In quasi-proportional auctions, each bidder receives a fraction of the allocation equal to the weight of their bid divided by the sum of weights of all bids, where each bids weight is determined by a weight function. We study the relationship between
the weight function, bidders private values, number of bidders, and the sellers revenue in equilibrium. It has been shown that if one bidder has a much higher private value than the others, then a nearly flat weight function maximizes revenue. Essentially, threatening the bidder who has the highest valuation with having to share the allocation maximizes the revenue. We show that as bidder private values approach parity, steeper weight functions maximize revenue by making the quasi-proportional auction more like a winner-take-all auction. We also show that steeper weight functions maximize revenue as the number of bidders increases. For flatter weight functions, there is known to be a unique pure-strategy Nash equilibrium. We show that a pure-strategy Nash equilibrium also exists for steeper weight functions, and we give lower bounds for bids at an equilibrium. For a special case that includes the two-bidder auction, we show that the pure-strategy Nash equilibrium is unique, and we show how to compute the revenue at equilibrium. We also show that selecting a weight function based on private value ratios and number of bidders is necessary for a quasi-proportional auction to produce more revenue than a second-price auction.
A stochastic combinatorial semi-bandit is an online learning problem where at each step a learning agent chooses a subset of ground items subject to combinatorial constraints, and then observes stochastic weights of these items and receives their sum
as a payoff. In this paper, we consider efficient learning in large-scale combinatorial semi-bandits with linear generalization, and as a solution, propose two learning algorithms called Combinatorial Linear Thompson Sampling (CombLinTS) and Combinatorial Linear UCB (CombLinUCB). Both algorithms are computationally efficient as long as the offline version of the combinatorial problem can be solved efficiently. We establish that CombLinTS and CombLinUCB are also provably statistically efficient under reasonable assumptions, by developing regret bounds that are independent of the problem scale (number of items) and sublinear in time. We also evaluate CombLinTS on a variety of problems with thousands of items. Our experiment results demonstrate that CombLinTS is scalable, robust to the choice of algorithm parameters, and significantly outperforms the best of our baselines.
Upcoming large imaging surveys will allow detailed studies of the structure and morphology of galaxies aimed at addressing how galaxies form and evolve. Computational approaches are needed to characterize their morphologies over large samples. We int
roduce an automatic method to quantify the outer structure of galaxies. The key to our approach is the division of a galaxy image into two sections delineated by the isophote which encloses half the total brightness of the galaxy. We call the central section the inner half-flux region (IHR) and the outer section the outer half-flux region (OHR). From this division, we derive two parameters: $A_{rm o}$, which measures the asymmetry of the OHR, and $D_{rm o}$, which measures the deviation of the intensity weighted centroid of the OHR from that of the IHR relative to the effective radius. We derive the two parameters from $HST$/ACS $z_{850}$-band images for a sample of 764 galaxies with $z_{850}<22$ mag and $0.35<z<0.9$ selected from GEMS and GOODS-South surveys. We show that the sample galaxies having strong asymmetric structures, in particular tidal tails, are well-separated from those with regular morphologies in the $A_{rm o}$-$D_{rm o}$ space. Meanwhile, the widely used $CAS$ and Gini-$M_{20}$ methods turn out to be insensitive to such morphological features. We stress that the $A_{rm o}$-$D_{rm o}$ method is an efficient way to select galaxies with significant asymmetric features like tidal tails and study galaxy mergers in the dynamical phase traced by these delicate features.
