Game-theoretic Solution Concepts Research Articles

Ascending Combinatorial Auctions with Allocation Constraints: On Game Theoretical and Computational Properties of Generic Pricing Rules

Combinatorial auctions are used in a variety of application domains, such as transportation or industrial procurement, using a variety of bidding languages and different allocation constraints. This flexibility in the bidding languages and the allocation constraints is essential in these domains but has not been considered in the theoretical literature so far. In this paper, we analyze different pricing rules for ascending combinatorial auctions that allow for such flexibility: winning levels and deadness levels. We determine the computational complexity of these pricing rules and show that deadness levels actually satisfy an ex post equilibrium, whereas winning levels do not allow for a strong game theoretical solution concept. We investigate the relationship of deadness levels and the simple price update rules used in efficient ascending combinatorial auction formats. We show that ascending combinatorial auctions with deadness level pricing rules maintain a strong game theoretical solution concept and reduce the number of bids and rounds required at the expense of higher computational effort. The calculation of exact deadness levels is a [Formula: see text]-complete problem. Nevertheless, numerical experiments show that for mid-sized auctions this is a feasible approach. The paper provides a foundation for allocation constraints in combinatorial auctions and a theoretical framework for recent Information Systems contributions in this field.

Efficiency with Linear Prices? A Game-Theoretical and Computational Analysis of the Combinatorial Clock Auction

Combinatorial auctions have been suggested as a means to raise efficiency in multi-item negotiations with complementarities among goods because they can be applied in procurement, energy markets, transportation, and the sale of spectrum auctions. The combinatorial clock (CC) auction has become very popular in these markets for its simplicity and for its highly usable price discovery, derived by the use of linear prices. Unfortunately, no equilibrium bidding strategies are known. Given the importance of the CC auction in the field, it is highly desirable to understand whether there are efficient versions of the CC auction providing a strong game theoretical solution concept. So far, equilibrium strategies have only been found for combinatorial auctions with nonlinear and personalized prices for very restricted sets of bidder valuations. We introduce an extension of the CC auction, the CC+ auction, and show that it actually leads to efficient outcomes in an ex post equilibrium for general valuations with only linear ask prices. We also provide a theoretical analysis on the worst case efficiency of the CC auction, which highlights situations in which the CC leads to highly inefficient outcomes. As in other theoretical models of combinatorial auctions, bidders in the field might not be able to follow the equilibrium strategies suggested by the game-theoretical predictions. Therefore, we complement the theoretical findings with results from computational and laboratory experiments using realistic value models. The experimental results illustrate that the CC+ auction can have a significant impact on efficiency compared to the CC auction.

On the Use of Games Theory in Financial Studies: The Case of Mixed Strategy Equilibrium

Copyright: © 2012 Teng J. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Since the 1980s, there has been an increasing use of games theory in the field of finance, including stock and forex trading [1]. The standard assumption of perfect competition was challenged. Analysis with the assumption of strategic interactions which uses games theory implicitly or explicitly becomes more popular. The use of games theory in the field of finance brings many new insights and should definitely be encouraged. But it also means that finance specialists need to be familiar with a new field of knowledge. Furthermore, one should also caution that games theory itself is a relatively new field and many of the game theoretic solution concepts are still being researched, refined and revised, and are not without controversy.

Cooperative institutions for sustainable common pool resource management: Application to groundwater

Beneficiaries of common pool resources (CPRs) may select available noncooperative and regulatory exogenous institutions for managing the resource, as well as cooperative management institutions. All these institutions may increase the long‐term gains, prolong the life of the resource, and help to escape the tragedy of the commons trap. Cooperative game theory approaches can serve as the backbone of cooperative CPR management institutions. This paper formulates and applies several commonly used cooperative game theoretic solution concepts, namely, the core, Nash‐Harsanyi, Shapley, and nucleolus. Through a numerical groundwater example, we show how CPR users can share the gains obtained from cooperation in a fair and efficient manner based on these cooperative solution concepts (management institutions). Although, based on their fairness rationales, various cooperative management institutions may suggest different allocations that are potentially acceptable to the users, these allocation solutions may not be stable as some users may find them unfair. This paper discusses how different methods, such as application of the plurality rule and power index, stability index, and propensity to disrupt concepts, can help identify the most stable and likely solutions for enforcing cooperation among the CPR beneficiaries. Furthermore, how the noncooperative managerial characteristics of the CPR users can affect the stability and acceptability of the different cooperative CPR management institutions is discussed, providing valuable policy insights for cooperative CPR management at community levels.

