Designing games to handle coupled constraints

The central goal in multiagent systems is to design local control laws for the individual agents to ensure that the emergent global behavior is desirable with respect to a given system level objective. In many systems, such as cooperative robotics or distributed power control, the design of these local control algorithms is further complicated by additional coupled constraints on the agents' actions. There are several approaches in the existing literature for designing such algorithms stemming from classical optimization theory; however, many of these approaches are not suitable for implementation in multiagent systems. This paper seeks to address the design of such algorithms using the field of game theory. Among other things, this design choice requires defining a local utility function for each decision maker in the system. This paper seeks to address the degree to which utility design can be effective for dealing with these coupled constraints. In particular, is it possible to design local agent utility functions such that all pure Nash equilibrium of the unconstrained game (i) optimize the given system level objective and (ii) satisfy the given coupled constraint. This design would greatly simplify the distributed control algorithms by eliminating the need to explicitly consider the constraints. Unfortunately, we illustrate that designing utility functions within the standard game theoretic framework is not suitable for this design objective. However, we demonstrate that by adding an additional state variable in the game environment, i.e., moving towards state based games, we can satisfy these performance criteria by utility design. We focus on the problem of consensus control to illustrate these results

How to use Rosen's normalised equilibrium to enforce a socially desirable Pareto efficient solution

We consider a situation, in which a regulator believes that constraining a complex good created jointly by competitive agents, is socially desirable. Individual levels of outputs that generate the constrained amount of the externality can be computed as a Pareto efficient solution of the agents' joint utility maximisation problem. However, generically, a Pareto efficient solution is not an equilibrium. We suggest the regulator calculates a Nash-Rosen coupled-constraint equilibrium (or a “generalised” Nash equilibrium) and uses the coupled-constraint Lagrange multiplier to formulate a threat, under which the agents will play a decoupled Nash game. An equilibrium of this game will possibly coincide with the Pareto efficient solution. We focus on situations when the constraints are saturated and examine, under which conditions a match between an equilibrium and a Pareto solution is possible. We illustrate our findings using a model for a coordination problem, in which firms' outputs depend on each other and where the output levels are important for the regulator.

Can planners control competitive generators?

Consider an electricity market populated by competitive agents using thermal generating units. Generation often emits pollution which a planner may wish to constrain through regulation. Furthermore, generators’ ability to transmit energy may be naturally restricted by the grid’s facilities. The existence of both pollution standards and transmission constraints can impose several restrictions upon the joint strategy space of the agents. We propose a dynamic, game-theoretic model capable of analysing coupled constraints equilibria (also known as generalised Nash equilibria). Our equilibria arise as solutions to the planner’s problem of avoiding both network congestion and excessive pollution. The planner can use the coupled constraints’ Lagrange multipliers to compute the charges the players would pay if the constraints were violated. Once the players allow for the charges in their objective functions they will feel compelled to obey the constraints in equilibrium. However, a coupled constraints equilibrium needs to exist and be unique for this modification of the players’ objective functions ..[there was a “to” here, incorrect?].. induce the required behaviour. We extend the three-node dc model with transmission line constraints described in [10] and [2] to utilise a two-period load duration curve, and impose multi-period pollution constraints. We discuss the economic and environmental implications of the game’s solutions as we vary the planner’s preferences.

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 ($n$-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

Can planners control competitive generators?

Consider an electricity market populated by competitive agents using thermal generating units. Generation often emits pollution which a planner may wish to constrain through regulation. Furthermore, generators’ ability to transmit energy may be naturally restricted by the grid’s facilities. The existence of both pollution standards and transmission constraints can impose several restrictions upon the joint strategy space of the agents. We propose a dynamic, game-theoretic model capable of analysing coupled constraints equilibria (also known as generalised Nash equilibria). Our equilibria arise as solutions to the planner’s problem of avoiding both network congestion and excessive pollution. The planner can use the coupled constraints’ Lagrange multipliers to compute the charges the players would pay if the constraints were violated. Once the players allow for the charges in their objective functions they will feel compelled to obey the constraints in equilibrium. However, a coupled constraints equilibrium needs to exist and be unique for this modification of the players’ objective functions ..[there was a “to” here, incorrect?].. induce the required behaviour. We extend the three-node dc model with transmission line constraints described in [10] and [2] to utilise a two-period load duration curve, and impose multi-period pollution constraints. We discuss the economic and environmental implications of the game’s solutions as we vary the planner’s preferences

Numerical solutions to coupled-constraint (or generalised Nash) equilibrium problems

Computational economics, Compliance problems, Coupled constraint games, Eeneralised Nash Equilibrium, Nikado–Isoda function, Relaxation algorithm (NIRA), Quasi-variational inequalities, C63, C72, C88, D78, E62, 91A, 91B, 90C,

Penalty methods for the solution of generalized Nash equilibrium problems and hemivariational inequalities with VI constraints

In this thesis we propose penalty methods for the solution of Generalized Nash Equilibrium Problems (GNEPs) and we consider centralized and distributed algorithms for the solution of Hemivariational Inequalities (HVIs) where the feasible set is given by the intersection of a closed convex set with the solution set of a lower-level monotone Variational Inequality (VI)

Penalty methods for the solution of generalized Nash equilibrium problems and hemivariational inequalities with VI constraints

In this thesis we propose penalty methods for the solution of Generalized Nash Equilibrium Problems (GNEPs) and we consider centralized and distributed algorithms for the solution of Hemivariational Inequalities (HVIs) where the feasible set is given by the intersection of a closed convex set with the solution set of a lower-level monotone Variational Inequality (VI)