We study a class of games in which a finite number of agents each controls a quantity of flow to be routed through a network, and are able to split their own flow between multiple paths through the network. Recent work on this model has contrasted the social cost of Nash equilibria with the best possible social cost. Here we show that additional costs are incurred in situations where a selfish ``leader'' agent allocates his flow, and then commits to that choice so that other agents are compelled to minimise their own cost based on the first agent's choice. We find that even in simple networks, the leader can often improve his own cost at the expense of increased social cost. Focusing on the 2-player case, we give upper and lower bounds on the worst-case additional cost incurred.
Deep Dive into The Price of Selfish Stackelberg Leadership in a Network Game.
We study a class of games in which a finite number of agents each controls a quantity of flow to be routed through a network, and are able to split their own flow between multiple paths through the network. Recent work on this model has contrasted the social cost of Nash equilibria with the best possible social cost. Here we show that additional costs are incurred in situations where a selfish ``leader’’ agent allocates his flow, and then commits to that choice so that other agents are compelled to minimise their own cost based on the first agent’s choice. We find that even in simple networks, the leader can often improve his own cost at the expense of increased social cost. Focusing on the 2-player case, we give upper and lower bounds on the worst-case additional cost incurred.
Imagine that two firms wish to route traffic from a source to a destination through a shared network. Any link suffers from a delay (also called latency) that increases with the amount of traffic that it attracts, and both firms want to minimise their own total delay. It is known (e.g. [10,4]) that the resulting social cost (sum of individuals' delays) is suboptimal, even for simple networks. If for example some but not all links are privately-owned, there is a tendency for both firms to over-use the shared link, in order to relieve pressure on the privately-owned links (Catoni and Pallottino [2], Cominetti et al. [4]).
Viewing this as a non-cooperative two-player game, suppose now one of the firms (player 1) is “forceful”, and the other one (player 2) is “pliant”. Player 1 may find that it pays to over-use a shared link even more than before, provided that player 2 responds by moving some of his own traffic away from the shared link and onto player 2’s private links. As a consequence, it turns out that player 1’s total delay may fall, but player 2’s total delay increases by a greater amount, thus increasing the social cost.
One way to model forceful and pliant players, is to let player 1 have Stackelberg leadership: player 1 selects his strategy, namely the split of his own flow between the links available to him. Then, player 2 chooses his strategy based on player 1’s choice, under the assumption that player 1 will not subsequently change his decision.
Thus there may be a social cost of Stackelberg leadership over and above the cost of selfish decentralised decision-making. In this paper we focus on a simple and well-known setting in which the players have access to a set of shared “parallel links”. Note that this is more restrictive than the scenario described above in that there are no private links. We give a simple example of how selfish stackelberg leadership (which we usually abbreviate to SSL) may nevertheless have a positive cost in this setting, and motivated by that example, we establish an upper bound on the price of SSL.
Our focus is on the 2-player atomic-splittable case, with parallel links having linear latency functions. In Subsection 2.1 we show that if there exists a player having Stackelberg leadership, then the social cost may be higher than in the Nash-Cournot setting. Furthermore, the remaining flow may even be disadvantaged as a direct result of being controlled by a single player, rather than a Wardrop flow. This situation arises in a very simple setting in which two players both have access to just two links having affine linear latency functions. This furnishes a lower bound on the price of selfish Stackelberg leadership, of a multiplicative factor 1.057. Subsection 2.5 gives our main result, a contrasting upper bound. We analyse games with two players each needing to route splittable flow through a shared network of parallel links having linear latency functions. If the latency functions are homogenous, there is no cost of SSL. However, for the case of affine latency functions, we show that the worst-case price of SSL is a multiplicative constant (thus, independent of the number of links), at most 1.322.
A large body of recent work (initiated mainly by Roughgarden and Tardos [20,19]) has studied from a game-theory perspective, how selfishness can degrade the overall performance of a system that has multiple (selfish) users. Much of this work has focused on situations where users have access to shared resources, and the cost of using a resource increases as the resource attracts more usage. Our focus here is on the “parallel links” network topology, also referred to as scheduling jobs to a set of load-dependent machines, which is one of the most commonly studied models (e.g. [6,11,12,13,14,18]). Papers such as [1,6,12] have studied the price of anarchy for these games in the “unsplittable flow” setting, where each user may only use a single resource. In contrast we study the “splittable flow” setting of [14]. This version (finitely many players, splittable flow) was shown in [14,15] to possess unique pure Nash equilibria (see Definition 2). Hayrapetyan et al. [10] study the cost of selfish behaviour in this model, and compare it with the cost of selfish behaviour in the Wardrop model (i.e. infinitely many infinitesimal users).
Stackelberg leadership refers to a game-theoretic situation where one player (the “leader”) selects his action first, and commits to it. The other player(s) then choose their own action based on the choice made by the leader. Recent work on Stackelberg scheduling in the context of network flow (e.g. [5,18,21]), has studied it as a tool to mitigate the performance degradation due to selfish users. The flow that is controlled by the leader is routed so as to minimise social cost in the presence of followers who minimise their own costs. In contrast, here we consider what happens when the leading flow is controlled by another selfish agent. We show here
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