Notice What Is The Dominant Ethnicity In Australia & User's Guide Manuals
Dominant Strategy Equilibrium
Ichiro Obara
UCLA
January 10, 2012
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
1 / 22
Dominant Action and Dominant Strategy Equilibrium
Dominant Action
Most of games are strategic in the sense that one player’s optimal
choice depends on other players’ choice.
For some games, however, there exists an action that is optimal
independent of other players’ choice. Such an optimal action is called
dominant action (or dominant strategy).
Games with dominant actions are easy. It is reasonable to assume
that a dominant action is played.
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
2 / 22
Dominant Action and Dominant Strategy Equilibrium
Strictly Dominant Action
Consider a strategic game G . There are different types of dominant actions.
An action is stictly dominant if it is “always” strictly optimal.
Strictly Dominant Action
ai∗ ∈ Ai is strictly dominant action for player i if
ui (ai∗ , a−i ) > ui (ai , a−i ) ∀ai 6= ai∗ , ∀a−i ∈ A−i .
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
3 / 22
Dominant Action and Dominant Strategy Equilibrium
Weakly Dominant Action
An action is weakly dominant if it is “always” optimal and every other
action is “sometimes” not optimal.
Weakly Dominant Action
ai∗ ∈ Ai is weakly dominant action for player i if
ui (ai∗ , a−i ) ≥ ui (ai , a−i ) ∀ai ∈ Ai , a−i ∈ A−i
with strict inequality for some a−i ∈ A−i for any ai 6= ai∗ .
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
4 / 22
Dominant Action and Dominant Strategy Equilibrium
Dominant Strategy Equilibrium
An action profile a∗ is a dominant strategy equilibrium if ai∗ is an
optimal action independent of the other players’ choice for every i.
Dominant Strategy Equilibrium
a∗ ∈ A is a dominant strategy equilibrium if for every i ∈ N,
ui (ai∗ , a−i ) ≥ ui (ai , a−i ) ∀ai ∈ Ai , ∀a−i ∈ A−i .
Note: When a∗ is a dominant strategy equilibrium, each ai∗ may not be even weakly
dominant.
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
5 / 22
Dominant Action and Dominant Strategy Equilibrium
Prisoner’s Dilemma
C
D
C
1,1
-1,2
D
2,-1
0,0
This is a common PD. C and D stand for “cooperate” and “defect”
respectively. D is a strictly dominant action, hence (D, D) is the
(only) dominant strategy equilibrium.
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
6 / 22
Applications
Second Price Auction
Second Price Auction
Consider n bidders with values vi ≥ 0, i = 1, ..., n, who are competing
for some object. If bidder i wins the object and pays p, then bidder
i’s payoff is vi − p.
The rule of second price auction is as follows.
◮
Bidders make bids b = (b1 , ..., bn ) simultaneously.
◮
The highest bidder wins the object and pays the second highest bid.
◮
If there are more than one highest bidders, then one is randomly
selected as the winner and pays own bid.
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
7 / 22
Applications
Second Price Auction
Second Price Auction
Theorem
In second price auction, bidding one’s own value (bi = vi ) is a weakly
dominant action. Hence b ∗ = v is a dominant strategy equilibrium.
Here is why.
◮
If someone else’s bid is higher than your value, then you have to pay
more than your value by winning. So it is optimal to announce your
true value and lose.
◮
If everyone else’s bid is lower than your value, then you can win the
object by bidding your true value and gets vi − maxj6=i bj . Since your
payment is always the same when winning, it is again optimal to
announce your true value and wins the object.
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
8 / 22
Applications
Second Price Auction
Remarks:
Second price auction is formally “equivalent” to english auction,
where the current highest bid is updated dynamically. So it is very
popular in real world. In fact, most internet auctions can be regarded
as a variant of second price auction.
We can also consider first and third price auction. Is it a dominant
action to bid your true value in these auctions?
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
9 / 22
Applications
Median Voter Theorem
Median Voter Theorem
Consider an election with two candidates A and B.
There are a continuum of citizens, whose most preferred policies are
distributed continuously on [0, 1] according to CDF F .
Candidates choose their policy simultaneously from [0, 1]. Each
citizen votes for the candidate whose policy is closer to his or her
most preferred policy. The candidate with majority votes wins.
