Given two functions f,g : {0,1}n → {0,1}, a mapping ψ : {0,1}n → {0,1}n is said to be a mapping from f to g if it is a bijection and f(z) = g(ψ(z)) for every z ∈ {0,1}n. In this paper we study Lipschitz mappings between Boolean functions.Our first result gives a construction of a C-Lipschitz mapping from the Majority function to the Dictator function for some universal constant C. On the other hand, there is no n/2-Lipschitz mapping in the other direction, namely from the Dictator function to the Majority function. This answers an open problem posed by Daniel Varga in the paper of Benjamini, Cohen and Shinkar (FOCS 2014 [1]).We also show a mapping from Dictator to XOR that is 3-local, 2-Lipschitz, and its inverse is O(log(n))-Lipschitz, where by L-local mapping we mean that each output bit of the mapping depends on at most L input bits.Next, we consider the problem of finding functions such that any mapping between them must have large average stretch, where the average stretch of a mapping φ is defined as $${\sf avgstretch}(\phi) = {\mathbb E}_{x,i}[{\sf dist}(\phi(x),\phi(x+e_i)].$$ We show that any mapping φ from XOR to Majority must satisfy avgStretch(φ) ≥ c$\sqrt{n}$ for some absolute constant c > 0. In some sense, this gives a ‘function analogue’ to the question of Benjamini, Cohen and Shinkar (FOCS 2014 [1]), who asked whether there exists a set A ⊆ {0,1}n of density 0.5 such that any bijection from {0,1}n−1 to A has large average stretch.Finally, we show that for a random balanced function f: {0,1}n → {0,1}n, with high probability there is a mapping φ from Dictator to f such that both φ and φ−1 have constant average stretch. In particular, this implies that one cannot obtain lower bounds on average stretch by taking uniformly random functions.

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Two classes of distributions that are widely used in the analysis of Bayesian auctions are the Monotone Hazard Rate (MHR) and Regular distributions. They can both be characterized in terms of the rate of change of the associated virtual value functions: for MHR distributions the condition is that for valuesIn this paper, we investigate five distinct auction settings for which good expected revenue bounds are known when the bidders' valuations are given by MHR distributions. In every case, we show that these bounds degrade gracefully when extended to α-SR distributions. For four of these settings, the auction mechanism requires knowledge of these distribution(s) (in the other setting, the distributions are needed only to ensure good bounds on the expected revenue). In these cases we also investigate what happens when the distributions are known only approximately via samples, specifically how to modify the mechanisms so that they remain effective and how the expected revenue depends on the number of samples.

We prove deviation bounds for the random variableis a Markov chain with stationary distribution and state space [N ], and f i :Our bound improves upon previously known bounds in that the dependence is on a 2 1 + · · · + a 2 n rather than max i {a i } √ n. We also prove deviation bounds for certain types of sums of vectorvalued random variables obtained from a Markov chain in a similar manner. One application includes bounding the expected value of the Schatten ∞-norm of a random matrix whose entries are obtained from a Markov chain.