Multiagent Systems
Mechanism Design
Reverse Game Theory
© Manfred Huber 2012
1
Social Choice
n 
Social choice and social welfare functions are
aimed at representing properties for the group of
agents
n 
n 
E.g. fairness
Voting schemes are one example of social choice
or welfare functions
n 
Plurality vote
n 
Plurality with instant runoff
n 
Pairwise eliminination
Etc.
n 
© Manfred Huber 2012
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Social Choice
n 
Game theory deals with rational decision making in
the context of a particular game
n 
All decisions by an agent are made purely in order to
increase its own expected utility
n 
n 
Other agents utilities only play a role when they cause
actions that influence the decision agent s utility
In many real application domains, there is a benefit
in achieving some social utility ( norms )
n 
n 
Social choices control interactions in a market/society
Social choices can make systems more robust to
sudden system changes
© Manfred Huber 2012
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Social Choice
n 
Social choice functions determine the preferred
social outcome for the group of agents given their
individual preferences
n 
A social choice function, C, is a function that maps a given
set of non-strict orders (preference) L _ i = {ok !ol | ok , ol ! O}
for the agents, to a preferred outcome for the society
C : L _n " O
n 
!
Social welfare functions, W, are a stronger notion
and determine the preferred social ordering
(preference relation) for the individual preferences
W : L _ n " L _ , W ((l! i ,…,l! n )) = l!W
© Manfred Huber 2012
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Properties for Social Welfare Functions
n 
n 
!
Pareto Efficiency : If all agents agree that outcome oj is
better than ok then the the social welfare function should
also prefer oj to ok
("i o j !i ok ) # o j !W ok
Independence of Irrelevant Alternatives : The relative
ordering of two outcomes under the social welfare
function should only depend on their ordering in the
individual preference relations (not on others).
("i (o j !'i ok # o j !"i ok )) $ (o j !W (!' ) ok # o j !W (!") ok )
n 
Non-Dictatorship : No single agent should always
determine the social ordering
¬"i #j,k (o j !i ok $ o j !W ok )
!© Manfred Huber 2012
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Social Welfare Functions
n 
Arrow s Theorem (1951)
n 
n 
In any game with at least 3 outcomes, any social
welfare function W that is Pareto efficient and
independent of irrelevant alternatives is dictatorial.
In general it is not possible to construct a
social welfare function that has all three
desirable traits
© Manfred Huber 2012
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Properties for Social Choice Functions
n 
n 
!
Weakly Pareto Efficiency : If all agents agree that
outcome oj is better than ok then the the social choice
function can not select ok
("i o j !i ok ) # C(l! ) $ ok
Monotonicity : Increased support for the best social choice
outcome through increased preference in individual agents
can not lead to a change in the social choice outcome.
(C(l! ) = o " #i,o' (o !i o'$ o !'i o')) $ C(l!' ) = o
n 
Non-Dictatorship : No single agent should always
determine the social choice
!
© Manfred Huber 2012
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Social Choice Functions
n 
Muller-Satterthwaite Theorem (1977)
n 
n 
In any game with at least 3 outcomes, any social
choice function C that is weakly Pareto efficient
and monotonic is dictatorial.
In general it is not even possible to construct
a social choice function that has all three
desirable traits
n 
Social choice functions are no simpler than social
welfare functions
© Manfred Huber 2012
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Social Choice
n 
While social choice/welfare functions can be
defined, they can not generally have all the
desirable properties under any individual set
of agent preferences
n 
Properties could still hold if the agents can only
have a limited range of preferences
© Manfred Huber 2012
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Mechanism Design
n 
n 
Social choice describes socially beneficial outcomes
To enforce social choice/welfare functions a
mechanism is needed that can force agents to
pursue the outcome/preference determined by the
function.
n 
n 
Agents have to collaborate or the rules of the game have
to be changed such that the social preference is in each
agent s self interest.
