A non-cooperative foundation for the continuous Raiffa solution

This paper provides a non-cooperative foundation for (asymmetric generalizations of) the continuous Raiffa solution. Specifically, we consider a continuous-time variation of the classic Ståhl–Rubinstein bargaining model, in which there is a finite deadline that ends the negotiations, and in which each player’s opportunity to make proposals is governed by a player-specific Poisson process, in that the rejecter of a proposal becomes proposer at the first next arrival of her process. Under the assumption that future payoffs are not discounted, it is shown that the expected payoffs players realize in subgame perfect equilibrium converge to the continuous Raiffa solution outcome as the deadline tends to infinity. The weights reflecting the asymmetries among the players correspond to the Poisson arrival rates of their respective proposal processes.     

Nash fuzzy equilibrium: Theory and application to a spatial duopoly

Consider a non-cooperative n-persons game. Each gambler has a set of mixed strategies at his disposal. The payoffs are some physical or immaterial objects. The game is a fuzzy game because (1) gamblers have more or less precise preferences for the payoffs and (2) the outcoming of payoffs is uncertain. The uncertainty can be expressed either by a distribution of possibility or by a distribution of probability. The product set of a gambler's mixed strategies is convex and compact and the payoff functions are continuous. Then a closed and convex fuzzy point-to-set mapping is defined on the product set of strategies and, by using a Butnariu theorem, the existence of a fixed point for this fuzzy point-to-set mapping is proved. The issue allows us to generalize a famous Nash result: a n-persons non-cooperative fuzzy game with mixed strategies has at least one equilibrium point. In the second part of the paper an economic application is devoted to the statement of the equilibrium existence conditions in a spatial duopoly. The model is not only more general than the classical ones, but also more relevant because new results are obtained.     

A Characterization of Nash Equilibrium for the Games with Random Payoffs

We consider a two-player random bimatrix game where each player is interested in the payoffs which can be obtained with certain confidence. The payoff function of each player is defined using a chance constraint. We consider the case where the entries of the random payoff matrix of each player jointly follow a multivariate elliptically symmetric distribution. We show an equivalence between the Nash equilibrium problem and the global maximization of a certain mathematical program. The case where the entries of the payoff matrices are independent normal/Cauchy random variables is also considered. The case of independent normally distributed random payoffs can be viewed as a special case of a multivariate elliptically symmetric distributed random payoffs. As for Cauchy distribution, we show that the Nash equilibrium problem is equivalent to the global maximization of a certain quadratic program. Our theoretical results are illustrated by considering randomly generated instances of the game.     

Preplay negotiations and the prisoner's dilemma

In order to improve outcomes of one shot noncooperative games a formal procedure for conducting preplay negotiations is proposed. For the prisoners' dilemma game it is shown that all the perfect equilibrium in the induced game (the game with the preplays) yield the cooperative pay-off. For another game it is shown that all perfect equilibrium payoffs converge to be Pareto optimal as the number of preplays increases.     

Local air pollutant emission reduction and ancillary carbon benefits of SO2 control policies: Application of AIM/CGE model to China

While a great deal of literature has been published in recent years on the ancillary benefits of greenhouse gas mitigation (e.g., reductions in local air pollution), less attention has been focused on the climate benefits of local air pollution strategies themselves. Local air pollution is, however, a more immediate issue now faced by developing countries. This study assesses the impacts on local air pollutant emission reduction and ancillary CO₂ emission reduction of SO₂ control policies in China, such as a sulphur tax, SO₂ total emissions control (TEC), and improvement of energy efficiency, based on the Asia-Pacific Integrated Model (AIM)/Computable General Equilibrium (CGE) country model. The simulation period is from 1997 to 2020. Major conclusions include the following: an SO₂ emission cap will help to control SO₂ emissions, but will result in a large GDP loss; the role of a SO₂ emission tax at the present level is very limited; and an ancillary carbon reduction benefit can be achieved through the introduction of SO₂ control policies in China.     

Loss of skills in coordination games

This paper deals with 2-player coordination games with vanishing actions, which are repeated games where all diagonal payoffs are strictly positive and all non-diagonal payoffs are zero with the following additional property: At any stage beyond r, if a player has not played a certain action for the last r stages, then he unlearns this action and it disappears from his action set. Such a game is called an r-restricted game. To evaluate the stream of payoffs we use the average reward. For r = 1 the game strategically reduces to a one-shot game and for r ≥ 3 in Schoenmakers (Int Game Theory Rev 4:119–126, 2002) it is shown that all payoffs in the convex hull of the diagonal payoffs are equilibrium rewards. In this paper for the case r = 2 we provide a characterization of the set of equilibrium rewards for 2 × 2 games of this type and a technique to find the equilibrium rewards in m × m games. We also discuss subgame perfection.     

