Frugal Path Mechanisms

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• Frugal Path Mechanisms
• by
• Aaron Archer and Eva TardosPresented by Ron
  Lavi at the seminarTopics on the border of CS,
  Game theory, and EconomicsCS dept., The Hebrew
  University, Jerusalem, Israel

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The Model
3
5
7
2
2
s
t
1
1
1

• Each edge is controlled by a selfish agent.The
  cost of the edge is known only to him.
• The utility of agent i is (his
  payment minus the cost he incurred)
• Problem How to find a low-cost s-t path ?

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Possible Solution VCG

• In VCG
• Each edge declares some cost.
• The shortest path according to the declared costs
  is chosen
• The payment to edge e in the chosen path(the
  cost of the shortest path in the graph without e)
  minus(the cost of the chosen path without
  counting e)
• VCG is truthful (you will be convinced later
  on).
• Problem The total payment increases when the
  chosen pathhas more edges. When C1, C2 are the
  1st, 2nd lowest costs,and the length of the
  shortest path is k, then

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• Main Question
• Is there a truthful mechanismthat pays less ??

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Threshold-based Mechanisms

• For all truthful mechanisms that pay 0 to
  loosing edges
• Observation If edge e wins when bidding be ,
  it must alsowin with a lower bid be lt be
• Conclusion 1 There is a threshold bid Te edge
  e wins withlower bids and looses with higher
  bids.
• Conclusion 2 The payment to edge e must be
  exactly Te ,its threshold bid.
• TheoremAny mechanism for the path selection
  problem istruthful if and only if it is a
  Threshold-based Mechanism.
• For example, VCG is Threshold-based (thus it is
  truthful).

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Min-Function Mechanisms

• Definition A mechanism is called a Min-Function
  Mechanismif it defines, for every s-t path P, a
  (positive real valued)function fP of the
  vector of bids bP , such that
• fP is monotonically strictly increasing, and
  continuous.
• The mechanism always selects the path P with the
  lowestvalue fP(bP).
• Notice that
• A Min-Function Mechanism is Threshold-based(and
  thus truthful).
• VCG (for the path selection problem) is a min
  functionmechanism.

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Costly Example for Min-Function Mechanisms

• Take any Min-Function mechanism, and any graph
  with twonode-disjoint s-t paths (P Q). The
  following scenario is costly
• Define , a vector of bids of the edges in
  path P each edgedeclares L/P , except the
  ith edge, who declares
• w.l.o.g
• Then, if P bids and Q bids then P
  wins. In this case, the cost of P is L, and the
  cost of Q is L(1d).
• Let us calculate the total payment in this case
• Each edge gets paid its threshold bid, .
• If an edge in P raises its bid by less than
  , P will still win.
• So, , and the
  total payment is ( e.g. for d1 the payment is
  (P1)L )

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• Conclusion All min-function mechanisms suffer
  from thesame drawback of the VCG mechanism we
  have seen.
• What remains to show In many cases, all
  truthfulmechanisms are min-function mechanisms.

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Reasonable Mechanism Properties

• 1. Edge Autonomy For any edge e, given the bids
  of the otheredges, e has a high enough bid that
  will ensure that no pathusing e will not win.
  (We say that e bids infinity)
• 2. Path Autonomy Given the bids of all edges
  outside P, thereis a bid bP such that P will be
  chosen.
• 3. Independence If path P wins, and an edge e
  not in P raisesits bid, then P will still win.
• Definition If path P wins, and there is an edge
  e such thatany (small) change in es bid causes
  another path Q to win,then P and Q are tied.
• 4. Sensitivity If P wins and Q is tied with P,
  then increasing thebid of any ,
  or decreasing the bid of any
  ,will cause P to lose.

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Comparing Bids

• Definition Suppose path P bids bP, path Q bids
  bQ, and all otheredges bid infinity. If P wins,
  then we writeIf either
  then they are comparable.
• Lemma Suppose path Q is node-disjoint from paths
  P and L.
• Proof Suppose P bids bP, Q bids bQ, L bids bL
  ,and allother edges bid infinity. Who wins?
• If Q, then after raising to infinity the bids of
  L\P, Q still wins(by independence),
  contradicting (The same for L)
• Since only P,Q, and L might win, then P wins.
• P still wins after raising to infinity the bids
  of Q.

