Open-loop Nash Equilibrium Research Articles

Transboundary pollution control and environmental absorption efficiency management

In this paper, we suggest a two-player differential game model of transboundary pollution that accounts for time-dependent environmental absorption efficiency, which allows the biosphere to switch from a carbon sink to a source. We investigate the impact of negative externalities resulting from a transboundary pollution non-cooperative game wherein countries are dynamically involved. Based on a linear-quadratic specification for the instantaneous revenue function, we assess differences related to both transient path and steady state between cooperative solution, open-loop and Markov perfect Nash equilibria (MPNE). Regarding the methodological contribution of the paper, we suggest a particular structure of the conjectured value function to solve MPNE problems with multiplicative interaction between state variables in one state equation, so that third-order terms that arise in the Hamilton–Jacobi–Bellman equation are made negligible. Using a collocation procedure, we confirm the validity of the particular structure of the conjectured value function. The results suggest unexpected contrasts in terms of pollution control and environmental absorption efficiency management: (i) in the long run, an open-loop Nash equilibrium (OLNE) allows equivalent emissions to the social optimum but requires greater restoration efforts; (ii) although an MPNE is likely to end up with lower emissions and greater restoration efforts than an OLNE, it has a much greater chance of falling in the emergency area; (iii) the absence of cooperation and or precommitment becomes more costly as the initial absorption efficiency decreases; (iv) more heavily discounted MPNE strategies are less robust than OLNE to prevent irreversible switching of the biosphere from a carbon sink to a source.

Systemic Risk and Stochastic Games with Delay

We propose a model of inter-bank lending and borrowing which takes into account clearing debt obligations. The evolution of log-monetary reserves of banks is described by coupled diffusions driven by controls with delay in their drifts. Banks are minimizing their finite-horizon objective functions which take into account a quadratic cost for lending or borrowing and a linear incentive to borrow if the reserve is low or lend if the reserve is high relative to the average capitalization of the system. As such, our problem is a finite-player linear–quadratic stochastic differential game with delay. An open-loop Nash equilibrium is obtained using a system of fully coupled forward and advanced-backward stochastic differential equations. We then describe how the delay affects liquidity and systemic risk characterized by a large number of defaults. We also derive a closed-loop Nash equilibrium using a Hamilton–Jacobi–Bellman partial differential equation approach.

Dynamic Border Carbon Taxes to Enhance Green Growth

This paper theoretically analyzes the growth and welfare impacts of dynamic Border Carbon Taxes (BCT) across trading countries. I build a trade model with dynamic investment decisions using a Ramsey growth model. The government in each country can invest either in costly nonpolluting capital or in cheap polluting capital. The model is solved numerically for an open loop Nash equilibrium to study different configurations of BCTs across countries. I find that a unilateral BCT is welfare enhancing for the country that applies it and an effective tool to shift the growth of the other country towards greener path even when countries are not concerned about the environment. Results show that a bilateral BCT becomes welfare enhancing for both countries if governments care sufficiently about the environmental quality of their consumers. Moreover, the asymmetry in initial development levels across countries induces a slower growth for the initially poorer country only if the other country is richer in the polluting capital. Furthermore, the model shows that trade openness should be achieved gradually along the development path of countries.

An Inverse Differential Game Approach to Modelling Bird Mid-Air Collision Avoidance Behaviours

In this paper, we investigate an inverse differential game approach to modelling the mid-air collision avoidance behaviours of birds. We propose a general method for estimating the cost-functional parameters of a noncooperative differential game from partial-state measurements of an open-loop Nash equilibrium. We apply the method to data of birds performing mid-air collision avoidance. Our analysis suggests that a differential game model provides a close description of the observed bird behaviours, and could provide new insights for the design of collision avoidance strategies for unmanned aircraft.

Non-cooperative dynamic games for general insurance markets

In the insurance industry, the number of product-specific policies from different companies has increased significantly. The strong market competition has boosted the demand for a competitive premium. In actuarial science, scant literature still exists on how competition actually affects the calculation and the cycles of company’s premiums. In this paper, we model premium dynamics via differential games, and study the insurers’ equilibrium premium dynamics in a competitive market. We apply an optimal control theory methodology to determine the open-loop Nash equilibrium premium strategies. The market power of each insurance company is characterized by a price sensitive parameter, and the business volume is affected by the solvency ratio. We study two models. Considering the average market premiums, the first model studies an exponential relation between premium strategies and volume of business. The second model initially characterizes the competition between any selected pair of insurers, and then aggregates all the paired competitions in the market. Numerical examples illustrate the premium dynamics, and show that premium cycles may exist in equilibrium.

