149 research outputs found
Approximate well-supported Nash equilibria in symmetric bimatrix games
The -well-supported Nash equilibrium is a strong notion of
approximation of a Nash equilibrium, where no player has an incentive greater
than to deviate from any of the pure strategies that she uses in
her mixed strategy. The smallest constant currently known for
which there is a polynomial-time algorithm that computes an
-well-supported Nash equilibrium in bimatrix games is slightly
below . In this paper we study this problem for symmetric bimatrix games
and we provide a polynomial-time algorithm that gives a
-well-supported Nash equilibrium, for an arbitrarily small
positive constant
Polylogarithmic Supports are required for Approximate Well-Supported Nash Equilibria below 2/3
In an epsilon-approximate Nash equilibrium, a player can gain at most epsilon
in expectation by unilateral deviation. An epsilon well-supported approximate
Nash equilibrium has the stronger requirement that every pure strategy used
with positive probability must have payoff within epsilon of the best response
payoff. Daskalakis, Mehta and Papadimitriou conjectured that every win-lose
bimatrix game has a 2/3-well-supported Nash equilibrium that uses supports of
cardinality at most three. Indeed, they showed that such an equilibrium will
exist subject to the correctness of a graph-theoretic conjecture. Regardless of
the correctness of this conjecture, we show that the barrier of a 2/3 payoff
guarantee cannot be broken with constant size supports; we construct win-lose
games that require supports of cardinality at least Omega((log n)^(1/3)) in any
epsilon-well supported equilibrium with epsilon < 2/3. The key tool in showing
the validity of the construction is a proof of a bipartite digraph variant of
the well-known Caccetta-Haggkvist conjecture. A probabilistic argument shows
that there exist epsilon-well-supported equilibria with supports of cardinality
O(log n/(epsilon^2)), for any epsilon> 0; thus, the polylogarithmic cardinality
bound presented cannot be greatly improved. We also show that for any delta >
0, there exist win-lose games for which no pair of strategies with support
sizes at most two is a (1-delta)-well-supported Nash equilibrium. In contrast,
every bimatrix game with payoffs in [0,1] has a 1/2-approximate Nash
equilibrium where the supports of the players have cardinality at most two.Comment: Added details on related work (footnote 7 expanded
Efficient parameterization of waverider geometries
This paper summarizes the results of investigations into the development of parametric waverider geometry models, with emphasis on their efficiency, in terms of their ability to cover a large feasible design space with a sufficiently small number of design variables to avoid the “curse of dimensionality.” The work presented here is focused on the parameterization of idealized waverider forebody geometries that provide the baseline shapes upon which more sophisticated and realistic hypersonic aircraft geometries can be built. Three different aspects of rationalizing the decisions behind the parametric geometry models developed using the osculating cones method are considered. Initially, three different approaches to the design method itself are discussed. Each approach provides direct control over different aspects of the geometry for which very specific shapes would be more complex to obtain indirectly, thus enabling the geometry to more efficiently meet any related design constraints. Then, a number of requirements and limitations are investigated that affect the available options for the parametric design-driving curves of the inverse design method. Finally, the performance advantages that open up with increasing flexibility of the design-driving curves in the context of a design optimization study are estimated. This allows one to reduce the risk of overparameterizing the geometry model, while still enabling a variety of meaningful shapes. Although the osculating cones method has mainly been used here, most of the findings also apply to other similar inverse design algorithms
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)
Approximate Well-supported Nash Equilibria below Two-thirds
In an epsilon-Nash equilibrium, a player can gain at most epsilon by changing
his behaviour. Recent work has addressed the question of how best to compute
epsilon-Nash equilibria, and for what values of epsilon a polynomial-time
algorithm exists. An epsilon-well-supported Nash equilibrium (epsilon-WSNE) has
the additional requirement that any strategy that is used with non-zero
probability by a player must have payoff at most epsilon less than the best
response. A recent algorithm of Kontogiannis and Spirakis shows how to compute
a 2/3-WSNE in polynomial time, for bimatrix games. Here we introduce a new
technique that leads to an improvement to the worst-case approximation
guarantee
Approximating Nash Equilibria and Dense Bipartite Subgraphs via an Approximate Version of Carathéodory's Theorem
We present algorithmic applications of an approximate version of Caratheodory's theorem. The theorem states that given a set of vectors X in R^d, for every vector in the convex hull of X there exists an ε-close (under the p-norm distance, for 2 ≤ p < ∞) vector that can be expressed as a convex combination of at most b vectors of X, where the bound b depends on ε and the norm p and is independent of the dimension d. This theorem can be derived by instantiating Maurey's lemma, early references to which can be found in the work of Pisier (1981) and Carl (1985). However, in this paper we present a self-contained proof of this result.
Using this theorem we establish that in a bimatrix game with n x n payoff matrices A, B, if the number of non-zero entries in any column of A+B is at most s then an ε-Nash equilibrium of the game can be computed in time n^O(log s/ε^2}). This, in particular, gives us a polynomial-time approximation scheme for Nash equilibrium in games with fixed column sparsity s. Moreover, for arbitrary bimatrix games---since s can be at most n---the running time of our algorithm matches the best-known upper bound, which was obtained by Lipton, Markakis, and Mehta (2003).
