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)
𝑁 = 1,2, … , 𝑛
Alternatives
𝑋 = 𝑥, 𝑦, 𝑏
We call 𝑋 as public alternative, because choosen alternative affects all
players in 𝑁
Basic Theory
Each player 𝑖 privately observes a signal or 𝑖 ‘s type 𝜃𝑖 ∈ Θ𝑖 determines
𝑖 ‘s preferences over outcome, and Θ (Set of Preferences) can be finite
or infinite.
All players type
𝜃 = (𝜃1, 𝜃2, … 𝜃𝑛)
Is called state of the word
The state 𝜃 is drawn randomly from state space 𝛩 ≡ 𝛩1 𝑥 𝛩2 𝑥 … 𝛩𝑛
Which is the set of all possible profiles of types 𝜃 𝜖 Θ
Each 𝜃 is according to some prior distribution (𝜙)
Basic Theory
• 𝜃𝑖 is player 𝑖 private information
• 𝜙 is common knowledge
Payoff
• Every alternative has a money equivalent value, and preferences are
additive in money
• Player 𝑖 is given amount of money equal to 𝑚𝑖 𝜖 ℝ
• As mechanism design allows transfer of money from every player
then if 𝑚𝑖 < 0 means the money is taken away from 𝑖
• If the public choosen alternative is 𝑥 , the player 𝑖’s payoff is
described
• 𝑣𝑖 𝑥, 𝑚𝑖, 𝜃 = 𝑢𝑖 𝑥, 𝜃𝑖 + 𝑚𝑖
• 𝑢𝑖 𝑥, 𝜃𝑖 means money equivalent value of alternative 𝑥 𝜖 𝑋
Outcome
Outcome = combination of choice of public alternatives together with
monetary amounts that each player gets or pays. Then outcome can be
represented as
𝑦 = 𝑥, 𝑚1, … , 𝑚𝑛
Mechanism Designer
• Assumption : mechanism designer doesnt have source of funds, then
the monetary payments that players make or receive have to be selffinanced
𝑛
𝑚𝑖 ≤ 0
𝑖=1
• Monetary transfer can be negative, means the designer can keep
some of the monet raised by players
Mechanism Design’s Objectives
• 𝑓 𝜃 = (𝑥 𝜃 , 𝑚1 𝜃 , … , 𝑚𝑛 𝜃 )
• 𝑥 𝜃 𝑖𝑠 𝑑𝑒𝑐𝑖𝑠𝑖𝑜𝑛 𝑟𝑢𝑙𝑒 and (𝑚1 𝜃 , … , 𝑚𝑛 𝜃 ) is transfer rule
Mechanism Game
• Mechanism designer desires to implement 𝑓 . , but he cannot do
this dorectly because the choice rule depends on unobserved state of
the world (𝜃)
• 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 𝜃𝑖 is only observed by player 𝑖 and not other’s players
(incomplete information), and players given prior over types (𝜙), then
this set up follows Bayesian Game
In formal game, we have :
Mechanism 𝑀 = Θ1, … , Θ𝑛, 𝑓 .
Strategy profile 𝑠 ∗ . = 𝑠 ∗ . , … , 𝑠 ∗ 𝑛 .
Outcome 𝑔 𝑠 ∗ .
= 𝑓(. )
,
Revelation Principle
• Mechanism 𝑀 = Θ1, … , Θ𝑛, 𝑓 . that implements social choice
function 𝑓 . using equilibrium strategy profile 𝑠 ∗ . =
𝑠 ∗ . , … , 𝑠 ∗ 𝑛 . , 𝑎𝑛𝑑 𝑔 𝑠 ∗ . = 𝑓(. )
• If the players are playing a mechanism that results in the
implementation of 𝑓(. ) then by construction of equilibrium beliefs
they must know that 𝑓(. ) will be implemented, and hence the
mechanism designer might as well implement it directly