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Title: Mechanism%20Design%20via%20Differential%20Privacy


1
Mechanism Design viaDifferential Privacy
  • Eric Shou
  • Stat/CSE 598B

2
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).

3
Example
Player 2 Player 2
Left Right
Player 1 Up 10,10 2,15
Player 1 Down 15, 2 5, 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.
4
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).

5
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.

6
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.

7
Pricing of Unlimited Supply Goods
Revenue
8
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.

9
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.

10
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.

11
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!

12
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?

13
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.

14
What is Differential Privacy?
  • A randomized function M gives e-differential
    privacy if for all data sets D1 and D2 differing
    on a single user, and all S ? Range(M),
  • PrM(D1) ? S exp(e) PrM(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.

15
Game Theory Implications
  • Differential Privacy implies many game theoretic
    properties
  • Approximate truthfulness
  • Collusion Resistance
  • Composability (Repeatability)

16
Approximate Truthfulness
  • For any mechanism M giving e-differential privacy
    and any non-negative function g of its range, for
    any D1 and D2 differing on a single input,
  • Eg(M(D1)) exp(e) Eg(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.

17
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 e-differential privacy
    and any non-negative function g of its range, for
    any D1 and D2 differing on at most t inputs,
  • Eg(M(D1)) exp(et) Eg(M(D2))

18
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

19
Composability
  • The sequential application of mechanismsMi,
    each giving ei-differential privacy, gives (Si
    ei)-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

20
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 Dn R -gt that assigns any a score to
    any pair (d,r) from Dn R given that higher
    scores are more appealing.
  • Goal of mechanism is to return an r ? R given d ?
    D such that q(d,r) is approximately maximized
    while guaranteeing differential privacy.
  • Example Revenue is q(d,r) r i di gt r.

R
21
General Differential Privacy Mechanism
  • Choose r with probability
    proportional to
  • exp(eq(d,r)) µ(r) probability
    measure
  • Let (d) output r with probability a
    exp(eq(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) a exp(e r i di gt r)

22
General Differential Privacy Mechanism
  • Let (d) output r with probability a
    exp(eq(d,r))
  • Higher scores are more probable because
    probability associated with a score increases as
    eeq(d,r) increases.
  • ex is an increasing function.
  • Thus in an auction with e-differential privacy,
    the expected revenue is close to the optimal
    fixed price revenue (OPT).

23
General Differential Privacy Mechanism
  • Two properties
  • Privacy
  • Accuracy

24
Privacy
  • (d) gives (2e?q)-differential privacy.
  • ?q 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 ?q ,
  • By a factor of at most exp(e?q) in the
    numerator and at least exp(-e?q) in the
    denominator.
  • exp(e?q) / exp(-e?q) exp(2e?q)
  • Example ?q 1

25
Accuracy
Good outcomes
Set value
  • Lemma Let St r q(d, r) gt OPT- t,
  • Pr(S2t) lt exp(-t)/µ(St)
  • Theorem (Accuracy)
  • For those t ln(OPT/tµ(St))/e,
  • Eq(d, eq?(d)) gt OPT - 3t
  • Size of µ(St) as a function of t defines how
    large t must before exponential bias can overcome
    small size of µ(St).

Bad outcomes
26
Graph of Price vs. Revenue
OPT
µ(St) width
Pr(S2t) lt exp(-t)/µ(St) small
Source Mcsherry, Talwar
27
Applications to Pricing and Auctions
  • Unlimited supply auctions
  • Attribute auctions
  • Constrained pricing problems

28
Unlimited Supply Auctions
R
  • Bidder has demand curve bi 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 pbi
    1 for all i, p.
  • q(b,p) pSibi(p) dollars in revenue
  • Mechanism gives 2e-differential privacy,
    and has expected revenue at least
  • OPT 3ln(e e2OPTm)/ e, where m is the number
    of items sold in OPT.

Cost of approximate truthfulness
29
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.
  • SEGk of different segmentations into k
    markets
  • OPTk 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
  • OPTk 3(ln(e ek1OPTkSEGkmk)/e

30
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.

31
Constrained Pricing Problem
  • Bidders bid on k different items
  • Demand curve bij 0,1 for each item j ? k
  • Demand non-increasing and bidders resources
    limited so that pbij(p) 1 for each i, j, p.
  • For each item j, at price p, revenue
  • q(b, (j, p)) pSibij(p)
  • Expected revenue at least
  • OPT - 3 ln(e e2OPTkm)/e

32
Comments
  • Tradeoff between approximate truthfulness and
    expected revenue.
  • Attribute auctions price discrimination?
  • Application of mechanism to other games?
  • Parallels with disclosure limitation?

33
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

34
Questions?
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