The Worst-Case Interference in DSL Systems Employing Dynamic Spectrum Management

This paper introduces a bottleneck game with finite sets of commuters and departing time slots as an extension of congestion games of Konishi et al. (J Econ Theory 72:225–237, 1997a). After characterizing Nash equilibrium of the game, we provide sufficient conditions for which the equivalence between Nash and strong equilibria holds. Somewhat surprisingly, unlike in congestion games, a Nash equilibrium in pure strategies may often fail to exist, even when players are homogeneous. In contrast, when there is a continuum of atomless players, the existence of a Nash equilibrium and the equivalence between the set of Nash and strong equilibria hold as in congestion games (Konishi et al. 1997a).

The most important analytical tool in non-cooperative game theory is the concept of a Nash equilibrium, which is a collection of possibly mixed strategies, one for each player, with the property that each player's strategy is a best reply to the strategies of the other players. If we do not go into normative game theory, which concerns itself with the recommendation of strategies, and focus instead entirely on the positive theory of prediction, two alternative interpretations of the Nash equilibrium concept are predominantly available.In the more traditional one, a Nash equilibrium is a prediction of actual play. A game may not have a Nash equilibrium in pure strategies, and a mixed strategy equilibrium may be difficult to incorporate into this interpretation if it involves the idea of actual randomization over equally good pure strategies. In another interpretation originating from Harsanyi (1973a), see also Rubinstein (1991), and Aumann and Brandenburger (1991), a Nash equilibrium is a ‘consistent’ collection of probabilistic expectations, conjectures, on the players. It is consistent in the sense that for each player each pure strategy, which has positive probability according to the conjecture about that player, is indeed a best reply to the conjectures about others.

The spectrum allotted for communications has been distributed to various service providers, who maintain and manage their portion of the spectrum by accommodating as many users as possible through spectral management techniques. Static spectrum management has been replaced by dynamic spectrum management to meet the vastly increasing number of users. Although dynamic spectrum management techniques have enabled the efficient assignment of channels to the user, it has been statistically observed that a huge part of the communications spectrum is scarcely used at all times. Cognitive radio is a useful tool for efficient dynamic spectrum management. The cognitive radio is adaptive in nature and is most effective because of its ability to monitor the spectrum, sense the channel occupancy and mobilise the secondary user to an unused primary user channel, as well as to relocate the secondary user if the primary user wishes to communicate. In this work, the energy-detection method and the cyclostationary method, which are the methods to sense the spectrum, are analysed and the results are compared. Simulation is done using a LabVIEW tool (National Instruments, Bangalore, India) and finally the implementation is done on a Field Programmable Gate Array.

The determination of wage rates and skill requirements for jobs is studied in a model where workers, who vary in ability, acquire skill through (costly) training. If firms' skill requirements are fixed, there exists a unique Nash equilibrium in wage rates, but the resulting allocation of labor is inefficient. The Nash equilibrium wage rates and allocation of labor approach the competitive equilibrium if and only if the number of job types increases on every skill range and no firm remains a monopsonist on any skill range. If firms choose both skill requirements and wage rates, no Nash equilibrium in pure strategies exists.

We investigate learning in a repeated bank choice game, where agents first choose a bank to deposit in and then decide to withdraw that deposit or not. This game has a single Nash equilibrium in pure strategies, characterized by all agents depositing in the bank that offers the highest return, even though it may be more vulnerable to bankruptcy if some agents withdraw early. We use an individual evolutionary learning algorithm to model under which circumstances and with which beliefs agents can learn the Nash equilibrium in the repeated game and compare the results to an experiment. We find participants coordinating on the Nash equilibrium in the presence of low or medium vulnerability banks, but efficient coordination fails when both banks are highly vulnerable (irrespective of whether participants have full or only partial information).

