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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) 1

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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 Sampling Solution 2: Profit Extraction 2

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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. 3

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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. 4

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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. 5

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Tradeoff The goal of prior-free mechanism design and this work therein is to sacrifice optimality to obtain prior freedom. 6

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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.) 7

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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). 8

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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 i x i (v) − p i (v) ≥ v i x i (z, v -i ) − p i (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. v i x i − p i ≥ v j x j − p j IC ≈ EF 9

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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 = 2 n 10

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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-it offer to each agent. ↓ v i < monopoly price → x i = 0 v i > monopoly price → x i = 1 This mechanism is envy free. 11

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Optimal mechanism not given an i.i.d distribution (in digital environment) 12

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Example- Selling a song to 5 people: AgentValue iiv (i) ↓ argmax i iv (i) = 3 → 13

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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. ivivi

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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. 15

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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 16

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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. 17

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Example AgentValue 110 …… 111 …… 1001 ↓ The price that will be offered for agents 1-10 is 1, and the value that will be offered to agents 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.) AgentValuev –i Price Offered Profit high valued 90 low valued high valued 89 low valued

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Theorem No anonymous, deterministic digital good auction is better than an n-approximation to the envy-free benchmark. 19

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Proof 20

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Proof cont. 21

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Proof cont. 23

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Proof cont. 24

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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. 25

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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′ S S S′ 26

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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. 27

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Theorem For all valuation proﬁles, the random sampling auction is at least a 15-approximation to the envy-free benchmark. 28

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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’’ ). 29

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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*
*

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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 k v k ≥ v k k/2 (multiply by v k ) 31

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Proof cont. EFO(v S’’ ) ≥ S k v k and S k v k ≥ v k k/2 ↓ (v k k/2 = EFO (2) (v)/2 by definition.) EFO(v S’’ ) ≥ EFO (2) (v)/2 32

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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) 33

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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. 34

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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’’ ). 35

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Proof cont. Like in the first part we define an event: ε= “ ∀ i, (i − S i ) ≥ S i /3” Let k′′ be index of the agent whose value is the monopoly price for the sample. ↓ v k’’ = η′′ and EFO(v S’’ )= S k’’ v k’’ (by definition) ↓ EFO(v S’’ )/3 = S k’’ v k’’ /3 (Divide the 2nd equation by 3) 36

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Proof cont. We combine the theorems: EFO(v S’’ ) ≥ EFO (2) (v)/2 and EFO(v S’’ )/3 = S k’’ v k’’ /3 ↓ If B and ε holds, then the expected revenue is at least EFO (2) (v)/ 6. 37

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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. 38

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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. 39

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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. 40

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Profit extractor The digital good profit extractor is dominant strategy incentive compatible. 41

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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! 42

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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.) 43

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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. 44

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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. 45

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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. 46

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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′′ 47

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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. 48

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Lemma Flip k > 1 coins then: E[min{#heads,#tails}] >= k/4 49

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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. 50

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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 51

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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 ↓ 52

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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) 53

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The End 54

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