Slope Takers in Anonymous Markets"

Slope Takers in Anonymous Markets

Marek Weretka,y

29th May 2011

Abstract

In this paper, we study interactions among large agents trading in anonymous markets. In such markets, traders have no information about other traders’payo¤s, identities or even submitted orders. To trade optimally, each trader estimates his own price impact from the available data on past prices as well as his own trades and acts optimally, given such a simple (residual supply) representation of the residual market. We show that in a Walrasian auction, which is known to have a multiplicity of Nash equilibria that if all traders independently estimate and re-estimate price impacts, market interactions converge to a particular Nash equilibrium. Consequently, anonymity and learning argument jointly provide a natural selection argument for the Nash equilibrium in a Walrasian auction. We then study the properties of such focal equilibrium. JEL clasi…cation: D43, D52, L13, L14 Keywords: Walrasian Auction, Anonymous Thin Markets, Price impacts

1 Introduction

Many markets, e.g., markets for …nancial assets are anonymous. In such markets information of each individual trader is restricted to own trades and commonly observed market price. Traders have no information about other traders’payo¤s, identities or submitted orders. The archetypical framework to study anonymous markets, a competitive equilibrium

I would like to thank Paul Klemperer, Margaret Meyer, Peyton Young and especially Marzena Rostek for helpful conversations.

yUniversity of Wisconsin-Madison, Department of Economics, 1180 Observatory Drive, Madison, WI 53706. E-mail: [email protected].

1 assumes the following preconditions 1) all traders freely and optimally adjust traded quantities to prices; 2) trade occurs in centralized anonymous markets; and 3) individual traders are negligible and hence treat prices parametrically. While the …rst two assumptions quite accurately describe crucial …nancial markets, the third assumption that all traders are price-takers appears at odds with empirical evidence from crucial …nancial markets. Even in markets as deep as NYSE or NASDAQ, institutional investor have signi…cant impacts on market prices. Market impacts are widely recognized as key determinants of trading costs and to minimize such costs institutional traders manage order executions with market impact models. Various versions of such models are o¤ered by Citigroup, EQ International, ITG, MCI Barra and OptiMark. In this paper we develop a framework to study market interactions in anonymous markets, in which non-negligible agents trading strategies rely on market impact modes. Attempts to develop a general equilibrium theory with price impacts has ended produ- cing only partial results. The work initiated by Negishi, Hahn, and Hart either required the imposition of very strong assumptions, or gave highly complex, intractable and inapplic- able models: in the Negishi’sapproach, traders were allowed to have subjective perceptions of the price impact, which implied a great deal of arbitrariness on the determination of equilibria, unless the subjective conjectures where exogenously restricted. The alternative approach models thin markets by imposing more structure. The model is endowed with explicit strategic game, such as a Walrasian auction. In such an auction traders strategies are demand schedules and trades and prices are determined by market clearing condition. Games based on Nash in demands have been introduced to IO by Grossman (1981) and Klemperer and Meyer (1989) and to …nancial microstructure by Kyle (1989). Modeling anonymous markets using Nash in demands results the following di¢ culties. In the deterministic setting equilibrium (outcome) is not determinate and the model does not give any testable predictions regarding prices or allocations. The proposed selection criteria e.g. Klemperer and Meyer (1989) or Kyle (1989) are applicable only to very particular settings.1 Secondly, under standard interpretation, playing Nash requires knowledge of equilibrium strategies of the trading partners. Given informational constraints imposed by anonymity, neither the classic justi…cation of Nash that relies on a common knowledge of the game nor the adaptive dynamic learning argument in which payers observe other

1 w. i’th quadratic with symmetric utilities. and additively separable noise

2 agents strategies and adjust their actions to myopically improve payo¤s, can explain how the economic system attains equilibrium. In this paper we model anonymous markets as a Walrasian auction but we depart from a standard game theoretic approach by making the following behavioral assumption that is motivated empirically. Instead of best responding to a priori known equilibrium strategies of other players, agents determine their trades using market impact models, i.e., they conceptualize a market as a supply curve, and use available data on own past trades and prices to estimate its slope. Thus instead of prices, traders take the slopes of their residual supplies as given, and thus we call them slope-takers. The paper demonstrates that the slope-taking behavior is rational, i.e., under general assumptions independent estimation and re-estimation of the market impact models by all traders and trading optimally relative to such models results in dynamics that eventually converges to a focal Nash equilibrium, called slope taking equilibrium. In large markets traders endogenously become price-takers and thus the framework naturally extends the Walrasian model to thin markets. The slope-taking model retains the following properties of the competitive theory:

Trader’s rationality–In the equilibrium slope-takers mutually best respond to their strategies in markets with an arbitrary number of traders;

Sharp predictions–Theslope-taking equilibrium exists, and its outcomes are generically locally unique under general assumptions about preferences;

Anonymity–Adopting slope-taking strategy only requires knowledge of one’s own trades and market prices.

Tractability–The slope-taking equilibrium has a reduced-form given by trades prices and price impacts, de…ned by the conditions of the competitive equilibrium adapted to thin markets.

