SUPPLEMENT TO STRATEGIC TRADING IN INFORMATIONALLY COMPLEX ENVIRONMENTS (Econometrica, Vol. 86, No. 4, July 2018, )

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1 Econometrica Spplementary aterial SUPPEENT TO STRATEGIC TRAING IN INFORATIONAY COPEX ENVIRONENTS Econometrica, Vol. 86, No. 4, Jly 208, 9 57 NICOAS S. ABERT Gradate School of Bsiness, Stanford Uniersity ICHAE OSTROVSKY Gradate School of Bsiness, Stanford Uniersity and NBER IKHAI PANOV epartment of Economics, New York Uniersity This spplement contains additional reslts and proofs omitted from the main paper. S.. ZERO AGGREGATE EAN IN EQUIIBRIU IN THISSECTION, we formally derie eqilibria in the special examples in which aggregate demand ends p being zero on the eqilibrim path footnote 2 in Section 2.2 and footnote 20 in Section 5. In Section S.., we work ot the example with one informed trader, corresponding to footnote 2, and in Section S..2, weworkottheexamplewith mltiple informed traders, corresponding to footnote 20. S... One Informed Trader EXAPE S.: The ale of the secrity,, is distribted normally with mean 0 and ariance. There is one strategic trader with signal θ who obseres the ale perfectly: θ =. The demand of liqidity traders is =. The market maker does not obsere any signals beyond the aggregate demand. This example satisfies Assmptions and 2 in Section 3, and ths we can se the closedform soltions deried in the proof of Theorem and presented in Section 3.2 of the paper for the special case k = 0. Becase we hae only one strategic trader in the example, many of the matrices become scalars, simplifying the calclation. Specifically, Σ θθ = Σ diag =, and, therefore, Λ = 2andΛ = /2. Next, Σ θ =, while Σ θ =. Ths, A = /2 anda = /2. The coefficients of the qadratic eqation in γ are a = /4, b = /2 4 /2 + = 0, and c = Var θ/2 = Var/2 = /4. Therefore, γ =, β = /γ =, and α = γa A =. Ths, on the eqilibrim path, aggregate demand is eqal to = αθ + = = 0. S..2. ltiple Informed Traders EXAPE S.2: The ale of the secrity,, is distribted normally with mean 0 and ariance. There are m strategic traders with the same signal θ =. The demand of Nicolas S. ambert: nlambert@stanford.ed ichael Ostrosky: ostrosky@stanford.ed ikhail Pano: pano@ny.ed 208 The Econometric Society

2 2 N. S. ABERT,. OSTROVSKY, AN. PANOV liqidity traders is =. The market maker does not obsere any signals beyond the aggregate demand. This example also satisfies Assmptions and 2 in Section 2, and ths we can se the closed-form soltions deried in the proof of Theorem and presented in Section 3.2 again, for the special case k = 0. We now hae mltiple strategic traders, so the calclations inole matrix maniplations. Specifically, Σ θθ is an m-dimensional matrix whose elements are all eqal to, while Σ diag is an m-dimensional identity matrix. We ths hae 2 m 2 Λ = and Λ = m m + 2 m Next, /m + /m + Σ θ = Σ θ = A = A = /m + /m + The coefficients of the qadratic eqation on γ are, therefore, a = m/m + 2, b = m /m + 2,andc = /m + 2, which in trn gies s γ = /m and β = m. Ths, α = γa A = /m + /m + m + = m /m + /m + and so on the eqilibrim path, aggregate demand is eqal to = α T θ + = /m m = 0. Ths, for eery m, the market price on the eqilibrim path is also always eqal to 0, not reealing any information contained in the signals of the strategic traders and in the demand coming from the liqidity traders. S.2. ZERO INTERCEPTS IN EQUIIBRIU In this section, we proe the statement made informally in footnote 3 in Section 3.2 that in or setting, linear eqilibria with nonzero intercepts do not exist. PROPOSITION S.: Sppose there exists an eqilibrim of the form d i θ i = δ i + α T i θ i, Pθ = β 0 + β T θ + β. Then β 0 = 0 and for all i, δ i = 0. PROOF: Consider a particlar realization of θ i, θ,and. Then in this eqilibrim, the realized price will be gien by P = Pθ = β 0 + β T θ + β n = β 0 + β T θ + β + β δ i + β i= n α T θ i i i= S.

3 TRAING IN INFORATIONAY COPEX ENVIRONENTS 3 By the definition of eqilibrim, for eery realization of θ and, the price set by the market maker is eqal to the expected ale of the secrity conditional on θ and : Pθ = E[ θ ] Integrating oer all possible realizations of θ and, we ths get, for the nconditional expectation of the price, E[P]=E[] Since by assmption, E[]=0, and also E[θ ], E[],andE[θ i ] for all i are eqal to 0, by taking the nconditional expectation of Eqation S., we get 0 = β 0 + β n δ i i= S.2 Now, as in Step 2 of the proof of Theorem, consider the expected payoff of strategic trader i from sbmitting demand d after obsering signal θ i.itiseqalto [ E d β 0 β T θ β d + δ j + ] α T θ j j + θ i = θ i j i j i Except for the presence of constants β 0 and δ j, this is the same expression as in Step 2 of the proof of Theorem. By the same logic as in that step, it has to be the case that β > 0 and the niqe optimal demand d is gien by d = [ ] E α T j 2β θ j + θ i = θ i = 2β β 0 β T θ β δ j + j i j i β 0 + β j i δ j + Σ T i 2β βt ΣT β i α T j ΣT + ij ΣT i j i By assmption, we also hae d = δ i + α T θ i i. Since the eqalities aboe hae to hold for all realizations θ i, it has to be the case that δ i = β 0 + β δ i 2β j i Σ ii θ i which can be rearranged as β 0 + β δ j + 2β δ i = 0 j i S.3 Combining Eqations S.2andS.3, we see that for eery i, β δ i = 0, and ths δ i = 0, and therefore we also hae β 0 = 0. Q.E.. We first proe the following lemma. S.3. PROOF OF PROPOSITION

