405 research outputs found
K-Implementation
This paper discusses an interested party who wishes to influence the behavior
of agents in a game (multi-agent interaction), which is not under his control.
The interested party cannot design a new game, cannot enforce agents' behavior,
cannot enforce payments by the agents, and cannot prohibit strategies available
to the agents. However, he can influence the outcome of the game by committing
to non-negative monetary transfers for the different strategy profiles that may
be selected by the agents. The interested party assumes that agents are
rational in the commonly agreed sense that they do not use dominated
strategies. Hence, a certain subset of outcomes is implemented in a given game
if by adding non-negative payments, rational players will necessarily produce
an outcome in this subset. Obviously, by making sufficiently big payments one
can implement any desirable outcome. The question is what is the cost of
implementation? In this paper we introduce the notion of k-implementation of a
desired set of strategy profiles, where k stands for the amount of payment that
need to be actually made in order to implement desirable outcomes. A major
point in k-implementation is that monetary offers need not necessarily
materialize when following desired behaviors. We define and study
k-implementation in the contexts of games with complete and incomplete
information. In the latter case we mainly focus on the VCG games. Our setting
is later extended to deal with mixed strategies using correlation devices.
Together, the paper introduces and studies the implementation of desirable
outcomes by a reliable party who cannot modify game rules (i.e. provide
protocols), complementing previous work in mechanism design, while making it
more applicable to many realistic CS settings
Dynamic Non-Bayesian Decision Making
The model of a non-Bayesian agent who faces a repeated game with incomplete
information against Nature is an appropriate tool for modeling general
agent-environment interactions. In such a model the environment state
(controlled by Nature) may change arbitrarily, and the feedback/reward function
is initially unknown. The agent is not Bayesian, that is he does not form a
prior probability neither on the state selection strategy of Nature, nor on his
reward function. A policy for the agent is a function which assigns an action
to every history of observations and actions. Two basic feedback structures are
considered. In one of them -- the perfect monitoring case -- the agent is able
to observe the previous environment state as part of his feedback, while in the
other -- the imperfect monitoring case -- all that is available to the agent is
the reward obtained. Both of these settings refer to partially observable
processes, where the current environment state is unknown. Our main result
refers to the competitive ratio criterion in the perfect monitoring case. We
prove the existence of an efficient stochastic policy that ensures that the
competitive ratio is obtained at almost all stages with an arbitrarily high
probability, where efficiency is measured in terms of rate of convergence. It
is further shown that such an optimal policy does not exist in the imperfect
monitoring case. Moreover, it is proved that in the perfect monitoring case
there does not exist a deterministic policy that satisfies our long run
optimality criterion. In addition, we discuss the maxmin criterion and prove
that a deterministic efficient optimal strategy does exist in the imperfect
monitoring case under this criterion. Finally we show that our approach to
long-run optimality can be viewed as qualitative, which distinguishes it from
previous work in this area.Comment: See http://www.jair.org/ for any accompanying file
Smooth Inequalities and Equilibrium Inefficiency in Scheduling Games
We study coordination mechanisms for Scheduling Games (with unrelated
machines). In these games, each job represents a player, who needs to choose a
machine for its execution, and intends to complete earliest possible. Our goal
is to design scheduling policies that always admit a pure Nash equilibrium and
guarantee a small price of anarchy for the l_k-norm social cost --- the
objective balances overall quality of service and fairness. We consider
policies with different amount of knowledge about jobs: non-clairvoyant,
strongly-local and local. The analysis relies on the smooth argument together
with adequate inequalities, called smooth inequalities. With this unified
framework, we are able to prove the following results.
First, we study the inefficiency in l_k-norm social costs of a strongly-local
policy SPT and a non-clairvoyant policy EQUI. We show that the price of anarchy
of policy SPT is O(k). We also prove a lower bound of Omega(k/log k) for all
deterministic, non-preemptive, strongly-local and non-waiting policies
(non-waiting policies produce schedules without idle times). These results
ensure that SPT is close to optimal with respect to the class of l_k-norm
social costs. Moreover, we prove that the non-clairvoyant policy EQUI has price
of anarchy O(2^k).
