Supply Chain Contracting Under Competition: Bilateral Bargaining vs. Stackelberg

We analyze contracting behaviors in a two‐tier supply chain system consisting of competing manufacturers and competing retailers. We contrast the contracting outcome of a Stackelberg game, in which the manufacturers offer take‐it‐or‐leave‐it contracts to the retailers, with that of a bargaining game, in which the firms bilaterally negotiate contract terms via a process of alternating offers. The manufacturers in the Stackelberg game possess a Stackelberg‐leader advantage in that the retailers are not entitled to make counteroffers. Our analysis suggests that whether this advantage would benefit the manufacturers depends on the contractual form. With simple contracts such as wholesale‐price contracts, which generally do not allow one party to fully extract the trade surplus, the Stackelberg game replicates the boundary case of the bargaining game with the manufacturers possessing all the bargaining power. In contrast, with sophisticated contracts such as two‐part tariffs, which enable full surplus extraction, the two games lead to distinct outcomes. We further show that the game structure being Stackelberg or bargaining critically affects firms' preferences over contract types and thus their equilibrium contract choices. These observations suggest that the Stackelberg game may not be a sufficient device to predict contracting behaviors in reality where bargaining is commonly observed.     

A Dynamic Reinterpretation of Nash Bargaining With Endogenous Threats

This paper concerns the two‐stage game introduced in Nash (1953). It formalizes a suggestion made (but not pursued) by Nash regarding equilibrium selection in that game, and hence offers an arguably more solid foundation for the “Nash bargaining with endogenous threats” solution. Analogous reasoning is then applied to an infinite horizon game to provide equilibrium selection in two‐person repeated games with contracts. In this setting, issues about enforcement of threats are much less problematic than in Nash's static setting. The analysis can be extended to stochastic games with contracts.     

Level‐k Auctions: Can a Nonequilibrium Model of Strategic Thinking Explain the Winner's Curse and Overbidding in Private‐Value Auctions?

This paper proposes a structural nonequilibrium model of initial responses to incomplete‐information games based on “level‐k” thinking, which describes behavior in many experiments with complete‐information games. We derive the model's implications in first‐ and second‐price auctions with general information structures, compare them to equilibrium and Eyster and Rabin's (2005) “cursed equilibrium,” and evaluate the model's potential to explain nonequilibrium bidding in auction experiments. The level‐k model generalizes many insights from equilibrium auction theory. It allows a unified explanation of the winner's curse in common‐value auctions and overbidding in those independent‐private‐value auctions without the uniform value distributions used in most experiments.     

Rushes in Large Timing Games

We develop a continuum player timing game that subsumes standard wars of attrition and pre‐emption games, and introduces a new rushes phenomenon. Payoffs are continuous and single‐peaked functions of the stopping time and stopping quantile. We show that if payoffs are hump‐shaped in the quantile, then a sudden “rush” of players stops in any Nash or subgame perfect equilibrium. Fear relaxes the first mover advantage in pre‐emption games, asking that the least quantile beat the average; greed relaxes the last mover advantage in wars of attrition, asking just that the last quantile payoff exceed the average. With greed, play is inefficiently late: an accelerating war of attrition starting at optimal time, followed by a rush. With fear, play is inefficiently early: a slowing pre‐emption game, ending at the optimal time, preceded by a rush. The theory predicts the length, duration, and intensity of stopping, and the size and timing of rushes, and offers insights for many common timing games.     

Definable and Contractible Contracts

This paper analyzes Bayesian normal form games in which players write contracts that condition their actions on the contracts of other players. These contracts are required to be representable in a formal language. This is accomplished by constructing contracts which are definable functions of the Godel code of every other player's contract. We provide a complete characterization of the set of allocations supportable as pure‐strategy Bayesian equilibria of this contracting game. When information is complete, this characterization provides a folk theorem. In general, the set of supportable allocations is smaller than the set supportable by a centralized mechanism designer.     

