This paper concerns a Bayesian game model of the Buyer's Bid Double Auction, which is a procedure for organizing trade that selects a market-clearing price from a list of offers/bids. Strategic misrepresentation by traders may make the outcome of trade inefficient. Satterthwaite and Williams (1989) showed that misrepresentation and inefficiency quickly converge to zero as the number of traders on each side of the market increases. This is extended here to cases in which the number of buyers may differ from the number of sellers. The existence of equilibria in a generic instance of the model is also proven.

In this paper we will show how a message process which "realizes" (or computes) a given social choice rule F can be used to construct a game which implements F in Nash equilibrium. Any efficient encoding of information that occurs in the message process causes a corresponding reduction in the size of the strategy space of the game which we will construct to implement F. Necessary and (stronger) sufficient conditions on the message process will be given for this construction.

How few traders constitute a bargaining problem or, alternatively how many traders constitute a market? Intuitively, people who meet to trade face a bargaining problem if several of them, with opposing interests, can influence the outcome of trade through their behavior. Traders constitute a competitive market only if the effect of any trader on the outcome of trade is insignificant. Analysis of bargaining using noncooperative game theory has been a lively research topic over the past ten years, while the formalization of perfectly competitive markets using a continuum of traders is now several decades old. In this paper I discuss a line of research that studies the transition between these two theories; in particular, a result from Mark Satterthwaite's and my paper (1989b) is presented that shows how a bargaining problem is transformed into a market as the number of traders increases. Incomplete information is an essential feature of the model that I discuss. In order to explain how traders achieve a competitive equilibrium, it is standard to assume that all potential gains from trade are commonly known at the outset. In contrast, each trader in the model that I describe privately knows his own preferences. A second feature is that a market with any finite number of traders on each side is modeled, rather than one with a continuum of traders. These two features together make strategic behavior seem especially likely. Price-taking behavior is not an axiom here; instead, the objective is to prove that a trader's equilibrium behavior is to increasingly act as a price taker as a market grows in size, and that only a small number of traders is needed on each side to compel each trader to act in this way. Though the background of this research will be discussed later in more detail, it is basically motivated by the intuition most of us share that a market can work pretty well despite having a relatively small number of traders who strategically act on their private information. While the axioms of perfect competition are, of course, never satisfied in the real world, I believe that this theory provides insight not only into immense markets but also into smaller markets. The research that I describe is another attempt to substantiate this belief.

A trader who privately knows his preferences may misrepresent them in order to influence the market price. This strategic behaviour may prevent realization of all gains from trade. In this paper, trade in a simple market with an explicit rule for price formation is modelled as a Bayesian game. We show that the difference between a trader's bid and his reservation value is maximally O(1/m) where m is the number of traders on each side of the market. Competitive pressure as m increases thus quickly overcomes the inefficiency private information causes and forces the market towards an efficient allocation.

Strategic behavior in a finite market can cause inefficiency in the allocation, and market mechanisms differ in how successfully they limit this inefficiency. A method for ranking algorithms in computer science is adapted here to rank market mechanisms according to how quickly inefficiency diminishes as the size of the market increases. It is shown that trade at a single market-clearing price in the k-double auction is worst-case asymptotic optimal among all plausible mechanisms: evaluating mechanisms in their least favorable trading environments for each possible size of the market, the k-double auction is shown to force the worst-case inefficiency to zero at the fastest possible rate.

A model of trade with m buyers and m sellers is considered in which price is set to equate revealed demand and supply. In a Bayesian Nash equilibrium, each trader acts not as a price-taker, but instead misrepresents his true demand/supply to influence price in his favor. This causes inefficiency. We show that in any equilibrium the amount by which a trader misreports is O(1/m) and the corresponding inefficiency is O(1/m2). The indeterminacy and the inefficiency that is caused by the traders' bargaining behavior in small markets thus rapidly vanishes as the market increases in size.