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Chapter 6: Prior-free Mechanisms

Roee and Ofir

(Also from “Envy Freedom and Prior-free Mechanism Design” by Devanur, Hartline, Yan)

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Talk overview

1. Introduction to prior free mechanisms and comparison with prior independent mechanisms.

2. Theorem: No anonymous, deterministic digital good auction is better than an n-approximation to the envy-free benchmark.

Solution 1: Random SamplingSolution 2: Profit Extraction

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The trouble with priors

• The prior can be inaccurate- For example, agents can lie during a market survey if they know that the results will affect their prices in the future.

• Prior dependent mechanisms are non robust- A mechanism that was designed to work on one distribution will probably not work on another.

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Prior free vs. Prior independent

Prior-independent mechanism can rely on there being a distribution where as the prior-free mechanism cannot.

↓The class of good prior-free mechanisms should

be smaller than the class of good prior-independent mechanisms.

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What is a good mechanism?

A good mechanism approximates the optimal mechanism for the distribution if there is a distribution; moreover, when there is no distribution this mechanism still performs well.

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Tradeoff

The goal of prior-free mechanism design and this work therein is to sacrifice optimality to obtain prior freedom.

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What is the objective?

The objective is profit maximization- we characterize (p,x) that gives the highest total revenue.

(p- payments vector, x – allocation vector.)

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How do we evaluate prior free mechanisms?

Envy-free optimal revenue benchmark:An outcome, allocation and payments, (x,p) , is

envy-free if no agent prefers to swap outcome (allocation and payment) with another agent.

(Similar in structure to incentive compatible mechanisms).

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Incentive Compatibility versus Envy Freedom

A mechanism is incentive compatible if no agent prefers the outcome when misreporting her value to the outcome when reporting the truth.

∀i, z, v. v ix i(v) − pi(v) ≥ vixi(z, v-i) − pi(z, v-i)

An allocation x with payments p is envy free for valuation profile v if no agent prefers the outcome of another agent to her own.

∀i, j. vixi − pi ≥ vjxj − pj

IC ≈ EF

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Digital Good Environment

There are n unit-demand agents denoted N = {1, . . . , n}and any subset of them can be served. I.e., X = 2n

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Optimal mechanism given a i.i.d distribution (in digital environment)

Post the monopoly price as a take-it-or-leave-itoffer to each agent. ↓v i < monopoly price → x i = 0

v i > monopoly price → x i = 1

This mechanism is envy free.

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Optimal mechanism not given an i.i.d distribution (in digital environment)

Without a prior there is no monopoly price. The upper bound on the revenue of any

monopoly price is maxi .

This is not incentive compatible, but it is envy free (optimal). (Denoted EFO(v))

v (i) - the i-th highest value.

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Example- Selling a song to 5 people:

Agent Value

1 50

2 40

3 30

4 20

5 10

i iv (i)

1 50

2 80

3 90

4 80

5 50

↓

argmax i iv (i) = 3 →

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Why is max i iv (i) not a good benchmark?

Not a good benchmark when the maximization is obtained at i=1.

↓

The envy-free (optimal) benchmark for digital goods is defined as EFO(2)(v) = max i≥2 iv (i)

This will be the benchmark.

i vi1 50000

2 5

3 4

4 2

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Intro to designing prior-free auctions

Deterministic auctions cannot give good prior-free approximation.

We will describe two approaches for designing prior-free auctions for digital goods.

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Deterministic Auctions

When figureing out a price to offer agent i we can use statistics from the values of all other agents v -i

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Deterministic optimal price auction

The deterministic optimal price auction offers each agent i the take-it-or-leave-it price of τ i equal to the monopoly price for v –i.

This mechanism is prior independent but not prior free.

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ExampleAgent Value

1 10

… …

10 10

11 1

… …

100 1

↓

The price that will be offered for agents 1-10 is 1, and the value that will be offered to agents 11-100 is 10. (Derived from v –i for each i.)

