Non-cooperative Equilibrium Research Articles

Non-Cooperative Equilibrium with Multiple Deviators

The paper suggests a non-cooperative simultaneous game, with a number of potential deviators is a parameter of the game. A definition of the game embeds mechanism design. The game has an equilibrium in mixed strategies. The equilibrium encompasses intra and inter group externalities and individual payoffs that make it different from a strong Nash, coalition-proof equilibrium and some other equilibrium concepts. We offer a non-cooperative stability criterion to describe a robustness of an equilibrium strategy profile to an increase in a number of deviators. The criterium may serve as a way to measure trust for the equilibrium in terms of a number of potential deviators.

Cooperative and Noncooperative Extraction in a Common Pool with Habit Formation

The present paper considers the exploitation of a common-property, nonrenewable resource, by individuals subject to habit formation. We formalize their behavior by means of a utility function, depending on the difference between the individuals’ current consumption and the consumption level which they aspire, the latter being a weighted average of past consumptions in the population. We derive and compare the benchmark cooperative solution and a noncooperative Markov-perfect Nash equilibrium of the differential game. We investigate how the intensity, persistence and initial level of habits shape the cooperative and noncooperative solutions. We prove that habit formation may either mitigate or worsen the tragedy of the commons.

On the Efficiency of Destination and Origin Commodity Taxation in the Presence of Consumption Generated Cross-Border Pollution

We combine consumption generated cross-border pollution and public pollution abatement in a theoretical framework to examine the efficiency of the destination and origin principles of commodity taxation. The key result of our study is that with public pollution abatement, the non-cooperative equilibrium origin-based consumption taxes are set efficiently. This result holds not only in the context of symmetric countries, but under certain conditions carry over also to asymmetric countries. The non-cooperative equilibrium destination-based consumption taxes are inefficient. In the absence of public pollution abatement, the Nash equilibrium commodity taxes under either taxation principle are inefficient.

Expected Worth for 2 2 Matrix Games with Variable Grid Sizes

We offer a detailed examination of a broad class of 2 × 2 matrix games as a first step toward considering measures of resource distribution and efficiency of outcomes. In the present essay, only noncooperative equilibria and entropic outcomes are considered, and a crude measure of efficiency employed. Other solution concepts and the formal construction of an efficiency index will be addressed in a companion paper.

The effects of green policy on the performance of green supply chains

This paper focuses on the performance of green supply chains and adding the green concept to it so as to emphasise the importance of environmental issues and the need of eco-friendly systems. The purpose of this study is to propose a model for evaluating the effects of green policy on the performance of green supply chains. We show that supply chain collaboration is benefit enhancing when the levels of green damage caused by pollution are low. However, when the level of damage is high, it turns out that suppliers anticipate tougher green policy when they share information than when they do not, and so do not share information. This distorts the R&D decisions in ways that make the non-cooperative equilibrium benefit enhancing.

Strategic risk in supply chain contract design

Supply chains facing asymmetric information can either operate in a cooperative mode with information and benefit sharing or can choose a non-cooperative form of interaction and align their incentives via screening contracts. In the cooperative mode, supply chain efficiency can be achieved, but high levels of trust and trustworthiness are required. In the non-cooperative mode, the contract mechanism guarantees a second best supply chain performance, but only if all parties choose their equilibrium strategies without trembles. Experimental evidence, however, shows that both operating modes often fail due to strategic risk. Cooperation is disrupted by deceptive signals and the lack of trust, whereas non-cooperative strategies suffer from persistent out-of-equilibrium behavior. We present two means to reduce strategic risk. First, a punishment mechanism leads to a better matching of trust and trustworthiness and supports the cooperative operating mode. Second, an enforcement of self-selection supports the non-cooperative equilibrium by increasing the attractiveness of screening contracts. We find that supply chain performance can benefit from reduced strategic risk in either operating mode.

Game Theoretic Design of MAC Protocols: Pricing Versus Intervention

In many wireless communication networks a common channel is shared by multiple users who must compete to gain access to it. The operation of the network by self-interested and strategic users usually leads to the overuse of the channel resources and to substantial inefficiencies. Hence, incentive schemes are needed to overcome the inefficiencies of non-cooperative equilibrium. In this work, we consider a slotted-Aloha-like random access protocol and two incentive schemes: pricing and intervention . We provide some criteria for the designer of the protocol to choose one scheme between them and to design the best policy for the selected scheme, depending on the system parameters. Our results show that intervention can achieve the maximum efficiency in the perfect monitoring scenario. In the imperfect monitoring scenario, instead, the performance of the system depends on the beliefs of the different entities and, in some cases, there exists a threshold for the number of users such that, for a number of users lower than the threshold, intervention outperforms pricing, whereas for a number of users higher than the threshold pricing outperforms intervention.