Bargaining,Reputation, and Equilibrium Selection in Repeated Games with Contracts

Consider a two‐person intertemporal bargaining problem in which players choose actions and offers each period, and collect payoffs (as a function of that period's actions) while bargaining proceeds. This can alternatively be viewed as an infinitely repeated game wherein players can offer one another enforceable contracts that govern play for the rest of the game. Theory is silent with regard to how the surplus is likely to be split, because a folk theorem applies. Perturbing such a game with a rich set of behavioral types for each player yields a specific asymptotic prediction for how the surplus will be divided, as the perturbation probabilities approach zero. Behavioral types may follow nonstationary strategies and respond to the opponent's play. In equilibrium, rational players initially choose a behavioral type to imitate and a war of attrition ensues. How much should a player try to get and how should she behave while waiting for the resolution of bargaining? In both respects she should build her strategy around the advice given by the “Nash bargaining with threats” (NBWT) theory developed for two‐stage games. In any perfect Bayesian equilibrium, she can guarantee herself virtually her NBWT payoff by imitating a behavioral type with the following simple strategy: in every period, ask for (and accept nothing less than) that player's NBWT share and, while waiting for the other side to concede, take the action Nash recommends as a threat in his two‐stage game. The results suggest that there are forces at work in some dynamic games that favor certain payoffs over all others. This is in stark contrast to the classic folk theorems, to the further folk theorems established for repeated games with two‐sided reputational perturbations, and to the permissive results obtained in the literature on bargaining with payoffs as you go.     

Robust Predictions in Games With Incomplete Information

We analyze games of incomplete information and offer equilibrium predictions that are valid for, and in this sense robust to, all possible private information structures that the agents may have. The set of outcomes that can arise in equilibrium for some information structure is equal to the set of Bayes correlated equilibria. We completely characterize the set of Bayes correlated equilibria in a class of games with quadratic payoffs and normally distributed uncertainty in terms of restrictions on the first and second moments of the equilibrium action–state distribution. We derive exact bounds on how prior knowledge about the private information refines the set of equilibrium predictions. We consider information sharing among firms under demand uncertainty and find new optimal information policies via the Bayes correlated equilibria. We also reverse the perspective and investigate the identification problem under concerns for robustness to private information. The presence of private information leads to set rather than point identification of the structural parameters of the game.     

Moral Hazard Contracting and Private Credit Markets

This paper studies the impact of credit markets on optimal contracting, when the agent's “interim preference” over upcoming contracts is private information because personal financial decisions affect it via the wealth effect. The main result is a severe loss of incentive provision: equilibrium contracts invariably cause the agent to shirk (i.e., exert minimal effort) if the agent's private financial decision precedes moral hazard contracting. The basic intuition is that committing on another private variable, other than the effort level, exposes the parties to further exploitation of efficient risk‐sharing by relaxing the incentive constraint that was binding ex ante, unless the risk‐sharing was fully efficient to begin with.     

Multilateral Contracting with Externalities

This paper proposes a model for multilateral contracting, where contracts are written and renegotiated over time, and where contracts may impose externalities on other agents. Equilibria always exist and the equilibrium value function is linear and monotonically increasing on the contracts. If the grand coalition, or contracting among all agents, is inefficient, we show that bargaining delays arise in positive‐externality games and equilibrium inefficiency may remain bounded away from zero even as bargaining frictions converge to zero. Otherwise, if the grand coalition is efficient, there are no bargaining delays, convergence to the grand coalition occurs in a finite number of contracting rounds, and the outcome becomes efficient as players become more patient.     

On the Advantage of Quantity Leadership When Outsourcing Production to a Competitive Contract Manufacturer

This study investigates a supply chain comprising an original equipment manufacturer (OEM) and a contract manufacturer (CM), in which the CM acts as both upstream partner and downstream competitor to the OEM. The two parties can engage in one of three Cournot competition games: a simultaneous game, a sequential game with the OEM as the Stackelberg leader, and a sequential game with the CM as the Stackelberg leader. On the basis of these three basic games, this study investigates the two parties' Stackelberg leadership/followership decisions. When the outsourcing quantity and wholesale price are exogenously given, either party may prefer Stackelberg leadership or followership. For example, when the wholesale price or the proportion of production outsourced to the CM is lower than a threshold value, both parties prefer Stackelberg leadership and, consequently, play a simultaneous game in the consumer market. When the outsourcing quantity and wholesale price are decision variables, the competitive CM sets a wholesale price sufficiently low to allow both parties to coexist in the market, and the OEM outsources its entire production to this CM. This study also examines the impact of the supply chain parties' bargaining power on contract outcomes by considering a wholesale price that is determined via the generalized Nash bargaining scheme, finding a Stackelberg equilibrium to be sustained when the CM's degree of bargaining power is great and the non‐competitive CM's wholesale price is high.     

Isotone Equilibrium in Games of Incomplete Information

An isotone pure strategy equilibrium exists in any game of incomplete information in which each player's action set is a finite sublattice of multidimensional Euclidean space, types are multidimensional and atomless, and each player's interim expected payoff function satisfies two “nonprimitive conditions” whenever others adopt isotone pure strategies: (i) single‐crossing in own action and type and (ii) quasi‐supermodularity in own action. Conditions (i), (ii) are satisfied in supermodular and log‐supermodular games given affiliated types,and in games with independent types in which each player's ex post payoff satisfies supermodularity in own action and nondecreasing differences in own action and type. This result is applied to provide the first proof of pure strategy equilibrium existence in the uniform price auction when bidders have multi‐unit demand, nonprivate values, and independent types.     

