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.

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

Multi-access edge computing (MEC) has been recently considered in challenging environments lacking available terrestrial infrastructures by extending the computing resources to the air for further enhancing the computation capability of the new aerial user equipment (AUE). Additionally, wireless power transfer (WPT) is a promising solution to prolong the battery lifetime of energy-constrained wireless devices like AUEs. In this paper, we investigate the integration of laser-beamed WPT in the high-altitude platform (HAP) aided MEC systems for the HAP-connected AUEs. By discretizing the three-dimensional coverage space of the HAP, we present a multi-tier tile grid-based spatial structure to provide aerial locations for laser charging. With this setup, we identify a new privacy vulnerability caused by the openness during the air-to-air transmission of WPT signaling messages in the presence of a terrestrial adversary. A privacy-aware laser-powered aerial MEC framework is developed that addresses this vulnerability and enhances the location privacy of AUEs for laser WPT. Specifically, the interaction between the HAP as a defender and the adversary in their tile grid allocation as charging locations to AUEs is formulated as a Colonel Blotto game, which models the competition of the players for limited resources over multiple battlefields for a finite time horizon. Moreover, we derive the mixed-strategy Nash equilibria of the tile grid allocation game for both symmetric and asymmetric tile grids between the defender and the adversary. Simulations 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.

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

Multi-access edge computing (MEC) has been recently considered in challenging environments lacking available terrestrial infrastructures by extending the computing resources to the air for further enhancing the computation capability of the new aerial user equipment (AUE). Additionally, wireless power transfer (WPT) is a promising solution to prolong the battery lifetime of energy-constrained wireless devices like AUEs. In this paper, we investigate the integration of laser-beamed WPT in the high-altitude platform (HAP) aided MEC systems for the HAP-connected AUEs. By discretizing the three-dimensional coverage space of the HAP, we present a multi-tier tile grid-based spatial structure to provide aerial locations for laser charging. With this setup, we identify a new privacy vulnerability caused by the openness during the air-to-air transmission of WPT signaling messages in the presence of a terrestrial adversary. A privacy-aware laser-powered aerial MEC framework is developed that addresses this vulnerability and enhances the location privacy of AUEs for laser WPT. Specifically, the interaction between the HAP as a defender and the adversary in their tile grid allocation as charging locations to AUEs is formulated as a Colonel Blotto game, which models the competition of the players for limited resources over multiple battlefields for a finite time horizon. Moreover, we derive the mixed-strategy Nash equilibria of the tile grid allocation game for both symmetric and asymmetric tile grids between the defender and the adversary. Simulations 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.

Privacy-Aware Wireless Power Transfer for Aerial Computation Offloading via Colonel Blotto Game

In this paper, a laser-powered aerial mobile edge computing (MEC) architecture is proposed, where a high-altitude platform (HAP) integrated with an MEC server transfers laser energy to charge aerial user equipments (AUEs) for offloading their computation tasks to the HAP. Particularly, we identify a new privacy vulnerability caused by the transmission of wireless power transfer (WPT) signaling in the presence of a malicious smart attacker (SA). To address this vulnerability, the interaction between the HAP and the SA in their allocation of tile grids as charging points to the AUEs in laser-enabled WPT is formulated as a Colonel Blotto game (CBG), which models the competition of two players for limited resources over multiple battlefields for a finite time horizon. Moreover, the utility function that each player receives over a battlefield is developed by identifying the tradeoff between privacy protection level and energy consumption of each AUE. We further obtain the mixed-strategy Nash equilibrium for the modified CBG with asymmetric players. Simulation results are presented to show the effectiveness of this game framework.