Characterizing Mechanism Design Over Discrete Domains

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Characterizing Mechanism Design Over Discrete
Domains
Ahuva Mualem and Michael Schapira
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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?

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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)
• Nisan, Ronens scheduling problem find a
  partition of the tasks T1..Tn to the machines
  that minimizes maxi costi (Ti ).

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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 i.
• 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).

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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 i.
• 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.
• - First vs. Second Price Auction.
• - Not all SCFs can be implemented e.g.,
  Majority vs. Minority between 2 alternatives.

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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 i.
• 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).
• Main Problem Which social choice functions
  are truthful?

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• Truthfulness and Monotonicity

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Truthfulness vs. Monotonicity Example 1 item
Auction with 2 bidders Myerson
1 wins
v1
?
?
v1
2 wins
?
v2
p2
v2
v'2
p2

• Mon. ? Truthfulness
• player 2 wins and pays p2.

?Mon. ? ?Truthfulness the curve is not monotone
- player 2 might untruthfully bid v2 v2.
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Truthfulness ? Monotonicity ?
Monotonicity refers to the social choice
function alone (no need to consider the payment
function). Problem Identify this class of
social choice functions.
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Truthfulness vs. Monotonicity

• Thm Roberts Every truthfully implementable f
  V ? A is Weak-Monotone.
• Thm Rochet f V ? A is truthfully
  implementable iff f is Cyclic-Monotone.
• Dfn V is called WM-domain if any social choice
  function on V satisfying Weak-Monotonicity is
  truthful implementable.

?
?
Cyclic-Monotonicity
Weak-Monotonicity
Simple-Monotonicity
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Truthfulness vs. Monotonicity

• Thm Roberts Every truthfully implementable f
  V ? A is Weak-Monotone.
• Thm Rochet f V ? A is truthfully
  implementable iff f is Cyclic-Monotone.
• Dfn V is called WM-domain if any social choice
  function on V satisfying Weak-Monotonicity is
  truthful implementable.

?
?
Cyclic-Monotonicity
Weak-Monotonicity
Simple-Monotonicity
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WM-Domains
Dfn V is called WM-domain if any social choice
function on V satisfying Weak-Monotonicity is
truthful implementable. Thm Bikhchandani,
Chatterji, Lavi, M, Nisan, Sen,Gui, Muller,
Vohra 2003 Combinatorial Auctions, Multi Unit
Auctions with decreasing marginal valuations, and
several other interesting domains (with linear
inequality constraints) are WM-Domains. Thm
Saks, Yu 2005 If V is convex, then V is a
WM-Domain. Thm Monderer 2007 If closure(V)
is convex and even if f is randomized, then
Weak-Monotonicity ? Truthfulness.
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WM-Domains
Dfn V is called WM-domain if any social choice
function on V satisfying Weak-Monotonicity is
truthful implementable. Thm Bikhchandani,
Chatterji, Lavi, M, Nisan, Sen,Gui, Muller,
Vohra 2003 Combinatorial Auctions, Multi Unit
Auctions with decreasing marginal valuations, and
several other interesting domains (with linear
inequality constraints) are WM-Domains. Thm
Saks, Yu 2005 If V is convex, then V is a
WM-Domain. Thm Monderer 2007 If closure(V)
is convex and even if f is randomized, then
Weak-Monotonicity ? Truthfulness.
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Cyclic-Monotonicity ?Truthfulness Rochet
? Convex Domains SaksYu
? Combinatorial Auctions with single minded
bidders LOS
? Essentially Convex Domains Monderer

• 1 item Auctions
• Myerson

? WM-Domains
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Cyclic-Monotonicity ?Truthfulness Rochet
? Convex Domains SaksYu
? Combinatorial Auctions with single minded
bidders LOS
? Essentially Convex Domains Monderer

• 1 item Auctions
• Myerson

Discrete Domains??
? WM-Domains
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? Monge Domains
? Integer Grid Domains
? 0/1 Domains
? WM-Domains
? Strong-Monotonicity ?Truthfulness
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• Weak / Strong / Cyclic Monotonicity

?
Cyclic-Monotonicity
Weak-Monotonicity
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Monotonicity Conditions

• Dfn1 f is Weak-Monotone if for any vi , ui
  and v-i
• f (vi , v-i) a and f (ui , v-i) b
• implies vi (a) ui (b) gt vi (b) ui (a).
• Dfn2 f is 3-Cyclic-Monotone if for any vi ,
  ui , wi and v-i
• f (vi , v-i) a , f (ui , v-i) b and f
  (wi , v-i) c
• implies vi (a) ui (b) wi (c) gt vi (b)
  ui (c) wi (a) .
• Dfn3 f is Strong-Monotone if for any vi , ui
  and v-i
• f (vi , v-i) a and f (ui , v-i) b
• implies vi (a) ui (b) gt vi (b) ui (a).

