players place their wallets on the table, then bid for the money contained in all allets. Each participant knows how much money she is carrying in her own wallet, but ignores the content of the other walletsOtherwise, valuations are interdependent; for instance, a pedagogically useful and con￾venient specification is vi(s) = P i∈N si . This is known as the “wallet game3” (note again the implicit symmetry assumption). The latter is a specification with pure common values. The expression “interdependent values” allows for intermediate cases such as vi(s) = aisi + P j6=i sj , for some ai > 1. In either case, the signals si , i ∈ N, may be independent (so that F(s) = Q i∈N Fi(si) for some collection of one-dimensional cdf’s Fi , i ∈ N) or correlated; for instance, suppose that there exist N + 1 i.i.d. random variables x0, x1, . . . , xN , uniform on [0, 1 2 ], such that si = x0 + xi for every i ∈ N. The key intuition is that, since bids will be functions of the signals, when signals are correlated, bidder i can make inferences about her opponents’ equilibrium bids based on her own signal. This is of course true regardless of whether values are private or interdependent. Finally, payment rules are relatively simple to specify. In a first-price auction, the winning bidder pays her bid. Thus, the payment rule m : A → RN + is defined by mi(a) =  ai i ∈ H(a) 0 i 6∈ H(a) In a second-price auction, she pays the highest non-winning bid: that is, mi(a) =  maxj6=i aj i ∈ H(a) 0 i 6∈ H(a) Auctions are modelled as (infinite) Bayesian games. In the simplified setting we are looking at, signals contain all payoff-relevant information (although other formulations could be considered). It is typical to let Ω = S and ti(s) = si × [0, 1]N−1 reflecting the assumption that players know their signal, and nothing else. Indeed, in this setting, the distinction between the signal si and the corresponding type si × [0, 1]N−1 is blurred, and I will follow conventional usage by adopting whatever term is most convenient. The cdf F is then used to define a common prior on Ω. As I have already noted, this has several implications for equilibrium analysis; one could entertain different hypotheses, but this is not the point of today’s lecture. Also note that the resulting game G = (N, Ω,(Ai , ui , ti)i∈N ) is infinite both because action spaces are (Ai = R+) and because the state space is. It should be noted that bidding 3The idea is as follows: players place their wallets on the table, then bid for the money contained in all wallets. Each participant knows how much money she is carrying in her own wallet, but ignores the content of the other wallets. 3