The approximate Carathéodory's theorem also leads to an additive approximation algorithm for the densest k-bipartite subgraph problem. Given a graph with n vertices and maximum degree d, the developed algorithm determines a k x k bipartite subgraph with density within ε (in the additive sense) of the optimal density in time n^O(log d/ε^2)
Observational study of chromospheric heating by acoustic waves
Aims. To investigate the role of acoustic and magneto-acoustic waves in
heating the solar chromosphere, observations in strong chromospheric lines are
analyzed by comparing the deposited acoustic-energy flux with the total
integrated radiative losses.
Methods. Quiet-Sun and weak-plage regions were observed in the Ca II 854.2 nm
and H-alpha lines with the Fast Imaging Solar Spectrograph (FISS) at the 1.6-m
Goode Solar Telescope (GST) on 2019 October 3 and in the H-alpha and H-beta
lines with the echelle spectrograph attached to the Vacuum Tower Telescope
(VTT) on 2018 December 11 and 2019 June 6. The deposited acoustic energy flux
at frequencies up to 20 mHz was derived from Doppler velocities observed in
line centers and wings. Radiative losses were computed by means of a set of
scaled non-LTE 1D hydrostatic semi-empirical models obtained by fitting
synthetic to observed line profiles.
Results. In the middle chromosphere (h = 1000-1400 km), the radiative losses
can be fully balanced by the deposited acoustic energy flux in a quiet-Sun
region. In the upper chromosphere (h > 1400 km), the deposited acoustic flux is
small compared to the radiative losses in quiet as well as in plage regions.
The crucial parameter determining the amount of deposited acoustic flux is the
gas density at a given height.
Conclusions. The acoustic energy flux is efficiently deposited in the middle
chromosphere, where the density of gas is sufficiently high. About 90% of the
available acoustic energy flux in the quiet-Sun region is deposited in these
layers, and thus it is a major contributor to the radiative losses of the
middle chromosphere. In the upper chromosphere, the deposited acoustic flux is
too low, so that other heating mechanisms have to act to balance the radiative
cooling.Comment: 11 pages, 10 figures, 3 table
Recommended from our members
Observational study of chromospheric heating by acoustic waves
Aims. Our aim is to investigate the role of acoustic and magneto-acoustic waves in heating the solar chromosphere. Observations in
strong chromospheric lines are analyzed by comparing the deposited acoustic-energy flux with the total integrated radiative losses.
Methods. Quiet-Sun and weak-plage regions were observed in the Ca ii 854.2 nm and H lines with the Fast Imaging Solar Spectrograph
(FISS) at the 1.6-m Goode Solar Telescope on 2019 October 3 and in the H and H lines with the echelle spectrograph
attached to the Vacuum Tower Telescope on 2018 December 11 and 2019 June 6. The deposited acoustic energy flux at frequencies
up to 20 mHz was derived from Doppler velocities observed in line centers and wings. Radiative losses were computed by means of
a set of scaled non-local thermodynamic equilibrium 1D hydrostatic semi-empirical models obtained by fitting synthetic to observed
line profiles.
Results. In the middle chromosphere (h = 1000–1400 km), the radiative losses can be fully balanced by the deposited acoustic energy
flux in a quiet-Sun region. In the upper chromosphere (h > 1400 km), the deposited acoustic flux is small compared to the radiative
losses in quiet as well as in plage regions. The crucial parameter determining the amount of deposited acoustic flux is the gas density
at a given height.
Conclusions. The acoustic energy flux is e ciently deposited in the middle chromosphere, where the density of gas is su ciently
high. About 90% of the available acoustic energy flux in the quiet-Sun region is deposited in these layers, and thus it is a major
contributor to the radiative losses of the middle chromosphere. In the upper chromosphere, the deposited acoustic flux is too low, so
that other heating mechanisms have to act to balance the radiative cooling
Inapproximability Results for Approximate Nash Equilibria.
We study the problem of finding approximate Nash equilibria that satisfy
certain conditions, such as providing good social welfare. In particular, we
study the problem -NE -SW: find an -approximate
Nash equilibrium (-NE) that is within of the best social
welfare achievable by an -NE. Our main result is that, if the
exponential-time hypothesis (ETH) is true, then solving -NE -SW for an
bimatrix game requires time. Building
on this result, we show similar conditional running time lower bounds on a
number of decision problems for approximate Nash equilibria that do not involve
social welfare, including maximizing or minimizing a certain player's payoff,
or finding approximate equilibria contained in a given pair of supports. We
show quasi-polynomial lower bounds for these problems assuming that ETH holds,
where these lower bounds apply to -Nash equilibria for all . The hardness of these other decision problems has so far only
been studied in the context of exact equilibria.Comment: A short (14-page) version of this paper appeared at WINE 2016.
Compared to that conference version, this new version improves the
conditional lower bounds, which now rely on ETH rather than RETH (Randomized
ETH
- …