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is sufficiently close to this published one will have truthful reporting as their best strat-
egy, while agents that consider the public distribution as grossly wrong may instead
be merely helpful by making reports that will drive the public poll closer to what they
consider to be the true distribution.
Such a mechanism was first shown in ([10]) for aggregating opinions about a hidden
signal that could be either good (g) or bad (b). At time
t
, the published polls shows the
average fraction
R
t
of good reports. An agent
A
i
has its own probability distribution
p
i
(
r
s
)
that characterizes the conditional probability distribution of a reference report
r
given its own observation
s
of the signal, where the reference report is filed by another
agent that observes the same signal and the same public poll. The mechanism compares
the report
s
filed by agent
A
i
to a reference report
r
filed by another agent B, and rewards
A
i
if the two reports match:
|
-
for matching a good report, the reward is
c
(
1
−
R
t
)
.
-
for matching a bad report, the reward is
cR
t
.
where
c
is a positive constant to scale the average reward, for example to ensure that it
compensates for the effort required to file it.
To analyze the incentives for agent
A
i
, we distinguish three cases:
a)
A
i
considers the current poll result reasonable, characterized by the fact that
p
i
(
g
g
)
.
b)
A
i
considers the poll result unreasonably high, characterized by the fact that
R
t
≥
p
i
(
g
|
b
)
<
R
t
<
p
i
(
g
|
g
)
, which means that no matter what
A
i
observes, it would always expect other
agents to observe a bad signal with a higher probability than the current poll result.
c)
A
i
considers the poll result unreasonably low, characterized by the fact that
p
i
(
g
|
|
b
)
≥
R
t
, symmetrically on the other side.
In the case where the poll result is reasonable, the agent is best off reporting truthfully.
Consider the case where it observes a good signal, then the expected rewards are:
-
for reporting good (truthful):
p
i
(
g
|
g
)
c
(
1
−
R
t
)
>
R
t
c
(
1
−
R
t
)
-
for reporting bad (non truthful):
p
i
(
b
|
g
)
cR
t
=(
1
−
p
(
g
|
g
))
cR
t
<
(
1
−
R
t
)
cR
t
Thus, the expected reward for reporting truthfully is strictly greater than the expected
reward for a non-truthful report. A symmetric analysis can be made for the case of a
bad observation.
As an example, consider that agents A and B both hire a plumber that according
to the public reputation scheme provides good service 90% of the time, based on 10
previous reports. Suppose that A sees the plumber at work and he does a good job.
Then A might consider that the current poll value is accurate or slightly too low and
report good service, expecting a payment of 10/9 with a probability of higher than 0.9,
so above 1 in expectation.