62 research outputs found
Truthful approximations to range voting
We consider the fundamental mechanism design problem of approximate social
welfare maximization under general cardinal preferences on a finite number of
alternatives and without money. The well-known range voting scheme can be
thought of as a non-truthful mechanism for exact social welfare maximization in
this setting. With m being the number of alternatives, we exhibit a randomized
truthful-in-expectation ordinal mechanism implementing an outcome whose
expected social welfare is at least an Omega(m^{-3/4}) fraction of the social
welfare of the socially optimal alternative. On the other hand, we show that
for sufficiently many agents and any truthful-in-expectation ordinal mechanism,
there is a valuation profile where the mechanism achieves at most an
O(m^{-{2/3}) fraction of the optimal social welfare in expectation. We get
tighter bounds for the natural special case of m = 3, and in that case
furthermore obtain separation results concerning the approximation ratios
achievable by natural restricted classes of truthful-in-expectation mechanisms.
In particular, we show that for m = 3 and a sufficiently large number of
agents, the best mechanism that is ordinal as well as mixed-unilateral has an
approximation ratio between 0.610 and 0.611, the best ordinal mechanism has an
approximation ratio between 0.616 and 0.641, while the best mixed-unilateral
mechanism has an approximation ratio bigger than 0.660. In particular, the best
mixed-unilateral non-ordinal (i.e., cardinal) mechanism strictly outperforms
all ordinal ones, even the non-mixed-unilateral ordinal ones
Consensus-Halving: Does It Ever Get Easier?
In the -Consensus-Halving problem, a fundamental problem in fair division, there are agents with valuations over the interval , and the goal is to divide the interval into pieces and assign a label "" or "" to each piece, such that every agent values the total amount of "" and the total amount of "" almost equally. The problem was recently proven by Filos-Ratsikas and Goldberg [2019] to be the first "natural" complete problem for the computational class PPA, answering a decade-old open question. In this paper, we examine the extent to which the problem becomes easy to solve, if one restricts the class of valuation functions. To this end, we provide the following contributions. First, we obtain a strengthening of the PPA-hardness result of [Filos-Ratsikas and Goldberg, 2019], to the case when agents have piecewise uniform valuations with only two blocks. We obtain this result via a new reduction, which is in fact conceptually much simpler than the corresponding one in [Filos-Ratsikas and Goldberg, 2019]. Then, we consider the case of single-block (uniform) valuations and provide a parameterized polynomial time algorithm for solving -Consensus-Halving for any , as well as a polynomial-time algorithm for ; these are the first algorithmic results for the problem. Finally, an important application of our new techniques is the first hardness result for a generalization of Consensus-Halving, the Consensus--Division problem. In particular, we prove that -Consensus--Division is PPAD-hard
Facility location with double-peaked preference
We study the problem of locating a single facility on a real line based on
the reports of self-interested agents, when agents have double-peaked
preferences, with the peaks being on opposite sides of their locations. We
observe that double-peaked preferences capture real-life scenarios and thus
complement the well-studied notion of single-peaked preferences. We mainly
focus on the case where peaks are equidistant from the agents' locations and
discuss how our results extend to more general settings. We show that most of
the results for single-peaked preferences do not directly apply to this
setting; this makes the problem essentially more challenging. As our main
contribution, we present a simple truthful-in-expectation mechanism that
achieves an approximation ratio of 1+b/c for both the social and the maximum
cost, where b is the distance of the agent from the peak and c is the minimum
cost of an agent. For the latter case, we provide a 3/2 lower bound on the
approximation ratio of any truthful-in-expectation mechanism. We also study
deterministic mechanisms under some natural conditions, proving lower bounds
and approximation guarantees. We prove that among a large class of reasonable
mechanisms, there is no deterministic mechanism that outperforms our
truthful-in-expectation mechanism
Heterogeneous Facility Location with Limited Resources
We initiate the study of the heterogeneous facility location problem with limited resources. We mainly focus on the fundamental case where a set of agents are positioned in the line segment [0,1] and have approval preferences over two available facilities. A mechanism takes as input the positions and the preferences of the agents, and chooses to locate a single facility based on this information. We study mechanisms that aim to maximize the social welfare (the total utility the agents derive from facilities they approve), under the constraint of incentivizing the agents to truthfully report their positions and preferences. We consider three different settings depending on the level of agent-related information that is public or private. For each setting, we design deterministic and randomized strategyproof mechanisms that achieve a good approximation of the optimal social welfare, and complement these with nearly-tight impossibility results
Two's Company, Three's a Crowd:Consensus-Halving for a Constant Number of Agents
We consider the -Consensus-Halving problem, in which a set of
heterogeneous agents aim at dividing a continuous resource into two (not
necessarily contiguous) portions that all of them simultaneously consider to be
of approximately the same value (up to ). This problem was
recently shown to be PPA-complete, for agents and cuts, even for very
simple valuation functions. In a quest to understand the root of the complexity
of the problem, we consider the setting where there is only a constant number
of agents, and we consider both the computational complexity and the query
complexity of the problem.
For agents with monotone valuation functions, we show a dichotomy: for two
agents the problem is polynomial-time solvable, whereas for three or more
agents it becomes PPA-complete. Similarly, we show that for two monotone agents
the problem can be solved with polynomially-many queries, whereas for three or
more agents, we provide exponential query complexity lower bounds. These
results are enabled via an interesting connection to a monotone Borsuk-Ulam
problem, which may be of independent interest. For agents with general
valuations, we show that the problem is PPA-complete and admits exponential
query complexity lower bounds, even for two agents
Achieving Diverse Objectives with AI-driven Prices in Deep Reinforcement Learning Multi-agent Markets
We propose a practical approach to computing market prices and allocations
via a deep reinforcement learning policymaker agent, operating in an
environment of other learning agents. Compared to the idealized market
equilibrium outcome -- which we use as a benchmark -- our policymaker is much
more flexible, allowing us to tune the prices with regard to diverse objectives
such as sustainability and resource wastefulness, fairness, buyers' and
sellers' welfare, etc. To evaluate our approach, we design a realistic market
with multiple and diverse buyers and sellers. Additionally, the sellers, which
are deep learning agents themselves, compete for resources in a common-pool
appropriation environment based on bio-economic models of commercial fisheries.
We demonstrate that: (a) The introduced policymaker is able to achieve
comparable performance to the market equilibrium, showcasing the potential of
such approaches in markets where the equilibrium prices can not be efficiently
computed. (b) Our policymaker can notably outperform the equilibrium solution
on certain metrics, while at the same time maintaining comparable performance
for the remaining ones. (c) As a highlight of our findings, our policymaker is
significantly more successful in maintaining resource sustainability, compared
to the market outcome, in scarce resource environments
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