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Ordinary and Prophet Planning under Uncertainty in Bernoulli Congestion Games

Roberto Cominetti, Marco Scarsini, Marc Schröder, Nicolás Stier-Moses

TL;DR

This work analyzes atomic congestion games under demand uncertainty where each player participates independently with probability $p_i$, introducing two social planners: an ordinary planner who commits to a routing before realizations and a prophet planner who adapts to the actual active set. The authors define ordinary and prophet Price of Anarchy (OPoA and PPoA) and derive tight, parameterized bounds as functions of the maximal participation probability $q=\max_i p_i$, showing worst-case instances occur with homogeneous participation. They develop a tailored $(\lambda,\mu, q)$-smoothness framework to bound the PPoA and provide explicit analytic expressions for affine cost functions, revealing three regimes in $q$ with valuable economic insights on information value: as participation grows, outcomes approach deterministic bounds; as participation shrinks, equilibria align more with social optimum. Extended analyses cover mixed and correlated equilibria, and generalizes to polynomial costs, including asymptotic and Bell-number-based bounds for the prophet case. Overall, the paper quantifies how the value of foresight and participation uncertainty shape efficiency in routing and congestion, with practical implications for real-time routing platforms and pricing schemes.

Abstract

We consider an atomic congestion game in which each player $i$ participates in the game with an exogenous and known probability $p_{i}\in(0,1]$, independently of everybody else, or stays out and incurs no cost. We compute the parameterized Price of Anarchy (PoA) to characterize the impact of demand uncertainty on the efficiency of selfish behavior, considering two different notions of a social planner. A prophet planner knows the realization of the random participation in the game; the ordinary planner does not. As a consequence, a prophet planner can compute an adaptive social optimum that selects different solutions depending on the players that turn out to be active, whereas an ordinary planner faces the same uncertainty as the players and can only minimize the expected social cost according to the player participation distribution. For both type of planners we obtain tight bounds for the PoA, by solving suitable optimization problems parameterized by the maximum participation probability $q=\max_{i} p_{i}$. In the case of affine costs, we find an analytic expression for the corresponding bounds.

Ordinary and Prophet Planning under Uncertainty in Bernoulli Congestion Games

TL;DR

This work analyzes atomic congestion games under demand uncertainty where each player participates independently with probability , introducing two social planners: an ordinary planner who commits to a routing before realizations and a prophet planner who adapts to the actual active set. The authors define ordinary and prophet Price of Anarchy (OPoA and PPoA) and derive tight, parameterized bounds as functions of the maximal participation probability , showing worst-case instances occur with homogeneous participation. They develop a tailored -smoothness framework to bound the PPoA and provide explicit analytic expressions for affine cost functions, revealing three regimes in with valuable economic insights on information value: as participation grows, outcomes approach deterministic bounds; as participation shrinks, equilibria align more with social optimum. Extended analyses cover mixed and correlated equilibria, and generalizes to polynomial costs, including asymptotic and Bell-number-based bounds for the prophet case. Overall, the paper quantifies how the value of foresight and participation uncertainty shape efficiency in routing and congestion, with practical implications for real-time routing platforms and pricing schemes.

Abstract

We consider an atomic congestion game in which each player participates in the game with an exogenous and known probability , independently of everybody else, or stays out and incurs no cost. We compute the parameterized Price of Anarchy (PoA) to characterize the impact of demand uncertainty on the efficiency of selfish behavior, considering two different notions of a social planner. A prophet planner knows the realization of the random participation in the game; the ordinary planner does not. As a consequence, a prophet planner can compute an adaptive social optimum that selects different solutions depending on the players that turn out to be active, whereas an ordinary planner faces the same uncertainty as the players and can only minimize the expected social cost according to the player participation distribution. For both type of planners we obtain tight bounds for the PoA, by solving suitable optimization problems parameterized by the maximum participation probability . In the case of affine costs, we find an analytic expression for the corresponding bounds.

Paper Structure

This paper contains 25 sections, 27 theorems, 182 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Related Work
  4. OPoA for nondecreasing costs and heterogeneous players.
  5. Prophet PoA for nondecreasing costs and heterogeneous players.
  6. Tight Bounds for Affine Cost Functions
  7. Other prior related work
  8. Organization of the Paper
  9. Bernoulli Congestion Games and the Price of Anarchy
  10. Atomic Congestion Games
  11. Bernoulli Congestion Games
  12. Social Optimum and Price of Anarchy with General Costs
  13. Tight Bounds for the OPoA
  14. Tight Bounds for the PPoA
  15. Extension to Mixed and Correlated Equilibria
  16. ...and 10 more sections

Key Result

Theorem 1

For each family $\mathcal{C}$ of nonnegative and nondecreasing cost functions we have $\mathop{\mathrm{\mathsf{PoA}}}\nolimits(\mathcal{C})=\gamma(\mathcal{C})$. Moreover, if $\mathcal{C}$ contains the zero cost function $c_{0}(\,\cdot\,)$, then the supremum in $\mathop{\mathrm{\mathsf{PoA}}}\noli

Figures (10)

  • Figure 1: Tight bounds for the OPoA and PPoA (thick lines in the figure) for Bernoulli congestion games with affine costs as a function of $q=\max_{i}p_{i}$. The dots corresponding to $q=1/n$ for $n\in\mathbb{N}_{+}$ in the OPoA curve, as well as the dashed segment corresponding to $q\ge 1/2$, mathematically coincide with previously known bounds for different models (see Section \ref{['suse:comparison-work']}). Our results fill in the gaps for that curve and make its regimes explicit. The vertical dashed lines depict the breakpoints where the OPoA changes regimes. The dot at $1/3$ does not coincide with the kink of the curve at $\bar{q}_{1}\sim 0.3774$ but they coincide at $\bar{q}_{0}=1/4$.
  • Figure 2: A Pigou network. The edges are annotated with their cost functions $c_{e}(x_{e})$.
  • Figure 3: The upper envelope gives the tight bound on $\mathop{\mathrm{\mathsf{OPoA}}}\nolimits(\mathcal{C}_{\mathsf{aff}},q)$ as a function of $q$.
  • Figure 4: The lower envelope $\Xi(q)$ gives a tight bound for $\mathop{\mathrm{\mathsf{PPoA}}}\nolimits(\mathcal{C}_{\mathsf{aff}},q)$.
  • Figure EC.1: The set $\mathcal{E}=\mathcal{E}_{1}\cup\mathcal{E}_{2}$ with two cycles of $n=8$ resources each. The strategies $s_{1}$ and $s_{1}'$ for player 1 are shown in blue and red, with $k=2$. For subsequent players these strategies are turned clockwise.
  • ...and 5 more figures

Theorems & Definitions (38)

  • Theorem 1: Rou:JACM2015
  • Definition 1
  • Proposition 1
  • Lemma 1
  • Corollary 1
  • Example 1
  • Definition 2
  • Theorem 2
  • Corollary 2
  • Example 2
  • ...and 28 more