Restricted cooperative games on metabolic networks reveal functionally important reactions

Understanding the emerging properties of complex biological systems is in the crux of systems biology studies. Computational methods for elucidating the role of each component in the synergetic interplay can be used to identify targets for genetic and metabolic engineering. In particular, we aim at determining the importance of reactions in a metabolic network with respect to a specific biological function. Therefore, we propose a novel game-theoretic framework which integrates restricted cooperative games with the outcome of flux balance analysis. We define productivity games on metabolic networks and present an analysis of their unrestricted and restricted variants based on the game-theoretic solution concept of the Shapley value. Correspondingly, this concept provides a characterization of the robustness and functional centrality for each enzyme involved in a given metabolic network. Furthermore, the comparison of two different environments – feast and famine – demonstrates the dependence of the results on the imposed flux capacities.

Social Tools for Networked Learning

Whereas the Web 1.0 was mainly driven by static content and web pages linked by hyperlinks, the social web, or Web 2.0, has opened up new ways of connecting to not just resources, but also to people. The connections that are made through the use of Social Media, contribute to a complex, but also promising network of people and resources. In an educational context, this is called a learning network, and both learning networks by themselves and the Social media by which they are constructed require that we rethink the ways we learn, our view on learners, tutors and learning, and the way we support learners and tutors. Numerous opportunities have emerged with the introduction of social media for learning, but so have numerous problems, ranging from awareness issues via (meta-)cognitive limitations and difficulties to affective and motivational problems. This paper addresses the above issues by providing an overview of the current research we perform. The research we do is based on three themes: Peer support, Learner support and Online Learner Identity. Peer support describes the way peers may help each other through effective and efficient use of a learning network. It uses 1) natural language processing to, for instance, analyse communication between learners, 2) mine learner profiles to adapt to their individual circumstances and identity and 3) social network analysis extended by game theoretic solution concepts to recommend suitable peers for cooperative learning. Learner support focuses on how we may lead the learner through the jungle of learning resources. It uses recommender techniques to filter out unnecessary learning resources and provides concise sets of candidate resources for learning. Finally, Online Learner Identity focuses on rethinking how we construct our online identity, how to analyse such, and how to profit from the differences with offline learning. It may use multi-agent systems technology to simulate the identity of learners and their interaction in a learning network, but also semantic technology to capture the meaning of online learner identities. The paper also describes the main techniques that we use in our research efforts to enhance networked learning. Furthermore, an overview of current projects within the themes is provided. We conclude that the results of our current research efforts will provide valuable insights to advance further on research and development of social tools for networked learning.

Exchange Economy in Two-User Multiple-Input Single-Output Interference Channels

We study the conflict between two links in a multiple-input single-output interference channel. This setting is strictly competitive and can be related to perfectly competitive market models. In such models, general equilibrium theory is used to determine equilibrium measures that are Pareto optimal. First, we consider the links to be consumers that can trade goods within themselves. The goods in our setting correspond to beamforming vectors. We utilize the conflict representation of the consumers in the Edgeworth box, a graphical tool that depicts the allocation of the goods for the two consumers, to provide closed-form solution to all Pareto optimal outcomes. Afterwards, we model the situation between the links as a competitive market which additionally defines prices for the goods. The equilibrium in this economy is called Walrasian and corresponds to the prices that equate the demand to the supply of goods. We calculate the unique Walrasian equilibrium and propose a coordination process that is realized by an arbitrator which distributes the Walrasian prices to the consumers. The consumers then calculate in a decentralized manner their optimal demand corresponding to beamforming vectors that achieve the Walrasian equilibrium. This outcome is Pareto optimal and dominates the noncooperative outcome of the systems. Thus, based on the game theoretic model and solution concept, an algorithm for a distributed implementation of the beamforming problem in multiple-input single-output interference channels is provided.