Decomposition algorithm of searching equilibria in a dynamic game

This paper considers a noncooperative game of several players (governments of neighboring countries) participating in emission reduction trading. A special emphasis is placed on the case of two players, one representing Eastern European countries and the other being the countries of the former Soviet Union. We perform statistical analysis of the model parameters based on real data under quadratic cost functions and logarithmic utility functions. The concepts of a noncooperative Nash equilibrium and cooperative Pareto maxima are introduced and connections between them are established. A new concept, i.e., a market equilibrium, which combines the properties of Nash and Pareto equilibria, is rigorously defined. An analytical solution of the market equilibrium problem is given. This analytical solution can serve for verification of numerical search algorithms. In addition, we propose a computational algorithm of market equilibrium search, which shifts a competitive Nash equilibrium to a cooperative Pareto maximum. The algorithm is interpreted as a repeated auction, where the auctioneer possesses no information about the cost functions and the functions of environmental effect from emission reduction of the participating countries. An auctioneer strategy leading to market equilibrium attainment is considered. From the game-theoretic viewpoint, a repeated auction describes a learning process in a noncooperative repeated game under uncertainty. We compare the results gained by the proposed computational algorithms with their analytical counterparts. And finally, numerical calculations of equilibrium and algorithm trajectories converging to the equilibrium are demonstrated.

BUNDLING COMPETITION BETWEEN MULTI-PRODUCT AND SINGLE-PRODUCT FIRMS

(ProQuest: ... denotes formulae omitted.)1. INTRODUCTIONMost price theories assume that a single price is placed on each product. The market for each product arrives at the equilibrium where the unique price equalizes the demand and the supply. Regarding how sensitive the demand for a product is, we take into account other goods such as complements or substitutes. Despite the existence of complementary or substitutable goods, the assumption of one-for-one relationship between product and price had been rarely challenged.In a real world, we often observe price discounts for multi-product purchases. If you buy more products of the same kind, you might get more discounts for an additional unit. Or if you buy several unrelated goods from the same vendor which supplies various types of products, you might be offered a discount on the bundle. In other words, the price of a bundle is placed on top of the stand-alone prices. We can theoretically devise (2n -1) prices for all possible combinations with n single products. For example, with two products, there are three prices: two stand-alone prices and one bundle price. As the marketing strategies become sophisticated, multiple intra -or inter- product purchases tend to be priced differently from the simple sum of component prices. Beyond the traditional strategy of component pricing, the bundle discount is utilized to entice customers into buying multiple products from the same vendor.Economics literature had paid scant attention to bundling strategy, a kind of nonlinear pricing, until Stigler (1963) studied the block booking in the US motion picture industry. It is suggested that a monopolist is willing to offer a bundle if consumer valuation on the products are heterogeneous. Following Stigler (1963), early literature of bundling mainly centers on the monopoly theory. Adams and Yellen (1976) illustrate prominent examples explaining why pure or mixed bundling benefits the monopolist more than component pricing. Negative correlation of consumer valuation on product space is a key factor that invites a bundling. McAfee, McMillan, and Whinston (1989) extend Adams and Yellen (1976) in continuous distribution of valuations. They derive a condition in which mixed bundling dominates other pricing strategies. Armstrong (1996) and Rochet and Chone (1998) study the bundling strategy in broader perspective of multidimensional nonlinear pricing. In the monopolist approach, bundling is an effective tool that sorts consumers into different groups for price discrimination. Offering a discounted bundle, the firm is able to distinguish the customers who prefer all the products from those who value only specific subset of the products. Thus a monopolistic multi-product firm would utilize the bundle offers to obtain the information of consumer preference.In practice, a multi-product firm usually competes with other suppliers. The competitors might contend in each product market, or they might contend employing a similar bundle offering. Compared to the monopoly bundling literature, there are few theoretical papers pertaining to competitive bundling. Spulber (1979) shows that there exists a unique non-cooperative equilibrium of price-discriminating firms. Yet nonlinear pricing with multi-product feature was not considered earlier than Armstrong and Vickers (2001), and Rochet and Stole (2002) investigated the non-linear pricing strategy of intra-product bundle under competition. Especially, Yin (2004) focuses on two-part tariffcompetition.Armstrong and Vickers (2009) build a pioneering two-stop model for a duopoly bundling. They assume there are two symmetric firms supplying two products each. Consumers who buy both products from the same firm pay only the bundle price. Others who buy the products, from different sellers so called mix-and-match, bear an additional shopping cost. One of the main results is that two multi-product firms would employ the mixed bundling strategy in equilibrium. …