→The revenue will be 10, which is much less then EFO(2)(v) = EFO(v) = 100. (Sell to the first 10 agents for 10.)

Agent Value v –i PriceOffered

Profit

1-10 10 9 high valued 90 low valued

1 10

11-100 1 10 high valued89 low valued

10 0

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Theorem

No anonymous, deterministic digital good auction is better than an n-approximation to the envy-free benchmark.

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Proof

First, we consider only the valuation profiles with values {1, h}.∈

(v): Number of high values (h).(v): Number of low values (1).

What is an anonymous auction?

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Proof cont.

• Let τ ((v-i), ) be the offer price for agent i. This means that the auction is anonymous (not a function of i).

• We show that this Auction cannot give a good approximation, by means of contradiction.

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• Without loss of generality: τ ()τ () < 1 – stupid, 1 < τ () < h – stupid,

• τ ()=1, τ ()= h (why??) • I.e., there exists a k* such that

τ ()=h}.

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Proof cont.

Now we will take n=m+1 and (v) = and (v) = m− +1.There are two cases:1. Low valued agents: τ (() , () ) = τ(,m − ) = h.2. For high-valued agents: τ (() , () ) = τ (− 1,m −

+ 1) = 1.

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Proof cont.

Now consider h=n. If =1 the benchmark is n.If > 1 the benchmark is also nTherefore, the auction profit is at best an n-approximation.

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Motivation

The problem with the deterministic optimal price auction is that it sometimes offers high-valued agents a low price and low-valued agents a high price.

Either of these prices would have been good if only it offered consistently to all agents.

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First solution: Random Sampling

1. Randomly partitions the agents into S and S ′ ′′(by flipping a fair coin for each agent)

2. Compute (empirical) monopoly prices η and η′ for S and S respectively′′ ′ ′′

3. Offers η to S and η to S′ ′′ ′′ ′

SSS′S′S′S′

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Example

v = (1.1, 1) → EFO(2)(v) = 2

With probability ½ both agents are in the same partition → revenue is 0.

With probability ½ each agent is in a different partition → revenue is 1.

So the expected revenue is ½, which is a 4-approximation to the benchmark.

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Theorem

For all valuation profiles, the random sampling auction is at least a 15-approximation to the envy-free benchmark.

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Proof

First, we assume that v (1) S and we call S’ the ∈ ′

Market and a S’’ the sample.

Now we want to prove two main theorems:• Show that EFO(v S’’) is close to EFO(2)(v).• Show that revenue from price η on S is close ′′ ′

to EFO(v S’’) .

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Proof cont.

Define:1. v (i) represent the i-th largest valued agent

2. X i is an indicator to the event that i S∈ ′′

3.Define S i = ∑ j<i X j

4.Define k to be number of winners in EFO(v)

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Proof cont.We will prove that with good probability EFO(v S’’) is close to the

benchmark, EFO(2)(v): Define the event B that S k ≥ k/2

Notice that EFO(v S’’ ) ≥ S k v k

(Optimal revenue≥ Revenue from price v k)

Now from Event B we can see:S kv k ≥ v k k/2 (multiply by v k )

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Proof cont.

EFO(v S’’ ) ≥ S k v k and S kv k ≥ v k k/2

↓ (v k k/2 = EFO(2)(v)/2 by definition.)

EFO(v S’’ ) ≥ EFO(2)(v)/2

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Proof cont.

But, we didn’t prove that event B happens in a good probability.

Therefore we now want to show that Pr(B)=1/2.(proof will be for even k, proof for odd k omited)

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Proof cont.

We assume in the beginning that v (1) S .∈ ′

Therefore, we need to divide k-1 (odd number) agents into the market and the sample (S’ and S’’).

At least one partition receives at least k/2 of these

agents and half the time it is the sample; therefore, Pr[B] = 1/2.

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Proof cont.

Now we want to prove the second part, that with good probability, revenue from price η

on S is close to EFO(v′′ ′ S’’).

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Proof cont.

