of the solution to the model. In particular, we do not wish to assume that such probabilities are commonly agreed upon among the players. Notice that OR follows the same approach The key elements are the set of states of the world, which also affect payoffs(otherwise there could be no payoff uncertainty! ) and the players' type partitions The idea is that, if the prevailing state is w E Q, each player i E N is only notified that the prevailing state is in the unique cell ti E Ti such that w E ti; that is, Player i rules out any w' ti, but regards any w'e ti as possible. We denote by ti(w) the set of states that Player i deems possible at w E S: that is, ti(w)is the unique ti E T such that w E Ti Indeed, in order to interpret the partition Ti correctly, one should think of states in each cell ti E Ti as virtually indistinguishable from the point of view of Player i. This will be clear from our use of the model To fix ideas, consider the following version of the"mineral rights " auction settin are two bidders, i=1, 2, and each receives a signal s; E 10,1,..., 10); the value of the object (concession)is v(s1, $2)=S1+S2 Thus,9={0,1 }×{0,1,…,10};T={{ k}:k=0, ui(ai, a-i,(S1, $2))=S1+S2-ai if ai>a-i,(S1+S2-ai)if ai=a-i, and 0 otherwise Si. On the other hand, both her signal and her opponent's signal determine payoffs e value The key observation is that each bidder i E N only observes her "part"of the tru Elements of the partition Ti are also referred to as types. Each type arguably captures a possible information state of Player i: it tells us all she knows about the prevailing state of the world Note well that, on the other hand, a type generally does not uniquely determine a player's payoff corresponding to each action profile: this is illustrated in the mineral rights auction example above. Perhaps more interestingly, one could think of different information states sayt,t∈T, such that, for all c,u’∈tu2t(a,a-1,)=t1(a1,a-;u) for all(a;,a-i)∈A: for example, this might be the case if, at w and w, player i attaches the same value to a painting, but holds different beliefs as to the prevailing state of the world(hence, about her opponents valuation). More on this point later Bayesian Nash Equilibrium The standard analysis of normal-form games with payoff uncertainty(suitably extended to accommodate our decision-theoretic view) is centered upon the notion of Bayesian Nash Equilibrium, due to John Harsany Definition 2 A Bayesian Nash Equilibrium of a normal-form game with payoff uncertaint G=(N,9,(A2,t1,T)eN) is a tuple(P,(at);∈n;)∈ v such that, for all i∈N (1)n2∈△()and, for all t∈T1,p1(t2)>0of the solution to the model. In particular, we do not wish to assume that such probabilities are commonly agreed upon among the players. Notice that OR follows the same approach. The key elements are the set of states of the world, which also affect payoffs (otherwise there could be no payoff uncertainty!), and the players’ type partitions. The idea is that, if the prevailing state is ω ∈ Ω, each player i ∈ N is only notified that the prevailing state is in the unique cell ti ∈ Ti such that ω ∈ ti ; that is, Player i rules out any ω 0 6∈ ti , but regards any ω 0 ∈ ti as possible. We denote by ti(ω) the set of states that Player i deems possible at ω ∈ Ω: that is, ti(ω) is the unique ti ∈ Ti such that ω ∈ Ti . Indeed, in order to interpret the partition Ti correctly, one should think of states in each cell ti ∈ Ti as virtually indistinguishable from the point of view of Player i. This will be clear from our use of the model. To fix ideas, consider the following version of the “mineral rights” auction setting: there are two bidders, i = 1, 2, and each receives a signal si ∈ {0, 1, . . . , 10}; the value of the object (concession) is v(s1, s2) = s1 + s2. Thus, Ω = {0, 1, . . . , 10} × {0, 1, . . . , 10}; Ti = {{(s1, s2) : si = k} : k = 0, 1, . . . , 10}; and ui(ai , a−i ,(s1, s2)) = s1 + s2 − ai if ai > a−i , 1 2 (s1 + s2 − ai) if ai = a−i , and 0 otherwise. The key observation is that each bidder i ∈ N only observes her “part” of the true value, si . On the other hand, both her signal and her opponent’s signal determine payoffs. Elements of the partition Ti are also referred to as types. Each type arguably captures a possible information state of Player i: it tells us all she knows about the prevailing state of the world. Note well that, on the other hand, a type generally does not uniquely determine a player’s payoff corresponding to each action profile: this is illustrated in the mineral rights auction example above. Perhaps more interestingly, one could think of different information states, say ti , t0 i ∈ Ti , such that, for all ω, ω0 ∈ ti∪t 0 i , ui(ai , a−i , ω) = ui(ai , a−i , ω0 ) for all (ai , a−i) ∈ A: for example, this might be the case if, at ω and ω 0 , player i attaches the same value to a painting, but holds different beliefs as to the prevailing state of the world (hence, about her opponents’ valuation). More on this point later. Bayesian Nash Equilibrium The standard analysis of normal-form games with payoff uncertainty (suitably extended to accommodate our decision-theoretic view) is centered upon the notion of Bayesian Nash Equilibrium, due to John Harsanyi: Definition 2 A Bayesian Nash Equilibrium of a normal-form game with payoff uncertainty G = (N, Ω,(Ai , ui , Ti)i∈N ) is a tuple (pi ,(αi,ti )ti∈Ti )i∈N such that, for all i ∈ N: (1) pi ∈ ∆(Ω) and, for all ti ∈ Ti , pi(ti) > 0; 3