For markets where the standard selection criteria for Nash equilibria proposed in the literature are applicable, i.e., quadratic symmetric cost or utility functions slope taking equilibrium coincides with a (Bayesian) Nash equilibria.2 Thus slope-taking framework,

2 In the model of oligopolistic industry with quadratic cost functions slope-taking equilibrium is equivalent to a unique equilibrium in Klemperer and Meyer (1989) model with unbounded noise. Slope-taking

3 jointly with Weretka (2007) and Rostek and Weretka (2009) bridges the general equilibrium theory with strategic literature based on Nash in demands games.3 The remainder of the paper is as follows. Section 2 explains the slope-taking idea and gives our main results: the convergence to a focal equilibrium and the characterization of the equilibrium. Section 3 relates our model to the existing competitive and strategic literature that re…nes the set of Nash in demands in a Walrasian auction. The proofs of all results are in the Appendix.

2 A Model with Slope-Takers

The classic model of anonymous markets, a Walrasian auction with competitive traders, was …rst proposed by Walras to rationalize the concept of a competitive equilibrium.4 In such an auction traders simultaneously submit demand schedule ti (p): R R that specify ! demands as a function of price. For any pro…le t ( ) = ti ( ) , the payo¤s are determined f gi as follows. Walrasian auctioneer aggregates all bids, …nds market clearing price p (s.t. i I ti (p) = 0), and determines allocation according to ti = ti (p) and m i =p ti. With 2 quasilinearP utility functions Ui (xi; mi) = ui (xi) + mi, where xi is a consumption of a commodity and mi is money, for any given pro…le of demands t ( ), the traders payo¤s are given by the utility evaluated at the equilibrium trade Ui (ei + ti; m i). The competitive solution is based on the paradigm that all traders are negligible and hence they are price-takers. The competitive demand schedules submitted in the W alras Walrasian auction maximize utility for any price, t (p) arg maxt U(ei + ti; pti). i i Equilibrium allocation and market clearing price pW alras; tW alras resulting from demands tW alras( ) = tW alras( ) , a competitive equilibrium, can be directly determined from the i i market clearing condition and optimization by all agents. The conditions can be written as a system of simultaneous equations which in turn allows to employ technics, e.g. dif- ferential topology or numerical methods to establish a number of equilibrium properties. The competitive theory is also consistent with informational constraints imposed by market anonymity. Unlike in the strategic models, price-takers only have to know their own equilibrium is well de…ned (exists and is determinate) for any number of traders, asymmetric, possibly nonquadratic utilities and many commodities. However slope-taking approach is more general than the Nash in demands literature nonquadratic environments with heterogenous traders and possibly many commodities. 3 Weretka and Rostek Weretka 4 In fact Tatonnemount

4 preferences and endowments. In particular traders can (best) respond to market conditions even if equilibrium strategies of trading partners are unknown. Competitive solution is not suitable for markets with atomic traders. When the number of traders is small, or even …nite, competitive bidding is not rational, i.e., tW alras( ) is not a Nash equilibrium. If traders j = i submit demands tW alras ( ), the residual supply 6 j faced by trader i has a strictly positive slope. Since schedule tW alras ( ) equalizes marginal i utility with price trader i can increase utility by shading his bid, i.e., submitting a demand schedule that is strictly below marginal utility. We now explain how competitive theory can be augmented to account for non-negligibility of traders in antonymous markets.

2.1 Slope-Takers: Example

Slope-Taking Equilibrium: Consider a market with I > 2 traders with utility functions 2 Ui (xi; mi) = xi 0:5(xi) + mi. Initial endowments ei are distributed according to c.d.f. Fi( ). The timing of the interactions is as follows. Each trader learns endowment ei (a type) and submits demand schedule. Equilibrium price and trades are determined by market clearing condition. Our cental behavioral assumption is that, instead of best responding to a priori known equilibrium strategies of other players, traders manage their portfolios using market impact models; they operate under the hypothesis that they face a residual supply5

pi(ti) = i + Miti + "i: (1)

Price impacts Mi are treated parametrically by all traders, and are estimated from the available data on past equilibrium prices and own trades. Traders who submit demand schedules that maximize expected utility given their market impact models Mi are called slope-takers. . Slope-taking behavior is optimal given strategies of trading partners if residual supply

(1) adequately summarizes other traders demands and the market impact model Mi re‡ects the true slope of residual supply. If trading partners j = i behave as slopes-takers, their 6 demand schedules de…ne a residual supply for each i, as in (1), even if market impact models of other traders Mj j=i are arbitrary (possibly incorrect). We now show that in f g 6 a Walrasian auction with statistically sophisticated traders strategies eventually converge to a focal Bayesian Nash equilibrium. 5 If they are sellers then they are monopolists who operate on a residual demand function.

5 Each trader places a collection of limit orders that form a demand schedule. By making choices contingent on prices, trader i can respond optimally to each of the realizations of the residual supply ("i), by equating marginal utility with marginal expenditure for any price. The construction of the optimal demand schedule is depicted in Figure 1. Schedule ti ( ) can be interpreted as a best response to a family of residual supplies with random intercept with arbitrary distribution and a deterministic slope Mi. Given market impact model Mi, the optimal demand schedule of trader i is given by

1 ei 1 ti (p) = p: (2) Mi + 1 Mi + 1 If traders j = i follow strategy (2) and their market impact models are (exogenously) given 6 6 by M i = Mj j=i, their demand schedules de…ne a residual supply for trader i, which f g 6 has a functional form as in (1). The implied true price impact of trader i is given by 1 M i = (Mj + 1 j = i) ; (3) I 1H j 6 where ( j = i) is the harmonic average of elements Mj + 1 for all traders but i. The in- H j 6 tercept of the residual supply is given by = M = 1 , and the stochastic component i i j 1+Mj ej is "i = M i= , where, "i, is independent from endowment ei. j 1+Mj P ScheduleP(2) is a best response to the strategies of slope-takers if the market impact model Mi coincides with the slope of the residual supply, resulting from aggregation of demand schedules submitted by trading partners, i.e., it is equal to Mi = M i. In a Bayesian Nash equilibrium traders mutually best respond to their trading strategies, which is the case with slope-takers when harmonic mean condition holds for all i. Consistency condition

