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Eric Shou Stat/CSE 598B

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What is Game Theory? Game theory is a branch of applied mathematics that is often used in the context of economics. Studies strategic interactions between agents. Agents maximize their return, given the strategies the other agents choose (Wikipedia).

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Example Player 2 LeftRight Player 1Up10,102,15 Down15, 25, 5 Dominant strategy for Player 1 is to choose down and the dominant strategy for Player 2 is to choose right. When Player 1 chooses down and Player 2 chooses right, they are in equilibrium because neither player will gain utility if he/she changes his/her position given the other players position.

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What is Mechanism Design? In economics, mechanism design is the art of designing rules of a game to achieve a specific outcome. Each player has an incentive to behave as the designer intends. Game is said to implement the desired outcome. strength of such a result depends on the solution concept used in the game (Wikipedia).

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Unlimited Supply Goods A seller is considered to have an unlimited supply of a good if the seller has at least as many identical items as the number of consumers, or the seller can reproduce items at a negligible marginal cost (Goldberg). Examples: digital audio files, pay-per-view television.

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Pricing of Unlimited Supply Goods Use market analysis and then set a fixed price. Fixed pricing often does not lead to optimal fixed price revenue due to inaccuracies in market analysis.

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Pricing of Unlimited Supply Goods Revenue

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Pricing of Unlimited Goods Use auctions to take input bids from bidders to determine what price to sell at and which bidders to give a copy of the item to. Assume bidders in the auction each have a private utility value, the maximum value they are willing to pay for the good. Assume each bidder is rational; each bidder bids so as to maximize their own personal welfare, i.e., the difference between their utility value and the price they must pay for the good.

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Digital Goods Auctions n bidders Each bidder has private utility of a good at hand Bidders submit bids in [0,1] Auctioneer determines who receives good and at what prices.

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Truthful Auctions Most common solution concept for mechanism design is truthfulness. Mechanism designed so that truthfully reporting ones value is dominant strategy. Bid auctions are considered truthful if each bidders personal welfare is maximized when he/she bids his/her true utility value.

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Truthful Mechanisms Mechanisms that are truthful simplifies analysis by removing need to worry about potential gaming users might apply to raise their utility. Thus, truthfulness as a solution concept is desired!

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Setting of Truthful Auctions Collusion among multiple players is prohibited. Utility functions of bidders are constrained to simple classes. Mechanisms are executed once. These strong assumptions limit domains in which these mechanisms can be implemented. How do you get people to truthfully bid the price they are willing to pay without the assumptions?

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Mechanism Design Differential Privacy Main idea of paper: Strong privacy guarantees, such as given by differential privacy, can inform and enrich the field of Mechanism Design. Differential privacy allows the relaxation of truthfulness where the incentive to misrepresent a value is non-zero, but tightly controlled.

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What is Differential Privacy? A randomized function M gives ε-differential privacy if for all data sets D1 and D2 differing on a single user, and all S Range(M), Pr[M(D1) S] exp(ε) × Pr[M(D2) S] Previous approaches focus on real valued functions whose values are insensitive to the change in data of a single individual and whose usefulness is relatively unaffected by additive perturbations.

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Game Theory Implications Differential Privacy implies many game theoretic properties: Approximate truthfulness Collusion Resistance Composability (Repeatability)

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Approximate Truthfulness For any mechanism M giving ε-differential privacy and any non-negative function g of its range, for any D1 and D2 differing on a single input, E[g(M(D1))] exp(ε) × E[g(M(D2))] Example: In an auction with.001-differential privacy, one bidder can change the sell price of the item so that the sell price if the bidder was truthful was at most exp(.001)=1.001 times the sell price if the bidder was untruthful.

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Collusion Resistance One fortunate property of differential privacy is that it degrades smoothly with the number of changes in the data set. For any mechanism M giving ε-differential privacy and any non-negative function g of its range, for any D1 and D2 differing on at most t inputs, E[g(M(D1))] exp(εt) × E[g(M(D2))]

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Example If a mechanism has.001-differential privacy, and there were a group of 10 bidders trying to improve their utility by underbidding, the 10 bidders can change the sell price of the item so that the sell price if they were truthful was at most exp(10*.001)=1.01 times the sell price if the bidders were untruthful. If the auctioned item was a music file, which was supposed to be sold at $1 if the bidders were truthful, the most the 10 bidders can lower it to is $.99. $1 / $.99 = 1.01

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Composability The sequential application of mechanisms{Mi}, each giving {ε i }-differential privacy, gives (Σ i ε i )-differential privacy. Example: If an auction with.001-differential privacy is rerun daily for a week, the seven prices of the week ahead can be skewed by at most exp(7*.001)=1.007 by a single bidder

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General Differential Privacy Mechanism Goal: randomly map a set of n inputs from a domain D to some output in a range R. Mechanism is driven by an input query function q: D n * R -> that assigns any a score to any pair (d,r) from D n * R given that higher scores are more appealing. Goal of mechanism is to return an r є R given d є D such that q(d,r) is approximately maximized while guaranteeing differential privacy. Example: Revenue is q(d,r) = r * #{i: d i > r}.

