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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.
We introduce the notion of a mathematical game. We give examples and classify them into various types, such as two-person games vs. n-person games (where n > 2), and zero-sum vs. constant-sum vs. variable-sum games. We carefully delineate the assumptions under which we operate in game theory. We illustrate how two-person games can be described by payoff matrices or by game trees. Using examples, including an analysis of the Battle of the Bismarck Sea from World War II, we develop the notions of a strategy, dominant strategy, and Nash equilibrium point of a game. Specializing to constant-sum games, we show the equivalence between Nash equilibrium and saddle point of a payoff matrix. We then consider games where the payoff matrix has no saddle point and develop the notion of a mixed strategy, after a quick review of some basic probability notions. Finally, we introduce the minimax theorem, which states that all constant-sum games have an optimal solution, and give a novel proof of the theorem in case the payoff matrix is 2 x 2.
Because both negotiators face the paradox, they are stuck in a dilemma, much like the prisoner’s dilemma. They need the other side to succeed, and indeed can achieve win–win with them. But greed and fear typically lead to a lose–lose outcome instead. We differentiate one-off interactions from repeated interactions to draw lessons from game theory, pondering the significance of metaphors in our decision-making as well as historical examples. We learn of a practical strategy that can help us negotiate in repeated interactions, changing the nature of the game to the “stag hunt game.”
A foundation of the economic analysis of policy instruments and human behaviour is game theory. The present chapter presents some basics of game theory in a nutshell. The first section formulates the social dilemma described in Chapter 9 in the language of game theory, introducing in particular the prisoner’s dilemma. The following section defines the concept of Nash equilibrium and argues that players in the prisoner’s dilemma are trapped in an unfavourable Nash equilibrium. Two other popular games relevant in the context of biodiversity conservation are presented: the coordination game and the chicken game. The final section of the chapter outlines evolutionary game theory which analyses the evolution of behaviour. Of particular interest within the context of social and prisoner’s dilemmas is the evolution and stability of cooperation among agents.
This chapter introduces the basic models used to study imperfect competition: Bertrand, Cournot, Stackleberg, and Hotelling. Applications of these models are also described. Integrated into the exposition is an introduction to game theory and the concept of Nash equilibrium.
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.
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.
This paper uses recent results on continuous-time finite-horizon optimal switching problems with negative switching costs to prove the existence of a saddle point in an optimal stopping (Dynkin) game. Sufficient conditions for the game's value to be continuous with respect to the time horizon are obtained using recent results on norm estimates for doubly reflected backward stochastic differential equations. This theory is then demonstrated numerically for the special cases of cancellable call and put options in a Black‒Scholes market.
A finite-element multigrid scheme for elliptic Nash-equilibrium multiobjective optimal control problems with control constraints is investigated. The multigrid computational framework implements a nonlinear multigrid strategy with collective smoothing for solving the multiobjective optimality system discretized with finite elements. Error estimates for the optimal solution and two-grid local Fourier analysis of the multigrid scheme are presented. Results of numerical experiments are presented to demonstrate the effectiveness of the proposed framework.
In this paper we study a reinsurance game between two insurers whose surplus processes are modeled by arithmetic Brownian motions. We assume a minimax criterion in the game. One insurer tries to maximize the probability of absolute dominance while the other tries to minimize it through reinsurance control. Here absolute dominance is defined as the event that liminf of the difference of the surplus levels tends to -∞. Under suitable parameter conditions, the game is solved with the value function and the Nash equilibrium strategy given in explicit form.
In this paper we study the Nash equilibrium in a smooth public goods economy, described as a non-cooperative game, where the set of players is a mixed measure space of consumers. We assume a finite number of private goods. We show that under certain conditions there exists a unique Nash equilibrium in the economy, where the public goods are produced with a linear technology. Moreover, we discuss the difference in market power between an atomless sector and an atom with the same utility function, and an atom with its split atomless sector, both in a pure exchange economy and a public goods economy.
Insecticide-treated nets (ITNs) are a major tool to control malaria. Over recent years increased ITN coverage has been associated with decreased malaria transmission. However, ITN ‘misuse’ has been increasingly reported and whether this emergent behaviour poses a threat to successful malaria control and elimination is an open question. Here, we use a game theory mathematical model to understand the possible roles of poverty and malaria infection protection by individual and emerging ‘community effects’ on the ‘misuse’ of malaria bednets. We compare model predictions with data from our studies in Lake Victoria Islands (LVI), Kenya and Aneityum, Vanuatu. Our model shows that alternative ITN use is likely to emerge in impoverished populations and could be exacerbated if ITNs become ineffective or when large ‘community effects’ emerge. Our model predicted patterns of ITN use similar to the observed in LVI, where ‘misuse’ is common and the high ITN use in Aneityum, more than 20 years after malaria elimination in 1990. We think that observed differences in ITN use may be shaped by different degrees of economic and social development, and educational components of the Aneityum elimination, where traditional cooperative attitudes were strengthened with the malaria elimination intervention and post-elimination surveillance.
Traffic flow is modeled by a conservation law describing the density of cars. It is
assumed that each driver chooses his own departure time in order to minimize the sum of a
departure and an arrival cost. There are N groups of drivers, The
i-th group consists of κi
drivers, sharing the same departure and arrival costs
For any given population sizes
we prove the existence of a Nash equilibrium solution, where no driver can lower his own
total cost by choosing a different departure time. The possible non-uniqueness, and a
characterization of this Nash equilibrium solution, are also discussed.
A two-server service network has been studied from the principal-agent perspective. In the model, services are rendered by two independent facilities coordinated by an agency, which seeks to devise a strategy to suitably allocate customers to the facilities and to simultaneously determine compensation levels. Two possible allocation schemes were compared — viz. the common queue and separate queue schemes. The separate queue allocation scheme was shown to give more competition incentives to the independent facilities and to also induce higher service capacity. In this paper, we investigate the general case of a multiple-server queueing model, and again find that the separate queue allocation scheme creates more competition incentives for servers and induces higher service capacities. In particular, if there are no severe diseconomies associated with increasing service capacity, it gives a lower expected sojourn time in equilibrium when the compensation level is sufficiently high.
We study a stochastic differential game between two insurance companies who employ reinsurance to reduce the risk of exposure. Under the assumption that the companies have large insurance portfolios compared to any individual claim size, their surplus processes can be approximated by stochastic differential equations. We formulate competition between the two companies as a game with a single payoff function which depends on the surplus processes. One company chooses a dynamic reinsurance strategy in order to maximize this expected payoff, while the other company simultaneously chooses a dynamic reinsurance strategy so as to minimize the same quantity. We describe the Nash equilibrium of this stochastic differential game and solve it explicitly for the case of maximizing/minimizing the exit probability.