Second, we consider the makespan (l_infty-norm) social cost by making
connection within the l_k-norm functions. We revisit some local policies and
provide simpler, unified proofs from the framework's point of view. With the
highlight of the approach, we derive a local policy Balance. This policy
guarantees a price of anarchy of O(log m), which makes it the currently best
known policy among the anonymous local policies that always admit a pure Nash
equilibrium.Comment: 25 pages, 1 figur
On the Approximation Performance of Fictitious Play in Finite Games
We study the performance of Fictitious Play, when used as a heuristic for
finding an approximate Nash equilibrium of a 2-player game. We exhibit a class
of 2-player games having payoffs in the range [0,1] that show that Fictitious
Play fails to find a solution having an additive approximation guarantee
significantly better than 1/2. Our construction shows that for n times n games,
in the worst case both players may perpetually have mixed strategies whose
payoffs fall short of the best response by an additive quantity 1/2 -
O(1/n^(1-delta)) for arbitrarily small delta. We also show an essentially
matching upper bound of 1/2 - O(1/n)
On the Structure of Equilibria in Basic Network Formation
We study network connection games where the nodes of a network perform edge
swaps in order to improve their communication costs. For the model proposed by
Alon et al. (2010), in which the selfish cost of a node is the sum of all
shortest path distances to the other nodes, we use the probabilistic method to
provide a new, structural characterization of equilibrium graphs. We show how
to use this characterization in order to prove upper bounds on the diameter of
equilibrium graphs in terms of the size of the largest -vicinity (defined as
the the set of vertices within distance from a vertex), for any
and in terms of the number of edges, thus settling positively a conjecture of
Alon et al. in the cases of graphs of large -vicinity size (including graphs
of large maximum degree) and of graphs which are dense enough.
Next, we present a new swap-based network creation game, in which selfish
costs depend on the immediate neighborhood of each node; in particular, the
profit of a node is defined as the sum of the degrees of its neighbors. We
prove that, in contrast to the previous model, this network creation game
admits an exact potential, and also that any equilibrium graph contains an
induced star. The existence of the potential function is exploited in order to
show that an equilibrium can be reached in expected polynomial time even in the
case where nodes can only acquire limited knowledge concerning non-neighboring
nodes.Comment: 11 pages, 4 figure
The Networked Common Goods Game
We introduce a new class of games called the networked common goods game
(NCGG), which generalizes the well-known common goods game. We focus on a
fairly general subclass of the game where each agent's utility functions are
the same across all goods the agent is entitled to and satisfy certain natural
properties (diminishing return and smoothness). We give a comprehensive set of
technical results listed as follows.
* We show the optimization problem faced by a single agent can be solved
efficiently in this subclass. The discrete version of the problem is however
NP-hard but admits an fully polynomial time approximation scheme (FPTAS).
* We show uniqueness results of pure strategy Nash equilibrium of NCGG, and
that the equilibrium is fully characterized by the structure of the network and
independent of the choices and combinations of agent utility functions.
* We show NCGG is a potential game, and give an implementation of best/better
response Nash dynamics that lead to fast convergence to an
-approximate pure strategy Nash equilibrium.
* Lastly, we show the price of anarchy of NCGG can be as large as
(for any ), which means selfish behavior
in NCGG can lead to extremely inefficient social outcomes
Metastability of Asymptotically Well-Behaved Potential Games
One of the main criticisms to game theory concerns the assumption of full
rationality. Logit dynamics is a decentralized algorithm in which a level of
irrationality (a.k.a. "noise") is introduced in players' behavior. In this
context, the solution concept of interest becomes the logit equilibrium, as
opposed to Nash equilibria. Logit equilibria are distributions over strategy
profiles that possess several nice properties, including existence and
uniqueness. However, there are games in which their computation may take time
exponential in the number of players. We therefore look at an approximate
version of logit equilibria, called metastable distributions, introduced by
Auletta et al. [SODA 2012]. These are distributions that remain stable (i.e.,
players do not go too far from it) for a super-polynomial number of steps
(rather than forever, as for logit equilibria). The hope is that these
distributions exist and can be reached quickly by logit dynamics.
