Introduction to Mechanism Design Thodoris Lykouris National - - PowerPoint PPT Presentation

Introduction Social welfare maximization Revenue maximization

Introduction to Mechanism Design

Thodoris Lykouris

National Technical University of Athens

May 16, 2013

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization

Contents

Introduction Motivation Game Theory Mechanism Design Social welfare maximization Single-item auction VCG Mechanims Examples Revenue maximization Bayesian setting Optimal mechanism

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Motivation

Algorithmic Game Theory

• 1. Theoretical Computer Science focuses on algorithms that are:

◮ fast ◮ (approximately) optimal Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Motivation

Algorithmic Game Theory

• 1. Theoretical Computer Science focuses on algorithms that are:

◮ fast ◮ (approximately) optimal

• 2. Economics take into consideration people’s incentives

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Motivation

Algorithmic Game Theory

• 1. Theoretical Computer Science focuses on algorithms that are:

◮ fast ◮ (approximately) optimal

• 2. Economics take into consideration people’s incentives

Intersection of those two!

◮ Algorithms are usually run on people that have incentives and

may not follow them if they can make things better for them by deviating (Algorithmic Game Theory)

◮ Thus we need to design the rules of the game so that we can

deal with this (Mechanism Design)

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Motivation

Examples

◮ Routing in roads

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Motivation

Examples

◮ Routing in roads (mechanism: tolls, odd/even numbers)

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Motivation

Examples

◮ Routing in roads (mechanism: tolls, odd/even numbers) ◮ Facility location

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Motivation

Examples

◮ Routing in roads (mechanism: tolls, odd/even numbers) ◮ Facility location (mechanism: median for 1-FL)

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Motivation

Examples

◮ Routing in roads (mechanism: tolls, odd/even numbers) ◮ Facility location (mechanism: median for 1-FL) ◮ Elections

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Motivation

Examples

◮ Routing in roads (mechanism: tolls, odd/even numbers) ◮ Facility location (mechanism: median for 1-FL) ◮ Elections (mechanism: Condorcet winner)

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Motivation

Examples

◮ Routing in roads (mechanism: tolls, odd/even numbers) ◮ Facility location (mechanism: median for 1-FL) ◮ Elections (mechanism: Condorcet winner) ◮ Scheduling

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Motivation

Examples

◮ Routing in roads (mechanism: tolls, odd/even numbers) ◮ Facility location (mechanism: median for 1-FL) ◮ Elections (mechanism: Condorcet winner) ◮ Scheduling (mechanism: use of payments)

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Motivation

Examples

◮ Routing in roads (mechanism: tolls, odd/even numbers) ◮ Facility location (mechanism: median for 1-FL) ◮ Elections (mechanism: Condorcet winner) ◮ Scheduling (mechanism: use of payments) ◮ Auctions

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Motivation

Examples

◮ Routing in roads (mechanism: tolls, odd/even numbers) ◮ Facility location (mechanism: median for 1-FL) ◮ Elections (mechanism: Condorcet winner) ◮ Scheduling (mechanism: use of payments) ◮ Auctions (mechanism: in the next slides)

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Game Theory

Prisoner’s dilemma

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Game Theory

Finite games

◮ n players ◮ Each player has a set of strategies Si ◮ Strategy profile is a vector of n strategies, one for each player:

σ = (σ1, σ2, . . . , σn)

◮ Pure strategy, when σi is deterministically chosen from Si ◮ Mixed strategy, when σi is a probability distribution on Si ◮ σ−i = (σ1, σ2, . . . , σi−1, σi+1, . . . , σn)

◮ For each player i, there is a payoff/utility function

ui : S1 × S2 × · · · × Sn → R

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Game Theory

Equilibria

Nash equilibrium: A strategy profile where no player has incentive to deviate unilaterally, given that no one else alters their strategy ∀ i, σ′

i : ui(σ) ≥ ui(σ−i, σ′ i)

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Game Theory

Equilibria

Nash equilibrium: A strategy profile where no player has incentive to deviate unilaterally, given that no one else alters their strategy ∀ i, σ′

i : ui(σ) ≥ ui(σ−i, σ′ i)

Dominant strategy equilibrium: A strategy profile in which every player’s strategy is at least as good as all other strategies, regardless of the actions of any other player ∀ i, σ′

i, σ′ −i : ui(σ′ −i, σi) ≥ ui(σ′ −i, σ′ i)

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Game Theory

Equilibria

Nash equilibrium: A strategy profile where no player has incentive to deviate unilaterally, given that no one else alters their strategy ∀ i, σ′

i : ui(σ) ≥ ui(σ−i, σ′ i)