Mechanism design or reverse game theory is trying to
change the rules of the game such that rational,
self interested agents achieve the social choice
© Manfred Huber 2012
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Mechanism Design
n 
n 
Mechanism design is important for a wide range of
real world multiagent settings, e.g.:
n 
Stock markets
n 
Sports games
n 
Legal system
Mechanism design in real world situations has
multiple ways to modify agent behavior, e.g.:
n 
Penalties and rewards
n 
Taxes and duties
n 
Modified selection (e.g. second price auctions)
© Manfred Huber 2012
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Mechanism Design in Bayesian Games
n 
Bayesian games can be divided into two components
n 
n 
n 
Bayesian game setting : defines the parts of the
Bayesian game that are fixed: (N,O,Θ,p,u)
Mechanism : defines the parts that can be modified by
the system designer in order to achieve the desired
social choice function: (A,M)
n 
A : Set of actions available
n 
M : AèΠ(O) determines the outcome distribution
The mechanism can change the outcome mapping
and action set but not the possible outcomes, the
agents preferences, or their type space
© Manfred Huber 2012
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Mechanism Design in Bayesian Games
n 
The goal of mechanism design is to pick a model that
optimizes the designer s own objective function (the
social choice function) when all agents play an
equilibrium strategy
n 
n 
n 
Agents hold private information in terms of their game type
If the equilibrium can be determined centrally according to
the utility functions, the agents action spaces can be
reduced to their type spaces
Mechanism design is aimed at designing the game that in its
equilibrium (and without knowledge of agents preferences)
implements the social choice given that agents might lie
© Manfred Huber 2012
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Mechanism Implementations
n 
Implementation in Dominant Strategies
n 
n 
A mechanism is an implementation in dominant strategies of
a social choice function C if for any vector of utility functions
u, the game has an equilibrium in dominant strategies, and
in any such equilibrium a∗ we have M (a∗ ) = C (u).
Bayes–Nash implementation
n 
A mechanism is an implementation in Bayes–Nash
equilibrium of a social choice function C if there exists a
Bayes–Nash equilibrium of the game of incomplete
information (N, A, Θ, p, u) such that for every θ ∈ Θ and
every action profile a ∈ A that can arise given type profile
θ in this equilibrium, we have that M (a) = C (u( ·, θ)).
© Manfred Huber 2012
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Mechanism Implementations
n 
Implementation in Dominant Strategies is a
stronger mechanism than Bayes-Nash
n 
n 
In general, if more than one Bayes-Nash equilibrium
exists the agent might play different equilibria
strategies and thus no equilibrium at all
Dominant strategies are harder to find and
significantly harder to ensure
© Manfred Huber 2012
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The Revelation Principle
n 
For any mechanism there is a truthful and direct
mechanism that has all its equilibria
n 
Instead of agents determining what information to
provide and how to use it to determine the equilibrium
strategy we can design a mechanism that given the
true information and makes the identical decision
n 
n 
n 
Mechanism lies for the agents
Mechanism incurs the computational cost of determining
the equilibrium and the corresponding agent strategies
The direct truthful mechanism can have additional
equilibria
© Manfred Huber 2012
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Mechanism Limitations
n 
n 
The goal of mechanism design is to achieve a
particular social choice function
Gibbard-Satterthwaite Theorem :
n 
n 
For any game with at least 3 outcomes, if choice function C
is onto (i.e. maintains citizen souvereignty) and dominantstrategy truthful, then C is dictatorial.
There is no mechanism that can implement
arbitrary social choice functions, but only if
n 
Agents can have arbitrary preferences
n 
Implementation is in dominant strategies
© Manfred Huber 2012
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Quasilinear Utilities
n 
n 
One way around the Gibbard-Satterthwaite
impossiblity is to limit agents to preferences that
can be represented by quasilinear utilities
Quasilinear preferences
n 
Agents have quasilinear preferences in an n-player
Bayesian game when:
n 
n 
© Manfred Huber 2012
Outcomes can be represented as O=X × Rn for a finite set X
Utility of agent i given joint type θ is the linear combination ui
(o, θ) = ui(x, θ) − fi (ρi ), where o=(x, ρ) is an element of
O, ui : X ×Θ → R is an arbitrary function and fi : R → R is a
strictly monotonically increasing function.
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Quasilinear Utilities
n 
Quasilinear utility allows to split the mechanism and
utility, ui(o,θ)=ui(x,θ) − fi (ρi ), into two parts:
n 
n 
Choice rule:
n 
x ∈ X is a discrete, non-monetary outcome
n 
Outcome utility is independent of agent wealth
Payment rule:
n 
n 
n 
© Manfred Huber 2012
ρi ∈ R is a monetary payment (possibly negative) that agent i
must make to the mechanism
Agent penalty/reward is independent of the game type and
does not affect other agents
fi addresses risk attitudes by representing the tradeoff between
the non-monetary utility and the monetary penalty
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Quasilinear Mechanisms
n 
A quasilinear mechanism is a mechanism for agents
with quasilinear preferences
n 
n 
n 
Bayesian game setting for quasilinear preferences
(N, O=X x Rn, Θ, p, u)
Quasilinear mechanism (A, χ,ρ)
n 
A=A1 x … x An is the set of action profiles
n 
χ: AèΠ(X) maps action profiles to discrete choices
n 
ρ : AèRn maps action profiles to payments
Direct quasilinear mechanism (χ,ρ)
n 
© Manfred Huber 2012
Ai = Θi actions of agents are revelations about their
game type (mechanism computes equilibrium actions)
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Conditional Utility Independence
n 
Conditional Utility Independence in Bayesian
games implies that an agent s utility does not
depend on other agents game types
"i,o,#,# ' ( # i = # 'i $ ui (o,# ) = ui (o,# ') )
n 
!