On discounted stochastic games with incomplete information on payoffs and a security application

This paper presents a robust optimization model for n$n$-person finite state/action stochastic games with incomplete information on payoffs. For polytopic uncertainty sets, we propose an explicit mathematical programming formulation for an equilibrium calculation. It turns out that a global optimal of this mathematical program yields an equilibrium point and epsilon-equilibria can be calculated based on this result. We briefly describe an incomplete information version of a security application that can benefit from robust game theory.     

Games with frequency-dependent stage payoffs

Games with frequency-dependent stage payoffs (FD-games), are infinitely repeated non-cooperative games played at discrete moments in time called stages. The stage payoffs depend on the action pair actually chosen, and on the relative frequencies with which all actions were chosen before. We assume that players wish to maximize their expected (limiting) average rewards over the entire time-horizon. We prove an analogy to, as well as an extension of the (perfect) Folk Theorem. Each pair of rewards in the convex hull of all individually-rational jointly-convergent pure-strategy rewards can be supported by an equilibrium. Moreover, each pair of rewards in same set giving each player strictly more than the threat-point-reward, can be supported by a subgame-perfect equilibrium. Under a pair of jointly-convergent strategies, the relative frequency of each action pair converges in the long run. Received: March 2002/Revised: January 2003     

纳入政策预期的国际气候博弈

本研究首先对Baumol和Oates构建的公共外部性模型的假设条件进行修正,从而构建起更符合实际的国际气候治理的数理模型;求解该数理模型,本研究推导出同时实现全球帕累托最优和自身财政收支平衡下,国际环境协议必须遵循的唯一政策规则;最后,以此为基础,本研究进一步构建起纳入政策预期的国际气候博弈模型,并通过数理分析论证,揭示了:如果世界各国都只考虑自身利益最大化,纳入政策预期下的气候博弈的均衡结果,将无法实现全球气候治理的帕累托最优。     

Delay in the alternating-offers model of bargaining

Alternating offers bargaining has been extensively used to model two-sided negotiations. The celebrated model of Rubinstein [Econometrica 50(1):97–109, 1982] has provided a formal justification for equitable payoff division. A typical assumption of these models under risk is that the termination event means a complete and irrevocable breakdown in negotiations. In this paper, the meaning of termination is reinterpreted as the imposition to finish negotiations immediately. Specifically, bargaining terminates when the last offer becomes definitive. While Rubinstein’s model predicts an immediate agreement with stationary strategies, we show that the same payoff allocation is attainable under non-stationary strategies. Moreover, the payoffs in delayed equilibria are potentially better for the proposer than those in which agreement is immediately reached.     

STABILITY OF AN SEIS EPIDEMIC MODEL WITH CONSTANT RECRUITMENT AND A VARYING TOTAL POPULATION SIZE

This paper considers an SEIS epidemic model with infectious force in the latent period and a general population-size dependent contact rate. A threshold parameter R is identified. If R≤1, the disease-free equilibrium O is globally stable. If R〉1, there is a unique endemic equilibrium and O is unstable. For two important special cases of bilinear and standard incidence ,sufficient conditions for the global stability of this endemic equilibrium are given. The same qualitative results are obtained provided the threshold is more than unity for the corresponding SEIS model with no infectious force in the latent period. Some existing results are extended and improved.     

Global attractivity in a population model with nonlinear death rate and distributed delays

We study the global attractivity of the unique positive equilibrium of a population model with distributed delays and nonlinear death rate. Both delay dependent and delay independent criteria are obtained which generalize, unify and improve known criteria. These results will be applied to some models with bounded and unbounded death functions.     

Correlation and unmediated cheap talk in repeated games with imperfect monitoring

This paper studies n-player \((n\ge 3)\) undiscounted repeated games with imperfect monitoring. We prove that all uniform communication equilibrium payoffs of a repeated game can be obtained as Nash equilibrium payoffs of the game extended by unmediated cheap talk. We also show that all uniform communication equilibrium payoffs of a repeated game can be reached as Nash equilibrium payoffs of the game extended by a pre-play correlation device and a cheap-talk procedure that only involves public messages; furthermore, in the case of imperfect public and deterministic signals, no cheap talk is conducted on the equilibrium path.     

