Download Mechanism Design: Dominant Strategies

Mechanism Design: Dominant Strategies
Jeffrey Ely
May 20, 2014
Jeffrey Ely
Mechanism Design: Dominant Strategies
Some Motivation
Previously we considered the problem of matching workers with firms
We considered some different institutions for tackling the incentive
problem arising from asymmetric information
What if we could design the institution? What is possible?
Formally an institution is a game.
Jeffrey Ely
Mechanism Design: Dominant Strategies
Social Choice Problems
A set of individuals i = 1, . . . , N.
Y : the set of alternatives,
Θi : the set of possible types for i ∈ N
v̂i (y , θ ): utility index of i if the type profile is θ ∈ ×j Θj .
I
For today and most of this class, we will specialize to private values:
v̂i (y , θ ) = v̂i (y , θi )
Jeffrey Ely
Mechanism Design: Dominant Strategies
Social Choice Functions
Definition
A social choice function is a mapping f : Θ → Y which describes the
desired outcome as a function of the agents types.
Jeffrey Ely
Mechanism Design: Dominant Strategies
Examples
Single object/single buyer (tioli) Auction setting. etc.
Examples of social choice functions.
Jeffrey Ely
Mechanism Design: Dominant Strategies
Game Forms
Consider any extensive-form where
The players are the individuals i = 1, . . . , N
Attached to terminal nodes are elements of Y .
We can then write g (σ) for the element of Y that is attached to the
terminal node reached by σ.
And consider the Bayesian extensive-form game in which the players are
first privately informed of their types and then they play out the extensive
form.
Jeffrey Ely
Mechanism Design: Dominant Strategies
Dominant Strategies
Definition
Given an extensive-form Γ, suppose that for each player i there is a
strategy profile σ
v̂i (g (σ(θ )), θi ) ≥ v̂i (g (σ̂i (θi ), σ−i (θ−i )), θi )
for all θ, and for all σ̂i (θi ). Then we say that σ is an (ex post)
dominant-strategy solution of Γ.
It is standard in mechanism design to use this definition of dominant
strategies.
Jeffrey Ely
Mechanism Design: Dominant Strategies
Implementation
Definition
If there is an extensive-form Γ with a dominant strategy solution σ such
that
f (θ ) = g (σ(θ )),
then we say that the social choice function f is implemented in dominant
strategies by Γ. We refer to Γ as the implementing mechanism. And we say
that the social choice function f is implementable in dominant strategies.
Jeffrey Ely
Mechanism Design: Dominant Strategies
Incentive Compatibility
Proposition
Suppose that the social choice function f is implementable in dominant
strategies. Then the dominant strategy incentive compatibility constraints
are satisfied:
v̂i (f (θ ), θi ) ≥ v̂i (f (θ̂i , θ−i ), θi )
for all θi , θ−i , θ̂i .
Jeffrey Ely
Mechanism Design: Dominant Strategies
Incentive Compatibility
Definition
If a social choice function satisfies the dominant-strategy
incentive-compatibility constraints we say that it is dominant strategy
incentive compatible, or DSIC.
Jeffrey Ely
Mechanism Design: Dominant Strategies
Examples Continued
Check incentive compatibility of the social choice functions from the
examples.
Jeffrey Ely
Mechanism Design: Dominant Strategies
Direct Revelation Mechanisms
Consider the simplest possible game form to implement a social choice
function f .
Simultaneous moves.
Each player’s action set Ai is simply Θi
I
I
I
Essentially, the players are simultaneously announcing types.
Of course nothing stops them from lying.
The truthful strategy for i is σi (θi ) = θi for all θi .
g ( θ ) = f ( θ ).
Note that this mechanism is uniquely defined for a given f . We call it the
direct-revelation mechanism associated with the social choice function f .
Jeffrey Ely
Mechanism Design: Dominant Strategies
Incentive Compatible Direct Revelation Mechanisms
When will the truthful strategies be a dominant strategy solution of a
direct revelation mechanism?
