The recent development of computational methods in repeated games has made it possible to study t... more The recent development of computational methods in repeated games has made it possible to study the properties of subgame-perfect equilibria in more detail. This paper shows that the lowest equilibrium payoffs may increase in pure strategies when the players become more patient and this may cause the set of equilibrium paths to be non-monotonic. A numerical example is constructed such that a path is no longer equilibrium when the players' discount factors increase. This property can be more easily seen when the players have different time preferences, since in these games the punishment strategies may rely on the differences between the players' discount factors. A sufficient condition for the monotonicity of equilibrium paths is that the lowest equilibrium payoffs do not increase, i.e., the punishments should not become milder. Keywords repeated games • minimum payoff • monotonicity • equilibrium path • unequal discount factors • subgame perfection
The recent development of computational methods in repeated games has made it possible to study t... more The recent development of computational methods in repeated games has made it possible to study the properties of subgame-perfect equilibria in more detail. This paper shows that the lowest equilibrium payoffs may increase in pure strategies when the players become more patient and this may cause the set of equilibrium paths to be non-monotonic. A numerical example is constructed such that a path is no longer equilibrium when the players' discount factors increase. This property can be more easily seen when the players have different time preferences, since in these games the punishment strategies may rely on the differences between the players' discount factors. A sufficient condition for the monotonicity of equilibrium paths is that the lowest equilibrium payoffs do not increase, i.e., the punishments should not become milder. Keywords repeated games • minimum payoff • monotonicity • equilibrium path • unequal discount factors • subgame perfection
Internet traffic volume is increasing and this causes scalability issues in content delivery. Thi... more Internet traffic volume is increasing and this causes scalability issues in content delivery. This problem can be addressed with different types of caching solutions. The incentives of different stakeholders to pay for these solutions are not known. However, it has been identified that Internet service providers (ISPs) need to be involved in the process of cache deployment due to their ownership of the network. This work evaluates a new business model where ISPs charge content providers (CPs) for a caching service because CPs benefit from more efficient content distribution. We provide conditions for sustainable paid in-network caching and their numerical evaluation in order to aid strategic decision-making by CPs, ISPs, and Cloud storage providers (CSPs). Although ISP caching as a paid service may not be an equilibrium, it turns out to be Pareto optimal at the right pricing. This encourages cooperation between CPs and ISPs. CSPs may choose cache friendly physical locations for their facilities in order The work of J.
We examine a specific class of bargaining problems where the golden and silver ratios appear in a... more We examine a specific class of bargaining problems where the golden and silver ratios appear in a natural way.
This paper characterizes the subgame-perfect pure-strategy equilibrium paths in discounted superg... more This paper characterizes the subgame-perfect pure-strategy equilibrium paths in discounted supergames with perfect monitoring. It is shown that all the equilibrium paths are composed of fragments called elementary subpaths. This characterization result is complemented with an algorithm for finding the elementary subpaths. By using these subpaths it is possible to generate equilibrium paths and payoffs. When there are finitely many elementary subpaths, all the equilibrium paths can be represented by a directed graph. These graphs can be used in analyzing the complexity of equilibrium outcomes. In particular, it is shown that the size and the density of the equilibrium set can be measured by the asymptotic growth rate of equilibrium paths and the Hausdorff dimension of the payoff set.
This paper examines the subgame perfect pure strategy equilibrium paths and payoff sets of discou... more This paper examines the subgame perfect pure strategy equilibrium paths and payoff sets of discounted supergames with perfect monitoring. The main contribution is to provide methods for computing and tools for analyzing the equilibrium paths and payoffs in repeated games. We introduce the concept of a first-action feasible path, which simplifies the computation of equilibria. These paths can be composed into a directed multigraph, which is a useful representation for the equilibrium paths. We examine how the payoffs, discount factors and the properties of the multigraph affect the possible payoffs, their Hausdorff dimension, and the complexity of the equilibrium paths. The computational methods are applied to the twelve symmetric strictly ordinal 2x2 games. We find that these games can be classified into three groups based on the complexity of the equilibrium paths.
This paper examines the pure-strategy subgame-perfect equilibrium payoffs in discounted supergame... more This paper examines the pure-strategy subgame-perfect equilibrium payoffs in discounted supergames with perfect monitoring. It is shown that the equilibrium payoffs can be identified as sub-self-affine sets or graph-directed iterated function systems. We propose a method to estimate the Hausdorff dimension of the equilibrium payoffs and relate it to the equilibrium paths and their graph presentation.
