Showing papers on "Stackelberg competition published in 1997"

Achieving network optima using Stackelberg routing strategies

Abstract: In noncooperative networks users make control decisions that optimize their individual performance objectives. Nash equilibria characterize the operating points of such networks. Nash equilibria are generically inefficient and exhibit suboptimal network performance. Focusing on routing, a methodology is devised for overcoming this deficiency, through the intervention of the network manager. The manager controls part of the network flow, is aware of the noncooperative behavior of the users and performs its routing aiming at improving the overall system performance. The existence of maximally efficient strategies for the manager, i.e., strategies that drive the system into the global network optimum, is investigated. A maximally efficient strategy of the manager not only optimizes the overall performance of the network, but also induces an operating point that is efficient with respect to the performance of the individual users (Pareto efficiency). Necessary and sufficient conditions for the existence of a maximally efficient strategy are derived, and it is shown that they are met in many cases of practical interest. The maximally efficient strategy is shown to be unique and it is specified explicitly.

A stochastic version of a Stackelberg-Nash-Cournot equilibrium model

Abstract: We consider a stochastic version of the Stackelberg-Nash-Cournot model proposed by Murphy et al. (1983). Ln the first stage, the leader chooses and announces his production level taking into account the reaction of the followers. The decision of the leader is taken when market demand is uncertain. In the second stage, the followers, knowing the leader's output, react to this level according to the Cournot assumption. At this stage, demand is known. We study the extension of the Murphy et al. model and give a numerical illustration of this model using the European gas market.

A study of optimal design under conflict using models of multi-player games

Abstract: In the broad context of modelling for system design, it is normally assumed that all decision-makers cooperate fully and thus avoid conflict. However, this is not always possible, in which case the design process is best modelled and studied as a multi-player game. The objective in this paper is to present and illustrate such a general game-theoretic framework for design modelling. In particular, explicit solutions are computed and interpreted for conservative or minmax, Pareto, Nash and Stackelberg games in a simple pressure vessel design example in which two players interact strategically, one of whom wants to minimize the weight and other wishes to maximize the volume. In another example in which a rotating disk is to be optimized for two objectives related to its design and manufacture, the Pareto solutions show how a novel manufacturing cost function can improve, under assumptions of a cooperative game, the optimal shape that was obtained under a single objective formulation related to design alone.

Interactive bilevel multi-objective decision making

Abstract: From a perspective of non-linear bilevel multi-objective decision-making problem (BMDMP) of the leader-follower Stackelberg game, this paper presents a two-person bilevel multi-objective decision-making model and an interactive algorithm for solving the problem. The algorithm simplifies a BMDMP by transforming it into separate multi-objective decision-making problems at the upper- and lower-levels, thereby avoiding the difficulty associated with non-convex mathematical programming to arrive at an optimal solution. In addition, the authors put forward the satisfactoriness concept as the upper level decision maker's preference. The algorithm facilitates the interactive nature of bilevel multi-objective decision making for non-inferior solutions. Thus, the algorithm provides a way to solve BMDMP.

Payoff dominance and the stackelberg heuristic

Abstract: Payoff dominance, a criterion for choosing between equilibrium points in games, is intuitively compelling, especially in matching games and other games of common interests, but it has not been justified from standard game-theoretic rationality assumptions. A psychological explanation of it is offered in terms of a form of reasoning that we call the Stackelberg heuristic in which players assume that their strategic thinking will be anticipated by their co-player(s). Two-person games are called Stackelberg-soluble if the players' strategies that maximize against their co-players' best replies intersect in a Nash equilibrium. Proofs are given that every game of common interests is Stackelberg-soluble, that a Stackelberg solution is always a payoff-dominant outcome, and that in every game with multiple Nash equilibria a Stackelberg solution is a payoff-dominant equilibrium point. It is argued that the Stackelberg heuristic may be justified by evidentialist reasoning.

Endogenous Stackelberg equilibria with incomplete information

Abstract: In this paper, the endogenous order of quantity decision is studied in a duopoly model with incomplete information. One firm knows the state of the demand curve while the other firm remains uninformed. Firms have to commit to a quantity in one out of two periods. While, a priori, simultaneous-move Cournot equilibria are possible, only Stackelberg equilibria, with either the informed or the uninformed firm moving first, emerge endogenously.

