Information Aggregation in Large Markets ú

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1 Information Aggregation in Large Markets ú Maximilian Mihm Lucas Siga May, 208 Abstract This paper studies when equilibrium prices can aggregate information in a large market. We consider an auction trading mechanism, where a large population of buyers and sellers receive private signals about the common value of an asset. Our main result identifies a property of information the betweenness property that is both necessary and su cient for aggregation. The characterization allows us to make predictions about equilibrium prices in complex information environments. As an application, we consider a multidimensional environment where the value of the asset depends on multiple inputs but signals are input specific. Information in the market is highly fragmented, yet equilibrium prices can aggregate information generically. Introduction When do prices aggregate information? This question is central to understanding a market economy, where prices are often the primary channel by which information about unknown fundamentals is transmitted to analysts and market participants. For example, the e cient market hypothesis embodies the idea that prices incorporate all private information held by traders in an asset market, and this idea is a cornerstone of modern macroeconomics and finance. In this paper, we provide a general answer to the information aggregation problem in a large market where the common-value of an asset depends on an unknown state, and traders receive private i.i.d. signals. We consider an auction trading mechanism that resembles the call market used to set daily opening prices on the New York Stock Exchange. After observing signals, traders submit sealed bid and ask prices. An auctioneer collects orders, and determines the market-clearing price. Our main result characterizes the environments where equilibrium prices aggregate information in this market when there is large (atomless) population of strategic buyers and sellers. Our model of the market combines key features of both strategic auction and competitive equilibrium models. Traders choose bids based solely on their private signals and cannot condition directly on prices. ú We thank Nageeb Ali, Vladimir Asriyan, Ayelen Banegas, Pablo Beker, Larry Blume, Vince Crawford, Mehmet Ekmekci, Vijay Krishna, Mark Machina, Larry Samuelson, Jeroen Swinkels, Juuso Välimäki, and Joel Watson for helpful comments and suggestions. Special thanks to Joel Sobel for his invaluable guidance at the various stages of this project. Division of Social Science, NYU Abu Dhabi. max.mihm@nyu.edu. Division of Social Science, NYU Abu Dhabi. lucas.siga@nyu.edu.

2 With their bids, traders determine the chances that they will trade. However, the large population implies that individual traders have negligible impact on prices and total trading volume. Accordingly, our model formalizes the key price-taking assumption of competitive equilibrium models, but with an explicit price formation process based on strategic auction models. The model allows us to address significant limitations in the previous literature on information aggregation in markets. We show that equilibrium prices can aggregate information even in complex information environments, where the previous auction literature makes no predictions about the information e ciency of prices. On the other hand, we establish limitations of market trading mechanisms by identifying the environments where Bayes-Nash equilibrium prices cannot implement a fully-revealing rational expectations equilibrium. To fix ideas, we start with two simple examples. Example. Consider a market for asset X, which depends on two independent inputs A and B (e.g., real returns from an investment in two di erent sectors, or yields of a commodity in two di erent locations). For simplicity, assume that the value of the asset is the sum of the two inputs. Traders are ex-ante identical, but receive specialized information (e.g, by industry or region). With equal probability, a trader receives a signal that is perfectly informative about one of the inputs but conveys no information about the other input. In a market with a large population of traders, public signals reveal the value of the asset almost surely because half of the population is perfectly informed about input A and the other half is perfectly informed about input B. The question is whether, in a market with private information, prices can aggregate the information dispersed over market participants. This market has a fully-revealing rational expectations equilibrium. But when traders condition directly on fully-revealing prices, they can ignore their private signals. It is therefore unclear where prices originate, or how they incorporate information (Hellwig, 980; Milgrom, 98). The auction literature addresses this problem by providing a complete description of the trading mechanism. However, in the market for asset X, signals do not satisfy the monotone likelihood-ratio property (MLRP). Previous results in the auction literature depend on the MLRP to establish an equilibrium in monotone bidding strategies, and nothing is known about whether auction prices can aggregate information when the MLRP is not satisfied. An auction with a large population of traders provides a new approach to the aggregation problem. For instance, it is straightforward to show that equilibrium prices can aggregate information in the market for asset X. To illustrate, assume that each input has a value of either or 2. As a result, there are four possible states {(, ), (, 2), (2, ), (2, 2)}, corresponding to the realization of the two inputs, and three possible values for the asset {2, 3, 4}. Suppose half of the traders are sellers who each own one unit of the asset, and the other half are buyers with unit demand. Now consider the following strategy. There are four signals in this environment, {L A, H A, L B, H B }, where a low signal L c conveys the information that input c œ{a, B} has the low realization and a high signal H c conveys the information that input c has the high realization 2. With a low signal, a trader submits a bid of 2 with probability 2 3 and 3 with probability 3 ; with a high signal, the trader submits a bid of We therefore follow Aumann (964, p.39), who argues that a mathematicalmodelappropriatetotheintuitivenotion of perfect competition must contain infinitely many participants and Milgrom (98, p.923), who argues that to address seriously such questions as how do prices come to reflect information...one needs a theory of how prices are formed. 2

