Notice What Is The Dominant Ethnicity In Australia & User's Guide Manuals

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Dominant Strategy Equilibrium

Ichiro Obara

UCLA

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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).

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Strictly Dominant Action

Consider a strategic gameG. There are different types of dominant actions. An action isstictly dominant if it is “always” strictly optimal.

Strictly Dominant Action

a∗

i ∈Ai isstrictly dominant actionfor playeri if

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Weakly Dominant Action

An action is weakly dominantif it is “always” optimal and every other action is “sometimes” not optimal.

Weakly Dominant Action a∗

i ∈Ai isweakly dominant action for player i if

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

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Dominant Strategy Equilibrium

An action profilea∗ _{is a} _{dominant strategy equilibrium}_if _a∗ i 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(a∗i,a−i)≥ui(ai,a−i) ∀ai ∈Ai, ∀a−i ∈A−i.

Note: Whena∗is a dominant strategy equilibrium, eacha∗

i may not be even weakly

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Prisoner’s Dilemma

C D

C 1,1 -1,2

D 2,-1 0,0

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Applications Second Price Auction

Second Price Auction

Considern bidders with values vi ≥0,i = 1, ...,n, who are competing for some object. If bidder i wins the object and paysp, then bidder

i’s payoff isvi −p.

The rule of second price auctionis as follows. ◮ Bidders make bidsb= (b

1, ...,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

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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 getsvi−maxj6=ibj. Since your

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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.

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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 CDFF.

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%.

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Median Voter Theorem

Pickx∗ _∈_[0,_{1] such that}_F₍_x∗_{) = 1}₋_F₍_x∗_{) = 0.5. A voter at}_x∗ _is calledmedian voter.

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Remark.

Location gameis 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

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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 choosingx∗ _{when the other candidate does} not choose x∗_.

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What if there are three candidates?

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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 isvi ∈[0,v]. But this information is private.

The cost of building a bridge isC >0, which must be equally shared. Resident i’s payoff isvi−C/n if a bridge is built and 0 if not. Assume thatv <C <(n−1)v.

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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 everyoneC/n. Thismechanisminduces ann-player game.

In this game, it is weakly dominant for anyi with vi >C/n to say that his or her benefit is vi and it is weakly dominant for anyi with

vi <C/n to say that his or her benefit is 0.

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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?

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General Quasi-linear Environment and Mechanism

Consider the following more general environment: ◮ Set of possible outcomes: X

◮ Agenti’s utility function isu

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Revelation Principle

Revelation Principle: We can assume that Mi= Ωi without loss of

generality. That is, if there is any mechanism that implements a particularx by a dominant strategy equilibrium, then there exists adirect mechanism to implement the samex by a dominant strategy equilibrium.

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VCG Mechanism

Now we defineVickrey-Clark-Gloves (VCG) mechanism.

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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,ωbi andω−i,

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

But this immediately follows from the definition of m∗ _because

ui(x∗(ωib, ω−i), ωi)−mi∗(ωib, ω−i) =

X

j

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

i’s message.

◮ IfPn

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

◮ IfPn

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

isC/n+ (n−1)C/n−P_j₆₌_ievj ≤0 .

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

ui(x, ωi) = vi−C_n

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VCG Mechanism

Remarks.

For this public good provision problem, truth-telling is a weakly dominant action (because every type can be pivotal by assumptionv <C<(n−1)v).

You can verify that the total payment is almost always less thanC. We can add arbitraryhi(ω−i) toi’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 thanC depending on trueω.

Second price auction is a VCG mechanism wherehi(v−i) = maxj6=ivj (verify