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To Travel Quickly or to Park Conveniently: Coupled Resource Allocations with Multi-Karma Economies

Ezzat Elokda, Andrea Censi, Saverio Bolognani, Florian Dörfler, Emilio Frazzoli

TL;DR

This paper addresses the challenge of coupling multiple public resources in cyber-physical-human systems using non-monetary karma economies. It extends the Dynamic Population Game framework to handle resource-specific karma accounts, introduces redistribution schemes and exchange-rate instruments, and proves the existence of a Stationary Nash Equilibrium ($SNE$). A real-world-inspired numerical study of highway express lanes and priority parking reveals that uniform redistribution to all with unit exchange often achieves near-maximal Nash welfare and Pareto improvements across designs. Overall, the results offer practical guidance for fair, scalable, and non-monetary coupling of multiple resources in interconnected CPHS.

Abstract

The large-scale allocation of public resources (e.g., transportation, energy) is among the core challenges of future Cyber-Physical-Human Systems (CPHS). In order to guarantee that these systems are efficient and fair, recent works have investigated non-monetary resource allocation schemes, including schemes that employ karma. Karma is a non-tradable token that flows from users gaining resources to users yielding resources. Thus far karma-based solutions considered the allocation of a single public resource, however, modern CPHS are complex as they involve the allocation of multiple coupled resources. For example, a user might want to trade-off fast travel on highways for convenient parking in the city center, and different users could have heterogeneous preferences for such coupled resources. In this paper, we explore how to optimally combine multiple karma economies for coupled resource allocations, using two mechanism-design instruments: (non-uniform) karma redistribution; and (non-unit) exchange rates. We first extend the existing Dynamic Population Game (DPG) model that predicts the Stationary Nash Equilibrium (SNE) of the multi-karma economies. Then, in a numerical case study, we demonstrate that the design of redistribution significantly affects the coupled resource allocations, while non-unit exchange rates play a minor role. To assess the allocation outcomes under user heterogeneity, we adopt Nash welfare as our social welfare function, since it makes no interpersonal comparisons and it is axiomatically rooted in social choice theory. Our findings suggest that the simplest mechanism design, that is, uniform redistribution with unit exchange rates, also attains maximum social welfare.

To Travel Quickly or to Park Conveniently: Coupled Resource Allocations with Multi-Karma Economies

TL;DR

This paper addresses the challenge of coupling multiple public resources in cyber-physical-human systems using non-monetary karma economies. It extends the Dynamic Population Game framework to handle resource-specific karma accounts, introduces redistribution schemes and exchange-rate instruments, and proves the existence of a Stationary Nash Equilibrium (). A real-world-inspired numerical study of highway express lanes and priority parking reveals that uniform redistribution to all with unit exchange often achieves near-maximal Nash welfare and Pareto improvements across designs. Overall, the results offer practical guidance for fair, scalable, and non-monetary coupling of multiple resources in interconnected CPHS.

Abstract

The large-scale allocation of public resources (e.g., transportation, energy) is among the core challenges of future Cyber-Physical-Human Systems (CPHS). In order to guarantee that these systems are efficient and fair, recent works have investigated non-monetary resource allocation schemes, including schemes that employ karma. Karma is a non-tradable token that flows from users gaining resources to users yielding resources. Thus far karma-based solutions considered the allocation of a single public resource, however, modern CPHS are complex as they involve the allocation of multiple coupled resources. For example, a user might want to trade-off fast travel on highways for convenient parking in the city center, and different users could have heterogeneous preferences for such coupled resources. In this paper, we explore how to optimally combine multiple karma economies for coupled resource allocations, using two mechanism-design instruments: (non-uniform) karma redistribution; and (non-unit) exchange rates. We first extend the existing Dynamic Population Game (DPG) model that predicts the Stationary Nash Equilibrium (SNE) of the multi-karma economies. Then, in a numerical case study, we demonstrate that the design of redistribution significantly affects the coupled resource allocations, while non-unit exchange rates play a minor role. To assess the allocation outcomes under user heterogeneity, we adopt Nash welfare as our social welfare function, since it makes no interpersonal comparisons and it is axiomatically rooted in social choice theory. Our findings suggest that the simplest mechanism design, that is, uniform redistribution with unit exchange rates, also attains maximum social welfare.

Paper Structure

This paper contains 17 sections, 1 theorem, 14 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation
  3. Multi-Karma Economy Model
  4. Karma-Based Multi-Resource Allocations
  5. Karma Design Instruments
  6. Karma Redistribution.
  7. Exchange Rates.
  8. Individual Strategic Problem
  9. Immediate Payoff Function $\sigma_\tau[x,b](d,\pi)$.
  10. State transition function $p_\tau[x^+ \mid x, b](d,\pi)$.
  11. Existence of SNE
  12. Numerical Analysis
  13. Individual and Social Welfare Measures
  14. Results and Discussion
  15. Redistribution.
  16. ...and 2 more sections

Key Result

Theorem 1

A $(d^*,\pi^*)$ is guaranteed to exist in the multi-karma economy.

Figures (3)

  • Figure 1: Example of coupled public resources.
  • Figure 2: Example of resource-urgency Markov chain for two user types. Type ${\texttt{S}}$ needs both resources every day, with urgency drawn independently at the beginning of the day (${\mathbb P}[u=1]=0.75$, ${\mathbb P}[u=9]=0.25$). Type ${\texttt{C}}$ is identical to ${\texttt{S}}$, except that it needs resource ${\texttt{H}}$ only on half of the days.
  • Figure 3: Resource utilization under different multi-karma designs.

Theorems & Definitions (2)

  • Definition 1
  • Theorem 1