Supermodular Games Research Articles

Application of super-modular game model on quality and safety management of supply chain based on process control

Application of super-modular game model on quality and safety management of supply chain based on process control

Supermodular Game-Based Distributed Joint Uplink Power and Rate Allocation in Two-Tier Femtocell Networks

This paper tackles the problem of joint users’ uplink transmission power and data rate allocation in multi-service two-tier femtocell networks. Each user—either macrocell (MUE) or femtocell user equipment (FUE)—is associated with a two-variable utility function that represents his perceived satisfaction with respect to his allocated resources (i.e., power and rate). User's utility function is differentiated based both on the tier that the user belongs to and the service he requests. The joint resource allocation problem is directly confronted as a two-variable optimization problem and formulated as a non-cooperative game. The theory of supermodular games is utilized towards treating the two-variable optimization problem and the inherent multidimensional competition that arises among the users. The existence of proposed game's Nash Equilibrium (NE) point is analytically shown, while game's convergence to its NE point is proven. A distributed and iterative algorithm for computing the desired NE is introduced, where the optimal values of each user's uplink transmission power and data rate are simultaneously updated at the same step. The performance of the proposed approach is evaluated via modeling and simulation and its superiority compared to other state of the art approaches is illustrated.

Abstracting Nash equilibria of supermodular games

Supermodular games are a well known class of noncooperative games which find significant applications in a variety of models, especially in operations research and economic applications. Supermodular games always have Nash equilibria which are characterized as fixed points of multivalued functions on complete lattices. Abstract interpretation is here applied to set up an approximation framework for Nash equilibria of supermodular games. This is achieved by extending the theory of abstract interpretation in order to cope with approximations of multivalued functions and by providing some methods for abstracting supermodular games, thus obtaining approximate Nash equilibria which are shown to be correct within the abstract interpretation framework.

Public Cloud Storage-Assisted Mobile Social Video Sharing: A Supermodular Game Approach

Mobile social video sharing enables mobile users to create ultra-short video clips and instantly share them with social friends, which poses significant pressure to the content distribution infrastructure. In this paper, we propose a public cloud-assisted architecture to tackle this problem. In particular, by motivating mobile users to upload videos to the local public cloud to serve requests, and, therefore, having a permission to access friends' videos stored in the cloud, our method can alleviate the traffic burden to the social service providers, while reducing the service latency of mobile users. First, we present a general framework to model the information diffusion and utility function of each user on the proposed architecture, and formulate the problem as a decentralized social utility maximization game. Second, we show that this problem is a supermodular game and there exists at least one socially aware Nash equilibrium (SNE). We then develop two decentralized algorithms to solve this problem. The first algorithm can find an SNE with less computation complexity, and the second algorithm can find the Pareto-optimal SNE with better performance. Finally, through extensive experiments, we demonstrate that the overall system performance can be significantly improved by exploiting the selflessness among social friends.

Computing Least Cores of Supermodular Cooperative Games

One of the goals of a cooperative game is to compute a valuedivision to the players from which they have no incentive todeviate. This concept is formalized as the notion of the core.To obtain a value division that motivates players to cooperate to a greater extent or that is more robust under noise, the notions of the strong least core and the weak least core have been considered. In this paper, we characterize the strong and the weak least cores of supermodular cooperative games using the theory of minimizing crossing submodular functions. We then apply our characterizations to two representative supermodular cooperative games, namely, the induced subgraph game generalized to hypergraphs and the airport game. For these games, we derive explicit forms of the strong and weak least core values, and provide polynomial-time algorithms that compute value divisions in the strong and weak least cores.

The Myopic Stable Set for Social Environments

We introduce a new solution concept for models of coalition formation, called the myopic stable set. The myopic stable set is defined for a very general class of social environments and allows for an infinite state space. We show that the myopic stable set exists and is non-empty. Under minor continuity conditions, we also demonstrate uniqueness. Furthermore, the myopic stable set is a super-set of the core and of the set of pure strategy Nash equilibria in non-cooperative games.Additionally, the myopic stable set generalizes and unifies various results from more specific environments. In particular, the myopic stable set coincides with the coalition structure core in coalition function form games if the coalition structure core is non-empty; with the set of stable matchings in the standard one-to-one matching model; with the set of pairwise stable networks and closed cycles in models of network formation; and with the set of pure strategy Nash equilibria in finite super-modular games, finite potential games, and aggregative games. We illustrate the versatility of our concept by characterizing the myopic stable set in a model of Bertrand competition with asymmetric costs, for which the literature so far has not been able to fully characterize the set of all (mixed) Nash equilibria.