4 4 N. S. ABERT,. OSTROVSKY, AN. PANOV EA S.: Consider a market with at least two strategic traders, and sppose the signals of traders and 2 can be represented as θ = θ ; θ C and θ 2 = θ 2 ; θ C, respectiely, in sch a way that random ector θ C has dimension of at least, random ector θ is orthogonal to θ C, and random ector θ 2 is also orthogonal to θ C. S. Represent trader s and trader 2 s eqilibrim strategies as ectors α = α ; α C and α 2 = α 2 ; α 2C, respectiely, with dimensions corresponding to those of θ ; θ C and θ 2 ; θ C. Then, α C = α 2C. PROOF: Consider eqilibrim condition 8 from the proof of Theorem, and rewrite it for trader as Var θ 0 α 2 = Co θ Co θ θ β 0 Varθ C α C β Coθ C Coθ C θ β Co θ θ 2 0 α 2 0 Varθ C α 2C Co θ θ j αj Co θ Coθ C θ j α j Coθ C j>2 Restricting attention to the bottom block of rows corresponding to the signal θ C, we get 2Varθ C α C = β CoθC Coθ C θ β Varθ C α 2C S.4 j>2 Coθ C θ j α j Coθ C The corresponding eqation for trader 2 is 2Varθ C α 2C = β CoθC Coθ C θ β Varθ C α C S.5 j>2 Coθ C θ j α j Coθ C Sbtracting Eqation S.5 from Eqation S.4, we get 2Varθ C α C α 2C = Varθ C α C α 2C and so Varθ C α C α 2C = 0. Since Varθ C is fll rank, we get α C = α 2C. Q.E.. We can now finish the proof of Proposition. Withot loss of generality, assme that θ A and θ B are orthogonal this can always be achieed by a change of basis. Slightly absing notation, let α A ; α B be the eqilibrim strategy of trader A and ths, by emma S., the eqilibrim strategy of trader B is α B. S. We maintain the assmption that matrices Varθ ; θ C and Varθ 2 ; θ C are fll rank.

5 TRAING IN INFORATIONAY COPEX ENVIRONENTS 5 The expected profit of trader A is eqal to β α T A Varθ Aα A + α T B Varθ Bα B and the expected profit of trader B is eqal to β α T B Varθ Bα B. Since matrix Varθ A is positie semidefinite, α T A Varθ Aα A 0 and, ths, the expected profit of trader A is at least as high as the expected profit of trader B. S.4. PROOF OF PROPOSITION 2 Some parts of the proof the lower bond on Varp, the lower bond on Var for class C, and the pper bond on the expected loss of liqidity traders for class C 3 rely on Theorem. Some parts the pper bond on Varp, the lack of pper bond on Var, the lower bond on Var for classes C 2 and C 3, the pper bond on β for classes C 2 and C 3, the lower bond on β, and the lower bond on the expected loss of liqidity traders rely on Theorem 2 or parts of its proof. The remaining parts of the proof of Proposition 2 are self-contained. Bonds on Varp. Asp = E[ ], wehae0 Varp σ. The fact that the pper bond can be approached arbitrarily closely in class C, and, therefore, also in classes C 2 and C 3 follows directly from Theorem 2 for the case Co θ θ = 0. Note also that the lack of pper bond on Var, the lower bond on β, and the lower bond on the expected loss of liqidity traders also follow directly from this case of Theorem 2. To see that the lower bond on Varp can also be approached arbitrarily closely, consider a market with two strategic traders and liqidity demand independent of all other ariables. Trader obseres + ε, whereε is distribted normally with mean zero and ariance σ εε, independently of and. Trader 2 obseres ε. The reslting coariance matrix is σ σ σ 0 σ Ω = σ + σ εε σ σ εε 0 σ σ σ εε σ + σ εε σ In this case, 2σ + σ Λ = Σ diag + Σ θθ = εε σ σ εε σ σ εε 2σ + σ εε Λ 2σ + σ = εε σ + σ εε 3σ 2 + 0σ σ εε + 3σ 2 σ + σ εε 2σ + σ εε εε A = Λ σ + 3σ Σ θ = εε σ σ 3σ 2 + 0σ = σ εε + 3σ 2 σ + 3σ εε σ 3σ εε + σ εε Next, sing the compact formla for the case when liqidity demand is independent of the other ariables in the model, we get β = A T Σ diaga σ = σ 3σ + σ εε 2σ + σ εε σ and α = β A σ = 2σ + σ εε

6 6 N. S. ABERT,. OSTROVSKY, AN. PANOV Finally, Varp = Var β α T θ + = β 2 α T Σ θθ α + σ 2 σ σ + σ εε σ = 2 3σ + σ εε σ 2σ + σ εε 4σ + σ σ 2 = 2 3σ + σ εε 2σ 2 + σ + σ εε = 2σ 2 3σ + σ εε Ths, Varp can be arbitrarily close to zero when σ εε is large. ower bonds on Var. Consider first class C. As the signals of the strategic traders and ths their demands are ncorrelated with, we hae Var σ, and ths it is sfficient to find a seqence of markets for which Var σ. Consider the same two-trader setp that we sed to establish the infimm for Varp. In that example, Var = Var α T θ + = α T Σ θθ α + σ = σ 2σ + σ εε 4σ + σ = σ 3σ + σ εε σ + σ εε and so as σ εε grows large, Var conerges to σ. To establish that inf Var = 0onclassC 2 and ths also C 3, consider a market with two grops of n strategic traders. The demand of liqidity traders is ncorrelated with the ale of the asset. In the first grop, all traders obsere θ = perfectly. In the second grop, all traders obsere θ 2 = perfectly. When a market contains seeral grops of identical traders, it is conenient to se the hat notation and closed-form expressions from the proof of the special case of Theorem 2. This proof is gien in Appendix B. Using that notation, we hae σ 0 Σ θθ = 0 σ and Σ diag = n σ 0 0 σ and then and  = Λ Σ θ = Λ = Σ θθ + Σ diag = n + n σ 0 0 σ n  n + 0 = Λ Σ θ = n 0 n +

