Title: THE THEORY OF MECHANISM DESIGN: APPLICATIONS TO VOTING AND AUCTIONS
1THE THEORY OF MECHANISM DESIGN APPLICATIONS TO
VOTING AND AUCTIONS
- Arunava Sen
- Indian Statistical Institute
Shaastra 2006 IIT ChennaiSeptember 29, 2006
2A SimpleVoting Model
- N voters
- Two alternatives, 1 (Change) and 0 (Status
Quo) - Each voter has an opinion either prefers 1 to
0 (written 10) or 0 to 1 (written 01). Each
voters opinion known only to herself. - Each voters votes, i.e. announces either 10 or
01. - Voting Rule count number of voters who have
voted 1 (n1) and 0 (n0). If n1 gt n0, 1 is
elected otherwise 0 is elected.
3A Simple Voting Model (contd)
- Question what should each voter announce i.e.
tell the truth or lie? - Note that the outcome depends not only on what a
voter announces but on what all voters announce. - Voters realize they are playing a Game.
- In order to decide how to vote, each voter must
have beliefs over how she thinks other voters
will vote.
4A Simple Voting Model (contd)
- Claim No voter can do better than by announcing
her true opinion (henceforth, the truth)
irrespective of how she believes other voters
will vote. - Suppose that she believes that (excluding
herself), n1 voters will vote 10 and n0 will vote
01. - Suppose n1-n0 gt 1. Then she does not believe
that her announcement can affect the outcome. So
she cannot do better by not telling the truth.
5A Simple Voting Model (contd)
- Suppose her true opinion is 10 and n1-n0 1,0 or
1. Then telling the truth yields outcomes 1,1
and 0 in the three cases respectively while lying
(i.e. announcing 01) will yield outcomes 0,0 and
0 respectively. Clearly lying does not help in
any case. - If her true opinion is 01, then truth-telling
yields outome 0 in every case while lying will
give 1,1 and 0 respectively. Lying doesnt help.
6A Simple Voting Model (contd)
- Are there other voting rules which have the same
property regarding the incentives to tell the
truth? - Yes.
- For Example (and this is almost the complete
class) Pick an integer (quota) q ? N. Elect 1
if n1 ? q otherwise elect 0. - An example of a rule which does not satisfy it
pick integers r and s such that 0 ? r lt s ? N and
elect 1 iff n1 lies between r and s.
7Voting 3 Candidates, 2 Voters
- Set of candidates A a,b,c
- Two voters 1 and 2
- Each voters opinon is a ranking of the
candidates, i.e. one of abc, acb, bac, bca, cab,
cba (here abc means a better than b better than
c). Known only to the voter.
8Voting 3 Candidates, 2 Voters (contd)
- Each voter votes by announcing her opinion, i.e.
one of abc, acb, bac, bca, cab or cab. - Voting rule Each voters top choice gets 2
points, middle 1 point and bottom 0 point. The
points for candidate is summed across voters
candidate with the highest no of points is
elected. Ties broken in the order a gt b gt c. - E.g. (abc, bca) ? b, (abc, cba) ? c etc.
- The voting rule can be represented as a matrix
whose entries are a, b or c.
9Voting 3 Candidates, 2 Voters (contd)
abc acb bac bca cab cba
abc a a a b a a
acb a a a a a c
bac a a b b a b
bca b a b b c b
cab a a a c c c
cba a c b b c c
10Voting 3 Candidates 2 Voters (contd)
- Suppose voter 1s opinion is abc and she believes
2 will announce bca. Truth-telling will yield b
while lying by announcing acb will yield a. Note
that a is better than b according to her true
opinion.This voting rule may not induce
truth-telling. - Is there a voting rule which will induce
truth-telling if there are 3 or more candidates? - Yes
11Voting Dictatorship
abc acb bac bca cab cba
abc a a a a a a
acb a a a a a a
bac b b b b b b
bca b b b b b b
cab c c c c c c
cba c c c c c c
12The Gibbard-Satterthwaite Theorem Informal
Statement
- Any other voting rules which induce
truth-telling? - NO!!
- The Gibbard-Satterthwaite Theorem (1973) The
only voting rule which has a range of at least
three which induces truth-telling is the
dictatorial one.
13A Formal Model
- A a,b,c, is a set of candidates. We assume
that A gt 2. - N 1,,N set of voters.