Two-Stage Game Theoretic Modelling of Airline Frequency and Fare Competition

Airlines make capacity and fare decisions in a competitive environment. Capacity decisions, encompassing decisions about frequency of service and seats-per-flight, affect both the operating costs and revenues of airlines. These decisions have significant implications for the performance of the air transportation system as a whole. Capacity and fare decisions of different airlines are interdependent, both serving as tools in an airlines competitive arsenal. This interdependency motivates a game theoretic approach to modeling the decision process. Capacity (especially frequency) decisions are typically made months in advance of flight departure, with only an approximate knowledge of what fares will be, while fare decisions are made weeks to minutes ahead of flight departure. Several studies have stressed the need to develop two-stage game theoretic models to account for the sequential nature of these decisions, but there are very few analytical, computational, or empirical results available for such models. In this article (working paper available at link in [1]), we develop a two-stage frequency and fare competition model, demonstrate its tractability across a wide range of assumptions, and validate its predictions against observed airline behavior. We take the payoff function of an airline operating in a set of markets to be the sum of the differences between revenues and costs in those markets, with costs as a linear function of flight frequency. To compute revenue, we explore two commonly used multinomial logit models of market share. Frequency decisions are made in the first stage of the game, while fare decisions are made in the second stage. We begin our analysis with a simplified version of this game: two airlines competing in a single market, with no connecting passengers, infinite seating capacity, and the absence of a nofly alternative. Under these assumptions, for either market share model, we are able to prove that (1) the second-stage fare game always has unique pure strategy Nash equilibrium (PSNE), (2) first-stage payoffs for each airline are concave with respect to that airline's frequency strategy across plausible utility parameter ranges, and (3) first-stage payoffs for each airline are submodular functions in the overall frequency strategy space. As the game is two-player, (3) means that by changing the sign of one player's strategy space, we can trivially convert the game into a supermodular game. These results demonstrate that subgame-perfect PSNE is a credible and tractable solution concept for our model. In particular, the existence and uniqueness results indicate the suitability of PSNE as a solution concept for the second-stage game. Concave payoffs ensure that individual first-stage payoff maximization problems are efficiently solvable, and supermodularity ensures that several iterative learning dynamics converge to equilibrium [2]. We then relax each of the assumptions made in this simplified model by computationally solving the second stage fare game, generating equilibrium fare decisions and profits for every set of frequency decisions for integer daily frequency values ranging from 1 to 20, for various numbers of players, seats per flight, and values of utility parameters (including the nofly option). Then, we fit quadratic approximations to these profits as functions of the frequencies of all players. We find an excellent fit (R2 > 0.9) in all cases. Additionally, the signs of all estimated coefficients are consistent with submodularity and concavity properties demonstrated earlier. We show that for an N-player game with such concave and submodular quadratic payoff functions, the myopic best response heuristic, where each player optimizes its payoff against fixed opponent strategies iteratively, converges to a PSNE. To test the tractability and predictive validity of our model in practice, we apply it to a 4-airline, 11-airport network in the Western U.S, using publicly available airline operations data. We use the quadratic functions of airline frequency fitted above and additionally enforce the aircraft availability constraints, and solve for equilibrium iteratively using the myopic best response heuristic. To calibrate the 11 free quadratic payoff coefficients of this model, we use a stochastic gradient approximation algorithm to minimize the absolute errors between observed and predicted frequency strategies. In practice, the game convergences to equilibrium quickly, and the calibrated model's frequency predictions approximate observed behavior both in-sample and out-ofsample, suggesting that refinements of the model could be pursued for use in scenario analysis, forecasting, planning, and policy making.

Energy Management Based on Demand-Side Pricing: A Supermodular Game Approach

This paper proposes a pricing strategy for energy management in the electricity market composed of one generation company, multiple competing utility companies, and consumers with heating, ventilation, and air conditionings, which participate in demand response to keep the electricity real-time balance. The utility company plays the role of an intermediary agent between the generation company and consumers. We model the cost of the utility companies from the demand-side regulation and give the condition of a unique optimal regulation price. For the multiple competing utility companies, we demonstrate that the retail pricing game belongs to the supermodular game and the game exists a pure Nash equilibrium. The uniqueness of the Nash equilibrium is also proved, and then a distributed algorithm is developed to search for the optimal equilibrium price. Furthermore, we study the influence of the wholesale price on the equilibrium. Finally, numerical results demonstrate the efficiency of the proposed pricing strategy.

