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An Introduction to Mechanism Design
Dinesh Garg
IBM India Research Lab, Bangalore
January 11, 2016
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Mother(Social Planner/ Mechanism
Designer)
Children 2(Rational & Intelligent)
Example 1: One Mother Two Child (Cake Cutting Problem)
Children 1(Rational & Intelligent)
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Mechanism Designer(Birbal/ Tenali Rama/ King Solomon)
Mother 1(Rational & Intelligent)
Baby
Example 2: Two Mothers One Child
Mother 2(Rational & Intelligent)
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Example 3: Voting
Set of Alternatives/ Candidates (aka universe)
Voter 1 Voter 2 Voter n
. . . . .
Preference of Voter 1
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Example 4: Auctions
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Mechanism Design Setup
1 2 n
2 n1
Xx
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1 2 n
2n
1 2 n
1
Xf n 1:
Xx nfx ,,1
Social Choice Function (SCF)
Planner ideally wants to aggregate preferences as per SCF
(had he known true types of all the agents)
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1 2 n
1c 2cnc
1
XCCg n 1:
1C
Xx nccgx ,,1
So What is Mechanism ?
22C n
nC
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1 2 n
1̂ 2̂ n̂
1
Xf n 1:
11 C
Xx nfx ˆ,,ˆ 1
Direct Revelation Mechanism (DRM)
222 C n
nnC
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Example 1: Cake Cutting Problem
X
1c2c
XCCg 21:
1
1 2
2
(Cut, Choose) (Cut, Choose)
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Example 2: Two Mothers One Child
XCCg 21:
1c2c
1 2
21
X
(Cut, Not Cut, Neutral) (Cut, Not Cut, Neutral)
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Example 3: Voting
Voter 1 Voter 2 Voter n
1 2 n
2 n
1̂ 2̂ n̂
111 C22 C nnC
Xf n 1:
Xx
Voting Rule
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1
2
n
11C
Xf n 1:
Xx
),,( 1 nf
XCCg n 1:
Xx
))(,),(( *
1
*
1 nnssg
)(*
111 sc
)(*
222 sc
)(*
nnn sc
x
),( 11 xu
),( 22 xu
),( nn xu
NiiCgM
(.),Mechanism
Implementing an SCF via Mechanism
22C
nnC
NiiNiiNii
b uCN
(.),,,, Induced Bayesian Game
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Equilibrium of Induced Bayesian Game
iiiiii
iiiiiiiiii
d
ii
SsSsNi
ssgussgu
,,,
))),(),((())),(),(((
A strategy profile is said to be dominant strategy
equilibrium if (.)(.),1
d
n
d ss
Dominant Strategy Equilibrium (DSE)
iiii
iiiiiiiiiiiiii
SsNi
ssguEssguEii
,,
]|))),(),((([]|))),(),((([ ***
)()(
A strategy profile is said to be Bayesian Nash
equilibrium
(.)(.), **
1 nss
Bayesian Nash Equilibrium (BNE)
Dominant Strategy-equilibrium Bayesian Nash- equilibrium
Observation
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Implementing an SCF
Dominant Strategy-implementation Bayesian Nash- implementation Observation
),,( ),,()(),( 11
*
1
*
1 nnnn fssg
We say that mechanism implements SCF
in Bayesian Nash equilibrium if
NiiCgM )((.), Xf :
Bayesian Nash Implementation
We say that mechanism implements SCF
in dominant strategy equilibrium if
),,( ),,()(),( 1111 nnn
d
n
d fssg
NiiCgM )((.), Xf :
Dominant Strategy Implementation
• Andreu Mas Colell, Michael D. Whinston, and Jerry R. Green, “Microeconomic Theory”,
Oxford University Press, New York, 1995.
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Properties of an SCF (Ex Post) Efficiency
For no profile of agents’ type there exists an
such that and for some
n ,,1 Xx
ifuxu iiii ),(, iiii fuxu ),(, i
Bayesian Incentive Compatibility (BIC)
Nis iiiii ,,)(*
If the direct revelation mechanism has a Bayesian
Nash equilibrium in which
NiifD )((.),
(.))(.),( **
1 nss
Dominant Strategy Incentive Compatibility (DSIC)
Nis iiii
d
i ,,)(
NiifD )((.),
(.))(.),( 1
d
n
d ss
If the direct revelation mechanism has a dominant
strategy equilibrium in which
(Deviation from SCF recommended outcome can’t make someone better-off without making anyone else worse-off)
(Irrespective of what others are doing, I must admit trust)
(If others are admitting truth, I must also do the same)
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Properties of an SCF
DictatorialFor every profile of agents’ type , we have
where is a particular agent known as dictator.
n ,,1
XyyuxuXxf dddd ),(),(|
d
(A special agent is favored all the times by the planner)
(Ex Post) Individual Rationality
where is the utility that agent receives by withdrawing from the
mechanism when his type is
),( )()),,(( iiiiiiii ufu
)( iiu
i
i
(Participation in the mechanism will never make anyone worse-off)
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GS Theorem: Suppose for a given SCF
(1) Range is finite and contains at least 3 elements
(2) Preference structure (aka type space) is rich
then, the SCF is DSIC iff it is dictatorial
Gibbard - Satterthwaite Impossibility Theorem
(By and large, DSIC and Non-dictatorship don’t co-exist)
Possible Ways Out
1. Relax the assumption on richness of preferences (e.g. single peaked preferences)
2. Relax the assumption on finite range by allowing transfer of payments
3. Relax the requirement of strong solution concept namely DSIC and instead work
with BIC
• A. Gibbard. Manipulation of voting schemes. Econometrica, 41:587-601, 1973.
