Mechanisms for Making Crowds
Truthful
Andrew Mao, Sergiy Nesterko
Improving Peer Prediction
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Weakness in the Miller et al. paper:
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Honest reporting is not a unique equilibrium (or even Paretooptimal)
Collusion is not limited to symmetric strategies,
nontransferable utility
Does not give a minimum bound on the payoff between lying
and truth-telling
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Players may be indifferent if difference in payoffs is less than ε
Scoring rules cannot be easily extended to accommodate new
constraints
Overview
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Address cases of collusion
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Improve payment mechanism by creating unique NE, or at
least Pareto-optimal NE
Use multiple reference raters (>= 4)
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"...By giving a higher reward for matching all but one of the reference
reports, it is possible to give a higher expected payoff to the truthful
reporting equilibrium..."
Symmetric and asymmetric strategies
Transferable / non-transferable utility
Automated mechanism design approach
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Payments computed by optimization, rather than closed form
scoring rules
Some Features
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Only pure strategies are considered
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Mixed strategy Bayes-Nash equilibria are too complicated to
compute
Initially, prove NE for truthful reporting, then extend to
different collusive cases
Payments to players for good or bad reports determine
best-response strategies
The Model
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Many buyers experience the same product with varying
levels of quality.
Define type as product quality, with a discrete
distribution.
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We'll use just two types - Good and Bad.
Buyers can rate what they get with either 1 (good) or 0
(bad). They get some reward for reporting.
In sequential games, respondent rewards are computed in
batches
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Apply this model repeatedly to achieve sequential play
Model continued
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Common prior among players, center
N respondents in each batch
Possible strategies: (0, n) and (1, n); for n = 0 … N-1
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n is the number of other players that submit a positive report
Probability that n positive reports are submitted by
remaining N-1 reviewers, given my signal oi:
Example of Incentive-Compatible Payments
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Plumber Bob has the following prior:
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P(G) = 0.8, P(B) = 0.2
P(1|G) = 0.9, P(1|B) = 0.15
Suppose Alice (customer) has a job done well. Then
P(G|1) = 0.96.
She is told: "the report is paid only if it matches the
reference report. A negative report is paid $2.62, while a
positive report is paid $1.54"
Then Alice expects the next user to get good service
from Bob with probability P(1|1) = P(1|G)P(G|1) +
P(1|B)P(B|1) = 0.87.
Example continued
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Alice wants to match the expected report of the next
customer
So, if she tells the truth, expected payoff is 0.87 * 1.54 +
0.13 * 0 = 1.34, if she lies 0.87 * 0 + 0.13 * 2.62 = 0.34.
So, no incentive to lie.
Note that if we let P(G) = 0.001 and P(B) = 0.999, this is
reversed!
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It is important that payoffs correspond to the right prior!
But even with smart payoffs, everyone 1 is still an
equilibrium!
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This is addressed in a later section
Automated Mechanism Design
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i.e., how did we magically compute payments to Alice?
First proposed by Conitzer and Sandholm (2003)
 In general, mechanisms are computed to satisfy specified
design goals, instead of deriving closed form rules
 Allows variations within a class of mechanisms to be
dynamically generated
 Mechanism can make use of specific available information
In this case:
 Computing payments by solving optimization problems
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Incentive-Compatible Payment Mechanisms
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Payment mechanism is incentive-compatible if honest
reporting is a Nash Equilibrium
How do you compute the payment scheme so as to
satisfy this?
Can you create a unique NE?
Is it efficient?
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We want:
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minimize expected payment to each player
reward margin between truthful and dishonest reports
all payments must be positive
Solving a Linear Program
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Simple case: no collusion resistance
For this to make sense, everyone must have the same
prior
Analytical Solution to the LP
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From constraints in the LP, we have two nonzero decision
variables (payments)
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Lemma: ratio of Pr[n|1]/Pr[n|0] is monotonically
increasing in n
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Must be for two separate reports: τ(0, n1) and τ(1, n2)
From the dual, expected payment depends on this ratio
Under cost minimization, incentive compatible payments
are driven to n1 = 0 and n2 = N - 1, respectively
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Result: only τ(0, 0) and τ(1, N-1) are positive payments
Satisfying Incentive Compatibility
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Consider the conditions for incentive compatibility, with
n1 = 0, n2 = N-1:
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τ(0, 0) > τ(1, 0); τ(1, N-1) > τ(0, N-1)
In the 2-player case, this becomes
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τ(0, 0) > τ(1, 0); τ(1, 1) > τ(0, 1)
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Obviously, this introduces the "all-report-high" and "allreport-low" equilibria
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Now, how do we fix this?
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Add more constraints to the optimization problem!
Extensions
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Coalition size (full coalition/fractional coalition)
Symmetric vs. asymmetric strategies
Transferable utility
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Some combinations of these conditions are unreasonable
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i.e. doesn’t make sense if colluders can make side payments
but not coordinate on asymmetric strategies
Achieving unique or Pareto-optimal Nash equilibria
Extension: Full coalition, symmetric
strategies, non-transferrable utilities
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We want to get rid of the “all-report-X” Nash
Equilibrium
Extending the plumber example to N = 4 agents, look at
probabilities
Note the differences in distributions!
Example continued
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Optimal payment scheme:
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Reporter is encouraged to "even out" the 0 distribution,
but the prior compensates
This gives the incentive for one person to switch when
everyone else is reporting the same
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Implicit collusion resistance to symmetric strategies
Extension: Partial Collusion, Asymmetric
Strategies, Nontransferable Utility
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Theorem: When more than half of the agents collude, no
incentive-compatible payment mechanism can make
truth-telling dominant strategy for the colluders
Cost of payments rises exponentially as the coalition
fraction increases
Extension: Partial Collusion, Asymmetric
Strategies, Transferable Utility
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Note that the normalized cost rises much faster than
before when participants can make side payments
Summary of Extensions
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Some conditions lead to MILPs, which are harder to solve
Unique vs. Pareto-optimal NE
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The latter is much cheaper
Partial collusion: payment cost increases dramatically
beyond a threshold of colluders
Improvements
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Extension to original peer prediction mechanism with
automated mechanism design
Dynamically generated payments, so rules don't have to
be in closed form
Expected payment from honest reporting better than
lying by some guaranteed threshold
Different conditions can generate Unique, Pareto-optimal,
or even Dominant NE, with corresponding different costs
Drawbacks
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Common prior still required for BNE
Report space is discrete (binary, in fact)
Sequential nature of reports submission is not considered
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Need at least a certain size group
Weird budget results if center has different prior from
users
Not necessarily incentivizing players to spend effort to
uncover information - why not just invent a report?
Discussion