Price or Quantity Competition? Oligopolistic Structures in International Commodity Markets

Abstract: International commodity markets may be characterized by price or quantity competition and by product differentiation. As an illustration, this paper presents a set of models of the Japanese market for imported beef. The models are evaluated using a non-nested econometric test. The one which best fits the data is a Stackelberg model with price leadership by Australia. This result provides evidence on the explicit nature of the game being played by exporters, unlike the applied conjectural variations approach which provides only an index of how competitive the market is.

Salience and focusing in pure coordination games

Abstract: This article is devoted to explaining why decision makers choose salient equilibria or focal points in pure coordination games - games in which players have identical preferences over the set of possible outcomes. Focal points, even when they arise as framing effects based on the labelling of options, are intuitively obvious choices, and experimental evidence shows that decision makers often coordinate successfully by choosing them. In response to arguments that focusing is not rationally justified, a psychological explanation and a conditional justification is offered in terms of a form of reasoning called the Stackelberg heuristic that has been used to explain the selection of payoff-dominant (Pareto-optimal) equilibria in common-interest games. Pure coordination games, if appropriately modelled, are shown to be reducible to common-interest games with payoff-dominant equilibria, and it is argued that focusing can therefore be explained by the Stackelberg heuristic.

Stackelberg leadership and transfers in private provision of public goods

Abstract: We consider transfers in a Stackelberg game of private provision of a public good. It turns out that the agent who is the follower in the process of making voluntary contributions to a public good may have an incentive to make monetary transfers to the Stackelberg leader even in a situation where neither has a comparative advantage in making contributions to the public good. The Stackelberg leader is willing to accept such transfers if the actual contribution game is fully non-cooperative because the transfer generates a Pareto superior outcome. If the contributions in the Stackelberg equilibrium is the threat point of a possible cooperative Nash bargaining game, the Stackelberg leader will refuse to accept the transfer if she can.

On "reputation" refinements with heterogeneous beliefs

Abstract: CONSIDER A REPEATED GAME with incomplete information in which a patient long run player, whose type is unknown, faces a sequence of short run opponents (as in Fudenberg and Levine (1989)). The standard "reputation" result is that the patient long run player can obtain an average long payoff almost equal to the Stackelberg payoff of the stage game by consistently playing as a Stackelberg leader. In the analysis, one generally takes as fundamental an assumption that the players have common prior beliefs on the states of the world and that behavior is consistent with the concept of Bayesian Nash equilibrium. However, these related suppositions have been called into question as unrealistic and too stringent (cf. Gul (1991)). In many games of incomplete information a player's prior probabilities on types (or states of the world) are best regarded as purely subjective psychological parameters, unknown to the modeler and to this player's opponents. Therefore, it is important to understand whether the standard reputation results (among others) are implied by weaker assumptions on the knowledge and behavior of the players. In fact, as Watson (1993) demonstrates, the reputation result does not require equilibrium. It is implied by a weak notion of rationalizability with some restrictions on the beliefs of the players. Here we qualify Watson's (1993) study and extend the line of inquiry of Watson (1993) and Battigalli (1994) concerning settings in which reputations are effective. As Watson shows, two main conditions on the beliefs of the players, along with weak rationalizability, imply the reputation result. First, there must be a strictly positive and uniform lower bound on the subjective probability that players assign to the "Stackelberg type." Second, the conditional beliefs of the short run players must not be too dispersed. Watson (1993) does not explicitly indicate on what the updated beliefs of the short run players are conditioned. We make this explicit and show that it is necessary to assume that the conditional beliefs of the short run players satisfy a stochastic independence property (cf. Battigalli (1996)). We also comment on the dispensability of equilibrium regarding the reputation result in games with two long run players.

Loss of Commitment? An Evolutionary Analysis of Bagwell’s Example

Abstract: In a recent paper Bagwell (1995) pointed out that only the Cournot outcome, but not the Stackelberg outcome, can be supported by a pure Nash equilibrium when actions of the Stackelberg leader are observed with the slightest error. The Stackelberg outcome, however, remains close to the outcome of a mixed equilibrium. We compare the predictions in various classes of evolutionary and learning processes in this game. Only the continuous best response dynamic uniquely selects the Stackelberg outcome under noise. All other dynamics analyzed allow for the Cournot equilibrium to be selected. In typical cases Cournot is the unique long run outcome even for vanishing noise in the signal.

