Multiple Battlefield Research Articles

Multi-battle contests over complementary battlefields

AbstractThis paper studies contests with complementary prizes where each agent simultaneously distributes a fixed budget over multiple battlefields. Each battlefield has a single prize which is divided among the competitors in proportion to an arbitrary power function of their investment levels. A unique pure strategy Nash equilibrium is shown to exist under arbitrarily sensitive battlefield success functions if objective functions exhibit constant subunitary elasticity of substitution between prize shares. In contrast, Blotto contests with linear objectives have only mixed strategy Nash equilibria if battlefield success functions are sufficiently sensitive to investment levels. Sufficient complementarity between prize shares allows pure strategy Nash equilibria to exist under arbitrarily sensitive battlefield success functions.

Privacy-Aware Laser Wireless Power Transfer for Aerial Multi-Access Edge Computing: A Colonel Blotto Game Approach

This article studies the integration of laser-beamed wireless power transfer (WPT) into high-altitude platform (HAP)-aided multiaccess edge computing (MEC) systems for the HAP-connected aerial user equipments (AUEs). By discretizing the 3-D coverage space of the HAP, we present a multitier tile grid-based spatial structure to provide the aerial locations in the form of tile grids to AUEs for laser charging. We identify a new privacy vulnerability caused by the openness during the WPT signaling transfer in the presence of a terrestrial adversary, which is able to launch the attacks by distributing the false tile grids to the AUEs. To address this vulnerability and enhance the location privacy of AUEs, we then propose a Colonel Blotto (CB) game framework to formulate the competitive tile grid allocation problem for the HAP and the adversary. The attack-defense interaction between the adversary and the HAP as a defender in their tile grid allocations to the AUEs is formulated as a CB game, which models the competition of two players for limited resources over multiple battlefields. Moreover, we derive the mixed-strategy Nash equilibria of the game for both symmetric and asymmetric tile grids between two players. Simulation results show that the proposed framework significantly outperforms the design baselines with a given privacy protection level in terms of system-wide expected total utilities.

Equilibrium characterizations of asymmetric Majoritarian contests*

The General Lotto game is a well-studied model where two opposing players strategically allocate their resources to multiple battlefields. In its classic setting, each player's objective is to secure as much value as possible by winning many individual battlefields. However, in many applications, a player must secure particular subsets of battlefields in order to succeed. The classic setting fails to capture a variety of adversarial interactions – for instance, ensuring the security of networks or cyber-security domains. In this paper, we focus on a particular alternate player objective known as the majoritarian objective, where a player needs to secure a majority of battlefields in order to succeed. Equilibrium characterizations for the majoritarian objective in the existing literature are limited to particular symmetric settings where both players place the same value in succeeding. Our contributions extend these equilibrium solutions to asymmetric cases in two different settings: 1) both players have fixed and asymmetric resource budgets, and 2) both players place different values in succeeding and pay costs for allocating resources.

Algorithm for Computing Approximate Nash Equilibrium in Continuous Games with Application to Continuous Blotto

Successful algorithms have been developed for computing Nash equilibrium in a variety of finite game classes. However, solving continuous games—in which the pure strategy space is (potentially uncountably) infinite—is far more challenging. Nonetheless, many real-world domains have continuous action spaces, e.g., where actions refer to an amount of time, money, or other resource that is naturally modeled as being real-valued as opposed to integral. We present a new algorithm for approximating Nash equilibrium strategies in continuous games. In addition to two-player zero-sum games, our algorithm also applies to multiplayer games and games with imperfect information. We experiment with our algorithm on a continuous imperfect-information Blotto game, in which two players distribute resources over multiple battlefields. Blotto games have frequently been used to model national security scenarios and have also been applied to electoral competition and auction theory. Experiments show that our algorithm is able to quickly compute close approximations of Nash equilibrium strategies for this game.

Computational Analyses of the Electoral College: Campaigning Is Hard But Approximately Manageable

In the classical discrete Colonel Blotto game—introduced by Borel in 1921—two colonels simultaneously distribute their troops across multiple battlefields. The winner of each battlefield is determined by a winner-take-all rule, independently of other battlefields. In the original formulation, each colonel’s goal is to win as many battlefields as possible. The Blotto game and its extensions have been used in a wide range of applications from political campaign—exemplified by the U.S presidential election—to marketing campaign, from (innovative) technology competition to sports competition. Despite persistent efforts, efficient methods for finding the optimal strategies in Blotto games have been elusive for almost a century—due to exponential explosion in the organic solution space—until Ahmadinejad, Dehghani, Hajiaghayi, Lucier, Mahini, and Seddighin developed the first polynomial-time algorithm for this fundamental gametheoretical problem in 2016. However, that breakthrough polynomial-time solution has some structural limitation. It applies only to the case where troops are homogeneous with respect to battlegruounds, as in Borel’s original formulation: For each battleground, the only factor that matters to the winner’s payoff is how many troops as opposed to which sets of troops are opposing one another in that battleground. In this paper, we consider a more general setting of the two-player-multi-battleground game, in which multifaceted resources (troops) may have different contributions to different battlegrounds. In the case of U.S presidential campaign, for example, one may interpret this as different types of resources—human, financial, political—that teams can invest in each state. We provide a complexity-theoretical evidence that, in contrast to Borel’s homogeneous setting, finding optimal strategies in multifaceted Colonel Blotto games is intractable. We complement this complexity result with a polynomial-time algorithm that finds approximately optimal strategies with provable guarantees. We also study a further generalization when two competitors do not have zerosum/ constant-sum payoffs. We show that optimal strategies in these two-player-multi-battleground games are as hard to compute and approximate as Nash equilibria in general noncooperative games and economic equilibria in exchange markets.

