For the case of non- players are completely reliable. This is particularly true in negligible failure probabilities, we provide a full character- ization of the maximal impact of failures on the social cost computerized and online environments, where agents may under worst-case equilibrium outcomes. The above example shows that the equilibrium outcomes can change considerably when agents may fail, and that lack of Introduction reliability may lead to a more socially desirable outcome.
Congestion games Rosenthal are a well-studied These observations highlight the importance of understand- model of strategic sharing of resource, and have been used to ing how failures affect the predicted outcomes of games. In the The characterization and computation of equilibrium out- absence of some agents, we compute the cost induced by the comes in congestion games have received much attention surviving agents, where each agent now aims to minimize see e.
Fabrikant, Papadimitriou, and Talwar ; her expected cost over all the realizations of the game. Ieong et al. In particular, Related work researchers focused on the Price of Anarchy, which is the Uncertainty in congestion games Though to the best of gap between the optimal cost and the cost under equilibrium our knowledge no previous work studies the effects of agent outcome Roughgarden and Tardos ; Christodoulou and failures on equilibria in congestion games, several works do Koutsoupias Nevertheless, an implicit assumption examine similar themes.
Penn et al. In their model uncertainty always has a hazardous effect, as In practice, however, agents may fail to follow their chosen it encourages the agents to overload the system. While our strategies, thereby utterly changing the costs of the game. Taxis al. The A different model of uncertainty was introduced by Bal- Copyright c , Association for the Advancement of Artificial can et al.
All rights reserved. Balcan et al. An allowed strategy is a path there is no clear interpretation for a failure of an agent. However, there are particular families of games where fail- ures do have a straightforward meaning. Messner and Pol- explicitly stated otherwise that all costs are non-zero. All congestion games are potential games, games with transferable utilities. They prove that as in our and thus admit a pure Nash equilibrium Rosenthal In case, failures in such games tend to have a beneficial ef- this work we restrict our attention to pure Nash equilibria.
This is since failures can expand the core of the original Types of congestion games We focus on games where game, thereby increasing its stability against collusion. Congestion Our contribution games where all Si are equal are called symmetric.
Our primary conceptual contribution is the introduction of In a resource selection game RSG , each agent i selects agent failures to congestion games. An example of an ior, where the survival probability goes to 1, and the case of SRTG without the costs is in Figure 1.
Interestingly, Price of Anarchy The Price of Anarchy PoA of a game we show that this no longer holds for Resource Selection G compares the social cost of the worst Nash equilibrium games with increasing costs. All omitted proofs can be found in the appendix. Agent failures Given a game G, we extend it with survival probabilities to every agent. Each the expected cost that each surviving agent experiences.
TheP P of a profile A is: social cost or total cost where Nx is the set of agents selecting resource x. For simplicity, 1 This technical assumption is required to avoid issues of divi- we assume all costs are non-negative integers, and unless sion by zero when computing a ratio between costs. Note that in the limit case strict. One General properties may wonder if this new game still has a pure Nash equilib- We first prove that any reliability extension of a congestion rium, since this is not guaranteed in other extensions such as game has a pure NE.
We emphasize that no restriction on the weighted congestion games Milchtaich Our model cost functions is required for this result. Somewhat surpris- Theorem 1. Let G be a congestion game, and p a probabil- ingly, we show that any reliability extension of G does admit ity vector.
Then Gp has a pure Nash equilibrium. Due to space constraints, we omit the full proof. However, it relies on the definition of the following function, which is Games with i. For this purpose it is convenient to focus istence of a weighted potential function, it is guaranteed that on i. Another important issue is whether properties of the orig- Effect of failures on the costs Failures change costs in inal game are conserved in Gp.
One property of interest two distinct but interrelated ways. Note that the direct effect applies to optimal implications on the PoA. It turns out that convexity is main- outcomes and to equilibrium outcomes alike. For example, tained in the perturbed game the proof is straightforward. Let cx be a convex [respectively, concave] the direct effect of failures is that agents will now face higher cost function in the game G, and p a probability vector.
Then cp j,x is also convex [resp. Indirect effect: the equilibria in the new game may In the remainder of this paper we assume that failures are change, leading to different payoffs. We We compute the total cost, summing over all the surviving do note however, that most of our results easily extend to the players. That is, X n X more general cases of distinct or correlated probabilities.
In such cases the in the limit case. Specifically, we want to know if the equilibrium costs in the game can change significantly with small fail- Effect of failures on the set of NE ure probabilities.
In each result, we specify whether the survival probability p is Proposition 3. Let G be a congestion game. There is some allowed to take any fixed value. To also an NE of G. For any Proof. Finally, on the costs. Since equalities can only disappear, Proposition 9. Let G b be a RSG with increasing costs. If there is more than one such deviation, then b not significantly increase. Denote by A1 the outcome where i plays b Proposition 6. As bounded PoA. Proof sketch. There- We argue that there are at most n steps until convergence.
In If an agent i moves from a to b in step t then no agent particular, there are no new bad equilibria. In particular, this means that each agent moves ligible for sufficiently small failure probabilities. We next show that for all t, By Proposition 6 the PoA cannot increase due to failures.
We get the following as a corollary, On the other hand, since i preferred b over a in G 1 Proposition 7. We next bound the two expressions. PoA due to failures in games with increasing costs. The Thus w. Thus ric singleton games, such a decrease is impossible.
That is, if either one is relaxed, then there is an example where the PoA can improve arbitrarily. We also show that the SPs' congestion experienced in different Nash equilibria is almost unique. Downloads from ePrints over the past year. Other digital versions may also be available to download e. This repository has been built using EPrints software , developed at the University of Southampton, but available to everyone to use.
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Use of this web site signifies your agreement to the terms and conditions. We investigate two models. The first model is congestion games with both resource and agent failures, where each agent chooses the same number of resources with the minimum expected cost.
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