Dynamic behavior of a delay cholera model with constant infectious period

In this paper, a delay cholera model with constant infectious period is investigated. By analyzing the characteristic equations, the local stability of a disease-free equilibrium and an endemic equilibrium of the model is established. It is proved that if the basic reproductive number $\mathcal{R}_0>1$, the system is permanent. If $\mathcal{R}_0<1$, by means of an iteration technique, sufficient conditions are obtained for the global asymptotic stability of the disease-free equilibrium. If $\mathcal{R}_0>1$, also by means of an iteration technique, sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium. Numerical simulations are carried out to illustrate the main theoretical results.     

Games with randomly disturbed payoffs: A new rationale for mixed-strategy equilibrium points

Equilibrium points in mixed strategies seem to be unstable, because any player can deviate without penalty from his equilibrium strategy even if he expects all other players to stick to theirs. This paper proposes a model under which most mixed-strategy equilibrium points have full stability. It is argued that for any gameΓ the players' uncertainty about the other players' exact payoffs can be modeled as a disturbed gameΓ ^*, i.e., as a game with small random fluctuations in the payoffs. Any equilibrium point inΓ, whether it is in pure or in mixed strategies, can “almost always” be obtained as a limit of a pure-strategy equilibrium point in the corresponding disturbed gameΓ ^* when all disturbances go to zero. Accordingly, mixed-strategy equilibrium points are stable — even though the players may make no deliberate effort to use their pure strategies with the probability weights prescribed by their mixed equilibrium strategies — because the random fluctuations in their payoffs willmake them use their pure strategies approximately with the prescribed probabilities.     

Dynamical analysis of a game network

In this paper, our purpose is to establish a direct dynamical relationship between payoffs of players in a game. A network dynamical model is first proposed to describe the direct dynamical relationship. A Markovian chain is also introduced in the given network model to show the mixed strategy property in a game. As a result, a direct dynamical relationship between payoffs of players in a game can be illustrated through a network model with a Markovian chain. In addition, a sufficient condition is addressed to guarantee that the discussed network model has a unique and stable Nash equilibrium. Finally, a numerical example is shown to illustrate the effectiveness of the obtained criterion.     

Contending “signals” in coalition choice 1

During negotiations for coalitions, each actor presumably searches his decision environment for organizing principles—signals—which, if present, would turn the negotiations in his favor. With only one prominent signal, we expect it to determine the nature of agreements, but things may be different with multiple signals. Data from four studies of a weighted‐majority game, two using only relative status of winnings in the game as the incentive (i.e., rank position of points accumulated) and two also using monetary reward as an incentive, are analyzed. Both relative status and resources (weights) were apparent to subjects in the studies. Results for coalitions and for payoffs show that a theory based on resources as the signal and one based on status as the signal both separately achieve success in the same bodies of data. As expected, the status‐signal theory better predicted coalitions in the two status‐reward studies, and the resource‐signal theory better predicted in the monetary‐reward studies. However, in all four studies, a two‐signal model which accounts for the reinforcing effects of signals achieves greater success. This indicates the importance of analyzing actors’ adoption of signals into their decision process, particularly in multisignal environments.     

HOW RAPID SHOULD EMISSION REDUCTION BE? A GAME‐THEORETIC APPROACH

Abstract In this paper, we search for multistage realization of international environmental agreements. To analyze countries' incentives and the results of their interactions, we mathematically represent players' strategic preferences and apply a game‐theoretic approach to make predictions about their outcomes. The initial decision on emissions reduction is determined by the Stackelberg equilibrium concept. We generalize Barrett's static “emission” model to a dynamic framework and answer the question “how rapid should the emission reduction be?” It appears that sharper abatement is desirable in the early term, which is similar to the conclusion of the Stern review. Numerical example demonstrates that abatement dynamics of the coalition and the free‐rider differ when discounting of the future payoffs increases. We show that without incentives from external organizations or governments, such pollution reduction path can actually lead to a decline in the agreement's membership size.     

Stability of a delayed SIRS epidemic model with a nonlinear incidence rate

In this paper, an SIRS epidemic model with a nonlinear incidence rate and a time delay is investigated. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease-free equilibrium is discussed. By comparison arguments, it is proved that if the basic reproductive number R₀<1, the disease-free equilibrium is globally asymptotically stable. If R₀>1, by means of an iteration technique, sufficient conditions are derived for the global asymptotic stability of the endemic equilibrium.