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"Differential games."
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ON THE CONVERGENCE OF CLOSED-LOOP NASH EQUILIBRIA TO THE MEAN FIELD GAME LIMIT
This paper continues the study of the mean field game (MFG) convergence problem: In what sense do the Nash equilibria of n-player stochastic differential games converge to the mean field game as n→∞? Previous work on this problem took two forms. First, when the n-player equilibria are open-loop, compactness arguments permit a characterization of all limit points of n-player equilibria as weak MFG equilibria, which contain additional randomness compared to the standard (strong) equilibrium concept. On the other hand, when the n-player equilibria are closed-loop, the convergence to the MFG equilibrium is known only when the MFG equilibrium is unique and the associated “master equation” is solvable and sufficiently smooth. This paper adapts the compactness arguments to the closed-loop case, proving a convergence theorem that holds even when the MFG equilibrium is nonunique. Every limit point of n-player equilibria is shown to be the same kind of weak MFG equilibrium as in the open-loop case. Some partial results and examples are discussed for the converse question, regarding which of the weak MFG equilibria can arise as the limit of n-player (approximate) equilibria.
Journal Article
Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equations
We study singular perturbations of optimal stochastic control problems and differential games arising in the dimension reduction of
system with multiple time scales. We analyze the uniform convergence of the value functions via the associated
Hamilton-Jacobi-Bellman-Isaacs equations, in the framework of viscosity solutions. The crucial properties of ergodicity and
stabilization to a constant that the Hamiltonian must possess are formulated as differential games with ergodic cost criteria. They are
studied under various different assumptions and with PDE as well as control-theoretic methods. We construct also an explicit example
where the convergence is not uniform. Finally we give some applications to the periodic homogenization of Hamilton-Jacobi equations with
non-coercive Hamiltonian and of some degenerate parabolic PDEs.
eBook
Differential game analysis of carbon emissions reduction and promotion in a sustainable supply chain considering social preferences
We incorporate consumer low-carbon awareness and social preferences, including relationship and status preferences, into a game of emissions reduction and promotion involving one manufacturer and one retailer using a long-term perspective. We investigate the channel members’ decision making and performance under three scenarios, including a decentralized scenario both with and without a cost-sharing contract as well as a centralized scenario. In addition, we examine the effects of some key parameters on the channel members’ decisions and performance. Our study finds that improving consumer low-carbon awareness is beneficial for carbon emissions reduction and both channel members’ utilities. A cost-sharing contract can incentivize the retailer to improve promotion efforts, and the manufacturer’s optimal emissions reduction effort is independent of the cost-sharing contract used. The cost-sharing proportion increases as the manufacturer gives more weight to the relationship and decreases as the retailer gives more weight to the relationship. A cost-sharing contract changes the effect of channel members’ social preferences and marginal profits on their decisions. Most importantly, the supply chain system can achieve Pareto improvement with a cost-sharing contract. If the manufacturer aims to optimize the supply chain’s total profit, then the supply chain system can achieve perfect coordination with a cost-sharing contract.
Journal Article
Stochastic Graphon Games: II. The Linear-Quadratic Case
In this paper, we analyze linear-quadratic stochastic differential games with a continuum of players interacting through graphon aggregates, each state being subject to idiosyncratic Brownian shocks. The major technical issue is the joint measurability of the player state trajectories with respect to samples and player labels, which is required to compute for example costs involving the graphon aggregate. To resolve this issue we set the game in a Fubini extension of a product probability space. We provide conditions under which the graphon aggregates are deterministic and the linear state equation is uniquely solvable for all players in the continuum. The Pontryagin maximum principle yields equilibrium conditions for the graphon game in the form of a forward-backward stochastic differential equation, for which we establish existence and uniqueness. We then study how graphon games approximate games with finitely many players over graphs with random weights. We illustrate some of the results with a numerical example.
Journal Article
ZERO-SUM PATH-DEPENDENT STOCHASTIC DIFFERENTIAL GAMES IN WEAK FORMULATION
We consider zero-sum stochastic differential games with possibly pathdependent volatility controls. Unlike the previous literature, we allow for weak solutions of the state equation so that the players’ controls are automatically of feedback type. In particular, we do not require the controls to be “simple,” which has fundamental importance for the possible existence of saddle-points. Under some restrictions, needed for the a priori regularity of the upper and lower value functions of the game, we show that the game value exists when both the appropriate path-dependent Isaacs condition, and the uniqueness of viscosity solutions of the corresponding path-dependent Isaacs-HJB equation hold. We also provide a general verification argument and a characterisation of saddle-points by means of an appropriate notion of second-order backward SDE.
