arXiv subjects

Marco Scarsini

Publications and source records attributed to Marco Scarsini.

36 records · Page 2Linked to original sources

arXiv2021

Social Learning in Nonatomic Routing Games

We consider a discrete-time nonatomic routing game with variable demand and uncertain costs. Given a routing network with single origin and destination, the cost function of each edge depends on some uncertain persistent state parameter. At every period, a random traffic demand is routed through the network according to a Wardrop equilibrium. The realized costs are publicly observed and the public Bayesian belief about the state parameter is updated. We say that there is strong learning when beliefs converge to the truth and weak learning when the equilibrium flow converges to the complete-information flow. We characterize the networks for which learning occurs. We prove that these networks have a series-parallel structure and provide a counterexample to show that learning may fail in non-series-parallel networks.

econ.TH↗

arXiv2021

The Price of Anarchy in Routing Games as a Function of the Demand

The price of anarchy has become a standard measure of the efficiency of equilibria in games. Most of the literature in this area has focused on establishing worst-case bounds for specific classes of games, such as routing games or more general congestion games. Recently, the price of anarchy in routing games has been studied as a function of the traffic demand, providing asymptotic results in light and heavy traffic. The aim of this paper is to study the price of anarchy in nonatomic routing games in the intermediate region of the demand. To achieve this goal, we begin by establishing some smoothness properties of Wardrop equilibria and social optima for general smooth costs. In the case of affine costs we show that the equilibrium is piecewise linear, with break points at the demand levels at which the set of active paths changes. We prove that the number of such break points is finite, although it can be exponential in the size of the network. Exploiting a scaling law between the equilibrium and the social optimum, we derive a similar behavior for the optimal flows. We then prove that in any interval between break points the price of anarchy is smooth and it is either monotone (decreasing or increasing) over the full interval, or it decreases up to a certain minimum point in the interior of the interval and increases afterwards. We deduce that for affine costs the maximum of the price of anarchy can only occur at the break points. For general costs we provide counterexamples showing that the set of break points is not always finite.

cs.GT↗

arXiv2021

Efficiency of equilibria in games with random payoffs

We consider normal-form games with $n$ players and two strategies for each player, where the payoffs are i.i.d. random variables with some distribution $F$ and we consider issues related to the pure equilibria in the game as the number of players diverges. It is well-known that, if the distribution $F$ has no atoms, the random number of pure equilibria is asymptotically Poisson$(1)$. In the presence of atoms, it diverges. For each strategy profile, we consider the (random) average payoff of the players, called Average Social Utility (ASU). In particular, we examine the asymptotic behavior of the optimum ASU and the one associated to the best and worst pure Nash equilibria and we show that, although these quantities are random, they converge, as $n\to\infty$ to some deterministic quantities.

math.PR↗

arXiv2020

The Buck-Passing Game

We consider a game in which players are the vertices of a directed graph. Initially, Nature chooses one player according to some fixed distribution and gives her a buck, which represents the request to perform a chore. After completing the task, the player passes the buck to one of her out-neighbors in the graph. The procedure is repeated indefinitely and each player's cost is the asymptotic expected frequency of times that she receives the buck. We consider a deterministic and a stochastic version of the game depending on how players select the neighbor to pass the buck. In both cases we prove the existence of pure equilibria that do not depend on the initial distribution; this is achieved by showing the existence of a generalized ordinal potential. We then use the price of anarchy and price of stability to measure fairness of these equilibria. We also study a buck-holding variant of the game in which players want to maximize the frequency of times they hold the buck, which includes the PageRank game as a special case.

cs.GT↗

arXiv2020

Pure Nash Equilibria and Best-Response Dynamics in Random Games

In finite games mixed Nash equilibria always exist, but pure equilibria may fail to exist. To assess the relevance of this nonexistence, we consider games where the payoffs are drawn at random. In particular, we focus on games where a large number of players can each choose one of two possible strategies, and the payoffs are i.i.d. with the possibility of ties. We provide asymptotic results about the random number of pure Nash equilibria, such as fast growth and a central limit theorem, with bounds for the approximation error. Moreover, by using a new link between percolation models and game theory, we describe in detail the geometry of Nash equilibria and show that, when the probability of ties is small, a best-response dynamics reaches a Nash equilibrium with a probability that quickly approaches one as the number of players grows. We show that a multitude of phase transitions depend only on a single parameter of the model, that is, the probability of having ties.

cs.GT↗

arXiv2020

Decomposition of games: some strategic considerations

Candogan et al. (2011) provide an orthogonal direct-sum decomposition of finite games into potential, harmonic and nonstrategic components. In this paper we study the issue of decomposing games that are strategically equivalent from a game-theoretical point of view, for instance games obtained via transformations such as duplications of strategies or positive affine mappings of of payoffs. We show the need to define classes of decompositions to achieve commutativity of game transformations and decompositions.

