16 research outputs found
Equilibria in Sequential Allocation
Sequential allocation is a simple mechanism for sharing multiple indivisible
items. We study strategic behavior in sequential allocation. In particular, we
consider Nash dynamics, as well as the computation and Pareto optimality of
pure equilibria, and Stackelberg strategies. We first demonstrate that, even
for two agents, better responses can cycle. We then present a linear-time
algorithm that returns a profile (which we call the "bluff profile") that is in
pure Nash equilibrium. Interestingly, the outcome of the bluff profile is the
same as that of the truthful profile and the profile is in pure Nash
equilibrium for \emph{all} cardinal utilities consistent with the ordinal
preferences. We show that the outcome of the bluff profile is Pareto optimal
with respect to pairwise comparisons. In contrast, we show that an assignment
may not be Pareto optimal with respect to pairwise comparisons even if it is a
result of a preference profile that is in pure Nash equilibrium for all
utilities consistent with ordinal preferences. Finally, we present a dynamic
program to compute an optimal Stackelberg strategy for two agents, where the
second agent has a constant number of distinct values for the items
Complexity of Manipulating Sequential Allocation
Sequential allocation is a simple allocation mechanism in which agents are
given pre-specified turns and each agents gets the most preferred item that is
still available. It has long been known that sequential allocation is not
strategyproof.
Bouveret and Lang (2014) presented a polynomial-time algorithm to compute a
best response of an agent with respect to additively separable utilities and
claimed that (1) their algorithm correctly finds a best response, and (2) each
best response results in the same allocation for the manipulator. We show that
both claims are false via an example. We then show that in fact the problem of
computing a best response is NP-complete. On the other hand, the insights and
results of Bouveret and Lang (2014) for the case of two agents still hold
Fair Allocation based on Diminishing Differences
Ranking alternatives is a natural way for humans to explain their
preferences. It is being used in many settings, such as school choice, course
allocations and residency matches. In some cases, several `items' are given to
each participant. Without having any information on the underlying cardinal
utilities, arguing about fairness of allocation requires extending the ordinal
item ranking to ordinal bundle ranking. The most commonly used such extension
is stochastic dominance (SD), where a bundle X is preferred over a bundle Y if
its score is better according to all additive score functions. SD is a very
conservative extension, by which few allocations are necessarily fair while
many allocations are possibly fair. We propose to make a natural assumption on
the underlying cardinal utilities of the players, namely that the difference
between two items at the top is larger than the difference between two items at
the bottom. This assumption implies a preference extension which we call
diminishing differences (DD), where X is preferred over Y if its score is
better according to all additive score functions satisfying the DD assumption.
We give a full characterization of allocations that are
necessarily-proportional or possibly-proportional according to this assumption.
Based on this characterization, we present a polynomial-time algorithm for
finding a necessarily-DD-proportional allocation if it exists. Using
simulations, we show that with high probability, a necessarily-proportional
allocation does not exist but a necessarily-DD-proportional allocation exists,
and moreover, that allocation is proportional according to the underlying
cardinal utilities. We also consider chore allocation under the analogous
condition --- increasing-differences.Comment: Revised version, based on very helpful suggestions of JAIR referees.
Gaps in some proofs were filled, more experiments were done, and mor
Algorithms for Manipulating Sequential Allocation
Sequential allocation is a simple and widely studied mechanism to allocate
indivisible items in turns to agents according to a pre-specified picking
sequence of agents. At each turn, the current agent in the picking sequence
picks its most preferred item among all items having not been allocated yet.
This problem is well-known to be not strategyproof, i.e., an agent may get more
utility by reporting an untruthful preference ranking of items. It arises the
problem: how to find the best response of an agent?
It is known that this problem is polynomially solvable for only two agents
and NP-complete for arbitrary number of agents.
The computational complexity of this problem with three agents was left as an
open problem. In this paper, we give a novel algorithm that solves the problem
in polynomial time for each fixed number of agents. We also show that an agent
can always get at least half of its optimal utility by simply using its
truthful preference as the response