Submodular Games Research Articles

Coverage Strategy for Small-Cell UAV-Based Networks in IoT Environment

In wireless communication, small cells are low-powered cellular base stations that can be used to enhance the coverage and capacity of wireless networks in areas where traditional cell towers may not be practical or cost-effective. Unmanned aerial vehicles (UAVs) can be used to quickly deploy and position small cells in areas that are difficult to access or where traditional infrastructure is not feasible. UAVs are deployed by telecommunication service providers to provide aerial network access in remote rural areas, disaster-affected areas, or massive-attendance events. In this paper, we focus on the scheduling of beaconing periods as an efficient means of energy consumption optimization. The conducted study provides a sub-modular game perspective of the problem and investigates its structural properties. We also provide a learning algorithm that ensures convergence of the considered UAV network to a Nash equilibrium operating point. Finally, we conduct extensive numerical investigations to assist our claims about the energy and data rate efficiency of the strategic beaconing policy (at Nash equilibrium).

Replication Robust Payoff Allocation in Submodular Cooperative Games

Submodular functions have been a powerful mathematical model for a wide range of real-world applications. Recently, submodular functions are becoming increasingly important in machine learning (ML) for modelling notions such as information and redundancy among entities such as data and features. Among these applications, a key question is payoff allocation, i.e., how to evaluate the importance of each entity towards the collective objective? To this end, classic solution concepts from cooperative game theory offer principled approaches to payoff allocation. However, despite the extensive body of game-theoretic literature, payoff allocation in submodular games are relatively under-researched. In particular, an important notion that arises in the emerging submodular applications is redundancy, which may occur from various sources such as abundant data or malicious manipulations where a player replicates its resource and act under multiple identities. Though many game-theoretic solution concepts can be directly used in submodular games, naively applying them for payoff allocation in these settings may incur robustness issues against replication. In this paper, we systematically study the replication manipulation in submodular games and investigate replication robustness, a metric that quantitatively measures the robustness of solution concepts against replication. Using this metric, we present conditions which theoretically characterise the robustness of semivalues, a wide family of solution concepts including the Shapley and Banzhaf value. Moreover, we empirically validate our theoretical results on an emerging submodular ML application, i.e., the ML data market.

Controlling Segregation in Social Network Dynamics as an Edge Formation Game

This paper studies controlling <i>segregation</i> in social networks via exogenous incentives. We construct an edge formation game on a directed graph. A user (node) chooses the probability with which it forms an inter- or intra- community edge based on a utility function that reflects the tradeoff between homophily (preference to connect with individuals that belong to the same group) and the preference to obtain an exogenous incentive. Decisions made by the users to connect with each other determine the evolution of the social network. We explore an <i>algorithmic recommendation mechanism</i> where the exogenous incentive in the utility function is based on <i>weak ties</i> which incentivizes users to connect across communities and mitigates the segregation. This setting leads to a submodular game with a unique Nash equilibrium. In numerical simulations, we explore how the proposed model can be useful in controlling segregation and echo chambers in social networks under various settings.

Characterizing robust solutions in monotone games

In game theory, p-dominance and its set-valued generalizations serve as important robust solution concepts. We show that in monotone games, (which include the broad classes of supermodular games, submodular games, and their combinations) these concepts can be characterized in terms of pure strategy Nash equilibria in an auxiliary game of complete information. The auxiliary game is constructed in a transparent manner that is easy to follow and retains a natural connection to the original game. Our results show explicitly how to map these concepts to corresponding Nash equilibria thereby identifying a new bijection between robust solutions in the original game and equilibrium notions in the auxiliary game. Moreover, our characterizations lead to new results about the structure of entire classes of such solution concepts. In games with strategic complements, these classes are complete lattices. More generally, they are totally unordered. We provide several examples to highlight these results.

Characterizing Robust Solutions to Monotone Games

In game theory, p-dominance and p-best response sets serve as important robustness solution concepts by allowing for deviations from the stringent common knowledge requirements of Nash equilibrium. However, solving for such sets remains largely intractable beyond the simplest of settings. The contributions of this paper are twofold: First, in monotone games, (which include the broad class of supermodular games, submodular games, and their combinations,) we show that these concepts can be characterized in terms of pure strategy Nash equilibria in an auxiliary game of complete information. This makes it considerably easier to compute such sets, facilitating their broader use. Second, these characterizations lead to new results about the structure of entire classes of such solution concepts, including minimal p-best response (p-MBR) sets, which generalize wellknown results for pure strategy Nash equilibria. In games with strategic complements, these classes are complete lattices. More generally, they are totally unordered. Several examples highlight the results.

