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 mar- ket. 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 im- pacts, 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 inform- ation 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 or- ders. 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 quant- ities 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 de- terministic 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 generic- ally 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 equi- valent 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 Pquasilinear 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 mar- ket 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 non- quadratic 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.