7 TRAING IN INFORATIONAY COPEX ENVIRONENTS 7 Then applying the closed-form expression from Step of Appendix B, we obtain β = n σ σ and the ector of mltipliers for the two grops α = Â/β Â = σ n σ n + n Ths, in this market, the aggregate demand is n = α T σ θ + = n n + σ n + + and so n Var = n + σ 2 + n + σ 2 = σ n + In particlar, Var conerges to zero as n grows. Upper bonds on E[ p]. First, we establish that sp E[ p] = σ σ /2 in classes C and C 2. Note that this is precisely the expected loss of the liqidity traders in the canonical one-trader Kyle 985model. Ths, it is enogh to show that for any other market in C and C 2, the expected loss of the liqidity traders cannot exceed this ale. To see this, note first that in sch markets, σ = 0. We hae p = E[ ]= Co. Var The expected loss of the liqidity traders is E[p ]=Co p = Co p = Co Co. Project on and : we then hae = a + b + cw for some coefficients a, b, andc, wherew is normally distribted with mean 0 and ariance, and is Var independent of and of. We then get E [ p ] σ σ ab = σ a 2 + σ b 2 + c 2 σ σ ab σ a 2 + σ b 2 = σ a/b + σ b/a 2 σ σ where the last ineqality follows from the fact that for any two real nmbers in this case, σ a/b and σ b/a, their arithmetic aerage is weakly higher than their geometric aerage. Second, we establish that sp E[ p] = σ σ in C 3.WehaeE[p ]= Co p σ Varp σ σ, where the first ineqality is jst the classical Cachy Schwarz ineqality and the second ineqality follows from the fact that E[ p p]=0, and ths Var = Var p + Varp Var p. To show that the bond is tight, consider a rescaled ersion of Example S. in Section S... Gien a normal ariable w with mean 0 and ariance, let the asset ale be = σ w and let the liqidity demand be = σ w, so that Var = σ, Var = σ, and the asset ale is perfectly negatiely correlated with liqidity demand.

8 8 N. S. ABERT,. OSTROVSKY, AN. PANOV There is one strategic trader who obseres signal θ = and ths also knows. The reslting coariance matrix is σ σ σ σ Ω = σ σ σ σ σ σ σ σ σ As there is only one strategic trader, many matrices become scalars. We hae Λ = Σ diag + Σ θθ = 2σ, Λ = /2σ, A = Λ Σ θ = /2, and A = Λ Σ θ = 2 σ σ. Using or closed-form characterization, we find β = σ σ and α = σ σ. Ths, in this market, the demand of the strategic trader is eqal to σ σ =, and the aggregate demand and the price are always eqal to 0. Since the strategic trader demands qantity and the price is 0, his expected profit is E[ ]= σ σ, which is also the expected loss of liqidity traders. Upper bonds on β. First, consider any market in class C.Asp = β,wehae E[p ] =Co p Co = Co p = β Co, and as the signals of the traders are ncorrelated with, Co = σ. Hence, β = E[p ]/σ. Using the pper bond obtained for the expected loss of liqidity traders, we immediately obtain the pper bond on β. Next, consider classes C 2 and C 3. Consider the two-grop market sed to establish the lower bond on Var in classes C 2 and C 3. Recall that, in this configration, we obtained β = n σ σ, which grows withot bond as n grows large. S.5. EXAPES In this section, we gie seeral additional examples that illstrate the general framework presented in the main paper and also help deelop intition for or information aggregation reslts. We first present a simple yet seemingly conterintitie example in which a trader informed abot the ale of the secrity trades in the direction opposite to his estimate of that ale. Next, we stdy what happens when one of the strategic traders is informed abot the demand of liqidity traders. We conclde by analyzing seeral examples in which the market maker possesses priate information abot the ale of the secrity, and stdy how this information gets incorporated into the price of the secrity and how it affects eqilibrim trading strategies and the sensitiity of eqilibrim prices to market demand. S.5.. Trading Against One s Own Signal We start with an example of information strctre nder which a trader who receies a signal abot the ale of the secrity trades in the opposite direction: if, based on his information, the expected ale of the secrity is positie, then he shorts the secrity; if it is negatie, then he bys it. Note that since or model is single period, there cannot be any dynamic incenties to maniplate prices, of the form I will try to mislead others first and then take adantage of the mispricing. S.2 S.2 For examples of settings in which sch dynamic incenties do arise, see Brnnermeier 2005 and Sadzik and Woolnogh 205.

9 TRAING IN INFORATIONAY COPEX ENVIRONENTS 9 EXAPE S.3: The ale of the secrity is distribted as N0. Therearetwo strategic traders. Trader obseres a noisy estimate of : θ = + ρ ξ,whereξ N0 is a random ariable independent of, andρ is a parameter that determines how accrate trader s signal is e.g., if ρ = 0, then trader obseres exactly, and if ρ is ery large, then trader s signal is not ery accrate. Trader 2 also obseres a noisy estimate of : θ 2 = + ρ 2 ξ, with the same drier of noise, ξ, as in trader s signal, bt with a potentially different magnitde of noise, ρ 2. Finally, there is demand from liqidity traders, N0, which is independent of all other random ariables. Formally, the reslting coariance matrix is 0 + ρ Ω = 2 + ρ ρ ρ ρ 2 + ρ From the analysis and closed-form characterization in the main paper, we know that in the niqe linear eqilibrim, the pricing rle is characterized by some β > 0, and the strategies of traders and 2 are characterized by α = Λ α S.6 2 β where Λ = 2+2ρ 2 +ρ ρ 2 +ρ ρ 2 2+2ρ. 2 2 Using the matrix inersion formla and setting δ = β which is positie, since Λ detλ is positie definite, we get α 2 + 2ρ 2 2 ρ = δ ρ 2 + 2ρ 2 α 2 ρ ρ ρ 2 = δ ρ 2 ρ 2 + 2ρ 2 ρ ρ 2 S.7 Ths, if ρ = 2ρ 2 +,traderneer trades,despiteθ ρ being informatie abot the ale 2 of the secrity, and for ρ > 2ρ 2 + > 0, trader always trades in the direction opposite ρ 2 to his signal θ,despiteθ being positiely correlated with the ale of the secrity,. Similarly, if ρ 2 is eqal to or greater than 2ρ +, then trader 2 does not trade or trades ρ in the direction opposite to his signal. To get the intition behind this seemingly pzzling behaior, consider a slight ariation of Example S.3. EXAPE S.4: The ale of the secrity is N0. There are two strategic traders. Trader obseres a noisy estimate of : θ = + ξ, whereξ N0, independent of. Trader2obseresξ: θ 2 = ξ. The demand from liqidity traders, N0, is independent of all other random ariables. The reslting coariance matrix is Ω =

