209 research outputs found

### On Nash Dynamics of Matching Market Equilibria

In this paper, we study the Nash dynamics of strategic interplays of n buyers
in a matching market setup by a seller, the market maker. Taking the standard
market equilibrium approach, upon receiving submitted bid vectors from the
buyers, the market maker will decide on a price vector to clear the market in
such a way that each buyer is allocated an item for which he desires the most
(a.k.a., a market equilibrium solution). While such equilibrium outcomes are
not unique, the market maker chooses one (maxeq) that optimizes its own
objective --- revenue maximization. The buyers in turn change bids to their
best interests in order to obtain higher utilities in the next round's market
equilibrium solution.
This is an (n+1)-person game where buyers place strategic bids to gain the
most from the market maker's equilibrium mechanism. The incentives of buyers in
deciding their bids and the market maker's choice of using the maxeq mechanism
create a wave of Nash dynamics involved in the market. We characterize Nash
equilibria in the dynamics in terms of the relationship between maxeq and mineq
(i.e., minimum revenue equilibrium), and develop convergence results for Nash
dynamics from the maxeq policy to a mineq solution, resulting an outcome
equivalent to the truthful VCG mechanism.
Our results imply revenue equivalence between maxeq and mineq, and address
the question that why short-term revenue maximization is a poor long run
strategy, in a deterministic and dynamic setting

### $f(R)$ gravity theories in the Palatini Formalism constrained from strong lensing

$f(R)$ gravity, capable of driving the late-time acceleration of the
universe, is emerging as a promising alternative to dark energy. Various $f(R)$
gravity models have been intensively tested against probes of the expansion
history, including type Ia supernovae (SNIa), the cosmic microwave background
(CMB) and baryon acoustic oscillations (BAO). In this paper we propose to use
the statistical lens sample from Sloan Digital Sky Survey Quasar Lens Search
Data Release 3 (SQLS DR3) to constrain $f(R)$ gravity models. This sample can
probe the expansion history up to $z\sim2.2$, higher than what probed by
current SNIa and BAO data. We adopt a typical parameterization of the form
$f(R)=R-\alpha H^2_0(-\frac{R}{H^2_0})^\beta$ with $\alpha$ and $\beta$
constants. For $\beta=0$ ($\Lambda$CDM), we obtain the best-fit value of the
parameter $\alpha=-4.193$, for which the 95% confidence interval that is
[-4.633, -3.754]. This best-fit value of $\alpha$ corresponds to the matter
density parameter $\Omega_{m0}=0.301$, consistent with constraints from other
probes. Allowing $\beta$ to be free, the best-fit parameters are $(\alpha,
\beta)=(-3.777, 0.06195)$. Consequently, we give $\Omega_{m0}=0.285$ and the
deceleration parameter $q_0=-0.544$. At the 95% confidence level, $\alpha$ and
$\beta$ are constrained to [-4.67, -2.89] and [-0.078, 0.202] respectively.
Clearly, given the currently limited sample size, we can only constrain $\beta$
within the accuracy of $\Delta\beta\sim 0.1$ and thus can not distinguish
between $\Lambda$CDM and $f(R)$ gravity with high significance, and actually,
the former lies in the 68% confidence contour. We expect that the extension of
the SQLS DR3 lens sample to the SDSS DR5 and SDSS-II will make constraints on
the model more stringent.Comment: 10 pages, 7 figures. Accepted for publication in MNRA

### On Revenue Maximization with Sharp Multi-Unit Demands

We consider markets consisting of a set of indivisible items, and buyers that
have {\em sharp} multi-unit demand. This means that each buyer $i$ wants a
specific number $d_i$ of items; a bundle of size less than $d_i$ has no value,
while a bundle of size greater than $d_i$ is worth no more than the most valued
$d_i$ items (valuations being additive). We consider the objective of setting
prices and allocations in order to maximize the total revenue of the market
maker. The pricing problem with sharp multi-unit demand buyers has a number of
properties that the unit-demand model does not possess, and is an important
question in algorithmic pricing. We consider the problem of computing a revenue
maximizing solution for two solution concepts: competitive equilibrium and
envy-free pricing.
For unrestricted valuations, these problems are NP-complete; we focus on a
realistic special case of "correlated values" where each buyer $i$ has a
valuation v_i\qual_j for item $j$, where $v_i$ and \qual_j are positive
quantities associated with buyer $i$ and item $j$ respectively. We present a
polynomial time algorithm to solve the revenue-maximizing competitive
equilibrium problem. For envy-free pricing, if the demand of each buyer is
bounded by a constant, a revenue maximizing solution can be found efficiently;
the general demand case is shown to be NP-hard.Comment: page2

### Pricing Ad Slots with Consecutive Multi-unit Demand

We consider the optimal pricing problem for a model of the rich media
advertisement market, as well as other related applications. In this market,
there are multiple buyers (advertisers), and items (slots) that are arranged in
a line such as a banner on a website. Each buyer desires a particular number of
{\em consecutive} slots and has a per-unit-quality value $v_i$ (dependent on
the ad only) while each slot $j$ has a quality $q_j$ (dependent on the position
only such as click-through rate in position auctions). Hence, the valuation of
the buyer $i$ for item $j$ is $v_iq_j$. We want to decide the allocations and
the prices in order to maximize the total revenue of the market maker.
A key difference from the traditional position auction is the advertiser's
requirement of a fixed number of consecutive slots. Consecutive slots may be
needed for a large size rich media ad. We study three major pricing mechanisms,
the Bayesian pricing model, the maximum revenue market equilibrium model and an
envy-free solution model. Under the Bayesian model, we design a polynomial time
computable truthful mechanism which is optimum in revenue. For the market
equilibrium paradigm, we find a polynomial time algorithm to obtain the maximum
revenue market equilibrium solution. In envy-free settings, an optimal solution
is presented when the buyers have the same demand for the number of consecutive
slots. We conduct a simulation that compares the revenues from the above
schemes and gives convincing results.Comment: 27page

### 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

### Smoothed and Average-Case Approximation Ratios of Mechanisms: Beyond the Worst-Case Analysis

The approximation ratio has become one of the dominant measures in mechanism design problems. In light of analysis of algorithms, we define the smoothed approximation ratio to compare the performance of the optimal mechanism and a truthful mechanism when the inputs are subject to random perturbations of the worst-case inputs, and define the average-case approximation ratio to compare the performance of these two mechanisms when the inputs follow a distribution. For the one-sided matching problem, Filos-Ratsikas et al. [2014] show that, amongst all truthful mechanisms, random priority achieves the tight approximation ratio bound of Theta(sqrt{n}). We prove that, despite of this worst-case bound, random priority has a constant smoothed approximation ratio. This is, to our limited knowledge, the first work that asymptotically differentiates the smoothed approximation ratio from the worst-case approximation ratio for mechanism design problems. For the average-case, we show that our approximation ratio can be improved to 1+e. These results partially explain why random priority has been successfully used in practice, although in the worst case the optimal social welfare is Theta(sqrt{n}) times of what random priority achieves.
These results also pave the way for further studies of smoothed and average-case analysis for approximate mechanism design problems, beyond the worst-case analysis

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