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

Mapping the complexity of higher education in the developing world

This repository item contains a single issue of Issues in Brief, a series of policy briefs that began publishing in 2008 by the Boston University Frederick S. Pardee Center for the Study of the Longer-Range Future.On October 27 and 28, 2009, a workshop of experts on higher education in developing countries was convened by the Boston University Frederick S. Pardee Center for the Study of the Longer-Range Future. The meeting was supported by a grant from the National Academies Keck Futures Initiative with additional support from the Pardee Center and the Office of the Boston University Provost. The meeting brought together experts in economics, public policy, education, development, university management, and quantitative modeling who had rich experiences across the developing world. These experts offered a variety of conceptual tools with which to look at the particular complexities associated with higher education in developing countries. The meeting was convened by the authors of this paper. This policy brief builds upon and reflects on the discussion at this meeting, but is not a meeting report, per se

Towards explaining the speed of kk-means

The $k$-means method is a popular algorithm for clustering, known for its speed in practice. This stands in contrast to its exponential worst-case running-time. To explain the speed of the $k$-means method, a smoothed analysis has been conducted. We sketch this smoothed analysis and a generalization to Bregman divergences

Spectral Sparsification via Bounded-Independence Sampling

We give a deterministic, nearly logarithmic-space algorithm for mild spectral sparsification of undirected graphs. Given a weighted, undirected graph $G$ on $n$ vertices described by a binary string of length $N$, an integer $k\leq \log n$, and an error parameter $\epsilon > 0$, our algorithm runs in space $\tilde{O}(k\log (N\cdot w_{\mathrm{max}}/w_{\mathrm{min}}))$ where $w_{\mathrm{max}}$ and $w_{\mathrm{min}}$ are the maximum and minimum edge weights in $G$, and produces a weighted graph $H$ with $\tilde{O}(n^{1+2/k}/\epsilon^2)$ edges that spectrally approximates $G$, in the sense of Spielmen and Teng [ST04], up to an error of $\epsilon$. Our algorithm is based on a new bounded-independence analysis of Spielman and Srivastava's effective resistance based edge sampling algorithm [SS08] and uses results from recent work on space-bounded Laplacian solvers [MRSV17]. In particular, we demonstrate an inherent tradeoff (via upper and lower bounds) between the amount of (bounded) independence used in the edge sampling algorithm, denoted by $k$ above, and the resulting sparsity that can be achieved.Comment: 37 page

A nearly-mlogn time solver for SDD linear systems

We present an improved algorithm for solving symmetrically diagonally dominant linear systems. On input of an $n\times n$ symmetric diagonally dominant matrix $A$ with $m$ non-zero entries and a vector $b$ such that $A\bar{x} = b$ for some (unknown) vector $\bar{x}$, our algorithm computes a vector $x$ such that $||{x}-\bar{x}||_A < \epsilon ||\bar{x}||_A$ {$||\cdot||_A$ denotes the A-norm} in time $${\tilde O}(m\log n \log (1/\epsilon)).$$ The solver utilizes in a standard way a `preconditioning' chain of progressively sparser graphs. To claim the faster running time we make a two-fold improvement in the algorithm for constructing the chain. The new chain exploits previously unknown properties of the graph sparsification algorithm given in [Koutis,Miller,Peng, FOCS 2010], allowing for stronger preconditioning properties. We also present an algorithm of independent interest that constructs nearly-tight low-stretch spanning trees in time $\tilde{O}(m\log{n})$, a factor of $O(\log{n})$ faster than the algorithm in [Abraham,Bartal,Neiman, FOCS 2008]. This speedup directly reflects on the construction time of the preconditioning chain.Comment: to appear in FOCS1

Enhancement of magnetocaloric effect in the Gd2Al phase by Co alloying

To understand the effect of Co doping on the magnetic entropy changes in Gd2 Al phase, a series of Gd2AlCox alloys with 0 ≤ x ≤ 0.6 were synthesized by arc-melting and the crystal structure was analyzed by XRD. The magnetic properties were investigated, and the entropychanges were calculated for a magnetic field change of 50 kOe. All the as-cast alloys dopedwith Co exhibited greater magnetic entropy changes than the original binary Gd2 Al phase. The main reasons attributed to this are the increase of ferromagnetic interaction indicated by the disappearance of cusp and sharp drop in magnetization and the reduction of the critical field required to trigger the field-induced transition below 50 K in Gd2 Al phase after Co alloying

Control influence on trust and relational governance in the client-contractor dyad

The construction industry has in recent years witnessed a paradigm shift towards the use of more collaborative contracting relationships and integrated processes in an attempt to improve construction project delivery. Trust is central to the success of these contracting approaches and although efforts are usually aimed at improving trust relations in client-contractor relationships, there has so far been mixed findings on how trust is influenced by formal control mechanisms discharged via formal contracts. In construction contracting, there is therefore the need to investigate how different governance modes and control mechanisms deployed on construction projects are perceived by those being controlled and how this in turn influences trust. Through a critique of the extant literature on trust and control in construction, this study reveals that the trust-control relationship which can be both complimentary and supplementary has far reaching implications on the measurement/assessment of trust in the construction project context. The orientation of governance and control mechanisms selected by clients and the behavioural consequences of these from contractors can thus be used as a measure of the degree of trust that exists in the dyad

First-principles study on the effective masses of zinc-blend-derived Cu_2Zn-IV-VI_4 (IV = Sn, Ge, Si and VI = S, Se)

The electron and hole effective masses of kesterite (KS) and stannite (ST) structured Cu_2Zn-IV-VI_4 (IV = Sn, Ge, Si and VI = S, Se) semiconductors are systematically studied using first-principles calculations. We find that the electron effective masses are almost isotropic, while strong anisotropy is observed for the hole effective mass. The electron effective masses are typically much smaller than the hole effective masses for all studied compounds. The ordering of the topmost three valence bands and the corresponding hole effective masses of the KS and ST structures are different due to the different sign of the crystal-field splitting. The electron and hole effective masses of Se-based compounds are significantly smaller compared to the corresponding S-based compounds. They also decrease as the atomic number of the group IV elements (Si, Ge, Sn) increases, but the decrease is less notable than that caused by the substitution of S by Se.Comment: 14 pages, 6 figures, 2 table