Pareto Optimality, Game Theory And Equilibria

In this chapter, we give an overview on the theory of noncooperative games. In the first part, we consider in detail zero-sum (constant-sum) games with two players having arbitrary strategy sets, under which necessary and sufficient conditions on the payoff function and the different (extended strategy) sets an equilibrium (saddle-point) strategy (for both players) exists. The existence of such an equilibrium strategy is equivalent to whether a so-called minimax theorem for the payoff function holds. The proof of such a result uses either the separation result for disjoint convex sets in finite dimensional linear spaces or the strong duality theorem for linear programming in combination with some elementary properties of compact sets. Both proof techniques are given together with a discussion of the most well-known minimax theorems that appeared in the literature. Also for the most famous minimax result given by Sion, we separately show an elementary proof avoiding the KKM lemma and using only the definition of connectedness. In the final part, we also consider n-person nonzero-sum noncooperative games. It is shown by a simple application of the same KKM lemma that a Nash equilibrium strategy (a generalization of a (saddle-point) equilibrium strategy for two players) exists under certain conditions on the payoff functions and the strategy sets. The main goal of this chapter is to discuss in detail and full generality the most elementary mathematical techniques for proving the existence of equilibrium points in noncooperative games (with an emphasis on two players).

This paper gives an overview of the existence and computation of equilibrium in nonlinear n-person games. After some introductory examples, sufficient existence results are presented in both cases of single-valued and multiple-valued best responses. The uniqueness of the equilibrium is also shown under general conditions. A special iterative method is discussed for the computation of the unique equilibrium based on a variational inequality, and a single-objective optimization model is introduced to provide the equilibria. An example of repeated oligopolies completes the paper.

The current chapter is devoted to the development of convex analysis concepts in the context of solving pursuit-evasion problems in differential games. Classic convex analysis is generalized; new concepts such as matrix-convex sets and H-convex sets are introduced and studied.With the help of these, it is shown possible to describe a rather wide class of differential games where players' strategies are produced in a comparatively constructive manner. The main attention is on studying those properties of matrix convexity that are required for the theory of differential games. Operational constructions for the initial positions sets, favorable to each player, for the derivation of the players' strategies are also described.

Efficiency and optimality are the two primary and generally conflicting goals in any auction design: the former focuses on the social welfare of the whole seller–bidder system, whereas the latter emphasizes revenue-maximizing on the seller side. In this chapter, we review the auctions design problem based on these two aspects in various information structures and circumstances. The most recent results are collected and analyzed. This chapter tends to complement the survey, Auction Theory: A Guide to the Literature, by Klemperer [58] in 1999. The main objective of this chapter is to provide a thorough survey on the current auctions design literature and to synthesize the developed theories underlying traditional auctions with the new elements and phenomena from the emerging and rapidly growing areas, such as online auctions.

This chapter explores models and algorithms applied to a class of Stackelberg games on networks. In these network interdictiongames, a network exists over which an operator wishes to execute some function, such as finding a shortest path, shipping a maximum flow, or transmitting a minimum cost multicommodity flow. The role of the interdictor is to compromise certain network elements before the operator acts, by (for instance) increasing the cost of flow or reducing capacity on an arc. We begin by reviewing the field of network interdiction and its related theoretical and mathematical foundations. We then discuss recent applications of stochastic models, valid inequalities, continuous bilinear programming techniques, and asymmetric analysis to network interdiction problems. Next, note that interdiction problems can be extended to a three-stage problem in which the operator fortifies the network (by increasing capacities, reducing flow costs, or defending network elements from the interdictor) before the interdictor takes action. We devote one section to ongoing research in this area and conclude by discussing areas for future research.

We consider time-discrete systems with a finite set of states. The starting and the final states of the dynamical system are fixed. We assume that the dynamics of the system is controlled by pactors (players), and each of them intends to optimize his own integral-time cost of the system's passages by a certain trajectory. Applying Nash and Pareto optimality principles for such a model, we obtain multiobjective control problems, solutions of which correspond with solutions of noncooperative and cooperative dynamic games, respectively. Necessary and sufficient conditions for the existence of Nash equilibrium and Pareto optimum in considered game control models are derived. Such conditions for stationary and nonstationary cases of the dynamic games are formulated. In the following, we extend dynamic programming technique for determining Nash equilibrium and Pareto optimum for dynamic games in positional form, especially for dynamic games on networks. Ef- ficient polynomial-time algorithms are elaborated for finding optimal strategies of players in dynamic games on networks. These algorithms are applied for studying and solving cyclic games. In addition, computational complexity of the proposed algorithms for the considered class of dynamic problems is discussed. Some extensions and generalizations of obtained results are suggested.

During the First World War, F.W. Lanchester published his book Aircraft in Warfare: The Dawn of the Fourth Arm[31] in which he proposed several mathematical models based on differential equations to describe combat situations. Since then, his work has been extensively modified to represent a variety of competitions, ranging from isolated battles to entire wars.

There exists a class of mathematical models known under the name of differential Lanchester type models. Such models have been studied from different points of view by many authors in hundreds of papers and unpublished reports. We note that Lanchester type models are used in the planning of optimal strategies, supply, and tactics.

In the last section, we will again consider Optimal Control by Viability, but in the set-valued case represented by differential inclusions.

In this paper, we develop static and dynamic models of global supply chains as networks with three tiers of decision-makers: manufacturers, retailers, and consumers associated with the demand markets who compete within a tier but cooperate between tiers. The decision-makers may be based in the same or in different countries, may transact in different currencies, and are faced with different degrees of environmental concern. Moreover, we allow for electronic transactions in the form of electronic commerce between the decision-makers. The proposed supernetworkframework formalizes the modeling and theoretical analysis of such global supply chains and also enables the dynamic tracking of the evolution of the associated prices and product transactions (as well as the emissions) to the equilibrium state. Moreover, it measures the impacts on the environment associated with the behaviors of the decision-makers. Finally, we propose a discrete-time algorithm that allows for the discretization of the continuous time trajectories and that results in closed form expressions at each iteration.