10 0 N. S. ABERT,. OSTROVSKY, AN. PANOV In this case, Λ = 4 2 and α = Λ α 2 β 0 = δ 2 2 = δ 4 0 S.8 for some δ>0, and ths trader 2 trades in the direction opposite to his signal. Note that in this example, trader 2 is not informed abot the ale of the secrity: his signal ξ is independent of. Howeer, he is informed abot the bias in trader s signal, and ths knows the direction in which trader is likely to err when sbmitting his demand. Ths, trader 2, by partly ndoing this error i.e., trading against it, can in expectation make a positie profit, despite not haing any direct information abot the ale of the secrity. In a sense, while trader trades on fndamental information, trader 2 trades on technical information: trader s ability to make money is de to his information abot the ale of the secrity, while trader 2 s ability to make a profit is de to his information abot the mistakes of other agents in the economy. S.3 In Example S.3, the intition is similar. If ρ 2 is large relatie to 2ρ + ρ, then the main part of trader 2 s information is abot the mistake that trader makes, and not abot the fndamental ale of the secrity. This cases trader 2 to want to ndo that mistake and trade against his signal, while trader contines to trade in a natral direction. When ρ 2 = 2ρ + ρ, the incenties of trader 2 to trade on fndamental information the positie correlation of his signal with the ale of the secrity and on the technical information the positie correlation of his signal with the mistake of trader cancel ot, and trader 2 ends p not trading. Examples S.3 and S.4 illstrate that a strategic trader s behaior in eqilibrim is drien not only by the correlation of his information and the ale of the asset, bt also by the informational content of his signals relatie to the information already contained in the signals and the reslting behaior of other agents potentially een to the point of reersing the direction of his trade. It is this flexibility that allows strategic traders information to get flly aggregated and incorporated in prices as market size grows, een for ery rich information strctres. In contrast, the behaior of liqidity traders is exogenos, and is not endogenosly affected by what information it contains. As a reslt, the information contained in liqidity demand is flly incorporated in market prices only nder appropriate correlation strctres see Section 5 for details. S.5.2. Information Abot iqidity emand In this section, we present an example that shows what happens when one of the strategic traders does not know anything abot the ale of the secrity, bt is informed abot the amont of liqidity trading. We then compare the eqilibrim to that of the standard model withot sch a trader. EXAPE S.5: The ale of the secrity is distribted as N0σ, and the demand from liqidity traders is distribted as N0σ, independently of. Therearetwo S.3 Formally, we say that trader i has fndamental information if Coθ i θ 0, and we say that trader i has technical information if Coθ i θ 0orCoθ i θ j θ 0forsomej i. If a trader has neither fndamental nor technical information, then in eqilibrim he does not trade and does not make any profit. In Section S.6, we formally state and proe this reslt Proposition S.2, and also explore in more detail the dependence and nondependence of eqilibrim trading strategies on arios types of information.

11 TRAING IN INFORATIONAY COPEX ENVIRONENTS strategic traders. Trader s signal is eqal to : θ =. He is flly informed abot the ale of the secrity, jst as in the standard Kyle model. Trader 2 is ninformed abot the ale of the secrity, bt has insider information abot the demand from liqidity traders: θ 2 =. Formally, the coariance matrix is σ σ 0 0 σ Ω = σ σ σ 0 0 σ σ The axiliary matrices in this example are 2σ 0 0 Λ = A 0 2σ = 2 and A = 2 0 Coefficient b in the qadratic eqation is eqal to 0 and, therefore, γ = c a = σ σ α = 2 γ = 2 α 2 = 2 σ σ For comparison, if the second strategic trader was not present, the model wold redce to the standard model of Kyle 985, and the eqilibrim wold be characterized by γ = 2 σ σ σ α = σ In other words, when the second strategic trader who is informed abot the demand from liqidity traders is present in the market, that trader takes away one-half of that liqidity demand. As a reslt, the first strategic trader, who knows the ale of the secrity, trades half as mch as he wold in the absence of that second trader, and the market maker s pricing rle is twice as sensitie. Therefore, for any realization of and, the price in the market with the second strategic trader will be exactly the same as that in the market withot that trader, and, ths, the informatieness of prices is not affected in either direction by whether there is a trader in that market who obseres the trading flow from liqidity traders. ikewise, the expected loss of liqidity traders is also naffected by the presence of a trader who obseres their demand. Since, by constrction, the market maker in expectation breaks een, it has to be the case that the profit of the second strategic trader comes ot of the first trader s pocket. In fact, the second trader takes away exactly onehalf of the first trader s profit. S.4 Also, as in Example S.4, the second trader is trading on S.4 To see this, note that the prices in the two markets are always the same, realization by realization, while the demand of the first strategic trader, in the presence of the second one, is exactly one-half of what it wold be in the absence of that trader.

12 2 N. S. ABERT,. OSTROVSKY, AN. PANOV technical information, and is only able to make a profit becase of the mistakes of other agents in the economy. Example S.5 shows that when liqidity demand is flly obsered by some strategic traders, they may hae an incentie to trade in the opposite direction, effectiely remoing part of that demand from the market. If the nmber of sch strategic traders grows large, they may end p remoing all liqidity demand from the market, potentially hindering information aggregation and possibly the existence of limit eqilibrim in large markets see, e.g., footnote 20 in the main paper. Ths, in Sections 5 and 6, we assme not only that the ariance of liqidity demand is positie, bt also that it remains positie when we condition it on the signals of large grops of strategic traders and the market maker. S.5.3. Informed arket aker In the preceding examples, the market maker does not receie any information other than the aggregate demand coming from strategic and liqidity traders. In this sbsection, we trn to examples in which the market maker does possess some additional information. We show how this information affects the strategies of other traders and illstrate the interplay between the weight the market maker places on this additional information and the weight she places on market demand. Or first two examples illstrate that the eqilibrim obtained when the market maker has priate information is generally not the same as when that information is pblicly aailable i.e., known both to the market maker and to all strategic traders. S.5 This difference trns ot to be important when we stdy the informatieness of prices as the sizes of some bt not all grops of strategic traders become large Section 6. In that setting, as the sizes of some of the grops become large, the market behaes as if the signals of those grops were obsered directly by the market maker and not as if the signals of those grops were obsered pblicly. EXAPE S.6: The ale of the secrity is N0. There is one strategic trader, who obseres signal θ = + ε. The market maker obseres signal θ = + ε 2.Variables ε and ε 2 are distribted normally with mean 0 and ariance, independently of each other and of all other ariables. The demand from liqidity traders is also independently distribted as N0. Formally, the coariance matrix that describes this information strctre is 0 Ω = Applying the formlas deried in Section 3, we get Σ diag = Σ θθ = Σ = 2andΣ θ = Σ θ = Σ =. Ths, Λ = /2 = 7/2, A = 0, and A = 2 /2/7 = /7. The S.5 Jain and irman 999 and o 200 stdy extensions of the Kyle 985 model with a partially informed market maker and with partially informatie pblic information, respectiely. The difference between the two cases can be seen by comparing their reslts setting σ 2 i = 0ino 200. Or Examples S.6 and S.7 are similar, thogh not identical, to the models of Jain and irman 999 and o 200. We present the examples to emphasize the distinction between the two cases within the same setp, as this distinction is important for or hybrid-market information aggregation reslts.

