
on Microeconomics 
By:  Hugo Hopenhayn; Maryam Saeedi 
Abstract:  This paper considers the design of an optimal rating system, in a market with adverse selection. We address two critical questions about rating design: First, given a number of categories, what are the criteria for setting the boundaries between them? Second, what are the gains from increasing the number of categories? A rating system helps reallocate sales from lower to higherquality producers, thus mitigating the problem of adverse selection. We focus on two main sources of market heterogeneity that determine the extent and effect of this reallocation: the distribution of firm qualities and the responsiveness of sellers' supply to prices. We provide a simple characterization for the optimal rating system as the solution to a standard kmeans clustering problem, and discuss its connection to supply elasticity and the skewness of firm qualities. Our results show that a simple twotier rating can achieve a large share of full information surplus. Additionally, we characterize the conflicting interests of consumers and producers in the design of a rating system. 
JEL:  D21 D60 D82 L11 
Date:  2019–09 
URL:  http://d.repec.org/n?u=RePEc:nbr:nberwo:26221&r=all 
By:  Ian Ball 
Abstract:  I introduce a model of scoring. A receiver tries to predict a latent characteristic of the sender. The sender has a vector of features, which can be distorted at a cost. An intermediary observes the features of the sender and assigns a linear score, which is passed to the receiver. The optimal scoring rule downweights features on which gaming is heterogeneous. Optimal scoring discourages sender manipulation and allows the receiver to make more accurate predictions than full transparency. 
Date:  2019–09 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:1909.01888&r=all 
By:  Yuval Heller (Bar Ilan University, Israel); Christoph Kuzmics (University of Graz, Austria) 
Abstract:  We define and characterize renegotiationproof equilibria of coordination games with preplay communication in which players have private preferences over the feasible coordinated outcomes. These are such that players never miscoordinate, players coordinate on their jointly preferred outcome whenever there is one, and players communicate only the ordinal part of their preferences. This set of renegotiation proof equilibrium strategies does not depend on the distribution of private preferences, and is thus robust to changes in players’ beliefs. Moreover, these equilibria are interim Pareto efficient and evolutionarily stable. 
Keywords:  Coordination games; Renegotiationproof; Equilibrium entrants; Secret handshake; Incomplete information; Evolutionary robustness 
JEL:  C72 C73 D82 
Date:  2019–09 
URL:  http://d.repec.org/n?u=RePEc:grz:wpaper:201910&r=all 
By:  Hitoshi Matsushima (University of Tokyo); Shunya Noda (University of British Columbia) 
Abstract:  We investigate the general mechanism design problems in which agents take exante hidden actions (or investments) that influence state distribution. We show that the variety of action choices drastically shrinks the set of mechanisms that induce targeted actions in the sense that there is a onetoone tradeoff between the dimensionality of action space and that of payment rules with which the targeted action profile is taken in an equilibrium. This result comprehends the observations made by previous works. When agents can take unilateral deviations to change the state distribution in various directions, we have equivalence properties with respect to expost payoffs, payments, and revenues. In particular, the pureVCG mechanism, the simplest form of the canonical VCG mechanism, becomes the only mechanism that induces an efficient action profile. Contrarily, the popular pivot mechanism generically fails to induce efficient actions, even when the action space is onedimensional. 
Date:  2019–09 
URL:  http://d.repec.org/n?u=RePEc:cfi:fseres:cf464&r=all 
By:  Dinard van der Laan; Zaifu Yang 
Abstract:  A seller has several heterogeneous indivisible items like tickets to sell over time before a deadline. These items become worthless after the deadline. Buyers arrive sequentially and randomly and have their own private valuations over items. Each buyer may acquire more than one item. We formulate this as an incentive compatible revenue maximization problem and characterize optimal allocation policies and derive various properties. 
