Eric Shou
Stat/CSE 598B
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).
Example
Player 2
Player 1
Left
Right
Up
10,10
2,15
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 player’s position.
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).
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.
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.
Pricing of Unlimited Supply Goods
Revenue
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.
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.
Truthful Auctions
 Most common solution concept for mechanism design
is “truthfulness.”
 Mechanism designed so that truthfully reporting one’s
value is dominant strategy.
 Bid auctions are considered truthful if each bidder’s
personal welfare is maximized when he/she bids
his/her true utility value.
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!
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?
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.
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.
Game Theory Implications
 Differential Privacy implies many game theoretic
properties:
 Approximate truthfulness
 Collusion Resistance
 Composability (Repeatability)
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.
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))]
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
Composability
 The sequential application of mechanisms{Mi}, each
giving {εi}-differential privacy, gives (Σi εi)-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
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 ->
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 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: di > r}.
General Differential Privacy
Mechanism

 Let
:= Choose r with probability proportional to
exp(εq(d,r)) * μ(r)
probability measure
(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: di > r})
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 eεq(d,r) increases.
 ex is an increasing function.
 Thus in an auction with ε-differential privacy, the
expected revenue is close to the optimal fixed price
revenue (OPT).
General Differential Privacy
Mechanism
 Two properties:
 Privacy
 Accuracy
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 user’s 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
Accuracy
Good
outcomes
Set value
 Lemma: Let St = {r : q(d, r) > OPT− t},
Pr(S2t) < exp(−t)/μ(St)
Bad outcomes
 Theorem (Accuracy):
For those t ≥ ln(OPT/tμ(St))/ε,
E[q(d, εqє(d))] > 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).
Graph of Price vs. Revenue
OPT
μ(St) = width
Pr(S2t) < exp(−t)/μ(St) = small
Source: Mcsherry, Talwar
Applications to Pricing and Auctions
 Unlimited supply auctions
 Attribute auctions
 Constrained pricing problems
Unlimited Supply Auctions
 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) = pΣibi(p) dollars in revenue
 Mechanism
gives 2ε-differential privacy, and has
expected revenue at least:
 OPT – 3ln(e + ε2OPTm)/ ε, where m is the number of
items sold in OPT.
Cost of approximate
truthfulness
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 + εk+1OPTkSEGkmk)/ε
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.
Constrained Pricing Problem
 Bidders bid on k different items
 Demand curve bij : [0,1] for each item j є [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)) = pΣibij(p)
 Expected revenue at least:
OPT − 3 ln(e + ε2OPTkm)/ε
Comments
 Tradeoff between approximate truthfulness and
expected revenue.
 Attribute auctions – price discrimination?
 Application of mechanism to other games?
 Parallels with disclosure limitation?
Conclusions
 General different privacy mechanism,
robust than truthful mechanisms.



Approximate truthfulness
Collusion resistance
Repeatability
 Properties


Privacy
Accuracy
 Applications



Unlimited supply auctions
Attribute auctions
Constrained pricing
, is more
Questions?