cs.GT↗

arXiv2019

Search for an Immobile Hider on a Stochastic Network

Harry hides on an edge of a graph and does not move from there. Sally, starting from a known origin, tries to find him as soon as she can. Harry's goal is to be found as late as possible. At any given time, each edge of the graph is either active or inactive, independently of the other edges, with a known probability of being active. This situation can be modeled as a zero-sum two-person stochastic game. We show that the game has a value and we provide upper and lower bounds for this value. Finally, by generalizing optimal strategies of the deterministic case, we provide more refined results for trees and Eulerian graphs.

cs.GT↗

arXiv2018

When is selfish routing bad? The price of anarchy in light and heavy traffic

This paper examines the behavior of the price of anarchy as a function of the traffic inflow in nonatomic congestion games with multiple origin-destination (O/D) pairs. Empirical studies in real-world networks show that the price of anarchy is close to 1 in both light and heavy traffic, thus raising the question: can these observations be justified theoretically? We first show that this is not always the case: the price of anarchy may remain a positive distance away from 1 for all values of the traffic inflow, even in simple three-link networks with a single O/D pair and smooth, convex costs. On the other hand, for a large class of cost functions (including all polynomials), the price of anarchy does converge to 1 in both heavy and light traffic, irrespective of the network topology and the number of O/D pairs in the network. We also examine the rate of convergence of the price of anarchy, and we show that it follows a power law whose degree can be computed explicitly when the network's cost functions are polynomials.

cs.GT↗

arXiv2018

Demand-Independent Optimal Tolls

Wardrop equilibria in nonatomic congestion games are in general inefficient as they do not induce an optimal flow that minimizes the total travel time. Network tolls are a prominent and popular way to induce an optimum flow in equilibrium. The classical approach to find such tolls is marginal cost pricing which requires the exact knowledge of the demand on the network. In this paper, we investigate under which conditions demand-independent optimum tolls exist that induce the system optimum flow for any travel demand in the network. We give several characterizations for the existence of such tolls both in terms of the cost structure and the network structure of the game. Specifically we show that demand-independent optimum tolls exist if and only if the edge cost functions are shifted monomials as used by the Bureau of Public Roads. Moreover, non-negative demand-independent optimum tolls exist when the network is a directed acyclic multi-graph. Finally, we show that any network with a single origin-destination pair admits demand-independent optimum tolls that, although not necessarily non-negative, satisfy a budget constraint.

cs.GT↗

arXiv2017

Location Games on Networks: Existence and Efficiency of Equilibria

We consider a game where a finite number of retailers choose a location, given that their potential consumers are distributed on a network. Retailers do not compete on price but only on location, therefore each consumer shops at the closest store. We show that when the number of retailers is large enough, the game admits a pure Nash equilibrium and we construct it. We then compare the equilibrium cost borne by the consumers with the cost that could be achieved if the retailers followed the dictate of a benevolent planner. We perform this comparison in term of the price of anarchy, i.e., the ratio of the worst equilibrium cost and the optimal cost, and the price of stability, i.e., the ratio of the best equilibrium cost and the optimal cost. We show that, asymptotically in the number of retailers, these ratios are bounded by two and one, respectively.

cs.GT↗

arXiv2017

Variance Allocation and Shapley Value

Motivated by the problem of utility allocation in a portfolio under a Markowitz mean-variance choice paradigm, we propose an allocation criterion for the variance of the sum of $n$ possibly dependent random variables. This criterion, the Shapley value, requires to translate the problem into a cooperative game. The Shapley value has nice properties, but, in general, is computationally demanding. The main result of this paper shows that in our particular case the Shapley value has a very simple form that can be easily computed. The same criterion is used also to allocate the standard deviation of the sum of $n$ random variables and a conjecture about the relation of the values in the two games is formulated.

math.PR↗

arXiv2016

Dynamic Atomic Congestion Games with Seasonal Flows

We propose a model of discrete time dynamic congestion games with atomic players and a single source-destination pair. The latencies of edges are composed by free-flow transit times and possible queuing time due to capacity constraints. We give a precise description of the dynamics induced by the individual strategies of players and of the corresponding costs, either when the traffic is controlled by a planner, or when players act selfishly. In parallel networks, optimal and equilibrium behavior eventually coincides, but the selfish behavior of the first players has consequences that cannot be undone and are paid by all future generations. In more general topologies, our main contributions are three-fold. First, we show that equilibria are usually not unique. In particular, we prove that there exists a sequence of networks such that the price of anarchy is equal to $n-1$, where $n$ is the number of vertices, and the price of stability is equal to 1. Second, we illustrate a new dynamic version of Braess's paradox: the presence of initial queues in a network may decrease the long-run costs in equilibrium. This paradox may arise even in networks for which no Braess's paradox was previously known. Third, we propose an extension to model seasonalities by assuming that departure flows fluctuate periodically over time. We introduce a measure that captures the queues induced by periodicity of inflows. This measure is the increase in costs compared to uniform departures for optimal and equilibrium flows in parallel networks.