3 3 with probability 3 and 4 with probability 2 3. When all traders follow this strategy, we can appeal (informally for now) to the law of large numbers to describe aggregate demand and supply. For each state, the aggregate demand D(p) represents the mass of buyers who submit a bid of p or above, and the aggregate supply S(p) represents the mass of sellers who submit an ask of p or below. When the value of the asset is 2, all traders receive a low signal; two-thirds then submit a bid of 2 and one-third submit a bid of 3 (Figure a). When the value of the asset is 3, half of the traders receive a high signal and the other half receive a low signal; one-third then submit a bid of 2, one-third submit a bid of 3, and one-third submit a bit of 4 (Figures b). When the value of the asset is 4, all traders receive high signals; one-third then submit a bid of 3, and two-thirds submit a bid of 4 (Figure c). As Figure illustrates, when supply and demand are aggregated, the market-clearing price equals the value of asset X in each state. Moreover, since individual traders cannot impact prices, there are no profitable deviations and the strategy is an equilibrium, where traders can rely on prices to aggregate information. 2 q S(p) q S(p) q S(p) D(p) p (a) Value of the asset is 2 3 D(p) p (b) Value of the asset is 3 3 D(p) p (c) Value of the asset is 4 Figure : Aggregate demand and supply. Example 2. Are there also environments where prices cannot aggregate information? Consider the market for an alternative asset Y, which has value 4 when both inputs are equal, and value 2 otherwise. The information that signals convey about states is the same as for asset X but the payo structure is di erent. In particular, inputs are substitutes for asset X and complementary for asset Y. It can be shown that equilibrium prices cannot aggregate information in the market for asset Y. To illustrate, consider any strategy-profile where aggregate supply and demand cross at p = 4 when all traders receive a low signal, and also when all traders receive a high signal. Suppose that, on aggregate, traders submit higher bids when they receive a high signal for input A than when they receive a low signal for input A. In order for the price to equal 4 in both states where the value is 4, itmustbe the case that (on aggregate) traders submit lower bids when they receive a high signal for input B than when they receive a low signal for input B. Now consider the state where the value of the asset is 2 and traders either receive a high signal on input A or a low signal on input B (i.e., in the state (2, )). Since aggregate bids are highest in this state, the price cannot be less than 4. As a result, there is no strategy where the market-clearing price is equal to the value in every state. There are strategies where the market-clearing price is di erent in every state, and prices therefore aggregate information. However, these strategies present traders with arbitrage opportunities. If traders predict 3

4 a price that is strictly less than the value in some state, buyers have an incentive to decrease their bids locally to increase their chances of trading, and sellers have an incentive to increase their asks locally to decrease their chances of trading. Likewise, if the price is strictly greater than the value, buyers have an incentive to increase their bids and sellers have an incentive to decrease their asks. As traders respond to these arbitrage opportunities, competitive forces apply upward pressure on prices in states where the asset is undervalued, and downward pressure on prices in states where the asset is overvalued. As equilibrium prices cannot equal values, the only escape is that equilibrium prices do not aggregate information. 2 The example of asset Y shows that some conditions are necessary for equilibrium prices to aggregate information. The previous auction literature has shown that the MLRP is su cient. But the example of asset X shows that the MLRP is not necessary. In fact, in environments with finite states and signals, we show that a much weaker condition which we call the betweenness property is both necessary and su cient. The condition is simple, and can be illustrated geometrically. A betweenness order is a ranking of distributions over signals, with the defining characteristic that level curves are linear. 2 The betweenness property is satisfied if there is a betweenness order that is monotone in values: higher value states generate higher ranked conditional distributions. To illustrate, consider the conditional probability that a trader receives one of the high signals in Examples and 2. In state (, ), the probability of receiving either signal H A or H B is 0; in state (2, ), the probability for H A is 2, and the probability for H B is 0; in state (, 2), the probability for H A is 0, and the probability for H B is 2 ; and in state (2, 2), the probability for either high signal is 2. Figure 2a illustrates this information structure. H B H B H B (2, ) (2, 2) (, ) (, 2) H A 2 3 H A 4 3 H A (a) Information structure (b) Asset X (c) Asset Y Figure 2: The betweenness property in Examples and 2. In Figure 2b, we replace states with the values of asset X. The dashed lines indicate level curves of a betweenness order that is monotone in values. As the figure illustrates, the betweenness property is satisfied, and this is why equilibrium prices can aggregate information. In Figure 2c, we replace states with the values of asset Y. The dashed lines indicate that the convex hull of high value states intersects the convex hull of low value states. In that case, there is no betweenness order that is monotone in values, and equilibrium prices cannot aggregate information. 2 Betweenness orders are a generalization of expected utility (where level curves are linear and parallel), introduced in Chew (983) and Dekel (986) as a model of decision-making under risk that can accommodate the Allais paradox. 4

