A Simplified Proof for Roberts Theorem

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A Simplified Proof for Roberts Theorem
Ron Lavi, Ahuva Mualem, and Noam Nisan
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Motivation

• Mechanisms elections, auctions (1st / 2nd
  price, double, combinatorial, ), resource
  allocations
• social goal vs. individuals strategic
  behavior.
• Main Problem Which social goals can be
  achieved ?
• For some settings, Roberts showed that only
  limited class of goals can be implemented (even
  if the society has unlimited amount of money).

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Social Choice Function (SCF(

• f V1 Vn ? A
• A is the finite set of possible alternatives.
• Each player has a valuation vi A ? R.
• f chooses an alternative from A for every v1 ,,
  vn.
• For CAs A all feasible partitions of the
  given items
• For CAs Vi vi that satisfy 1, 2, 3(1)
  no externalities (2) free disposal (3)
  normalization
• For CAs a partition S1..Sn that maximizes ?i
  vi(Si).
• Alternatively for CAs S1..Sn that maximizes
  mini vi (Si ).

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Social Choice Function (SCF(

• f V1 Vn ? A
• A is the finite set of possible alternatives.
• Each player has a valuation vi A ? R.
• f chooses an alternative from A for every v1 ,,
  vn.
• 1 item Auction A player i wins, i1..n,
  Vi R, f (v) argmax(vi)
• Combinatorial Auction a partition of items
  S1..Sn that maximizes
• ?i vi(Si).
• Nisan, Ronens problem a partition of tasks
  S1..Sn that minimizes maxi ci (Si ).

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Truthful Implementation of SCFs

• Dfn A Mechanism m(f, p) is a pair of a SCF f
  and a payment function pi for every player.
• Dfn A Mechanism is truthful (in dominant
  strategies) if rational players tell the truth
  ? vi , v-i , wi vi ( f(vi , v-i)) pi(vi ,
  v-i) vi ( f(wi , v-i)) pi(wi , v-i).
• - If the mechanism m(f, p) is truthful we also
  say that m implements f.
• - 1st vs. 2nd Price Auction.
• - Not all SCFs can be implemented Majority
  vs. Minority between 2 alternatives.

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What SCFs are implementable?

• Positive Result VCG ? A and ? V
  f(v)argmaxa? A Si vi (a).
• (Also all affine maximizers f(v) argmaxa? A
  Si wi vi (a) ?a .)

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What SCFs are implementable?

• Positive Result VCG ? A and V f(v)argmaxa?
  A Si vi (a).
• (Also all affine maximizers f(v) argmaxa? A
  Si wi vi (a) ?a .)
• Negative Result Roberts 79 If A 3 and V
  is unrestricted (RnA), then only affine
  maximizers are implementable.

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What SCFs are implementable?

• Positive Result VCG ? A and V f(v)argmaxa?
  A Si vi (a).
• (Also all affine maximizers f(v) argmaxa? A
  Si wi vi (a) ?a .)
• Negative Result Roberts 79 If A 3 and V
  is unrestricted (RnA), then only affine
  maximizers are implementable.
• Single Parameter Domains (essentially Ai 2 )
  Many other SCFs are implementable.
• CAs with single minded bidders Lehmann,
  OCallaghan, Shoham, Scheduling Related Parallel
  Machines Archer, Tardos, Profit Maximization
  of Digital goods Fiat, Goldberg, Hartline,
  Karlin

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What SCFs are implementable?

• Positive Result VCG ? A and V f(v)argmaxa?
  A Si vi (a).
• (Also all affine maximizers f(v) argmaxa? A
  Si wi vi (a) ?a .)
• Negative Result Roberts 79 If A 3 and V
  is unrestricted (RnA), then only affine
  maximizers are implementable.
• Single Parameter Domains Many other SCFs are
  implementable.
• e.g. LOS99
• OPEN

severely restricted domains with non-affine
maximizers
unrestricted domains with only affine
maximizers
Multiparameter Domains
CAs ?
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Comparison with the non-quasi-linear case
Arrow 63 Gibbard-Satterthwaite73-5
Single-Peaked Domains
Saturated
Domains must be dictatorial
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• Monotonicity

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In 1 item Auctions Truthfulness ? Monotonicity
This can be generalized to single parameter
domains LOS99. Monotonicity refers to the
social choice function alone (no need to consider
the payment function). - .
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Truthfulness ? Cyclic -Monotonicity

• Thm1 RochetRoz A SCF f V ? A is
  truthfully implementable iff f is
  Cyclic-Monotone.
• Thm2 If f is truthfully implementable then f
  satisfies W-MON.

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Truthfulness ? Cyclic -Monotonicity

• Thm1 RochetRoz A social choice function f
  V ? A is truthfully implementable iff f is
  Cyclic-Monotone.
• Thm2 If f is truthfully implementable then f
  satisfies W-MON.
• Dfn1 f satisfies W-MON if for any vi , ui
  and v-i
• f (vi , v-i) a and f (ui , v-i) b
• implies ui (b) - ui (a) gt vi (b) - vi (a).

vi (a) vi (b)
ui (a)
ui (b)
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• Example single player,
• 2 alternatives a, and b,
• 2 possible valuations v, and u.
• Majority satisfies W-MON.
• Minority doesnt.