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The edge (s,t) is in G

• Theorem If G contains the edge (s,t) then any
  truthful mechanismsatisfying the above
  properties is a min function mechanism.
• Proof Let R be the edge (s,t). Define the
  functions
• Choose some P and raise all bids besides P and R
  to infinity.Notice that now, fP(bP) is exactly
  Rs threshold bid.
• Therefore, R wins if fR(bR) is minimal, and
  looses if not.
• For any P,Q (besides R), if fP(bP) lt fQ(bQ) then
  Q will not winchoose some c, fP(bP) lt c lt
  fQ(bQ) , so

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(proof continued)

• Claim fP() is strictly increasing.
• Fix some P, e in P, and some bid bP. If R bids
  fP(bP) then Rand P are tied. Thus any decrease
  in es bid causes P to win(by sensitivity).
• Consider a new bid bP, in which e decreases its
  bid by d.We need to show that fP(bP) lt fP(bP).
• But if fP(bP) fP(bP) bR then R and P are
  tied again.Thus increasing es bid by d/2 will
  cause P to lose - contradiction.
• Claim fP() is continuous.
• Otherwise let (bP-e ,be) be a discontinuity point
  for fP() ,jumping from x to y. Suppose R bids
  (xy)/2.
• Thus R and P are tied. If R wins then small
  increase in Rs bidmakes P the winner -
  contradiction. (the same if P wins).

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Three s-t connected components

• Theorem If G contains an s-t connected component
  and twoother s-t paths (disjoint from the rest),
  then any truthfulmechanism satisfying the above
  properties is a min functionmechanism.
• Proof Let R,S be the two separate s-t paths.
  Define
• By defining we can show
  that
• is strictly increasing and continuous, similar to
  before.
• For other bids of R, we define

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(proof continued)

• Claim
• Case 1 Neither P nor Q is RTake c, fP(bP) lt c
  lt fQ(bQ) and so
• Case 2a Q R and P is not STake bS such that
  fP(bP) lt fS(bS) lt fR(bR) andBy case 1,
  and the claim follows.
• for case 2b, P R and Q is not S, the proof is
  similar
• Case 3a P S and Q RChoose some other path
  L. By autonomy and continuityof fL() there is a
  bid bL such that fS(bS) lt fL(bL) lt fR(bR).By
  case 1 and by case 2 ,
  as needed.
• for case 3b, P R and Q S, the proof is
  similar

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• Conclusion Any truthful mechanism on a graph
  that containseither an s-t edge or three
  edge-disjoint s-t paths, and thatsatisfies the
  desired properties, can be forced to pay times
  the cost of the winning path, where k is the
  lengthof the winning path.
• Remark There is a graph with two disjoint s-t
  paths anda truthful mechanism for this graph,
  that is not a minfunction mechanism. (the next
  slides, if time permits)

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Two s-t disjoint paths
P
x1
x2
s
t
y1
y2
Q

• The following mechanism
• is a counter example
• Given bids bP (x1,x2 ) and bQ (y1,y2 ), draw a
  line in the x1-x2plane, with x1-intercept
  y1(y2 / 2), and x2-intercept y2(y1 / 2).
• If bP is strictly above the line, Q wins,
  otherwise P wins.
• Verify all four bidders have threshold values.
  Autonomy,independence and sensitivity hold.

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Two s-t disjoint paths (continue)

• To see that this
• mechanism is not
• min-function
• Take two Q bidsb1Q (2,1 ), b2Q (1,2 )
• The dashed/solid line is Ps threshold if Q bids
  b1Q / b2Q
• From the diagram
  and they arent tied
• If the mechanism were a min function, we would
  have fP(b1P) lt fQ(b1Q) lt fP(b2P) lt fQ(b2Q) lt
  fP(b1P) - a contradiction