This paper proposes a theoretical insurance model to explain well-documented loss underreporting and to study how strategic underreporting affects insurance demand. We consider a utility-maximizing insured who purchases a deductible insurance contract and follows a barrier strategy to decide whether she should report a loss. The insurer adopts a bonus-malus system with two rate classes, and the insured will move to or stay in the more expensive class if she reports a loss. First, we fix the insurance contract (deductibles) and obtain the equilibrium reporting strategy in semi-closed form. A key result is that the equilibrium barriers in both rate classes are strictly greater than the corresponding deductibles, provided that the insured economically prefers the less expensive rate class, thereby offering a theoretical explanation to underreporting. Second, we study an optimal deductible insurance problem in which the insured strategically underreports losses to maximize her utility. We find that the equilibrium deductibles are strictly positive, suggesting that full insurance, often assumed in related literature, is not optimal. Moreover, in equilibrium, the insured underreports a positive amount of her loss. Finally, we examine how underreporting affects the insurer’s expected profit.

Recently, in their 2019 paper, Poyago-Theotoky and Yong consider a managerial Cournot duopoly with pollution externalities and emission taxes and propose an explicit environmental incentive in a managerial compensation contract. The authors compare several exogenous equilibria emerging in the symmetric sub-games in which the owner offers either the environmental delegation contract or the standard sales delegation contract: abatement and social welfare (resp. emission taxes) under environmental delegation are higher (resp. lower) than under sales delegation. The present work extends their model using a game-theoretic approach to analyse the asymmetric sub-games, in which only one firm adopts the environmental contract, and adds the contract decision stage. Results show that the environmental contract never emerges as the unique sub-game perfect Nash equilibrium of this non-cooperative managerial decision game. Indeed, if the green R&D technology is efficient, the sales contract emerges as the unique Pareto-inefficient Nash equilibrium. Otherwise, if the green R&D technology is inefficient, multiple Nash equilibria in pure strategies exist (coordination game). Our findings offer direct policy implications.

In the context of propositional logics, we apply semantics modulo satisfiability—a restricted semantics which comprehends only valuations that satisfy some specific set of formulas—with the aim to efficiently solve some computational tasks. Three possible such applications are developed.

We begin by studying the possibility of implicitly representing rational McNaughton functions in Łukasiewicz Infinitely-valued Logic through semantics modulo satisfiability. We theoretically investigate some approaches to such representation concept, called representation modulo satisfiability, and describe a polynomial algorithm that builds representations in the newly introduced system. An implementation of the algorithm, test results and ways to randomly generate rational McNaughton functions for testing are presented. Moreover, we propose an application of such representations to the formal verification of properties of neural networks by means of the reasoning framework of Łukasiewicz Infinitely-valued Logic.

Then, we move to the investigation of the satisfiability of joint probabilistic assignments to formulas of Łukasiewicz Infinitely-valued Logic, which is known to be an NP-complete problem. We provide an exact decision algorithm derived from the combination of linear algebraic methods with semantics modulo satisfiability. Also, we provide an implementation for such algorithm for which the phenomenon of phase transition is empirically detected.

Lastly, we study the game theory situation of observable games, which are games that are known to reach a Nash equilibrium, however, an external observer does not know what is the exact profile of actions that occur in a specific instance; thus, such observer assigns subjective probabilities to players actions. We study the decision problem of determining if a set of these probabilistic constraints is coherent by reducing it to the problems of satisfiability of probabilistic assignments to logical formulas both in Classical Propositional Logic and Łukasiewicz Infinitely-valued Logic depending on whether only pure equilibria or also mixed equilibria are allowed. Such reductions rely upon the properties of semantics modulo satisfiability. We provide complexity and algorithmic discussion for the coherence problem and, also, for the problem of computing maximal and minimal probabilistic constraints on actions that preserves coherence.

Abstract prepared by Sandro Márcio da Silva Preto.

E-mail: spreto@ime.usp.br

URL:https://doi.org/10.11606/T.45.2021.tde-17062021-163257

We study an N-player game where a pure action of each player is to select a nonnegative function on a Polish space supporting a finite diffuse measure, subject to a finite constraint on the integral of the function. This function is used to define the intensity of a Poisson point process on the Polish space. The processes are independent over the players, and the value to a player is the measure of the union of her open Voronoi cells in the superposition point process. Under randomized strategies, the process of points of a player is thus a Cox process, and the nature of competition between the players is akin to that in Hotelling competition games. We characterize when such a game admits Nash equilibria and prove that when a Nash equilibrium exists, it is unique and consists of pure strategies that are proportional in the same proportions as the total intensities. We give examples of such games where Nash equilibria do not exist. A better understanding of the criterion for the existence of Nash equilibria remains an intriguing open problem.