If the candidates choose the same policy, each candidate wins the
election with 50%.
This is a strategic game between A and B.
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
10 / 22
Applications
Median Voter Theorem
Median Voter Theorem
Pick x ∗ ∈ [0, 1] such that F (x ∗ ) = 1 − F (x ∗ ) = 0.5. A voter at x ∗ is
called median voter.
50% of citizens are more “liberal” than the median voter and 50% of
citizens are more “conservative” than the median voter.
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
11 / 22
Applications
Median Voter Theorem
Remark.
Location game is a similar game in Economics.
◮
Two restaurants A and B.
◮
A continuum of consumers are distributed on a “street” [0, 1].
◮
Restaurants choose where to locate simultaneously. Consumers go to a
closer restaurant.
◮
The objective of the restaurants is to maximize the (expected) number
of customers.
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
12 / 22
Applications
Median Voter Theorem
Median Voter Theorem
Theorem
In this two candidate election model, it is weakly dominant for each
candidate to choose x ∗ .
Proof.
A candidate can win by choosing x ∗ when the other candidate does
not choose x ∗ .
A candidate wins with 50% by choosing x ∗ when the other candidate
chooses x ∗ . but he would surely lose if he chooses x 6= x ∗ .
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
13 / 22
Applications
Median Voter Theorem
What if there are three candidates?
What if there are three candidates and two are elected?
What if the policy space is two dimensional?
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
14 / 22
Applications
VCG mechanism
Public Good Problem
Consider the following situation.
There is a plan to build a bridge in some village with n residents.
Each resident’s benefit from the bridge is vi ∈ [0, v ]. But this
information is private.
The cost of building a bridge is C > 0, which must be equally shared.
Resident i’s payoff is vi − C /n if a bridge is built and 0 if not.
Assume that v < C < (n − 1)v .
The goal is to build a bridge in a socially efficient way, i.e. building a
P
bridge if and only if the total social benefit i vi is at least as large
as C .
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
15 / 22
Applications
VCG mechanism
An Example of Mechanism
Suppose that you ask everyone what his/her benefit is, build a bridge
if and only it looks socially efficient to build a bridge. Then you tax
everyone C /n. This mechanism induces an n-player game.
In this game, it is weakly dominant for any i with vi > C /n to say
that his or her benefit is v i and it is weakly dominant for any i with
vi < C /n to say that his or her benefit is 0.
This mechanism may generate an inefficient allocation.
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
16 / 22
Applications
VCG mechanism
Question: Can you build a tax scheme in which each resident reveals
his or her preference truthfully as a dominant strategy equilibrium and
an efficient allocation is implemented whatever each agent’s benefit
is?
The answer is yes. In fact, there exists such a mechanism in general
with private value and quasi-linear payoffs.
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
17 / 22
Applications
VCG mechanism
General Quasi-linear Environment and Mechanism
Consider the following more general environment:
◮
Set of possible outcomes: X
◮
Agent i’s utility function is ui : X × Ωi → ℜ. If x ∈ X is implemented
and player i pays mi , then the payoff of type ωi ∈ Ωi is ui (x, ωi ) − mi .
A mechanism is a triple (x, (Mi ), (mi )), which is defined by:
◮
Mi : a set of “messages” for agent i
◮
x : M → X : allocation function
◮
mi : M → ℜ: payment function for agent i
A mechanism induces a strategic game.
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
18 / 22
Applications
VCG mechanism
Revelation Principle
Revelation Principle: We can assume that Mi = Ωi without loss of
generality. That is, if there is any mechanism that implements a particular x
by a dominant strategy equilibrium, then there exists a direct mechanism
to implement the same x by a dominant strategy equilibrium.
Proof: For any mechanism, consider a direct mechanism that takes each
resident’s type as an input and generates the optimal message in the original
mechanism as an output, which is then used to generate the allocation and
the payments in the original mechanism. Clearly it is a dominant strategy
equilibrium for agents to announce their true type and the same allocation is
implemented.
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
19 / 22
Applications
VCG mechanism
VCG Mechanism
Now we define Vickrey-Clark-Gloves (VCG) mechanism.