Multi-Stakeholder Road Pricing Game: Solution Concepts

A road pricing game is a game where various stakeholders and/or regions with different (and usually conflicting) objectives compete for toll setting in a given transportation network to satisfy their individual objectives. We investigate some classical game theoretical solution concepts for the road pricing game. We establish results for the road pricing game so that stakeholders and/or regions playing such a game will beforehand know what is obtainable. This will save time and argument, and above all, get rid of the feelings of unfairness among the competing actors and road users. Among the classical solution concepts we investigate is Nash equilibrium. In particular, we show that no pure Nash equilibrium exists among the actors, and further illustrate that even “mixed Nash equilibrium” may not be achievable in the road pricing game. The paper also demonstrates the type of coalitions that are not only reachable, but also stable and profitable for the actors involved.

A formal methodology for aggregating multiple market views

We describe a formal aggregation procedure that allows investors to combine disparate market views, including the reasons that support them, to produce a single collective view. The methodology uses a many-valued logic to represent individual chains of reasoning and the sum absolute difference as a way of measuring disagreement between individuals. Group level views are derived by selecting the individual views that have the lowest sum absolute difference from the other views. In cases where more than one individual view possesses the minimum sum absolute difference, a variant of the game-theoretic solution concept known as the Shapley value is employed as a way to generate a compromise between the candidate views. In this way a single group-level market view can always be derived that is both formally consistent and a genuine balance between individual judgements.

A note on solution concepts for nontransferable utility games

While many game-theoretic solution concepts pertain to conditions of transferable utility (TU), market transactions and other exchanges generally occur under nontransferable utility (NTU). Extant solutions translate TU solution concepts to the NTU case by utilizing endogenously determined utility weight vectors, which can violate notions of equity. A new solution procedure is developed that resolves this longstanding problem, and an existence proof is provided.

PAYOFF LEVELS, LOSS AVOIDANCE, AND EQUILIBRIUM SELECTION IN GAMES WITH MULTIPLE EQUILIBRIA: AN EXPERIMENTAL STUDY

Game theorists typically assume that changing a game's payoff levels—by adding the same constant to, or subtracting it from, all payoffs—should not affect behavior. Although this invariance is an implication of the theory when payoffs mirror expected utilities, it is an empirical question when “payoffs” are actually money amounts. Loss avoidance is a phenomenon where payoff‐level changes matter when they change the signs of payoffs: gains become losses or vice versa. We report the results of a human‐subjects experiment designed to test for two types of loss avoidance: certain‐loss avoidance (avoiding a strategy leading to a sure loss, in favor of an alternative that might lead to a gain) and possible‐loss avoidance (avoiding a strategy leading to a possible loss, in favor of an alternative that leads to a sure gain). Subjects in the experiment play three versions of Stag Hunt, which are identical up to the level of payoffs, under a variety of treatments. We find strong evidence of behavior consistent with certain‐loss avoidance in the experiment. We also find evidence of possible‐loss avoidance, although weaker than that for certain‐loss avoidance. Our results carry implications for theorists modeling real‐life situations with game theory and for experimenters attempting to test theory and interpret observed behavior in terms of theory. (JEL D81, C72, C73)

Iterated regret minimization: A new solution concept

For some well-known games, such as the Travelerʼs Dilemma or the Centipede Game, traditional game-theoretic solution concepts—most notably Nash equilibrium—predict outcomes that are not consistent with empirical observations. We introduce a new solution concept, iterated regret minimization, that exhibits the same qualitative behavior as that observed in experiments in many games of interest, including Travelerʼs Dilemma, the Centipede Game, Nash bargaining, and Bertrand competition. As the name suggests, iterated regret minimization involves the iterated deletion of strategies that do not minimize regret.

The State of Solving Large Incomplete‐Information Games, and Application to Poker

Game‐theoretic solution concepts prescribe how rational parties should act, but to become operational the concepts need to be accompanied by algorithms. I will review the state of solving incomplete‐information games. They encompass many practical problems such as auctions, negotiations, and security applications. I will discuss them in the context of how they have transformed computer poker. In short, game‐theoretic reasoning now scales to many large problems, outperforms the alternatives on those problems, and in some games beats the best humans.

Iterative information update and stability of strategies

In this paper, we investigate processes involving iterative information updating due to van Benthem (Int Game Theory Rev 9:13–45, 2007), who characterized existent game-theoretic solution concepts by such processes in the framework of Plaza’s public announcement logic. We refine this approach and clarify the relationship between stable strategies and information update processes. After extending Plaza’s logic, we then derive the conditions under which a stable outcome is determined independently of the order of the iterative information updates. This result gives an epistemic foundation for the order independence of iterated elimination of disadvantageous strategies.

