dominant strategy equilibrium - ucla econ · applications second price auction second price auction...

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.



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

a∗i ∈ Ai is strictly dominant action for player i if

ui (a∗i , a−i ) > ui (ai , a−i ) ∀ai 6= a∗i , ∀a−i ∈ A−i .



Weakly Dominant Action

An action is weakly dominant if it is “always” optimal and every other

action is “sometimes” not optimal.

Weakly Dominant Action

a∗i ∈ Ai is weakly dominant action for player i if

ui (a∗i , 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= a∗i .




An action profile a∗ is a dominant strategy equilibrium if a∗i is an

optimal action independent of the other players’ choice for every i .


a∗ ∈ A is a dominant strategy equilibrium if for every i ∈ N,

ui (a∗i , a−i ) ≥ ui (ai , a−i ) ∀ai ∈ Ai , ∀a−i ∈ A−i .

Note: When a∗ is a dominant strategy equilibrium, each a∗i may not be even weakly

dominant.



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.


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.

I Bidders make bids b = (b1, ..., bn) simultaneously.

I The highest bidder wins the object and pays the second highest bid.

I If there are more than one highest bidders, then one is randomly

selected as the winner and pays own bid.



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.

I 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.

I If everyone else’s bid is lower than your value, then you can win the

object by bidding your true value and gets vi −maxj 6=i bj . Since your

payment is always the same when winning, it is again optimal to

announce your true value and wins the object.



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?


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.




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.



Remark.

Location game is a similar game in Economics.

I Two restaurants A and B.

I A continuum of consumers are distributed on a “street” [0, 1].

I Restaurants choose where to locate simultaneously. Consumers go to a

closer restaurant.

I The objective of the restaurants is to maximize the (expected) number

of customers.




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∗.



What if there are three candidates?

What if there are three candidates and two are elected?

What if the policy space is two dimensional?


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

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


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.



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.



General Quasi-linear Environment and Mechanism

Consider the following more general environment:

I Set of possible outcomes: X

I Agent i ’s utility function is ui : X × Ωi → <. 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:

I Mi : a set of “messages” for agent i

I x : M → X : allocation function

I mi : M → <: payment function for agent i

A mechanism induces a strategic game.



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.



VCG Mechanism

Now we define Vickrey-Clark-Gloves (VCG) mechanism.

VCG Mechanism

(x∗, (Ωi ), (m∗i )) is a Vickrey-Clark-Gloves (VCG) mechanism if it

satisfies

x∗(ω) ∈ maxx∈X∑

j uj(x , ωj)

m∗i (ω) = −∑

j 6=i uj(x∗(ω), ωj) + hi (ω−i ) for some hi : Ω−i → <.



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 , ωi and ω−i ,

ui (x∗(ωi , ω−i ), ωi )−m∗i (ωi , ω−i ) ≥ ui (x

∗(ωi , ω−i ), ωi )−m∗i (ωi , ω−i )

But this immediately follows from the definition of m∗ because

ui (x∗(ωi , ω−i ), ωi )−m∗i (ωi , ω−i ) =

∑j

uj(x∗(ωi , ω−i ), ωj)− hi (ω−i )

for every ωi , ωi and ω−i .



Here is one such mechanism for the public good problem. Let vi be

i ’s message.

I If∑n

j=1 vj < C , then no bridge and no payment.

I If∑n

j=1 vj ≥ C , then a bridge is built. Each individual’s total payment

is C/n + (n − 1)C/n −∑

j 6=i vj ≤ 0 .

For this particular mechanism, ωi = vi , x ∈ 0, 1,

ui (x , ωi ) =(vi − C

n

)x , and hi (ω−i ) = 0.



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 ) = maxj 6=i vj (verify

this yourself).


dominant strategy equilibrium - ucla econ · applications second price auction second price auction...

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