13 TRAING IN INFORATIONAY COPEX ENVIRONENTS 3 coefficients in the qadratic eqation for γ are a = 2/49, b = 0, and c =, and ths β = γ = 2 7 Hence, the strategic trader s behaior is gien by α = A = 2 β 2 and the market maker s sensitiity to her own signal is β = Σ Σ Σ T A 3 θ = 7 Consider now a ariation of Example S.6, in which the market maker s signal is pblic information i.e., known to both the market maker and the strategic trader. EXAPE S.7: The ale of the secrity is N0. The market maker obseres signal θ = + ε 2. The strategic trader now obseres two signals, θ = + ε and θ 2 = + ε 2. Both ε and ε 2 are normally distribted with mean 0 and ariance, independently of each other and of all other ariables. The demand from liqidity traders is independently distribted as N0. The coariance matrix that describes this information strctre is now Ω = We now hae Σ diag = Σ θθ = 2 2, Σ = 2, Σ θ = 2, Σθ =,andσ =. Ths, Λ = 7/2 2, Λ = /6 2 7/2, A = 0,andA = Λ 0 /2 2 = /6 /2. The coefficients of the qadratic eqation on γ are now T 2 a = /36 = /24 b= 0 and c = /2 2 /2 and, ths, β = γ = 6 2 the strategic trader s behaior is gien by 6 α = A = 3 β 6 6

14 4 N. S. ABERT,. OSTROVSKY, AN. PANOV and the market maker s sensitiity to her own signal is now gien by β = Σ Σ Σ T A θ = 2 The eqilibria in these two examples are sbstantiely different: the sensitiities of the market maker to the aggregate demand and to her own signal are different, and the sensitiity of the strategic trader s demand to signal θ is different as well. We can also compte the expected profits that the strategic trader makes in these two markets and ths the losses of liqidity traders: in the first example, the expected profit is 2/7, while in the second one it is greater: 6/2. These differences illstrate the point that haing the market maker obsere a signal is sbstantiely different from haing that signal obsered pblicly. Or next example considers the case in which a strategic trader s information is strictly worse than the information aailable to the market maker. EXAPE S.8: et ν, ν 2, ε, ε 2,and be independent random ariables, each distribted normally with mean 0 and ariance. The ale of the secrity is = ν + ν 2. The demand from liqidity traders is. There are two partiallyinformed strategic traders and a partially informed market maker. Trader s signal is θ = ν + ε. Trader 2 s signal is θ 2 = ν 2 + ε 2. The market maker s signal is θ = ν 2. Note that while trader possesses some exclsie information abot the ale of the secrity, trader 2 does not becase ν 2 is obsered by the market maker and ε 2 is pre noise. Formally, the coariance matrix is Ω = The axiliary matrices in this example are Λ = A 0 3 = and A 0 = 4 0 Therefore, in this case, we hae α α 2 = β 4 0 and so α 2 = 0. Ths, trader 2 does not trade in eqilibrim. This illstrates a more general phenomenon: in eqilibrim, a strategic trader cannot make a positie profit and does not trade if his information is the same as or worse than in the information-theoretic sense that of the market maker. S.6 Or final example considers a seqence of markets indexed by the nmber of strategic traders, m. All traders receie the same information, which is imperfectly correlated with both the ale of the asset and the market maker s information. S.6 See Proposition S.2 in Section S.6 for a formal statement and proof of this reslt.

15 TRAING IN INFORATIONAY COPEX ENVIRONENTS 5 EXAPE S.9: The ale of the secrity,, the demand from liqidity traders,, and two information shocks, ε and ε 2, are all distribted normally with mean 0 and ariance, independently of each other. There are m identically informed strategic traders and a partially informed market maker. Each strategic trader obseres a signal θ = + ε. The market maker obseres a signal θ = + ε 2. Formally indexing all matrices by the nmber of strategic traders in the market, m, the coariance matrix is Ω m = The axiliary matrices are Λ m = m + 3 6m + 8 6m m + so that Λ m 6m + 8 6m + 8 = 3 3 6m + 8 6m m + 8 6m A m = and A m = 3m m m m + 8 3m + 6m m m m m + 8 3m + 6m + 8 Coefficient b in the qadratic eqation is eqal to 0, and so γ m = c a = 3m + 4 2m

16 6 N. S. ABERT,. OSTROVSKY, AN. PANOV α m i = γ m A m i = 2m β m = 2 m = 2m + 4 3m + 4 6m + 8 = m + 2 3m + 4 Note that the weight β that the market maker places on her own signal is not constant in m. If there were no strategic traders at all, and only noise traders m = 0, althogh strictly speaking that case is not allowed by or general setp, it wold be eqal to = Coθ.Asm grows, this weight is monotonically decreasing conerging to in 2 Varθ 3 the limit. Intitiely, as m grows, an increasingly large fraction of the market maker s information abot the ale of the secrity is also contained in the strategic demand, and can be extracted from it by the market maker ths leaing a smaller part for the signal θ that the market maker obseres directly. The second obseration concerns the informatieness of prices. Take any m, and consider a realization of θ, θ, and. In this realization, demand is eqal to mα mθ i + = m 2m θ +, and the market price P set by the market maker is eqal to β m + βm θ = m θ 3m+4 + m+2 θ 3m+4 + 2m. Now fix the realization of random ariables, 3m+4 and let the nmber of strategic traders, m, grow to infinity. Then price P conerges to θ 3 + θ 3. Bt notice that this expression is precisely the expected ale of the asset,, conditional on the information aailable in the market: is ninformatie, becase it is independent of all other random ariables, and T θ θ θ E[ θ θ ]=Co Var θ θ θ = 2 θ 2 θ = 3 θ + 3 θ Hence, as the nmber of strategic traders becomes large, their information and the information of the market maker get incorporated into the market price with precisely the weights that a Bayesian obserer with access to all information aailable in the market wold assign. In other words, as the nmber of strategic traders becomes large, all information aailable in the market is aggregated and reealed by the market price. Of corse, this is not a coincidence: as we show in Theorem 2, the information aggregation reslt holds ery generally. S.6. THE IPACT OF INFORATIVE AN UNINFORATIVE SIGNAS ON TRAING STRATEGIES In this section, we proe the statement made in footnote S.3 in Section S.5. and also following Example S.8 in Section S.5.3 that a trader who possesses neither fndamental nor technical information and who is ths less informed than the market maker does not trade in eqilibrim and ths does not make a profit. We also show that the presence of sch a trader does not affect the trading behaior of other agents or the pricing behaior of the market maker. We then present two examples showing that the interaction of different types of information can, in general, be qite sbtle: a trader who possesses only