Algorithms for Closed Under Rational Behavior (CURB) Sets

We provide a series of algorithms demonstrating that solutions according to the fundamental game-theoretic solution concept of closed under rational behavior (CURB) sets in two-player, normal-form games can be computed in polynomial time (we also discuss extensions to n-player games). First, we describe an algorithm that identifies all of a player’s best responses conditioned on the belief that the other player will play from within a given subset of its strategy space. This algorithm serves as a subroutine in a series of polynomial-time algorithms for finding all minimal CURB sets, one minimal CURB set, and the smallest minimal CURB set in a game. We then show that the complexity of finding a Nash equilibrium can be exponential only in the size of a game’s smallest CURB set. Related to this, we show that the smallest CURB set can be an arbitrarily small portion of the game, but it can also be arbitrarily larger than the supports of its only enclosed Nash equilibrium. We test our algorithms empirically and find that most commonly studied academic games tend to have either very large or very small minimal CURB sets.

A unified approach to comparative statics puzzles in experiments

The paper shows that several game-theoretic solution concepts provide similar comparative statics predictions over a wide class of games. I start from the observation that, in many experiments, behavior is affected by parameter shifts that leave the Nash equilibrium unchanged. I explain the direction of change with a heuristic structural approach, using properties such as strategic complementarities and increasing differences. I show that the approach is consistent with general comparative statics results for (i) the Nash equilibrium of a game with perturbed payoff functions, (ii) the quantal response equilibrium, (iii) level- k reasoning. I also relate the structural approach to equilibrium selection concepts.

Computing Shapley's saddles

Game-theoretic solution concepts, such as Nash equilibrium, are playing an ever increasing role in the study of systems of autonomous agents. A common criticism of Nash equilibrium is that its existence relies on the possibility of randomizing over actions, which in many cases is deemed unsuitable, impractical, or even infeasible.

Ranking games

The outcomes of many strategic situations such as parlor games or competitive economic scenarios are rankings of the participants, with higher ranks generally at least as desirable as lower ranks. Here we define ranking games as a class of n-player normal-form games with a payoff structure reflecting the players' von Neumann–Morgenstern preferences over their individual ranks. We investigate the computational complexity of a variety of common game-theoretic solution concepts in ranking games and deliver hardness results for iterated weak dominance and mixed Nash equilibrium when there are more than two players, and for pure Nash equilibrium when the number of players is unbounded but the game is described succinctly. This dashes hope that multi-player ranking games can be solved efficiently, despite their profound structural restrictions. Based on these findings, we provide matching upper and lower bounds for three comparative ratios, each of which relates two different solution concepts: the price of cautiousness, the mediation value, and the enforcement value.

Everything Else Being Equal: A Modal Logic for Ceteris Paribus Preferences

This paper presents a new modal logic for ceteris paribus preferences understood in the sense of “all other things being equal”. This reading goes back to the seminal work of Von Wright in the early 1960’s and has returned in computer science in the 1990’s and in more abstract “dependency logics” today. We show how it differs from ceteris paribus as “all other things being normal”, which is used in contexts with preference defeaters. We provide a semantic analysis and several completeness theorems. We show how our system links up with Von Wright’s work, and how it applies to game-theoretic solution concepts, to agenda setting in investigation, and to preference change. We finally consider its relation with infinitary modal logics.

Endogenous formation of political parties

We consider a population of individuals characterized by their ideal points in the space of political programs. Each individual joins some political party, and the party program is determined according to a certain rule depending on the ideal points of its members. The utility function of any individual increases in the size of his party, and decreases as the distance between the party program and his ideal point increases. In contrast to the previous literature on endogenous formation of coalitions we, examine a model without side payments, do not fix the number of parties and consider a large (formally infinite) population. The latter assumption permits us to consider continuous distributions of individual ideal points and facilitates the formal analysis. We use game-theoretic solution concepts and study existence, uniqueness and computation problems for Nash equilibria. We also examine coalitional equilibria, that is such party structures where no new party can provide greater utilities for all its members. We find out the properties, in particular, the number of different coalitional equilibria and the number of parties at each equilibrium, depending on parameters of the individual utility function.