Pricing strategies for competitive firms with indirect network effects

In this paper, we analyze the impact of indirect network effects and competition on two firms each offering two complementary products. We first formulate the pricing game with general network effects and show that it is neither a supermodular nor a submodular game. We then derive the closed-form Nash equilibrium with linear demand functions. Furthermore, we examine a model with two symmetric firms each having a major product and a minor product, where the major product of one firm is the minor one of the other. We find that the impacts of network effects of major and minor products on the equilibrium prices are not symmetric. Moreover, as the substitution rate increases, the firm might partially sacrifice the major product market to boost the minor product market when the substitution rate is low and the network effect of the minor product is quite weak compared to that of the major product. Our numerical analysis shows that strong externalities between the two complementary products will benefit the firms. Finally, the models with symmetric product structures and generalized customer preference are analyzed to verify the robustness of the main results.

Data Offloading in UAV-Assisted Multi-Access Edge Computing Systems: A Resource-Based Pricing and User Risk-Awareness Approach.

Unmanned Aerial Vehicle (UAV)-assisted Multi-access Edge Computing (MEC) systems have emerged recently as a flexible and dynamic computing environment, providing task offloading service to the users. In order for such a paradigm to be viable, the operator of a UAV-mounted MEC server should enjoy some form of profit by offering its computing capabilities to the end users. To deal with this issue in this paper, we apply a usage-based pricing policy for allowing the exploitation of the servers’ computing resources. The proposed pricing mechanism implicitly introduces a more social behavior to the users with respect to competing for the UAV-mounted MEC servers’ computation resources. In order to properly model the users’ risk-aware behavior within the overall data offloading decision-making process the principles of Prospect Theory are adopted, while the exploitation of the available computation resources is considered based on the theory of the Tragedy of the Commons. Initially, the user’s prospect-theoretic utility function is formulated by quantifying the user’s risk seeking and loss aversion behavior, while taking into account the pricing mechanism. Accordingly, the users’ pricing and risk-aware data offloading problem is formulated as a distributed maximization problem of each user’s expected prospect-theoretic utility function and addressed as a non-cooperative game among the users. The existence of a Pure Nash Equilibrium (PNE) for the formulated non-cooperative game is shown based on the theory of submodular games. An iterative and distributed algorithm is introduced which converges to the PNE, following the learning rule of the best response dynamics. The performance evaluation of the proposed approach is achieved via modeling and simulation, and detailed numerical results are presented highlighting its key operation features and benefits.

A Stackelberg game approach for multiple energies trading in integrated energy systems

In this paper, we propose a novel game model based on the hierarchical Stackelberg game for analyzing the multiple energies trading (MET) problem in integrated energy systems (IESs). In the proposed game model, a number of distributed energy stations (DESs) lead the game deciding the unit prices of electricity and cooling energies they generated, while multiple energy users (EUs) perform as followers determining the amounts of energies to consume. In addition to maximizing the profit of each DES, the proposed hierarchical game model also considers the benefit of each EU, who actively participates in the MET. We prove that, for the first time, there exists a unique Stackelberg Equilibrium (SE) in the MET, so that the existence of an equilibrium strategy optimizing the objectives of all participants can be guaranteed. Moreover, we found that the price setting game played by DESs is a submodular game when energy dispatching is performed during their generation process, which indicates that the behaviors of DESs are strategic substitutes. Furthermore, the SE is obtained in a closed-form, by which the effects of coupling generation of multiple energies are analyzed. Finally, a best response algorithm is provided to obtain the SE in an iterative way. Numerical studies demonstrate the convergency of the proposed best response algorithm, corroborate the jointly effects of market scale and exogenous parameter on the SE and verify the practicability of the proposed game-theoretic approach.

A green strategic activity scheduling for UAV networks: A sub-modular game perspective

Unmanned aerial vehicles (UAVs) were initially developed for military monitoring and surveillance tasks but found several interesting applications in the civilian domain. A promising application/technology is to use drone small cells (DSCs) to expand wireless communication coverage on demand. Rapid deployment along with limited operating costs are key factors that boost the development of DSCs for both military and civilian utilizations. DSCs are rapidly deployable to provide connectivity for temporary users (e.g. attendees of festivals, sporting events, or seminars), or over disaster areas to replace damaged communication infrastructure. UAVs are battery-powered, which makes energy consumption optimization a critical issue for acceptable performance, high availability, and an economically viable DCS deployment. In this article we focus on the scheduling of beaconing periods as an efficient means of energy consumption optimization. The conducted study provides a sub-modular game perspective of the problem and investigates its structural properties. We also provide a learning algorithm that ensures convergence of the considered UAV network with its unique Nash equilibrium operating point. Finally, we conduct extensive numerical investigations to assist our claims about the energy efficiency of the strategic beaconing policy (at Nash equilibrium).

