This study investigates the optimal weighted reinsurance contract from the perspectives of reinsurance demand and supply within the framework of Stackelberg–Nash differential game. In this paper, both the insurer and reinsurer are allowed to negotiate and select their preferred ceded/accepted proportions and reinsurance premiums. The ultimate reinsurance contract is established by their optimal strategies and bargaining powers. It is assumed that both the insurer and reinsurer aim to maximize the expected utility of their terminal wealth. Based on game-theoretic arguments and the dynamic programming approach, we derive the explicit Stackelberg–Nash equilibrium strategies. To better understand how bargaining power affects the optimal reinsurance contract, we also examine three degenerate optimization problems. Our findings indicate that to successfully establish a reinsurance contract, the bargaining powers of the insurer and reinsurer must not be excessively unbalanced. Furthermore, under suitable parameter conditions, the Stackelberg–Nash contract may improve the expected utilities of both parties relative to the classical Stackelberg model. Additional theoretical analyses and numerical examples are provided to illustrate the economic intuition and insights underlying the results.
This paper focuses on a fluid model for a non-Markovian many-server queueing network with abandonment, where externally arrived and internally routed customers are served under the non-idling First-Come-First-Serve (FCFS) discipline at each station of many parallel servers. The routing follows a Markovian mechanism. Externally arrived and internally routed customers in each queue may have different service time distributions, as well as different patience time distributions, and all these distributions may be station-dependent. The fluid model dynamics is described by the fluid contents of externally arrived customers and internally routed customers in each queue (both waiting and receiving service) and a set of four measure-valued processes, tracking the amount of service time each externally arrived customer in service has received, the amount of service time each internally routed customer in service has received, the waiting times of externally arrived customers and the waiting times of internally routed customers in queue. Under mild conditions on the service and patience time distributions, we prove the existence and uniqueness of a solution to the fluid model equations. We also characterize the invariant states of this fluid model when the arrival rates are constant. The convergence of the properly scaled stochastic processes describing the network dynamics to the unique solution to fluid model equations is also established.
A TU-game with coalition structure is a combination of a TU-game and a partition of the player set which represents the cooperation restrictions among the players. This paper introduces and axiomatically characterizes the block-solidarity value, a novel solution concept for this class of games which is defined as the Shapley value of the solidarity restricted game. Given a TU-game with coalition structure, the solidarity restricted game assumes that each element of the coalition structure can be represented by any of its members and the worth assigned to a coalition is the worth of the union of the elements of the partition represented in that coalition.
We consider the problem of assigning the entries of a household data set to real-world address data. This household assignment problem occurs in the geo-referencing step of spatial microsimulation models. The resulting combinatorial optimization model is a maximum weight matching problem with additional side constraints. Even for real-world instances of medium size, such as the city of Trier in Germany, the number of binary variables in the model exceeds 10^9 and the resulting instances are computationally intractable for standard solvers for mixed-integer linear optimization. Hence, we derive two methods to compute feasible points of good quality, one based on the Lagrangian relaxation of the side constraints and the other one based on problem-tailored decomposition strategies. We theoretically analyze the feasible points obtained by both methods. Moreover, we extensively test the two proposed methods on both real-world and synthetic data sets. Our results show that the proposed methods are significantly faster than Gurobi applied to the original mixed-integer problem while computing near-optimal points. The methods are more efficient in terms of memory usage, which is extremely high for solvers such as Gurobi when applied to realistic instances, rendering their application to real-world instances infeasible. Finally, our results for the city of Trier show a realistic demographic distribution, which illustrates the applicability of our approach in practice.
We consider an insurance company endowed with an initial capital and a surplus driven by a mean-avoiding Ornstein-Uhlenbeck (OU) process. The company’s target is to maximise the expected discounted unrestricted dividends. The parameters of the considered OU process are crucial for finding the optimal strategy. To solve the problem, we distinguish between three different cases. In the first case, the absolute value of the speed rate of the OU process is assumed to be strictly smaller than the constant discounting rate. Here, we are able to find both the optimal strategy – a bang-bang strategy with a constant barrier – and the value function. If the absolute value of the speed is equal to the discounting rate, we find an explicit expression for the value function and prove that the optimal strategy does not exist. In the last case, when the absolute value of the speed rate exceeds the discounting rate, the value function is infinite.