Stable and Coordinating Contracts for a Supply Chain with Multiple Risk‐Averse Suppliers

We analyze a decentralized supply chain with a single risk‐averse retailer and multiple risk‐averse suppliers under a Conditional Value at Risk objective. We define coordinating contracts and show that the supply chain is coordinated only when the least risk‐averse agent bears the entire risk and the lowest‐cost supplier handles all production. However, due to competition, not all coordinating contracts are stable. Thus, we introduce the notion of contract core, which reflects the agents' “bargaining power” and restricts the set of coordinating contracts to a subset which is “credible.” We also study the concept of contract equilibrium, which helps to characterize contracts that are immune to opportunistic renegotiation. We show that, the concept of contract core imposes conditions on the share of profit among different agents, while the concept of contract equilibrium provide conditions on how the payment changes with the order quantity.     

Competition for a Majority

We define the class of two‐player zero‐sum games with payoffs having mild discontinuities, which in applications typically stem from how ties are resolved. For such games, we establish sufficient conditions for existence of a value of the game, maximin and minimax strategies for the players, and a Nash equilibrium. If all discontinuities favor one player, then a value exists and that player has a maximin strategy. A property called payoff approachability implies existence of an equilibrium, and that the resulting value is invariant: games with the same payoffs at points of continuity have the same value and ɛ‐equilibria. For voting games in which two candidates propose policies and a candidate wins election if a weighted majority of voters prefer his proposed policy, we provide tie‐breaking rules and assumptions about voters' preferences sufficient to imply payoff approachability. These assumptions are satisfied by generic preferences if the dimension of the space of policies exceeds the number of voters; or with no dimensional restriction, if the electorate is sufficiently large. Each Colonel Blotto game is a special case in which each candidate allocates a resource among several constituencies and a candidate gets votes from those allocated more than his opponent offers; in this case, for simple‐majority rule we prove existence of an equilibrium with zero probability of ties.     

Managing the Performance Trade‐Offs from Partner Integration: Implications of Contract Choice in R&D Projects

Formal contracts represent an important governance instrument with which firms exercise control of and compensate partners in R&D projects. The specific type of contract used, however, can vary significantly across projects. In some, firms' govern partnering relationships through fixed‐price contracts, whereas in others, firms' use more flexible time and materials or performance‐based contracts. How do these choices affect the costs and benefits that arise from greater levels of partner integration? Furthermore, how are these relationships affected when the choice of contract is misaligned with the scope and objectives of the partnering relationship? Our study addresses these questions using data from 172 R&D projects that involve partners. We find that, (i) greater partner integration is associated with higher project costs for all contract types; (ii) greater partner integration is associated with higher product quality only in projects that adopt more flexible time and materials or performance‐based contracts; and (iii) in projects where the choice of contract is misaligned with the scope and objectives of the partnering relationship, greater partner integration is associated with higher project costs, but not with higher product quality. Our results shed light on the subtle interplay between formal and relational contracting. They have important implications for practice, with respect to designing optimal governance structures in partnered R&D projects.     

Incentive Contracts in Projects with Unforeseeable Uncertainty

Designing incentive contracts that constructively guide employee efforts is a particularly difficult challenge in novel innovation initiatives, where unforeseen events may occur. Empirical studies have observed a variety of incentive structures in innovation settings: “time and material contracts” (compensation for executing orders), “downside protection” (target‐driven incentives with protection from unexpected risks), and “upside rewards” (additional remuneration for pursuing opportunities). This paper develops a model of incentives in presence of unforeseen events and offers a theoretical prediction of which of the empirically observed incentive structures should be used under which circumstances. The combination of three key influences drives the shape of the best incentive contract. First, the presence of unforeseeable uncertainty, or the occurrence of events that cannot possibly be foreseen at the outset. These may force a change in the project's plan, making pure target setting insufficient. Second, fairness concerns dictate that the employee's expected compensation cannot be shifted downward by unforeseen events, because it would cause demotivation, hostility, and defection. Third, management may not be able to observe the detailed actions of the employee (moral hazard) nor whether a positive or negative unforeseen event has occurred (asymmetric information).     

