This paper examines the subgame-perfect equilibria in the symmetric 2× 2 supergames. We extend th... more This paper examines the subgame-perfect equilibria in the symmetric 2× 2 supergames. We extend the folk theorem by solving the smallest discount factor values when the players obtain all the feasible and individually rational payoffs. This enables us to determine all the equilibrium payoffs for high discount factor values, which is in general a difficult task since the payoff sets are complicated for patient players. We study how the different assumptions affect the set of equilibria by comparing the payoff sets in pure, randomized and correlated strategies. Moreover, we analyze how exactly the stage game’s payoffs affect the required level of patience and organize the games into few classes. We find that the bounds generally depend on how large the payoff set is compared to the set of feasible payoffs and that the bounds are quite moderate for many games. We also observe discontinuities in the bounds, which means that small changes in the stage game’s payoffs may affect dramaticall...
This paper examines how to construct subgame-perfect mixed-strategy equilibria in 1 discounted re... more This paper examines how to construct subgame-perfect mixed-strategy equilibria in 1 discounted repeated games with perfect monitoring. We introduce a relatively simple class of strategy 2 profiles that are easy to compute and may give rise to a large set of equilibrium payoffs. These sets 3 are called self-supporting sets, since the set itself provides the continuation payoffs that are required 4 to support the equilibrium strategies. Moreover, the corresponding strategies are simple as the players 5 face the same augmented game on each round but they play different mixed actions after each realized 6 pure-action profile. We find that certain payoffs can be obtained in equilibrium with much lower 7 discount factor values compared to pure strategies. The theory and the concepts are illustrated in 8 2 × 2 games.
This paper introduces a new solution concept for games with incomplete preferences. The concept i... more This paper introduces a new solution concept for games with incomplete preferences. The concept is based on rationalizability and it is more general than the existing ones based on Nash equilibrium. In rationalizable strategies, we assume that the players choose nondominated strategies given their beliefs of what strategies the other players may choose. Our solution concept can also be used, e.g., in ordinal games where the standard notion of rationalizability cannot be applied. We show that the sets of rationalizable strategies are the maximal mutually nondominated sets. We also show that no new rationalizable strategies appear when the preferences are refined, i.e., when the information gets more precise. Moreover, noncooperative multicriteria games are suitable applications of incomplete preferences. We apply our framework to such games, where the outcomes are evaluated according to several criteria and the payoffs are vector-valued. We use the sets of feasible weights to represent the relative importance of the criteria. We demonstrate the applicability of the new solution concept with an ordinal game and a bicriteria Cournot game.
This paper examines how to construct subgame-perfect mixed-strategy equilibria in 1 discounted re... more This paper examines how to construct subgame-perfect mixed-strategy equilibria in 1 discounted repeated games with perfect monitoring. We introduce a relatively simple class of strategy 2 profiles that are easy to compute and may give rise to a large set of equilibrium payoffs. These sets 3 are called self-supporting sets, since the set itself provides the continuation payoffs that are required 4 to support the equilibrium strategies. Moreover, the corresponding strategies are simple as the players 5 face the same augmented game on each round but they play different mixed actions after each realized 6 pure-action profile. We find that certain payoffs can be obtained in equilibrium with much lower 7 discount factor values compared to pure strategies. The theory and the concepts are illustrated in 8 2 × 2 games.
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.
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 to provide the necessary capacity to the ISPs. However, the required amounts are in all likelihood too small to be an incentive for the CSPs. ISP caching as a paid service can be an equilibrium when future benefits are considered and when the ISPs terminate caching-related improvements of service quality for clients who do not pay for caching.
We present a complete algorithm for finding an ϵ-Nash equilibrium, for arbitrarily small ϵ, in ga... more We present a complete algorithm for finding an ϵ-Nash equilibrium, for arbitrarily small ϵ, in games with more than two players. The best prior complete algorithm has significantly worse complexity and has, to our knowledge, never been implemented. The main components of our tree-search-based method are a node-selection strategy , an exclusion oracle, and a subdivision scheme. The node-selection strategy determines the next region to be explored—based on the region's size and an estimate of whether the region contains an equilibrium. The exclusion oracle provides a provably correct sufficient condition for there not to exist an equilibrium in the region. The subdivision scheme determines how the region is split if it cannot be excluded. Unlike well-known incomplete methods, our method does not need to proceed locally, which avoids it getting stuck in a local minimum that may be far from any actual equilibrium. The run time grows rapidly with the game size, and this suggests a hybrid scheme where one of the relatively fast prior incomplete algorithms is run, and if it fails to find an equilibrium , then our method is used.
