1. Mechanisms for Making Crowds Truthful Andrew Mao, Sergiy Nesterko

2. Improving Peer Prediction  Weakness in the Miller et al. paper:  Honest reporting is not a unique equilibrium (or even Pareto- optimal)  Collusion is not limited to symmetric strategies, nontransferable utility  Does not give a minimum bound on the payoff between lying and truth-telling  Players may be indifferent if difference in payoffs is less than ε  Scoring rules cannot be easily extended to accommodate new constraints

3. Overview  Address cases of collusion  Improve payment mechanism by creating unique NE, or at least Pareto-optimal NE  Use multiple reference raters (>= 4)  "...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  Payments computed by optimization, rather than closed form scoring rules

4. Some Features  Only pure strategies are considered  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

5. The Model  Many buyers experience the same product with varying levels of quality.  Define type as product quality, with a discrete distribution.  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  Apply this model repeatedly to achieve sequential play

6. Model continued  Common prior among players, center  N respondents in each batch  Possible strategies: (0, n) and (1, n); for n = 0 … N-1  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 o i :

7. Example of Incentive-Compatible Payments  Plumber Bob has the following prior:  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.

8. Example continued  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!  It is important that payoffs correspond to the right prior!  But even with smart payoffs, everyone 1 is still an equilibrium!  This is addressed in a later section

9. Automated Mechanism Design  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

10. Incentive-Compatible Payment Mechanisms  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?  We want:  minimize expected payment to each player  reward margin between truthful and dishonest reports  all payments must be positive

11. Solving a Linear Program  Simple case: no collusion resistance  For this to make sense, everyone must have the same prior

12. Analytical Solution to the LP  From constraints in the LP, we have two nonzero decision variables (payments)  Must be for two separate reports: τ (0, n 1 ) and τ (1, n 2 )  Lemma: ratio of Pr[n|1]/Pr[n|0] is monotonically increasing in n  From the dual, expected payment depends on this ratio  Under cost minimization, incentive compatible payments are driven to n 1 = 0 and n 2 = N - 1, respectively  Result: only τ (0, 0) and τ (1, N-1) are positive payments

13. Satisfying Incentive Compatibility  Consider the conditions for incentive compatibility, with n 1 = 0, n 2 = N-1:  τ (0, 0) > τ (1, 0); τ (1, N-1) > τ (0, N-1)  In the 2-player case, this becomes  τ (0, 0) > τ (1, 0); τ (1, 1) > τ (0, 1)  Obviously, this introduces the "all-report-high" and "all- report-low" equilibria  Now, how do we fix this?  Add more constraints to the optimization problem!

14. Extensions  Coalition size (full coalition/fractional coalition)  Symmetric vs. asymmetric strategies  Transferable utility  Some combinations of these conditions are unreasonable  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

15. Extension: Full coalition, symmetric strategies, non-transferrable utilities  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!

16. Example continued  Optimal payment scheme:  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  Implicit collusion resistance to symmetric strategies

17. Extension: Partial Collusion, Asymmetric Strategies, Nontransferable Utility  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

18. Extension: Partial Collusion, Asymmetric Strategies, Transferable Utility  Note that the normalized cost rises much faster than before when participants can make side payments

19. Summary of Extensions  Some conditions lead to MILPs, which are harder to solve  Unique vs. Pareto-optimal NE  The latter is much cheaper  Partial collusion: payment cost increases dramatically beyond a threshold of colluders

20. Improvements  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

21. Drawbacks  Common prior still required for BNE  Report space is discrete (binary, in fact)  Sequential nature of reports submission is not considered  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?

22. Discussion