Imputation Distribution Procedure Research Articles

Cooperative differential games with the utility function switched at a random time moment

This paper describes a differential game of $n$ persons in which the utility functions of the players have a hybrid form, namely, they are changed at a random moment in time. With the help of integration in parts, the form of the payoff functional is simplified. For the cooperative scenario the problem of time-consistency of the optimality principle chosen by the players is studied and a solution is proposed in the form of an adapted imputation distribution procedure. The differential investment game is considered as an example.

Dynamic Cost-Sharing Game with Spanning Arborescence

This paper presents the dynamic Shapley value for cost-sharing game with spanning arborescence. The cooperative behaviour of players is determined, and a two-stage directed network game is considered. At each stage, a cost matrix associated with the directed network is defined by players adopting strategies, and a minimum cost spanning arborescence on the directed network is determined. After the first stage, a particular player will leave the game with a certain probability, which depends on all players' behaviours in the first stage. The characteristic function is defined. Using the Imputation Distribution Procedure(IDP), the dynamic Shapley value in the game is constructed.

Two-stage network games modeling the Belt and Road Initiative

Inspired by the Belt and Road Initiative, we introduce a model of two-stage network games, when players first form a directed network (the first stage) and then they may reconsider the actions made at the first stage and choose controls to influence other players in a positive or negative way (the second stage). At both stages players get their payoffs. Considering a cooperative version of the game, we examine the problem of subgame network consistency and design an imputation distribution procedure as a new system of stage payments to guarantee the long-term cooperation. The Shapley value with exogenous directed graph constraint is used as a cooperative solution concept. Finally, we prove that a cooperative subgame is convex which ensures the non-emptiness of the core.

Time-Consistency of an Imputation in a Cooperative Hybrid Differential Game

This work is aimed at studying the problem of maintaining the sustainability of a cooperative solution in an n-person hybrid differential game. Specifically, we consider a differential game whose payoff function is discounted with a discounting function that changes its structure with time. We solve the problem of time-inconsistency of the cooperative solution using a so-called imputation distribution procedure, which was adjusted for this general class of differential games. The obtained results are illustrated with a specific example of a differential game with random duration and a hybrid cumulative distribution function (CDF). We completely solved the presented example to demonstrate the application of the developed scheme in detail. All results were obtained in analytical form and illustrated by numerical simulations.

Subgame Consistent Cooperative Behavior in an Extensive form Game with Chance Moves

We design a mechanism of the players’ sustainable cooperation in multistage n-person game in the extensive form with chance moves. When the players agreed to cooperate in a dynamic game they have to ensure time consistency of the long-term cooperative agreement. We provide the players’ rank based (PRB) algorithm for choosing a unique cooperative strategy profile and prove that corresponding optimal bundle of cooperative strategies satisfies time consistency, that is, at every subgame along the optimal game evolution a part of each original cooperative trajectory belongs to the subgame optimal bundle. We propose a refinement of the backwards induction procedure based on the players’ attitude vectors to find a unique subgame perfect equilibrium and use this algorithm to calculate a characteristic function. Finally, to ensure the sustainability of the cooperative agreement in a multistage game we employ the imputation distribution procedure (IDP) based approach, that is, we design an appropriate payment schedule to redistribute each player’s optimal payoff along the optimal bundle of cooperative trajectories. We extend the subgame consistency notion to extensive-form games with chance moves and prove that incremental IDP satisfies subgame consistency, subgame efficiency and balance condition. An example of a 3-person multistage game is provided to illustrate the proposed cooperation mechanism.

IDP-Core: Novel Cooperative Solution for Differential Games

IDP-core is a new cooperative solution for dynamic and differential games. A novel approach of constructing solutions for dynamic and differential games was employed in which the time consistency property was used as the main axiom property for the cooperative solution. Another new and important approach used for constructing IDP-core is the IDP dominance, which allows to select undominated imputation distribution procedures and construct the cooperative solution or imputation set. This approach shows the potential of using the time consistency property as the main axiom for solutions in various fields such as Social Choice and Mechanism Design. The overall procedure for defining the cooperative solution is also new since IDP-core was constructed using imputation distribution procedures but not by using imputations directly.

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.

Looking forward approach with random horizon in cooperative differential games

In the paper authors present a new approach to determination and computation of a solution for differential games with prescribed duration in the case when players lack certain information about the dynamical system and payoff function on the whole time interval on which the game is played. At each time instant players receive information about dynamical system and payoff functions, however the duration of the period of this information is unknown and can be represented as a random variable with known parameters. At certain periods of time the information is updated. A novel solution is described as a combination of imputation sets in the truncated subgames that are analyzed using Looking Forward Approach with random horizon. A resource extraction game serves as an illustration in order to compare a cooperative trajectory, imputations, and imputation distribution procedure in the game with Looking Forward Approach and in the original game with prescribed duration. Looking Forward Approach is used for constructing game theoretical models and defining solutions for conflict-controlled processes where information about the process updates dynamically.

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.