(3) for all i de…nes a system of I non-linear equations and I unknowns M = Mi . The f gi system has the unique (symmetric) solution M i = 1= (I 2). Slope-taking strategies with consistent market impact models are mutual best responses (state by state): demand I 2 schedules ti (p) = I1 (1 ei p) for all i constitute a Bayesian Nash equilibrium. We refer to this Bayesian Nash equilibrium a slope-taking equilibrium.

6 Trader i’sresidual supply gives all pairs (ti; p) that are consistence with market clearing given demand schedules of other traders. For any pro…le tj ( ) j=i, the residual supply is implicitly by market clearing f g 6 condition j=i tj (p) + ti = 0. Given smooth demand schedules of traders j = i, the slope of residual 6 6 supply at theP equilibrium price is given by a harmonic average of individual demands of other traders, and discounted by 1=(I 1) elements. Since (inverse of) competitive demands coincide with marginal utilities, their slopes coincide are given by the second derivative of the utility function and hence is strictly negative. Thus the slope of residual supply is always strictly positive.

6 In the context of oligopolistic industry, Klemperer and Meyer (1989) demonstrate that, when the uncertainty faced by traders is bounded, there is a continuum of (Bayesian) Nash equilibria. In the considered example, the slope-taking equilibrium is unique, and hence our criterion is stronger than the one implied by the Bayesian Nash conditions. The slope- taking equilibrium selection derives from the fact that the traders using market impact model equalize marginal utility with marginal expenditure for all prices, including the ones not realized in equilibrium. This need not be the case in the Nash equilibrium. The parts of demand schedules for prices outside of equilibrium support are not pinned down by Nash condition while they do a¤ect the bidding environment of the trading partners. These parts determine the slopes of the residual supplies in equilibrium, and their indeterminacy gives rise to a multiplicity of equilibrium outcomes. Equalizing marginal utility with revenue for all prices is natural in anonymous markets. Traders do not have su¢ cient data to estimate the distribution of other traders endowments and hence to deduce the support of the equilibrium prices. Relying on market impact model restricts the traders information about the residual market to parameter Mi, and to robustly best respond to residual supply with arbitrary distribution of "i have to bid optimally for all prices. Convergence: Now, using simple learning argument, we argue that the interactions among slope-taking agents in anonymous markets eventually converge to an equilibrium in which traders play the focal Bayesian Nash equilibrium. We consider a Walrasian auction repeated in…nite number of times. The timeline is divided into periods indexed by T each period is in turn divided into subperiods t. At the start of each subperiod, traders learn t their endowments ei, simultaneously submit demand schedules and market clearing price and equilibrium trades are determined. Traders enter period T with market impact models T Mi and act optimally relative to their models throughout the whole period. At the end of T slope-takers use available time series collected within period T to re-estimate their T +1 price impacts and enter period T + 1 with new market impact models Mi . Observe t that endowments are not correlated with "i, and hence ei provide an instrument for ti, and t t t market impact model is identi…able in time series p ; ti; ei t T . 2 We …rst consider dynamics assuming that within each period traders have su¢ ciently many observations to precisely estimate their true price impacts. In this case the estimates of price impacts follow a deterministic path determined by a system of equations 1 M T +1 = M T + 1 j = i for all i: (4) i I 1H j j 6 7 0 Suppose that all agents initially are perfectly competitive (Mi = 0). With identical initial estimates and the symmetric system of di¤erence equations, the market impact models T +1 1 T follow the same trajectory for all i, given by linear equation Mi = I 1 Mi + 1 . With T I > 2, the coe¢ cient multiplying Mi is smaller than one and the price impacts approach monotonically the unique steady state M i. In the limit, traders play a Bayesian Nash equilibrium. In our exercise, although in each period traders precisely learn their price impacts, they do not attain Bayesian Nash equilibrium after the …rst round of estimation. This is because at the end of period T , traders estimate price impact for T and enter T + 1 T +1 with revised market impact models Mi , which steepens submitted demand schedules, and consequently increases the price impact for trader i in T +1. The mutual reinforcement of price impacts is a feature that is peculiar to games based on Nash in demands and is not present in e.g., Cournot in which one round of perfect estimation su¢ ces for convergence to an equilibrium. The speed of convergence to a steady state depends positively on the number of traders. This is because, in more competitive markets, the mutual reinforcement of price impacts is smaller. Percentage of a gap closed in one round of estimation is given T T by M =(M i M ) = (I 2) = (I 1) 100%. With four and ten traders, 66% and 89% i i of the gap, respectively, is closed within one round of estimation. With the heterogenous initial estimates, the market impact dynamics is generated by a nonlinear dynamic system with the unique, globally stable, steady state M . The slope- taking strategies eventually (pointwise) converge to mutual best responses and, in the limit, traders behave as strategic Bayesian players. The model generates interesting transitional dynamics. After each estimation round the order of traders’price impacts reverses; that is, the trader whose price impact in T was the highest, has the lowest price impact at T + 1. If, in each period, a sample has the same …nite number of observations, then the estimation is not exact and price impact dynamics follow a nonlinear stochastic process. The estimates in the limit have some distribution around the slope-taking equilibrium. Therefore, in our analysis we assume that in each subsequent estimation round, traders (simultaneously) increase the sample size to make their estimation more accurate. Since, in our model learning does not occur through experimentation with perturbing optimal trades, but rather from a natural experiment resulting from randomness of endowments, the information about price impacts accumulates through trading even in the limit slope taking equilibrium. This is why, unlike in some other models, slope-taking traders cannot be locked in with inaccurate estimates because they stop experimenting too fast. The only

