Strong Time Consistency Research Articles

Strong Time-Consistency of a Core: An Application to the Pollution Control Problem in Eastern Siberia

In this paper, we study the (strong) time-consistency property of the core for a linear-quadratic differential game of pollution control with nonzero absorption coefficient and real values of the model parameters. The values of parameters are evaluated based on the data for the largest aluminum enterprises of Eastern Siberia region of the Russian Federation for the year 2016. The obtained results are accompanied with illustrations.

Emission taxation, green innovations and inverted-U aggregate R&D efforts in a linear state differential game

We revisit a well known differential Cournot game with polluting emissions, to propose a version of the model in which environmental taxation is levied on emissions rather than the environmental damage. This allows to attain strong time consistency under open-loop information, and yields two main results which can be summarized as follows: (i) to attain a fully green technology in steady state, the regulator may equivalently adopt an appropriate tax rate (for any given number of firms) or regulate market access (for any given tax rate); (ii) if the environmental damage depends on emissions only (i.e., not on industry output) then the aggregate green R&D effort takes an inverted-U shape, and the industry structure maximising aggregate green innovation also minimises individual and aggregate emissions. This calls for a coordination of environmental and merger regulation so as to create the industry structure most favourable to green innovation.

Multistage Games with Pairwise Interactions on Complete Graph

This paper is devoted to multistage games with pairwise interactions. The case of complete graph is considered, the vertices of which are players, and the edges are the connections between them. The characteristic function is introduced and its supermodularity is proved for single-stage game. A new approach for construction of characteristic function of multistage game is proposed, based on the use of values of characteristic functions of single-stage games. On the basis of the newly constructed characteristic function, the optimality principle is introduced, which is an analogue of the core, and its strong time-consistency proved. The work is illustrated by an example.

Time (in)consistency of multistage distributionally robust inventory models with moment constraints

Recently, there has been a growing interest in developing inventory control policies which are robust to model misspecification. One approach is to posit that nature selects a worst-case distribution for any relevant stochastic primitives from some pre-specified family. Several communities have observed that a subtle phenomena known as time inconsistency can arise in this framework. In particular, it becomes possible that a policy which is optimal at time zero may not be optimal for the associated optimization problem in which the decision-maker recomputes her policy at each point in time, which has implications for implementability. If there exists a policy which is optimal for both formulations, we say that the policy is time consistent, and the problem is weakly time consistent. If every optimal policy is time consistent, we say that the problem is strongly time consistent. We study these phenomena in the context of managing an inventory over time, when only the mean, variance, and support are known for the demand at each stage. We provide several illustrative examples showing that here the question of time consistency can be quite subtle. We complement these observations by providing simple sufficient conditions for weak and strong time consistency. Although a similar phenomena was previously identified by Shapiro for the setting in which only the mean and support of the demand are known, here our model is rich enough to exhibit a variety of additional interesting behaviors.

Strong Time-Consistent Core for a Class of Linear-State Games

Time consistency is an important property of any solution to a cooperative dynamic game. If the solution satisfies this property, players do not need to revise it and break a cooperative agreement. Strong time consistency is a stricter property which is applicable to cooperative set solutions. In this paper, the authors examine a class of linear-state games which come into use in many applications of dynamic games. Considering the core as the solution of the game, the authors provide sufficient conditions for its strong time consistency. In case of its inconsistency, the authors show how core elements can be realized using a strong time-consistent imputation distribution procedure.

Linearly Solvable Mean-Field Traffic Routing Games

We consider a dynamic traffic routing game over an urban road network involving a large number of drivers in which each driver selecting a particular route is subject to a penalty that is affine in the logarithm of the number of drivers selecting the same route. We show that the mean-field approximation of such a game leads to the so-called linearly solvable Markov decision process, implying that its mean-field equilibrium (MFE) can be found simply by solving a finite-dimensional linear system backward in time. Based on this backward-only characterization, it is further shown that the obtained MFE has the notable property of strong time-consistency. A connection between the obtained MFE and a particular class of fictitious play is also discussed.

On the attainment of the maximum sustainable yield in the Verhulst–Lotka–Volterra model

We reformulate the Verhulst–Lotka–Volterra model of natural resource extraction under the alternative assumptions of Cournot behaviour and perfect competition, to revisit the tragedy of commons vs the possibility of sustainable harvesting. After a brief layout of the open-loop solution including the Ramsey rule, we rely on the state-redundancy property and the consequent strong time consistency of the static equilibrium output to investigate the different impact of demand elasticity on the regulator’s possibility of driving industry harvest to the maximum sustainable yield in the two settings. The presence of a flat demand function offers the authority a fully effective regulatory tool in the form of the exogenous price faced by perfectly competitive firms, to drive their collective harvest rate to the maximum sustainable yield. The same cannot happen under Cournot competition, as in this case the price is endogenous and the regulator’s policy is confined to limiting access to the common pool.

Strong Strategic Support of Cooperation in Multistage Games

The problem of cooperation in repeated and multistage games is considered. The strong equilibrium (equilibrium stable against deviations of coalitions) with payoffs which can be attained under cooperation is constructed for a wide class of such games. The new solution concept based on solutions of stage games is introduced and in some cases this solution is a subset of the core defined for repeated and multistage games in a classical way. It is also proved that this newly introduced solution concept is strongly time consistent. The strong time consistency of the solution is a very important property since in case it does not take place players in reality in some time instant in subgame on cooperative trajectory may switch from the previously selected optimal solution to any other optimal solution in the subgame and as result realize the solution which will not be optimal in the whole game.