Pollution accumulation and abatement policy in a supply chain

This paper seeks to determine how the double marginalization phenomenon affects the tradeoff between polluting emissions and abatement activities related to pollution accumulation in a supply chain composed of one manufacturer and one retailer. The environmental consequence of this inefficiency, which emerges in a non-cooperative vertical setting governed by a single-parameter contract, is overlooked in the literature on pollution control. In the setup of a two-stage game, we investigate the impact of double marginalization for non-cooperative equilibrium. To check whether there are differences between dynamic and strategic effects of double marginalization on pollution accumulation, both the open-loop and feedback Nash equilibria are derived over a finite time horizon, with the cooperative solution as a benchmark.

The Dynamics of Climate Agreements

I develop a dynamic model of private provision of public bads allowing investments in technologies. The analysis is tractable and the MPE unique. The framework is used to derive optimal incomplete contracts in a dynamic setting. While the noncooperative equilibrium is very inefficient, short-term contracts can be worse due to hold-up problems. The optimal long-term contract is more ambitious if its length is relatively short and the technological spillover large. The optimal length increases in this externality. With renegotiation, the outcome is first best. The results have several implications for how to design a climate treaty.

A solution for the non-cooperative equilibrium problem of two person via fixed point theory

In this paper, we investigate the non-cooperative equilibrium problem of two person games in the setting of game theory and propose a solution via coupled fixed point results in the context of partial metric spaces. We also realize that our coupled fixed point results can be applied to get a solution of a class of nonlinear Fredholm type integral equations.

Cooperative Control of Power System Load and Frequency by Using Differential Games

Electric power systems are evolving toward smart grids all over the world, which are generally regarded as the next-generation power systems with extensive application of advanced control, communications, and computation techniques. More and more distributed intelligent controllers will be employed in smart grids. Load frequency control (LFC) is a key issue in the traditional interconnected power systems, which constantly requires different control areas (CAs) to share the regulation burden of the CA that lacks regulation capacity by providing the power supports via the tie-line. Such a process imposes extra regulation costs on the helping CAs, for example, the wear and tear of generating units, which may result in the unfairness and the CAs’ deviation from the LFC command. This situation becomes even more serious with the integration of intermittent renewable energy such as wind and solar power. In this paper, cooperative control by using differential games (DGs) is proposed as a possible solution to this problem. With a two-area and a three-area LFC models, a noncooperative equilibrium solution, and two cooperative equilibrium solutions with different time-consistency properties are derived. The case studies compare the proposed three DG-based LFC schemes with the traditional approaches based on proportional-integral control and the optimal control, and the results show that the cooperative control schemes allocate the regulation burden more rationally and ensure the CAs’ sticking to the LFC command, which will be favorable to the stability of power system operation. The capability of conflict solution of DG-based cooperative control makes it a very promising scheme for smart grid control.

Optimal monetary policy in an economy with vertical production and trade: An analysis based on the perspective of local currency pricing

By incorporating the factor of firms’ asymmetric price setting behavior into a two-country model with vertical production and trade, we analyze the issues of optimal monetary policies in non-cooperative and cooperative equilibriums. The three main results to be stressed are as follows. First, we show that if the home and foreign countries have identical ratios with respect to intermediate goods firms that set their export prices in the local currency, the optimal monetary stance in a non-cooperative equilibrium is eased in response to an increase in the ratios. Second, under such an optimal monetary stance, we show that welfare is maximized in both the home and foreign countries if all home and foreign intermediate goods firms set their export prices in the local currency. Third, in a scenario where not all home and foreign intermediate goods firms adopt this pricing strategy, we show that there is always a welfare gain from cooperation.

Long-Term Relationships: Static Gains and Dynamic Inefficiencies

Do contractual frictions matter when firms are engaged in repeated interactions? This paper argues that long-term relationships, which allow firms to (partly) overcome the static costs associated with low contractibility, will under certain circumstances create dynamic inefficiencies. We consider the repeated interaction between final good producers and intermediate input suppliers, where the provision of the intermediate input is noncontractible. A producer/supplier pair can be a good match or a bad match, with bad matches featuring lower productivity. This allows us to build a cooperative equilibrium where producers can switch suppliers and start cooperation immediately with new suppliers. Every period, one supplier has the opportunity to innovate, and in the baseline model, innovations are imitated after one period. We show that (i) innovations need to be larger to break up existing relationships in the cooperative equilibrium than in either a set-up where the input is contractible or when we preclude cooperation in long-term relationships, (ii) the rate of innovation in the cooperative equilibrium is lower than in the contractible case, and may even be lower than in the non-cooperative equilibrium and (iii) cooperation may reduce welfare. Next, we assume that the frontier technology diffuses slowly to suppliers (instead of after one period). In that case, for sufficiently slow diffusion, the innovation rate in the cooperative equilibrium may be higher than even in the contractible case. Finally, we show that cooperation may also increase relationship specific innovations.

