502 research outputs found
The Network Improvement Problem for Equilibrium Routing
In routing games, agents pick their routes through a network to minimize
their own delay. A primary concern for the network designer in routing games is
the average agent delay at equilibrium. A number of methods to control this
average delay have received substantial attention, including network tolls,
Stackelberg routing, and edge removal.
A related approach with arguably greater practical relevance is that of
making investments in improvements to the edges of the network, so that, for a
given investment budget, the average delay at equilibrium in the improved
network is minimized. This problem has received considerable attention in the
literature on transportation research and a number of different algorithms have
been studied. To our knowledge, none of this work gives guarantees on the
output quality of any polynomial-time algorithm. We study a model for this
problem introduced in transportation research literature, and present both
hardness results and algorithms that obtain nearly optimal performance
guarantees.
- We first show that a simple algorithm obtains good approximation guarantees
for the problem. Despite its simplicity, we show that for affine delays the
approximation ratio of 4/3 obtained by the algorithm cannot be improved.
- To obtain better results, we then consider restricted topologies. For
graphs consisting of parallel paths with affine delay functions we give an
optimal algorithm. However, for graphs that consist of a series of parallel
links, we show the problem is weakly NP-hard.
- Finally, we consider the problem in series-parallel graphs, and give an
FPTAS for this case.
Our work thus formalizes the intuition held by transportation researchers
that the network improvement problem is hard, and presents topology-dependent
algorithms that have provably tight approximation guarantees.Comment: 27 pages (including abstract), 3 figure
Backward Stackelberg Differential Game with Constraints: a Mixed Terminal-Perturbation and Linear-Quadratic Approach
We discuss an open-loop backward Stackelberg differential game involving
single leader and single follower. Unlike most Stackelberg game literature, the
state to be controlled is characterized by a backward stochastic differential
equation (BSDE) for which the terminal- instead initial-condition is specified
as a priori; the decisions of leader consist of a static terminal-perturbation
and a dynamic linear-quadratic control. In addition, the terminal control is
subject to (convex-closed) pointwise and (affine) expectation constraints. Both
constraints are arising from real applications such as mathematical finance.
For information pattern: the leader announces both terminal and open-loop
dynamic decisions at the initial time while takes account the best response of
follower. Then, two interrelated optimization problems are sequentially solved
by the follower (a backward linear-quadratic (BLQ) problem) and the leader (a
mixed terminal-perturbation and backward-forward LQ (BFLQ) problem). Our
open-loop Stackelberg equilibrium is represented by some coupled
backward-forward stochastic differential equations (BFSDEs) with mixed
initial-terminal conditions. Our BFSDEs also involve nonlinear projection
operator (due to pointwise constraint) combining with a Karush-Kuhn-Tucker
(KKT) system (due to expectation constraint) via Lagrange multiplier. The
global solvability of such BFSDEs is also discussed in some nontrivial cases.
Our results are applied to one financial example.Comment: 38 page
Bertrand-Edgeworth competition with sequential capacity choice
Duopoly;Capacity;microeconomics
Feedback Stackelberg-Nash equilibria in mixed leadership games with an application to cooperative advertising
In this paper we characterize the feedback equilibrium of a general infinite-horizon Stackelberg-Nash differential game where the roles of the players are mixed. By mixed we mean that one player is a leader on some decisions and a follower on other decisions. We prove a verification theorem that reduces the task of finding equilibrium strategies in functional spaces to two simple steps: First solving two static Nash games at the Hamiltonian level in a nested version and then solving the associated system of Hamilton-Jacobi-Bellman equations. As an application, we study a novel manufacturer-retailer cooperative advertising game where, in addition to the traditional setup into which the manufacturer subsidizes the retailer's advertising effort, we also allow the reverse support from the retailer to the manufacturer. We find an equilibrium that can be expressed by a solution of a set of quartic algebraic equations. We then conduct an extensive numerical study to assess the impact of model parameters on the equilibrium
Game Theoretic Model Predictive Control for Autonomous Driving
This study presents two closely-related solutions to autonomous vehicle control problems in highway driving scenario using game theory and model predictive control. We first develop a game theoretic four-stage model predictive controller (GT4SMPC). The controller is responsible for both longitudinal and lateral movements of Subject Vehicle (SV) . It includes a Stackelberg game as a high level controller and a model predictive controller (MPC) as a low level one. Specifically, GT4SMPC constantly establishes and solves games corresponding to multiple gaps in front of multiple-candidate vehicles (GCV) when SV is interacting with them by signaling a lane change intention through turning light or by a small lateral movement. SVâs payoff is the negative of the MPCâs cost function , which ensures strong connection between the game and that the solution of the game is more likely to be achieved by a hybrid MPC (HMPC). GCVâs payoff is a linear combination of the speed payoff, headway payoff and acceleration payoff. . We use decreasing acceleration model to generate our prediction of TVâs future motion, which is utilized in both defining TVâs payoffs over the prediction horizon in the game and as the reference of the MPC. Solving the games gives the optimal gap and the target vehicle (TV). In the low level , the lane change process are divided into four stages: traveling in the current lane, leaving current lane, crossing lane marking, traveling in the target lane. The division identifies the time that SV should initiate actual lateral movement for the lateral controller and specifies the constraints HMPC should deal at each step of the MPC prediction horizon. Then the four-stage HMPC controls SVâs actual longitudinal motion and execute the lane change at the right moment. Simulations showed the GT4SMPC is able to intelligently drive SV into the selected gap and accomplish both discretionary land change (DLC) and mandatory lane change (MLC) in a dynamic situation. Human-in-the-loop driving simulation indicated that GT4SMPC can decently control the SV to complete lane changes with the presence of human drivers. Second, we propose a differential game theoretic model predictive controller (DGTMPC) to address the drawbacks of GT4SMPC. In GT4SMPC, the games are defined as table game, which indicates each players only have limited amount of choices for a specific game and such choice remain fixed during the prediction horizon. In addition, we assume a known model for traffic vehicles but in reality driversâ preference is partly unknown. In order to allow the TV to make multiple decisions within the prediction horizon and to measure TVâs driving style on-line, we propose a differential game theoretic model predictive controller (DGTMPC). The high level of the hierarchical DGTMPC is the two-player differential lane-change Stackelberg game. We assume each player uses a MPC to control its motion and the optimal solution of leadersâ MPC depends on the solution of the follower. Therefore, we convert this differential game problem into a bi-level optimization problem and solves the problem with the branch and bound algorithm. Besides the game, we propose an inverse model predictive control algorithm (IMPC) to estimate the MPC weights of other drivers on-line based on surrounding vehicleâs real-time behavior, assuming they are controlled by MPC as well. The estimation results contribute to a more appropriate solution to the game against driver of specific type. The solution of the algorithm indicates the future motion of the TV, which can be used as the reference for the low level controller. The low level HMPC controls both the longitudinal motion of SV and his real-time lane decision. Simulations showed that the DGTMPC can well identify the weights traffic vehiclesâ MPC cost function and behave intelligently during the interaction. Comparison with level-k controller indicates DGTMPCâs Superior performance
- âŠ