Altruism Improves Congestion in Series-Parallel Nonatomic Congestion Games

TL;DR

This work investigates when altruistic routing—where agents consider the overall congestion—guarantees improved total latency in nonatomic congestion games. Focusing on series-parallel networks with two agent types (altruistic and selfish), it introduces Braess-resistance and symmetry as key network properties and proves that, under these conditions, increasing the altruist fraction $r^{\rm{a}}$ monotonically improves total congestion via a subgradient argument on the latency functional $\mathcal{L}(x)$. Conversely, it shows that removing either Braess-resistance or symmetry can create perversity, where altruism degrades welfare. The results provide a complete characterization of when altruism is guaranteed to help in this setting and offer guidance for network design and policy when deploying altruistic routing policies.

Abstract

Self-interested routing polices from individual users in a system can collectively lead to poor aggregate congestion in routing networks. The introduction of altruistic agents, whose goal is to benefit other agents in the system, can seemingly improve aggregate congestion. However, it is known in that in some network routing problems, altruistic agents can actually worsen congestion compared to that which would arise in the presence of a homogeneously selfish population. This paper provides a thorough investigation into the necessary conditions for altruists to be guaranteed to improve total congestion. In particular, we study the class of series-parallel non-atomic congestion games, where one sub-population is altruistic and the other is selfish. We find that a game is guaranteed to have improved congestion in the presence of altruistic agents (even if only a small part of the total population) compared to the homogeneously selfish version of the game, provided the network is symmetric, where all agents are given access to all paths in the network, and the series-parallel network for the game does not have sub-networks which emulate Braess's paradox -- a phenomenon we refer to as a Braess-resistant network. Our results appear to be the most complete characterization of when behavior that is designed to improve total congestion (which we refer to as altruism) is actually guaranteed to do so.

Figures (2)

• Figure 1: Figure \ref{['subfig:NBR']} is an example of the perversity that arises if a network is not Braess-resistant. The path set is $\mathcal{P} = \{(e_1, e_3), (e_1, e_4), (e_2, e_4), e_5\}$, so traffic travelling along $e_2$ cannot use $e_3$. In this example, two units of traffic are routed; when all traffic is selfish, half of the population routes along $(e_1, e_3)$, and the other half routes along $(e_2, e_4)$, for total latency $\mathcal{L}(\bar{x}) = 4$. Now, when half of traffic is altruistic, and the other half is selfish, the selfish agents will use $(e_1, e_4)$, and altruistic traffic will prefer $e_5$, resulting in total latency $\mathcal{L}(x) = 5$. Figure \ref{['subfig:NI']} provides an example of the perversity that can arise if the path set is not symmetric. Again, two units of traffic traverse the network, and altruists have access to the top two edges while selfish agents have access to all three. If all traffic is selfish, the bottom two edges are used resulting in total latency $\mathcal{L}(\bar{x}) = 2$. Now, if half the population becomes altruistic, altruists now prefer the top edge, and the total latency becomes $\mathcal{L}(x) = 2 + d \geq 2 = \mathcal{L}(\bar{x})$.
• Figure 2: Geometric interpretation for the proof of Lemma \ref{['lem:Path_Ordering']}. Figure \ref{['subfig:Case_1']} represents the network that arises in Case 1. Here, there exists a path, used by selfish agents in $x$, $\rho$, that has every edge strictly decreasing from $x$ to $\tilde{x}$, and so the cost of the path goes down. However, we know from Corollary \ref{['cor:p_a_AND_p_s']} that there is a min-cost path, $p_{\rm{s}}$, in $\tilde{x}$, such that each edge in the path is non-decreasing, so the cost of the path is non-decreasing. This implies that the cost of $p_{\rm{s}}$ is strictly greater that the cost of $\rho$, contradicting that $p_{\rm{s}}$ is a min-cost path. Figure \ref{['subfig:Case_2']} presents a similar argument. Here, selfish agents using $p_{\rm{s}} = \sigma_1 + \sigma_3$ are better off using $\sigma + \sigma_3$ or $\sigma + \sigma_2$, both of which are possible since the path set is Braess-resistant. Finally, figure \ref{['subfig:Case_3']} shows the [sub]-network that arises if $p_{\rm{s}}$ shares no common vertices with $\rho$. Because the flow is strictly decreasing on $\sigma$, and non-decreasing on $\sigma_2$, there must exist a sub-path $\sigma_1$ that shares incident vertices with $\sigma$. It can be deduced that altruists are using $\sigma_1 + \sigma_2$, and that their cost in $\tilde{x}$ has either strictly gone up from $x$, or that the path $p^\prime = \sigma_1 + \sigma_2$ and $p_{\rm{a}}$ are constant. Either case produces a contradiction.

Theorems & Definitions (6)

• Definition 1
• Definition 2
• Theorem III.1
• Lemma IV.1
• Corollary IV.2
• Lemma IV.3