An Approach Based on Generalized Nash Games and Shared Constraints for Discrete Time Dynamic Games

In this paper, we provide new results on conditions for the existence of open-loop Nash equilibria in discrete time dynamic games (DTDGs) using the theory of generalized Nash games. We use the state conjecture formulation of DTDGs introduced in Abraham and Kulkarni (Under Rev Math Methods Oper Res, 2017) for the analysis, which transforms a DTDG into an equivalent generalized Nash game. The existence of a Nash equilibrium for a generalized Nash game is intractable in general, without strong assumptions in the problem structure. We introduce a new and related game called the DTDG with consistent conjectures, which has a shared constraint structure unlike the original game. We show that an open-loop Nash equilibrium of the original DTDG is a Nash equilibrium of this new game, and hence, the existence of a Nash equilibrium for the DTDG with consistent conjectures becomes necessary for the existence of an open-loop Nash equilibrium in the original DTDG. Then we show that when the cost function of the game admits a potential function, the existence of a Nash equilibrium in the DTDG with consistent conjectures follows under mild assumptions. Finally, we show a reverse relation, wherein under certain conditions, a Nash equilibrium of the DTDG with consistent conjectures is shown to be an \(\epsilon \)-Nash equilibrium of the original game. We use an example from dynamic duopoly competition with sticky prices to illustrate the concepts and results.

Non-Cooperative Dynamic Games for General Insurance Markets

In the insurance industry, the number of product-specific policies from different companies has increased significantly. The strong market competition has boosted the demand for a competitive premium. In actuarial science, scant literature still exists on how competition actually affects the calculation and the cycles of company’s premiums. In this paper, we model premium dynamics via differential games, and study the insurers’ equilibrium premium dynamics in a competitive market. We apply an optimal control theory methodology to determine the open-loop Nash equilibrium premium strategies. The market power of each insurance company is characterized by a price sensitive parameter, and the business volume is affected by the solvency ratio. We study two models. Considering the average market premiums, the first model studies an exponential relation between premium strategies and volume of business. The second model initially characterizes the competition between any selected pair of insurers, and then aggregates all the paired competitions in the market. Numerical examples illustrate the premium dynamics, and show that premium cycles may exist in equilibrium.

Inverse Noncooperative Dynamic Games

We consider the problem of computing parameters of player cost functions in discrete-time nonzero-sum noncooperative dynamic games from open-loop Nash equilibria. Although similar inverse problems have been investigated in the optimal control literature where there is a single player (or decision maker), there has been limited attention given to the inverse dynamic game problem with multiple (competing) players. By exploiting the minimum principle of optimal control, we propose a method of inverse dynamic games for when the information structure of the game is open-loop. Our method involves solving a system of linear equations and is able to recover the true unknown parameters (up to an unknown scaling factor) whenever a testable rank condition holds. We illustrate our method in an example two-player game.

A differential game in a duopoly with instantaneous incentives

In this paper, we study a differential game in which two competing firms exploit a public renewable resource that is relevant from a landscape point of view. We consider a policy maker that provides an instantaneous incentive to the firms in order to prevent the resource exhaustion during the whole extraction period, which coincides with the harvesting license period. We compute an open-loop Nash equilibrium of the differential game, showing that it coincides with a linear feedback Nash equilibrium. Finally, we compute the value of the incentive that leads to the maximization of social welfare considering the incentive both as a pure transfer and as a cost.