Robust Pairwise n-Person Stochastic Duel Game

This paper introduces an extended version of a stochastic game under the antagonistic duel-type setup. The most flexible multiple person duel game is analytically solved. Moreover, the explicit formulas are solved to determine the time-dependent duel game model using the first exceed theory in multiple game stages. Unlike conventional stochastic duel games, multiple battlefields are firstly introduced and each battlefield becomes a shooting ground of pairwise players in a multiperson game. Each player selects different targets in different game stages. An analogue of this new theory was designed to find the best shooting time within multiple battlefields. This model is fully mathematically explained and is the basis with which to apply a stochastic duel-type game in various practical applications.

Tech Wars: Return of the Conglomerate - Throwback or Dawn of a New Series for Competition in the Digital Era?

There was once a time when conglomerates roamed supreme in the realms of the market. These large and colorful beasts were recognized as the dominant species of the corporate form, and were the subject of both praise and fear. Then beginning in the 1980s, their fate suddenly changed for the worse, and under the assault of so-called “bust up takeovers” these creatures receded from the corporate scene. This led commentators to observe by the turn of the century that the once vaunted conglomerate form was under the risk of becoming an “endangered” species. Or so they believed. The conglomerate model has begun to stage a comeback in a seemingly unlikely place for what many have relegated as an old-fashioned (and even ill-conceived) relic of the past – none other than the high-flying and perennially evolving tech industry. This astonishing return of the conglomerate model has coincided with an intriguing change in the nature of competition for the high-tech industry, particularly where platform businesses that harness network effects are involved. To put it bluntly, competition in the tech industry in the digital era has morphed into a multi-contact war – a war between conglomerate-like businesses, with clashes ensuing on multiple battlefields. What does this then portend for antitrust law and analysis? Both the change in the competitive nature of the tech industry and the concomitant resurgence of the conglomerate model in today’s digital era have garnered little, if any, substantive attention so far within antitrust circles. This does not mean that the multi-dimensional characteristics of market competition involving the tech industry has been completely overlooked. Much ink has been spilt over the attributes of multi-sided markets, and the network effects that course through platforms straddling such multiple dimensions of the market. Nor is the consideration of multi-contact competition completely novel for antitrust. Antitrust has grappled with aspects of multi-contact competition going back to as far as more than half a century ago, mostly in the context of conglomerate mergers. But, there has not yet been a proper analysis of the recent rise of multi-contact competition waged by conglomerate businesses (either in substance or true form) in today’s tech industry. This essay is a preliminary foray into certain key aspects of multi-contact competition waged by conglomerate businesses in the tech industry, and its implications for antitrust law and analysis. Through the discussion, we will observe how insights from the field of strategic management can prove invaluable for the analysis, and that the careful consideration of the firm’s strategy is a prerequisite to properly analyzing this phenomenon, as it should be for antitrust analysis in general.

Distributed Parameter Estimation for Mobile Wireless Sensor Network Based on Cloud Computing in Battlefield Surveillance System

The construction of a battlefield surveillance system is very important for monitoring the attack of enemy aircrafts and missiles, which integrates various sensors and mobile devices. Then, multiple battlefield surveillance systems can be connected together to form a battlefield surveillance network. The mobile nodes can be deployed in a certain region to monitor enemy aircrafts and missiles. Thus, some important issues have to be solved efficiently, including the cooperation across the administrative domains of a cloud network, the direction-of-arrival (DOA), and a polarization estimation algorithm for a mobile wireless sensor network (MWSN). In this paper, the architecture of a battlefield surveillance system is constructed based on mobile cloud computing and 5G link. The root multiple signal classification (Root-MUSIC)-like algorithm is proposed for estimating the 1-D DOA and a polarization parameter with a uniform linear array. The Root-MUSIC algorithm is replaced by the Fourier transform, the former algorithm that can be extended to an arbitrary topology structure of a MWSN. Then, the proposed algorithm is extended to the 2-D DOA and a polarization estimation in further. Based on the deployment of different MWSNs, the estimation results of DOA and polarization parameters are fused in order to improve the estimation performance. Finally, the parameter information (DOA and polarization parameter) of enemy aircrafts and missiles can be achieved. The computer simulation verifies the effectiveness of the proposed algorithm. The proposed algorithm ensures the parameter estimation accuracy with a low computational complexity.

Conflicts with Multiple Battlefields

Conflicts with Multiple Battlefields