5 The intuition for our characterization result comes from three important insights about large markets. First, if prices aggregate information, they must equal values; otherwise traders have arbitrage opportunities (as in the market for asset Y ). Second, the law of large numbers provides a powerful representation of aggregate bidding behavior (as in the market for asset X). In particular, cumulative bid distributions (for both buyers and sellers) are separable in a component that depends only on strategic behavior and a component that depends only on information primitives. Finally, the strategic component of a bid distribution has a dual representation as a betweenness order, and vice versa. For prices to equal values, the betweenness order must be monotone in values, which is exactly what the betweenness property requires. The betweenness property also has a remarkable connection with the MLRP. In fact, we show that the MLRP is equivalent to a unanimity betweenness property where every betweenness order must be monotone in values. 3 The betweenness property is therefore much weaker than the MLRP because it drops the unanimity requirement, replacing the order on signals imposed by the MLRP with a single order on distributions over signals. As such, we show that the MLRP is related to the betweenness property exactly in the same way as first-order stochastic dominance is related to expected utility. While the MLRP is very restrictive in environments with a large number of states, we show that the betweenness property is generic as long as the number of signals is greater than the number of states. As a result, there are many environments where the previous literature makes no predictions about the information e ciency of auction prices, yet we show that equilibrium prices can aggregate information almost surely. On the other hand, in environments where the number of states is large relative to the number of signals, the betweenness property is also restrictive. While a fully-revealing rational expectations equilibrium always exists in these environments, it generally cannot be implemented in a Bayes-Nash equilibrium where traders cannot condition directly on equilibrium prices. This highlights limitations of the market when prices must distinguish between many values with limited signals. By replacing orders on signals with orders on distributions over signals, our approach can accommodate arbitrary multidimensional environments, where little is currently known about the aggregation properties of markets. As an illustration, we consider a broad class of multidimensional environments, where states have multiple inputs and signals are specific to inputs (as in the markets for assets X and Y ). For instance, one could think of the market for an asset that bundles returns for real assets in multiple sectors and traders who have access to di erent sector reports; or the market for a commodity with yields that depend on weather conditions in multiple locations and traders who have access to di erent weather forecasts. A signal then conveys information that is relevant for only one dimension of the asset s value, and traders must rely on prices to aggregate the fragmented information di used in the marketplace. We show that the MLRP is never satisfied in such environments. On the other hand, when the value is separable in inputs, the betweenness property is generic whenever there are at least as many signals as states for each input. 4 This finding is more powerful than our first genericity result because, even when there are far fewer signals than values, the separability of inputs can be su cient for equilibrium prices to aggregate information almost surely. 3 The unanimity applies for betweenness orders with the ranking over degenerate distributions implied by the MLRP. 4 For instance, the value of asset X is separable in the two inputs while the value of asset Y is not. 5

6 Contribution to the literature. Our work primarily contributes to a literature on information aggregation in common-value auctions (Wilson, 977; Milgrom, 98; Pesendorfer and Swinkels, 997; Kremer, 2002; Reny and Perry, 2006). This literature studies the aggregation problem by looking at asymptotic equilibrium prices in a finite auction, as the population of traders grows. The MLRP is crucial to overcome the significant challenge of solving for an equilibrium when each trader has market power and conditions on the pivotal event where she determines the price. We depart from the auction literature by focussing on a market with an atomless population of traders. Competitive forces rather than pivotal considerations are therefore the main driving force behind individual and aggregate behavior. This allows us to study the aggregation problem without imposing strong order properties on signals. To establish a closer connection with the auction literature, we also consider approximate equilibria in finite auctions as the population grows and competition intensifies. In an approximate equilibrium, traders ignore their vanishing impact on prices but still respond to arbitrage opportunities. As in the large market, we show that information can be aggregated asymptotically if and only if the betweenness property is satisfied. Accordingly, when the betweenness property is satisfied, prices aggregate information as long as traders eventually disregard their market power. When the betweenness property is not satisfied, equilibrium prices cannot aggregate information, regardless of whether traders internalize their impact on prices or not. Our work also contributes to the literature on rational expectations equilibrium (Grossman and Stiglitz, 976; Radner, 979). In the standard rational expectations model, assets are divisible and traders submit supply and demand schedules to a clearing house. To distinguish from the auction model, we call this a Walrasian market. In a fully-revealing rational expectations equilibrium of a Walrasian market, traders condition the quantity they are willing to trade both on the price and on the state, which they can infer from the equilibrium price. We depart from the rational expectations literature by studying a market with an explicit protocol for the price formation process. Traders condition their orders on their private signals, rather than the information conveyed by equilibrium prices. This allows us to show, constructively, how equilibrium prices incorporate the private information dispersed over market participants. To establish a closer connection with the rational expectations literature, we also extend our analysis to a Walrasian market. A fully-revealing rational expectations equilibrium always exist, but we show that fully-revealing prices can be implemented in a Bayes-Nash equilibrium if and only if the betweenness property is satisfied. Accordingly, when the betweenness property is satisfied, our analysis provides microfoundations for the fully-revealing rational expectations equilibrium. When the betweenness property is not satisfied, the fully-revealing rational expectations equilibrium cannot be implemented in a Walrasian market where traders are unable to make inferences directly from equilibrium prices. The paper is organized as follows. Section 2 defines the betweenness property and describes the market. Section 3 presents our main aggregation result and establishes the direct connection between equilibrium prices and information primitives. Section 4 studies the betweenness property in detail, establishing its relation to the MLRP and our genericity and multi-input analysis. Section 5 presents (i) the approximation by finite auctions, and (ii) the analysis of a Walrasian market. Section 6 discusses related literature in more detail. Section 7 concludes. Proofs are given in the Appendix. 6

