CSCE 475/875
Handout 9: Application: A Social Ranking System
September
29, 2011
(Based
on Shoham and Leyton-Brown 2011)
Introduction
We now turn to a specialization of the social choice
problem that has a computational flavor, and in which some interesting progress
can be made. Specifically, consider a setting in which the set of agents is the
same as the set of outcomes—agents
are asked to vote to express their opinions about each other, with the goal of
determining a social ranking.
Such settings have great practical importance. For
example, search engines rank Web pages
by considering hyperlinks from one
page to another to be votes about the importance of the destination pages.
Similarly, online auction sites employ reputation
systems to provide assessments of agents’ trustworthiness based on ratings
from past transactions.
Our setting is characterized by two assumptions.
·
First,
: the set of
agents is the same as the set of outcomes.
·
Second,
agents’ preferences are such that each agent divides the other agents into a
set that he likes equally, and a set that he dislikes equally (or,
equivalently, has no opinion about). Formally, for each 
the outcome set 
(equivalent to 
) is partitioned
into two sets 
and 
, with 
, and with 
.
We call this the ranking
systems setting, and call a
social welfare function in this setting a ranking rule.
Now, consider an example in which Alice votes only for
Bob, Will votes only for Liam, and Liam votes only for Vic. Who should be ranked highest? Three of the
five kids have received votes (Bob, Liam, and Vic); these three should
presumably rank higher than the remaining two. But of the three, Vic is
special: he is the only one whose voter (Liam) himself received a vote. Thus,
intuitively, Vic should receive the highest rank. This intuition is captured by the idea of transitivity.
Definition 9.5.2 (Strong transitivity) Consider a preference profile in which outcome 
receives
at least as many votes as 
, and it is possible to pair up all the voters for with
voters from so
that each voter for is
weakly preferred by the ranking rule to the corresponding voter for . Further
assume that o2 receives more votes than and/or
that there is at least one pair of voters where the ranking rule strictly
prefers the voter for to
the voter for . Then the ranking rule satisfies strong transitivity if it always strictly prefers to .
Further, consider an example in which Vic votes for
almost all the kids, whereas Ray votes only for one. If Vic and Ray are ranked
the same by the ranking rule, strong transitivity requires that their votes
must count equally. However, we might feel that Ray has been more decisive, and
therefore feel that his vote should be counted more strongly than Vic’s.
Definition 9.5.3 (Weak transitivity) Consider a preference profile in which outcome o2 receives at least as many votes as o1, and it is possible to pair up all the voters for o1 with voters for o2 who
have both voted for exactly the same number of outcomes so that each
voter for o2 is weakly preferred by the ranking rule to the
corresponding voter for o1. Further assume that o2 receives more votes than o1 and/or that there is at least one pair of voters where
the ranking rule strictly prefers the voter for o2 to the voter for o1.
Then the ranking rule satisfies weak
transitivity if it always strictly prefers o2 to o1.
Definition 9.5.4 (RIIA, informal) A ranking rule satisfies ranked independence of irrelevant alternatives (RIIA)
if the relative rank between pairs of outcomes is always determined
according to the same rule, and this rule depends only on
1. the number of votes each outcome received; and
2. the relative ranks of these voters.
Note that this definition prohibits the ranking rule from caring about the identities
of the voters, which is allowed by IIA.
Despite the fact that Arrow’s theorem does not apply
in this setting, it turns out that another, very different impossibility result
does hold.
Theorem 9.5.5 There is no ranking system that always satisfies both weak transitivity
and RIIA.
What hope is there then for ranking systems? The
obvious way forward is to consider relaxing one axiom and keeping the other.
Indeed, progress can be made both by
relaxing weak transitivity and by relaxing RIIA. For example, the famous
PageRank algorithm (used originally as the basis of the Google search engine)
can be understood as a ranking system that satisfies weak transitivity but not RIIA.
Theorem 9.5.6 Approval voting is the only ranking rule that satisfies RIIA, positive
response, and anonymity.
Definition 9.5.7 (Strong quasi-transitivity) Consider a preference profile in which outcome o2 receives at least as many votes as o1, and it is possible to pair up all the voters for o1 with voters from o2 so
that each voter for o2 is weakly preferred by the ranking rule to the
corresponding voter for o1. Then the ranking rule satisfies strong quasi-transitivity if it weakly prefers o2 to o1, and
strictly strong prefers o2 to o1 if either o1 received
no votes or each paired voter for o2 is
strictly preferred by the ranking rule to the corresponding voter for o1.
forall 
do 
repeat
forall do
if 
then
/
else
until 
converges
Figure 9.3: A ranking
algorithm that satisfies strong quasi-transitivity and RIIA.