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Understanding Police Force Resource Allocation using Adversarial Optimal Transport with Incomplete Information

Yinan Hu, Juntao Chen, Quanyan Zhu

TL;DR

This paper constructs the concept of Bayesian equilibrium and design a distributed algorithm that achieve those equilibria, making the model applicable to large-scale networks.

Abstract

Adversarial optimal transport has been proven useful as a mathematical formulation to model resource allocation problems to maximize the efficiency of transportation with an adversary, who modifies the data. It is often the case, however, that only the adversary knows which nodes are malicious and which are not. In this paper we formulate the problem of seeking adversarial optimal transport into Bayesian games. We construct the concept of Bayesian equilibrium and design a distributed algorithm that achieve those equilibria, making our model applicable to large-scale networks. Keywords: game theory, crime control, Markov games

Understanding Police Force Resource Allocation using Adversarial Optimal Transport with Incomplete Information

TL;DR

This paper constructs the concept of Bayesian equilibrium and design a distributed algorithm that achieve those equilibria, making the model applicable to large-scale networks.

Abstract

Adversarial optimal transport has been proven useful as a mathematical formulation to model resource allocation problems to maximize the efficiency of transportation with an adversary, who modifies the data. It is often the case, however, that only the adversary knows which nodes are malicious and which are not. In this paper we formulate the problem of seeking adversarial optimal transport into Bayesian games. We construct the concept of Bayesian equilibrium and design a distributed algorithm that achieve those equilibria, making our model applicable to large-scale networks. Keywords: game theory, crime control, Markov games
Paper Structure (20 sections, 1 theorem, 30 equations, 8 figures)

This paper contains 20 sections, 1 theorem, 30 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Formulation of source allocation without adversaries
  4. Discrete regularized optimal transport over networks
  5. The dispatcher's utility function
  6. Adversarial regularized optimal transport
  7. The adversary's information structure
  8. Beliefs/types
  9. Action spaces
  10. Stage Utility functions
  11. Analysis of Static Game
  12. Existence of a solution
  13. Bayesian equilibrium
  14. Dynamic adversarial optimal transport
  15. Formulation of dynamic Bayesian game
  16. ...and 5 more sections

Key Result

Proposition 1

Let $\mathcal{G}$ be the Bayesian games mentioned in definition def:Bayesian_games. Let $(x^*, \{\xi^*(\theta)\}_{\theta \in \Theta})$ be a strategy profile meeting the requirements of Bayesian equilibrium in definition def:Bayesian_equilibrium. Then such an equilibrium exists in the game $\mathcal{

Figures (8)

  • Figure 1: Possible thresholding function $\phi(\xi^t;\xi^{t-1})$. Where $\tau = \xi^{t-1}$ as the criminal's action in the last stage plays the role of threshold.
  • Figure 2: The NYC crime heat map expressed in regions in different time periods. Left: May 2022, Right: June 2022.
  • Figure 3: The scheme of resource allocation problem with $m = 3$ police stations (source nodes) and $n = 4$ neighborhoods (target nodes).
  • Figure 4: The optimal transport plan of police resource $x^*$ without the fairness term. The dispatcher's utility function is chosen to be \ref{['eq:planner_utility']} with $\lambda=0$. We choose the perception matrix $M$ to be in \ref{['eq:perception_matrix']}.
  • Figure 5: The optimal plan of police resource allocation $x$ with fairness term taken into account. The perception coefficient $M$ is set in \ref{['eq:perception_matrix']}. The dispatcher's utility is set in \ref{['eq:planner_utility']} with the parameter $\lambda = 3$.
  • ...and 3 more figures

Theorems & Definitions (5)

  • Definition 1: Bayesian game of adversarial transport
  • Definition 2: Bayesian equilibrium for adversarial OT
  • Proposition 1
  • Definition 3: History of the criminal's actions and the dispatcher's plans
  • Definition 4: Perfect Bayesian Nash equilibrium (PBNE)