7 2 Model We study a double-sided auction with a large population of traders. The common value of an asset depends on an unknown state, and traders receive private signals that are i.i.d. conditional on the state. In this market, we are interested in the information that equilibrium prices convey about values. 2. The information environment The environment has a finite set of states W={Ê,...,Ê M } and signals S={s,...,s K }, with a probability distribution P on W S. In state Ê, an asset has value v(ê) and the conditional distribution over signals is P Ê. To simplify exposition, we assume that P has full support and states with di erent values generate di erent conditional distributions over signals (i.e., v(ê) = v(ê Õ ) implies P Ê = P Ê Õ). The key primitives are the value function v : W æ R ++ and information structure {P Ê : Ê œ W}. The starting point for our analysis is a new perspective on the information environment. As we show in the introduction, some property of information is necessary for aggregation: values must be related in some way to the information structure, so that competitive forces can guide aggregate behavior and ensure that equilibrium prices aggregate information. The previous auction literature generally imposes an order on signals that is strongly correlated with values, and uses this order to obtain an equilibrium in monotone bidding strategies. We depart from this approach. Instead of imposing an order on signals directly, we consider a continuous weak order on the set of distributions over signals D(S). We denote the asymmetric part of this weak order by º, and the symmetric part by. 5 The following definition recalls two prominent classes of continuous weak orders. Definition. The continuous weak order is (i) a linear order if, for all œ (0, ) and, Õ, ÕÕ œ D(S), Õ implies +( ) ÕÕ Õ +( ) ÕÕ ;(ii)abetweenness order if º Õ implies º +( ) Õ º Õ, and Õ implies +( ) Õ Õ. The defining characteristic of linear orders is that level curves can be represented by parallel hyperplanes. Betweenness orders are a generalization where level curves are also represented by hyperplanes but not necessarily by parallel ones (Figure 3). 6 The following monotonicity properties formalize the intuitive idea that better states generate better conditional distributions. Definition 2. An environment satisfies the betweenness (resp., linear) property if there is a betweenness (resp., linear) order on D(S) such that v(ê) >v(ê Õ ) implies P Ê º P Ê Õ. 5 The binary relation on D(S) is a continuous weak order if it is (i) complete and transitive; (ii) º Õ for some, Õ œ D(S); and (iii) º Õ º ÕÕ implies +( ) ÕÕ Õ for some œ (0, ). Continuous weak orders are studied extensively in the literature on decision-making under risk, where S is interpreted as a finite set of prizes, is a lottery over prizes, and is a preference relation over lotteries. 6 von Neumann and Morgenstern (944) show that a preference relation over lotteries has an expected utility representation if and only if it is a linear order. Linear orders are therefore central in the theory of decision-making under risk. Betweenness orders are a generalization of expected utility that can accommodate well-known behavioral anomalies such as the Allais paradox (Chew, 983; Dekel, 986). 7

8 The betweenness property is central for our information aggregation result; the linear property is useful as a reference and in our genericity analysis. As betweenness orders are more general, the linear property implies the betweenness property and not vice versa (Figures 3 and 4). We provide a detailed analysis of these information properties in Section 4, but first establish the central role of the betweenness property in markets. M M H 4 (a) Linear property L 3 H L (b) Betweenness property Figure 3: Linear and betweenness properties. There are four states with values {, 2, 3, 4}, and the point labeled m represents the conditional distribution over signals in the state with value m. TheinformationstructureinFigure3asatisfiesthelinearproperty: there is a linear order where better states induce better conditional distributions. The information structure in Figure 3b does not satisfy the linear property because the translation of any level curve with the correct ranking over P and P 2 has the wrong ranking over P 3 and P 4 ;itsatisfiesthebetweenness property, where moving to higher level curves can also involve rotation. M M H L H L (a) Non-separation (b) Non-nesting Figure 4: Failure of the betweenness property. In Figure 4a, the convex hulls of {P, P 2 } and {P 3, P 4 } intersect and so a hyperplane cannot strictly separate {P, P 2 } from {P 3, P 4 }.Figure4billustratesadi erentfailureofthebetweennessproperty: hyperplanes can separate high from low states, but a hyperplane that separates P from {P 2, P 3, P 4 } and a hyperplane that separates P 4 from {P, P 2, P 3 },mustintersectinsidethesimplex. 8

9 2.2 The market There is an infinite set of traders I endowed with a non-atomic probability distribution. 7 The auction format provides an explicit protocol for the price formation process, and the large population ensures that individual traders have negligible impact on prices. Nature chooses a state Ê according to the marginal distribution on W. Traders do not observe the state, but receive a private signal drawn independently from the conditional distribution P Ê. After receiving their signals, each trader submits a sealed bid from a compact interval B [0, b], which contains v(w). The traders are divided into a set of buyers with mass Ÿ œ (0, ) and a set of sellers with mass Ÿ. Each seller owns a unit of an indivisible asset, and each buyer has unit demand. For a buyer, a bid represents the maximum price at which they are willing to trade; for a seller, it represents the minimum price at which they willing to trade. Given a bid-profile a : IæB, where a(i) represents the bid for trader i, the auctioneer determines a price and an allocation of assets. 8 The price p(a) is the lowest bid at which the mass of sellers willing to trade exceeds the mass of buyers, and all trade occurs at this price. A buyer trades if her bid is strictly above the price and does not trade if her bid is strictly below the price, and vice versa for sellers. To clear the market, the auctioneer uniformly randomizes over bids equal to the price in order to maximize total trading volume. The payo the payo for a seller is p(a) v(ê) if she trades and 0 otherwise. 9 for a buyer is v(ê) p(a) if she trades and 0 otherwise; A strategy-profile : I S æbis a mapping from types to Borel probability distributions over bids, where (i, s) is the (mixed) bidding strategy for trader i when they receive signal s. A strategy-profile and conditional distribution P Ê generate a unique probability measure P Ê over bid-profiles in state Ê. 0 The expected payo for type (i, s) is P i ( s) q Ê P i( Ê)P s (Ê), wherep s (Ê) is the probability of state Ê conditional on signal s, P i ( Ê) s A fi i(a Ê)dP Ê is the expected payo conditional on state Ê, and fi i (a Ê) is trader i s payo in state Ê for the bid-profile a. A strategy-profile is a Bayes-Nash equilibrium (henceforth, equilibrium) if each type maximizes their expected payo of other types. given the strategy In principle, a state Ê and strategy-profile generate a distribution over prices derived from the distribution P Ê over bid-profiles. However, in our market, the Strong Law of Large Numbers (SLLN) implies that the price is almost surely constant. Proposition. For every strategy-profile there exists a unique price-function p : W æbsuch that, in state Ê, the price is equal to p (Ê) almost surely. 2 7 Our formal model of the large population follows Al-Najjar (2008), where I is countably infinite and endowed with a finitely-additive probability measure on the power-set. This population model overcomes significant challenges with measurability and the law of large numbers in continuum agent models (see, e.g., Judd 985). We discuss the population model in detail in Appendix A.2.. For intuition, there is no loss in suspending problems related to measurability and the law of large numbers, and thinking of the population as a continuum endowed with Lesbegue measure. 8 The set of bid-profiles A = {a : IæB} is endowed with the -algebra A generated by cylinder sets of the form {a : a(i) =b} for some i œi and b œ B. 9 A more detailed description of the auction format is given in Appendix A A unique countably-additive measure P Ê on (A, ) is guaranteed by the Hahn-Kolmogorov Extension Theorem. Our result also applies if we define equilibrium as a strategy-profile where almost all types play a best-response. 2 Formally, this means that there is a measurable subset A µasuch that P Ê (A) = and p(a) =p (Ê) for all a œ A. 9