• M. A. Satterthwaite. Strategy-proofness and arrow's conditions: Existence and correspondence theorem
for voting procedure and social welfare functions. Journal of Economic Theory, 10:187-217, 1975.
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Quasi-Linear Environment
011
i
iin tnitKkttkX ,,, ,|),,,(
11111 tkvxu ),(),(
project choice Monetary transfer
to agent i
No outside source of funding
Valuation function of agent 1
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Properties of an SCF in Quasi-Linear Environment
Allocative Efficiency (AE)
An SCF is AE if for each , satisfies(.)),(.),(.),((.) nttkf 1 )(k
n
i
iiKk
kvk1
),(maxarg)(
An SCF is BB if for each , we have
(Strong) Budget Balance
(.)),(.),(.),((.) nttkf 1
01
n
i
it )(
Lemma
An SCF is (ex post) efficient in quasi-linear
environment (having no outside source of funding) iff it is (AE + BB)
(.)),(.),(.),((.) nttkf 1
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Vickrey (1961) Clarke (1971) Groves (1973)
Vickrey-Clarke-Groves (VCG) Mechanisms
• T. Groves. Incentives in teams. Econometrica, 41:617-631, 1973.
• E. Clarke. Multi-part pricing of public goods. Public Choice, 11:17-23, 1971.
• W. Vickrey. Counterspeculation, auctions, and competitive sealed tenders. Journal of Finance, 16(1):8-
37, March 1961.
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Under some mild conditions on preference structure, VCG are
the only mechanisms in quasi-linear environment satisfying
AE+DSIC
Vickrey Clarke Groves
Vickrey-Clarke-Groves (VCG) Mechanisms
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Groves (Possibility) TheoremIn any quasi-linear environment, there exists an SCF which is
AE + DSIC
The dAGVA (Possibility) TheoremIn any quasi-linear environment, there exists a social
choice function which is AE+BB+BIC
(Im) Possibility Theorems in Quasi-Linear Environments
Green- Laffont (Impossibility) TheoremIn any quasi-linear environment, if preference structure (aka type
space) is rich then there is no SCF which is AE + BB+ DSIC
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Myerson-Satterthwaite (Impossibility) Theorem
In the quasi-linear environment, there is no SCF which
is AE + BB+ BIC +IR
Myerson’s (Possibility) Theorem for Optimal Mechanism
In the quasi-linear environment, if the type is one
dimensional, then there exist SCF which are BIC+ IIR
and maximize the earning (or surplus) of the planner
(Im) Possibility Theorems in Quasi-Linear Environments
• R. B. Myerson. Optimal Auction Design. Math. Operations Res., 6(1): 58 -73, Feb. 1981.
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References
Andrew Mas-Collel, Michael D. Whinston,
and Jerry R. Green (1995),
“Microeconomic Theory”, Oxford
University Press
Y. Narahari, Dinesh Garg, Ramasuri
Narayanam, and Hastagiri Prakash
(2009), “Game Theoretic Problems in
Network Economics and Mechanism
Design”, Springer
Noam Nisan, Tim Roughgarden, Eva
Tardos, and Vijay Vazirani (2007),
“Algorithmic Game Theory”, Cambridge
University Press
Jorg Rothe (2015), “Economics and
Computation: An Introduction to
Algoritmic Game Theory, Computational
Social Choice, and Fair Division”,
Springer
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Two Fundamental Design Aspects
Preference Aggregation
Information Revelation (Elicitation)
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1 2 n
2n
1̂ 2̂ n̂
1
Xf n 1:
Xx nfx ˆ,,ˆ 1
Information Elicitation Problem
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SCFs not implementable
by any indirect mechanism in DSE
SCFs implementable
by an indirect mechanism in DSE
DSEI DSEI
Set of SCFs
F
Revelation Principle for DSE
SCFs implementable
by a direct mechanism in DSE
SCFs not implementable
by any direct mechanism in DSE
DSED
DSED SCFs truthfully implementable by
a direct mechanism in DSE
DSIC
Set of SCFs
F
DSICIDDSIC DSEDSE
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Revelation Principle for BNE
Set of SCFs
SCFs not implementable
by any indirect mechanism in BNE
SCFs implementable
by an indirect mechanism in BNE
BNEI BNEI F
SCFs implementable
by a direct mechanism in BNE
SCFs not implementable
by any direct mechanism in BNE
BNED
BNED SCFs truthfully implementable by
a direct mechanism in BNE
BIC
Set of SCFs
F
BICIDBIC BNEBNE
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Absence of Dictatorial SCF in Quasi-Linear Environments
Lemma:
In quasi-linear environment (having no source of outside funding), the utility
of an agent can be made arbitrary high and thereby no SCF is a dictatorial
SCF in this environment
(Thus, GS impossibility theorem does not bite us here)
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Space of SCFs in Quasi-linear Environment
Allocatively
Efficient
Budget
Balance
Dominant Strategy
Incentive Compatible
Ex post efficient SCF
Groves Mechanisms
dAGVA Mechanisms
Bayesian Incentive Compatible