This paper examines the subgame-perfect equilibria in discounted stochastic games with finite sta... more This paper examines the subgame-perfect equilibria in discounted stochastic games with finite state and action spaces. The fixed-point characterization of pure-strategy equilibria is generalized to unobservable mixed strategies. It is also shown that the pure-strategy equilibria consist of elementary subpaths, which are repeating fragments that give the acceptable action plans in the game. The developed methodology offers a novel way of computing and analyzing equilibrium strategies that need not be stationary nor Markovian.
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 equilib-ria 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.
We present a complete algorithm for finding an ϵ-Nash equilibrium, for arbitrarily small ϵ, in ga... more We present a complete algorithm for finding an ϵ-Nash equilibrium, for arbitrarily small ϵ, in games with more than two players. The best prior complete algorithm has significantly worse complexity and has, to our knowledge, never been implemented. The main components of our tree-search-based method are a node-selection strategy , an exclusion oracle, and a subdivision scheme. The node-selection strategy determines the next region to be explored—based on the region's size and an estimate of whether the region contains an equilibrium. The exclusion oracle provides a provably correct sufficient condition for there not to exist an equilibrium in the region. The subdivision scheme determines how the region is split if it cannot be excluded. Unlike well-known incomplete methods, our method does not need to proceed locally, which avoids it getting stuck in a local minimum that may be far from any actual equilibrium. The run time grows rapidly with the game size, and this suggests a hybrid scheme where one of the relatively fast prior incomplete algorithms is run, and if it fails to find an equilibrium , then our method is used.
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 to provide the necessary capacity to the ISPs. However, the required amounts are in all likelihood too small to be an incentive for the CSPs. ISP caching as a paid service can be an equilibrium when future benefits are considered and when the ISPs terminate caching-related improvements of service quality for clients who do not pay for caching.
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 2×2 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.
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 examines the subgame-perfect equilibria in the symmetric 2× 2 supergames. We extend th... more This paper examines the subgame-perfect equilibria in the symmetric 2× 2 supergames. We extend the folk theorem by solving the smallest discount factor values when the players obtain all the feasible and individually rational payoffs. This enables us to determine all the equilibrium payoffs for high discount factor values, which is in general a difficult task since the payoff sets are complicated for patient players. We study how the different assumptions affect the set of equilibria by comparing the payoff sets in pure, randomized and correlated strategies. Moreover, we analyze how exactly the stage game’s payoffs affect the required level of patience and organize the games into few classes. We find that the bounds generally depend on how large the payoff set is compared to the set of feasible payoffs and that the bounds are quite moderate for many games. We also observe discontinuities in the bounds, which means that small changes in the stage game’s payoffs may affect dramaticall...
This paper examines how to construct subgame-perfect mixed-strategy equilibria in 1 discounted re... more This paper examines how to construct subgame-perfect mixed-strategy equilibria in 1 discounted repeated games with perfect monitoring. We introduce a relatively simple class of strategy 2 profiles that are easy to compute and may give rise to a large set of equilibrium payoffs. These sets 3 are called self-supporting sets, since the set itself provides the continuation payoffs that are required 4 to support the equilibrium strategies. Moreover, the corresponding strategies are simple as the players 5 face the same augmented game on each round but they play different mixed actions after each realized 6 pure-action profile. We find that certain payoffs can be obtained in equilibrium with much lower 7 discount factor values compared to pure strategies. The theory and the concepts are illustrated in 8 2 × 2 games.
This paper introduces a new solution concept for games with incomplete preferences. The concept i... more This paper introduces a new solution concept for games with incomplete preferences. The concept is based on rationalizability and it is more general than the existing ones based on Nash equilibrium. In rationalizable strategies, we assume that the players choose nondominated strategies given their beliefs of what strategies the other players may choose. Our solution concept can also be used, e.g., in ordinal games where the standard notion of rationalizability cannot be applied. We show that the sets of rationalizable strategies are the maximal mutually nondominated sets. We also show that no new rationalizable strategies appear when the preferences are refined, i.e., when the information gets more precise. Moreover, noncooperative multicriteria games are suitable applications of incomplete preferences. We apply our framework to such games, where the outcomes are evaluated according to several criteria and the payoffs are vector-valued. We use the sets of feasible weights to represent the relative importance of the criteria. We demonstrate the applicability of the new solution concept with an ordinal game and a bicriteria Cournot game.
