# Mechanism Design

```Mechanism Design
Mechanism design is the sub-field of microeconomics and game theory
that considers how to implement good system-wide solutions to
problems that involve multiple self-interested agents, each with private
information about their preferences. In recent years mechanism design
has found many important applications; e.g., in electronic market
design, in distributed scheduling problems, and in combinatorial
resource allocation problems.
Basic Theory
The Number of Players (Agents)
= 1,2, … ,
Alternatives
= , ,
We call  as public alternative, because choosen alternative affects all
players in
Basic Theory
Each player  privately observes a signal or  ‘s type  ∈ Θ determines
‘s preferences over outcome, and Θ (Set of Preferences) can be finite
or infinite.
All players type
= (1, 2, … )
Is called state of the word
The state  is drawn randomly from state space  ≡ 1  2  …
Which is the set of all possible profiles of types   Θ
Each  is according to some prior distribution ()
Basic Theory
•  is player  private information
•  is common knowledge
Payoff
• Every alternative has a money equivalent value, and preferences are
• Player  is given amount of money equal to   ℝ
• As mechanism design allows transfer of money from every player
then if  < 0 means the money is taken away from
• If the public choosen alternative is  , the player ’s payoff is
described
•  , ,  =  ,  +
•  ,  means money equivalent value of alternative
Outcome
Outcome = combination of choice of public alternatives together with
monetary amounts that each player gets or pays. Then outcome can be
represented as
= , 1, … ,
Mechanism Designer
• Assumption : mechanism designer doesnt have source of funds, then
the monetary payments that players make or receive have to be selffinanced

≤ 0
=1
• Monetary transfer can be negative, means the designer can keep
some of the monet raised by players
Mechanism Design’s Objectives
•   = (  , 1  , … ,   )
•      and (1  , … ,   ) is transfer rule
Mechanism Game
• Mechanism designer desires to implement  . , but he cannot do
this dorectly because the choice rule depends on unobserved state of
the world ()
• In order to implement state-continget choice rule, designer has to ask
players to reveal their types, however players may have not incentives
to reveal their true types.
Bayesian Game
• Because  is only observed by player  and not other’s players
(incomplete information), and players given prior over types (), then
this set up follows Bayesian Game
In formal game, we have :
Mechanism  = Θ1, … , Θ,  .
Strategy profile  ∗ . =  ∗ . , … ,  ∗  .
Outcome   ∗ .
= (. )
,
Revelation Principle
• Mechanism  = Θ1, … , Θ,  . that implements social choice
function  . using equilibrium strategy profile  ∗ . =
∗ . , … ,  ∗  . ,    ∗ . = (. )
• If the players are playing a mechanism that results in the
implementation of (. ) then by construction of equilibrium beliefs
they must know that (. ) will be implemented, and hence the
mechanism designer might as well implement it directly
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