8 steady state in the model with increasing sample size is the slope-taking equilibrium. The outcome of numerical simulations (…rst, second, and third quantile) for the model with increasing samples is depicted in Figure 2. All the paths converge to a unique steady state. The dynamic system has downward bias relative to its steady state— relative to a deterministic model with in…nite samples it results in (on average lower values). This is because a harmonic mean puts greater weight on the smaller values and hence the heterogeneity induces the slower convergence rate. The reported results are robust to our assumptions.

2.2 Quadratic Model

Consider an economy with L 1 commodities and traders preferences ui (xi) = i xi 1 xi vix where vi is a positive de…nite L L matrix. In a multi-good Walrasian auction 2 individual demand schedules specify demands for all commodities for any price vector, i.e., L L ti : R R . The price impact models are L L positive semide…nite matrices, and ! L prices p and individual trades ti are vectors in R . A slope-taking equilibrium is de…ned as a (Bayesian) Nash equilibrium t ( ) in a Walrasian auction for which each ti ( ) is optimal given (some) market impact model Mi. The tractability of the framework with slope-takers derives from the fact that the outcome of slope-taking equilibrium, (p; t; M ), can be characterized by the conditions of competitive equilibrium, augmented to thin markets: market clearing, optimization and consistency of market impact models.

Lemma 1 (Necessary Conditions). In a Walrasian auction an outcome of slope-taking equilibrium (p; t; M ) satis…es 1) i ti = 0; and, 2) DuP (ei + ti) =p + Miti for all i; and, 7 3) M i = [1=(I 1)] M j + vj j = i for all i. H j 6 Conditions (1)-(3) de…ne a system of equations, each for one endogenous variable and standard technics developed in the general equilibrium theory can be applied to characterize equilibrium properties. The slope-taking equilibrium with L = 2 can be represented geometrically in an Edge- worth box. Figure 2 depicts equilibrium outcome (p; t; M ) for the k replica of the economy 7 With L > 1; ( j = i) is a harmonic average of I 1 positive de…nite matrices. H j 6

9 with two types of traders i = 1; 2; with identical quadratic preferences and endowment e1 = (1; 0) and e2 = (0; 1). Given linear price impact models, budget constraints are quadratic in non-numeraire consumption and budget sets.The set of all a¤ordable consumption vectors (ti p(ti) + mi 0) have shapes of circles. With non-negligible price impacts, and the slopes budget line (or more curves) given by marginal payments do not coincide with prices. In the Walrasian equilibrium, the consumption of each trader is equal to xi;W alras = (0:5; 0:5), and is independent from k. The equilibrium consumption is given by a convex combination xi = xi;W alras +(1 ) ei where = (2k 3) = (2k 2). For k = 2, the slope taking equilibrium is depicted in Figure 2. The consumption of trader 1 is given by x1 = (0:75; 0:25). At optimum a budget curve of each i is tangent to the indi¤erence curve. Their equilibrium slopes are determined by marginal payments (and not prices). Since they are di¤erent for two types of traders, the indi¤erence curves are not tangent to one other. Thus allocation in slope-taking equilibrium is not Pareto e¢ cient. In panel b) we show how equilibrium allocation converges to a Walrasian one. As k approaches in…nity, the weight put on the Walrasian consumption is closer to one and hence the equilibrium becomes competitive. We now give existence and convergence result for a slope taking-equilibrium in a quadratic model with heterogenous traders and multiple commodities. The …rst result demonstrates the existence of slope taking equilibrium in a quadratic model with many commodities and arbitrary, possibly heterogenous preferences8

Proposition 1. (Existence) In an economy with L 1 commodities slope taking equilibrium exists if and only if I 3. With two traders the consistency condition implies that the two market impact models mutually reinforce without discounting. In the re-estimation process, for any initial market impact models increase to in…nity the demand schedules become in…nitely steep, and the whole system converges to a no-trade equilibrium. The second proposition shows that in economy with one commodity, when traders use price impact models, in repeated interactions in anonymous markets, the slope-taking equilibrium will prevail in a long run, regardless of the of initially assumed about price impacts. We consider a Walrasian auction repeated in…nite number of times. The process

8 This result complements Weretka (2007) existence result as follows. In this section we consider economy L with many commodities, consumption domain is R rather than R+, and Inada conditions do not hold.

10 of statistical inference is as described in the previous section. We also assume that in each period traders have su¢ ciently many observations to (perfectly) accurately estimate price impact models.

Proposition 2. (Convergence) In an economy with L = 1, for any initial price impacts 0 I T M R+, the system of market impact models M converges to mutually consistent one 2 M and equilibrium demand schedules (pointwise) converge to the slope-taking equilibrium.