n 
Agents utilities can be expressed purely in terms of
their own game type (rather than the joint game type)
ui(o,θi)
Agents can determine their utility without knowledge of
the (private) game type information of the other
agents
© Manfred Huber 2012
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Quasilinear Mechanisms with
Conditional Utility Independence
n 
Valuation : an agent s valuation, ν, for a choice
is the maximum amount it would be willing to
pay to receive this choice
νi(x) = ui(x,θi)
n 
Direct mechanism for situations with conditional
utility independence can be redefined in terms of
agents declared valuations as actions
Ai = {"ˆ i }
n 
Declared valuations can be different from the actual
valuations
© Manfred Huber 2012
!
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Properties for Quasilinear Mechanisms
with Conditional Utility Independence
n 
n 
!
Truthfulness : a quasilinear mechanism is truthful if
in equilibrium each agent declares its actual valuation
"i,# i #ˆ i = # i
Strict Pareto Efficiency : a quasilinear mechanism is
strictly Pareto efficient if in its equilibrium strategy
profile it selects a choice, x, that maximizes the
actual joint valuations of all agents
"# , x'
%# (x) $ %# (x')
i
i
i
i
n 
All pareto efficient outcomes involve the same choice
n 
Social-welfare maximization, also called economic efficiency
© Manfred Huber 2012
!
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Properties for Quasilinear Mechanisms
with Conditional Utility Independence
n 
Strict Budget Balance : a quasilinear mechanism is budget
balanced if in its equilibrium, s, the mechanism does not
loose or gain money
"#
$ (s(# )) = 0
%
n 
i
Weak Budget Balance : a quasilinear mechanism is weakly
budget balanced if in its equilibrium, s, the mechanism
does not loose money
!
"#
$ (s(# )) & 0
%
n 
i
i
i
Budget Balance : a quasilinear mechanism is budget
balanced if in equilibrium the mechanism does not expect
to loose or gain money
!
E " $ # i (s(" )) = 0
[
© Manfred Huber 2012
i
]
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Properties for Quasilinear Mechanisms
with Conditional Utility Independence
n 
n 
!
Ex-Interim Individual Rationality : a quasilinear
mechanism is ex-interim individual rational if in
equilibrium no agent expects to loose by participating
in the game given only its own type information
"i,# i E # $ i |# i [# i ( % (si (# i ),s$i (# $i ))) $ & i (si (# i ),s$i (# $i ))] ' 0
Ex-Post Individual Rationality : a quasilinear
mechanism is ex-post individual rational if in
equilibrium no agent looses by participating
"i,# (# i ( $ (s(# ))) % & i (s(# ))) ' 0
© Manfred Huber 2012
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Properties for Quasilinear Mechanisms
with Conditional Utility Independence
n 
Tractability : a quasilinear mechanism is tractable
if χ and ρ can be computed in polynomial time
n 
Tractability makes a mechanism computationally
feasible to implement
© Manfred Huber 2012
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Objectives for Quasilinear Mechanisms
n 
Revenue Maximization : A mechanism is revenue
maximizing if among all possible mechanisms
that meet the other constraints, it is the one that
in equilibrium maximizes the expected profit for
the mechanism
argmax E$
" ,#
!© Manfred Huber 2012
[% # (s($))]
i
i
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Objectives for Quasilinear Mechanisms
n 
Revenue Minimization : A mechanism is revenue
minimizing if among all possible mechanisms that
meet the other constraints, it is the one that in
equilibrium minimizes the worst case profit for
the mechanism
argmin max$
" ,#
!© Manfred Huber 2012
[% # (s($ ))]
i
i
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Objectives for Quasilinear Mechanisms
n 
Fairness is a desirable property but very
difficult to define in general
n 
n 
Does fairness require equality across agents ?
