Simple near-optimal auctions Posted price auctions Recap Last - - PowerPoint PPT Presentation

Algorithmic game theory

Ruben Hoeksma December 17, 2018

Simple near-optimal auctions

Posted price auctions

Recap

Last week:

◮ Optimal revenue single item auction (Myerson’s auction) ◮ Virtual valuations ◮ For any feasible (DSIC, IR) allocation rule a we have

E

 

i∈N

πi

  = E  

i∈N

via(v)

 

Today:

◮ Near-optimal auctions ◮ Posted price auctions ◮ VCG auctions

Optimal auction - realistic?

Advantages of the optimal auction:

◮ Provable maximum expected revenue ◮ Computations on a per bidder basis ◮ Nothing structurally changes when adding/removing bidders

Disadvantages of the optimal auction:

◮ Necessity to have a good knowledge of bidders valuation

distribution

◮ Bidders need to know the whole auction structure ◮ Payments can seem confusing ◮ Highest bid not always wins (counter intuitive)

Posted price auction

Simple auction

◮ Set one threshold t at which we sell the item ◮ Sell to anyone with virtual value higher than t

Auction is based on The prophet inequality.

Game

◮ In n stages, n prizes are offered ◮ In each stage i, you can accept the prize pi or forfeit it forever ◮ n prizes have known independent distributions G1, . . . , Gn

Q: How well can you do compared to a prophet who knows all the realizations?

The prophet inequality

Theorem (Prophet inequality)

For every sequence of independent distributions G1, . . . , Gn, there is a threshold t such that accepting any prize pi ≥ t guarantees expected reward 1

2Ep[maxi pi].

• Proof. Lower bound on threshold t strategy.

◮ q(t) = Ep[pi < t, ∀i], probability of not accepting any prize ◮ Simple lower bound: Ep[maxi pi] ≥ q(t) · 0 + (1 − q(t)) · t ◮ Improvement: bound the amount that we get extra ◮ If exactly one pi ≥ t then we get an extra pi − t ◮ If more prizes are at least t, we get one of them: bound as if pi = t

(1−q(t)) · t + n

i=1 P[pi ≥ t, pj < t, ∀j = i]E[pi − t|pi ≥ t]

= (1 − q(t)) · t + n

i=1 P[pj < t, ∀j = i]

• ≥q(t)

P[pi ≥ t]E[pi − t|pi ≥ t]

• =E[(pi−t)+]

The prophet inequality

Proof Cont. So, E[Rev(t-threshold)] ≥ (1 − q(t)) · t + q(t)

n

• i=1

E[(pi − t)+] Upper bound on the prophet’s strategy E[max

i

pi] = E[t + max

i

(pi − t)] ≤ t + E[max

i

(pi − t)+] ≤ t +

n

• i=1

E[(pi − t)+] Compare to the lower bound: (1 − q(t)) · t + q(t) n

i=1 E[(pi − t)+]

Set t such that q(t) = 1

2: 1 2 · t + 1 2 n

• i=1

E[(pi − t)+] ≥

1 2E[max i

pi]

Posted price auction

Single item auction:

◮ Single item ◮ n bidders N with distributions ϕ1(·), . . . , ϕn(·) ◮ Assume regular distributions

Equivalence between prizes pi and virtual valuations vi. E[maxi pi] and E [

i∈N via(v)] = E[maxi(vi)+]

Posted price auction:

◮ Set t such that P[maxi vi ≥ t] = 1 2. ◮ For each bidder i find the smallest valuation vi such that vi ≥ t,

set ri = vi.

◮ Offer bidder i the item at price ri. If they accept: sell the item to

them; otherwise offer the item to bidder i + 1. E[Rev(posted price auction)] ≥ 1

2E[max i

(vi)+] = 1

2E[Rev(OPT)]

Posted price auction

Advantages of posted price auctions:

◮ Extremely simple for the players ◮ No need to know the whole auction ◮ No strategizing possible ◮ Constant factor of the optimal auction revenue ◮ Posted price mechanisms exist that achieve e−1 e

fraction of the

• ptimal auction revenue

Disadvantages:

◮ Computation of virtual valuations still necessary ◮ Reserve price for each bidder individual ◮ Reliant on the correctness of assumed distributions

Simple near-optimal auctions

Prior-independent auctions (Bulow-Klemperer Theorem)

Prior independent auctions

Prior independent

Independent on any extra information that may or may not be know before finding the solution.

◮ Looking for a single item auction ◮ Should not depend on anything except that we want to allocate a

single item

◮ Not even on number of players

Q: What can we do? Compare different auctions and give an analysis that ensures us that

• ne is the better choice.

Assumption: All values are distributed equally (for the analysis)

Bulow-Klemperer

Bulow-Klemperer Theorem

Let F be a regular distribution and n a positive integer, then Ev1,...,vn+1∼F[Rev(SPA(n + 1))] ≥ Ev1,...,vn∼F[Rev(OPTF(n))] , where OPTF(n) is the optimal auction for distribution F with n bidders and SPA(n + 1) is the Second price auction / Vickrey auction with n + 1 bidders. Note: Analysis can use the distribution as long as we don’t base our choices on it.

• Proof. Claim: Vickrey auction maximizes revenue among all auctions

that always allocate the item (exercise).

Bulow-Klemperer

Proof cont. Consider the following auction with n + 1 bidders, with v1, . . . , vn+1 ∼ F:

◮ Optimal auction between first n bidders ◮ If the auction of n bidders does not provide a winner, give away

the item to bidder n + 1 for free Properties of this auction:

◮ Revenue is exactly equal to the revenue of OPTF(n) ◮ It always allocates the item

Thus, SPA(n + 1) has revenue at least as large as this auction (by the Claim) and thus at least as large as OPTF(n).

Prior independent auctions

Corollary

Ev1,...,vn∼F[Rev(SPA(n))] ≥ n−1

n

· Ev1,...,vn∼F[Rev(OPTF(n))]

• Proof. Consider the n − 1 bidder auction that simulates OPTF(n) and
• nly considers the n − 1 bidders with largest expected payments.

This auction achieves at least n−1

n

fraction of OPTF(n). Yet, OPTF(n − 1) achieves higher expected revenue. Thus, Ev1,...,vn∼F[Rev(SPA(n))] ≥ Ev1,...,vn−1∼F[Rev(OPTF(n − 1))] ≥ n−1

n

· Ev1,...,vn∼F[Rev(OPTF(n))] . Take away: It is better to spend resources on getting more bidders, than

• n improving the estimation of those bidders’ valuation distributions.