The convergence in a quadratic model can be explained as follows. While consistency I condition is not a contraction on the whole domain R+, with I 3, it becomes a con- I traction when the domain is restricted to arbitrarily compact set in R+. It follows that the consistent price impacts are unique. The next result demonstrates the existence of slope-taking equilibrium in markets with many commodities and heterogenous quadratic utilities.

2.3 Non-quadratic Model

The framework with slope-takers can be applied to economies with non-quadratic quasilinear utility functions and (approximately) deterministic endowments. The convexity of utilities in equilibrium, given by the second derivatives are function of equilibrium allocation and in an economy with slope-takers and stochastic endowments, true price impacts are random (even when conditioned on equilibrium price). Thus market impact models that assume deterministic price impact (1) are not adequate representations of the slope- taking strategies of trading partners. In this section we argue that when randomness of the endowments is small, the loss of utility associated with inaccurate representation of the market is minimal is likely to be outweighed by the bene…t of having a simple representation of anonymous market in which parameters are identi…able from the available data. In particular, in the deterministic settings, the slope-taking behavior is mutually optimal. In this section we consider an economy with L 1 non-numeraire commodities and nonquadratic, quasilinear utility functions ui (xi) + mi where nonlinear components ui ( ) satisfy standard assumptions (di¤erentiability, monotonicity, strict convexity and Inada).

We also assume deterministic endowments. Given non-quadratic utility ui (xi), the demand schedule ti ( ) optimal relative to market impact model Mi is nonlinear and is determined from the condition Dui (ei + ti) = p + Miti for all prices. Given such demand schedules for traders j = i, in a Walrasian auction the true price impact of trader i at equilibrium 6 11 trade ti is given by a harmonic average condition M i = [1=(I 1)] M j + vj j = i where H j 6 2 convexities vj j=i are de…ned as vj D uj (ej + tj). f g 6 Lemma 1 extends to non-quadratic economies with deterministic endowments considered in this section, i.e., conditions (1)-(3) are necessary for p; t; M to be an outcome of a slope-taking equilibrium. As shown in Weretka (2007), in any nonquardatic economy with L = 1 there exists a triple p; t; M satisfying the three conditions and generically it is locally unique (and hence typically there can be at most …nite number of them). Our next proposition demonstrates that under additional mild conditions any such p; t; M can be rationalized as an outcome of a slope-equilibrium in a Walrasian auction. Let demand schedules ti ( ) be optimal relative to market impact models M from p; t; M satisfying f gi conditions (1)-(3). Lemma 2 (Sufficient Condition). If at equilibrium schedules tj( ) j=i de…ne convex f g 6 optimization problem of i, for all i, then demand schedules ti ( ) are mutual best responses f gi and hence constitute slope-taking equilibrium in a Walrasian auction.

The su¢ cient condition for ti ( ) to be a Nash equilibrium is that the implied residual f gi supplies for each i de…ne globally convex optimization problems for all traders. The problem of a trader facing a supply function may fail to be convex when the supply is very concave, which in turn might endogenously occur in models with a small number of traders, in which the marginal utility function of some of them is very convex. In particular such condition is trivially satis…ed in economies with quadratic utilities considered in the previous section. We now explain how the traders can learn their true price impacts. We interpret our deterministic model with nonquadratic utilities as an approximation of a more realistic environment in which endowment randomness in small. In such markets, the slopes of the residual supplies are almost deterministic. Still they can be estimated even with small variation of prices and trades. Our result says that the bene…t from having a more complex (and accurate) model of residual markets for each trader is negligible. Figure 2 demonstrates convergence of market impact models

2.4 Tatonnement process (Ascending Auction)

The slope-taking approach can be applied to a class of dynamic games with strategy spaces of su¢ cient reach to allow the traders to make choices contingent on prices. An example of such a game is the ascending double auction (tatonnement process) in which an auctioneer

12 adjusts the price in the direction of the excess aggregate demand, and the traders respond to the announced price by specifying the quantities they are willing to trade. The trade takes place only after the price attains the level that clears the market. Our argument is based on the observation that the slope-taking Bayesian Nash equilibrium is an ex post equilibrium, i.e., traders have no incentives to change their slope-taking strategy, even if they learn the realizations of types of other traders. In the ascending auction, by observing the history of price adjustments, traders obtain new information about the types of other traders, but such additional information does not a¤ect the bidding incentives as long as other traders stick to their strategies. The slope-taking equilibrium is an ex post equilibrium for the following reasons. In the independent private values settings, the realization of types of other traders a¤ects trader i payo¤ only up to a residual supply such realization de…nes, given the equilibrium strategies of other traders. By submitting a demand schedule, trader i can chose a di¤erent quantity for each possible realization of the residual supply, and the chosen quantity for one realization does not a¤ect the utility in any other realization. The trader solves independent optimization problems, which allows him to attain optimal outcome in each realization even if, in the static auction, he decides about his bid at the interim stage of the game. This also explains why, in the dynamic versions of an auction, traders have no incentives to revise (??); even though their posterior distributions of the types of other traders evolve. Consider a continuous-time version of the ascending auction. In the initial bargaining stage, a market clearing price is determined and then the trade takes place. Let H be a set of all bidding histories of di¤erent lengths de…ned on open time intervals (0; x); where x R+ . Each history hx summarizes bidding by all traders up to x. The strategy of 2 [ 1 trader i is a function ti:H R, that speci…es a demand at x after any history hx. The ! auctioneer, at x; adjusts the price according to

p = ti (hx) : (5) i X History hx induces a price path on the (open) time interval, and the end of history price at x is given as a limit p (hx). The bargaining stage ends for any hx such that i ti (hx) = 0 and hence for which the price is in a steady state. The equilibrium priceP is given by p (hx) ; the trades are given by ti (hx), and the payo¤s of all the participants are given by a utility function evaluated at the price and trades determined at the end on the game. Proposition 3 provides a result for the model discussed in Section 2.2 but can be easily

13 extended to nonquadratic deterministic setting. Let ti( ) be a slope-taking strategy in the static Walrasian auction.

