THE WAIT-AND-SEE OPTION IN ASCENDING PRICE AUCTIONS

Cake-cutting protocols aim at dividing a ``cake'' (i.e., a divisible resource) and assigning the resulting portions to several players in a way that each of the players feels to have received a ``fair'' amount of the cake. An important notion of fairness is envy-freeness: No player wishes to switch the portion of the cake received with another player's portion. Despite intense efforts in the past, it is still an open question whether there is a \emph{finite bounded} envy-free cake-cutting protocol for an arbitrary number of players, and even for four players. We introduce the notion of degree of guaranteed envy-freeness (DGEF) as a measure of how good a cake-cutting protocol can approximate the ideal of envy-freeness while keeping the protocol finite bounded (trading being disregarded). We propose a new finite bounded proportional protocol for any number n \geq 3 of players, and show that this protocol has a DGEF of 1 + \lceil (n^2)/2 \rceil. This is the currently best DGEF among known finite bounded cake-cutting protocols for an arbitrary number of players. We will make the case that improving the DGEF even further is a tough challenge, and determine, for comparison, the DGEF of selected known finite bounded cake-cutting protocols.Comment: 37 pages, 4 figure

A Cryptographic Moving-Knife Cake-Cutting Protocol

This paper proposes a cake-cutting protocol using cryptography when the cake is a heterogeneous good that is represented by an interval on a real line. Although the Dubins-Spanier moving-knife protocol with one knife achieves simple fairness, all players must execute the protocol synchronously. Thus, the protocol cannot be executed on asynchronous networks such as the Internet. We show that the moving-knife protocol can be executed asynchronously by a discrete protocol using a secure auction protocol. The number of cuts is n-1 where n is the number of players, which is the minimum.Comment: In Proceedings IWIGP 2012, arXiv:1202.422

Self-assembly of diblock molecular polymer brushes in the spherical confinement of nanoemulsion droplets

Understanding the self-assembly behavior of polymers of various topologies is key to a reliable design of functional polymer materials. Self-assembly under confinement conditions emerges as a versatile avenue to design polymer particles with complex internal morphologies while simultaneously facilitating scale-up. However, only linear block copolymers have been studied to date, despite the increasing control over macromolecule composition and architecture available. This study extends the investigation of polymer self-assembly in confinement from regular diblock copolymers to diblock molecular polymer brushes (MPBs). Block-type MPBs with polystyrene (PS) and polylactide (PLA) compartments of different sizes are incorporated into surfactant-stabilised oil-in-water (chloroform/water) emulsions. The increasing confinement in the nanoemulsion droplets during solvent evaporation directs the MPBs to form solid nano/microparticles. Microscopy studies reveal an intricate internal particle structure, including interpenetrating networks and axially-stacked lamellae of PS and PLA, depending on the PS/PLA ratio of the brushes.Australian Research Council. Grant Number: DE180100007 endowed professorship. Grant Number: 2016‐2022 German Research Foundation (DFG). Grant Numbers: 2017‐2022, 37692067

Cutting the same fraction of several measures

We study some measure partition problems: Cut the same positive fraction of $d+1$ measures in $\mathbb R^d$ with a hyperplane or find a convex subset of $\mathbb R^d$ on which $d+1$ given measures have the same prescribed value. For both problems positive answers are given under some additional assumptions.Comment: 7 pages 2 figure

Solving kk-means on High-dimensional Big Data

In recent years, there have been major efforts to develop data stream algorithms that process inputs in one pass over the data with little memory requirement. For the $k$-means problem, this has led to the development of several $(1+\varepsilon)$-approximations (under the assumption that $k$ is a constant), but also to the design of algorithms that are extremely fast in practice and compute solutions of high accuracy. However, when not only the length of the stream is high but also the dimensionality of the input points, then current methods reach their limits. We propose two algorithms, piecy and piecy-mr that are based on the recently developed data stream algorithm BICO that can process high dimensional data in one pass and output a solution of high quality. While piecy is suited for high dimensional data with a medium number of points, piecy-mr is meant for high dimensional data that comes in a very long stream. We provide an extensive experimental study to evaluate piecy and piecy-mr that shows the strength of the new algorithms.Comment: 23 pages, 9 figures, published at the 14th International Symposium on Experimental Algorithms - SEA 201

Confinement Assembly of ABC Triblock Terpolymers for the High-Yield Synthesis of Janus Nanorings

Block copolymers are versatile building blocks for the self-assembly of functional nanostructures in bulk and solution. While spheres, cylinders, and bilayer sheets are thermodynamically preferred shapes and frequently observed, ring-shaped nanoparticles are more challenging to realize due to energetic penalties that originate from their anisotropic curvature. Today, a handful of concepts exist that produce core-shell nanorings, while more complex (e.g. patchy) nanorings are currently out of reach and have only been predicted theoretically. Here, we demonstrate that confinement assembly of properly designed ABC triblock terpolymers is a general route to synthesize Janus nanorings in high purity. The triblock terpolymer self-assembles in the spherical confinement of nanoemulsion droplets into prolate ellipsoidal microparticles with an axially-stacked lamellar-ring (lr)-morphology. We clarified and visualized this complex, yet well-ordered, morphology with transmission electron tomography (ET). Blocks A and C formed stacks of lamellae with the B microdomain sandwiched in-between as nanorings. Cross-linking of the B-rings allows disassembly of the microparticles into Janus nanorings (JNRs) carrying two strictly separated polymer brushes of A and C on top and bottom. Decreasing the B volume leads to Janus spheres and rods, while an increase of B results in perforated and filled Janus disks. The confinement assembly of ABC triblock terpolymers is a general process that can be extended to other block chemistries and will allow to synthesize a large variety of complex micro- and nanoparticles that inspire studies in self-assembly, interfacial stabilization, colloidal packing, and nanomedicine

Multiagent Negotiation for Fair and Unbiased Resource Allocation

This paper proposes a novel solution for the n agent cake cutting (resource allocation) problem. We propose a negotiation protocol for dividing a resource among n agents and then provide an algorithm for allotting portions of the resource. We prove that this protocol can enable distribution of the resource among n agents in a fair manner. The protocol enables agents to choose portions based on their internal utility function, which they do not have to reveal. In addition to being fair, the protocol has desirable features such as being unbiased and verifiable while allocating resources. In the case where the resource is two-dimensional (a circular cake) and uniform, it is shown that each agent can get close to l/n of the whole resource