The reforms in spectrum management aim towards more efficient, flexible and shared use of spectru... more The reforms in spectrum management aim towards more efficient, flexible and shared use of spectrum, while fostering innovation and competition in mobile communications. The goal is to have affordable wireless broadband and make Internet access a human right. To achieve this goal, the regulators need to identify, evaluate and decide between the different spectrum access models. This article presents a taxonomy of the models, which is based on priorities between the different users of spectrum. For example, the low priority systems may be allowed to use the spectrum only if the high priority systems are not occupying the frequency. The hierarchy of the systems can be implemented using different technologies like centralized databases or distributed sensing devices. The taxonomy clarifies the regulatory alternatives and emphasizes the differences in ownership and usage rights. The new models can be incorporated in the auction design in order to estimate their value to the companies, and this reflects the recent trend of supporting spectrum management with market-based methods.
The demand for spectrum is growing rapidly as data intensive mobile communications and high-defin... more The demand for spectrum is growing rapidly as data intensive mobile communications and high-definition television are getting more popular. The spectrum regulators are preparing for flexible spectrum use and the implementation of new technologies. To make the appropriate decisions, the regulators need to identify and evaluate the different alternatives. In this paper, we examine a systematic classification of the spectrum access models that can be used together with suitable auctions in order to evaluate and decide between the different models. The basic idea of the taxonomy is to distinguish the different forms of spectrum sharing. For example, multiple systems may coexist in the same frequency band so that the low priority systems may transmit only if the high priority systems are not occupying the band. The hierarchy of the systems can be implemented by using a variety of technologies like centralized databases or distributed sensing devices. The taxonomy clarifies the regulatory alternatives and highlights the differences in ownership and usage rights. We also discuss how to incorporate the new spectrum access models in the auction design and how to support spectrum management using economic modeling and mechanism design framework.
This article examines potential deployment strategies for mobile operators to satisfy the high an... more This article examines potential deployment strategies for mobile operators to satisfy the high anticipated data traffic volume in their mobile networks. The mobile network offloading is analyzed with the traditional evolution of macrocellular radio access networks. Cognitive radio is considered as a technology that may augment these deployment strategies. A duopoly model is developed to examine where and when mobile offloading is likely to happen. The results show how much offloading there will be and the exact conditions when the offloading is profitable based on the estimated cost parameters and the future demands. An illustrative case study in Helsinki metropolitan area is examined for year 2015.
Cognitive radio provides an opportunity for more efficient use of spectrum and features that impr... more Cognitive radio provides an opportunity for more efficient use of spectrum and features that improve and increase services in mobile business. This paper analyzes the economic impact of cognitive radio with a three-player oligopoly model. The three players represent the three markets of the industry: devices, connectivity and services. We examine how the new technology affects the markets, and how the changes in one market affect the other markets. With the model, it is possible to estimate the players' utilities under different parameters and scenarios. We also present a method to estimate the parameter changes in the model from the technological improvements of cognitive radio. The model can be extended to incorporate more specific issues like the running out of spectrum and the effects of coalitions, which are studied in this paper.
We examine a contracting problem with asymmetric information in a monopoly pricing setting. Tradi... more We examine a contracting problem with asymmetric information in a monopoly pricing setting. Traditionally, the problem is modeled as a one-period Bayesian game, where the incomplete information about the buyers' preferences is handled with some subjective probability distribution. Here we suggest an iterative online method to solve the problem. We show that, when the buyers behave myopically, the seller can learn the optimal tariff by selling the product repeatedly. In a practical modification of the method the seller offers linear tariffs and adjusts them until optimality is reached. The adjustment can be seen as gradient adjustment, and it can be done with limited information and so that it benefits both the seller and the buyers. Our method uses special features of the problem and it is easily implementable.
Optimal screening is one of the basic models of contracting under incomplete information, and we ... more Optimal screening is one of the basic models of contracting under incomplete information, and we study the problem in a quality pricing application. We present a simple numerical method for solving the pricing problem when the firm has limited information about the buyers’ utility functions. In the method, the firm learns the optimal price schedule as the demand data is collected. We examine what the firm can learn about the preferences by observing the sales, and how the revealed information can be used in adjusting the quality-price bundles to increase the profit. We analyze the properties of the solution and derive the first-order optimality conditions under different assumptions. We show that the problem can be solved by making use of these optimality conditions together with the buyers’ marginal valuations. The firm can estimate the marginal valuations either by offering linear tariffs or by selling test bundles near the current solution.
Optimal screening is one of the basic models of contracting under incomplete information, and we ... more Optimal screening is one of the basic models of contracting under incomplete information, and we study the problem in a quality pricing application. We present a numerical method for solving the pricing problem when the firm is uncertain about the buyers' utility functions. In the method, the firm learns the optimal price schedule online as the demand data is collected. We examine what the firm can learn about the preferences by observing the sales, and how the revealed information can be used in adjusting the quality-price bundles to increase the profit. We derive the first-order optimality conditions for the problem, and show that the firm can evaluate these conditions by estimating the buyers' marginal valuations. The firm can estimate the marginal valuations either by offering linear tariffs or by selling test bundles near the current solution. We introduce an interval approach, where each iteration updates either the lower or the upper bound of the optimal quality. By narrowing down the interval, an approximately optimal bundle is found. Though, the model is one-dimensional and the standard single-crossing condition is assumed, the method gives guidelines for solving a multidimensional screening problem without the single-crossing condition.