Proposition 3. (Tatonnement process) Bidding ti (hx) ti(p (hx)) for all i constitutes a Subgame Perfect Nash Equilibrium in an ascending auction.

Also, in ascending auction, the only information the traders have to have, to bid according to ti(hx) is a currently announced price, which by assumption is observed, and individual price impact. Since the outcome in the game is the same as in the static auction, by identical argument, traders can learn their price impact from past equilibrium prices and trades in repeated dynamic Walrasian auctions.

3 Application

We now discuss how the slope-taking model bridges the following four strands of literature: Walrasian model with price-takers; Klemperer and Meyer (1989) model of Nash in demands; and strategic literature based on a Linear Bayesian Nash equilibrium proposed by Kyle (1989).

3.1 Competitive Equilibrium and Tatonnement Process

It is often claimed that the competitive equilibrium theory is a good predictor of the market interactions if the following three preconditions are satis…ed: 1) there are many traders; 2) if these traders freely and optimally adjust their traded quantities to prices; and 3) if they trade in centralized anonymous markets. The slope-taking approach from this paper formalizes this claim using modern, game theoretic language. Slope taking equilibrium constitutes a (Bayesian) Nash equilibrium in a game de…ned by a Walrasian auction, and it is a limit point of a learning process that is empirically observed in anonymous markets. Weretka (2007) shows that, in large economies consistent price impact models become negligible and the demands of the slope-taking traders, t( ) (pointwise), converge to the competitive demands tW alras ( ). Thus, in large markets, slope-taking equilibrium becomes equivalent to the tatonnement process proposed by Walras to rationalize a competitive equilibrium, and an outcome p; t; M coincides with the equilibrium with price-takers W alras W alras p ; t ; 0 . Thus, slope taking approach provides natural game theoretic found- ation for price taking behavior in anonymous markets with many traders.

14 3.2 Nash in Supplies

In their classic paper, Klemperer and Meyer (1989) study a model of oligopolistic industry with I identical producers who face a stochastic demand and whose strategies are supply functions. To facilitate comparison, we focus on an equivalent model of oligopsonistic industry with I buyers with identical utilities u( ) and identical deterministic endowments ei who face a supply function S(p; "). The argument that allows Klemperer and Meyer to characterize Nash equilibria is as follows. Let tKM ( ) be a demand function in the symmetric equilibrium in the Klemperer i and Meyer setting. Given one-dimensional uncertainty ", for any p observed in equilibrium there exists a unique corresponding realization of "; and hence, conditional on price, all endogenous variables are deterministic. Bidding strategy tKM ( ) speci…es the quantity i conditional on p; therefore, it solves a standard monopsony problem, given the deterministic KM residual supply, which can be found by solving S (p; ") = (I 1)t (p) + ti for p. The i KM slope of the residual supply, Mi is a function of @t ( ) =@p and hence the …rst-order i optimality condition

u0 (ei + ti) =p + Miti (6)

KM implicitly de…nes ti as a function of @t ( ) =@p. The …rst-order condition holds for all i prices in the equilibrium support, and hence on such a set, (6) de…nes a di¤erential equation in ti. The set of all ti ( ) that simultaneously solve a di¤erential equation, and also for which the second order conditions hold for any price, constitutes the set of Nash equilibrium in demands. For the case when uncertainty " has unbounded support Klemperer and Meyer establish the following properties of the equilibrium set : Nash equilibrium exists; it is symmetric; and the set of all equilibria is connected (i.e., a demand schedule ti( ) that is bracketed by any two equilibrium demand schedules tKM ( ) is also an equilibrium bid). i They also show in the model with quadratic cost functions and linear demand with additive unbounded noise that the equilibrium is unique. The slope-taking approach complements these results in the following ways. In the quadratic example, the unique equilibrium coincides with the slope-taking equilibrium from this paper and the learning story based on re-estimation of the price impacts is applicable to equilibrium tKM ( ). The arguments in Klemperer and Meyer rely on the assumption that uncertainty is one- dimensional, which guarantees that the endogenous variables, conditional on prices, are