10 3 Main result We are interested in strategy-profiles where prices convey the same information about the unknown value as would obtain in a counterfactual environment where all signals are public. By the SLLN, the proportion of bidders who receive signal s in state Ê is almost surely equal to P Ê (s). Public signals would therefore reveal the value of the asset almost surely. As such, a strategy-profile conveys the same information as public signals if there is a one-to-one mapping between values and prices. Definition 3. Strategy-profile aggregates information if v(ê) = v(ê Õ ) implies p (Ê) = p (Ê Õ ). It is always possible to construct a strategy-profile that aggregates information. However, we are interested in strategies where traders respond to incentives generated by the competition for assets. While an individual trader has negligible impact on the price and total trading volume, she can a ect her allocation through her bids and thereby influence her expected payo. In an equilibrium, traders will therefore try to exploit arbitrage opportunities based on their predictions about prices and values. Accordingly, the aggregate supply and demand for assets depends on the incentives of the traders, and equilibrium requires that these competitive forces are resolved. Our main result characterizes when equilibrium prices convey the same information about values as would obtain if signals were public. Theorem. There is an equilibrium strategy-profile that aggregates information if and only if the betweenness property is satisfied. By connecting the aggregation problem directly with the information primitives, the result allows us to distinguish between two types of environments. When the betweenness property is satisfied, there are equilibrium prices that aggregate all private information in the market. This highlights the potential of the market. Even if individual traders are poorly informed about the value, competitive forces can coordinate individual behavior so that prices are perfectly informative. On the other hand, when the betweenness property is not satisfied, information aggregation necessarily fails. This highlights the limitations of the market. Even if the population as a whole is perfectly informed, the market cannot guide traders behavior so that prices reveal their collective information. Remark (Existence and uniqueness). The market always has a no-trade equilibrium where prices are completely uninformative. To illustrate, consider the following strategy-profile: regardless of their signals, all sellers ask for b and all buyers bid 0. In that case, the price is equal to 0 in every state. Buyers would like to trade at these prices but there is no supply, and so they cannot increase their chances of trading by submitting a higher bid. Sellers do not want to trade, and so have no incentive to ask for a lower price. It follows that this strategy-profile is a no-trade equilibrium. 2 Remark 2 (Risk preferences). The assumption that traders are risk neutral simplifies exposition, but the result extends to a market where traders have heterogenous risk preferences. Suppose that each trader i œihas a strictly-increasing utility function u i : R æ R, where marginal utilities are uniformly bounded away from 0. Given a bid-profile a : IæB, the payo for buyer x in state Ê is then fi x (a Ê) =w x (a Ê)u x (v(ê) p(a)) + ( w x (a Ê)) u x (0), wherew x (a Ê) is the probability 0

