THE THEORY OF MECHANISM DESIGN: APPLICATIONS TO VOTING AND AUCTIONS

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THE THEORY OF MECHANISM DESIGN APPLICATIONS TO
VOTING AND AUCTIONS

• Arunava Sen
• Indian Statistical Institute

Shaastra 2006 IIT ChennaiSeptember 29, 2006
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A 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.

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A 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.

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A 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.

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A 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.

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A 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.

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Voting 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.

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Voting 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.

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Voting 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
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Voting 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

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Voting 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
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The 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.

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A 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.

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A 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.

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The 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.

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Single 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.

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Single 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.

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Single 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.

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Single 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.

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Single 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.

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The 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.

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Optimal 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.

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Open 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.