Lasse Peters

Contingency Games

Contingency Games for Multi-Agent Interaction

    title     = {Contingency Games for Multi-Agent Interaction},
    author    = {Peters, Lasse and Bajcsy, Andrea and Chiu, Chih-Yuan, and Fridovich-Keil, David and Laine, Forrest, and Ferranti, Laura and Alonso-Mora, Javier},
    booktitle = {arXiv e-prints},
    year      = {2023},
    url       = {}


Contingency planning, wherein an agent generates a set of possible plans conditioned on the outcome of an uncertain event, is an increasingly popular way for robots to act under uncertainty. In this work we take a game-theoretic perspective on contingency planning, tailored to multi-agent scenarios in which a robot’s actions impact the decisions of other agents and vice versa. The resulting contingency game allows the robot to efficiently interact with other agents by generating strategic motion plans conditioned on multiple possible intents for other actors in the scene. Contingency games are parameterized via a scalar variable which represents a future time when intent uncertainty will be resolved. By estimating this parameter online, we construct a game-theoretic motion planner that adapts to changing beliefs while anticipating future certainty. We show that existing variants of game-theoretic planning under uncertainty are readily obtained as special cases of contingency games, and we offer an efficient method for solving N-player contingency games with nonlinear dynamics and non-convex costs and constraints. Through a series of simulated autonomous driving scenarios, we demonstrate that plans generated via contingency games provide quantitative performance gains over game-theoretic motion plans that do not account for future uncertainty reduction.

Interactive Demo

Contingency games are best understood via an interactive example. Below, we provide an interactive version of the example given in the video above: an autonomous robot encountering a jaywalking pedestrian.

By varying the sliders controlling the main parameters of the contingency game, we can build intuition for how the robot’s behavior changes as a function of the uncertainty in the pedestrian’s intent. In particular, observe that:

  • The robot balances the cost of each hypothesis according to the likelihood under the current belief.
  • The robot exploits the anticipated certainty to balance safety and efficiency.
  • At extreme values of the branching time, we recover established game-theoretic planning paradigms as a special case:
    • When the branching time is zero, the robot assumes full certainty and solves the certainty-equivalent problem.
    • When the branching time is equal to the planning horizon, the robot develops a single plan that minimizes expected cost under fixed uncertainty.


This project has produced a range of software packages which we will link here as they become available:

  • ParametricMCPs.jl: The mixed-complementary problem backend for our contingency game solver.

Released upon publication of the manuscript:

Last Updated: 2023-05-13