17 TRAING IN INFORATIONAY COPEX ENVIRONENTS 7 technical information may be able to make a positie profit een if he is the only strategic trader in the market, and a trader who has some signals less informatie than those of the market maker may neertheless se those signals for trading if he also has some other, more informatie signals. We conclde by showing that if a sbector of strategic traders signals is ninformatie rather than simply being less informatie than the signal of the market maker, then the traders do not se this sbector of signals in eqilibrim, and the presence of these signals has no impact on the agents trading behaior or on the pricing behaior of the market maker. Formally, we say that trader i is weakly less informed than the market maker if, conditional on the market maker s signal θ,traderi s signal θ i does not contain any additional information abot other random ariables in the model,,andθ j for j i: Coθ i θ = Coθ i θ = Coθ i θ j θ = 0 Note that this is eqialent to the trader haing neither fndamental nor technical information, as defined in footnote S.3 in Section S.5.. PROPOSITION S.2: Sppose trader i is less informed than the market maker. Then in eqilibrim, α i = 0, and the profit of trader i is 0. The eqilibrim strategies of all other traders and the pricing behaior of the market maker are the same as in the economy in which trader i is not present. PROOF: For notational simplicity, sppose i = and sppose the dimensionality of trader s signal is k i.e., θ R k. Note that by the maintained Assmption of Section 3, at least one strategic trader receies at least some information abot the ale of the secrity that is not contained in the market maker s signal, and, ths, there are at least two strategic traders in this market: n 2. From the closed-form soltion gien in Sections 3. and 3.2, the ector α that stacks all trading strategy ectors α j on top of each other is gien by α = β A A = Λ β Coθ θ Coθ θ Note first that the assmption that trader is less informed than the market maker implies that the top k elements of ector β Coθ θ Coθ θ are all 0. Next, consider matrix Λ = Σ diag + Varθ θ, Σ Σ 22 0 Λ = 0 0 Σ nn S.9 Varθ θ Varθ 2 θ Coθ 2 θ n θ + 0 Coθ n θ 2 θ Varθ n θ

18 8 N. S. ABERT,. OSTROVSKY, AN. PANOV Σ + Varθ = θ 0 0 where is an inertible matrix. Ths, Λ = Σ + Varθ θ 0 0 and, therefore, in ector α = Λ Coθ θ Coθ θ β S.0 S. the first k elements are all 0 and they are precisely the elements that describe the eqilibrim trading strategy of trader, α. The reslt abot the zero profit of trader is then immediate. Next, the first k elements of ector A = Λ Coθ θ and ector A = Λ Coθ θ are all 0, and so it is immediate that the coefficients a, b, andc of the qadratic eqation that pins down market depth γ = /β are the same as in the corresponding qadratic eqation for the market in which trader is not present see Section 3.2 for the closed-form soltion formlas. Ths, the presence of trader does not affect β. Finally, from the block-diagonal formlas in Eqations S.9, S.0, and S., it follows that the elements of ectors A and A after the first k are exactly the same as in the corresponding ectors for the market in which trader is not present. This obseration, combined with the obserations that the first k elements of A and A are all 0 and that β is not affected by the presence of trader, imply that β is not affected by the presence of trader and that the elements of ector α after the first k are exactly the same as in the corresponding ector for the market in which trader is not present. Q.E.. The statement of Proposition S.2 is intitie, bt it is important to note that the interaction of information and trading can, in general, be qite sbtle. To illstrate the sbtlety, we present two examples. The first example shows that a trader who possesses only technical information may still be able to make a positie profit een if he is the only strategic trader in the market. S.7 In the example, the reason for the ability of the technical trader to make a profit is that liqidity demand is informatie abot the ale of the secrity, and so the market maker s pricing rle is sensitie to aggregate demand. The technical strategic trader, in trn, is informed abot the bias in liqidity demand. EXAPE S.0: The ale of the secrity is distribted as N0. iqidity demand is distribted as = + ε, whereε N0. Random ariables and ε are independent. There is a single strategic trader in the market, whose signal is gien by θ = ε. The market maker does not directly obsere any signals k m = 0. To compte a linear eqilibrim, sppose the sensitiity of the market maker s pricing rle is β > 0 and sppose the strategic trader s trading rle is gien by mltiplier α.the S.7 Strictly speaking, the setting of the example iolates Assmption of or model, which states that at least one strategic trader in the market mst possess some fndamental information. Ths, or general reslts are not applicable to this setting. Neertheless, as we show, there still exists a niqe linear eqilibrim in this example.