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General Differential Privacy Mechanism := Choose r with probability proportional to exp(εq(d,r)) * μ(r) probability measure Let (d) output r with probability α exp(εq(d,r)) A change to q(d,r) caused by a single participant has a small multiplicative influence on the density of any output, thus guaranteeing differential privacy. Example: p(r) α exp(ε r * #{i: d i > r})

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General Differential Privacy Mechanism Let (d) output r with probability α exp(εq(d,r)) Higher scores are more probable because probability associated with a score increases as e εq(d,r) increases. e x is an increasing function. Thus in an auction with ε-differential privacy, the expected revenue is close to the optimal fixed price revenue (OPT).

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General Differential Privacy Mechanism Two properties: Privacy Accuracy

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Privacy (d) gives (2εΔq)-differential privacy. Δq is the largest possible difference in the query function when applied to two inputs that differ only on a single users value, for all r. Proof: Letting μ be a base measure, the density of at r is equal to: exp(q(d, r))μ(r) / exp(q(d, r))μ(r)dr Single change in d can change q by at most Δq, By a factor of at most exp(εΔq) in the numerator and at least exp(-εΔq) in the denominator. exp(εΔq) / exp(-εΔq) = exp(2εΔq) Example: Δq = 1

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Accuracy Lemma: Let S t = {r : q(d, r) > OPT t}, Pr(S2t) < exp(t)/μ(St) Theorem (Accuracy): For those t ln(OPT/tμ(S t ))/ε, E[q(d, ε q є (d))] > OPT 3t Size of μ(S t ) as a function of t defines how large t must before exponential bias can overcome small size of μ(S t ). Good outcomes Bad outcomes Set value

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OPT Graph of Price vs. Revenue μ(St) = width Pr(S2t) < exp(t)/μ(St) = small Source: Mcsherry, Talwar

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Applications to Pricing and Auctions Unlimited supply auctions Attribute auctions Constrained pricing problems

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Unlimited Supply Auctions Bidder has demand curve b i : [0,1] + describing how much of an item they want at a given price, p. Demand is non-increasing with price, and resources of a bidder are limited such that pb i 1 for all i, p. q(b,p) = pΣ i b i (p) dollars in revenue Mechanism gives 2ε-differential privacy, and has expected revenue at least: OPT – 3ln(e + ε 2 OPTm)/ ε, where m is the number of items sold in OPT. Cost of approximate truthfulness

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Attribute Auctions Introduce public attributes to each of the bidders (e.g. age, gender, state of residence). Attributes can be used to segment the market. By offering different prices to different segments and leading to larger optimal revenue. SEG k = # of different segmentations into k markets OPT k = optimal revenue using k market segments Taking q to be the revenue function over segmentations into k markets and their prices, has expected revenue at least: OPT k – 3(ln(e + ε k+1 OPT k SEG k m k )/ε

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Constrained Pricing Problem Limited set of offered prices that can go to bidders. Example: A movie theater must decide which movie to run. Solicit bids from patrons on different films. Theater only collects revenue from bids for one film.

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Constrained Pricing Problem Bidders bid on k different items Demand curve b i j : [0,1] for each item j є [k] Demand non-increasing and bidders resources limited so that pb i j (p) 1 for each i, j, p. For each item j, at price p, revenue q(b, (j, p)) = pΣ i b i j (p) Expected revenue at least: OPT 3 ln(e + ε 2 OPTkm)/ε

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Comments Tradeoff between approximate truthfulness and expected revenue. Attribute auctions – price discrimination? Application of mechanism to other games? Parallels with disclosure limitation?

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Conclusions General different privacy mechanism,, is more robust than truthful mechanisms. Approximate truthfulness Collusion resistance Repeatability Properties Privacy Accuracy Applications Unlimited supply auctions Attribute auctions Constrained pricing

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