Special Lecture: Incentive Compatible Protocols

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Special LectureIncentive Compatible Protocols

• Prof. Xiaotie Deng
• Department of Computer Science

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Part I The Second Price Auction
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Second Price Auction

• William VickreyCounterspeculation, Auctions and
  Sealed TendersJournal of Finance, 1961, 16, 8-37
• Also known as Vickrey Auction

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Second Price Auction Basic Rules

• Let there be 1 computer and n bidders want to buy
  that computer
• Each bidder i thinks that the computer worth vi
  dollars, but it keeps its value as a secret
• Each bidder i declares that the computer worth
  Bidi dollars for it, and submit it as its bid
• The bidders submit their bids simultaneously
• The auctioneer announces the result

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Second Price Auction Winner Rules

• The bidder with highest bid win the computer
• It pays the price of the highest bid among all
  other bidders
• If there are more than one highest bid, choose
  any one.

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Second Price Auction Example 1

• One computer
• 3 bidders
• Bid1 US2500
• Bid2 US1200
• Bid3 US1100
• The true values are private information and are
  kept unknown
• Bidder 1 wins the computer and pays US1200 as
  its price

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Second Price Auction Example 2

• One computer
• 3 bidders
• Bid1 US1200
• Bid2 US1200
• Bid3 US1100
• The true values are private information and are
  kept unknown
• Bidder 1 (XOR 2) wins the computer and pays
  US1200 as its price

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Second Price Auction Analysis Assumptions

• The winners utility is its true value of the
  item minus the price it pays
• For example, you pay US1200 to buy a computer
  which worth US2500 to you, then your utility is
  US1300
• Another example, you pay US1200 to buy a
  computer which worth US200 to you, then your
  utility is -US1000 (negative)
• Any losers utility is 0

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Second Price Auction Analysis Definitions

• Dominant Strategy
• A bid is called a dominant strategy is bidding it
  maximizes the bidders utility, no matter what
  other bidders bids are.
• Truthful
• Also called Incentive Compatible
• An auction is truthful, if and only if reporting
  the true value vi as the bid bidi is every
  bidders dominant strategy

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Second Price Auction Analysis Theorem

• Theorem The Second Price Auction Is Truthful
• Proof. Considering any agent i. Let the highest
  bid among all other agents be x.
• If valuei lt x, then bidding no less than x
  would result in negative utility, while all other
  cases (including bidding valuei) results in 0
  utility.
• If valuei gt x, then bidding smaller than x
  would result in 0 utility, while all other cases
  (including bidding valuei) results in valuei
  x gt0 utility.
• In either case, bidding true value maximizes the
  bidders utility. Thus bidding true value is
  dominant strategy for it. The above arguments
  applies to every agent.

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Second Price Auction Generalizations
Multi-Units

• m Indifferent Computers
• n Bidders, ngtm
• The highest m bidders win a computer each.
• Each winner pay the highest bid among all losers
  as its price.
• If there are any ties, choosing any would be ok.

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Second Price Auction Generalizations
Multi-Units Example

• 2 computers
• 4 bidders
• Bid1 US2000
• Bid2 US1100
• Bid3 US1100
• Bid4 US1000
• Bidder 1 is winner, pay 1100
• Bidder 2 (XOR 3) is winner, pay 1100

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Second Price Auction Generalizations
Multi-Units Truthfulness

• Question Is the Vickrey auction with multiple
  units of items truthful? How to mathematically
  prove it?

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Part II On Incentive Compatible Selection
Protocol

• Prof. Xiaotie Deng
• With CHEN Xi, Becky LIU Jie

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Outline

• Tournament in sports
• Examples and Results
• Limits and Solutions
• Strong Incentive Compatible Model
• Weak Incentive Compatible Model
• Conclusions and Future Works

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Tournament in Sports
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Incentive Compatible Competitive Selection

• Model n agents, with an unknown linear order.
• Objective select k of them as winners according
  the linear order
• Discovery methodology Pair-wise comparison
• Could candidate cheat?

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Swiss Team

• Players are assigned to groups according to of
  wins of their pairwise games.
• Players in the same group play against each
  other.
• End of Game after log n games and depends on the
  number of winners, usually until there are a
  group of k players who have more wins than the
  rest of n-k players.

