584 research outputs found
On the Existence of Pure Strategy Nash Equilibria in Integer-Splittable Weighted Congestion Games
We study the existence of pure strategy Nash equilibria (PSNE) in integer–splittable weighted congestion games (ISWCGs), where agents can strategically assign different amounts of demand to different resources, but must distribute this demand in fixed-size parts. Such scenarios arise in a wide range of application domains, including job scheduling and network routing, where agents have to allocate multiple tasks and can assign a number of tasks to a particular selected resource. Specifically, in an ISWCG, an agent has a certain total demand (aka weight) that it needs to satisfy, and can do so by requesting one or more integer units of each resource from an element of a given collection of feasible subsets. Each resource is associated with a unit–cost function of its level of congestion; as such, the cost to an agent for using a particular resource is the product of the resource unit–cost and the number of units the agent requests.While general ISWCGs do not admit PSNE [(Rosenthal, 1973b)], the restricted subclass of these games with linear unit–cost functions has been shown to possess a potential function [(Meyers, 2006)], and hence, PSNE. However, the linearity of costs may not be necessary for the existence of equilibria in pure strategies. Thus, in this paper we prove that PSNE always exist for a larger class of convex and monotonically increasing unit–costs. On the other hand, our result is accompanied by a limiting assumption on the structure of agents’ strategy sets: specifically, each agent is associated with its set of accessible resources, and can distribute its demand across any subset of these resources.Importantly, we show that neither monotonicity nor convexity on its own guarantees this result. Moreover, we give a counterexample with monotone and semi–convex cost functions, thus distinguishing ISWCGs from the class of infinitely–splittable congestion games for which the conditions of monotonicity and semi–convexity have been shown to be sufficient for PSNE existence [(Rosen, 1965)]. Furthermore, we demonstrate that the finite improvement path property (FIP) does not hold for convex increasing ISWCGs. Thus, in contrast to the case with linear costs, a potential function argument cannot be used to prove our result. Instead, we provide a procedure that converges to an equilibrium from an arbitrary initial strategy profile, and in doing so show that ISWCGs with convex increasing unit–cost functions are weakly acyclic
Linear Regression from Strategic Data Sources
Linear regression is a fundamental building block of statistical data
analysis. It amounts to estimating the parameters of a linear model that maps
input features to corresponding outputs. In the classical setting where the
precision of each data point is fixed, the famous Aitken/Gauss-Markov theorem
in statistics states that generalized least squares (GLS) is a so-called "Best
Linear Unbiased Estimator" (BLUE). In modern data science, however, one often
faces strategic data sources, namely, individuals who incur a cost for
providing high-precision data.
In this paper, we study a setting in which features are public but
individuals choose the precision of the outputs they reveal to an analyst. We
assume that the analyst performs linear regression on this dataset, and
individuals benefit from the outcome of this estimation. We model this scenario
as a game where individuals minimize a cost comprising two components: (a) an
(agent-specific) disclosure cost for providing high-precision data; and (b) a
(global) estimation cost representing the inaccuracy in the linear model
estimate. In this game, the linear model estimate is a public good that
benefits all individuals. We establish that this game has a unique non-trivial
Nash equilibrium. We study the efficiency of this equilibrium and we prove
tight bounds on the price of stability for a large class of disclosure and
estimation costs. Finally, we study the estimator accuracy achieved at
equilibrium. We show that, in general, Aitken's theorem does not hold under
strategic data sources, though it does hold if individuals have identical
disclosure costs (up to a multiplicative factor). When individuals have
non-identical costs, we derive a bound on the improvement of the equilibrium
estimation cost that can be achieved by deviating from GLS, under mild
assumptions on the disclosure cost functions.Comment: This version (v3) extends the results on the sub-optimality of GLS
(Section 6) and improves writing in multiple places compared to v2. Compared
to the initial version v1, it also fixes an error in Theorem 6 (now Theorem
5), and extended many of the result
On the robustness of learning in games with stochastically perturbed payoff observations
Motivated by the scarcity of accurate payoff feedback in practical
applications of game theory, we examine a class of learning dynamics where
players adjust their choices based on past payoff observations that are subject
to noise and random disturbances. First, in the single-player case
(corresponding to an agent trying to adapt to an arbitrarily changing
environment), we show that the stochastic dynamics under study lead to no
regret almost surely, irrespective of the noise level in the player's
observations. In the multi-player case, we find that dominated strategies
become extinct and we show that strict Nash equilibria are stochastically
stable and attracting; conversely, if a state is stable or attracting with
positive probability, then it is a Nash equilibrium. Finally, we provide an
averaging principle for 2-player games, and we show that in zero-sum games with
an interior equilibrium, time averages converge to Nash equilibrium for any
noise level.Comment: 36 pages, 4 figure
Congestion pricing using a raffle-based scheme
We propose a raffle-based scheme for the decongestion of a shared resource. Our scheme builds on ideas from the economic literature on incentivizing contributions to a public good. We formulate a game-theoretic model for the decongestion problem in a setup with a finite number of users, as well as in a setup with an infinite number of non-atomic users. We analyze both setups, and show that the former converges toward the latter when the number of users becomes large. We compare our results to existing results for the public good provision problem. Overall, our results establish that raffle-based schemes are useful in addressing congestion problems.National Science Foundation (U.S.) (Grant CNS-0910711)National Science Foundation (U.S.) (Grant CCF-0424422)United States. Air Force Office of Scientific Research (FA9550-06-1-0244
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