- Each voter i has an opinion over the candidates
which can be represented by an ordering Pi over
the elements of A. Thus aPib signifies that i
prefers a to b when her opinion is Pi. - Let IP denote the set of all orderings over A.
- A profile P ? (P1,,PN) ? IPN.
14A Formal Model (contd)
- A voting rule f is a map f IPN ? A.
- For every profile P, f elects f (P) ? A.
- The voting rule f induces truth-telling if for
all voters i, all profiles P and orderings PI,
we have either f (P) f (P1,..,Pi-1, Pi,
Pi1,,PN) or f (P) Pi f (P1,..,Pi-1, Pi,
Pi1,,PN). - Suppose Pi represents is true opinion. If f
induces truth-telling then i cannot do better
than announcing Pi no matter what she believes
other voters will announce. - The voting rule f is dictatorial if there exists
a voter i such that for all profiles P, f (P)
max Pi.
15The Gibbard-Satterthwaite Theorem
- Theorem (Gibbard- Satterthwaite)
- Let f be a voting rule with Range f gt 2. Then
induces truth-telling if and only if it is
dictatorial.
16Single Object Auctions
- Single object one seller and N potential buyers
called bidders. - Each bidder i has a valuation vi for the object.
- Bidder is valuation is known only to himself.
Assume that vis are i.i.d random variables with
distribution function F and associated density
function f. - If bidder i with valuation vi gets the object his
payoff is vi pi where pi is his payment
otherwise his payoff is zero.
17Single Object Auctions (contd)
- Each bidder i makes a bid bi, bi ? 0.
- An auction is an N1 tuple of functions (d,
p1,,pN) where d IRN ? 1,, N and pi IRN ?
IR. Here d (b1,,bN) specifies who gets the
object and pi (b1,,bN) the amount paid by the i
th bidder as a function of the bids. We will let
di (b1,,bN) 1 if d (b1,,bN) i and 0
otherwise.
18Single Object Auctions (contd)
- The auction (d, p1,,pN) induces truth-telling
(is incentive compatible) if vi. di (b1,..vi,bN)
- pi (b1,..vi,bN) ? vi. di (b1,..bi,bN) - pi
(b1,..bi,bN) for all vi, bi, b1,,bN and i. - In other words, bidder i cannot do better than
revealing her true valuation irrespective of her
beliefs about the bids of other bidders.
19Single Object Auctions (contd)
- The auction (d, p1,. .,pN) is individually
rational if vi. di (v1,..vi,vN) - pi
(v1,..vi,vN) ? 0 for all v1,vN and i. - If the auction is individually rational, then
bidders will participate voluntarily. In
particular, bidders cannot be charged if they do
not receive the object.
20Single Object Auctions (contd)
- The sellers expected revenue from the auction
(d, p1,.pN) is given by - ?v1..?vN ?i pi (v1..vN) f (v1).f (vN).
dv1dvN. - The problem is to find the auction (d, p1,.pN)
which maximizes the sellers expected revenue
subject to incentive compatibility and individual
rationality.
21The Optimal Single Object Auction
- Let c (vi) vi (1 F (vi))/ f(vi)
- Assume that c (.) is a monotone increasing
function. Satisfied by most common distributions
(e.g. uniform). - Myerson (1981) shows that there exists an optimal
auction of the following kind give the object
to the highest bidder i provided that c (bi) ? 0
and make him pay max c-1 (0), bj where b is the
maximum of bids of all bidders other than i. If
c(bi) lt 0, object stays with seller. - Myersons result is actually more general.
22Optimal Auctions (contd)
- This is a second price auction with a reserve
price c-1 (0). - In a second price auction, the highest bidder
gets the object but pays the second highest bid.
In this case the seller also makes a bid even
though he has no value for the object. This
increases his expected revenue. - It is is easy to show in these auctions that
bidders are induced to reveal their valuations
truthfully.
23Open Questions
- Combinatorial Auctions Several objects to be
sold. Bidders have valuations for all subsets of
objects (bundles). Examples spectrum auctions,
airport privatization? - What is the revenue optimal auction? Efficiency?
VCG mechanisms. Important complexity
considerations. Dynamic auctions. - In voting, characterization of
incentive-compatible voting rules with special
domains, randomized voting rules etc.