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An example of n4 k2

• Divide into two pairs competing against each
  other
• Winners play against each other, and losers play
  against each other.
• Player with two wins wins and two loses loses,
  the other two plays against each other and the
  winner is also a winner

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Cheating Possibility

• Rank 1 and Rank 3 cooperate to get both selected.
• Group rank 1 and rank 2 together, rank 3 and rank
  4 together rank 3 and rank 1 win first round
• Rank 1 loses to rank 3 in the second round
• Rank 2 and rank 1 compete in the round 3
• Result rank 1 and rank 3 selected.

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Cheating step

• Rank 1 agent intentionally loses to rank 3 agent
  in the above protocol.

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Swiss Team Protocol
Winner
Loser
Loser
Winner
gt
gt
gt
Round 1
vs.
vs.
Winner
Loser
Loser
Winner
Round 2
vs.
vs.
Loser
Winner
Round 3
vs.
Losers
Winners
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Cheating Strategy
Winner
Loser
Loser
Winner
gt
gt
gt
Round 1
vs.
vs.
Winner
Loser
Loser
Winner
vs. cheat
Round 2
vs.
Winner
Loser
Round 3
vs.
Losers
Winners
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Design problem

• Is it possible to design a protocol so that
  participants will play true according to their
  own knowledge and capacity
• i.e., rank 1 player will not intentionally lose
  to rank 3 player?
• Such a property is called incentive compatibility.

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Examples and Results
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Proposition

• There is an incentive compatible strategy if we
  want to select 2 players out of 4 players in
  NEITHER the following two cases
• if agents would help without sacrificing
  themselves
• if agents would sacrifice for their group.
• I will leave it as a mental exercise

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Proposition

• Existence of an incentive compatible protocol to
  select 2 players out of 5 players, if agents
  would help without sacrificing themselves

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Protocol

• Let each pair fight once
• If there is no cycle, choose the top three
• If there is a cycle, then there must be a cycle
  of length 3. delete it and select the other two.

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Idea of Proof

• Trivial if there is no cycle
• Otherwise, consider the simple case rank 2 is not
  selected. Then, it must be eliminated on the
  cycle.
• However, to form the cycle containing rank 2
  player, one of the player in the cycle must be
  rank 1 player.
• Therefore, rank 1 player is also not selected. IT
  sacrificed itself. Contradiction to assumption.

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Theorems

• If agents would help without sacrificing
  themselves, then
• there exists an incentive compatible protocol to
  select k best players from n players with n-kgt3
• Not if n-k1, 2.
• Idea extend the previous two cases.

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Theorems

• There is no incentive compatible protocol for
  competitive selection if agents would sacrifice
  for their group.

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Limitations and Solutions
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Problem Definition

• Suppose a tournament is held between n players Pn
  p1,p2,,pn, m highest ranked players will be
  selected as winners.
• Assign each player an ID in Nn 1, 2,,n
• Arrange a match between each pair of players
• Represent results as a directed graph G
• Set of winners W fn,m(G).

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Symbols Definitions

• Indexing function I
• Is a 1-1 function between Pn Nn.
• Result graph G (Nn, E)
• Represents pairwise competition results
• Use Kn to denote the set of all such Gs.
• Selection protocol fn,m
• Selects m winners out of n players given G
• Ranking function R
• Represents the true ranking of players in Pn.

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Symbols Definitions (cont.)

• A tournament Tn (R, B)
• Where B is the set of bad players.
• Benefit of bad group
• Ben(fn,m,Tn,I,G) i?fn,m(G), I-1(i)?G-pi?B,
  R(pi)?m
• Cheating Strategy GS
• Is a result graph in Kn where Ben(fn,m,Tn,I,GS)gt0

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Symbols Definitions (cont.)

• A result graph G is s-feasible if
• For any players pi,pj?B, if rank R(pi)ltR(pj),
  then edge I(pi)I(pj)?E
• For any players pi?B pj?B, if rank R(pi)ltR(pj),
  then edge I(pi)I(pj)?E.
• A result graph G is w-feasible if
• It is s-feasible, and
• For any player p?B such that R(p)?m, we have
  I(p)?fn,m(G).

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Strong Incentive Compatible Model

• Goal have more players in the bad group to be
  selected as winners.
• Characteristic willing to sacrifice themselves
  to gain group benefit.
• Question Is there a selection function such that
  no cheating strategy exists under strong model?

