4 research outputs found
Mixed Strategy Constraints in Continuous Games
Equilibrium problems representing interaction in physical environments
typically require continuous strategies which satisfy opponent-dependent
constraints, such as those modeling collision avoidance. However, as with
finite games, mixed strategies are often desired, both from an equilibrium
existence perspective as well as a competitive perspective. To that end, this
work investigates a chance-constraint-based approach to coupled constraints in
generalized Nash equilibrium problems which are solved over pure strategies and
mixing weights simultaneously. We motivate these constraints in a discrete
setting, placing them on tensor games (-player bimatrix games) as a
justifiable approach to handling the probabilistic nature of mixing. Then, we
describe a numerical solution method for these chance constrained tensor games
with simultaneous pure strategy optimization. Finally, using a modified
pursuit-evasion game as a motivating examples, we demonstrate the actual
behavior of this solution method in terms of its fidelity, parameter
sensitivity, and efficiency
Computing Algorithm for an Equilibrium of the Generalized Stackelberg Game
The generalized Stackelberg game (single-leader multi-follower game) is
intricately intertwined with the interaction between a leader and followers
(hierarchical interaction) and the interaction among followers (simultaneous
interaction). However, obtaining the optimal strategy of the leader is
generally challenging due to the complex interactions among the leader and
followers. Here, we propose a general methodology to find a generalized
Stackelberg equilibrium of a generalized Stackelberg game. Specifically,
we first provide the conditions where a generalized Stackelberg equilibrium
always exists using the variational equilibrium concept. Next, to find an
equilibrium in polynomial time, we transformed the generalized
Stackelberg game into a Stackelberg game whose Stackelberg equilibrium is
identical to that of the original. Finally, we propose an effective computation
procedure based on the projected implicit gradient descent algorithm to find a
Stackelberg equilibrium of the transformed Stackelberg game. We validate
the proposed approaches using the two problems of deriving operating strategies
for EV charging stations: (1) the first problem is optimizing the one-time
charging price for EV users, in which a platform operator determines the price
of electricity and EV users determine the optimal amount of charging for their
satisfaction; and (2) the second problem is to determine the spatially varying
charging price to optimally balance the demand and supply over every charging
station.Comment: 37 pages, 10 figure
On the polyhedral homotopy method for solving generalized Nash equilibrium problems of polynomials
The generalized Nash equilibrium problem (GNEP) is a kind of game to find
strategies for a group of players such that each player's objective function is
optimized. Solutions for GNEPs are called generalized Nash equilibria (GNEs).
In this paper, we propose a numerical method for finding GNEs of GNEPs of
polynomials based on the polyhedral homotopy continuation and the Moment-SOS
hierarchy of semidefinite relaxations. We show that our method can find all
GNEs if they exist, or detect the nonexistence of GNEs, under some genericity
assumptions. Some numerical experiments are made to demonstrate the efficiency
of our method.Comment: 25 pages, Version to appear in Journal of Scientific Computin