VCG Mechanism
(x ∗ , (Ωi ), (mi∗ )) is a Vickrey-Clark-Gloves (VCG) mechanism if it
satisfies
P
x ∗ (ω) ∈ maxx∈X j uj (x, ωj )
P
mi∗ (ω) = − j6=i uj (x ∗ (ω), ωj ) + hi (ω−i ) for some hi : Ω−i → ℜ.
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
20 / 22
Applications
VCG mechanism
VCG Mechanism
Under VCG mechanism, it is a dominant strategy equilibrium for
agents to report their signals truthfully.
To show this, we just need to show that for every ωi , ω
bi and ω−i ,
ωi , ω−i )
ωi , ω−i ), ωi ) − mi∗ (b
ui (x ∗ (ωi , ω−i ), ωi ) − mi∗ (ωi , ω−i ) ≥ ui (x ∗ (b
But this immediately follows from the definition of m∗ because
ωi , ω−i ) =
ui (x ∗ (b
ωi , ω−i ), ωi ) − mi∗ (b
X
uj (x ∗ (b
ωi , ω−i ), ωj ) − hi (ω−i )
j
for every ωi , ω
bi and ω−i .
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
21 / 22
Applications
VCG mechanism
Here is one such mechanism for the public good problem. Let vei be
i’s message.
◮
◮
If
If
Pn
ej
j=1 v
< C , then no bridge and no payment.
Pn
ej
j=1 v
≥ C , then a bridge is built. Each individual’s total payment
P
is C /n + (n − 1)C /n − j6=i vej ≤ 0 .
For this particular mechanism, ωi = vi , x ∈ {0, 1},
ui (x, ωi ) = vi − Cn x, and hi (ω−i ) = 0.
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
22 / 22
Applications
VCG mechanism
VCG Mechanism
Remarks.
For this public good provision problem, truth-telling is a weakly dominant
action (because every type can be pivotal by assumption v < C < (n − 1)v ).
You can verify that the total payment is almost always less than C . We can
add arbitrary hi (ω−i ) to i’s payment in order to cover the cost of building a
bridge. However, it is known that the budget cannot be balanced in general.
That is, the total payment may be more or less than C depending on true ω.
Second price auction is a VCG mechanism where hi (v−i ) = maxj6=i vj (verify
this yourself).
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
23 / 22
Ichiro Obara
UCLA
January 10, 2012
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
1 / 22
Dominant Action and Dominant Strategy Equilibrium
Dominant Action
Most of games are strategic in the sense that one player’s optimal
choice depends on other players’ choice.
For some games, however, there exists an action that is optimal
independent of other players’ choice. Such an optimal action is called
dominant action (or dominant strategy).
Games with dominant actions are easy. It is reasonable to assume
that a dominant action is played.
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
2 / 22
Dominant Action and Dominant Strategy Equilibrium
Strictly Dominant Action
Consider a strategic game G . There are different types of dominant actions.
An action is stictly dominant if it is “always” strictly optimal.
Strictly Dominant Action
ai∗ ∈ Ai is strictly dominant action for player i if
ui (ai∗ , a−i ) > ui (ai , a−i ) ∀ai 6= ai∗ , ∀a−i ∈ A−i .
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
3 / 22
Dominant Action and Dominant Strategy Equilibrium
Weakly Dominant Action
An action is weakly dominant if it is “always” optimal and every other
action is “sometimes” not optimal.
Weakly Dominant Action
ai∗ ∈ Ai is weakly dominant action for player i if
ui (ai∗ , a−i ) ≥ ui (ai , a−i ) ∀ai ∈ Ai , a−i ∈ A−i
with strict inequality for some a−i ∈ A−i for any ai 6= ai∗ .
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
4 / 22
Dominant Action and Dominant Strategy Equilibrium
Dominant Strategy Equilibrium
An action profile a∗ is a dominant strategy equilibrium if ai∗ is an
optimal action independent of the other players’ choice for every i.
Dominant Strategy Equilibrium
a∗ ∈ A is a dominant strategy equilibrium if for every i ∈ N,
ui (ai∗ , a−i ) ≥ ui (ai , a−i ) ∀ai ∈ Ai , ∀a−i ∈ A−i .
Note: When a∗ is a dominant strategy equilibrium, each ai∗ may not be even weakly
dominant.
