Bayesian-Nash games ∗

A finite group of players (economic agents), denoted by N = {1, . . . , n}, interact. Any interaction can be represented as a simultaneous non-cooperative choice of individual plans of action.1 A set of possible decisions or choices (a set of all possible contingent plans of action) is a set of strategies. Denote Si to be a set of strategies available to player i (si ∈ Si is a generic element); s = (s1, . . . , sn) = {si}i∈N is a profile of strategies of all the players, s ∈ S = ×i∈N Si; s−i = sr {si}, S−i = S \ Si, s = (si, s−i). A set of alternatives (allocations), A, is a set of all possible outcomes. A mechanism is a rule that for any collection of strategies selects a probability distribution over the set of alternatives A, M : S ∆(A). → Example 1: Voting. A set of alternatives A is a set of possible candidates to be chosen from. Players are individuals with a right to vote. Sets of strategies are determined by a voting procedure. In a simple ballot voting, for example, each player submits a ballot for some candidate, so Si = A. In a multistage voting, like a procedure to select a city to hold Olympics, a strategy has to name a city in the first round of voting, a city in the second round of voting conditional on results of the first round, and so on, for each round conditional on previous results. A voting mechanism specifies how the winner is selected. For instance, in a simple majority voting, bar ties, M(s) = argmaxa∈A #i∈N {si = a}. Example 2: Auctions. A set of alternatives A is a set of all possible allocations of goods for sale and transfers involved. For example, with one object for sale and only the winner paying, a set of alternatives is A = N × R+, a pair (w,m) ∈ A says that player w ∈ N is the winner and has to pay m. A set of strategies depends on an auction format. In sealed-bid auctions, a strategy is a bid, Si = R+. In open or dynamic auctions, like an English (a usual ascending price) auction, a strategy has to specify how to act (bid) for any possible scenario that can occur in the auction. The winner is determined according to the rules of the specific format. In the first-price sealed-bid auction the winner is the highest bidder and has to pay own bid. Thus, bar ties, w = argmaxi∈N si and m = sw. In the second-price sealed-bid auction the winner is also the highest bidder but pays the highest bid among the rest of the participants, m = maxj=w sj . 6 Example 3: Public good provision (discrete setting). There is a public project that can be built if sufficient funds are collected from citizens (to cover a cost c ∈ R+). Thus, an allocation (b,m1, . . .mn) ∈ ∗To appear as part of “Secure direct implementation” by Sergei Izmalkov, Matt Lepinsky, and Silvio Micali. 1 This is called a normal or strategic form representation. For a detailed coverage of game theory and of incomplete information games in particular the reader is referred to two excellent textbooks on the subject, Fudenberg and Tirole (1991) and Osborne and Rubinstein (1997).