Proposition
Given a social choice function f , if the dominant strategy incentive
compatibility constraints are satisfied then the truthful strategies are a
dominant strategy solution of the direct-revelation mechanism associated
with f .
Jeffrey Ely
Mechanism Design: Dominant Strategies
The Revelation Principle
The Revelation Principle says that in order to implement a social choice
function f , we don’t need to consider games that are any more complicated
than direct-revelation mechanisms.
Proposition
A social choice function is implementable in dominant strategies if and only
if it is DSIC
Jeffrey Ely
Mechanism Design: Dominant Strategies
Monetary Transfers and Quasi-Linear Utilities
Today we will examine a setup that is typical in economic applications.
Monetary transfers: Y = X × RN , so y = (x, t ) where
I x ∈ X is an alternative that affects all agents.
I t = (t , . . . , t ) is a monetary transfer scheme where t is the payment
1
i
N
made by agent i.
Quasi-linear utilities:
v̂i (y , θi ) = vi (x, θi ) − ti
For example, the single object/single buyer example and the auction
example.
Jeffrey Ely
Mechanism Design: Dominant Strategies
DSIC Mechanisms For This Setting
A social choice function has two components
α : Θ → X is allocation rule,
τ : Θ → RN is the set of transfers.
In this setting, DSIC means that ∀i ∈ N, ∀θ−i ∈ Θ−i ,
vi (α(θi , θ−i ), θi ) − τi (θi , θ−i ) ≥ vi (α(θ̂i , θ−i ), θi ) − τi (θ̂i , θ−i ),
for all θ̂i ∈ Θi .
The pair (α, τ ) is also referred to as a mechanism.
Jeffrey Ely
Mechanism Design: Dominant Strategies
Efficient Mechanisms
Definition
An efficient mechanism is a mechanism Γ = (α, τ ), where the allocation
rule is such that
N
α(θ ) ∈ arg max x ∈X
∑ vj (x, θj ).
j =1
Jeffrey Ely
Mechanism Design: Dominant Strategies
The Efficient Allocation for the Auction Example
Jeffrey Ely
Mechanism Design: Dominant Strategies
The Vickrey-Clarke-Groves Mechanism
The VCG mechanism (Vickrey-Clarke-Groves mechanism) is a mechanism
Γ = (α, τ ), where α is the efficient allocation rule and the transfer rule τ is
defined as follows.
τi (θ ) =
∑ vj (α(θ̄i , θ−i ), θj ) − ∑ vj (α(θ ), θj )
i 6 =j
i 6 =j
where θ̄ = (θ̄1 , . . . , θ̄n ) is a pre-specified profile of default types and
Jeffrey Ely
Mechanism Design: Dominant Strategies
Important
The VCG mechanism is a direct-revelation game. All of the θ’s in the
formula are the announced θ’s of the players, as opposed to their actual
θ’s. The rules of the mechanism can never depend on the actual θ’s of the
players.
Jeffrey Ely
Mechanism Design: Dominant Strategies
VCG Mechanism For The Auction Example
Jeffrey Ely
Mechanism Design: Dominant Strategies
VCG is DSIC
Proposition
For any profile of default types the VCG mechanism is dominant strategy
incentive compatible.
Jeffrey Ely
Mechanism Design: Dominant Strategies
Further Specialization
Linear values:
vi (x, θi ) = θi vi (x ).
Θi = [0, 1].
Jeffrey Ely
Mechanism Design: Dominant Strategies
Example: Bilateral Trade
Two individuals, buyer and seller.
Seller possesses an indivisible object which the buyer is potentially
interested in.
The alternatives are X = ∆{trade, notrade}.
Both have zero value from no trade:
vb (notrade) = vs (notrade) = 0.