On the Complementarity between Law and Social Norms

AbstractThis article aims to clarify the relationship between the law and social norms and examine how they interact – whether the law completely replaces, or coexists with, the pre-existing social norms. We model a close-knit society consisting of injurers and victims, assuming that both the law and social norms maximize social welfare and that the costs of enforcing legal penalties are greater than those of enforcing social norms. We find that social norms completely replace the law, even in a non-cooperative Nash equilibrium; we then develop exceptions to this result. In particular, when community sanctions for violating social norms are calibrated without consideration of its marginal social benefit due to a reduction in law enforcement cost, our model can have multiple equilibria, including cases in which legal sanctions persist even in the long run, in spite of their inefficiently high enforcement costs. However, we also show that this possibility of an inefficient non-cooperative equilibrium can be eliminated if the government behaves as a Stackelberg leader for the formation of social norms.

Approximation of zero-sum continuous-time Markov games under the discounted payoff criterion

We deal with a two-person zero-sum continuous-time Markov game \(\mathcal {G}\) with denumerable state space, general action spaces, and unbounded payoff and transition rates. We consider noncooperative equilibria for the discounted payoff criterion. We are interested in approximating numerically the value and the optimal strategies of \(\mathcal {G}\). To this end, we propose a definition of a sequence of game models \(\mathcal {G}_n\) converging to \(\mathcal {G}\), which ensures that the value and the optimal strategies of \(\mathcal {G}_n\) converge to those of \(\mathcal {G}\). For numerical purposes, we construct finite state and actions game models \(\mathcal {G}_n\) that can be explicitly solved, and we study the convergence rate of the value of the games. A game model based on a population system illustrates our results.

Cooperative Action on Greenhouse Gas Emissions and the Distribution of Global Output and damage

This paper compares the performance of two allowance allocation rules in an international climate change treaty. I construct a model in which countries differ according to both GDP and an idiosyncratic damage parameter that links global emissions to damage for an individual country. Allowances are allocated to treaty members according to an allocation rule based on a single allocation parameter. The model can be solved analytically to determine upper and lower bounds on this allocation parameter that ensure that a treaty of any given composition is internally and externally stable. I focus on grand coalition treaties. The first treaty examined uses a simple proportional rule in which allocations are set as some fraction of emissions in the non-cooperative equilibrium. The second treaty adds a coverage-contingent element to the allocation rule such that the emissions reduction required of each treaty member is weighted by the fraction of global emissions that treaty members as a whole emit in the non-cooperative equilibrium. I show that the coverage-contingent treaty outperforms the simpler treaty except when all countries have the same emissions intensity in the non-cooperative equilibrium (in which case neither treaty can reduce emissions).

Search duplication in research and design spaces — Exploring the role of local competition

A lack of sufficient diversification in research strategies has been identified as an important problem for delegated research. We show that this problem can be solved by local competition (such as bribery, lobbying, rent seeking, competition at the patent office) among players who apply the same search strategies or develop the same design. Such competition can restore full efficiency in the non-cooperative equilibrium. Local competition interacts with the choice of whether to cluster or diversify, and rather than adding a further inefficiency to the existing ones, it eliminates inefficiency. The results are robust and hold under simultaneous search strategy choices as well as for sequential choices.

Non‐Governmental Public Norm Enforcement in Large Societies as a Two‐Stage Game of Voluntary Public Good Provision

AbstractIn small groups, norm enforcement is achieved through mutual punishment and reward. In large societies, norms are enforced by specialists such as government officials. However, not every public cause is overseen by states, for instance those organized at the international level. This paper shows how nongovernmental norm enforcement can emerge as a decentralized equilibrium. As a first stage, individuals voluntarily contribute to a nongovernmental agency that produces an incentive system. The second stage is the provision of a public good on the basis of private contributions. The incentive system increases contributions by means of public approval or disapproval of behavior. It is shown that, even in large populations, nongovernmental norm enforcement can be supported in a noncooperative equilibrium of utility‐maximizing individuals. This result is in sharp contrast to those obtained in the standard situation of voluntary provision of an intrinsic public good which—without altruism or related motives—is eroded by free‐riding. Reliance on altruistic behavior is not required in supplying the second‐order public good “norm enforcement” in large societies.