19 TRAING IN INFORATIONAY COPEX ENVIRONENTS 9 aggregate demand is then eqal to + + α ε, and the expected ale of the secrity conditional on aggregate demand is then gien by E[ ]= + + α 2 Ths, β =. ++α 2 On the other hand, sppose the strategic trader obseres a realization θ of his signal. If he sbmits demand d, his expected profit is eqal to E [ d β + θ + d ] = β θd + d 2 Ths, the optimal demand is d = θ/2, and so α = /2, which in trn implies β = 4/5 and a positie expected profit of trader conditional on obsering realization θ of his signal, trader s expected profit is eqal to θ 2 /5. Or next example shows that for Proposition S.2 to hold, it is important that a trader s entire ector of signals is less informatie than the signal of the market maker. If only some of trader i s signals are less informatie, while others contain sefl information that the market maker does not obsere, then the trader may end p ptting nonzero weights on the less informatie signals in eqilibrim. EXAPE S.: The ale of the secrity is distribted as N0. There is a single strategic trader in the market who obseres a ector of signals θ ; θ 2 R 2. The first component of the ector is gien by θ = + δ, whereδ N0 is a random ariable independent of. The second component is θ 2 = δ. The market maker s signal is gien by θ = δ. Finally, the demand from liqidity traders is N0, distribted independently of and δ. Note that the second component of the strategic trader s ector of signals is weakly less informatie than the market maker s signal: it contains no additional information. By Proposition S.2, if that were the only information that the trader obsered, he wold not trade based on it. In Example S., howeer, that is not the case. In the niqe eqilibrim, the strategic trader will pt negatie weight on component θ 2 : his demand is gien by d = θ θ 2 = and so α = ;. The market maker ignores her signal β = 0 and pts weight β = /2 on aggregate demand. Note that this eqilibrim is essentially the same as the eqilibrim of the standard Kyle 985 model and that trader haing access to signal θ 2 is essential for this eqialence: if trader had access only to signal θ, the eqilibrim wold be sbstantiely different, with, e.g., β 0. The last reslt of this section shows that if some elements of traders ectors of signals are ninformatie abot all other releant random ariables inclding the information of the market maker, then traders will indeed not trade on that information: it will hae no impact on their trading. We se this reslt in Section S0 in the proof of the general case of Theorem 3, where in some cases we add axiliary ninformatie signals to aoid haing to deal with zero coariance matrices. Formally, sppose each strategic trader i s signal θ i can be represented as a pair of signals θ ; i θ in sch a way that the combined ectors i θ = θ ; ; θ and n θ = θ ; ; θ hae the property that ector n θ is ninformatie: Co θ = Co θ = Co θ θ = Co θ θ = 0

20 20 N. S. ABERT,. OSTROVSKY, AN. PANOV Note that for some traders i, θ i or θ i may be empty, bt we assme that both θ and θ hae at least one element. et α and α be the ectors of components of eqilibrim demand mltiplier α corresponding to θ and θ. PROPOSITION S.3: Sppose θ is ninformatie. Then α = 0. oreoer, the eqilibrim is not affected by the presence of signals in θ : pricing mltipliers β and β are the same as those in the market where only signals in θ are obsered, and the ector of strategy mltipliers α is the same as the ector of strategy mltipliers in the market where only signals in θ are obsered. PROOF: The proof is similar to that of Proposition S.2. From the closed-form soltion gien in Sections 3. and 3.2, the ector α is gien by α = β A A = Λ β Coθ θ Coθ θ Since by assmption, θ is ninformatie, it is independent of,, andθ, and so the rows corresponding to elements of θ in the matrix β Coθ θ Coθ θ are all 0. Call this obseration. Next, consider matrix Λ = Σ diag + Varθ θ = Σ diag + Σ θθ Σ θ Σ Σ T.Takeany θ element i from ector θ and any element j from ector θ these elements can be parts of the same trader s signal or can be obsered by different traders. Since by assmption, θ is independent of both θ and θ, the entries in matrix Λ in cells i j and j i are both 0 becase the corresponding entries are all 0 in matrices Σ diag, Σ θθ,andσ θ Σ Σ T θ. This, in trn, implies that the entries in matrix Λ in cells i j and j i are also both 0. S.8 Call this obseration 2. Combining obseration and obseration 2, we find that the entries in ector α = Λ β Coθ θ Coθ θ corresponding to the elements of θ are all eqal to 0. The proof of the statement that the pricing mltipliers β and β are the same as those in the market where only signals in θ are obsered, and that the ector of strategy mltipliers α is the same as the ector of strategy mltipliers in the market where only signals in θ are obsered, is completely analogos to the proof of the statement in Proposition S.2 that the eqilibrim strategies of traders j i and the pricing behaior of the market maker are the same as in the economy in which trader i less informed than the market maker is not present. Q.E.. S.8 To see this, note that by rearranging the rows and colmns of matrix Λ to first list the entries corresponding to θ and then the entries corresponding to θ, one gets a block-diagonal matrix Σ diag + Var θ θ 0 0 Σ diag + Var θ the inerse of which is also block diagonal: Σ diag + Var θ θ 0 0 Σ diag + Var θ

21 TRAING IN INFORATIONAY COPEX ENVIRONENTS 2 S.7. RESCAING IQUIITY EAN In this section, we formally show that eqilibrim prices are not affected by the scale of liqidity demand. Take an original market as defined in Section 2 and consider an alternatie market in which the demand from liqidity traders, alt, is eqal to the demand from liqidity traders in the original market mltiplied by a constant factor ρ>0: alt = ρ. For notational conenience, we append the sperscript alt to ariables associated with the alternatie market, while ariables associated with the original markets hae the same notation as before. The total demand obsered by the market maker in the eqilibrim of the alternatie market is ths alt = d alt i + alt i where d alt i = α alt T θ i is the demand of trader i. The price in the eqilibrim of this alternatie market is then a random ariable gien by p alt = β alt i α alt T θ i + ρ + β alt T θ PROPOSITION S.4: In the niqe linear eqilibrim of the alternatie market, β alt β alt = β ρ = β α alt = ρα In particlar, eqilibrim price does not depend on the scale of liqidity demand: p alt = p PROOF: The proof follows directly from the eqilibrim characterization of Section 3.2. First note that Σ alt θθ = Coθ θ = Σ θθ and, similarly, Σ alt = Σ diag diag, Σ alt Coθ θ = Σ, Σ alt θ that Σ alt θ and σ alt = Coθ ρ = ρσ θ, Σ alt = Coθ = Σ θ,andσ alt = Co ρ = ρσ. This immediately yields A alt a, b alt = ρb, andc alt = ρ 2 c. Hence, γ alt = ργ, β alt and, therefore, p alt = p. = = Coθ = Σ.Nextnote = Coθ ρ = ρσ, σ alt = Varρ = ρ 2 σ, = A, A alt = ρa alt, a alt = = β /ρ, β alt = β,andα alt = ρα, Q.E.. S.8. PROOF OF THEORE 2 GENERA CASE The proof of Theorem 2 in the main paper applies to the special case in which the coariance matrix of random ector θ; θ ; is fll rank. In this section, we proe Theorem 2 for the general case, imposing only Assmptions and 2: Co θ θ 0and Var θ θ >0. Before proceeding to the proof, we first proe the following lemma.

