Leader-follower Differential Game Research Articles

A Differential Game for Optimal Water Price Management

This article studies a leader–follower differential game with a finite horizon, where a single buyer reacts to the selling price set by an agency (water supplier). The Open-Loop Stackelberg equilibrium is calculated, assuming that the user demand is fully satisfied (that is, the interior solution is considered), and the following different tariff schemes are analyzed: linear scheme, increasing block tariff, and convex tariff. Numerical simulations highlight how tariff convexity and seasonality in buyer’s preferences affect water price and demand, and the dynamics of the basin over time. The study shows that synchrony or asynchrony between basin recharge and buyer cyclical demand can dramatically affect the dynamics and basin levels observed at the end of the time period considered. Additionally, the presence of a large number of fluctuations in buyer preferences affects basin fluctuations, while natural recharge may help in maintaining acceptable levels of future water demands.

Inter-temporal decisions, optimal taxation and non-compliant behaviors in groundwater management

A sizable part of the water extracted is unauthorized. This phenomenon may contribute to exacerbate the problem of groundwater over-exploitation. To consider both issues, we study the interaction between the water agency and farmers through a leader–follower differential game in which both agents are perfect foresight. Since the farmers have to pay a tax on individual withdrawals imposed by the water agency to manage the groundwater, illegal behaviors may arise to save this cost. However, if discovered, the farmers are punished with an administrative sanction. The game is solved using feedback Stackelberg solution. Moreover, to enrich the model’s policy suggestions, we also consider an alternative context in which farmers can adopt a trigger strategy. Finally, we perform numerical simulations based on the western La Mancha (Spain) aquifer data to better understand both the analytical results and the effects of the sanction mechanism on non-compliant behaviors.

Optimal control of stochastic singular affine systems with Markovian jumps

We consider an optimal control problem for a class of stochastic singular affine systems with Markovian jumps. We establish the existence and uniqueness of the solution to stochastic singular affine systems with Markovian jumps for the first time. Via square completion technique and the generalized Itô’s formula, we derive new kinds of generalized differential Riccati equations (GDREs) and generalized backward differential equations (GBDEs), which give sufficient conditions for the well-posedness of the optimal control problem, and present an explicit representation of optimal control. Also, we discuss the solvability of the GDREs in two cases. As an application, we present a leader-follower differential game to demonstrate the practicability of our results.

Groundwater Exploitation and Illegal Behaviors in a Differential Game

This paper studies groundwater management through a leader–follower differential game between a water agency and farmers. The access to the common pool is not free and farmers have to pay a water tax to withdraw. To save this cost, we assume that farmers may not declare all the water pumped and face the risk of being sanctioned. The water agency, on the other hand, knows that unauthorized water extraction may occur and chooses the water tax that maximizes only the legal social welfare. The value of the farmers’ evasion share is unknown ex-ante by the water agency. The game is solved using feedback Nash solution. To understand how the water agency may counter illegal behaviors, we perform numerical simulations based on the Western La Mancha (Spain) aquifer data. It emerges that the optimal path of the water table is always above the minimum level, although it is sensitive to the level of the ecosystem costs. Moreover, increasing the administrative sanction reduces the unauthorized water extraction and raises the social welfare.

ON A NEW PARADIGM OF OPTIMAL REINSURANCE: A STOCHASTIC STACKELBERG DIFFERENTIAL GAME BETWEEN AN INSURER AND A REINSURER

AbstractThis paper proposes a new continuous-time framework to analyze optimal reinsurance, in which an insurer and a reinsurer are two players of a stochastic Stackelberg differential game, i.e., a stochastic leader-follower differential game. This allows us to determine optimal reinsurance from joint interests of the insurer and the reinsurer, which is rarely considered in the continuous-time setting. In the Stackelberg game, the reinsurer moves first and the insurer does subsequently to achieve a Stackelberg equilibrium toward optimal reinsurance arrangement. Speaking more precisely, the reinsurer is the leader of the game and decides on an optimal reinsurance premium to charge, while the insurer is the follower of the game and chooses an optimal proportional reinsurance to purchase. Under utility maximization criteria, we study the game problem starting from the general setting with generic utilities and random coefficients to the special case with exponential utilities and constant coefficients. In the special case, we find that the reinsurer applies the variance premium principle to calculate the optimal reinsurance premium and the insurer's optimal ceding/retained proportion of insurance risk depends not only on the risk aversion of itself but also on that of the reinsurer.

The Differential Game Analysis on the Construction Problem of the Public Libraries

Using the differential game theory, we studied the investment problem on public libraries, and established the Leader-follower differential game model, which is between the governments and the private enterprises (social organization). By building the Hamiltonian solving the Stackelberg balanced optimal solutions in the main two game players when they choose their strategies. By adjusting the strategy of government and social utility’s coefficient of dynamic to increase the comprehensive social utility of libraries.