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
5 / 22
Dominant Action and Dominant Strategy Equilibrium
Prisoner’s Dilemma
C
D
C
1,1
-1,2
D
2,-1
0,0
This is a common PD. C and D stand for “cooperate” and “defect”
respectively. D is a strictly dominant action, hence (D, D) is the
(only) dominant strategy equilibrium.
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
6 / 22
Applications
Second Price Auction
Second Price Auction
Consider n bidders with values vi ≥ 0, i = 1, ..., n, who are competing
for some object. If bidder i wins the object and pays p, then bidder
i’s payoff is vi − p.
The rule of second price auction is as follows.
◮
Bidders make bids b = (b1 , ..., bn ) simultaneously.
◮
The highest bidder wins the object and pays the second highest bid.
◮
If there are more than one highest bidders, then one is randomly
selected as the winner and pays own bid.
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
7 / 22
Applications
Second Price Auction
Second Price Auction
Theorem
In second price auction, bidding one’s own value (bi = vi ) is a weakly
dominant action. Hence b ∗ = v is a dominant strategy equilibrium.
Here is why.
◮
If someone else’s bid is higher than your value, then you have to pay
more than your value by winning. So it is optimal to announce your
true value and lose.
◮
If everyone else’s bid is lower than your value, then you can win the
object by bidding your true value and gets vi − maxj6=i bj . Since your
payment is always the same when winning, it is again optimal to
announce your true value and wins the object.
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
8 / 22
Applications
Second Price Auction
Remarks:
Second price auction is formally “equivalent” to english auction,
where the current highest bid is updated dynamically. So it is very
popular in real world. In fact, most internet auctions can be regarded
as a variant of second price auction.
We can also consider first and third price auction. Is it a dominant
action to bid your true value in these auctions?
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
9 / 22
Applications
Median Voter Theorem
Median Voter Theorem
Consider an election with two candidates A and B.
There are a continuum of citizens, whose most preferred policies are
distributed continuously on [0, 1] according to CDF F .
Candidates choose their policy simultaneously from [0, 1]. Each
citizen votes for the candidate whose policy is closer to his or her
most preferred policy. The candidate with majority votes wins.
If the candidates choose the same policy, each candidate wins the
election with 50%.
This is a strategic game between A and B.
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
10 / 22
Applications
Median Voter Theorem
Median Voter Theorem
Pick x ∗ ∈ [0, 1] such that F (x ∗ ) = 1 − F (x ∗ ) = 0.5. A voter at x ∗ is
called median voter.
50% of citizens are more “liberal” than the median voter and 50% of
citizens are more “conservative” than the median voter.
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
11 / 22
Applications
Median Voter Theorem
Remark.
Location game is a similar game in Economics.
◮
Two restaurants A and B.
◮
A continuum of consumers are distributed on a “street” [0, 1].
◮
Restaurants choose where to locate simultaneously. Consumers go to a
closer restaurant.
◮
The objective of the restaurants is to maximize the (expected) number
of customers.
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
12 / 22
Applications
Median Voter Theorem
Median Voter Theorem
Theorem
In this two candidate election model, it is weakly dominant for each
candidate to choose x ∗ .
Proof.
A candidate can win by choosing x ∗ when the other candidate does
not choose x ∗ .
A candidate wins with 50% by choosing x ∗ when the other candidate
chooses x ∗ . but he would surely lose if he chooses x 6= x ∗ .
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
13 / 22
Applications
Median Voter Theorem
What if there are three candidates?
What if there are three candidates and two are elected?
What if the policy space is two dimensional?
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
14 / 22
Applications
VCG mechanism
Public Good Problem
Consider the following situation.
There is a plan to build a bridge in some village with n residents.
Each resident’s benefit from the bridge is vi ∈ [0, v ]. But this
information is private.
The cost of building a bridge is C > 0, which must be equally shared.
Resident i’s payoff is vi − C /n if a bridge is built and 0 if not.
Assume that v < C < (n − 1)v .
The goal is to build a bridge in a socially efficient way, i.e. building a
P
bridge if and only if the total social benefit i vi is at least as large
as C .
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
15 / 22
Applications
VCG mechanism
An Example of Mechanism
Suppose that you ask everyone what his/her benefit is, build a bridge
if and only it looks socially efficient to build a bridge. Then you tax
everyone C /n. This mechanism induces an n-player game.
