1. Mediators Slides by Sherwin Doroudi Adapted from “Mediators in Position Auctions” by Itai Ashlagi, Dov Monderer, and Moshe Tennenholtz

2. Bayesian & Pre-Bayesian Games Consider a game where every player has private information regarding his/her “type” A player’s strategy maps types to actions Ex: You are either type A or type B and you have actions “play” and “pass”; one strategy might be A→ “play” and B→ “pass”; we can write this as (A, B) → (“play”, “pass”)

3. Bayesian & Pre-Bayesian Games These are games of incomplete information In a Bayesian Game there is a commonly known prior probability measure on the profile of types Ex: You are either type A or B, I am either type X or Y, and we know that it is common knowledge that our types are equality likely to be (A, X), (A, Y), (B, X), or (B, Y)

4. Bayesian & Pre-Bayesian Games In a pre-Bayesian game, there is no prior probability over they types the players can take Ex: An auction setting in which there is no known distribution with which the players value the goods We will be concerned only with pre-Bayesian games

5. Equilibria in Pre-Bayesian Games When priors regarding types are not known (i.e. in pre-Bayesian game) we are primarily concerned with ex post equilibrium “A profile of strategies, one for each player, such that no player has a profitable deviation independently of the types of the other players” Requiring dominant strategies is stricter

6. Ex: Pre-Bayesian Game Game H: Assume player I has only one type but player II is either type A or type B 5 2 3 0 0 0 4 2 I II α β α β Type(II) = A 2 2 0 0 3 3 5 2 I II α β α β Type(II) = B

7. Ex: Pre-Bayesian Game Game H: The ex post equilibrium is I plays β and II plays (A, B) →(β, α) 5 2 3 0 0 0 4 2 I II α β β α β β Type(II) = A 2 2 0 0 3 3 5 2 I II α β β α α β Type(II) = B

8. Question In general, do all pre-Bayesian games have at least one ex post equilibrium?

9. Another Pre-Bayesian Game Game G: As before, player I has only one type but player II is either type A or type B 5 2 3 0 0 0 2 2 I II α β α β Type(II) = A 2 2 0 0 3 0 5 2 I II α β α β Type(II) = B

10. Another Pre-Bayesian Game Game G: Player I has different strictly dominant strategies depending on II’s type 5 2 3 0 0 0 2 2 I II α α β α β Type(II) = A 2 2 0 0 3 0 5 2 I II α β β α β Type(II) = B

11. Another Pre-Bayesian Game Game G: BUT! Player I has no idea which type II is, so no ex post equilibrium exists 5 2 3 0 0 0 2 2 I II α α β α b Type(II) = A 2 2 0 0 3 0 5 2 I II α b b α β Type(II) = B

12. Some Bad News… So we have seen that not all pre-Bayesian games have an ex post equilibrium. What can we do to rectify this situation? Will allowing for mixed strategies help? Can we modify or “transform” these “problematic” games such as G, so as to “give” such games nicer properties?

13. Mediators We introduce a mediator, “a reliable entity that can interact with the players and perform on their behalf actions in a given game” Players need not make use of the mediator’s services Now players can choose to play a strategy or ask the mediator to play for them (by revealing to the mediator their type)

14. Mediators Since a mediator acts reliably and deterministically given what information the mediator is given, the mediator should not be thought of as a player (even in the simple sense where “Nature” is often considered a player) Rather, the mediator expands the strategies available to the existing players

15. Mediators In games such as auctions where players’ actions are equal to their possible types (i.e. where actions are tantamount to revealing a type), strategies involve revealing a type given a true type, so adding a mediator turns the action space into two copies of itself, a revelation of type, or a telling of this type to the mediator

16. Revisiting Game G Consider this mediator: If both players request mediator services, (α, α) or (β, β) is played when II reports A or B respectively If only I seeks the mediator, α is played for I If only II seeks the mediator, α (resp. β) is played if A (resp. B) is reported 5 2 3 0 0 0 2 2 I II α β α β Type(II) = A 2 2 0 0 3 0 5 2 I II α β α β Type(II) = B

17. Mediated Game G Let m denote requesting mediator services: 5 2 2 2 3 0 5 2 m α m-Am-B 5 3 0 5 2 3 0 2 2 0 0 0 0 2 2 β αβ 2 Type(II)=A I II

18. Mediated Game G (contd.) Let m denote requesting mediator services: 2 2 5 2 0 0 2 2 m α m-Am-B 2 0 0 2 2 0 0 2 2 3 0 3 0 5 2 β αβ 2 Type(II)=B I II

19. What Did We Accomplish? We used a mediator to “transform” the problematic no-equilibrium game G into one with an ex post equilibrium The new ex post equilibrium is simply the one that calls for both players to make use of the mediator’s services and to report their true type (if applicable) to the mediator “Incentive compatibility”

20. Where Can We Go With This? We can “improve” many pre-Bayesian games by adding a mediator, though we must choose carefully how this mediator acts We can turn mechanisms that do not promote truthful direct revelation into mechanisms that do promote truthful revelations When is “fixing” mechanisms really necessary, though?

21. In Particular… We can consider position auctions that do not necessarily have equilibrium and try to transform them into VCG auctions Many existing auctions are not VCG, even though VCG has nice (at least theoretical) properties, such as ex post equilibrium existence Why not just change the auction to VCG?

22. Ex: A Two-Player Self-Price Auction Consider a self-price auction with two players and one good (or one position with a click- through rate of 1) For c ≥ 1, the mediator m[c], seeing player I report value u and player II report value v bids v for I and 0 for II if u ≥ v and 0 for I and u for II If only one player uses the mediator, reporting w, the mediator bids cw on their behalf

23. Ex: A Two-Player Self-Price Auction Using a T-strategy (truthfully reporting ones type to the mediator) implements a VCG outcome For c > 1 this is T-strategy is not dominant, and could cause negative utilities, so c = 1 is the natural choice here We’ve transformed a self-price auction into a next-price (VCG w/ 2 players)!

24. Valid Mediators In the previous example the mediator seemingly tries to “punish” the player who doesn’t use the mediator by submitting a high bid for the player using the mediator—this hurts the player using the mediator more; this is OK, but… we don’t want negative utilities Valid mediators implement only nonnegative utilities for players using the T-strategy

25. What Else Can We Do? Implementing a VCG outcome via mediators in some special cases of: Generalized next-price position auctions K-next-price positions auctions Weighted next-price positions auctions Google-like position auctions Self-position auctions (with > 1 positions)???

26. Conclusion We have shown that mediators allow us to transform pre-Bayesian games to allow for useful properties (such as equilibrium existence) In particular we examined that some position auctions can have a VCG-outcome implemented via the use of such mediators

27. Shortcomings of VCG Mechanisms Adapted from “Thirteen Reasons Why the Vickrey-Clarke-Groves Process Is Not Practical” by Michael H. Rothkopf

28. Some Problems w/ VCG Dominant strategy equilibria are weak; other weak equilibria may exist Exponential growth of effort NP completeness of winner determination problem Process can be revenue deficient

29. VCG Allows for Cheating Conspiracies by competing bidders Sequence of strategy proof auctions need not be strategy-proof False-name bids by single bidders False-name bids by the auctioneer What if our mediator is not reliable?

30. Concluding Question What is the real advantage in transforming designed mechanisms into VCG mechanisms if they were not designed to be VCG in the first place?