Keywords:  Revenue Maximization, Random and Sequential Assignment, Heterogeneity 
JEL:  C61 D21 D82 
Date:  2019–09 
URL:  http://d.repec.org/n?u=RePEc:yor:yorken:19/13&r=all 
By:  Seungjin Han 
Abstract:  This paper studies the class of robust equilibria in a general competing mechanism game for decentralized markets with frictions in which nondeviating sellers punish a deviator with dominant strategy incentive compatible (DIC) direct mechanisms. Given onedimensional, independent, and private types, the lower bound of a seller's payoff in such equilibria is his minmax value over all DIC direct mechanisms if a seller can deviate to a contract that determines a menu of any complex mechanisms conditional on buyers' messages and he chooses a mechanism he wants from it. In applications, the number of sellers is endogenized given a number of buyers and fixed entry costs. As the number of buyer increases, a unique equilibrium emerges and the equilibrium ratio of buyers to sellers converges to the point where a seller's net profit is zero with the monopoly terms of trade. 
Keywords:  competing mechanisms, dominant strategy incentive compatibility, frictions, implicit collusion 
JEL:  C72 D82 
Date:  2019–09 
URL:  http://d.repec.org/n?u=RePEc:mcm:deptwp:201909&r=all 
By:  Silvia Albrizio (Yonsei University) 
Abstract:  In this paper, we study decision making and games with vector outcomes. We provide a general framework where outcomes lie in a real topological vector space and the decision makerâ€™s preferences over outcomes are described by a preference cone, which is defined as a convex cone satisfying a continuity axiom. Further, we define a notion of utility representation and introduce a duality between outcomes and utilities. We provide conditions under which a preference cone is represented by a utility and is the dual of a set of utilities. We formulate a decisionmaking problem with vector outcomes and study optimal choices. We also consider games with vector outcomes and characterize the set of equilibria. Lastly, we discuss the problem of equilibrium selection based on our characterization. 
Keywords:  Decision making, Duality, Games, Incomplete preferences, Utility representation, Vector outcomes. 
JEL:  C02 C72 D01 
Date:  2019–09 
URL:  http://d.repec.org/n?u=RePEc:yon:wpaper:2019rwp146&r=all 
By:  Azad Gholami, Reza (Dept. of Business and Management Science, Norwegian School of Economics); Sandal, Leif K. (Dept. of Business and Management Science, Norwegian School of Economics); Ubøe, Jan (Dept. of Business and Management Science, Norwegian School of Economics) 
Abstract:  Supply channels typically face uncertain and timevarying demand. Nonetheless, timedependent channel optimization while addressing uncertain demand has received limited attention due to the high level of complexity of the ensuing nested equilibrium problems. The level of complexity rises when demand is dependent on current and previous prices. We consider a decentralized supply channel whose two members, a manufacturer and a retailer, must address the demand for a perishable commodity within a multiperiod time horizon. Using a general (additivemultiplicative) stochastic model for the pricedependent demand, the purpose of this paper is to provide the channel members with analytic tools to devise optimal pricing and supply strategies at different times. In the first part of the paper, we propose a constructive theorem providing an explicit solution algorithm to obtain equilibrium states for bilevel optimization in decentralized supply channels. We also prove that the resulting equilibria are subgame perfect. In the second part, we allow the retailer to postpone her supply and pricing decisions until demand uncertainty is resolved at each period. Using subgame perfectness of the equilibria, we propose solution algorithms that use the extra information obtained by postponement. Finally, in a number of comparison theorems, we show that postponement strategies are always beneficial for a centralized channel (whose revenue structure is identical to that of a retailer). Whereas for a decentralized channel, due to vertical competitions, there may be scenarios wherein postponement strategies, i.e. access to extra information, turn out to be detrimental to the manufacturer and even to the whole channel. 
Keywords:  Stochastic optimization; bilevel programming; game theory; pricing theory; stochastic demand; timedependent demand; pricedependent demand 
JEL:  C61 C73 D81 
Date:  2019–09–09 
URL:  http://d.repec.org/n?u=RePEc:hhs:nhhfms:2019_009&r=all 
By:  Azad Gholami, Reza (Dept. of Business and Management Science, Norwegian School of Economics); Sandal, Leif K. (Dept. of Business and Management Science, Norwegian School of Economics); Ubøe, Jan (Dept. of Business and Management Science, Norwegian School of Economics) 
Abstract:  We analyze the problem of timedependent channel coordination in the face of uncertain demand. The channel, composed of a manufacturer and a retailer, is to address a timevarying and uncertain pricedependent demand. The decision variables of the manufacturer are wholesale and (possibly zero) buyback prices, and those of the retailer are order quantity and retail price. Moreover, at each period, the retailer is allowed to postpone her retail price until demand uncertainty is resolved. In order to place emphasis on the pricedecadent nature of demand, we embed a class of memory effects in demand structure, such that current demand at each period demand is affected by pricing history as well as current price. The ensuing equilibria problems, thus, become highly nested in time. We then propose our memorybased solution algorithm which coordinates the channel with optimal buyback contracts at each period. We show that, contrary to the conventional belief, too generous buyback prices may not only be suboptimal to the manufacturer, but also decrease the expected profit for the retailer and thus for the whole channel. 