Buyer has a value θb for the good and seller has cost θs for selling the
good:
vb (x ) = x
vs (x ) = −x
so that the buyer’s utility from trading with probability x and paying
tb is
θb vb (x ) − tb = x θb − tb
and the seller’s utility from trading with probabilty x and receiving ts is
θs vs (x ) + ts = ts − x θs
Jeffrey Ely
Mechanism Design: Dominant Strategies
Efficient Trade
The efficient allocation rule is
(
∗
α (θ ) =
trade
if θb > θs
notrade otherwise
We know that the VCG mechanism is a DSIC efficient mechanism. Let’s
calculate the transfers τ (θ ). We will pick the default types to be θb = 0,
θs = 1.
When there is no trade, τb (θ ) = τs (θ ) = 0. When there is trade, i.e. when
θb > θs ,
τbVCG (θ ) = θs (buyer pays seller’s cost),
τsVCG (θ ) = −θb (and seller receives buyers value).
There is a deficit since θb > θs .
Jeffrey Ely
Mechanism Design: Dominant Strategies
A Convenient Characterization of DSIC
Call Ui (θ ) the ex-post (or indirect) utility of the mechanism.
Ui (θ ) = θi vi (α(θ )) − τ (θ ).
DSIC can be expressed using Ui :
Ui (θ ) ≥ θi vi (α(θ̂i , θ−i )) − τ (θ̂i , θ−i )
= θ̂i vi (α(θ̂i , θ−i )) + (θi − θ̂i )vi (α(θ̂i , θ−i )) − τ (θ̂i , θ−i )
= Ui (θ̂i , θ−i ) + (θi − θ̂i )vi (α(θ̂i , θ−i ))
This is a way of expressing the constraint that θi does not want to
misreport his type to be θ̂i .
Jeffrey Ely
Mechanism Design: Dominant Strategies
A Convenient Characterization of DSIC
Similarly, DSIC means that type θ̂i does not want to misreport his type to
be θi :
Ui (θ̂i , θ−i ) ≥ Ui (θ ) + (θ̂i − θi )vi (α(θ )).
From these two conditions, assuming WLOG that θi > θ̂i it follows that
∀ θ −i ∈ Θ −i
vi (α(θ )) ≥
Ui (θ ) − Ui (θ̂i , θ−i )
≥ vi (α(θ̂i , θ−i )).
θi − θ̂i
Jeffrey Ely
Mechanism Design: Dominant Strategies
Monotonicity
Lemma
If (α, τ ) is DSIC, then
θi > θ̂i
⇒
vi (α(θ )) ≥ vi (α(θ̂i , θ−i )).
That is, vi (α(·, θ−i )) is an increasing function, ∀θ−i .
Jeffrey Ely
Mechanism Design: Dominant Strategies
Smoothness
Since a monotonic function is continuous almost everywhere,
lim vi (α(θ̂i , θ−i )) = vi (α(θ )),
θ̂i →θi
almost everywhere, therefore recalling that
vi (α(θ )) ≥
Ui (θ ) − Ui (θ̂i , θ−i )
≥ vi (α(θ̂i , θ−i )).
θi − θ̂i
we obtain
Ui (θ ) − Ui (θ̂i , θ−i )
∂Ui (θi , θ−i )
= lim
= vi (α(θ )).
∂θi
θi − θ̂i
θ̂i →θi
Thus Ui is differentiable almost everywhere.
Jeffrey Ely
Mechanism Design: Dominant Strategies
Payoff Equivalence
Monotonicity implies that this derivative is increasing, thus Ui is a convex
differentiable function. A convex differentiable function is the integral of its
derivative:
Lemma
The indirect utility function from a DSIC mechanism satisfies
Ui (θ̄i , θ−i ) = Ui (0, θ−i ) +
Z θ̄i
0
vi (α(s, θ−i ))ds,
∀θ̄i .