Dominant strategy equilibrium: A strategy profile in which every player’s strategy is at least as good as all other strategies, regardless of the actions of any other player ∀ i, σ′

i, σ′ −i : ui(σ′ −i, σi) ≥ ui(σ′ −i, σ′ i)

In Prisoner’s Dilemma, (Confess,Confess) is both a NE and a DSE

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Game Theory

Battle of the sexes

◮ Alice prefers going to ballet to going to soccer. ◮ Bob prefers going to soccer to going to ballet ◮ Both prefer doing anything together than doing anything

alone. Alice \ Bob Ballet Soccer Ballet (10,3) (2,2) Soccer (0,0) (3,10)

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Game Theory

Battle of the sexes

◮ Alice prefers going to ballet to going to soccer. ◮ Bob prefers going to soccer to going to ballet ◮ Both prefer doing anything together than doing anything

alone. Alice \ Bob Ballet Soccer Ballet (10,3) (2,2) Soccer (0,0) (3,10) Here there are 2 NE [(Ballet, Ballet),(Soccer, Soccer)] but no DSE.

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Game Theory

Battle of the sexes (altered)

Bob can change the game in his profit. Alice \ Bob Ballet with mum Soccer Ballet with mum (-10,0) (-10,2) Soccer (0,0) (3,10)

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Game Theory

Battle of the sexes (altered)

Bob can change the game in his profit. Alice \ Bob Ballet with mum Soccer Ballet with mum (-10,0) (-10,2) Soccer (0,0) (3,10) Now there is just one NE (Soccer, Soccer).

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Game Theory

Battle of the sexes (altered)

Bob can change the game in his profit. Alice \ Bob Ballet with mum Soccer Ballet with mum (-10,0) (-10,2) Soccer (0,0) (3,10) Now there is just one NE (Soccer, Soccer).

◮ Generally, in Mechanism Design, the designer (in this case,

Bob) tries to change the game

◮ to maximize one objective function (in this case, Bob’s

happiness).

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Mechanism Design

Single-item auction: The setting

◮ n possible buyers ◮ possible outcomes: buyer i/no bidder gets it ◮ every possible buyer has a valuation for the painting

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Mechanism Design

Single-item auction: The setting

◮ n possible buyers ◮ possible outcomes: buyer i/no bidder gets it ◮ every possible buyer has a valuation for the painting

We need to find:

◮ Who will get the painting, if any? ◮ How much will he pay?

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Mechanism Design

Mechanism Design

◮ A set of players I = {1, 2, . . . , n} ◮ A set of outcomes A ◮ For every player i a set of possible valuations (functions):

Vi = {υi : A → R} υi(α):how much player i values outcome α

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Mechanism Design

Mechanism Design

◮ A set of players I = {1, 2, . . . , n} ◮ A set of outcomes A ◮ For every player i a set of possible valuations (functions):

Vi = {υi : A → R} υi(α):how much player i values outcome α Mechanism:

◮ An outcome function χ : V1 × V2 × · · · × Vn → A ◮ A payment function p = (p1, p2, . . . , pn), where

pi : V1 × V2 × · · · × Vn → R

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Mechanism Design

Induced game

◮ Players I = {1, 2, . . . , n} ◮ Strategies of player i : Vi ◮ Utility of player i, given strategy profile (v1, v2, . . . , vn):

ui(v1, v2, . . . , vn) = υi(χ(v1, v2, . . . , vn)) − pi(v1, v2, . . . , vn)

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Mechanism Design

Objective

Lead to an outcome that maximizes some objective function.

◮ Social welfare: Combined happiness of all players

n

i=1 υi(α) ◮ Revenue: The total profit of the auctioneer

n

i=1 pi(α)

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization

Contents

Introduction Motivation Game Theory Mechanism Design Social welfare maximization Single-item auction VCG Mechanims Examples Revenue maximization Bayesian setting Optimal mechanism

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Single-item auction

Single-item auction: The setting

◮ n possible buyers ◮ Each of them has a valuation for the product υi ◮ Each of them makes a bid bi

We need to find:

◮ Outcome function? ◮ Payment function?

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Single-item auction

First-price auction

The bidder i∗ with the highest bid bi takes the product (outcome) and pays b∗

i . ◮ Social welfare equal to max υi, so it is maximized ◮ Not stable ◮ Players have incentive to lie and misreport their true

valuations (bi = υi)

◮ Revenue, at best, equal to maxj=i∗ υj

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Single-item auction

Second-price auction (Vickrey auction)

The bidder with the highest bid takes the product and pays the second highest bid.

