311 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Are Equivariant Equilibrium Approximators Beneficial?
Recently, remarkable progress has been made by approximating Nash equilibrium
(NE), correlated equilibrium (CE), and coarse correlated equilibrium (CCE)
through function approximation that trains a neural network to predict
equilibria from game representations. Furthermore, equivariant architectures
are widely adopted in designing such equilibrium approximators in normal-form
games. In this paper, we theoretically characterize benefits and limitations of
equivariant equilibrium approximators. For the benefits, we show that they
enjoy better generalizability than general ones and can achieve better
approximations when the payoff distribution is permutation-invariant. For the
limitations, we discuss their drawbacks in terms of equilibrium selection and
social welfare. Together, our results help to understand the role of
equivariance in equilibrium approximators.Comment: To appear in ICML 202
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
On Supermodular Contracts and Dense Subgraphs
We study the combinatorial contract design problem, introduced and studied by
Dutting et. al. (2021, 2022), in both the single and multi-agent settings.
Prior work has examined the problem when the principal's utility function is
submodular in the actions chosen by the agent(s).
We complement this emerging literature with an examination of the problem
when the principal's utility is supermodular.
In the single-agent setting, we obtain a strongly polynomial time algorithm
for the optimal contract.
This stands in contrast to the NP-hardness of the problem with submodular
principal utility due to Dutting et. al. (2021).
This result has two technical components, the first of which applies beyond
supermodular or submodular utilities.
This result strengthens and simplifies analogous enumeration algorithms from
Dutting et. al. (2021), and applies to any nondecreasing valuation function for
the principal.
Second, we show that supermodular valuations lead to a polynomial number of
breakpoints, analogous to a similar result by Dutting et. al. (2021) for gross
substitutes valuations.
In the multi-agent setting, we obtain a mixed bag of positive and negative
results.
First, we show that it is NP-hard to obtain any finite multiplicative
approximation, or an additive FPTAS.
This stands in contrast to the submodular case, where efficient computation
of approximately optimal contracts was shown by Dutting et. al. (2022).
Second, we derive an additive PTAS for the problem in the instructive special
case of graph-based supermodular valuations, and equal costs.
En-route to this result, we discover an intimate connection between the
multi-agent contract problem and the notorious k-densest subgraph problem.
We build on and combine techniques from the literature on dense subgraph
problems to obtain our additive PTAS.Comment: 31 pages, 2 figure
Computing equilibria by minimizing exploitability with best-response ensembles
In this paper, we study the problem of computing an approximate Nash
equilibrium of a continuous game. Such games naturally model many situations
involving space, time, money, and other fine-grained resources or quantities.
The standard measure of the closeness of a strategy profile to Nash equilibrium
is exploitability, which measures how much utility players can gain from
changing their strategy unilaterally. We introduce a new equilibrium-finding
method that minimizes an approximation of the exploitability. This
approximation employs a best-response ensemble for each player that maintains
multiple candidate best responses for that player. In each iteration, the
best-performing element of each ensemble is used in a gradient-based scheme to
update the current strategy profile. The strategy profile and best-response
ensembles are simultaneously trained to minimize and maximize the approximate
exploitability, respectively. Experiments on a suite of benchmark games show
that it outperforms previous methods
Essays on the economics of networks
Networks (collections of nodes or vertices and graphs capturing their linkages) are a common object of study across a range of fields includ- ing economics, statistics and computer science. Network analysis is often based around capturing the overall structure of the network by some reduced set of parameters. Canonically, this has focused on the notion of centrality. There are many measures of centrality, mostly based around statistical analysis of the linkages between nodes on the network. However, another common approach has been through the use of eigenfunction analysis of the centrality matrix. My the- sis focuses on eigencentrality as a property, paying particular focus to equilibrium behaviour when the network structure is fixed. This occurs when nodes are either passive, such as for web-searches or queueing models or when they represent active optimizing agents in network games. The major contribution of my thesis is in the applica- tion of relatively recent innovations in matrix derivatives to centrality measurements and equilibria within games that are function of those measurements. I present a series of new results on the stability of eigencentrality measures and provide some examples of applications to a number of real world examples
Approximation Algorithms for Envy-Free Cake Division with Connected Pieces
Cake cutting is a classic model for studying fair division of a heterogeneous, divisible resource among agents with individual preferences. Addressing cake division under a typical requirement that each agent must receive a connected piece of the cake, we develop approximation algorithms for finding envy-free (fair) cake divisions. In particular, this work improves the state-of-the-art additive approximation bound for this fundamental problem. Our results hold for general cake division instances in which the agents\u27 valuations satisfy basic assumptions and are normalized (to have value 1 for the cake). Furthermore, the developed algorithms execute in polynomial time under the standard Robertson-Webb query model.
Prior work has shown that one can efficiently compute a cake division (with connected pieces) in which the additive envy of any agent is at most 1/3. An efficient algorithm is also known for finding connected cake divisions that are (almost) 1/2-multiplicatively envy-free. Improving the additive approximation guarantee and maintaining the multiplicative one, we develop a polynomial-time algorithm that computes a connected cake division that is both (1/4 +o(1))-additively envy-free and (1/2 - o(1))-multiplicatively envy-free. Our algorithm is based on the ideas of interval growing and envy-cycle elimination.
In addition, we study cake division instances in which the number of distinct valuations across the agents is parametrically bounded. We show that such cake division instances admit a fully polynomial-time approximation scheme for connected envy-free cake division
Solutions in multi-actor projects with collaboration and strategic incentives
This dissertation focuses on the mathematical analysis of projects involving decisions by multiple players. These players all have their own capabilities, requirements, and incentives, but their (monetary) outcome is dependent on the decisions of other players as well. Game theory is a mathematical tool to analyze the interactive decision-making process, generally paired with a method to ‘resolve’ the conflict situation. The way in which players interact in such a situation is commonly divided in two categories, distinguishing between cooperative and competitive (non-cooperative) behavior. This dissertation first studies two models within a cooperative framework, starting with the definition and analysis of a new influence measure for general, collaborative projects. The second model applies to situations where players cooperate on the construction of a new joint infrastructure, with a specific focus on cost allocation for CO2 transport infrastructure. Next, two-stage models are considered, in which a noncooperative first stage is followed by a cooperative second stage. Subsequently, social welfare loss in auctions with a corrupt auctioneer is studied. Finally, a new solution concept is presented that refines the notion of Nash equilibria for a general class of non-cooperative games
On algorithmically boosting fixed-point computations
This paper is a thought experiment on exponentiating algorithms. One of the
main contributions of this paper is to show that this idea finds material
implementation in exponentiating fixed-point computation algorithms. Various
problems in computer science can be cast as instances of computing a fixed
point of a map. In this paper, we present a general method of boosting the
convergence of iterative fixed-point computations that we call algorithmic
boosting, which is a (slight) generalization of algorithmic exponentiation. We
first define our method in the general setting of nonlinear maps. Secondly, we
restrict attention to convergent linear maps and show that our algorithmic
boosting method can set in motion exponential speedups in the convergence rate.
Thirdly, we show that algorithmic boosting can convert a (weak) non-convergent
iterator to a (strong) convergent one. We then consider a variational approach
to algorithmic boosting providing tools to convert a non-convergent continuous
flow to a convergent one. We, finally, discuss implementations of the
exponential function, an important issue even for the scalar case
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