This is called payoff equivalence (also envelope condition or Mirrlees
condition)
Jeffrey Ely
Mechanism Design: Dominant Strategies
Payoff Equivalence
This means that any two DSIC mechanisms implementing the same
allocation rule α give the same indirect utility function up to a family of
constants (the numbers Ui (0, θ−i ).) In particular, for any pair of types
θi , θi0 of i and any profile θ−i of types of the other agents,
Ui (θi , θ−i ) − Ui (θi0 , θ−i ) = Ûi (θi , θ−i ) − Ûi (θi0 , θ−i )
where U and Û represent the indirect utility functions of the two
mechanisms.
Jeffrey Ely
Mechanism Design: Dominant Strategies
Revenue Equivalence
Since
Ui (θ ) = θi vi (α(θ )) − τi (θ )
another way of saying this is that the transfer rule is unique up to a family
of constants. That is
τi (θi , θ−i ) − τi (θi0 , θ−i ) = τ̂i (θi , θ−i ) − τ̂i (θi0 , θ−i )
This is the revenue equivalence theorem.
Jeffrey Ely
Mechanism Design: Dominant Strategies
Applying Payoff Equivalence To The Bilateral Trade
Example
Let τbVCG (·) and τsVCG (·) be the transfer functions in the VCG mechanism
for the bilateral trade problem. By payoff equivalence, any DSIC efficient
mechanism has a transfer rule τ̂ satisfying
τ̂b (θ ) − τ̂b (0, θs ) = τbVCG (θ ) − τbVCG (0, θs )
τ̂s (θ ) − τ̂s (θb , 1) = τsVCG (θ ) − τsVCG (θb , 1).
Substituting the expressions for the VCG transfers in this problem and
rearranging, we obtain
τ̂b (θ ) = θs + τ̂s (0, θs )
τ̂s (θ ) = −θb + τ̂s (θb , 1).
Jeffrey Ely
Mechanism Design: Dominant Strategies
Individual Rationality
Typically we assume that individuals cannot be compelled to participate in
the mechanism. If we normalize their reservation utility to be zero, then we
can express this individual rationality constraint:
Ui ( θ ) ≥ 0
for all θ. This is an ex post individual rationality constraint as it is required
to hold for all θ which is in the spirit of DSIC.
Jeffrey Ely
Mechanism Design: Dominant Strategies
Individual Rationality in The Bilateral Trade Example
If the mechanism is ex-post individually rational then
τ̂b (0, θs ) ≤ 0
τ̂s (θb , 1) ≤ 0
because types θb = 0 and θs = 1 will have a zero value (regardless of the
mechanism.)
Jeffrey Ely
Mechanism Design: Dominant Strategies
Budget Surpluses and Deficits
Definition
The (ex-post) budget surplus of a mechanism at a type profile θ is given by
S (θ ) =
∑ τi (θ )
i
If S (θ ) ≥ 0 for all θ then we say that the mechanism never runs a
deficit.
If S (θ ) = 0 for all θ then we say that the transfer rule and the
associated mechanism are ex post budget balanced.
Jeffrey Ely
Mechanism Design: Dominant Strategies
Budget Surpluses in The Bilateral Trade Example
Consider any efficient DSIC mechanism in the bilateral trade example. The
ex-post budget surplus at a profile θ where α(θ ) = trade will be equal to
τbVCG (θ ) + τsVCG (θ ) + τ̂ (0, θs ) + τ̂ (θb , 0).
and since τ̂ (0, θs ) ≤ 0 and τ̂ (θb , 0) ≤ 0, any efficient DSIC mechanism has
a smaller ex-post budget surplus than the VCG mechanism (with default
type profile θ̄ = (0, 1).)
Jeffrey Ely
Mechanism Design: Dominant Strategies
Budget Balance in the Bilateral Trade Example
Recall that the VCG mechanism runs a deficit at every profile where
θb > θs . Hence
Proposition
There does not exist an efficient, DSIC, ex-post IR, budget-balancing
mechanism for the bilateral trade problem.
Jeffrey Ely
Mechanism Design: Dominant Strategies