◮ If a bidder gets the product, what he pays doesn’t depend on

his bid.

◮ No reason to lie about his true valuation ◮ If he doesn’t have the highest valuation but he reports a

higher bid, he risks taking the product and have negative utility υi − b.

◮ If he declares a lowest bid, he risks not taking the product,

whilst what he pays, if he takes it, doesn’t change Hence, Vickrey auction is truthful.

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Single-item auction

Truthfulness

A mechanism is truthful (incentive compatible/strategyproof) iff no player has an incentive to misreport his true valuation and increase his utility. Formally, ∀ player i,∀ strategy profile (b1, b2, . . . , bn), bi ∈ Vi: υi(χ(b−i, υi)) − pi(b−i, υi) ≥ υi(χ(b−i, bi)) − pi(b−i, bi) This means that telling the truth is Dominant Strategy Equilibrium.

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization VCG Mechanims

Vickrey auction

◮ Goal: Maximizing the social welfare ◮ Single-item auction ◮ ALLOCATION (player with highest bid):

i = arg maxn

i=1 bi ◮ PAYMENT (second highest bid)

p∗

i = maxj=i∗ bj

• 1. Does not depend on bi∗
• 2. but only on b−i∗ and outcome

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization VCG Mechanims

Groves Auctions

◮ ALLOCATION (outcome that maximizes the social welfare):

χ(v1, v2, . . . , vn) = arg maxα∈A n

i=1 vi(α) ◮ PAYMENT (second highest bid)

p(v1, v2, . . . , vn) = hi(v−i) −

j=i vj(χ(v1, v2, . . . , vn))

• 1. hi(v−i): does not depend on vi

2.

j=i vj(χ(v1, v2, . . . , vn)): sum of valuations of other players

in given outcome

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization VCG Mechanims

Clarke pivot rule

hv−i = maxα′∈A

• j=i vi(α′)

◮ Maximum social welfare when player i does not participate

Intuition: Each player pays the damage they cause to others by their presence to the auction

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization VCG Mechanims

Observations

Every Groves Mechanism is truthful:

◮ Payment:

p(v1, v2, . . . , vn) = hi(v−i) −

j=i vj(χ(v1, v2, . . . , vn)) ◮ Utility of player i:

ui(v−i, vi) = vi(χ(v−i, vi)) − pi(v−i, vi) = vi(χ(v−i, vi)) − [hi(v−i) −

j=i vj(χ(v−i, vi))] = −hi(v−i) + j vj(χ(v−i, vi)) ◮ Hence, Something that does not take into account his bid +

Social welfare

◮ Mechanism maximizes social welfare, so player has no reason

to lie Clarke pivot rule satisfies individual rationality:

◮ For every player i: ui(v1, v2, . . . , vn) ≥ 0

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Examples

Connecting nodes in a graph

We want to connect A and B. Each edge has a cost (valuation is the negative cost). Implement VCG to maximize social welfare in this setting. A D C B CAD = 2 CAC = 1 CBC = 4 CCD = 3 CBD = 5 Which edges will we use and how much will we pay them?

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Examples

Connecting nodes in a graph (solution)

◮ players: edges ◮ outcomes: paths from A to B. ◮ If edge e is used, player e has a valuation −ce.

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Examples

Connecting nodes in a graph (solution)

◮ players: edges ◮ outcomes: paths from A to B. ◮ If edge e is used, player e has a valuation −ce. ◮ Maximize social welfare: Pick shortest path (ACB,

−1 − 4 = −5)

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Examples

Connecting nodes in a graph (solution)

◮ players: edges ◮ outcomes: paths from A to B. ◮ If edge e is used, player e has a valuation −ce. ◮ Maximize social welfare: Pick shortest path (ACB,

−1 − 4 = −5)

◮ We pay each player the damage they cause

(pAD = pBD = pCD = 0 as they are not used) pAC = (2 + 5) − 4 = 3, uAC = −1 + 3 = 2 pCB = (2 + 5) − 1 = 6, uAC = −4 + 6 = 2

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Examples

Combinatorial auctions

◮ Two items for sale. ◮ Every player i has a valuation υi,1 for the first, a valuation υi,2

for the second and a valuation υi,12 for both.