cs.GT↗

arXiv2016

On the Price of Anarchy of Highly Congested Nonatomic Network Games

We consider nonatomic network games with one source and one destination. We examine the asymptotic behavior of the price of anarchy as the inflow increases. In accordance with some empirical observations, we show that, under suitable conditions, the price of anarchy is asymptotic to one. We show with some counterexamples that this is not always the case. The counterexamples occur in very simple parallel graphs.

cs.GT↗

arXiv2013

Existence of equilibria in countable games: an algebraic approach

Although mixed extensions of finite games always admit equilibria, this is not the case for countable games, the best-known example being Wald's pick-the-larger-integer game. Several authors have provided conditions for the existence of equilibria in infinite games. These conditions are typically of topological nature and are rarely applicable to countable games. Here we establish an existence result for the equilibrium of countable games when the strategy sets are a countable group and the payoffs are functions of the group operation. In order to obtain the existence of equilibria, finitely additive mixed strategies have to be allowed. This creates a problem of selection of a product measure of mixed strategies. We propose a family of such selections and prove existence of an equilibrium that does not depend on the selection. As a byproduct we show that if finitely additive mixed strategies are allowed, then Wald's game admits an equilibrium. We also prove existence of equilibria for nontrivial extensions of matching-pennies and rock-scissors-paper. Finally we extend the main results to uncountable games.

cs.GT↗

arXiv2012

A Colonel Blotto Gladiator Game

We consider a stochastic version of the well-known Blotto game, called the gladiator game. In this zero-sum allocation game two teams of gladiators engage in a sequence of one-to-one fights in which the probability of winning is a function of the gladiators' strengths. Each team's strategy consists of the allocation of its total strength among its gladiators. We find the Nash equilibria and the value of this class of games and show how they depend on the total strength of teams and the number of gladiators in each team. To do this, we study interesting majorization-type probability inequalities concerning linear combinations of Gamma random variables. Similar inequalities have been used in models of telecommunications and research and development.

cs.GT↗

arXiv2012

On the Core of Dynamic Cooperative Games

We consider dynamic cooperative games, where the worth of coalitions varies over time according to the history of allocations. When defining the core of a dynamic game, we allow the possibility for coalitions to deviate at any time and thereby to give rise to a new environment. A coalition that considers a deviation needs to take the consequences into account because from the deviation point on, the game is no longer played with the original set of players. The deviating coalition becomes the new grand coalition which, in turn, induces a new dynamic game. The stage games of the new dynamical game depend on all previous allocation including those that have materialized from the deviating time on. We define three types of core solutions: fair core, stable core and credible core. We characterize the first two in case where the instantaneous game depends on the last allocation (rather than on the whole history of allocations) and the third in the general case. The analysis and the results resembles to a great extent the theory of non-cooperative dynamic games.

cs.GT↗

arXiv2011

Stochastic comparisons of stratified sampling techniques for some Monte Carlo estimators

We compare estimators of the (essential) supremum and the integral of a function $f$ defined on a measurable space when $f$ may be observed at a sample of points in its domain, possibly with error. The estimators compared vary in their levels of stratification of the domain, with the result that more refined stratification is better with respect to different criteria. The emphasis is on criteria related to stochastic orders. For example, rather than compare estimators of the integral of $f$ by their variances (for unbiased estimators), or mean square error, we attempt the stronger comparison of convex order when possible. For the supremum, the criterion is based on the stochastic order of estimators.

math.ST↗

arXiv2010

Lowest Unique Bid Auctions

We consider a class of auctions (Lowest Unique Bid Auctions) that have achieved a considerable success on the Internet. Bids are made in cents (of euro) and every bidder can bid as many numbers as she wants. The lowest unique bid wins the auction. Every bid has a fixed cost, and once a participant makes a bid, she gets to know whether her bid was unique and whether it was the lowest unique. Information is updated in real time, but every bidder sees only what's relevant to the bids she made. We show that the observed behavior in these auctions differs considerably from what theory would prescribe if all bidders were fully rational. We show that the seller makes money, which would not be the case with rational bidders, and some bidders win the auctions quite often. We describe a possible strategy for these bidders.

math.PR↗