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An Cheating Example

• P9 p1, p2 p9 with capacity p1 gt p2 gt p9
  B p4, p6, p7 .
• For any fn,m, we can find an I such that there
  exists a cheating strategy.

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An Cheating Example

• First, set I(p1)1, I(p2)2, I(p3)3.
• For any selection protocol fn,m, let Wfn,m(G) be
  the set of winners.
• Let
• k1,k2 kc W - 1,2,3 (cgt2)
• W k1,k2
• Consider the
• following 3 cases

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Case 1
p1
p7
p6
p2
p8
p4
p3
p5
p9
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Case 2
p1
p6
p4
p2
p7
p8
p5
p3
p9
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Case 3
p1
p6
p4
p2
p9
p5
p3
p8
p7
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General Case Under Strong Model

• For any fn,m, if Tn satisfies
• 1. R-1(m1), R-1(m2)?B R-1(m)?B 2. ?p?B
  with R(p) lt mThen cheating strategy always
  exists.
• .

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Weak Incentive Compatible Model

• Goal have more players in the bad group to be
  selected as winners.
• Characteristic WONT sacrifice themselves to
  gain group benefit.
• Question Is there a selection function such that
  no cheating strategy exists under weak model?

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Nonexistence of ideal f4,2

• Let P4p1, p2, p3, p4 with capacity p1gtp2gtp3gtp4
  and B p1, p3.
• For any f4,2, we can find an I such that G below
  is a cheating strategy.

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Case 1
p1
p2
p3
p4
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Case 2
p1
p2
p4
p3
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Selection protocol for 3n-m
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Proof

• Let U be the set of good player p satisfying R(p)
  ? m.
• Goal for any indexing function I and w-feasible
  graph G, all good players in U will be chosen as
  winners.
• Therefore, no effective cheating strategy exists.

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Case 1

• Protocol stops after deleting k cycles where
  nm3k, and all remaining players are selected as
  winners.
• Proof
• Suppose p?U is deleted, then there must exist one
  bad player p with R(p)ltR(p)?m who is deleted
  along with p in the same cycle.
• Contradicts with our assumption that G is
  w-feasible.

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Case 2

• Protocol stops after deleting c cycles where cltk,
  then the top m remaining players are selected as
  winners.
• Proof
• Previous discussion shows all players in U
  remains.
• For any p?U, since R(p)?m, at most m-1 players
  can win p. Therefore, p must be selected as a
  winner.

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Conclusions and Future Works
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Conclusion

• Strong collaboration (sacrificing) complicates
  games.
• Weak collaboration allows for incentive
  compatible protocols

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Improvement???

• With a bad players and b truthful players in the
  top k players, what is the optimal expected
  number of truthful players elected to the top k
  players by a randomized protocol?
• Randomized Protocol should reduce the number of
  round to O(log n).
• Can we improve the number of rounds for
  deterministic protocols?
• Using expanders ???
• How can we handle non-linear order ?

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Applications

• Can we design an email protocol that limits spam
  emails?
• Honestly state an email is an advertisement.
• Can we design a protocol to evaluate overall
  trustworthiness of a virtual identity?
• Can we design an protocol in friends of friends
  network to force people to reveal their own true
  private trust level of their own friends for
  network trustworthiness evaluations?

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More general models?

• In the protocol, we require good players always
  play true, which may not be necessary when the
  bad players cheat.
• Can we handle such more general models
• What happens when there are more collaborative
  sub-groups?

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New Directions

• Selection by an outside committee
• Conferences where program committee members are
  not allowed to submitting papers.
• Output truthful
• Players may cheat but the outcome is the same as
  all the players play truthfully.

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An Example

• Problem Selection of one winner out of two
  candidates by a committee of 2k1 members.
• Each member favors one to another.
• Outcome the winner is the one favored by the
  majority
• Protocol
• Members who voted for the winner will receive a
  bonus of X dollars.
• Players all will vote for the one favored by the
  majority.

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Selection among members or by outsiders?

• Can we establish a general theorem stating one is
  better than another?

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Recent works of others on Output Truthful

• Li XiangYang, Kao Ming-Yang, et al., worked on
  several other important problems on the design of
  output truthful mechanisms.
• It would often improve efficiencies while
  maintaining truthfulness of the outcome.

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THANKS
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Summary Of The Lecture

• Second Price Auction
• On Incentive Compatible Selection Protocol