Journal Article
Differential game of product–service supply chain considering consumers’ reference effect and supply chain members’ reciprocity altruism in the online-to-offline mode
Supply chain members’ reciprocal altruism and consumers’ quality and service reference effects are important behavioral factors that affect the decision-making of supply chain members. This article incorporates these factors into a product–service supply chain consisting of a manufacturer and a retailer in the online-to-offline (O2O) environment. Based on the inherent dynamics of the model, we construct a differential game model between the manufacturer and the retailer. Based on the Bellman continuous dynamic programming theory, this study analyzes the quality strategy of the manufacturer, the service level strategy of the retailer, and the performance of the supply chain system under three decision-making patterns (decentralization, centralization, and reciprocal altruism) within the O2O framework. The results show that compared with the decentralized decision-making model, reciprocal altruism helps members develop higher quality and service levels, improve brand goodwill, and obtain greater utility. The results are verified by numerical examples, and sensitivity analysis of consumer quality and the service reference effect, channel preference, and members’ reciprocal altruism behavior on the supply chain performance is carried out. The results show: (1) Consumers’ reference effects cause an “anchoring mentality” among consumers, which leads the manufacturer to lower the quality level and the retailer to lower the service level. This hurts the performance of the product–service supply chain. Consumers’ channel preference has an important impact on supply chain members’ strategies and performance. (2) Retailers should encourage consumers to purchase products online and use offline channel services as sales assistance measures to satisfy consumers’ experience utility. (3) As a positive social preference, the supply chain performance under the members’ reciprocal altruism decision-making model is Pareto-improved and receives additional social benefits. (4) Only when the manufacturer and the retailer have pure altruistic preference, that is, minimum return and maximum altruism, the total profit of the supply chain can reach that of the centralized decision-making scenario.
Journal Article
Dynamic Pricing of Perishable Assets Under Competition
We study dynamic price competition in an oligopolistic market with a
mix
of substitutable and complementary perishable assets. Each firm has a fixed initial stock of items and competes in setting prices to sell them over a finite sales horizon. Customers sequentially arrive at the market, make a purchase choice, and then leave immediately with some likelihood of no purchase. The purchase likelihood depends on the time of purchase, product attributes, and current prices. The demand structure includes time-variant linear and multinomial logit demand models as special cases. Assuming deterministic customer arrival rates, we show that any equilibrium strategy has a simple structure, involving a finite set of
shadow prices
measuring capacity externalities that firms exert on each other: equilibrium prices can be solved from a one-shot price competition game under the
current-time
demand structure, taking into account capacity externalities through the
time-invariant
shadow prices. The former reflects the transient demand side at every moment, and the latter captures the aggregate supply constraints over the sales horizon. This simple structure sheds light on dynamic revenue management problems under competition, which helps capture the essence of the problems under demand uncertainty. We show that the equilibrium solutions from the deterministic game provide precommitted and contingent heuristic policies that are asymptotic equilibria for its stochastic counterpart, when demand and supply are sufficiently large.
This paper was accepted by Yossi Aviv, operations management
.
Journal Article
Algorithm for Overcoming the Curse of Dimensionality For Time-Dependent Non-convex Hamilton–Jacobi Equations Arising From Optimal Control and Differential Games Problems
In this paper we develop a parallel method for solving possibly non-convex time-dependent Hamilton–Jacobi equations arising from optimal control and differential game problems. The subproblems are independent so they can be implemented in an embarrassingly parallel fashion, which usually has an ideal parallel speedup. The algorithm is proposed to overcome the curse of dimensionality (Bellman in Adaptive control processes: a guided tour. Princeton University Press, Princeton,
1961
; Dynamic programming. Princeton University Press, Princeton,
1957
) when solving HJ PDE . We extend previous work Chow et al. (Algorithm for overcoming the curse of dimensionality for certain non-convex Hamilton–Jacobi equations, Projections and differential games, UCLA CAM report, pp 16–27,
2016
) and Darbon and Osher (Algorithms for overcoming the curse of dimensionality for certain Hamilton–Jacobi equations arising in control theory and elsewhere, UCLA CAM report, pp 15–50,
2015
) and apply a generalized Hopf formula to solve HJ PDE involving time-dependent and perhaps non-convex Hamiltonians. We suggest a coordinate descent method for the minimization procedure in the Hopf formula. This method is preferable when even the evaluation of the function value itself requires some computational effort, and also when we handle higher dimensional optimization problem. For an integral with respect to time inside the generalized Hopf formula, we suggest using a numerical quadrature rule. Together with our suggestion to perform numerical differentiation to minimize the number of calculation procedures in each iteration step, we are bound to have numerical errors in our computations. These errors can be effectively controlled by choosing an appropriate mesh-size in time and the method does not use a mesh in space. The use of multiple initial guesses is suggested to overcome possibly multiple local extrema in the case when non-convex Hamiltonians are considered. Our method is expected to have wide application in control theory and differential game problems, and elsewhere.
Journal Article