Like in the first part we define an event:ε= “∀i , (i − Si) ≥ Si /3”

Let k ′′ be index of the agent whose value is the

monopoly price for the sample. ↓vk’’ = η and EFO(v′′ S’’)= Sk’’ vk’’ (by definition)

↓EFO(vS’’)/3 = Sk’’ vk’’/3 (Divide the 2nd equation by 3)

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Proof cont.

We combine the theorems:EFO(v S’’ ) ≥ EFO(2)(v)/2 and EFO(vS’’)/3 = Sk’’ vk’’/3

↓If B and ε holds, then the expected revenue is at

least EFO(2)(v)/ 6.

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Proof cont.

From the Balanced Sampling Lemma (no proof) we assume Pr[ε] ≥ 0.9

B and ε holds = Pr[ε B] = 1−Pr[∧ ￢ ε]−Pr[ ￢ B] ≥ 0.4. Therefore, the random sampling auction is a 15-

approximation to the envy-free benchmark.

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Solution 2: Profit extractor

We design a mechanism that obtains profit at least R on any input v with EFO(v) ≥ R. We call this mechanism a profit extractor.

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Profit extractor

The digital good profit extractor for target R and valuation profile v finds the largest k such that v(k) ≥ R/k, sells to the top k agents at price R/k, and rejects all other agents. If no such set exists, it rejects all agents.

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Profit extractor

The digital good profit extractor is dominant strategy incentive compatible.

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Proof:

We need to design a mechanism that makes profit R from n agents by selling them a digital good.

Try #1:1. Offer price R to the agents- sell if 1 agent accepts.2. If not, offer price R/2- sell if 2 agents accept.3. And so on…

This mechanism is not DSIC!

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Proof cont. (Ascending price mechanism)

Solution:1. Offer price R/n to all agents. Sell if all n agents

accept. 2. If not, offer price R/(n-k) to the n-k agents who

accepted the last offer. Sell if all n-k agents left accept.

3. And so on…

This mechanism is DSIC. (Agents drop out when the price rises above their valuation.)

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Profit extractor

For all valuation profiles v, the digital good profit extractor for target R obtains revenue R if R ≤ EFO(v) and zero otherwise.

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Proof

EFO(v) = kv (k) for some k

If R <= EFO(v) → R/k <= v (k) and the profit extractor will find this k.

If R > EFO(v) → R > EFO(v) = max k kv (k) then there is no k for which R/k <= v (k)

→ The mechanism has no winners and no revenue.

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Approximate Reduction to Decision Problem

We use random sampling to approximately reduce the mechanism design problem of optimizing profit to profit extraction.

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Random Sampling profit extraction auction

The random sampling profit extraction auction works as follows:

1. Randomly partition the agents by flipping a fair coin for each agents and assigning her to S or S .′ ′′

2. Calculate R = EFO(v′ s’) and R = EFO(v′′ s’’), the benchmark profit for each part.

3. Profit extract R from S and R from S′′ ′ ′ ′′

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Random Sampling profit extraction auction

The revenue of this mechanism is: min(R’, R’’) . ↓(Profit extractor is DSIC.) ↓Random sampling profit extraction auction is

DSIC.

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Lemma

Flip k > 1 coins then: E[min{#heads,#tails}] >= k/4

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For digital good environments and all valuation profiles, the revenue of the random sampling profit extraction auction is a 4-approximation to the envy-free benchmark.

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Proof

Define:REF: Envy-free benchmark and its revenue.APX: Random sampling profit extraction

auction and its expected revenue. (= E[min(R,R )])′ ′′

Assume k >=2

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Proof cont.

Assume that the envy-free benchmark sells to k agents at price p. → REF = kp

Of the k Winners in REF let k’ be the number of them that are in S’, and k’’ the number in S’’.

↓R’ >= k’p , R’’ >= k’’p ↓

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Proof cont.

APX/REF = E[min(R ,R )]/kp′ ′′ ≥ E[min(k p,k p)]/kp′ ′′ = E[min(k ,k )]/k ′ ′′ ≥ ¼ (from the lemma)

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The End