This paper examines how to construct subgame-perfect mixed-strategy equilibria in 1 discounted re... more This paper examines how to construct subgame-perfect mixed-strategy equilibria in 1 discounted repeated games with perfect monitoring. We introduce a relatively simple class of strategy 2 profiles that are easy to compute and may give rise to a large set of equilibrium payoffs. These sets 3 are called self-supporting sets, since the set itself provides the continuation payoffs that are required 4 to support the equilibrium strategies. Moreover, the corresponding strategies are simple as the players 5 face the same augmented game on each round but they play different mixed actions after each realized 6 pure-action profile. We find that certain payoffs can be obtained in equilibrium with much lower 7 discount factor values compared to pure strategies. The theory and the concepts are illustrated in 8 2 × 2 games.
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.
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 to provide the necessary capacity to the ISPs. However, the required amounts are in all likelihood too small to be an incentive for the CSPs. ISP caching as a paid service can be an equilibrium when future benefits are considered and when the ISPs terminate caching-related improvements of service quality for clients who do not pay for caching.
We present a complete algorithm for finding an ϵ-Nash equilibrium, for arbitrarily small ϵ, in ga... more We present a complete algorithm for finding an ϵ-Nash equilibrium, for arbitrarily small ϵ, in games with more than two players. The best prior complete algorithm has significantly worse complexity and has, to our knowledge, never been implemented. The main components of our tree-search-based method are a node-selection strategy , an exclusion oracle, and a subdivision scheme. The node-selection strategy determines the next region to be explored—based on the region's size and an estimate of whether the region contains an equilibrium. The exclusion oracle provides a provably correct sufficient condition for there not to exist an equilibrium in the region. The subdivision scheme determines how the region is split if it cannot be excluded. Unlike well-known incomplete methods, our method does not need to proceed locally, which avoids it getting stuck in a local minimum that may be far from any actual equilibrium. The run time grows rapidly with the game size, and this suggests a hybrid scheme where one of the relatively fast prior incomplete algorithms is run, and if it fails to find an equilibrium , then our method is used.
This paper examines the subgame-perfect equilibria in discounted stochastic games with finite sta... more This paper examines the subgame-perfect equilibria in discounted stochastic games with finite state and action spaces. The fixed-point characterization of pure-strategy equilibria is generalized to unobservable mixed strategies. It is also shown that the pure-strategy equilibria consist of elementary subpaths, which are repeating fragments that give the acceptable action plans in the game. The developed methodology offers a novel way of computing and analyzing equilibrium strategies that need not be stationary nor Markovian.
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 equilib-ria 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.
We present a complete algorithm for finding an ϵ-Nash equilibrium, for arbitrarily small ϵ, in ga... more We present a complete algorithm for finding an ϵ-Nash equilibrium, for arbitrarily small ϵ, in games with more than two players. The best prior complete algorithm has significantly worse complexity and has, to our knowledge, never been implemented. The main components of our tree-search-based method are a node-selection strategy , an exclusion oracle, and a subdivision scheme. The node-selection strategy determines the next region to be explored—based on the region's size and an estimate of whether the region contains an equilibrium. The exclusion oracle provides a provably correct sufficient condition for there not to exist an equilibrium in the region. The subdivision scheme determines how the region is split if it cannot be excluded. Unlike well-known incomplete methods, our method does not need to proceed locally, which avoids it getting stuck in a local minimum that may be far from any actual equilibrium. The run time grows rapidly with the game size, and this suggests a hybrid scheme where one of the relatively fast prior incomplete algorithms is run, and if it fails to find an equilibrium , then our method is used.
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 to provide the necessary capacity to the ISPs. However, the required amounts are in all likelihood too small to be an incentive for the CSPs. ISP caching as a paid service can be an equilibrium when future benefits are considered and when the ISPs terminate caching-related improvements of service quality for clients who do not pay for caching.
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 2×2 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.
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.
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Papers by Kimmo Berg
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 2×2 games. We find that these games can be classified into three groups based on the complexity of the equilibrium paths.
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 2×2 games. We find that these games can be classified into three groups based on the complexity of the equilibrium paths.