Auctions with Heterogeneous Items and Budget Limits

We study individual rational, Pareto optimal, and incentive compatible mechanisms for auctions with heterogeneous items and budget limits. For multi-dimensional valuations we show that there can be no deterministic mechanism with these properties for divisible items. We use this to show that there can also be no randomized mechanism that achieves this for either divisible or indivisible items. For single-dimensional valuations we show that there can be no deterministic mechanism with these properties for indivisible items, but that there is a randomized mechanism that achieves this for either divisible or indivisible items. The impossibility results hold for public budgets, while the mechanism allows private budgets, which is in both cases the harder variant to show. While all positive results are polynomial-time algorithms, all negative results hold independent of complexity considerations

Vickrey Auctions for Irregular Distributions

The classic result of Bulow and Klemperer \cite{BK96} says that in a single-item auction recruiting one more bidder and running the Vickrey auction achieves a higher revenue than the optimal auction's revenue on the original set of bidders, when values are drawn i.i.d. from a regular distribution. We give a version of Bulow and Klemperer's result in settings where bidders' values are drawn from non-i.i.d. irregular distributions. We do this by modeling irregular distributions as some convex combination of regular distributions. The regular distributions that constitute the irregular distribution correspond to different population groups in the bidder population. Drawing a bidder from this collection of population groups is equivalent to drawing from some convex combination of these regular distributions. We show that recruiting one extra bidder from each underlying population group and running the Vickrey auction gives at least half of the optimal auction's revenue on the original set of bidders

Constant Rank Bimatrix Games are PPAD-hard

The rank of a bimatrix game (A,B) is defined as rank(A+B). Computing a Nash equilibrium (NE) of a rank-$0$, i.e., zero-sum game is equivalent to linear programming (von Neumann'28, Dantzig'51). In 2005, Kannan and Theobald gave an FPTAS for constant rank games, and asked if there exists a polynomial time algorithm to compute an exact NE. Adsul et al. (2011) answered this question affirmatively for rank-$1$ games, leaving rank-2 and beyond unresolved. In this paper we show that NE computation in games with rank $\ge 3$, is PPAD-hard, settling a decade long open problem. Interestingly, this is the first instance that a problem with an FPTAS turns out to be PPAD-hard. Our reduction bypasses graphical games and game gadgets, and provides a simpler proof of PPAD-hardness for NE computation in bimatrix games. In addition, we get: * An equivalence between 2D-Linear-FIXP and PPAD, improving a result by Etessami and Yannakakis (2007) on equivalence between Linear-FIXP and PPAD. * NE computation in a bimatrix game with convex set of Nash equilibria is as hard as solving a simple stochastic game. * Computing a symmetric NE of a symmetric bimatrix game with rank $\ge 6$ is PPAD-hard. * Computing a (1/poly(n))-approximate fixed-point of a (Linear-FIXP) piecewise-linear function is PPAD-hard. The status of rank-$2$ games remains unresolved

Budget Feasible Mechanisms for Experimental Design

In the classical experimental design setting, an experimenter E has access to a population of $n$ potential experiment subjects $i\in \{1,...,n\}$, each associated with a vector of features $x_i\in R^d$. Conducting an experiment with subject $i$ reveals an unknown value $y_i\in R$ to E. E typically assumes some hypothetical relationship between $x_i$'s and $y_i$'s, e.g., $y_i \approx \beta x_i$, and estimates $\beta$ from experiments, e.g., through linear regression. As a proxy for various practical constraints, E may select only a subset of subjects on which to conduct the experiment. We initiate the study of budgeted mechanisms for experimental design. In this setting, E has a budget $B$. Each subject $i$ declares an associated cost $c_i >0$ to be part of the experiment, and must be paid at least her cost. In particular, the Experimental Design Problem (EDP) is to find a set $S$ of subjects for the experiment that maximizes V(S) = \log\det(I_d+\sum_{i\in S}x_i\T{x_i}) under the constraint $\sum_{i\in S}c_i\leq B$; our objective function corresponds to the information gain in parameter $\beta$ that is learned through linear regression methods, and is related to the so-called $D$-optimality criterion. Further, the subjects are strategic and may lie about their costs. We present a deterministic, polynomial time, budget feasible mechanism scheme, that is approximately truthful and yields a constant factor approximation to EDP. In particular, for any small $\delta > 0$ and $\epsilon > 0$, we can construct a (12.98, $\epsilon$)-approximate mechanism that is $\delta$-truthful and runs in polynomial time in both $n$ and $\log\log\frac{B}{\epsilon\delta}$. We also establish that no truthful, budget-feasible algorithms is possible within a factor 2 approximation, and show how to generalize our approach to a wide class of learning problems, beyond linear regression

