Transcript Mediators in Position Auctions

Mediators
Slides by Sherwin Doroudi
Adapted from “Mediators in Position
Auctions” by Itai Ashlagi, Dov
Monderer, and Moshe Tennenholtz
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”)
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)
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
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
Ex: Pre-Bayesian Game
• Game H: Assume player I has only one type
but player II is either type A or type B
II
I
2
α
5
2
2
4
Type(II) = A
2
α
3
β
β
α
I
0
0
0
II
β
α
0
0
3
2
β
3
5
Type(II) = B
Ex: Pre-Bayesian Game
• Game H: The ex post equilibrium is I plays β
and II plays (A, B) →(β, α)
II
I
2
α
5
2
2
4
Type(II) = A
2
α
3
β
β
α
I
0
0
0
II
β
α
0
0
3
2
β
3
5
Type(II) = B
Question
• In general, do all pre-Bayesian games have at
least one ex post equilibrium?
Another Pre-Bayesian Game
• Game G: As before, player I has only one type
but player II is either type A or type B
II
I
2
α
5
2
2
2
Type(II) = A
2
α
3
β
β
α
I
0
0
0
II
β
α
0
0
0
2
β
3
5
Type(II) = B
Another Pre-Bayesian Game
• Game G: Player I has different strictly
dominant strategies depending on II’s type
II
I
2
α
5
2
2
2
Type(II) = A
2
α
3
β
β
α
I
0
0
0
II
β
α
0
0
0
2
β
3
5
Type(II) = B
Another Pre-Bayesian Game
• Game G: BUT! Player I has no idea which type
II is, so no ex post equilibrium exists
II
I
2
α
5
2
2
2
Type(II) = A
2
α
3
β
β
α
I
0
0
0
II
b
α
0
0
0
2
b
3
5
Type(II) = B
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?
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)
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
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
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
II
II
β
α
I
2
0
α
5
2
0
α
2
3
0
β
α
I
0
0
2
2
β
β
0
2
Type(II) = A
3
5
Type(II) = B
Mediated Game G
• Let m denote requesting mediator services:
II
I
m-A
m-B
2
m
5
2
3
2
2
2
0
3
0
0
0
3
5
5
0
2
2
α
β
5
2
0
β
α
0
Type(II)=A
2
2
Mediated Game G (contd.)
• Let m denote requesting mediator services:
II
I
m-A
m-B
2
m
2
2
0
2
2
2
0
0
0
0
3
0
2
2
0
2
2
α
β
2
5
0
β
α
3
Type(II)=B
2
5
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”
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?
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?
Ex: A Two-Player Self-Price Auction
• Consider a self-price auction with two players
and one good (or one position with a clickthrough 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
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)!
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
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)???
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
Shortcomings of VCG
Mechanisms
Adapted from “Thirteen Reasons
Why the Vickrey-Clarke-Groves
Process Is Not Practical” by Michael
H. Rothkopf
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
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?
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?