◮ υi,12 is not necessarily υi,1 + υi,2. ◮ Goal: Give the items with an objective to maximize social

welfare

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Examples

Combinatorial auctions (solution)

• 1. Outcome:

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Examples

Combinatorial auctions (solution)

• 1. Outcome:

◮ Compute:

i∗∗ = arg max υi,12 (i∗, j∗) = arg maxi=j(υi,1 + υj,2)

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Examples

Combinatorial auctions (solution)

• 1. Outcome:

◮ Compute:

i∗∗ = arg max υi,12 (i∗, j∗) = arg maxi=j(υi,1 + υj,2)

◮ Offer it to:

1.1 i∗∗ if max υi,12 > maxi=j(υi,1 + υj,2) 1.2 (i∗, j∗) else

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Examples

Combinatorial auctions (solution)

• 1. Outcome:

◮ Compute:

i∗∗ = arg max υi,12 (i∗, j∗) = arg maxi=j(υi,1 + υj,2)

◮ Offer it to:

1.1 i∗∗ if max υi,12 > maxi=j(υi,1 + υj,2) 1.2 (i∗, j∗) else

• 2. Payment:

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Examples

Combinatorial auctions (solution)

• 1. Outcome:

◮ Compute:

i∗∗ = arg max υi,12 (i∗, j∗) = arg maxi=j(υi,1 + υj,2)

◮ Offer it to:

1.1 i∗∗ if max υi,12 > maxi=j(υi,1 + υj,2) 1.2 (i∗, j∗) else

• 2. Payment:

2.1 pi∗∗ = max(maxi=i∗∗ υi,12, maxi=j=i∗∗(υi,1 + υj,2)) 2.2

◮ pi∗ = max(maxi=i∗ υi,12, maxi=j=i∗(υi,1 + υj,2)) − υj∗,2 ◮ pj∗ = max(maxi=j∗ υi,12, maxi=j=j∗(υi,1 + υj,2)) − υi∗,1 Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Examples

Substitute products

◮ Suppose now that there are two products A and B (laptops)

and the players won’t to take just one.

◮ A is better than B and everybody prefers it but the valuations

varies among bidders.

◮ The players have valuations for the products υi,A and υi,B

(υi,A ≥ υi,B))

◮ Goal is to find who will get each product and at what price. ◮ Objective: Maximize social welfare

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Examples

Substitute products (solution)

• 1. Outcome:

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Examples

Substitute products (solution)

• 1. Outcome:

◮ Offer it to:

(i∗, j∗) = arg maxi=j(υi,A + υj,B)

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Examples

Substitute products (solution)

• 1. Outcome:

◮ Offer it to:

(i∗, j∗) = arg maxi=j(υi,A + υj,B)

• 2. Payment:

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Examples

Substitute products (solution)

• 1. Outcome:

◮ Offer it to:

(i∗, j∗) = arg maxi=j(υi,A + υj,B)

• 2. Payment:

◮ pi∗ = maxi=j=i∗(υi,A + υj,B)) − υj∗,B ◮ pj∗ = maxi=j=j∗(υi,A + υj,B)) − υi∗,A Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Examples

Generalized Second Price Auction

◮ Similar setting as in search engines ◮ Auction behind each keyword

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Examples

Generalized Second Price Auction

◮ Similar setting as in search engines ◮ Auction behind each keyword ◮ However VCG is not used there

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Examples

Generalized Second Price Auction

◮ Similar setting as in search engines ◮ Auction behind each keyword ◮ However VCG is not used there ◮ Yahoo! used First-Price Auction

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Examples

Generalized Second Price Auction

◮ Similar setting as in search engines ◮ Auction behind each keyword ◮ However VCG is not used there ◮ Yahoo! used First-Price Auction ◮ Google used Generalized Second Price Auction (everybody

pays the price of the person below him)

◮ Not truthful but offers more revenue.

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization

Contents

Introduction Motivation Game Theory Mechanism Design Social welfare maximization Single-item auction VCG Mechanims Examples Revenue maximization Bayesian setting Optimal mechanism

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Bayesian setting

Relaxations of the setting

◮ Generally, impossible to find an optimal mechanism that

maximizes revenue

◮ Two relaxations:

• 1. Suppose that the distribution of agents’ preferences is common

knowledge (Bayesian setting)

• 2. Try to approximate optimal mechanism’s solution

Bayesian auctions:

◮ For each player, the seller knows a probability distribution

fi(vi), vi ∈ [ℓi, ri]

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Bayesian setting

Single-item auction, 2 players

◮ Vickrey: E[min{x1, x2}] = 1 3 ◮ Anything better?