15 deterministic. The arguments cannot be extended in an obvious way to the settings with multidimensional shocks such as the one discussed in Section 2.2. With private information and nonlinear strategies tj(p; ej), di¤erent realizations of endowments ej j=i de…ne a f g 6 family of stochastic residual supplies for i, even conditional on equilibrium price p. Since slope Mi is stochastic, condition (6) does not de…ne a di¤erential equation, which is essential to characterize the equilibrium set. The slope-taking approach pins down equilibrium in the quadratic models with private information and does so even without unbounded uncertainty. Such a unique slope-taking equilibrium inherits properties from the quadratic example considered by Klemperer and Meyer, as t( ) is invariant to adding a stochastic component to the model. The additional advantage of the slope-taking approach is its tractability. In settings with identical traders, tKM ( ) can be found by solving a nonlinear di¤erential equation i with an endogenously determined boundary condition which makes the characterization of an equilibrium challenging. For example, even in the simple model with three identical producers with non-quadratic cost functions, the uniqueness (or even determinacy) of an equilibrium has yet to be established.9 The model complicates even more with heterogenous traders as an equilibrium is given by the solution to I nonlinear di¤erential equations and there are no results that characterize it even with two heterogenous traders. The slope- taking equilibrium is well de…ned and can be easily characterized in quite general settings. Finally, in non-quadratic settings, the Klemperer and Meyer equilibrium is demanding in terms of the level of rationality it requires from the traders. To play tKM ( ), the least i the traders have to know is their true price impacts for any possible realizations of the residual supply, including the ones associated with extreme ". Since extreme realizations are infrequent, the price impacts for such realizations are not easily estimable from the data available in anonymous markets. Given the limited information, traders are likely to extrapolate the price impact from the typical realizations to the extreme ones, which the slope-taking approach assumes. Such an approximation is without a signi…cant loss of utility for the trader as long as the preferences are close to quadratic and/or the trades are concentrated around typical values. This is why we view our framework as a less elegant but more realistic and tractable model of anonymous makers.

9 Laussel (1992) shows the existence and uniqueness of the Nash equilibrium in a model of oligopolistic industry with 2 …rms.

16 3.3 Linear Bayesian Nash (LBN)

In an alternative approach, Kyle (1989) studies a version of a Walrasian auction in which traders have identical quadratic preferences and he restricts attention to the symmetric Bayesian Nash equilibrium in which submitted schedules are linear in quantities and types The literature based on symmetric linear Bayesian Nash equilibrium (LBN) primarily focused on the question of aggregation of information and, therefore, it assumes common values with jointly normally distributed signals. In the slope-taking model from Section 2.2, we have assumed independent private values, and this is the environment in which we compare the two approaches. The slope-taking strategies are linear in types and, hence, fall into a category of LBN equilibria. We now show that in fact in the quadratic setting slope taking strategies and LBN are equivalent.

Proposition 4. (Equivalence with LBN) In the quadratic independent private value model, LBN equilibrium is equivalent to the slope-taking equilibrium.

The main weakness of the symmetric LBN approach relative to, for example, Klem- perer and Meyer (1989), is that it imposes ad hoc restrictions on the strategies played in equilibrium which re…nes the set of Nash equilibria. The linearity assumption is sometimes justi…ed by the tractability and the fact that simplicity of LBN makes it a focal point of the game. The slope-taking approach o¤ers an alternative, behavioral justi…cation of a LBN based on the anonymity of the markets. The LBN is the only steady state of the learning process in which traders use market impact models and estimate and re-estimate their parameters. Thus LBN is a plausible solution concept in models of nontransparent …nancial markets. In addition, the slope-taking approach explains the transitional dynamics of the markets towards an LBN. Another criticism of the symmetric LBN is that the equilibrium is well-de…ned only in a particular, quadratic, framework with symmetric utilities and hence it does not permit robustness analysis outside of this setting. The slope-taking approach is more general as it de…nes equilibrium (in the deterministic setting) in nonquadratic models and, when viewed as an extension of LBN to nonquadratic settings, allows for such robustness analysis. Another implication of equivalence result is that an outcome of LBN is equivalently characterized by the conditon from Lemma 1. It allows one to characterize a LBN in fairly complex models. Given equivalence result, the following result is immediate

17 Corollary 1. (Existence of LBN) In the quadratic multi-good model with independent private values LBN equilibrium exists if and only if I 3. Corollary establishes existence of a LBN in an economy with L commodities and quadratic but otherwise arbitrary (possibly asymmetric) preferences. Rostek and Weretka (1989) study precise relationship between Nash (in particular linear Bayesian Nash) and equlibria with slope-takers in abstract, possibly non-market games.

4 Conclusions

Our model allows one to study the directions and magnitudes of biases in the predicted prices and trades resulting from the assumption of perfect competition in markets with …nitely many traders. It bridges the general equilibrium literature, which mainly focused on the “reduced-form”competitive equilibrium, with the existing strategic literature based on Nash in demands equilibrium. For example, Weretka and Carvajal (2007) extend the theory of complete markets to the anonymous thin markets. For example Rostek and Weretka (2008a) study dynamics of the symmetric LBN in double auctions with symmetric traders and many trading periods.

18 References

[1] Akgun, U., (2004), Mergers in Supply Functions, Journal of Industrial Economics, LII,4

[2] Carvajal, A. and Weretka, M., (2007), State Prices and Arbitrage in Thin Fin- ancial Markets, Working Paper, University of Wisconsin, Madison.

[3] Grossman, S., (1981), Nash Equilibrium and the Industrial Organization of Markets with Large Fixed Costs, Econometrica, 49, pp.1149-1172

[4] Klemperer, P., and Meyer, M.A., (1989), Supply Function Equilibria in Oligopoly under Uncertainty, Econometrica, 57, pp. 1243-1277

[5] Kyle, A., (1989) Informed Speculation and Imperfect Competition Review of Eco- nomic Studies, 56, pp. 517-556

[6] Laussel, D. (1992) Strategic commercial policy revisited: a supply-function equilibrium model American Economic Review 89, pp. 84-99

[7] Rostek, M., and Weretka, M., (2008a) Nash a Games with Contingent Strategies.-

[8] Rostek, M., and Weretka, M., (2008b) Dynamic Thin Markets, Working Paper, University of Wisconsin, Madison.-

[9] Weretka, M., (2006b), Endogenous Market Power, Working Paper, University of Wisconsin, Madison.