Traffic management through link tolls - An approach utilizing side constrained traffic equilibrium models

Abstract: We propose a systematic means for achieving a set of overall traffic management or planning goals with respect to the performance of the traffic network through the use of link tolls. The primary goals are defined by a set of link flow restrictions. The tolls that achieve these goals are obtained by solving a generalization of the classical user equilibrium model which includes a set of side constraints on the link flows. The set of toll prices obtained is not necessarily unique; this fact enables the traffic planner to choose a toll scheme which satisfies exogenous constraints and which may optimize a secondary goal, such as with respect to the toll itself. The overall model is derived as a special case of a mathematical program with equilibrium constraints (MPEC) describing a Stackelberg game involving the traffic manager and the users of the network. The model is shown to yield valuable information also in the case where the management goals and exogenous toll constraints are inconsistent with each other or with the underlying network. We give several examples of possible applications of the model, including the achievement of a system optimal flow and the derivation of actions for making public transport more attractive, and propose a conceptual algorithm for solving it. The paper is hoped to provoke continued research in both theoretical and applied directions.

A Stochastic Version of a Stackelberg-Nash-Cournot

Abstract: We consider a stochastic version of the Stackelberg-Nash-Cournot model proposed by Murphy et al. (1983). In the first stage, the leader chooses and announces his production level taking into account the reaction of the followers. The decision of the leader is taken when market demand is uncertain. In the second stage, the followers, knowing the leader’s output, react to this level according to the Cournot assumption. At this stage, demand is known. We study the extension of the Murphy et al. model and give a numerical illustration of this model using the European gas market. 1I ntroduction The supply side of an oligopolistic market is modeled as follows. There are N +1 firms which supply a homogeneous product in a noncooperative manner. One of the firms, hereafter named the leader, has to decide now what its future output will be. We can imagine that this firm has not yet installed production capacities. We suppose that this firm chooses its future output level taking into account the reaction of the other producers. The others firms, hereafter named the followers, then react after knowing the leader’s output level. We assume that the followers arrive at a Nash Equilibrium, i.e. each firm chooses its output level maximizing its profit under the assumption that the other firms will hold their production at the

A Graphical Analysis of the Cournot-Nash and Stackelberg Models.

Abstract: (1997). A Graphical Analysis of the Cournot-Nash and Stackelberg Models. The Journal of Economic Education: Vol. 28, No. 1, pp. 48-57.

Discrete time Fuggati equations in open loop Nash and Stackelberg games

Abstract: We study the asymptotic behavior of difference equations appearing in the necessary optimality conditions of noncooperative open loop Nash and Stackelberg games. Moreover we study also the properties of the solutions of the corresponding algebraic Riccati-type equations.

A Dynamic Stackelberg Model with Production-adjustment Costs

Abstract: We investigate the (dynamic) stability of a stackelberg oligopoly model of a market of a homogeneous good, with output competition, one Stackelberg leader and a number of identical followers. We assume that each firm incurs quadratic production-adjustment costs if it changes its output. We present a simple necessary and sufficient condition for stability of the model. Using the condition, we compare the stability of this model with the stability of two related Cournot models in which all firms present are followers. It turns out that the Stackelberg model is “more stable” than these two Cournot models.

Leader-follower dynamic game of new product diffusion

Abstract: The objective of this paper is to study optimal pricing strategies in a duopoly, under an asymmetric information structure, where the appropriate solution concept is the feedback Stackelberg equilibrium. In order to take into account effects such as imitation (e.g., word of mouth) and saturation, the demand (state equation) is assumed to depend on past cumulative sales, market potential, and both players' prices. We assume also that the unit production cost decreases with cumulative production (learning effects). Each player maximizes his total discounted profit over the planning horizon. The problem is formulated as a two-player discrete-time finite-horizon game. Existence results are first obtained under rather mild conditions. Since the solution of this problem is intractable by analytical methods, we use a numerical approach. Thus, we design a numerical algorithm for the computation of feedback Stackelberg equilibria and use it to obtain strategies in various representative cases. The numerical results presented are intented to give some insights into the optimal pricing strategies in the context of an asymmetrical feedback information structure.

Conflict and cooperation in designing international telecommunication networks

Abstract: In this paper, we study the problem of designing an international telecommunication network when multiple countries are involved. Each Country has the option of cooperating in the design of the overall international network, or simply forming bilateral arrangements with individual countries. Most countries will gain by having a truly international network which allows routing via transit countries; this takes advantage of distributed peak traffic hours, and improves the network resource utilization. However there are many situations when countries may obtain greater gains by either behaving pre-emptively or by waiting for the major countries to make their decisions, and then being followers in a Stackelberg game. Using mathematical programming models, we examine conditions under which each of these behaviours are optimal.

