Download Mechanism Design

Recent contributions to
Implementation/Mechanism Design Theory
E. Maskin
Mechanism Design/Implementation
• part of economic theory devoted to “reverse engineering”
• usually we take mechanism, game, or economy as given
– try to predict the outcomes it generates in equilibrium
• in MD, we (the “planner”) start with outcome(s) a we want
as a function of underlying state    (describes utility
utility functions ui ,  , technologies, endowments, etc.)
social choice function f :   A
– difficulty: we may not know state
– try to design a mechanism (game, outcome function,
tax schedule) whose equilibrium outcomes same as that
prescribed by social choice function
mechanism implements SCF
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• Goes back (at least) to 19th century Utopians
– can one design “humane” alternative to laissezfaire capitalism?
• Socialist Planning Controversy 1920s-40s
– can one construct a centralized planning
mechanism that replicates or improves on
competitive markets?
O. Lange and A. Lerner: yes
L. von Mises and F. von Hayek: no
– brought to fore 2 major themes
incentives
information
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Formal mechanism-design theory dates from
3 papers
• L. Hurwicz (1960)
– introduced basic concepts
• mechanism
• informational decentralization
• informational efficiency
• W. Vickrey (1961)
– exhibited a particular but important mechanism: 2nd
price auction
• J. Mirrlees (1971)
– developed standard analytic techniques
– derived standard properties (e.g., “no distortion at top”)
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Since then, field has expanded dramatically
• vast literature, ranging from
– very general
possible outcomes  abstract set of social alternatives
(at least 10 major survey articles and books in last dozen years
or so)
– quite particular
design of bilateral contracts between buyer and seller
(several recent books on contract theory, including BoltonDewatripont (2005) and Laffont-Martimort (2002))
design of auctions for allocating a good among competing
bidders
(several recent books - - Krishna (2002), Milgrom (2004),
Klemperer (2004))
– far too much recent work to survey properly here
– will pick 3 specific developments (both general and
particular)
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• interdependent values in auction design
• robustness of mechanisms
• indescribable states, renegotiation and
incomplete contracts
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Interdependent values in auction
design
• seller has 1 good
• n potential buyers
• how to allocate good efficiently?
(to buyer who values good the most)
i.e., how to implement SCF that selects
efficient allocation
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In private values case (each buyer’s valuation is
independent of others’ information),
Vickrey (1961) answered question:
• 2nd price auction is efficient
– buyers submit bids
– winner is high bidder
– winner pays 2nd highest bid
• if vi is buyer i’s valuation, optimal for him to bid
bi  vi
• winner will have highest valuation
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What if values are interdependent?
• each buyer i gets private signal si (onedimensional)
• buyer i’s valuation is vi  si , si 
• buyer i no longer knows own valuation
– so can’t bid valuation in equilibrium
– might bid expected valuation, but this not
enough for efficiency: might have
Es vi  si , si   Es v j  s j , s j 
but
vi  si , si   v j  s j , s j 
i
j
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•
consider auction in which
– each buyer i makes contingent bid
bi  vi   i's bid if other buyers' valuations revealed to be vi
– calculate fixed point  v1 , , vn  such that
  for all i
vi  bi v i
– winner is buyer i such that
()
vi  max v j
j i
– winner pays
 
 
bi v i  max vj , where vj  b j v j
•
j i
ji
v
If (i) vi  0 and (ii) vi  j whenever vi  v j and i  j
si
si
si
then, in equilibrium buyer i with signal si bids true contingent valuation:
()
bi  v i  si , s i    vi  si , si  for all si


– idea: i pays lowest constant bid max b j  v j  that would win
•
•
•
j i
– in ordinary 2nd price auction, i pays lowest bid
from , true valuations are fixed point
from  , winner has highest valuation
auction efficient
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• dynamic auctions like English auction easier on
buyers than one-shot mechanisms like 2nd –price
auction
• Perry and Reny (2005) develop dynamic auction
– even works for multiple goods, if substitutes
• open problem: How to handle multiple goods with
complementarities in dynamic auction
• Jehiel and Moldovanu (2002) and Jehiel,
Moldovanu, Meyer-Ter-Vehn, and Zame (2005)
establish fundamental limitation on how far one
can go with multiple goods and multidimensional
signals
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Robust Mechanism Design
auction in which buyer i bids bi  vi  is “robust” or
“independent of detail” in sense that
• it doesn’t matter whether auction designer knows
buyers’ signal spaces or functional forms vi  si , si 
• it doesn’t matter what buyer i believes about the
distribution of s i
– optimal for buyer i to set
bi  vi  si , si    vi  si , si  for all si
regardless of i’s belief about s i
– i.e., bidding truthfully is an ex post equilibrium
(remains equilibrium even if i knows s i )
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Why is robustness important?
• common in Bayesian mechanism design to identify buyer i’s possible
types with his possible preferences (common more generally than
justified)
set of possible types  set of possible preferences 
• but this has extreme implication: if you know i’s preferences, know his
beliefs over other’s types
– no reason why this should hold
– very strong consequences:
in auction model above, if signals correlated, auctioneer can attain
efficiency and extract all buyer surplus, even without conditions such as
vi v j