Maxmin Fairness : A quasilinear mechanism is
maxmin fair if among all possible mechanisms
that meet the other constraints it is the one that
maximizes the expected minimum utility among
all agents
argmax E v [min i ($ i ( " (s($ ))) % # i (s($ )))]
" ,#
© Manfred Huber 2012
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Objectives for Quasilinear Mechanisms
n 
Price-of-anarchy Minimization : A quasilinear
mechanism minimizes the price of anarchy if
among all possible mechanisms that meet the
other constraints it is the one that minimizes the
worst-case ratio between optimal social welfare
and the one achieved by the mechanism in its
worst equilibrium s(ν)
argmin
" ,#
© Manfred Huber 2012
max x % $ i (x)
i
% $ (" (s($ )))
i
i
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Groves Mechanism
n 
The Groves mechanism is a mechanism for
quasilinear utilities that is:
n 
Dominant strategy truthful
n 
Pareto efficient
but not generally
n 
Budget balanced
n 
Individual rational
© Manfred Huber 2012
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Groves Mechanism
n 
The Groves mechanism is a direct quasilinear
mechanism (χ,ρ)
n 
Choice rule
" (#ˆ ) = argmax x $ #ˆ i (x)
i
n 
n 
In a truthful and efficient mechanism the choice should be the
one that maximizes the joint valuation
Payment rule
!
" i (#ˆ ) = hi (#ˆ $i ) $ ' #ˆ j ( % (#ˆ ))
j&i
n 
© Manfred Huber 2012
!
Payment is a constant cost that is independent of its declared
valuation and receives the sum of the other agents valuations
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Groves Mechanism
n 
Truthfulness is a dominant strategy under the
Groves mechanism
n 
A rational choice for agent I requires to declare a
valuation that maximizes the utility
argmax"ˆ i (" i ( # ("ˆ )) $ % i ("ˆ )) = argmax x " i (x) + ' "ˆ j (x)
(
n 
n 
!
Groves mechanism chooses
argmax x # "ˆ i (x) = argmax x "ˆ i (x) + # "ˆ j (x)
i
!
)
j&i
(
j$i
)
For these to end in the same choice the agent should
be truthful and since it does not depend on the other
agents declarations it is dominant
© Manfred Huber 2012
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Groves Mechanism
n 
Green-Laffont Theorem : An efficient social
choice function C : RXn → X × Rn can be
implemented in dominant strategies for agents with
unrestricted quasilinear utilities only with the
payment rule of the Groves mechanism.
n 
n 
The Groves mechanism is the only way to implement
efficient and truthful mechanism that implements
arbitrary efficient social choice functions in dominant
strategies
The theorem also holds for Bayes-Nash incentivecompatible efficient mechanisms
© Manfred Huber 2012
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Clarke Tax
n 
n 
The Groves mechanism does not require a
particular constant cost (allocation function) h
Clarke Tax is an allocation function for the
Groves mechanism
hi ("ˆ #i ) = & "ˆ j ( $ ("ˆ #i ))
j%i
n 
Cost is the sum of agent i independent valuations for all
other agents
!
© Manfred Huber 2012
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Vickrey-Clarke-Groves Mechanism
n 
The Vickrey-Clarke-Groves (VCG) mechanism is a
direct quasilinear mechanism (χ,ρ)
n 
Choice rule
" (#ˆ ) = argmax x $ #ˆ i (x)
i
n 
n 
In a truthful and efficient mechanism the choice should be the
one that maximizes the joint valuation
Payment rule
!
" i (#ˆ ) = ' #ˆ j ( $ (#ˆ %i )) % ' #ˆ j ( $ (#ˆ ))
j&i
n 
© Manfred Huber 2012
!
j&i
Payment is the social cost which is a cost equal to the Clarke
tax plus a reward equal to the sum of the other agents
valuations
36
Vickrey-Clarke-Groves Mechanism
n 
The Vickrey-Clarke-Groves mechanism has the
following properties
n 
Dominant strategy truthful
n 
Pareto efficient
n 
n 
Ex-post individual rational when the choice set
monotonicity and no negative externalities properties
hold
Weakly budget-balanced when the no single-agent
effect property holds.
© Manfred Huber 2012
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Mechanism Design
n 
Other mechanisms have been designed
n 
Weighted Groves mechanisms
n 
D Aspremont, Gerard-Varet (AGV)
n 
n 
Efficient and strictly budget balanced in Nash equilibrium
strategies
In constrained mechanisms for situations where the
mechanism can not modify the strategy space or
utility functions, the mechanims can:
n 
Establish contracts
n 
Offer bribes
n 
Offer to act as a mediator
© Manfred Huber 2012
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Mechanism Design
n 
Mechanism design is aimed at modifying the “rules”
of the game such that agents’ rational decisions have
particular properties
n 
n 
n 
Mechanisms can generate outcomes based on agents’
actions
Mechanisms can modify agents’ actual payoffs through
payment rules
In the case of quasilinear utilities the Vickery-ClarkGroves mechanism provides a framework:
n 
Agents can be truthful and have incentive to participate
Result is pareto efficient and weakly budget balanced
n 
© Manfred Huber 2012
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