Strong Time-Consistent Subset of the Core in Cooperative Differential Games with Finite Time Horizon

Time consistency is one of the most important properties of solutions in cooperative differential games. This paper uses the core as a cooperative solution of the game. We design a strong time-consistent subset of the core. The design method of this subset is based on a special class of imputation distribution procedures (IDPs).

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 regularization of a cooperative solution in a multistage game with random time horizon

In this paper, we consider a general class of cooperative multistage games with random time horizon and discuss the problem of implementing a cooperative solution. It is known that in many cases a cooperative solution can be time-inconsistent and hence not realizable. To solve this problem, the imputation distribution procedure was proposed. However, the computed payment distribution scheme may result in negative payments which are not feasible. In this case, one has to carry out a regularization procedure as described in the paper. We describe a general regularization scheme and apply it both to the core and to the Shapley value. It is shown that for the mentioned two cases the regularization can be carried out in two alternative ways thus providing a basis for developing efficient numerical schemes. For the Shapley value the regularization procedure was elaborated and described in the form of an algorithm. The obtained results are illustrated with two numerical examples.

Payoff Distribution in a Multi-Company Extraction Game with Uncertain Duration

A nonrenewable resource extraction game model is analyzed in a differential game theory framework with random duration. If the cumulative distribution function (c.d.f.) of the final time is discontinuous, the related subgames are differentiated based on the position of the initial instant with respect to the jump. We investigate properties of optimal trajectories and of imputation distribution procedures if the game is played cooperatively.

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.

About the Looking Forward Approach in Cooperative Differential Games with Transferable Utility

This paper presents a complete description and the results of the Looking Forward Approach for cooperative differential games with transferable utility. The approach is used for constructing game theoretical models and defining solutions for conflict-controlled processes where information about the process updates dynamically or for differential games with dynamic updating. It is supposed that players lack certain information about the dynamical system and payoff function over the whole time interval on which the game is played. At each instant, information about the game structure updates, players receive new updated information about the dynamical system and payoff functions. A resource extraction game serves as an illustration in order to compare a cooperative trajectory, imputations, and the imputation distribution procedure in a game with the Looking Forward Approach and in the original game with a prescribed duration.

Cost allocation for the problem of pollution reduction: a dynamic cooperative game approach

This paper studies CO2 emissions at a global level. The authors use Dynamic Optimisation to derive the minimum penalty cost on countries every single time. They then use an Imputation Distribution Procedure to allocate the minimum penalty cost among countries. Their work provides the extension of the Shapley value cost allocation as a penalty to reduce CO2 emissions. The paper has implications for how to provide initiatives to improve cooperation on reducing CO2 emissions at an international level. Results show that a reduction in cost of only one country can be harmful for other countries. In this way, some countries can end up or worse off in a case where all countries experience a uniform decrease in their penalty cost. Therefore, the findings of this work suggest a low penalty-cost scenario that helps the countries fight for pollution reduction and provide fruitful links for policy-makers. They show that the Clean Development Mechanism (CDM) of the Kyoto Protocol could be implemented by the Shapley value cost allocation.

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.

Promotion of cooperation when benefits come in the future: A water transfer case

This paper presents a two-regime differential game, with a first period in which two countries cooperate in a joint investment project to construct a specific infrastructure. This period ends when the infrastructure is finished, which serves to increase each player's welfare in a subsequent non-cooperative game played by the two countries thereafter. We define an imputation distribution procedure (IDP) to share the investment costs during cooperation according to each player’ future benefits. We prove that the IDP is time consistent if at any time within the cooperative period each country's share on the surplus to go is equal to or converges towards the country's relative gains from the existence of the infrastructure (realized in the subsequent non-cooperative period). Furthermore, we obtain the instantaneous side-payment scheme which makes the IDP feasible. The mechanism is studied for a joint investment project to build a water canal to transfer water between a surplus and a deficit river basin.

Time consistency of the interval Shapley-like value in dynamic games

The paper focuses on the problem of time consistency of an interval Shapley-like value in cooperative interval dynamic games. This problem has not been investigated in the game theory literature for the considered class of dynamic games. It is proved that the interval Shapley-like value is time inconsistent, and to satisfy the time consistency condition for the interval value, it is necessary to introduce a new control variable, so-called imputation distribution procedure. The imputation distribution procedure is proposed in an explicit form.

Strategically supported cooperation in dynamic games with coalition structures

the problem of strategic stability of long-range cooperative agreements in dynamic games with coalition structures is investigated. Based on imputation distribution procedures, a general theoretical framework of the differential game with a coalition structure is proposed. A few assumptions about the deviation instant for a coalition are made concerning the behavior of a group of many individuals in certain dynamic environments. From these, the time-consistent cooperative agreement can be strategically supported by $\varepsilon$-Nash or strong $\varepsilon$-Nash equilibria. while in games in the extensive form with perfect information, it is somewhat surprising that without the assumptions of deviation instant for a coalition, Nash or strong Nash equilibria can be constructed.