Cursed Equilibrium

There is evidence that people do not fully take into account how other people's actions depend on these other people's information. This paper defines and applies a new equilibrium concept in games with private information, cursed equilibrium, which assumes that each player correctly predicts the distribution of other players' actions, but underestimates the degree to which these actions are correlated with other players' information. We apply the concept to common‐values auctions, where cursed equilibrium captures the widely observed phenomenon of the winner's curse, and to bilateral trade, where cursedness predicts trade in adverse‐selections settings for which conventional analysis predicts no trade. We also apply cursed equilibrium to voting and signalling models. We test a single‐parameter variant of our model that embeds Bayesian Nash equilibrium as a special case and find that parameter values that correspond to cursedness fit a broad range of experimental datasets better than the parameter value that corresponds to Bayesian Nash equilibrium.     

Reputation in Continuous‐Time Games

We study reputation dynamics in continuous‐time games in which a large player (e.g., government) faces a population of small players (e.g., households) and the large player's actions are imperfectly observable. The major part of our analysis examines the case in which public signals about the large player's actions are distorted by a Brownian motion and the large player is either a normal type, who plays strategically, or a behavioral type, who is committed to playing a stationary strategy. We obtain a clean characterization of sequential equilibria using ordinary differential equations and identify general conditions for the sequential equilibrium to be unique and Markovian in the small players' posterior belief. We find that a rich equilibrium dynamics arises when the small players assign positive prior probability to the behavioral type. By contrast, when it is common knowledge that the large player is the normal type, every public equilibrium of the continuous‐time game is payoff‐equivalent to one in which a static Nash equilibrium is played after every history. Finally, we examine variations of the model with Poisson signals and multiple behavioral types.     

Who's Who in Networks. Wanted: The Key Player

Finite population noncooperative games with linear‐quadratic utilities, where each player decides how much action she exerts, can be interpreted as a network game with local payoff complementarities, together with a globally uniform payoff substitutability component and an own‐concavity effect. For these games, the Nash equilibrium action of each player is proportional to her Bonacich centrality in the network of local complementarities, thus establishing a bridge with the sociology literature on social networks. This Bonacich–Nash linkage implies that aggregate equilibrium increases with network size and density. We then analyze a policy that consists of targeting the key player, that is, the player who, once removed, leads to the optimal change in aggregate activity. We provide a geometric characterization of the key player identified with an intercentrality measure, which takes into account both a player's centrality and her contribution to the centrality of the others.     

Will More Purchasing Options Benefit Customers?

This paper considers the sale of a seasonal product in the face of strategic customers. At the beginning of the selling season, the retailer announces both the price p_h at which the product will be sold during the selling season and the post‐season clearance price p<p_h for unsold items. We analyze two operating regimes: The “no reservation regime” allows a buyer either to purchase the product at price p_h when he arrives or to enter a lottery to purchase at price p if the product remains unsold. The “reservation regime” offers each buyer one extra option than the no reservation regime: reserve the product for purchase at the clearance price p. If the buyer reserves the product under the reservation regime and if it remains unsold at the end of the selling season, then he is obligated to purchase it at price p. We consider a situation in which heterogeneous customers with probabilistic valuation arrive in accord with a Poisson process. We characterize the rational purchasing behavior wherein each arriving customer is strategic; each customer takes other customers' purchasing behavior into consideration. By considering the Nash equilibrium of this game, we show that strategic customer behavior can render the customer to be worse off and the retailer to be better off under the reservation regime, despite the fact that this regime offers one extra option (reservation) to a customer. Hence, more purchasing options do not necessarily benefit customers.     

Line Balancing in Parallel M/M/1 Lines and Loss Systems as Cooperative Games

We consider production and service systems that consist of parallel lines of two types: (i) M/M/1 lines and (ii) lines that have no buffers (loss systems). Each line is assumed to be controlled by a dedicated supervisor. The management measures the effectiveness of the supervisors by the long run expected cost of their line. Unbalanced lines cause congestion and bottlenecks, large variation in output, unnecessary wastes and, ultimately, high operating costs. Thus, the supervisors are expected to join forces and reduce the cost of the whole system by applying line‐balancing techniques, possibly combined with either strategic outsourcing or capacity reduction practices. By solving appropriate mathematical programming formulations, the policy that minimizes the long run expected cost of each of the parallel‐lines system, is identified. The next question to be asked is how to allocate the new total cost of each system among the lines' supervisors so that the cooperation's stability is preserved. For that sake, we associate a cooperative game to each system and we investigate its core. We show that the cooperative games are reducible to market games and therefore they are totally balanced, that is, their core and the core of their subgames are non‐empty. For each game a core cost allocation based on competitive equilibrium prices is identified.