si si
(Crémer and McLean (1985))
– As Neeman (2001) and Heifetz and Neeman (2004) show, CrémerMcLean result goes away for suitably rich type spaces (preference
corresponds to multiple possible beliefs)
• no reason why auction designer should know what buyers’ type spaces
are (how preferences related to beliefs)
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Given SCF f :    , can we find mechanism for
which, regardless of type space associated with
preference space  , there always exists f-optimal
equilibrium?
(robust partial implementation)
• sufficient condition: f partially implementable in
ex post equilibrium, i.e., there exists mechanism
that always has f-optimal ex post equilibrium (may
be other equilibria)
– ex post equilibrium reduces to dominant strategy
equilibrium with private values
• interestingly, Bergemann and Morris (2004) show
that condition not necessary
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But ex post partial implementability is
necessary for robust partial implementation
if
• outcome space takes form
X  Y1 
"common"
 Yn
"private transfers"
outcome
(public good)
agent i cares just about  x, yi 
• satisfied in above auction model (and, more
generally, in quasilinear models)
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• So far have concentrated on partial
implementation (not all equilibria have to
be f-optimal)
• But unless planner sure that agents will play
f-optimal equilibrium, more appropriate
concept is full implementation: all
equilibria of mechanism must be f-optimal
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• key to full implementation is some species of monotonicity
– full implementation in Nash equilibrium (agents have complete
information) requires standard monotonicity:
for all,  ,     if, for all j,
then
u j  f   ,   u j  b,  implies u j  f   ,   u j b, 
 for all b
f    f  
equivalently: for all  :    and   for which
f       f    (here       ),
there exist i and

a  A such that


ui f      ,      ui  a,      and ui  a,    ui f      , 

analogous condition for Bayesian implementation (agents have incomplete
information)
- - Postlewaite and Schmeidler (1986), Palfrey and Srivastava (1989),
Jackson (1991)
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• standard monotonicity: for all  :    and   
for which f       f   
there exist i and a  A such that



ui f      ,      ui  a,      and ui  a,    ui f      , 
• ex post monotonicity key to ex post full implementabilty (Bergemann
and Morris 2005):
for all   1 , ,  n   j :  j   j and
   i, i  for which f       f    , there exist i and a such that






ui f i ,  i  i   , i ,  i  i    ui a, i ,  i  i   for all i
and
ui  a,    ui f      , 


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
• in economic settings with n > 3, f is ex post fully
implementable if and only if it satisfies ex post
monotonicity and ex post incentive compatibility
ui  f    ,    ui  f i ,  i  ,   with all i, i ,  
• ex post equilibrium is refinement of Nash
equilibrium but ex post monotonicity doesn’t
imply standard monotonicity (nor is it implied)
– although ex post equilibrium is more demanding
solution concept, makes ruling out equilibria easier
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• Notable SCF where ex post monotonicity but not
standard monotonicity satisfied: efficient
generalized second-price allocation rule in
interdependent values auction model when n > 3
• ex post monotonicity satisfied because truthful
equilibrium is unique ex post equilibrium
– Berliun (2003) shows that hypothesis n > 3 is
important: there exist inefficient ex post equilibria in
case n = 2.
• efficient second-price allocation rule not
standardly monotonic: if lower losers’ signal
values, monotonicity requires that same allocation
still chosen - - but winner’s payment will fall
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• But ex post full implementation not quite enough
– ensures ex post equilibria optimal
– but other equilibria could be nonoptimal
• really need robust full implementation:
f :    implementable by mechanism such that,
regardless of type space associated with Θ, all
equilibria are f-optimal
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• robust monotonicity is key:
for all  there exist i and i such that f  i,i    f i,i  for all i
and
for all i and all probability distributions  i on i1 i  there exists a  A
such that
ui  f i ,  i i   , i ,  i i     ui  a, i ,  i i    for all i
and
 i i  ui a, i ,i   i i  ui f i , i  , i , i
  i
 