Dynamic Budget Allocation in Competitive Search Advertising

Search advertising marketplace is highly competitive in nature. As such, advertisers have to consider their competitors' responses when making advertising decisions, which concern both bid determination in individual keyword auction instances, and budget allocation over time at the aggregate market level. Although bidding decisions have been extensively treated in competitive auction settings in the literature, competitive budgeting models are yet to be developed. Applying the differential game framework, we propose a dynamic budget allocation model in a duopolistic competitive setting. We first examine the steady-state Nash equilibrium policies for the advertisers, and then investigate the optimal paths of open-loop Nash equilibrium. In addition, we provide an algorithm to compute the optimal path of budget policies. Computational experiments were conducted to derive additional managerial insights. One important finding of this study is that the effects of two advertising factors, quality score and potential market revenue, on the steady-state budget levels are non-monotonic. This finding to some extent challenges the commonly-held view that higher quality scores lead to lower optimal advertising expenditures. Another interesting finding suggests that advertisers should set a budget cap even in a competitive environment to avoid the potential risk of over-spending in sponsored search advertising.

A note on differential games with Pareto-optimal NASH equilibria: Deterministic and stochastic models<sup>†</sup>

Pareto optimality and Nash equilibrium are two standard solution concepts for cooperative and non-cooperative games, respectively. At the outset, these concepts are incompatible-see, for instance, [ 7 ] or [ 10 ]. But, on the other hand, there are particular games in which Nash equilibria turn out to be Pareto-optimal [ 1 ], [ 4 ], [ 6 ], [ 18 ], [ 20 ]. A class of these games has been identified in the context of discrete-time potential games [ 13 ]. In this paper we introduce several classes of deterministic and stochastic potential differential games [ 12 ] in which open-loop Nash equilibria are also Pareto optimal.

Dynamic modeling of nontargeted and targeted advertising strategies in an oligopoly

With the growing collection of sales and marketing data and depth of detailed knowledge of consumer habits and trends, firms are gaining the capability to discern customers of other firms from the potential market of uncommitted consumers. Firms with this capability will be able to implement a strategy where the advertising effort towards customers of competing firms may differ from that towards uncommitted consumers. In this work, dynamic models for advertising in an oligopoly setting with fixed total market size and sales decay are presented. Two models are described in detail: a nontargeted model in which the advertising effort is the same for both categories of prospective customers, and a targeted model that gives firms the capability to allocate effort across the two categories differently. In the differential game setting, open-loop and closed-loop Nash equilibrium strategies are derived for both models. Several strategic questions that a firm may face when practicing targeted advertising on a fixed budget are discussed and addressed.

Dynamic R&D with spillovers: A comment

Cellini and Lambertini [2009. Dynamic R&D with spillovers: competition vs cooperation. J. Econ. Dyn. Control 33, 568–582] study a dynamic R&D game with spillovers. This comment demonstrates that, contrary to what is claimed in their paper, the game is not state redundant and the open-loop Nash equilibrium is not subgame perfect.

Dynamic Budget Allocation in Competitive Search Advertising: A Differential Game Approach

Over the past decade, sponsored search advertising has evolved into a major advertising outlet on the Internet. Given the competitive and dynamic nature of the search advertising market, it has long been a great challenge for advertisers to identify their optimal advertising strategies. Using a differential game framework, we propose a dynamic budget allocation model to explore the optimal advertising strategies for advertisers engaged in a duopolistic competition. The proposed model captures the dynamic nature of the search advertising market by incorporating two important advertising variables, namely, the time-varying quality score and the potential market revenue of advertisers, which are specific to the unique context of sponsored search advertising. In this model, we first identify and examine the steady-state Nash equilibrium strategies of the advertisers, and then investigate the optimal paths for the open-loop Nash equilibrium. In addition, we provide an algorithm to compute the optimal path of budget and the corresponding trajectory of market share for the advertisers, and conduct computational experiments to validate our analytical findings. One important finding in this study is that the effects of the two advertising variables on the equilibrium budget levels are non-monotonic. Moreover, our results indicate that as an advertiser improves her quality scores, her market share in equilibrium will increase accordingly; in the meantime, however, the advertiser may need to escalate her advertising expenditure. This finding to some extent challenges the commonly-held view that higher quality scores lead to lower optimal advertising expenditures. Another interesting finding suggests that advertisers should set a budget cap even in a competitive environment to avoid the potential risk of over-spending in sponsored search advertising.

