6 research outputs found
Minority Becomes Majority in Social Networks
It is often observed that agents tend to imitate the behavior of their
neighbors in a social network. This imitating behavior might lead to the
strategic decision of adopting a public behavior that differs from what the
agent believes is the right one and this can subvert the behavior of the
population as a whole.
In this paper, we consider the case in which agents express preferences over
two alternatives and model social pressure with the majority dynamics: at each
step an agent is selected and its preference is replaced by the majority of the
preferences of her neighbors. In case of a tie, the agent does not change her
current preference. A profile of the agents' preferences is stable if the
preference of each agent coincides with the preference of at least half of the
neighbors (thus, the system is in equilibrium).
We ask whether there are network topologies that are robust to social
pressure. That is, we ask if there are graphs in which the majority of
preferences in an initial profile always coincides with the majority of the
preference in all stable profiles reachable from that profile. We completely
characterize the graphs with this robustness property by showing that this is
possible only if the graph has no edge or is a clique or very close to a
clique. In other words, except for this handful of graphs, every graph admits
at least one initial profile of preferences in which the majority dynamics can
subvert the initial majority. We also show that deciding whether a graph admits
a minority that becomes majority is NP-hard when the minority size is at most
1/4-th of the social network size.Comment: To appear in WINE 201
Maximum Nash Welfare and Other Stories About EFX
We consider the classic problem of fairly allocating indivisible goods among agents with additive valuation functions and explore the connection between two prominent fairness notions: maximum Nash welfare (MNW) and envy-freeness up to any good (EFX). We establish that an MNW allocation is always EFX as long as there are at most two possible values for the goods, whereas this implication is no longer true for three or more distinct values. As a notable consequence, this proves the existence of EFX allocations for these restricted valuation functions. While the efficient computation of an MNW allocation for two possible values remains an open problem, we present a novel algorithm for directly constructing EFX allocations in this setting. Finally, we study the question of whether an MNW allocation implies any EFX guarantee for general additive valuation functions under a natural new interpretation of approximate EFX allocations
Stable Fractional Matchings
We study a generalization of the classical stable matching problem that allows for cardinal preferences (as opposed to ordinal) and fractional matchings (as opposed to integral). After observing that, in this cardinal setting, stable fractional matchings can have much higher social welfare than stable integral ones, our goal is to understand the computational complexity of finding an optimal (i.e., welfare-maximizing) or nearly-optimal stable fractional matching. We present simple approximation algorithms for this problem with weak welfare guarantees and, rather unexpectedly, we furthermore show that achieving better approximations is hard. This computational hardness persists even for approximate stability. To the best of our knowledge, these are the first computational complexity results for stable fractional matchings. En route to these results, we provide a number of structural observations