The Myopic Stable Set for Social Environments

We introduce a new solution concept for models of coalition formation, called the myopic stable set. The myopic stable set is defined for a very general class of social environments and allows for an infinite state space. We show that the myopic stable set exists and is non-empty. Under minor continuity conditions, we also demonstrate uniqueness. Furthermore, the myopic stable set is a superset of the core and of the set of pure strategy Nash equilibria in noncooperative games. Additionally, the myopic stable set generalizes and unifies various results from more specific environments. In particular, the myopic stable set coincides with the coalition structure core in coalition function form games if the coalition structure core is non-empty; with the set of stable matchings in the standard one-to-one matching model; with the set of pairwise stable networks and closed cycles in models of network formation; and with the set of pure strategy Nash equilibria in finite supermodular games, finite potential games, and aggregative games. We illustrate the versatility of our concept by characterizing the myopic stable set in a model of Bertrand competition with asymmetric costs, for which the literature so far has not been able to fully characterize the set of all (mixed) Nash equilibria.

LeChatelier–Samuelson principle in games and pass-through of shocks

The LeChatelier–Samuelson principle states that, as a reaction to a shock, an agent's short-run adjustment of an affected action is smaller than its long-run adjustment (when the agent can also adjust other related actions). We extend the principle to strategic environments where the long-run adjustment also accounts for other players adjusting their strategies. We show that the principle holds for supermodular games (strategic complements) satisfying monotone comparative statics and provide sufficient conditions for the principle to hold in games of strategic substitutes/heterogeneity. We discuss the principle's implications for cost pass-through of multiproduct firms.

Efficient coalitional bargaining with noncontingent offers

A new feature pertaining to proposer's ability to implement offers is introduced in the extensive form bargaining mechanism studied in Okada (1996). This mechanism is used to analyze the coalitional setting of strictly supermodular games. The new feature in the mechanism is that the proposer has a choice to implement his proposal with any subset of responders who have accepted it. Thus the institutional feature of ‘every responder has veto power’ is relaxed here. It is shown that for all sufficiently high discount factors δ, there exists an efficient subgame perfect equilibrium in pure stationary strategies (SSPE) whose limiting outcome is the core-constrained Nash Bargaining Solution. Moreover, all efficient SSPE are payoff-equivalent in the limit as δ→1.

On the robustness and strategic implications of a parsimonious advertising – inventory competitive model with extensions to pricing competition

Using a market share attraction structure of advertising competition, this research introduces first, a parsimonious inventory-advertising model for which competition is limited to two symmetric firms thus obtaining its related comparative statics. Following a supermodular game approach, the derived comparative statics remain robust upon considering non-dominant firms (each with a market share of less than 50%) competing in an asymmetric oligopoly of N rivals where each firm controls the inventory/advertising decision variables. Furthermore, changes in model parameters are investigated in terms of their relative impact on equilibrium measures. In this regard, the findings indicate that for the analyzed model, the marketing parameters possess much more influence on the equilibrium measures than the operations parameters. The generalization appears applicable to both non-dominated oligopolies as well as dominated oligopolies (where one firm has a market share equal to or greater than 50%) where each firm controls the inventory/advertising decision variables in addition to price. However, changes in shift parameters elicit a response from the dominant firm which will differ from its non-dominant rivals for products and services. Additionally, while the literature shows that the non-dominant firm will pursue a defensive strategy of decreasing both the equilibrium price and the amount of equilibrium advertising in response to competitive entry, our findings show that results for a fixed market potential may not continue to be robust once inclusion of the operations variables of order quantity are considered. Finally, the study explores the implications of the free - entry equilibrium for which the number of firms in the industry is determined endogenously.

How does socio-economic environment influence the distribution of altruism?

This paper analyzes how socio-economic environment affects the extent of individual altruism and its distribution within a society. We show that if the socio-economic institution that structures interactions between individuals can be represented by a supermodular game, individuals tend to become more homogeneous in terms of altruism as spillover effects increase. This implies that people living close to one another and engaging in joint production, such as irrigation agriculture in a rural village, are more likely to exhibit similar levels of altruism than those living apart and engaging in more or less independent work.

Comments on: Remarkable polyhedra related to set functions, games and capacities

A set function assigns a real number to any subset of a universal set. Set functions arise in many fields of mathematics and applied mathematics, but this survey focusses on cooperative games with transferable utility and on capacities where, formally speaking, the latter are a subclass of the former. Within these applications, the paper focusses on (i) the structure of polyhedra generated by capacities and several special classes of capacities; (ii) the structure of polyhedra generated by supermodular games; (iii) the structure of the core and variations on the core of capacities and games.

Remarkable polyhedra related to set functions, games and capacities

Set functions are widely used in many domains of operations research (cooperative game theory, decision under risk and uncertainty, combinatorial optimization) under different names (TU-game, capacity, nonadditive measure, pseudo-Boolean function, etc.). Remarkable families of set functions form polyhedra, e.g., the polytope of capacities, the polytope of p-additive capacities, the cone of supermodular games, etc. Also, the core of a set function, defined as the set of additive set functions dominating that set function, is a polyhedron which is of fundamental importance in game theory, decision-making and combinatorial optimization. This survey paper gives an overview of these notions and studies all these polyhedra.

