chapter 6: prior-free mechanisms roee and ofir (also from “envy freedom and prior-free mechanism...
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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