19 4.1 Appendix

Proof. Theorem ?? (Convergence): Under perfect estimation the dynamics of the model is given by M T +1 = H M T . Let J ( ) be a Jacobian of H ( ) The Jacobian is I I matrix with zeros on its diagonal and ij (> 0) o¤ diagonal elements

0 12 : : : 1I 0 : : : J ( ) = 0 21 2I 1 : (7) B :::::::::::: C B C B I1 I2 ::: 0 C B C @ A For i = j; the typical element ij is given by 6 1 1 ij = @ (Mj + vj) =@Mj = (8) 00 1 1 j=i X6 2 @@ 1 A A (Mj + vj) = 2 : 1 j=i (Mj + vj) 6 P Note that ij > 0 and j i;j < 1; for any M by strict convexity of quadratic operator T T T and vj > 0. Consider arbitrary M and M and let = M M: Then P T +1 = M T +1 M = H M T M = J M~ M T M = J M~ T T ; (9) where M~ T is some point in the line segment M T between M , and third equality follows I from the Mean Value Theorem. For x = xi R+ de…ne x = maxi xi and f gi 2 k k j j I 0 B = M R+ M M < ; (10) 2 j n o and = max sup i;j (M) : (11) i M B j 2 X Since B is compact, the maximum is attained on the set and hence < 1: Then

T +1 = J M~ T T T T max i;j M~ (12) i j X T ;

20 where we used the fact that M~ 0 B and above equation and induction M~ T B for all T: 2 2 But then T T 0 ( ) ; and hence in the limit

lim T = lim M T M = 0; T T !1 !1

T and hence limT M = M . !1 Proof. Proposition ?? (Existence): The existence specializes the argument from Weretka I (2006) to a quadratic setting. Let v = vi and M = Mi : For any v R++ solution f gi f gi 2 I I M is a …xed point of a function H : R+ R+ de…ned as !

H( ) = (H1( );:::;HI ( )); (13) where each component Hi( ) is given by 1 Hi(M) = (M j + vj j = i); (14) I 1H j 6 De…ne as follows 1 max vi: (15) I 2 i I With I > 2, exists and 0 < < and let 0 M R+ v 1 where 1 is a unit 1 M f 2 j g I I I vector in R : De…ne an auxiliary function A : R+ R+ as !

A( ) = (A1( );:::;AI ( )); (16) where each component An( ; ) is given by 1 1 Ai(M) = (Mj + vj) : (17) I 1 2I 1 3 j=i X6 4 5 For any M 2 M0 1 1 I 2 Ai(M) + max vi + = ; (18) I 1 i I 1 I 1

21 where the last step holds by (15). The ith component of function H( ) (A( )) is a harmonic (arithmetic) mean of elements Mj + vj for all j = i, discounted by factor 1=(I 1) < 1. 6 By the standard harmonic-arithmetic mean inequality

Hi(M) Ai(M) ; (19) I and hence H : 0 0 : 0 is a non-empty convex and compact box in R+ and since H M !M M I is a continuous function given v R++, by the Brouwer …xed point theorem there exists M 2 such that H M = M: Nonexistence with I = 2 follows from the fact that M i = M j + vj for i; j = 1; 2 has no solution. Finally, for uniqueness, suppose there is more than one solution M 1 and M 2: Since both are …xed points

M 1 M 2 = H M 1 H M 2 = J M~ M 1 M 2 ; 1 2 But this is only possible when M = M given that for any i j i;j < 1.

Proof. Proposition 1: (Reduced Form Representation):PTo establish that p; t; M is a noncompetitive equilibrium, it su¢ ces to verify three conditions: 1) market clearing; 2) optimization; and 3) consistency of price impacts. By de…nition, ti = ti (p) where p is de…ned by market clearing price and hence ti = 0: Strategy ti ( ) for any price i equalizes marginal utility with marginal expenditure,P in particular at p and hence ti are optimal. Finally, Weretka (2006) shows that the consistency of price impacts is equivalent to M being a …xed point of the harmonic average condition and hence M are consistent and p; t; M is a noncompetitive equilibrium. Proof. Proposition ?? (Nonquadratic utilities):In the deterministic setting, ti ( ) ; is a best response to tj ( ) if it selects a point on the residual supply de…ned by tj ( ) that f gj f gj maximizes utility: ti ( ) crosses the residual supply at the trade ti for which marginal utility is equalized with marginal expenditure and hence the …rst-order condition is satis…ed. If the problem is globally convex then ti is optimal and, hence, ti ( ) is a best response. Proof. Proposition 3 (Ascending Auction): Follows directly from the ex post prop- erty of the slope-taking equilibrium.

Proof. Proposition 4 (Equivalence with LBN):Since, in the quadratic setting slope- taking equilibrium is linear, it belongs to the class of LBN. Now, we show that any LBN is

22 a slope-taking equilibrium. In LBN strategies of the traders are linear in types and prices LBN 0 e p 0 e p t (p) = c c ei c p where c ; c ; c are some constants. The strategies of other i j i i i i LBN traders tj ( ) de…ne a stochastic residual supply for i as in ?? with price impact j=i given byn o 6 1 M i = cp (20) 0 j 1 j X @ A LBN LBN ti ( ) is a best response to tj ( ) only if it is a best response to residual supply j i n LBNo with M : Since this is true for any i; tj ( ) is a slope-taking equilibrium. j n o