Frugality in Mechanism Design

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Frugality in Mechanism Design
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How bad can mechanism's payment be?
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A Payment Minimization Problem

• Each agent owns one edge. Her private cost is ce
• The goal is acquire a path pay as less as
  possible
• Required IC IR

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VCG payments
c1
ci

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cn
Q Is VCG good or bad?

• Bad It might be that c1 ltlt c2
• Good But at least P ? c2

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VCG payments (cont)
Q is it always that P ? c2 ?
?(k) overpayment, despite competition in the
market!
Q Can we do better? Ideas?
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Archer Tardos
Main question Is frugality unavoidable? Their
answer Yes! At least under additional conditions
on the mechanism
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Characterization of truthful mechanisms

• Thm A path selection mechanism is truthful iff it
  has thresholds.
• That is, ? pe(b-e) s.t.
• if be lt pe, e wins (and gets paid pe)
• if be gt pe, e loses.

• Payment functions are completely determined by
  path selection rule!
• Truthfulness is due to the single edge per agent

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Path Terminology

• We denote paths by P, Q
• Note that path can intersect each other
• We denote bids on paths by bP, bQ We will only
  speak about bids which agree on P?Q

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P
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Min Function Mechanisms

• For each path P, define a function fP(bP)
• Select path with lowest function value.
• Functions fP should satisfy
• fP(bP) is continuous, strictly increasing in be
  (?e ? P)
• fP(bP) -gt ? as be -gt ? (?e ? P)
• fP(bP) -gt 0 as bP -gt 0

These mechanism are reasonable (exercise) Q What
is the function of VCG? Q How can we prefer short
paths?
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every min function mechanism is frugal
Q Ideas for a costly example?
Be

• From all Be let emax denote the edge which
  causes f() to be maximal.
• P must win and each edge gets at least 1/k ?
• Thus, frugality is at least (1 k?) / (1 ?)

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Our goal show that for large classes of graphs
any reasonable mechanism is equivalent to a min
function mechanism
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Reasonable properties

• 1) Path autonomy if all edges along P lower bids
  enough, P wins.
• 2) Independence if P wins, and an edge not in P
  raises its bid, then P still wins.

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• Def Paths P and Q are tied (for the lead) if P
  wins, but we can cause Q to win by either
  arbitrarily small increases to be for some e?P-Q,
  or
• decreases to bf for some f?Q-P.
• 3) Sensitivity if P and Q are tied, then
  arbitrarily small increases to any be , e?P-Q, or
  
• decreases to any bf , f?Q-P cause Q to win.

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Q When in VCG path are tied?
e
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Thm If the graph contains the edge s-t, then any
reasonable mechanism is equivalent to a min
function mechanism
A similar theorem for another class of graphs
or