11 that buyer x will trade in state Ê given bid-profile a. Likewise, the payo for seller y in state Ê is fi y (a Ê) =w y (a Ê)u y (p(a) v(ê)) + ( w y (a Ê)) u y (0). We can adjust the definition of equilibrium accordingly, and our main result applies as stated. 2 Proof sketch. An important advantage of modeling the trading mechanism explicitly is that it allows us to show where prices originate, and why the betweenness property is necessary and su to aggregate information. Our proof is constructive and consists of three key steps. We provide a sketch of the argument and illustrate the equilibrium construction with an example. The first step in the argument identifies the restrictions that competition imposes in our environment. If an equilibrium strategy-profile aggregates information, then prices must equal values almost surely (i.e., p = v). To see why, consider a strategy-profile that aggregates information and suppose there is a state Ê such that p (Ê) <v(ê). Since the price is strictly less than the value, it would be good for buyers to trade in state Ê, and bad for sellers to trade. In general, there could be another state Ê Õ where the price is strictly higher than the value, and it is bad for buyers to trade and good for sellers. However, because aggregates information, p (Ê Õ ) = p (Ê), and so a buyer who submits a bid equal to p (Ê) can decrease their bid marginally below p (Ê), thereby guaranteeing that they trade in state Ê (where trading is good) without changing the likelihood that they trade in state Ê Õ (where trading is bad). Likewise, a seller who submits a bid equal to p (Ê) can increase their bid marginally above p (Ê), thereby guaranteeing that they do not trade in state Ê (where trading is bad) without changing the likelihood that they trade in state Ê Õ (where trading is good). As buyers and sellers respond to these opposing arbitrage opportunities, competitive forces exert upward pressure on the price in state Ê, and downward pressure on the price in state Ê Õ. These competitive pressures are only resolved when prices are equal to values in every state. The second step in our argument uses the SLLN to characterize aggregate bidding behavior. For a strategy-profile let B and S denote, respectively, the restriction to buyers and sellers. We use the SLLN to show that the aggregate bidding behavior of sellers can be characterized by a vector a vector of 2 cumulative distribution functions F S F S s,...,f S s K,whereF S s k (b) represents the normalized share of sellers who submit an ask price less than b when they receive signal s k. The total mass of sellers who submit an ask price less than b depends on the strategy-profile (chosen by traders) and the distribution over signals (chosen by nature). In particular, the mass of ask prices less b in state Ê is (almost surely) equal to ( Ÿ)F S Ê (b) ( Ÿ)F S (b) P Ê. Similarly, the mass of buyers who submit a bid strictly greater than b is described by Ÿ( F B Ê (b)) Ÿ( F B(b)) P Ê. Accordingly, aggregate supply and demand first cross in state Ê at the lowest price where Ÿ( F B Ê (p)+( Ÿ)F S Ê cient (p)) Æ ( Ÿ)F S Ê (p); (p) F Ê (p). As a result, the market-clearing price is given by that is, Ÿ Æ ŸF B Ê the Ÿ-quantile of a cumulative distribution functions, F Ê, which is additively separable in terms of a component F ŸF B +( Ÿ)F S, which depends only on strategic behavior, and another component P Ê, which depends only on the primitive information structure. 3 The final step in the argument establishes a duality between bidding strategies and betweenness orders: the quantiles of any bidding strategy can be approximated by a betweenness order, and 3 The formal proof for these heuristic arguments establishes Proposition.

12 vice versa. This step of the argument is geometric. Let i : S æbbe bidding strategy for trader 2 i, and F i F i s,...,f i s K denote the trader s bidding strategy in cumulative form. Given a bid b, we can interpret the vector F i(b) as the norm of a hyperplane in R K. By varying the bid, we obtain a collection of hyperplanes that provides a geometric characterization of the bidding strategy. Moreover, we show that (i) any quantile of the cumulative bidding strategy can be represented as the intersection of these hyperplanes with the unit simplex, and (ii) when we look at the intersection of these hyperplanes with the simplex D(S) they have essentially the same properties as the level curves of a betweenness order. When we apply this duality to the aggregate bidding strategy F obtained in step, it follows that a strategy-profile induces a price-function that is monotone in values if and only if it can be represented by a betweenness order that is also monotone in values. These three steps allow us to show the following. If there is an equilibrium strategy-profile that aggregates information, equilibrium prices must equal values (by step ); the hyperplanes that represent the aggregate bidding strategy are therefore monotone in values (by step 2); and so there is a betweenness order that is also monotone in values (by step 3). This establishes that the betweenness property is necessary for information aggregation. On the other hand, when the betweenness property is satisfied, we can use the level curves of the betweenness order to construct a symmetric strategy profile so that p = v. Clearly, this strategy-profile aggregates information. Moreover, since individual traders have negligible market power, the expected payo for each bidder is zero regardless of their own strategy, and so every type is playing a best-response. As such, is also an equilibrium. To illustrate the equilibrium construction, consider an environment with three states W = {Ê, Ê 2, Ê 3 }, three signals S = {s L, s M, s H }, and a value function where v(ê m )=m for each state. In Figure 5a, the vectors Õ l and Õ m are norms of two hyperplanes, H( Õ l, cõ l ) and H( Õ m, c Õ m), that represent level curves of a betweenness order. 4 Because higher values generate better conditional distributions, the betweenness property is satisfied. M M 2 l 2 F () m F (2) 3 3 H (a) Betweenness property L H (b) Equilibrium strategy L Figure 5: Duality of bidding strategies and betweenness orders. In Figure 5a, vectors l and m are norms of hyperplanes that represent level curves of a betweenness order. As higher values generate better conditional distributions,the betweenness property is satisfied. In Figure 5b, the vectors F () and F (2) are norms of hyperplanes that represent the aggregate bidding strategy. As higher values generate higher Ÿ-quantiles, the strategy-profile aggregates information. 4 We denote by H(, c) {z œ R K : z = c} a hyperplane in R K, defined by the norm œ R K and constant c œ R, with upper and lower half-spaces denoted H + (, c) and H (, c), respectively. 2