Long-term implementation of the cooperative solution in a multistage multicriteria game

In order to find an optimal and time consistent cooperative path in multicriteria multistage game the minimal sum of relative deviations rule is introduced. Using this rule one can construct a vector-valued characteristic function that is weakly superadditive. The sustainability of the cooperative agreement is ensured by using an imputation distribution procedure (IDP) based approach.We formulate the conditions an IDP should satisfy to guarantee that the core is strongly time consistent (STC). Namely, if the imputation distribution procedure for the Shapley value satisfies the efficiency condition, the strict balance condition and the strong irrational-behavior-proof condition, given that the Shapley value belongs to the core of each subgame along the cooperative path, it can be used as a “supporting imputation” which guarantees that the whole core is STC. We discuss three payment schedules and check whether they can be used as supporting imputation distribution procedures for the considered multicriteria game.

Strongly Time-Consistent Core in Differential Games with Discrete Distribution of Random Time Horizon

In this paper we investigate the problem of strong time-consistency of the core for a particular class of differential games with random time horizon, namely, it is assumed that there exists a set of probabilities of the end of the game at discrete time instants. A condition guaranteeing the strong time consistency of the core is presented.

Sustainable cooperation in multicriteria multistage games

We use the imputation distribution procedure approach to ensure sustainable cooperation in a multistage game with vector payoffs. In order to choose a particular Pareto optimal and time consistent strategy profile and the corresponding cooperative trajectory we suggest a refined leximin algorithm. Using this algorithm we design a characteristic function for a multistage multicriteria game. Furthermore, we provide sufficient conditions for strong time consistency of the core.

On the Strong Time Consistency of the Core

Time consistency is one of desirable properties for any solution of a cooperative dynamic game. If a solution is time-consistent, the players do not need to break a cooperative agreement. In this paper, we consider the core as the solution and establish conditions for its strong time consistency. When the core is not strongly time-consistent, we show that in some cases its elements can be yielded using a strongly time-consistent imputation distribution procedure. An explicit form of the procedure is given.

Emission Taxation, Green Innovations and Inverted-U Aggregate R&D Efforts in a Linear State Oligopoly Game

We revisit the well known differential Cournot game with polluting emissions dating back to Benchekroun and Long (1998), proposing a version of the model in which environmental taxation is levied on emissions rather than the environmental damage. This allows to attain strong time consistency under open-loop information, and yields two main results which can be summarized as follows: (i) to attain a fully green technology in steady state, the regulator may equivalently adopt an appropriate tax rate (for any given number of firms) or regulate market access (for any given tax rate); (ii) if the environmental damage depends on emissions only (i.e., not on industry output) then the aggregate green R&D effort takes an inverted-U shape, in accordance with Aghion et al. (2005), and the industry structure maximising aggregate green innovation also minimises individual and aggregate emissions.

Cooperation in two-stage games on undirected networks

In the paper, cooperative two-stage network games are studied. At the first stage of the game, players form a network, while at the second stage players choose their behaviors according to the network realized at the first stage. As a cooperative solution concept in the game, the core is considered. It is proved that some imputations from the core are time inconsistent, whereas one can design for them a time-consistent imputation distribution procedure. Moreover, the strong time consistency problem is also investigated.

Stable cooperation in dynamic vehicle routing problems

This paper explores transportation cost minimization in the vehicle routing problem under carriers' cooperation on large-scale transport networks. The underlying model is a dynamic cooperative game with transferable utility (TU) functions. We propose a heuristic iterative algorithm for constructing the subadditive characteristic function of the corresponding static game. The algorithm is modified in order to construct the characteristic function of the dynamic routing game. The Shapley value and the subcore concept serve to form the optimal solutions. And finally, we describe a cost distribution procedure in the dynamic game, which guarantees the strong time consistency of the subcore and the dynamic stability of the cooperation agreement.

Strongly Time-consistent Solutions for Two-stage Network Games

In the paper, cooperative two-stage network games are studied. On the first stage of the game players form a network, while on the second stage players choose their behaviors according to the network realized on the first stage. As a cooperative solution concept in the game the core is considered. It is proved that some imputations are time-inconsistent, whereas one can design time-consistent imputation distribution procedure for them. Moreover, strong time-consistency is also considered.

Tail VaR Measures in a Multi-period Setting

This paper studies a coherent acceptability measure which is a negative coherent risk measure, in a multi-period model. When a coherent acceptability measure changes according to new information in the market, a time consistency plays an important role. The usual strong time consistency gives too severe a multi-period Tail Value at Risk (Tail VaR) from a practical viewpoint. We study a weak type of time consistency and propose new multi-period Tail VaR measures.

Weak and Strong Time Consistency

We motivate and provide proofs of Ba?ar and Olsder’s (1995) theorems on the subject. The context is the increasing appreciation that the neoclassical framework is not the only model of the economy.

The Leitmann–Schmitendorf advertising game with n players and time discounting

The extension of the Leitmann–Schmitendorf advertising game to n players and positive time discounting is investigated. We show that the strong time consistency of the open-loop Nash equilibrium is preserved. As to optimal controls, while the boundary solution is unaffected by the number of firms as well as discounting, the inner solution depends on industry structure. The fully symmetric version of the game allows us to identify the parameter regions wherein both solutions are sustainable.

Weak and Strong Time Consistency in a Differential Oligopoly Game with Capital Accumulation

We illustrate a differential oligopoly game with capital accumulation where the accumulation dynamics of productive capacity is modelled à la Ramsey. The model is solved under the open-loop information structure, to show that it admits an open-loop Nash equilibrium which is indeed a degenerate feedback one and therefore strongly time consistent, even if, by construction, the problem under consideration is not a linear state game.