We identify a class of potential games, called asymptotically well-behaved,
for which the behavior of the logit dynamics is not chaotic as the number of
players increases so to guarantee meaningful asymptotic results. We prove that
any such game admits distributions which are metastable no matter the level of
noise present in the system, and the starting profile of the dynamics. These
distributions can be quickly reached if the rationality level is not too big
when compared to the inverse of the maximum difference in potential. Our proofs
build on results which may be of independent interest, including some spectral
characterizations of the transition matrix defined by logit dynamics for
generic games and the relationship of several convergence measures for Markov
chains
Privacy and Truthful Equilibrium Selection for Aggregative Games
We study a very general class of games --- multi-dimensional aggregative
games --- which in particular generalize both anonymous games and weighted
congestion games. For any such game that is also large, we solve the
equilibrium selection problem in a strong sense. In particular, we give an
efficient weak mediator: a mechanism which has only the power to listen to
reported types and provide non-binding suggested actions, such that (a) it is
an asymptotic Nash equilibrium for every player to truthfully report their type
to the mediator, and then follow its suggested action; and (b) that when
players do so, they end up coordinating on a particular asymptotic pure
strategy Nash equilibrium of the induced complete information game. In fact,
truthful reporting is an ex-post Nash equilibrium of the mediated game, so our
solution applies even in settings of incomplete information, and even when
player types are arbitrary or worst-case (i.e. not drawn from a common prior).
We achieve this by giving an efficient differentially private algorithm for
computing a Nash equilibrium in such games. The rates of convergence to
equilibrium in all of our results are inverse polynomial in the number of
players . We also apply our main results to a multi-dimensional market game.
Our results can be viewed as giving, for a rich class of games, a more robust
version of the Revelation Principle, in that we work with weaker informational
assumptions (no common prior), yet provide a stronger solution concept (ex-post
Nash versus Bayes Nash equilibrium). In comparison to previous work, our main
conceptual contribution is showing that weak mediators are a game theoretic
object that exist in a wide variety of games -- previously, they were only
known to exist in traffic routing games
Efficient Equilibria in Polymatrix Coordination Games
We consider polymatrix coordination games with individual preferences where
every player corresponds to a node in a graph who plays with each neighbor a
separate bimatrix game with non-negative symmetric payoffs. In this paper, we
study -approximate -equilibria of these games, i.e., outcomes where
no group of at most players can deviate such that each member increases his
payoff by at least a factor . We prove that for these
games have the finite coalitional improvement property (and thus
-approximate -equilibria exist), while for this
property does not hold. Further, we derive an almost tight bound of
on the price of anarchy, where is the number of
players; in particular, it scales from unbounded for pure Nash equilibria ( to for strong equilibria (). We also settle the complexity
of several problems related to the verification and existence of these
equilibria. Finally, we investigate natural means to reduce the inefficiency of
Nash equilibria. Most promisingly, we show that by fixing the strategies of
players the price of anarchy can be reduced to (and this bound is tight)
Budgeted personalized incentive approaches for smoothing congestion in resource networks
Abstract. Congestion occurs when there is competition for resources by selfish agents. In this paper, we are concerned with smoothing out congestion in a network of resources by using personalized well-timed incentives that are subject to budget constraints. To that end, we provide: (i) a mathematical formulation that computes equilibrium for the resource sharing congestion game with incentives and budget constraints; (ii) an integrated approach that scales to larger problems by exploiting the factored network structure and approximating the attained equilibrium; (iii) an iterative best response algorithm for solving the unconstrained version (no budget) of the resource sharing congestion game; and (iv) theoretical and empirical results (on an illustrative theme park problem) that demonstrate the usefulness of our approach.
- …