13 To construct the equilibrium strategy-profile, we first need to manipulate the hyperplanes H( Õ l, cõ l ) and H( Õ m, c Õ m) in way that does not change their intersection with the unit simplex. By the manipulations, the new hyperplanes H( l, c l ) and H( m, c m ) still represent the same betweenness order. However, the manipulation ensures that the new constants satisfy c l = c m = Ÿ, and the norms satisfy l, m œ [, 0] 3 and l >> m. In particular, guaranteeing that l strictly dominates m uses the properties of the betweenness order. It is di cult to provide intuition for this step of the construction, and we refer the reader to the formal arguments developed in Lemmas and 2 in Appendix A.. However, to indicate how we manipulate hyperplanes without changing the underlying weak order, it is useful to consider the simpler case of a linear order represented by a collection of parallel hyperplanes {H(, ) : œ D(S)} for œ R K. The linear order can also be represented by a collection of non-parallel hyperplanes. For instance, for each distribution, define the hyperplane H(,0). Then Õ œ H + (,0) if and only if Õ Ø 0, i.e., Õ. Thus the collection of hyperplanes {H(,0) : œ D(S)} also represents the linear order, but these hyperplanes are not parallel, they have the same constants, and the norms are strictly ordered. We use the new hyperplanes H( l, Ÿ) and H( m, Ÿ) to construct a bidding strategy i : S æbfor trader i, where, for each signal, the trader randomizes over the finite set of values {, 2, 3}. As a result, i is fully described by a 2 3 matrix, A F i () F i(2) B A F i s L () F i s L (2) F i s M () F i s M (2) F B i s H () F. i s H (2) In particular, because l (s), m (s) œ [0, ] and l (s) < m (s), we can choose i so that F i() = l and F i(2) = m. Hence, we construct the bidding strategy from the underlying betweenness order given by the betweenness property. It remains to show that the symmetric strategy-profile, where every trader follows i (i.e., (i, s) = i (s) for every type (i, s) œi S) ensures that, almost surely, the price is equal to the value in every state. This follows because the SLLN implies that aggregate bidding strategy F derived in step of the proof sketch is (almost surely) equal to the cumulative distribution function F i from the betweenness order. As a result: derived (a) As P is in the strict upper half-space of H( l, Ÿ), it follows that F () P >Ÿ. In state Ê, the mass of bids less or equal to is therefore (almost surely) strictly greater than Ÿ, and so the price can be no higher than. On the other hand, no trader submits a bid strictly lower than, and so the price can be no lower than. Therefore, p (Ê )= (Figure 6a). (b) As P 2 is in the strict lower half-space of H( l, Ÿ), it follows that F () P 2 <Ÿ. In state Ê 2, the mass of bids less than or equal to is therefore (almost surely) strictly less than Ÿ, and so the price must be strictly greater than. On the other hand, P 2 is in the strict upper half-space of H( m, Ÿ), and so F (2) P 2 <Ÿ. As a result, the mass of bids less than or equal to 2 is (almost surely) greater than Ÿ, and so price can be no higher than 2. Because no trader submits a bid in the interval (, 2), it follows that p (Ê 2 )=2 (Figure 6b). (c) As P 3 is in the strict lower half-space of H( m, g), it follows that F (2) P 3 <Ÿ. In state 3

14 Ê 3, the mass of bids less than or equal to 2 is therefore (almost surely) strictly less than Ÿ, and so the price must be strictly greater than 2. On the other hand, no trader submits a bid greater than 3, and so the price can be no higher than 3. Because no trader submits a bid in the interval (2, 3), it follows that p (Ê 3 )=3 (Figure 6c). F F 2 F 3 Ÿ Ÿ Ÿ 2 3 bids 2 3 bids 2 3 bids (a) CDF in state (b) CDF in state 2 (c) CDF in state 3 Figure 6: Cumulative bid distributions. For each state, Figure 6 illustrates the cumulative bid distribution generated (a.s.) by the symmetric strategy-profile. The cumulative distribution functions are step functions because of the finite support of i.thepriceisequaltotheÿ-quantile of the cumulative bid distribution, and the strategy-profile therefore aggregates information. 4 The betweenness property In this section, we provide a detailed analysis of the betweenness property. We first show that the MLRP can also be characterized by betweenness orders but requires a unanimity property, which the betweenness property does not impose. This illustrates how our aggregation result generalizes insights from the previous auction literature, where the MLRP is imposed to simplify equilibrium analysis. We also provide an approach to quantify the likelihood that an arbitrary information structure will satisfy the betweenness property. Our conclusions are reassuring about the scope for equilibrium prices to aggregate information in competitive markets, and are in stark contrast to our comparative findings for the MLRP. We first provide a simple condition that is necessary and su cient to ensure that the betweenness property is generic in an environment with many states, where the MLRP has arbitrarily small measure. We then consider a class of multidimensional environments where the MLRP is never satisfied, and provide necessary and su cient conditions to ensure that the betweenness property is satisfied almost surely. 4. Betweenness and the MLRP We use the symbol D to denote a weak order on signals, where B is the asymmetric part. Given a weak order D on signals and a state Ê, letf Ê be the conditional cumulative distribution function over signals, defined by F Ê (s) q s Õ Es P Ê(s Õ ), and let L Ê be the conditional likelihood-ratio function, defined by L Ê (s, s Õ ) P Ê (s)/p Ê (s Õ ). 4

