Derandomization of auctions

We study the role of randomization in seller optimal (i.e., profit maximization) auctions. Bayesian optimal auctions (e.g., Myerson, 1981) assume that the valuations of the agents are random draws from a distribution and prior-free optimal auctions either are randomized (e.g., Goldberg et al., 2006) or assume the valuations are randomized (e.g., Segal, 2003). Is randomization fundamental to profit maximization in auctions? Our main result is a general approach to derandomize single-item multi-unit unit-demand auctions while approximately preserving their performance (i.e., revenue). Our general technique is constructive but not computationally tractable. We complement the general result with the explicit and computationally-simple derandomization of a particular auction. Our results are obtained through analogy to hat puzzles that are interesting in their own right

Computing optimal strategies for a cooperative hat game

We consider a `hat problem' in which each player has a randomly placed stack of black and white hats on their heads, visible to the other player, but not the wearer. Each player must guess a hat position on their head with the goal of both players guessing a white hat. We address the question of finding the optimal strategy, i.e., the one with the highest probability of winning, for this game. We provide an overview of prior work on this question, and describe several strategies that give the best known lower bound on the probability of winning. Upper bounds are also considered here

Optimal Competitive Auctions

We study the design of truthful auctions for selling identical items in unlimited supply (e.g., digital goods) to n unit demand buyers. This classic problem stands out from profit-maximizing auction design literature as it requires no probabilistic assumptions on buyers' valuations and employs the framework of competitive analysis. Our objective is to optimize the worst-case performance of an auction, measured by the ratio between a given benchmark and revenue generated by the auction. We establish a sufficient and necessary condition that characterizes competitive ratios for all monotone benchmarks. The characterization identifies the worst-case distribution of instances and reveals intrinsic relations between competitive ratios and benchmarks in the competitive analysis. With the characterization at hand, we show optimal competitive auctions for two natural benchmarks. The most well-studied benchmark $\mathcal{F}^{(2)}(\cdot)$ measures the envy-free optimal revenue where at least two buyers win. Goldberg et al. [13] showed a sequence of lower bounds on the competitive ratio for each number of buyers n. They conjectured that all these bounds are tight. We show that optimal competitive auctions match these bounds. Thus, we confirm the conjecture and settle a central open problem in the design of digital goods auctions. As one more application we examine another economically meaningful benchmark, which measures the optimal revenue across all limited-supply Vickrey auctions. We identify the optimal competitive ratios to be $(\frac{n}{n-1})^{n-1}-1$ for each number of buyers n, that is $e-1$ as $n$ approaches infinity

A tree formulation for signaling games

The paper has as a starting point the work of the philosopher Professor D. Lewis. We provide a detailed presentation and complete analysis of the sender/receiver Lewis signaling game using a game theory extensive form, decision tree formulation. It is shown that there are a number of Bayesian equilibria. We explain which equilibrium is the most likely to prevail. Our explanation provides an essential step for understanding the formation of a language convention. The informational content of signals is discussed and it is shown that a correct action is not always the result of a truthful signal. We allow for this to be reflected in the payoff of the sender. Further, concepts and approaches from neighbouring disciplines, notably economics, suggest themselves immediately for interpreting the results of our analysis (rational expectations, self-fulfilling prophesies)