Age-structured linear-state differential games

In this paper we search for conditions on age-structured differential games to make their analysis more tractable. We focus on a class of age-structured differential games which show the features of ordinary linear-state differential games, and we prove that their open-loop Nash equilibria are sub-game perfect. By means of a simple age-structured advertising problem, we provide an application of the theoretical results presented in the paper, and we show how to determine an open-loop Nash equilibrium.

Domestic environmental policy and international cooperation for global commons

The paper analyzes the strategic behavior of several countries engaged in capital accumulation, pollution mitigation, and environmental adaptation in the context of an environmental common good. Both cooperative and non-cooperative strategies are discussed. The non-cooperative strategy is a dynamic game in which each country makes its own environmental decision following the open-loop Nash equilibrium. The cooperative social planner problem assumes an international environmental agreement in force. The non-cooperative and cooperative solutions are compared in the symmetric case of two countries and extended to several identical countries. It is shown that the non-cooperative strategy in multi-country world leads to over-production, over-consumption, over-pollution, and over-adaptation.

A numerical solution of open-loop Nash equilibrium in nonlinear differential games based on Chebyshev pseudospectral method

In general, the applications of differential games for solving practical problems have been limited, because all calculations had to be done analytically. In this investigation, a simple and efficient numerical method for solving nonlinear nonzero-sum differential games with finite- and infinite-time horizon is presented. In both cases, derivation of open-loop Nash equilibria solutions usually leads to solving nonlinear boundary value problems for a system of ODEs. The proposed numerical method is based on a combination of minimum principle of Pontryagin and expanding the required approximate solutions as the elements of Chebyshev polynomials. Applying Chebyshev pseudospectral method, two-point boundary value problems in differential games are reduced to the solution of a system of algebraic equations. Finally, several examples are given to demonstrate the accuracy and efficiency of the proposed method and a comparison is made with the results obtained by fourth order Runge–Kutta method.

Open-Loop Nash Equilibria in a Class of Linear-Quadratic Difference Games With Constraints

We study a class of $N$ -player finite-horizon linear-quadratic difference games with linear constraints. We introduce constrained open-loop information structure and derive necessary conditions for the existence of constrained open-loop Nash equilibria. We show that these conditions lead to a weakly coupled system of parametric two-point boundary value problem and a set of linear complementarity problems. By restricting the costate variables to be affine in the state variable, we show that these necessary conditions can be reformulated as a single large-scale linear complementarity problem. Then we provide sufficient conditions under which a solution of the linear complementarity problem constitutes a constrained open-loop Nash equilibrium.

Battle for Climate and Scarcity Rents: Beyond the Linear-Quadratic Case.

Industria imports oil, produces final goods and wishes to mitigate global warming. Oilrabia exports oil and buys final goods from the other country. Industria uses the carbon tax to impose an import tariff on oil and steal some of Oilrabia’s scarcity rent. Conversely, Oilrabia has monopoly power and sets the oil price to steal some of Industria’s climate rent. We analyze the relative speeds of oil extraction and carbon accumulation under these strategic interactions for various production function specifications and compare these with the efficient and competitive outcomes. We prove that for the class of HARA production functions, the oil price is initially higher and subsequently lower in the open-loop Nash equilibrium than in the efficient outcome. The oil extraction rate is thus initially too low and in later stages too high. The HARA class includes linear, loglinear and semi-loglinear demand functions as special cases. For non-HARA production functions, Oilrabia may in the open-loop Nash equilibrium initially price oil lower than the efficient level, thus resulting in more oil extraction and climate damages. We also contrast the open-loop Nash and efficient outcomes numerically with the feedback Nash outcomes. We find that the optimal carbon tax path in the feedback Nash equilibrium is flatter than in the open-loop Nash equilibrium. It turns out that for certain demand functions using the carbon tax as an import tariff may hurt consumers’ welfare as the resulting user cost of oil is so high that the fall in welfare wipes out the gain from higher tariff revenues.

Conflict in the Profit-Led Growth Model

Abstract We model the interaction between capitalists and entrepreneurs as a dynamic game. The open-loop Nash equilibrium and the closed-loop Nash equilibrium are distinguished. The purpose is to answer some questions that have arisen in the development of profit-led versus wage-led growth models. We find that the rate of profit and the discount rate as well as the responsiveness of the wage rate or aggregate consumption to the accumulation of capital are critical to explaining the change in regimes.