Vickrey-Clark-Groves auctions First price vs second price Recap - - PowerPoint PPT Presentation

Algorithmic game theory

Ruben Hoeksma December 3, 2018

Vickrey-Clark-Groves auctions

First price vs second price

Recap

Last week:

◮ Introduction mechanism design ◮ Single item auctions ◮ First price auction ◮ Vickrey (second price) auction ◮ Truthfulness ◮ Individual rationality ◮ Revelation principle

Today:

◮ Bayes-Nash equilibrium ◮ Revenue equivalence ◮ Vickrey-Clarke-Groves auctions

Equilibria in mechanism design

Mechanism design

◮ Players have private types. ◮ Player i’s strategy depends on their type. ◮ NE is not well suitable

Instead,

◮ let Fi be the distribution of player i’s type ti ◮ utility functions ui(ti, s(t))

Definition (Bayes-Nash equilibrium (BNE))

The strategy vector s(t) is a Bayes-Nash equilibrium (BNE) if Es−i∼F−i[ui(ti, si(ti), s−i(t−i)] ≥ Es−i∼F−i[ui(ti, xi, s−i(t−i)] .

First price vs second price auctions

Single item auction

◮ Alice, valuation a ∈ [0, 1] ◮ Bob, valuation b ∈ [0, 1] ◮ a, b ∼ U(0, 1): P[a ≤ x] = x for all x ∈ [0, 1]

Claim

In a first price auction where both players have valuation distribution U(0, 1), it is a Bayes-Nash equilibrium when both players bid half their valuation.

Proof.

Let x denote Alice’s bid and y Bob’s bid. TP: if y = b

2 then x = a 2 is a best response (visa versa by symmetry).

First price vs second price auctions

Proof cont.

uA(x, b

2) =

• if x ≤ b

2

a − x

• therwise

P[x > b

2] = P[b ≤ 2x] =

• 2x

if 0 ≤ x ≤ 1

2

1 if 1

2 < x ≤ 1

E[uA(a, x, b)] = 2x(a − x) = 2xa − 2x2 Minimum at 0 =

d dx (2xa − 2x2) = 2a − 4x

⇒ x =

a 2 .

Revenue equivalence

Remember:

Myerson’s revelation principle

Let M be a mechanism such that there is an equilibrium strategy vector for the players. Then, there exists a mechanism M′ in which the strategies of the players are just to report a type, and M′ has an equilibrium in which all players report their type truthfully.

Revenue equivalence

Two auctions that have the same allocation in BNE, for any player, if they have a type for which the expected payment is equal in both auctions, then the expected payment is equal for each type of that

• player. If this is true for each player, the expected revenue of the

auctions is equal.

Revenue equivalence

Example: Two player first price and second price auction with U(0, 1).

◮ Both allocate the item to the player with highest val. in BNE.

Analysis: E[max{a, b}] =

2 3 and E[min{a, b}] = 1 3 for a, b ∼ U(0, 1).

When a = 0 (or b = 0), the expected payment is 0. E[revenue(SPA(A, B))] = E[min{a, b}] =

1 3

E[revenue(FPA(A, B))] = E[max{ a

2, b 2}] = 1 2E[max{a, b}] = 1 3

Vickrey-Clark-Groves auctions

Sponsored search auctions

Sponsored search

Model:

◮ A search has k sponsored slots ◮ Each slot j has a click trough rate (CTR) αj ◮ n bidders have value vi for a click ◮ Valuation of bidder i for slot j is viαj ◮ Each bidder is assigned at most one slot ◮ Price are set per click

Question: Can we achieve an auction similar to the Vickrey auction?

◮ truthtelling ◮ maximizes the social welfare in equilibrium ◮ individually rational

Sponsored search - welfare maximization

Welfare maximization: max

i∈N viαs(i)

Assume truthfulness. What is the optimal allocation? Claim: Assigning greedily highest vi to highest αj is optimal.