A Note on Regulating Oligopolistic Industries: A Hierarchical Model.

Abstract: In a 1994 paper, extending the well-known incremental surplus subsidy scheme of Sappington and Sibley, Schwermer presented a non-Bayesian incentive scheme for regulating a Cournot oligopoly industry. This note designs an oligopolistic incentive scheme for a hierarchical Stackelberg model in which firms choose outputs sequentially.

A Two-Stage Cournot Duopoly with Inventory Costs

Abstract: This paper analyses the Cournot duopoly model which has two production periods before the market clears. As shown by Saloner (1987), if inventory costs are zero, many outcomes including both Cournot and Stackelberg outcomes are subgame perfect equilibrium outcomes. However, if small inventory costs are introduced, the Cournot outcome is no longer found in equilibrium and the equilibrium outcomes are only of Stackelberg-type. This suggests that more attention should be paid to the Stackelberg model than to the Cournot model.

Comparison of different Stackelberg solutions in a deterministic dynamic pollution control

Abstract: We study different dynamic Stackelberg solutions within a pollution control problem framework. This study is made under the assumption of different information structures, mainly we assume open-loop, feedback and closed-loop structures of information. Some of the numerical results may appear counterintuitive. Hence, there exists some situations where the time consistent solution is optimal in comparison of the time inconsistent one. Moreover, the perfect discretionary solution is advantageous for everyone then to stay committed to the initial one.

Asymmetric multistage models of R&D : technology adoption, contracts and protection

Abstract: This thesis consists of three individual models on technology adoption, contracts and protection. The first model is motivated by the inconsistency between empirical results and theoretical models regarding the firm size effects upon the timing of adoption. By proposing a two-stage, endogenous learning, Stackelberg model, we conclude that in a pure strategy equilibrium, the large firm may or may not tacitly delay its adoption to capture the information advantage, depending on cost and belief parameters. The welfare analysis provides a justification for government interventions in firms’ adoption decisions. The second model is motivated by the fact that although more and more resources have been devoted to R&D activities, there is little theoretical discussion regarding R&D funding issues. Chapter 3 derives the optimal funding contract, which happens to be a cost-plus-fixed-fee contract in the literature. After considering the adverse selection problem, the optimal contract induces no efficiency loss under both discrete and continuous settings and the principal will be more conservative in funding. The optimal auction maintains both allocation and production efficiency, and bidding the principal’s reservation price will be a dominant strategy in a second price auction. Neither the revenue equivalence nor the separation property will hold. With symmetric beliefs, the optimal funding length is shorter than that of contractible effort. Under some assumptions, the lock-in effect persists and the principal will prefer short-term contracts to long-term contracts. The third model decides the optimal protection forms, protection rates and protection lengths under various cost and revenue circumstances. Since the incentive scheme will be affected by the target firm’s future profits, we show that in the context of incomplete information, screening protection schemes can sometimes coincide with the efficient schemes. In R&D area, our result suggests that optimal patent length need not necessarily be increasing in firm’s investment efficiency.

On the Unique D1 Equilibrium in the Stackelberg Model with Asymmetric Information

Abstract: This note studies a version of the Stackelberg model in which the Leader has more information about demand than the Follower. We show that there exists a unique D1 equilibrium and that this equilibrium is perfectly revealing. We also give a full characterization of the equilibrium in terms of the posterior beliefs of the Follower and show under which condition there is first mover disadvantage.

Existence of a feedback equilibrium for two-stage Stackelberg games

Abstract: In this paper we consider a two-stage feedback game in which two players solve a Stackelberg problem at each stage and each player knows the state of the game at every level of play. In this kind of game, the leader does not have the ability to announce his strategy at all levels of play prior to the start of the game. Without assuming that at each stage the Stackelberg problem has a unique solution, we define a concept of "feedback Stackelberg solution" and give sufficient conditions to get existence of such a solution.

Diagrammatic demonstration of the cournot equilibrium

Abstract: This note further develops Ikema's (1990, 1991) diagrammatic demonstrati on of the Cournot-oligopoly equilibrium on the price- quantity plane. It is shown that Ikema's method can be used to illustrate the Stackelberg and Bertrand equilibria on the price- quantity plane. Furthermore, we algebraically derive the ``monopoly- equilibrium'' curve and show that the Cournot- oligopoly equilibrium is stable if the monopoly-equilibrium curves for all producers are upward sloping.