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.

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.

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

We show that the LeChatelier-Samuelson principle holds in many strategic environments: the adjustment to changes in parameters (shocks) in the short run (only by one player) is smaller in magnitude than the adjustment in the long run (allowing other players to adjust as well). We show that the principle holds for supermodular games (strategic complements) regardless of whether the shock directly affects only one player's action (idiosyncratic shock). The principle holds for submodular games (strategic substitutes) if the shock is idiosyncratic and there are only two players. We provide a simple sufficient condition for the principle to hold in submodular games with more than two players. The principle might fail to hold in submodular games if the shock is not idiosyncratic. We show that our findings explain the breakdown of the principle in non-strategic environments as well, apply our results to multiproduct monopoly and oligopoly cost pass-through, and argue that the principle might explain the empirical findings of overshifting of cost and tax changes.

Nonspecific Networking

A new model of strategic networking is developed and analyzed, where an agent’s investment in links is nonspecific. The model comprises a large class of games which are both potential and super- or submodular games. We obtain comparative statics results for Nash equilibria with respect to investment costs for supermodular as well as submodular networking games. We also study supermodular games with potentials. We find that the set of potential maximizers forms a sublattice of the lattice of Nash equilibria and derive comparative statics results for the smallest and the largest potential maximizer. Finally, we provide a broad spectrum of applications from social interaction to industrial organization.

A Conjectural Cooperative Equilibrium in Strategic Form Games

This paper presents a new cooperative equilibrium for strategic form games, denoted Conjectural Cooperative Equilibrium (CCE). This concept is based on the expectation that joint deviations from any strategy profile are followed by an optimal and noncooperative reaction of non deviators. We show that CCE exist for all symmetric supermodular games. Furthermore, we discuss the existence of a CCE in specific submodular games employed in the literature on environmental agreements.

The value of information in oligopoly with demand uncertainty

This paper discusses the value of information in supermodular and submodular games, using a simple duopoly model where the level of demand is uncertain. It is shown that the value of information issuperadditive (resp.,subadditive) between players if the game issupermodular (resp.,submodular). For example, in a Bertrand (resp., Cournot) market with (possibly imperfect) substitute products, one firm's information acquisition increases (resp., decreases) the other firm's incentive to acquire the same information. Furthermore, when the game is either supermodular or submodular, the value of information is higher when the player isexpected to be informed according to the opponent's belief than when the player is expected to be uninformed; this result is reversed when the game has asymmetric modularity (i.e., one player's action is substitutional to the other's, and the latter's action is complemental to the former's). These qualitative observations have a potential to be applied to a larger class of games with uncertainty where payoffs are smooth (e.g., twice continuously differentiable) in actions and states.

The Strategic Advantage of Negatively Interdependent Preferences

We study certain classes of supermodular and submodular games which are symmetric with respect to material payoffs but in which not all players seek to maximize their material payoffs. Specifically, a subset of players have negatively interdependent preferences and care not only about their own material payoffs but also about their payoffs relative to others. We identify sufficient conditions under which members of the latter group have a strategic advantage in the following sense: at all intragroup symmetric equilibria of the game, they earn strictly higher material payoffs than do players who seek to maximize their material payoffs. The conditions are satisfied by a number of games of economic importance. We discuss the implications of these findings for the evolutionary theory of preference formation and the theory of strategic delegation. Journal of Economic Literature Classification Numbers: C72, D62.

Equilibrium Points in Nonzero-Sum n-Person Submodular Games

A submodular game is a finite noncooperative game in which the set of feasible joint decisions is a sublattice and the cost function of each player has properties of submodularity and antitone differences. Examples of submodular games include 1) a game version of a system with complementary products; 2) an extension of the minimum cut problem to a situation where players choose from different sets of nodes and perceive different capacities, with special cases being a game with players choosing whether or not to participate in available economic activities and a game version of the selection problem; 3) the pricing problem of competitors producing substitute products; 4) a game version of the facility location problem; and 5) a game with players determining their optimal usage of available products. A fixed point approach establishes the existence of a pure equilibrium point for certain submodular games. Two algorithms which correspond to fictitious play in dynamic games generate sequences of feasible joint decisions converging monotonically to a pure equilibrium point. Bounds show these algorithms to be very efficient when the set of feasible decisions is finite. An optimal decision for each player is an isotope function of the decisions of other players.