Proof

Suppose not. Let s be an optimal allocation of bidders to slots. Then, there are two bidders i, h ∈ N such that vh > vi and αs(h) < αs(i). We compare the objective of s to the objective when the allocations of i and h are switched. The difference in objective value is viαs(i) + vhαs(h) − (viαs(h) + vhαs(i)) = vi(αs(i) − αs(h)) + vh(αs(h) − αs(i)) = (vi − vh)(αs(i) − αs(h)) < 0 So, such two bidders cannot exist in an optimal allocation.

Sponsored search - A payment rule

Payment rule idea: The ℓ-th highest bidder pays the (ℓ + 1)-st highest

• bid. (Generalization of second price auction)

Observation: Individual rationality holds Truthtelling?: No!

Proof

bi > vi: Not advantageous bi < vi: Example: 2 slots, 3 bidders; α1 = 1

10, α2 = 1 20, v1 = 10,

v2 = 9, v3 = 6. Suppose b2 = v2, b3 = v3. Best response for bidder 1: Utility of bidder 1; if b1 > v2: u1(b1, v2, v3) = α1(v1 − v2) = 1

10(10 − 9) = 1 10

if v2 > b1 > v3: α2(v1 − v3) = 1

20(10 − 6) = 4 20 = 2 10

Vickrey auction - another interpretation

Social welfare (= total value of the allocation) = v1 Social welfare of bidders 2, . . . , n = 0 Social welfare of bidders 2, . . . , n if Bidder 1 did not participate = v2 Bidder 1 imposes a “cost” of v2 on the other players by their presence.

Definition (Externalities)

The externalities of a player are all costs imposed and/or benefits gained by others from that player’s actions. In the SPA, the payment of the highest bidder is equal to (an approximation of) their externalities..

Vickrey-Clarke-Groves (VCG) auction

Idea: payments are equal to externalities.

Definition (Vickrey-Clarke-Groves (VCG) auction)

For a set of possible allocations A, the VCG auction is the following

• 1. Bidders “bid”, bi(a), ∀a ∈ A, value for each possible allocation.
• 2. Allocation, a∗, maximizes the total reported value

a∗ = argmax

a∈A

• i∈N

bi(a) .

• 3. Bidder i’s payment:

max

a∈A

  

• ℓ=i

bℓ(a)

   −

• ℓ=i

bℓ(a∗) .

Sponsored search - VCG auction

Let v1 ≥ v2 ≥ . . . ≥ vn and α1 ≥ α2 . . . ≥ αk. Welfare:

k

• j=1

vjαj Without Bidder i:

i−1

• j=1

vjαj +

k+1

• j=i+1

vjαj−1 Player i’s externalities:

i−1

• j=1

vjαj +

k+1

• j=i+1

vjαj−1 −

 

i−1

• j=1

vjαj +

k+1

• j=i+1

vjαj

  =

k+1

• j=i+1

vj(αj−1 − αj)

Sponsored search - VCG auction

VCG auction for Sponsored search

• 1. Bidders submit bids, order them b1 ≥ b2 ≥ . . . ≥ bn
• 2. Assign slots 1, . . . , k to bidders 1, . . . , k, respectively
• 3. Bidder i ∈ {1, . . . , k} pays

1 αi

k+1

• j=i+1

bj(αj−1 − αj) per click, where αk+1 ≡ 0, and bj ≡ 0 for all j > n. Note: 1 αi

k+1

• j=i+1

αj−1 − αj = αi − αk+1 αi = αi − 0 αi = 1

Sponsored search - VCG auction

Theorem

The VCG auction for sponsored search is truthtelling.

• Proof. Consider bidder i. Let si(b) be the slot of bidder i for bid

vector b. Bidder i maximizes αsi(bi,b−i)

 vi −

1 αsi(bi,b−i)

k+1

• j=i+1

bj

• αsi(0,b−i) − αsi(bi,b−i)
• 



= αsi(bi,b−i)vi +

k+1

• j=i+1

bjαsi(bi,b−i) −

k+1

• j=i+1

bjαsi(0,b−i) = αsi(bi,b−i)vi +

• j=i

bjαsi(bi,b−i) −

• j=i

bjαsi(0,b−i)

• Constant w.r.t. bi

. Player i maximizes the social welfare minus a constant.