15 Definition 4. An environment satisfies first-order stochastic dominance (FOSD) if there is a weak order D on signals such that v(ê) >v(ê Õ ) implies F Ê (s) Æ F Ê Õ(s) for all s. It satisfies the monotone likelihood-ratio property (MLRP) if there is a weak order D on signals such that v(ê) >v(ê Õ ) implies L Ê (s, s Õ ) ØL Ê Õ(s, s Õ ) for all s D s Õ. M F (M) F (L) M L (H, M) L (M, L) H L H L (a) FOSD (b) MLRP Figure 7: FOSD and MLRP. Signals are ordered s H B s M B s L. In Figure 7a, F (L) indicates distributions with the same probability on s L as P,andF (M) indicates distributions with the same cumulative probability for s L and s M.FOSDimpliesthatP 2 must be to the left of these lines. In Figure 7b, L (M, L) indicates distributions with the same likelihood-ratio between s M and s L as distribution P, and L (H, M) indicates distributions with the same likelihood-ratio between s H and s M.TheMLRPimpliesthatP 2 must be to the left of these lines, which is more restrictive than FOSD. It is well-known that FOSD implies the MLRP and not vice versa (Figure 7). However, it is less immediate how FOSD and the MLRP are related to the betweenness property, which requires a weak order on distributions over signals rather than a weak order on signals. To establish a formal connection, consider a weak order D on S and a continuous weak order on D(S). Then is monotone (with respect to D) ifs B s Õ implies s º s Õ,where s denotes the distribution over signals with probability on signal s. It is well-known that a distribution first-order stochastic dominates distribution Õ if and only if every monotone linear order ranks higher than Õ (Hadar and Russell, 969). As such, there is a characterization of FOSD in terms of a unanimity property of monotone linear orders. Proposition 2. An environment satisfies FOSD if and only if it satisfies a unanimity linear property: for every monotone linear order, v(ê) >v(ê Õ ) implies P Ê º P Ê Õ. While FOSD requires that better states generate better conditional distributions in terms of every monotone linear order, the linear property in Definition only requires that better states generate better conditionals for some linear order. As such, FOSD implies the linear property and not vice versa (Figure 8a). The following theorem establishes an analogous relation for the MLRP and the betweenness property. Theorem 2. An environment satisfies the MLRP if and only if it satisfies a unanimity betweenness property: for every monotone betweenness order, v(ê) >v(ê Õ ) implies P Ê º P Ê Õ. 5

16 While the MLRP requires that better states generate better conditional distributions in terms of every monotone betweenness order, the betweenness property only requires that better states induce better conditionals for some betweenness order. As a result, the MLRP implies the betweenness property and not vice versa (Figure 8b). M l M h l h Ê Ê H L H L (a) FOSD ULP (b) MLRP UBP Figure 8: Illustration of Proposition 2 and Theorem 2. In Figure 8a, a line passing through Ê represents a level curve of a monotone linear order if and only if it is strictly between l and h.theshadedregionthusrepresentstheintersectionoftheuppercontour sets of all monotone linear orders; comparing to 7a this shaded area is equivalent to the restriction imposed by FOSD. In Figure 8b, a line passing through Ê represents a level curves of a monotone betweenness order if and only if it is strictly between l and h.theshadedregionthusrepresentsthe intersection of the upper contour sets of all monotone betweenness orders; comparing to 7b this shaded area is equivalent to the restriction imposed by the MLRP. The preceding discussion can be summarized as follows. The MLRP and FOSD are characterized by a unanimity condition on weak orders over distributions: for every monotone order in a certain class, better states generate better conditional distributions. Proposition 2 shows that the relevant class for FOSD are linear orders. As the MLRP implies FOSD, the relevant class for the MLRP must be larger. Theorem 2 shows that the required generalization is exactly the class of betweenness orders. One natural way to generalize FOSD or the MLRP is to dispense with the unanimity requirements. For the case of FOSD, this leads to the linear property; for the MLRP, it leads to the betweenness property. Our information aggregation result demonstrates how these observations can lead to useful new insights on strategic interactions in environments with incomplete information (Figure 9). MLRP FOSD UBP ULP BP LP Figure 9: Relations between information properties. 6

17 4.2 Genericity We also provide a way to quantify information properties. As previously, we consider environments with M states and K signals. To simplify exposition, we assume that the value-function v : W æ R ++ is injective, so that the set of values v(w) also has cardinality M. We fix the states, signals and value function. 5 The information structure can then be represented by a real matrix of dimension K M, where column m represents the distribution over signals conditional on state Ê m. As a result, we can quantify information structures with the Lebesgue measure on R (K )M. The set of all such information structures is open (by full support) thus measurable, and has Lebesgue measure one. The subset of information structures that satisfy the betweenness property is also open, thus measurable. The boundary of the set of information structures that satisfy the MLRP is a Lebesgue null-set, and so the information structures satisfying the MLRP are also measurable. The following theorem characterizes when the betweenness property is satisfied almost surely. Theorem 3. The betweenness property has full measure if and only if K Ø M. Together with our aggregation result, Theorem 3 establishes when information aggregation is a generic equilibrium property in a competitive double-sided auction. In particular, information aggregation does not require any particular order on the set of signals. As long as the cardinality of signals is larger than the cardinality of states, the betweenness property is generic and there is an equilibrium strategy-profile that aggregates information (Figure 0). On the other hand, in environments where the number of states is strictly greater than the number of signals, there is always a strictly positive measure of information structures where the betweenness property does not hold, and equilibrium prices cannot aggregate information. Moreover, from the proof, it is evident that the measure of information structures where the betweenness property is satisfied vanishes if the number of signals is held constant and the number of values increases. To provide intuition for Theorem 3, suppose K = M. Let P W =(P Ê,...,P ÊK ) be the K K matrix that represents an information structure with K signals and K states. It is well-known that the set of real K K matrices that are invertible has full measure. For an invertible matrix, the system of equations P W = has a solution for every œ R K. Now choose a such that (m) > (m Õ ) whenever v(ê m ) >v(ê m Õ). Then, the solution for this defines an expected utility function œ R K such that the linear property is satisfied, which implies that the betweenness property is satisfied. A similar argument can be applied when K>Mby completing rectangular matrices appropriately. On the other hand, when K<Mwe show that there is a strictly positive measure of information structures where a state with a high value is in the convex hull of states with lower values. This is inconsistent with the betweenness property. By contrast, the following proposition shows that, with many states, it is highly unlikely that an information structure will satisfy the MLRP, regardless of the number of signals. 5 The marginal over states is not relevant for the results because the information properties depend only on the information structure and value function. 7