In this game, it is weakly dominant for any i with vi > C /n to say
that his or her benefit is v i and it is weakly dominant for any i with
vi < C /n to say that his or her benefit is 0.
This mechanism may generate an inefficient allocation.
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
16 / 22
Applications
VCG mechanism
Question: Can you build a tax scheme in which each resident reveals
his or her preference truthfully as a dominant strategy equilibrium and
an efficient allocation is implemented whatever each agent’s benefit
is?
The answer is yes. In fact, there exists such a mechanism in general
with private value and quasi-linear payoffs.
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
17 / 22
Applications
VCG mechanism
General Quasi-linear Environment and Mechanism
Consider the following more general environment:
◮
Set of possible outcomes: X
◮
Agent i’s utility function is ui : X × Ωi → ℜ. If x ∈ X is implemented
and player i pays mi , then the payoff of type ωi ∈ Ωi is ui (x, ωi ) − mi .
A mechanism is a triple (x, (Mi ), (mi )), which is defined by:
◮
Mi : a set of “messages” for agent i
◮
x : M → X : allocation function
◮
mi : M → ℜ: payment function for agent i
A mechanism induces a strategic game.
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
18 / 22
Applications
VCG mechanism
Revelation Principle
Revelation Principle: We can assume that Mi = Ωi without loss of
generality. That is, if there is any mechanism that implements a particular x
by a dominant strategy equilibrium, then there exists a direct mechanism
to implement the same x by a dominant strategy equilibrium.
Proof: For any mechanism, consider a direct mechanism that takes each
resident’s type as an input and generates the optimal message in the original
mechanism as an output, which is then used to generate the allocation and
the payments in the original mechanism. Clearly it is a dominant strategy
equilibrium for agents to announce their true type and the same allocation is
implemented.
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
19 / 22
Applications
VCG mechanism
VCG Mechanism
Now we define Vickrey-Clark-Gloves (VCG) mechanism.
VCG Mechanism
(x ∗ , (Ωi ), (mi∗ )) is a Vickrey-Clark-Gloves (VCG) mechanism if it
satisfies
P
x ∗ (ω) ∈ maxx∈X j uj (x, ωj )
P
mi∗ (ω) = − j6=i uj (x ∗ (ω), ωj ) + hi (ω−i ) for some hi : Ω−i → ℜ.
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
20 / 22
Applications
VCG mechanism
VCG Mechanism
Under VCG mechanism, it is a dominant strategy equilibrium for
agents to report their signals truthfully.
To show this, we just need to show that for every ωi , ω
bi and ω−i ,
ωi , ω−i )
ωi , ω−i ), ωi ) − mi∗ (b
ui (x ∗ (ωi , ω−i ), ωi ) − mi∗ (ωi , ω−i ) ≥ ui (x ∗ (b
But this immediately follows from the definition of m∗ because
ωi , ω−i ) =
ui (x ∗ (b
ωi , ω−i ), ωi ) − mi∗ (b
X
uj (x ∗ (b
ωi , ω−i ), ωj ) − hi (ω−i )
j
for every ωi , ω
bi and ω−i .
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
21 / 22
Applications
VCG mechanism
Here is one such mechanism for the public good problem. Let vei be
i’s message.
◮
◮
If
If
Pn
ej
j=1 v
< C , then no bridge and no payment.
Pn
ej
j=1 v
≥ C , then a bridge is built. Each individual’s total payment
P
is C /n + (n − 1)C /n − j6=i vej ≤ 0 .
For this particular mechanism, ωi = vi , x ∈ {0, 1},
ui (x, ωi ) = vi − Cn x, and hi (ω−i ) = 0.
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
22 / 22
Applications
VCG mechanism
VCG Mechanism
Remarks.
For this public good provision problem, truth-telling is a weakly dominant
action (because every type can be pivotal by assumption v < C < (n − 1)v ).
You can verify that the total payment is almost always less than C . We can
add arbitrary hi (ω−i ) to i’s payment in order to cover the cost of building a
bridge. However, it is known that the budget cannot be balanced in general.
That is, the total payment may be more or less than C depending on true ω.
Second price auction is a VCG mechanism where hi (v−i ) = maxj6=i vj (verify
this yourself).
Obara (UCLA)
Dominant Strategy Equilibrium
January 10, 2012
23 / 22