  i



– stronger than both ex post monotonicity and standard monotonicity
– together with ex post incentive compatibility, necessary and
sufficient for robust full implementation in economic environments
• For n > 3, generalized 2nd price allocation rule robustly fully virtually
implementable as long as not “too much” interdependence, i.e.,
vi
not too big
s j
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• so far, “robustness” requirement pertains to
mechanism designer
– may not know agents’ type spaces
• also recent contributions in which
robustness pertains to agents playing
mechanism
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• large literature considering implementation in
various refinements of Nash equilibrium
– allows implementation of nonmonotonic SCFs
• any species of Nash equilibrium entails that agents
have common knowledge of preferences, i.e.,
state 
• but what if agents are (slightly ) uncertain about
?
– which SCFs are robust to this uncertainty?
– answer depends on nature of uncertainty
• for a natural form of uncertainty, only monotonic
SCFs robustly implementable in this sense (Chung
and Ely 2003)
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Example (Jackson and Srivastava (1996))
n2
A  a, b, c


1
c
a
b
   , 
2
a
b
c
f implemented in
undominated Nash
equilibrium by
1
c
a
b
f    a
2
a
c
b
f    c
m2
m2
m1
a
a
m1
b
c
not monotonic
 m1, m2  unique u.d. equil in 
 m1, m2  unique u.d. equil in  
• m2 is dominated in state   if player 2 sure that state is  
• If 2 attaches small probability to  , m2 not dominated , so
mechanism no longer implements f
• in fact, no mechanism can implement f because nonmonotonic
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Open problems:
(1) Implications of ex post monotonicity and
robust monotonicity for applications
(2) implications of other sorts of uncertainty
for implementability
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Indescribable States, Renegotiation and
Incomplete Contracts
• incomplete contracts literature studies how
assigning ownership (or control) of
productive assets affects efficiency of
outcome
• For efficiency to be in doubt, must be some
constraint on contracting
(i.e., on mechanism design)
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• In this literature, constraint is
incompleteness of contract
– contract not as fully contingent on state of
world as parties would like
• Reason for incompleteness
– parties plan to trade some good in future
– do not know characteristics of good (state) at
the time of contracting (although common
knowledge at time of trade)
– contract cannot even describe set of possible
states (too vast)
– so contract cannot be fully contingent
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Nevertheless, we have:
Irrelevance Theorem:
If parties are risk averse and can assign probability
distribution to their future payoffs, then can
achieve same expected payoffs as with fully
contingent contract (even though cannot describe
possible states in advance)
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Idea:
• design contracts that specify payoff contingencies
• later, when state of world realized, can fill in
physical details
• possible problem: incentive compatibility
– will it be in parties interest to specify physical details
truthfully?
– but if
different states  different preferences,
can design mechanism to ensure incentive compatibility
• make mechanism part of contract
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Where does risk aversion come in?
Answer: helps with incentive compatibility
• if parties are supposed to play  m1 , m2  in 
but 1 plays m1 instead, must be punished
• but if  m1, m2  is equilibrium play in  ,
then not clear from  m1, m2  who has
deviated
• resolution: punish them both with
inefficient outcome a.
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But what if parties can renegotiate outcome ex post ?
• not an issue when designer is third party; here parties
themselves design contract
• why settle for a ?
• will renegotiate a to get something Pareto optimal
• renegotiation interferes with effective punishment: can’t
punish both parties
• in Segal (1999) and Hart and Moore (1999), renegotiation
is so constraining that mechanisms are useless
Risk aversion
• Pareto frontier (in utility space) is strictly concave
• so if randomize between 2 Pareto optimal points, generate
point in interior (bad outcome)
• so can punish both parties after all.
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Why isn’t randomization renegotiated away?
• randomization occurs only out of equilibrium
• ex ante, parties have no incentive to renegotiate
(expect equilibrium
– have incentive not to renegotiate: renegotiation
interferes with getting equilibrium outcome
• what about renegotiation ex post?
– create randomizing device so that as soon as a party
deviates from equilibrium, randomization realized
– no time to renegotiate
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Open problem:
How to provide fully satisfactory
foundation for incomplete contracts?
• possible answer: bounded rationality (inability
to perform dynamic programming)
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