games exhibit serious discontinuities, so that the existence of Nash equilibria, and even best replies(see below) is not guaranteed by the standard machinery Quasi-linearity completes the specification of payoffs: fixing a payment rule m, for any profile a∈A, mot()-m(o)i∈H(a) i g H(a) nalysIs We now consider a few even more special illustrative cases. Before we begin, consider the following definition Definition 1 Fix a bidding game g defined as above, and a player iE N. a bid function for bidder i is a function ai: Si-A That is a bid function is a degenerate signal-contingent randomized action Second-Price Auctions with Private values Let us assume for notational simplicity that vi(s)=si. These bidding games are traditionally analyzed using the notion of weak dominance. An action ai weakly dominates an alternative action a; for type s; of bidder i iff, for all opponent bid functions(a,)i#i and for all realization of the opponents' signals t(a,(a3(s;)≠,sS-)≥t(a1,(a(s)≠i,S,-) and moreover there exists a profile of opponents' bid functions and a profile of signal real- izations for which the above inequality is strict p Lith private values, however, this reduces to the requirement that, for all profiles of onents'actions a_i(not bid tions!) ui( vs-∈0,1 and that the inequality be strict for at least one profile of opponents'actions; this is a consequence of the assumption that s-i does not affect vi(s But this leads to the conclusion that, for any signal realization s E 0, 1, bidder iE N and pair of actions ai, ah, ai weakly dominates a for type si of bidder i iff ai weakly dominates a; in the game without payoff uncertainty defined by G(s)=N, (Ai, ui( s))ieN. This is the game you analyzed in Problem Set 1, so you have already proved the following resultgames exhibit serious discontinuities, so that the existence of Nash equilibria, and even best replies (see below) is not guaranteed by the standard machinery. Quasi-linearity completes the specification of payoffs: fixing a payment rule m, for any profile a ∈ A, ui(a, s) =  1 |H(a)| [vi(s) − mi(a)] i ∈ H(a) 0 i 6∈ H(a) Analysis We now consider a few even more special illustrative cases. Before we begin, consider the following definition: Definition 1 Fix a bidding game G defined as above, and a player i ∈ N. A bid function for bidder i is a function ai : Si → Ai . That is, a bid function is a degenerate signal-contingent randomized action. Second-Price Auctions with Private Values Let us assume for notational simplicity that vi(s) = si . These bidding games are traditionally analyzed using the notion of weak dominance. An action ai weakly dominates an alternative action a 0 i for type si of bidder i iff, for all opponent bid functions (aj )j6=i and for all realization of the opponents’ signals, ui(ai ,(aj (sj ))j6=i , si , s−i) ≥ ui(a 0 i ,(aj (sj ))j6=i , si , s−i) and moreover there exists a profile of opponents’ bid functions and a profile of signal real￾izations for which the above inequality is strict. With private values, however, this reduces to the requirement that, for all profiles of opponents’ actions a−i (not bid functions!), ui(ai , a−i , si , s−i) ≥ ui(a 0 i , a−i , si , s−i) ∀s−i ∈ [0, 1]N−1 and that the inequality be strict for at least one profile of opponents’ actions; this is a consequence of the assumption that s−i does not affect vi(s). But this leads to the conclusion that, for any signal realization s ∈ [0, 1]N , bidder i ∈ N and pair of actions ai , a0 i , ai weakly dominates a 0 i for type si of bidder i iff ai weakly dominates a 0 i in the game without payoff uncertainty defined by G(s) = {N,(Ai , ui(·, s))i∈N }. This is the game you analyzed in Problem Set 1, so you have already proved the following result: 4