Social Dilemmas and Cooperation in Complex Networks

In this paper we extend the investigation of cooperation in some classical evolutionary games on populations were the network of interactions among individuals is of the scale-free type. We show that the update rule, the payoff computation and, to some extent the timing of the operations, have a marked influence on the transient dynamics and on the amount of cooperation that can be established at equilibrium. We also study the dynamical behavior of the populations and their evolutionary stability.Comment: 12 pages, 7 figures. to appea

Efficiency Guarantees in Auctions with Budgets

In settings where players have a limited access to liquidity, represented in the form of budget constraints, efficiency maximization has proven to be a challenging goal. In particular, the social welfare cannot be approximated by a better factor then the number of players. Therefore, the literature has mainly resorted to Pareto-efficiency as a way to achieve efficiency in such settings. While successful in some important scenarios, in many settings it is known that either exactly one incentive-compatible auction that always outputs a Pareto-efficient solution, or that no truthful mechanism can always guarantee a Pareto-efficient outcome. Traditionally, impossibility results can be avoided by considering approximations. However, Pareto-efficiency is a binary property (is either satisfied or not), which does not allow for approximations. In this paper we propose a new notion of efficiency, called \emph{liquid welfare}. This is the maximum amount of revenue an omniscient seller would be able to extract from a certain instance. We explain the intuition behind this objective function and show that it can be 2-approximated by two different auctions. Moreover, we show that no truthful algorithm can guarantee an approximation factor better than 4/3 with respect to the liquid welfare, and provide a truthful auction that attains this bound in a special case. Importantly, the liquid welfare benchmark also overcomes impossibilities for some settings. While it is impossible to design Pareto-efficient auctions for multi-unit auctions where players have decreasing marginal values, we give a deterministic $O(\log n)$-approximation for the liquid welfare in this setting

Emergence of a measurement basis in atom-photon scattering

The process of quantum measurement has been a long standing source of debate. A measurement is postulated to collapse a wavefunction onto one of the states of a predetermined set - the measurement basis. This basis origin is not specified within quantum mechanics. According to the theory of decohernce, a measurement basis is singled out by the nature of coupling of a quantum system to its environment. Here we show how a measurement basis emerges in the evolution of the electronic spin of a single trapped atomic ion due to spontaneous photon scattering. Using quantum process tomography we visualize the projection of all spin directions, onto this basis, as a photon is scattered. These basis spin states are found to be aligned with the scattered photon propagation direction. In accordance with decohernce theory, they are subjected to a minimal increase in entropy due to the photon scattering, while, orthogonal states become fully mixed and their entropy is maximally increased. Moreover, we show that detection of the scattered photon polarization measures the spin state of the ion, in the emerging basis, with high fidelity. Lastly, we show that while photon scattering entangles all superpositions of pointer states with the scattered photon polarization, the measurement-basis states themselves remain classically correlated with it. Our findings show that photon scattering by atomic spin superpositions fulfils all the requirements from a quantum measurement process

Experimental realization of a quantum game on a one-way quantum computer

We report the first demonstration of a quantum game on an all-optical one-way quantum computer. Following a recent theoretical proposal we implement a quantum version of Prisoner's Dilemma, where the quantum circuit is realized by a 4-qubit box-cluster configuration and the player's local strategies by measurements performed on the physical qubits of the cluster. This demonstration underlines the strength and versatility of the one-way model and we expect that this will trigger further interest in designing quantum protocols and algorithms to be tested in state-of-the-art cluster resources.Comment: 13 pages, 4 figure