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Bayesian setting

Single-item auction, 2 players

◮ Vickrey: E[min{x1, x2}] = 1 3 ◮ Anything better? ◮ Reserved price at 1 2 ◮ Then the revenue’s expectation is 1 4 × 0 + 1 4 × 2 3 + 1 2 × 1 2 = 5 12

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Optimal mechanism

Truthfulness

Given the valuation trajectory of other players x−i,

◮ the probability ai(xi, x−i) that player i gets the product,

denoting valuation xi should be a monotone function of xi

◮ the price he pays should be xiαi(xi, x−i) −

xi

0 αi(t, x−i)dt

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Optimal mechanism

Optimal revenue (Myerson)

We want to maximize:

• i

1 1

0 . . .

1

0 pi(xi, x−i)fn(xn) . . . f1(x1)dxn . . . dx1

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Optimal mechanism

Optimal revenue (Myerson)

We want to maximize:

• i

1 1

0 . . .

1

0 pi(xi, x−i)fn(xn) . . . f1(x1)dxn . . . dx1

Expected revenue from player i given utilities x−i: 1

0 pi(xi, x−i)fi(xi)dxi = . . .

1

0 ai(xi, x−i)[xi − 1−Fi(xi) fi(xi) ]fi(xi)dxi

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Optimal mechanism

Virtual values as the solution

Virtual value: φi(xi) = [xi − 1−Fi(xi)

fi(xi) ]

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Optimal mechanism

Virtual values as the solution

Virtual value: φi(xi) = [xi − 1−Fi(xi)

fi(xi) ]

Hence, we need to maximize: 1

0 . . .

1 ai(xi)φi(xi)f (x1) . . . f (xn)dx1 . . . dxn

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Optimal mechanism

Virtual values as the solution

Virtual value: φi(xi) = [xi − 1−Fi(xi)

fi(xi) ]

Hence, we need to maximize: 1

0 . . .

1 ai(xi)φi(xi)f (x1) . . . f (xn)dx1 . . . dxn Weighted VCG on φi(xi). If i has the highest φi(xi) and j the second highest,

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Optimal mechanism

Virtual values as the solution

Virtual value: φi(xi) = [xi − 1−Fi(xi)

fi(xi) ]

Hence, we need to maximize: 1

0 . . .

1 ai(xi)φi(xi)f (x1) . . . f (xn)dx1 . . . dxn Weighted VCG on φi(xi). If i has the highest φi(xi) and j the second highest,

• 1. Outcome: i gets the item

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Optimal mechanism

Virtual values as the solution

Virtual value: φi(xi) = [xi − 1−Fi(xi)

fi(xi) ]

Hence, we need to maximize: 1

0 . . .

1 ai(xi)φi(xi)f (x1) . . . f (xn)dx1 . . . dxn Weighted VCG on φi(xi). If i has the highest φi(xi) and j the second highest,

• 1. Outcome: i gets the item
• 2. Payment: and pays φ−1

i

(φj(xj))

◮ φi monotone if Fi regular ◮ Ironing it otherwise Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Optimal mechanism

Virtual values as the solution

Virtual value: φi(xi) = [xi − 1−Fi(xi)

fi(xi) ]

Hence, we need to maximize: 1

0 . . .

1 ai(xi)φi(xi)f (x1) . . . f (xn)dx1 . . . dxn Weighted VCG on φi(xi). If i has the highest φi(xi) and j the second highest,

• 1. Outcome: i gets the item
• 2. Payment: and pays φ−1

i

(φj(xj))

◮ φi monotone if Fi regular ◮ Ironing it otherwise

• 3. Solution for all single-parameter settings

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Optimal mechanism

Example

◮ f1(x1) = 1 2, x1 = [0, 2] and 0 elsewhere ◮ f2(x2) = 1 2, x2 = [1, 3] and 0 elsewhere ◮ φ1(x1) = 2x1 − 2, x1 = [0, 2] ◮ φ2(x2) = 2x2 − 3, x2 = [1, 3]

Introduction to Mechanism Design National Technical University of Athens

Introduction Social welfare maximization Revenue maximization Optimal mechanism

Example

◮ f1(x1) = 1 2, x1 = [0, 2] and 0 elsewhere ◮ f2(x2) = 1 2, x2 = [1, 3] and 0 elsewhere ◮ φ1(x1) = 2x1 − 2, x1 = [0, 2] ◮ φ2(x2) = 2x2 − 3, x2 = [1, 3] ◮ If player 1 declares 2 and player 2 declares 2.4 for instance,

player 2 will lose the product.

◮ We incentivize player 2 not to denote just 2 + ǫ and we win

more than 2 from him!

Introduction to Mechanism Design National Technical University of Athens