22 22 N. S. ABERT,. OSTROVSKY, AN. PANOV EA S.2: et φ be a random ector normally distribted with mean 0 and coariance matrix Σ. et α be a ector of the same dimensionality, and let {α k } be a seqence of ectors sch that lim α k T φ = α T φ k for eery realization φ of random ector φ except possibly on a set of probability 0. Then the limit also holds in the 2 sense, that is, lim E [ α k T φ α T φ 2] = 0 k PROOF: etn be the rank of matrix Varφ and take an orthogonal matrix sch that 0 T Varφ = 0 0 where is a symmetric positie definite matrix of size N if matrix Varφ is itself positie definite, and ths fll rank, then we can simply take to be the identity matrix and, ths, = Varφ. et ψ = T φ and, ths, since is orthogonal, φ = ψ. If, for almost all realizations φ of φ, wehaeα k T φ α T φ, then for almost all realizations ψ of ψ, wealsohae α k T ψ α T ψ, which can be rewritten as T α k T ψ T α T ψ S.2 Since ψ is distribted normally with mean 0 and coariance matrix 0 0 0,where is a positie definite matrix of rank N, the fact that the conergence in Eqation S.2holds for almost all realizations ψ of random ector ψ implies that the first N components of ector T α k conerge to the first N components of ector T α.then E [ α k T φ α T φ 2] = E [ αk T ψ α T ψ 2] = E [ T α k T α T ψψ T T α k T α ] = T α k T α T Varψ T α k T α = T α k T α T 0 00 T α k T α Since the first N components of T α k conerge to the first N components of T α,weget that the first N components of T α k T α conerge to 0, which in trn implies by the last eqation that E[α k T φ α T φ 2 ] 0. Q.E.. We are now ready to proe Theorem 2. The proof proceeds in fie steps. Step. Step is identical to Step in the proof of the special case in Appendix B and is therefore omitted. The remaining steps are different from those in the proof of the special case. Step 2. et s now consider the entire seqence of markets, and restore sperscript m for the ariables. We introdce, for each market m, a random ector θ m,whichis

23 TRAING IN INFORATIONAY COPEX ENVIRONENTS 23 independent of the other random ariables of the model, and is distribted normally with m mean 0 and coariance matrix Σ. diag Applying the sal projection formlas for jointly normal random ariables, we note that the expressions for Λ m,  m,andâm that were obtained in Step can be written as Λ m = Var θ + ξ m + θ m θ  m  m = [ Var θ + ξ m + θ m θ ] Co θ + ξ m + θ m θ = [ Var θ + ξ m + θ m θ ] Co θ + ξ m + θ m θ Note that, nlike the special case considered in Appendix B, the limit of Λ m is not garanteed to be inertible, and so we cannot directly consider the limits of Âm and  m. Instead, we in essence will perform a change of basis, and will work in the new system of coordinates. Formally, let be an orthogonal matrix sch that 0 T Varθ θ = 00 where is a symmetric positie definite matrix whose size is the rank of Varθ θ and if Varθ θ is itself positie definite, we can simply take to be the identity matrix and = Varθ θ. Note that Assmption Co θ θ 0 implies that Varθ θ 0; ths, the size of matrix is greater than 0. et θ ; θ be the random ector defined as T θ, where the dimensionality of θ is eqal to the rank of. Note that Varθ θ = and that Varθ θ = 0. In a similar fashion, we let θ m ; θ m = T θ m and ξ m ; ξ m = T ξ m. We will first show that the following limits hold: Var ξ m + θ m ξ m + θ m 0 Var  m T ξ m + θ m 0 Var  m T ξ m + θ m 0 Var  m T θ m 0 Co  m Co  m Co  m Co  m Var  m Var  m Var  m T θ m 0 T ξ m 0 T ξ m 0 T θ m θ m 0 T θ m θ m 0 T θ m ξ m 0 T θ m ξ m 0

24 24 N. S. ABERT,. OSTROVSKY, AN. PANOV Co  m T θ m  m T θ m 0 Step 2a. We show that Varξ m + θ m ξ m + θ m 0. Note that Var ξ m + θ m ξ m + θ m i i Var ξ m + θ m i i where ξ m + θ m i denotes the ith element of random ector i ξm + θ m. We then obsere that, sing the ariance coariance ineqality, Co ξ m + θ m ξ m + θ m ξ m + θ m 2 i i j j Var ξ m + θ m ξ m + θ m i i Var ξ m + θ m ξ m + θ m j j Var ξ m + θ m i i Var ξ m + θ m j j Since Varξ m + θ m i = i Varξm + Var θ m i i 0foralli, we conclde that eery element of matrix Varξ m + θ m ξ m + θ m conerges to 0. Step 2b. We now show that Var m T ξ m + θ m 0andVar m T ξ m + θ m 0. We focs on the first limit; the second one can be obtained ia a similar deriation. For any ector θ m of the same dimensionality as θ m and ξ m,wecanrewrite  m T θ m as E [ θ + ξ m + θ m = θ m θ = 0 ] = E [ T θ + T ξ m + T θ m = T θ m θ = 0 ] = E [ θ + ξ m + θ m = θ m ξ m + θ m = θ m θ = 0 ] where we define θ m ; θ m = T θ m and the second eqality makes se of the fact that, conditional on θ = 0, it is the case that θ = 0 almost srely. et χ m = ξ m + θ m E[ξ m + θ m ξ m + θ m ], that is, χ m is the residal of the projection of ξ m + θ m on ξ m + θ m.asξ m and θ m are independent of, ξ m + θ m is also independent of, and noting that ξ m + θ m is independent of θ, χ m,andθ,weget Âm T θ m = E [ θ + ξ m + θ m = θ m ξ m + θ m = θ m θ = 0 ] = E [ θ + χ m = f θ m θ = 0 ] where we define f θ m = θ m E[ξ m + θ m ξ m + θ m = θ m ].Next, E [ θ + χ m = f θ m θ = 0 ] = Co θ + χ m θ [ Var θ + χ m θ ] f θ m Hence, since fξ m + θ m = χ m, Var  m T ξ m + θ m = Var Co [ ] θ + χ m θ Var θ + χ m θ χ m Therefore, to show that Var m T ξ m + θ m 0, it is enogh to show that Var χ m 0