Keywords:  Stochastic optimization; bilevel programming; game theory; channel coordination; buyback contracts; price postponement; pricing theory; contract theory 
JEL:  C61 C73 D81 
Date:  2019–09–09 
URL:  http://d.repec.org/n?u=RePEc:hhs:nhhfms:2019_010&r=all 
By:  Azad Gholami, Reza (Dept. of Business and Management Science, Norwegian School of Economics); Sandal, Leif K. (Dept. of Business and Management Science, Norwegian School of Economics); Ubøe, Jan (Dept. of Business and Management Science, Norwegian School of Economics) 
Abstract:  Almost every vendor faces uncertain and timevarying demand. Inventory level and price optimization while catering to stochastic demand are conventionally formulated as variants of newsvendor problem. Despite its ubiquity in potential applications, the timedependent (multiperiod) newsvendor problem in its general form has received limited attention in the literature due to its complexity and the highly nested structure of its ensuing optimization problems. The complexity level rises even more when there are more than one decision maker in a supply channel, trying to reach an equilibrium. The purpose of this paper is to construct an explicit and e cient solution procedure for multiperiod pricesetting newsvendor problems in a Stackelberg framework. In particular, we show that our recursive solution algorithm can be applied to standard contracts such as buy back contracts, revenue sharing contracts, and their generalizations. 
Keywords:  Stochastic demand; timedependent demand; pricedependent demand; memory functions; market engineering; demand manipulation; prescriptive analytics; pricing theory 
JEL:  C61 C73 D81 
Date:  2019–09–09 
URL:  http://d.repec.org/n?u=RePEc:hhs:nhhfms:2019_008&r=all 
By:  James Albrecht; Xiaoming Cai; Pieter A. Gautier; Susan Vroman 
Abstract:  We consider a labor market with search frictions in which workers make multiple applications and firms can post and commit to general mechanisms that may be conditioned both on the number of applications received and on the number of offers received by its candidate. When the contract space includes application fees, there exists a continuum of equilibria of which only one is socially efficient. In the inefficient equilibria, firms have market power that arises from the fact that the value of a worker’s application portfolio depends on what other firms offer, which allows individual firms to free ride and offer workers less than their marginal contribution. Finally, by allowing for general mechanisms, we are able to examine the sources of inefficiency in the multiple applications literature. 
Keywords:  multiple applications, directed search, competing mechanisms, efficiency, market power 
JEL:  C78 D44 D83 
Date:  2019 
URL:  http://d.repec.org/n?u=RePEc:ces:ceswps:_7805&r=all 
By:  Kerem Ugurlu 
Abstract:  We give explicit solutions for utility maximization of terminal wealth problem $u(X_T)$ in the presence of Knightian uncertainty in continuous time $[0,T]$ in a complete market. We assume there is uncertainty on both drift and volatility of the underlying stocks, which induce nonequivalent measures on canonical space of continuous paths $\O$. We take that the uncertainty set resides in compact sets that are time dependent. In this framework, we solve the robust optimization problem with logarithmic, power and exponential utility functions, explicitly. 
Date:  2019–09 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:1909.05335&r=all 
By:  Tommaso Lando; Lucio BertoliBarsotti 
Abstract:  We study a generalized family of stochastic orders, semiparametrized by a distortion function H, namely Hdistorted stochastic dominance, which may determine a continuum of dominance relations from the first to the secondorder stochastic dominance (and beyond). Such a family is especially suitable for representing a decision maker's preferences in terms of risk aversion and may be used in those situations in which a strong order does not have enough discriminative power, whilst a weaker one is poorly representative of some classes of decision makers. In particular, we focus on the class of power distortion functions, yielding powerdistorted stochastic dominance, which seems to be particularly appealing owing to its computational simplicity and some interesting statistical interpretations. Finally, we characterize distorted stochastic dominance in terms of distortion functions yielding isotonic classes of distorted expectations. 