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Truthfulness ? Cyclic -Monotonicity

• Thm1 RochetRoz A social choice function f
  V ? A is truthfully implementable iff f is
  Cyclic-Monotone.
• Thm2 If f is truthfully implementable then f
  satisfies W-MON.
• Dfn1 f is W-MON if for any vi , ui and v-i
  
• f (vi , v-i) a and f (ui , v-i) b
• implies vi (a) - vi (b) ui (b) - ui (a) gt
  0.
• Dfn2 f is Cyclic-Monotone if for any k, vi 1,
  vi 2, , vi k , v-i
• ?j1k vi j (f(vi j, v-i )) - vi j (f(vi j1
  , v-i )) gt 0.
• Remark If A2, then Dfn1Dfn2.

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• Example
• single player
• A a, b, c.
• V1 v, u, w.
• f(v)a, f(u)b, f(w)c.
• f satisfies W-MON, but not Cyclic-Monotonicity
  
• v(a) v(b) u(b) u(c) w(c) w(a) lt 0.

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Monotonicity what is really needed for our proof

• Def dabi (v-i ) inf vi (a) - vi (b) vi
  ? Vi s.t. f (vi , v-i ) a .
• In particular, if f(v) a, then
• vi (a) - vi (b) dabi (v-i ) for every b ?
  a.

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Monotonicity what is really needed for our proof

• Def dabi (v-i ) inf vi (a) - vi (b)
  vi ? Vi s.t. f (vi , v-i ) a .
• W-MON essentially ? dabi (v-i )
  dbai (v-i ) 0
• Cyclic-Monotonicity essentially ?
• dabi (v-i ) dbci (v-i )dcai (v-i ) 0

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• In Unrestricted Domains the Monotonicity
  condition can be strengthened w.l.o.g

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• Dfn f satisfies S-MON if for any vi , ui and
  v-i
• f (vi , v-i) a and f (ui , v-i) b
• implies ui (b) - ui (a) gt vi (b) - vi (a).
• Intuition S-MON stability
• Prop If V is unrestricted then for every f
  there exists f
• if f satisfies W-MON ? f satisfies
  S-MON.
• if f is affine maximizer ? f is affine
  maximizer.

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• Weaker version of Roberts Thm
• Let A 3, and V RnA. If f is
  decisive and
• truthful implementable,
• then f must be affine maximizer (with wj gt 0,
  j 1..n) .
• Equivalent Thm
• Let A 3, and V RnA. If f is
  decisive and
• satisfies cyclic-monotonicity and S-MON,
• then f must be affine maximizer (with wj gt 0,
  j 1..n) .

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• (For every a, b, c ? A, and v-i )
• Claim1 -? lt dabi (v-i ) lt ? .
• Claim2 dabi (v-i ) dbai (v-i
  ) 0 .
• Claim3 dabi (v-i ) dbci (v-i ) dcai (v-i )
  0 .
• Claim4 dabi (v-i ) dab i (v-i - L 1j,c ),
  for every L ? R.
• Claim5 dabi (v-i ) dab i (v-i (a) - v-i (b))
  .
• Conclusion dabi (r ) dbai (-r ) 0 ,
• dabi (r ) dbci (t ) dcai (-r - t ) 0
  , for every r, t ? Rn-1
• dabi (0 ) dbai (0 ) 0 .
• dabi (0 ) dbci (0 ) dcai (0 ) 0 .

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• (For every a, b, c ? A, r, t, s? Rn-1 )
• Claim6 dabi (rt ) - dabi (r ) dcbi (st )
  - dcbi (s ) .
• Claim7 dabi (r ) - Sj? i wj rj dabi (0
  ) .
• (the same wj gt 0 for every a and b).
• Finally Fix some c. If f (v) a, then for
  every b we have
• vi (a) - vi (b) gt dabi (v-i (a) - v-i (b))
  
• - Sj? i wj (vj (a) - vj (b)) dabi (0 )
• - Sj? i wj (vj (a) - vj (b)) - dbci (0 ) -
  dcai (0 )
• - Sj? i wj (vj (a) - vj (b)) - dbci (0 )
  daci (0 ).
• And so f(v) argmaxa ? A vj (a) Sj? i
  wj vj (a) - daci (0 )

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• Thank you!

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W-MON ? Truthfulness?

• Thm Bikhchandani, Chatterji, Lavi, M, Nisan,
  Sen, Muller, Vohra
• W-Mon ? Truthfulness for Combinatorial Auctions,
• Multi Unit Auctions with decreasing marginal
  valuations, and several other interesting
  domains.
• Thm Saks, Yu
• If V is convex, then W-Mon ? Truthfulness.

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Impossiblities for Restricted Domains

• Thm Lavi, M, Nisan
• Every player-decisive, non-degenerate
  implementable SCF for Combinatorial Auctions that
  satisfies S-MON must be an almost affine
  maximizer.

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Monotonicity one more thing

• Many other implementation concepts imply similar
  monotonicity conditions. E.g., truthfulness in
  expectation Lavi, Swamy, M, Schapira.

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• Thank you!
• !!!