1,084 research outputs found
How Discontinuous is Computing Nash Equilibria? (Extended Abstract)
We investigate the degree of discontinuity of several solution concepts from non-cooperative game theory. While the consideration of Nash equilibria forms the core of our work, also pure and correlated equilibria are dealt with. Formally, we restrict the treatment to two player games, but results and proofs extend to the -player case. As a side result, the degree of discontinuity of solving systems of linear inequalities is settled
How Incomputable is Finding Nash Equilibria?
We investigate the Weihrauch-degree of several solution concepts from noncooperative game theory. While the consideration of Nash equilibria forms the core of our work, also pure and correlated equilibria, as well as various concepts of iterated strategy elimination, are dealt with. As a side result, the Weihrauch-degree of solving systems of linear inequalities is settled
A Hydraulic Approach to Equilibria of Resource Selection Games
Drawing intuition from a (physical) hydraulic system, we present a novel
framework, constructively showing the existence of a strong Nash equilibrium in
resource selection games (i.e., asymmetric singleton congestion games) with
nonatomic players, the coincidence of strong equilibria and Nash equilibria in
such games, and the uniqueness of the cost of each given resource across all
Nash equilibria. Our proofs allow for explicit calculation of Nash equilibrium
and for explicit and direct calculation of the resulting (unique) costs of
resources, and do not hinge on any fixed-point theorem, on the Minimax theorem
or any equivalent result, on linear programming, or on the existence of a
potential (though our analysis does provide powerful insights into the
potential, via a natural concrete physical interpretation). A generalization of
resource selection games, called resource selection games with I.D.-dependent
weighting, is defined, and the results are extended to this family, showing the
existence of strong equilibria, and showing that while resource costs are no
longer unique across Nash equilibria in games of this family, they are
nonetheless unique across all strong Nash equilibria, drawing a novel
fundamental connection between group deviation and I.D.-congestion. A natural
application of the resulting machinery to a large class of
constraint-satisfaction problems is also described.Comment: Hebrew University of Jerusalem Center for the Study of Rationality
discussion paper 67
Constant Rank Bimatrix Games are PPAD-hard
The rank of a bimatrix game (A,B) is defined as rank(A+B). Computing a Nash
equilibrium (NE) of a rank-, i.e., zero-sum game is equivalent to linear
programming (von Neumann'28, Dantzig'51). In 2005, Kannan and Theobald gave an
FPTAS for constant rank games, and asked if there exists a polynomial time
algorithm to compute an exact NE. Adsul et al. (2011) answered this question
affirmatively for rank- games, leaving rank-2 and beyond unresolved.
In this paper we show that NE computation in games with rank , is
PPAD-hard, settling a decade long open problem. Interestingly, this is the
first instance that a problem with an FPTAS turns out to be PPAD-hard. Our
reduction bypasses graphical games and game gadgets, and provides a simpler
proof of PPAD-hardness for NE computation in bimatrix games. In addition, we
get:
* An equivalence between 2D-Linear-FIXP and PPAD, improving a result by
Etessami and Yannakakis (2007) on equivalence between Linear-FIXP and PPAD.
* NE computation in a bimatrix game with convex set of Nash equilibria is as
hard as solving a simple stochastic game.
* Computing a symmetric NE of a symmetric bimatrix game with rank is
PPAD-hard.
* Computing a (1/poly(n))-approximate fixed-point of a (Linear-FIXP)
piecewise-linear function is PPAD-hard.
The status of rank- games remains unresolved
On the Inefficiency of the Uniform Price Auction
We present our results on Uniform Price Auctions, one of the standard
sealed-bid multi-unit auction formats, for selling multiple identical units of
a single good to multi-demand bidders. Contrary to the truthful and
economically efficient multi-unit Vickrey auction, the Uniform Price Auction
encourages strategic bidding and is socially inefficient in general. The
uniform pricing rule is, however, widely popular by its appeal to the natural
anticipation, that identical items should be identically priced. In this work
we study equilibria of the Uniform Price Auction for bidders with (symmetric)
submodular valuation functions, over the number of units that they win. We
investigate pure Nash equilibria of the auction in undominated strategies; we
produce a characterization of these equilibria that allows us to prove that a
fraction 1-1/e of the optimum social welfare is always recovered in undominated
pure Nash equilibrium -- and this bound is essentially tight. Subsequently, we
study the auction under the incomplete information setting and prove a bound of
4-2/k on the economic inefficiency of (mixed) Bayes Nash equilibria that are
supported by undominated strategies.Comment: Additions and Improvements upon SAGT 2012 results (and minor
corrections on the previous version
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.Comment: 29 pages, 7 figure
Efficiency and Equilibria in Games of Optimal Derivative Design
In this paper the problem of optimal derivative design, profit maximization and risk minimization under adverse selection when multiple agencies compete for the business of a continuum of heterogenous agents is studied. In contrast with the principal-agent models that are extended within, here the presence of ties in the agents' best-response correspondences yields discontinuous payoff functions for the agencies. These discontinuities are dealt with via efficient tie-breaking rules. The main results of this paper are a proof of existence of (mixed-strategies) Nash equilibria in the case of profit-maximizing agencies, and of socially efficient allocations when the firms are risk minimizers. It is also shown that in the particular case of the entropic risk measure, there exists an efficient "fix-mix" tie-breaking rule, in which case firms share the whole market over given proportions.Adverse selection, Nash equilibria, Pareto optimality, risk transfer, socially efficient allocations, tie-breaking rules
Double Oracle Algorithm for Computing Equilibria in Continuous Games
Many efficient algorithms have been designed to recover Nash equilibria of
various classes of finite games. Special classes of continuous games with
infinite strategy spaces, such as polynomial games, can be solved by
semidefinite programming. In general, however, continuous games are not
directly amenable to computational procedures. In this contribution, we develop
an iterative strategy generation technique for finding a Nash equilibrium in a
whole class of continuous two-person zero-sum games with compact strategy sets.
The procedure, which is called the double oracle algorithm, has been
successfully applied to large finite games in the past. We prove the
convergence of the double oracle algorithm to a Nash equilibrium. Moreover, the
algorithm is guaranteed to recover an approximate equilibrium in finitely-many
steps. Our numerical experiments show that it outperforms fictitious play on
several examples of games appearing in the literature. In particular, we
provide a detailed analysis of experiments with a version of the continuous
Colonel Blotto game
The El Farol Bar Problem Revisited: Reinforcement Learning in a Potential Game
We revisit the El Farol bar problem developed by Brian W. Arthur (1994) to investigate how one might best model bounded rationality in economics. We begin by modelling the El Farol bar problem as a market entry game and describing its Nash equilibria. Then, assuming agents are boundedly rational in accordance with a reinforcement learning model, we analyse long-run behaviour in the repeated game. We then state our main result. In a single population of individuals playing the El Farol game, learning theory predicts that the population is eventually subdivided into two distinct groups: those who invariably go to the bar and those who almost never do. In doing so we demonstrate that learning theory predicts sorting in the El Farol bar problem.
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