Technology adoption: Hysteresis and absence of lock-in

We introduce a simple model of technology adoption with overlapping generations of players. Technologies generate network effects, and players are both backward- and forward-looking. We use results from the supermodular games literature to guarantee equilibrium existence and uniqueness. In line with the empirical literature, the equilibrium adoption path exhibits hysteresis and technologies cannot lock-in. We characterize the expected time of adoption, which can be seen as a measure of technology dominance.

Reconciling normative and behavioural economics

The aim of this thesis is to address from a methodological, philosophical and theoretical perspective the problem of how to reconcile normative and behavioural economics — the “reconciliation problem”. The first part develops a methodological assessment of behavioural welfare economics (and more specifically of libertarian paternalism), which currently constitutes the most accepted approach to deal with the reconciliation problem. We argue that behavioural welfare economics is in the direct continuation of neoclassical welfare economics, and that its conception of individual behaviour is tightly connected to Pareto’s reductionist model of the Homo economicus. It is assumed that an individual can be defined by “true” preferences — distinct from her revealed preferences — and that those preferences are consistent and context-independent. The satisfaction of those true preferences is taken as the normative criterion. The existence of true preferences requires accepting that people have access to a latent mode of reasoning able to generate consistent and context-independent preferences, and to treat individual decision-making as if it was the result of an optimisation problem faced by an inner rational agent trapped in an outer psychological shell. We argue on the contrary that we cannot define unambiguously such true preferences. We then argue that the normative challenge raised by behavioural economics is that individuals may lack of autonomy: rather than focusing on the satisfaction of one’s preferences, we argue that what matters is the ability to choose one’s own preferences. Since we suggest that the difficulties of behavioural welfare economics are related to its commitment to an implausible model of individual preferences, we provide in the second part of the thesis a model of preferences compatible with our methodological analysis: we take Bacharach’s variable frame theory as the primitive of our analysis, and build a model of endogenous preferences, in which individuals can choose to some extent their own preferences. The main contributions of this model are that (i) it does not require the existence of latent true preferences, and (ii) the individuals are allowed to team reason. We assume that collective preferences are strategically chosen, and show that team reasoners tend to be aggressive (cooperative) with other teams in submodular (supermodular) games. We finally show that team reasoning can be empirically justified as an ecologically rational heuristic.

Book Reviews

Benjamin Williams of George Washington University reviews “Structural Econometric Models”, by Eugene Choo and Matthew Shum. The Econlit abstract of this book begins: “Twelve papers explore recent developments in the use of structural econometric models in empirical economics. Papers discuss Euler equations for the estimation of dynamic discrete choice structural models; approximating high-dimensional dynamic models—sieve value function iteration; identifying dynamic games with serially correlated unobservables; partial identification in two-sided matching models; identification of matching complementarities—a geometric viewpoint; comparative static and computational methods for an empirical one-to-one transferable utility matching model; a test for monotone comparative statics; estimating supermodular games using rationalizable strategies; estimation of the loan spread equation with endogenous bank-firm matching; the collective marriage matching model— identification, estimation, and testing; deflation in durable goods markets—an empirical model of the Tokyo condominium market; and a dynamic analysis of the U.S. cigarette market and antismoking policies. Choo is with the Department of Economics at the University of Calgary. Shum is with the Division of Humanities and Social Sciences at the California Institute of Technology.”

Learning in monotone bayesian games

This paper studies learning in monotone Bayesian games with one-dimensional typesand finitely many actions. Players switch between actions at a set of thresholds. A learning algorithm under which players adjust theirstrategies in the direction of better ones using payoffs received at similar signalsto their current thresholds is examined. Convergence to equilibrium is shown in the case of supermodular games and potentialgames.

Lechatelier-Samuelson Principle in Games and Pass-Through of Shocks

The LeChatelier-Samuelson principle ("the principle") states that as a reaction to a shock, an agent's short-run adjustment of an action is smaller than the long-run adjustment of that action when the other related actions can also be adjusted. We extend the principle to strategic environments and to shocks that affect more than one action directly. We define long run as an adjustment that also includes the affected player adjusting its other actions and other players adjusting their strategies. We show that the principle holds for 1) supermodular games (strategic complements), 2) submodular games (strategic substitutes) for shocks that affect only one player's action directly and when the players' payoffs depend only on their own strategies and the sum of the rivals' strategies (for example, homogeneous Cournot oligopoly). We also provide other sufficient conditions for the principle to hold in games of strategic substitutes. Our results imply that when the principle holds a multiproduct oligopoly might have lower cost pass-through in the short run than in the long run. Hence, we argue that the principle might explain the empirical findings of overshifting of cost and unit tax by multiproduct firms.