Date:  2019–09 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:1909.04767&r=all 
By:  J\"orn Sass; Dorothee Westphal 
Abstract:  In this paper we investigate a utility maximization problem with drift uncertainty in a continuoustime BlackScholes type financial market. We impose a constraint on the admissible strategies that prevents a pure bond investment and we include uncertainty by means of ellipsoidal uncertainty sets for the drift. Our main results consist in finding an explicit representation of the optimal strategy and the worstcase parameter and proving a minimax theorem that connects our robust utility maximization problem with the corresponding dual problem. Moreover, we show that, as the degree of model uncertainty increases, the optimal strategy converges to a generalized uniform diversification strategy. 
Date:  2019–09 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:1909.01830&r=all 
By:  Nasimeh Heydaribeni; Achilleas Anastasopoulos 
Abstract:  We consider a nonzerosum linear quadratic Gaussian (LQG) dynamic game with asymmetric information. Each player observes privately a noisy version of a (hidden) state of the world $V$, resulting in dependent private observations. We study perfect Bayesian equilibria (PBE) for this game with equilibrium strategies that are linear in players' private estimates of $V$. The main difficulty arises from the fact that players need to construct estimates on other players' estimate on $V$, which in turn would imply that an infinite hierarchy of estimates on estimates needs to be constructed, rendering the problem unsolvable. We show that this is not the case: each player's estimate on other players' estimates on $V$ can be summarized into her own estimate on $V$ and some appropriately defined public information. Based on this finding we characterize the PBE through a backward/forward algorithm akin to dynamic programming for the standard LQG control problem. Unlike the standard LQG problem, however, Kalman filter covariance matrices, as well as some other required quantities, are observationdependent and thus cannot be evaluated offline through a forward recursion. 
Date:  2019–09 
URL:  http://d.repec.org/n?u=RePEc:arx:papers:1909.04834&r=all 
By:  Tetsuya Shinkai (School of Economics, Kwansei Gakuin University); Ryoma Kitamura (Faculty of Economics, Otemon Gakuin University) 
Abstract:  We consider the product line strategies of duopolistic firms, each of which can supply two verticallydifferentiated products under nonnegative output constraints and expectations of their rival's product line reaction. Considering a game of firms with heterogeneous (homogeneous) unit costs for high (low) quality products, we derive the equilibria for the game and conduct comparative statics of the equilibria outcomes on the relative superiority of the highquality product and relative cost efficiency. In two of the equilibria, we find that where the costinefficient firm supplies a highquality good, social welfare can worsen as its unit cost decreases. We also characterize the result using the production substitution of differentiated goods within a firm and the highquality good between firms. Further, by comparing social welfare in the firstbest equilibria with those in the Cournot duopoly equilibria, we find that the social welfare of the market worsens in the multiproduct Cournot duopoly equilibria as the relative superiority of the highquality good increases. 
Keywords:  Multiproduct firm; Duopoly; Production substitution; Vertical product differentiation 
JEL:  D21 D43 L13 L15 
Date:  2019–09 
URL:  http://d.repec.org/n?u=RePEc:kgu:wpaper:197&r=all 
By:  Satoru Fujishige; Zaifu Yang 
Abstract:  We study a general barter market in which every agent is initially endowed with several inherently indivisible items and wishes to exchange with other agents. There is no medium of exchange like money. Agents have general preferences over bundles of items and may acquire several items. It is wellknown that the core of such an economy is typically empty. We propose a new but more general notion of core called a Markovian core. A Markovian core allocation is individually rational, but Paretoefficient and stable against any possible coalition deviation by comparison with their current assignments instead of their initial endowments. We show that the market has always a nonempty strict Markovian core through a decentralized Paretoimprovement process. 
Keywords:  Decentralized market, barter market, indivisibility, efficiency, stability, Markovian core. 
JEL:  C62 D72 
Date:  2019–09 
URL:  http://d.repec.org/n?u=RePEc:yor:yorken:19/11&r=all 