In this paper, I revisit the question of how and in what sense can individuals comprising a group be held responsible for morally reprehensible behaviour by that group. The question is tackled by posing a counterfactual: what would happen if selfish individuals became moral creatures? A game called the Samaritan’s Curse is developed, which sheds light on the dilemma of group moral responsibility, and raises new questions concerning ‘conferred morality’ and self-fulfilling morals, and also forces us to question some implicit assumptions of game-theory.

In 2015, Guglielmi and Badia discussed optimal strategies in a particular type of service system with two strategic servers. In their setup, each server can be either active or inactive and an active server can be requested to transmit a sequence of packets. The servers have varying probabilities of successfully transmitting when they are active, and both servers receive a unit reward if the sequence of packets is transmitted successfully. Guglielmi and Badia provided an analysis of optimal strategies in four scenarios: where each server does not know the other’s successful transmission probability; one of the two servers is always inactive; each server knows the other’s successful transmission probability and they are willing to cooperate.

Unfortunately, the analysis by Guglielmi and Badia contained some errors. In this paper we correct these errors. We discuss three cases where both servers (I) communicate and cooperate; (II) neither communicate nor cooperate; (III) communicate but do not cooperate. In particular, we obtain the unique Nash equilibrium strategy in Case II through a Bayesian game formulation, and demonstrate that there is a region in the parameter space where there are multiple Nash equilibria in Case III. We also quantify the value of communication or cooperation by comparing the social welfare in the three cases, and propose possible regulations to make the Nash equilibrium strategy the socially optimal strategy for both Cases II and III.

Sufficient conditions are given under which ratifiable acts exist.

We consider a polling system with two queues, exhaustive service, no switchover times, and exponential service times with rate µ in each queue. The waiting cost depends on the position of the queue relative to the server: it costs a customer c per time unit to wait in the busy queue (where the server is) and d per time unit in the idle queue (where there is no server). Customers arrive according to a Poisson process with rate λ. We study the control problem of how arrivals should be routed to the two queues in order to minimize the expected waiting costs and characterize individually and socially optimal routeing policies under three scenarios of available information at decision epochs: no, partial, and complete information. In the complete information case, we develop a new iterative algorithm to determine individually optimal policies (which are symmetric Nash equilibria), and show that such policies can be described by a switching curve. We use Markov decision processes to compute the socially optimal policies. We observe numerically that the socially optimal policy is well approximated by a linear switching curve. We prove that the control policy described by this linear switching curve is indeed optimal for the fluid version of the two-queue polling system.

In this paper we establish a new connection between a class of two-player nonzero-sum games of optimal stopping and certain two-player nonzero-sum games of singular control. We show that whenever a Nash equilibrium in the game of stopping is attained by hitting times at two separate boundaries, then such boundaries also trigger a Nash equilibrium in the game of singular control. Moreover, a differential link between the players' value functions holds across the two games.

We consider nonzero-sum games where multiple players control the drift of a process, and their payoffs depend on its ergodic behaviour. We establish their connection with systems of ergodic backward stochastic differential equations, and prove the existence of a Nash equilibrium under generalised Isaac's conditions. We also study the case of interacting players of different type.

We present solutions to nonzero-sum games of optimal stopping for Brownian motion in [0, 1] absorbed at either 0 or 1. The approach used is based on the double partial superharmonic characterisation of the value functions derived in Attard (2015). In this setting the characterisation of the value functions has a transparent geometrical interpretation of 'pulling two ropes' above 'two obstacles' which must, however, be constrained to pass through certain regions. This is an extension of the analogous result derived by Peskir (2009), (2012) (semiharmonic characterisation) for the value function in zero-sum games of optimal stopping. To derive the value functions we transform the game into a free-boundary problem. The latter is then solved by making use of the double smooth fit principle which was also observed in Attard (2015). Martingale arguments based on the Itô–Tanaka formula will then be used to verify that the solution to the free-boundary problem coincides with the value functions of the game and this will establish the Nash equilibrium.

In this paper we study a stochastic differential game between two insurers whose surplus processes are modelled by quadratic-linear diffusion processes. We consider an exit probability game. One insurer controls its risk process to minimize the probability that the surplus difference reaches a low level (indicating a disadvantaged surplus position of the insurer) before reaching a high level, while the other insurer aims to maximize the probability. We solve the game by finding the value function and the Nash equilibrium strategy in explicit forms.