We study trading behavior and the properties of prices in informationally complex markets. Our model is based on the single-period version of the linear-normal framework of Kyle (1985). We allow for essentially arbitrary correlations among the random variables involved in the model: the value of the traded asset, the signals of strategic traders and competitive market makers, and the demand from liquidity traders. We show that there always exists a unique linear equilibrium, characterize it analytically, and illustrate its properties in a series of examples. We then use this characterization to study the informational efficiency of prices as the number of strategic traders becomes large. If liquidity demand is positively correlated (or uncorrelated) with the asset value, then prices in large markets aggregate all available information. If liquidity demand is negatively correlated with the asset value, then prices in large markets aggregate all information except that contained in liquidity demand.

This supplement contains additional results and proofs omitted from the main paper. S.1. ZERO AGGREGATE DEMAND IN EQUILIBRIUMIN THIS SECTION, we formally derive equilibria in the special examples in which aggregate demand ends up being zero on the equilibrium path (footnote 12 in Section 2.2 and footnote 20 in Section 5). In Section S.1.1, we work out the example with one informed trader, corresponding to footnote 12, and in Section S.1.2, we work out the example with multiple informed traders, corresponding to footnote 20. S.1.1. One Informed Trader EXAMPLE S.1: The value of the security, v, is distributed normally with mean 0 and variance 1. There is one strategic trader with signal θ 1 who observes the value perfectly: θ 1 = v. The demand of liquidity traders is u = −v. The market maker does not observe any signals beyond the aggregate demand.This example satisfies Assumptions 1 and 2 in Section 3, and thus we can use the closedform solutions derived in the proof of Theorem 1 and presented in Section 3.2 of the paper (for the special case k M = 0). Because we have only one strategic trader in the example, many of the matrices become scalars, simplifying the calculation.Specifically, Σ θθ = Σ diag = 1, and, therefore, Λ = 2 and Λ −1 = 1/2. Next, Σ θv = 1, while This example also satisfies Assumptions 1 and 2 in Section 2, and thus we can use the closed-form solutions derived in the proof of Theorem 1 and presented in Section 3.2 (again, for the special case k M = 0). We now have multiple strategic traders, so the calculations involve matrix manipulations.Specifically, Σ θθ is an m-dimensional matrix whose elements are all equal to 1, while Σ diag is an m-dimensional identity matrix. We thus haveThe coefficients of the quadratic equation on γ are, therefore2 , and c = 1/(m + 1) 2 , which in turn gives us γ = 1/m and β D = m. Thus,and so on the equilibrium path, aggregate demand is equal toThus, for every m, the market price on the equilibrium path is also always equal to 0, not revealing any information contained in the signals of the strategic traders and in the demand coming from the liquidity traders. S.2. ZERO INTERCEPTS IN EQUILIBRIUMIn this section, we prove the statement made informally in footnote 13 in Section 3.2 that in our setting, linear equilibria with nonzero intercepts do not exist. PROPOSITION S.1: Suppose there exists an equilibrium of the formThen β 0 = 0 and for all i, δ i = 0.PROOF: Consider a particular realization of θ i , θ M , and u. Then in this equilibrium, the realized price will be given byBy the definition of equilibrium, for every realization of θ M and D, the price set by the market maker is equal to the expected value of the security conditional on θ M and D:Integrating over all possible realizations of θ M and D, we thus get, for the unconditional expectation of the price,Since by assumption, E[v] = 0, and also E[θ M ], E[u], and E[θ i ] (for all i) are equal to 0, by taking the unconditional expectation of Equation (S.1), we getNow, as inStep 2 of the proof of Theorem 1, consi...

We study trading behavior and the properties of prices in informationally complex markets. Our model is based on the single-period version of the linear-normal framework of [Kyle 1985]. We allow for essentially arbitrary correlations among the random variables involved in the model: the true value of the traded asset, the signals of strategic traders, the signals of competitive market makers, and the demand coming from liquidity traders. We first show that there always exists a unique linear equilibrium, characterize it analytically, and illustrate its properties in a series of examples. We then use this equilibrium characterization to study the informational efficiency of prices as the number of strategic traders becomes large. If the demand from liquidity traders is uncorrelated with the true value of the asset or is positively correlated with it (conditional on other signals), then prices in large markets aggregate all available information. If, however, the demand from liquidity traders is negatively correlated with the true value of the asset, then prices in large markets aggregate all available information except that contained in liquidity demand. Categories and Subject Descriptors: J.4 [Social and Behavioral Sciences]: Economics Extended AbstractWhether and how dispersed information enters into market prices is one of the central questions of information economics. A key obstacle to full information revelation and aggregation in markets is the strategic behavior of informed traders. A trader who has private information about the value of an asset has an incentive to trade in the direction of that information. However, the more he trades, the more he reveals his information, and the more he moves the prices closer to the true value of an asset. Thus, to maximize his profits, an informed trader may stop short of fully revealing his information, and thus the informational efficiency of market prices may fail.Thus, an important and natural question is when we should expect market prices to in fact reflect all information available to market participants. One intuition proposed in the literature is that we should expect such outcomes when the number of informed traders is large, and each one of them is informationally small. In that case, each of the informed traders has limited impact on market prices, but their aggregate behavior does in fact reflect the aggregate information available in the market. As a result, market prices are close to those that would prevail if all private information were publicly available, and all trades happened at those prices.Non-strategic explorations of this intuition go back to [Hayek 1945], [Grossman 1976], and[Radner 1979]. Subsequently, a line of research has considered strategic foundations for this intuition, studying strategic behavior of informed agents in finite markets, and then considering the properties of prices as the number of these agents becomes large. This stream of work, however, imposes very strict assumptions on how