Optimal screening has been studied in economics, game theory, and recently computer science. We s... more Optimal screening has been studied in economics, game theory, and recently computer science. We study the problem in a nonlinear pricing application, where a monopoly designs a price schedule from which the buyers self-select the quality they wish to consume. We formulate a multidimensional model with buyers' utility functions that need not satisfy the standard single-crossing assumption. We characterize the solution with the first-order optimality conditions and present a framework for analyzing the solution. With the framework, the structure of the solution is easily illustrated and the sensitivity analysis can be done. We give numerical examples that demonstrate the properties of the solution. With these observations, we discuss the complexity of the problem and solving the problem under limited information. We examine what information the monopoly needs when adjusting the price schedule to increase the profit. This paper applies, e.g., to pricing situations in electronic commerce where the seller may have limited information available, and the seller learns about the buyers' preferences online when doing the business.
This paper characterizes and enumerates the possible solution structures in nonlinear pricing pro... more This paper characterizes and enumerates the possible solution structures in nonlinear pricing problem when the number of buyer types is given. It is shown that the single-crossing property, which is a standard assumption in the literature, reduces the complexity of solving the problem dramatically. The number of possible solution structures is important when the pricing problem is solved under limited information.
This paper studies a monopoly pricing problem in a situation where the firm has uncertainty about... more This paper studies a monopoly pricing problem in a situation where the firm has uncertainty about the buyers' preferences. We examine continuous learning dynamics, where the firm sells some quality-price bundles and adjusts them using only local information about the buyers' preferences. The learning dynamics define different learning paths, and we compute how much profit the paths make, how long it takes to learn the optimal tariff, and how the buyers' utilities change during the learning period. We also compute the optimal learning path in terms of discounted profit with dynamic programming and complete information. We examine how close the optimal path is to the learning dynamics that use only limited information. Numerical examples show that the optimal path may involve jumps, where the buyer types switch from one bundle to another, and these are something the learning dynamics cannot discover. The learning dynamics have, however, the benefit that they can be generalized to pricing problems with many buyer types and qualities.
We present well-known interpretations of Lagrange multipliers in physical and economic applicatio... more We present well-known interpretations of Lagrange multipliers in physical and economic applications, and introduce a new interpretation in nonlinear pricing problem. The multipliers can be interpreted as a network of directed flows between the buyer types. The structure of the digraph and the fact that the multipliers usually have distinctive values can be used in solving the optimization problem more efficiently. We also find that the multipliers satisfy a conservation law for each node in the digraph, and the non-uniqueness of the multipliers are connected to the stability of the solution structure.
We give a new approach in modeling the incomplete information in a buyer-seller game. We assume t... more We give a new approach in modeling the incomplete information in a buyer-seller game. We assume that the seller does not know the buyer's utility function at all. Usually the problem is solved by determining the Bayesian Nash equilibrium of the game, where it is assumed that the buyer's utility function has only some parameters unknown to the seller, and the seller knows the distribution of these parameters. Instead, we assume that the seller faces different types of buyers repeatedly, and the seller learns the buyers' preferences. We present an adjustment process that leads to Bayesian Nash equilibrium by using linear tariffs to extract enough information. This approach motivates the Bayesian Nash equilibrium of the buyer-seller game.
In this paper we develop an on-line method for reaching an optimal solution for interacting agent... more In this paper we develop an on-line method for reaching an optimal solution for interacting agents in a market situation. Specifically we study a buyer-seller game with a monopolistic seller and many buyers. At each round of the repeated game the seller adjusts a piecewise linear tariff the limit of which defines the Bayesian Nash equilibrium of the stage game with one seller and one buyer with several types. Our scheme will require only a small part of the buyers' preferences to be elicited; and hence requires only a small amount of computation and communication.
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Papers by Kimmo Berg
It is shown that all the equilibrium paths are composed of fragments called elementary subpaths. This characterization result is complemented with an algorithm for finding the elementary subpaths. By using these subpaths it is possible to generate equilibrium paths and payoffs. When there are finitely many elementary subpaths, all the equilibrium paths can be represented by a directed graph. These graphs can be used in analyzing the complexity of equilibrium outcomes. In particular, it is shown that the size and the density of the equilibrium set can be measured by the asymptotic growth rate of equilibrium paths and the Hausdorff dimension of the payoff set.