Multiplicity, Instability and Sunspots in Games

A version of the Monty Hall problem is presented where the players are permitted to select quantum strategies. If the initial state involves no entanglement the Nash equilibrium in the quantum game offers the players nothing more than can be obtained with a classical mixed strategy. However, if the initial state involves entanglement of the qubits of the two players, it is advantageous for one player to have access to a quantum strategy while the other does not. Where both players have access to quantum strategies there is no Nash equilibrium in pure strategies, however, there is a Nash equilibrium in quantum mixed strategies that gives the same average payoff as the classical game.

In this paper we deal with linear production situations in which there is a limited common-pool resource, managed by an external agent. The profit that a producer, or a group of producers, can attain depends on the amount of common-pool resource obtained through a certain procedure. We contemplate a competitive process among the producers or groups of producers and study the corresponding non cooperative games, describing their (strict) Nash equilibria in pure strategies. It is shown that strict Nash equilibria form a subset of strong Nash equilibria, which in turn form a proper subset of Nash equilibria.

A consignment contract with revenue sharing between an app developer and a distribution platform

On extremal pure Nash equilibria for mixed extensions of normal-form games

While single-level Nash equilibrium problems are quite well understood nowadays, less is known about multi-leader multi-follower games. However, these have important applications, e.g., in the analysis of electricity and gas markets, where often a limited number of firms interacts on various subsequent markets. In this paper, we consider a special class of two-level multi-leader multi-follower games that can be applied, e.g., to model strategic booking decisions in the European entry-exit gas market. For this nontrivial class of games, we develop a solution algorithm that is able to compute the complete set of Nash equilibria instead of just individual solutions or a bigger set of stationary points. Additionally, we prove that for this class of games, the solution set is finite and provide examples for instances without any Nash equilibria in pure strategies. We apply the algorithm to a case study in which we compute strategic booking and nomination decisions in a model of the European entry-exit gas market system. Finally, we use our algorithm to provide a publicly available test library for the considered class of multi-leader multi-follower games. This library contains problem instances with different economic and mathematical properties so that other researchers in the field can test and benchmark newly developed methods for this challenging class of problems.

A full game theoretic analysis of the two-person game with incomplete information is not available in the literature. It might be helpful to know something about the set of the equilibrium points of this game in order to make a prediction of the experimental results and the shape of the strategies. Therefore, we shall solve the game for Nash equilibria in pure strategies. A refinement of the equilibrium concept, to sequential equilibria for example, would be a narrowing from a behavioral point of view. It is not obvious that subjects follow the concept of sequential rationality. Since a large variety of possible behavior in equilibrium should be captured by the analysis we shall concentrate on the set of pure Nash equilibria. The restriction to pure strategies was chosen in order to simplify the computations. In this respect it is of interest that in the strategy experiment only pure strategies are observed.

There are many situations in which a customer's proclivity to buy the product of any firm depends not only on the classical attributes of the product such as its price and quality, but also on who else is buying the same product. We model these situations as games in which firms compete for customers located in a “social network”. Nash Equilibrium (NE) in pure strategies exist and are unique. Indeed there are closed-form formulae for the NE in terms of the exogenous parameters of the model, which enables us to compute NE in polynomial time. An important structural feature of NE is that, if there are no a priori biases between customers and firms, then there is a cut-off level above which high cost firms are blockaded at an NE, while the rest compete uniformly throughout the network. We finally explore the relation between the connectivity of a customer and the money firms spend on him. This relation becomes particularly transparent when externalities are dominant: NE can be characterized in terms of the invariant measures on the recurrent classes of the Markov chain underlying the social network.

A pure-strategy equilibrium existence theorem is extended to include games with non-expected utility players. It is shown that to guarantee the existence of a Nash equilibrium in pure strategies, the linearity of preferences in the probabilities can be replaced by the weaker requirement of quasiconvexity in the probabilities.

We propose a new fixed point theorem that completely characterizes the existence of fixed points for multivalued maps on finite sets. Our result can be seen as a generalization of Abian’s fixed point theorem. In the context of finite games, our result can be used to characterize the existence of a Nash equilibrium in pure strategies and can therefore distinguish pure strategy equilibria from mixed strategy equilibria in the celebrated Nash theorem.