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• Example A single player,
• 2 alternatives a, and b, and
• 2 possible valuations v1, and v2.
• Majority satisfies Weak-Mon.
• f(v1) a, f(v2) b.
• Minority doesnt.
• f(v1) b, f(v2) a.

v1 v2
a 1 0
b 0 1
v1 v2
a 1 0
b 0 1
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Monotonicity Conditions

• Dfn1 f is Weak-Monotone if for any vi , ui
  and v-i
• f (vi , v-i) a and f (ui , v-i) b
• implies vi (a) ui (b) gt vi (b) ui (a).
• Dfn2 f is 3-Cyclic-Monotone if for any vi ,
  ui , wi and v-i
• f (vi , v-i) a , f (ui , v-i) b and f
  (wi , v-i) c
• implies vi (a) ui (b) wi (c) gt vi (b)
  ui (c) wi (a) .
• Dfn3 f is Strong-Monotone if for any vi , ui
  and v-i
• f (vi , v-i) a and f (ui , v-i) b
• implies vi (a) ui (b) gt vi (b) ui (a).

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• Example
• single player
• A a, b, c.
• V1 v1, v2, v3.
• f(v1)a, f(v2)b, f(v3)c.

v1 v2 v3
a 0 1 -2
b -2 0 1
c 1 -2 0
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• Example
• single player
• A a, b, c.
• V1 v1, v2, v3.
• f(v1)a, f(v2)b, f(v3)c.
• f satisfies Weak-Monotonicity , but not
  Cyclic-Monotonicity

v1 v2 v3
a 0 1 -2
b -2 0 1
c 1 -2 0
v1 v2
a 0 1
b -2 0
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• Discrete Domains
• Integer Grids and Monge

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Integer Grid Domains are SM-Domains but not
WM-Domains
PropYu 2005 Integer Grid Domains are not
WM-Domains. Thm Any social choice function on
Integer Grid Domain satisfying
Strong-Monotonicity is truthful
implementable. Similarly Prop 0/1-Domains
are SM-Domains, but not WM-Domains.

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Monge Matrices
Dfn Bbr,c is a Monge Matrix if for
every r lt r and c lt c br, c br, c gt
br, c br, c.
Example 4X5 Monge Matrix
1 2 2 0 0
0 1 5 4 4
-2 0 8 8 8
-1 1 9 9 10
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Monge Domains

• Dfn V V1 . . .Vn is a Monge Domain if for
  every i?n
• there is an order over the alternatives in A
  a1, a2, . . .
• and an order over the valuations in Vi vi
  1, vi 2, . . . ,
• such that the matrix Bibr,c
• in which br,c vi c( ar)
• is a Monge matrix.
• Examples
• Single Peaked Preferences
• Public Project(s)

vi 1 vi 2 vi 3 vi 4 vi 5
a1 1 2 2 0 0
a2 0 1 5 4 4
a3 -2 0 8 8 8
a4 -1 1 9 9 10
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Monotonicity on Monge Domains
Dfn f is Weak-Monotone if for any vi , ui
and v-i f (vi , v-i) a and f (ui ,
v-i) b implies vi (a) ui (b) gt vi (b)
ui (a). There are two cases to consider
vi 1 vi 2 vi 3 vi 4 vi 5
a1 1 2 2 0 0
a2 0 1 5 4 4
a3 -2 0 8 8 8
a4 -1 1 9 9 10
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• A simplified Congestion Control Example
• Consider a single communication link with
  capacity C gt n.
• Each player i has a private integer value di
  that represents the number of packets it wishes
  to transmit through the link.
• For every vector of declared values d d1, d2,
  . . . , dn, the capacity of the link is shared
  between the players in the following recursive
  manner (known as fair queuing Demers, Keshav,
  and Shenker) If di gt ?C / n? then allocate
  a capacity of ?C / n? to each player.
• Otherwise, perform the following steps Let dk
  be the lowest declared value. Allocate a capacity
  of dk to player k. Apply fair queuing to
  share the remaining capacity of C - dk between
  the remaining players.

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A simplified Congestion Control Example
(cont.) Assume the capacity C5, then Vi

vi 1 vi 2 vi 3 vi 4 vi 5
a1 1 1 1 1 1
a2 1 2 2 2 2
a3 1 2 3 3 3
a4 1 2 3 4 4
a5 1 2 3 4 5
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A simplified Congestion Control Example
(cont.) Clearly, a player i cannot get a
smaller capacity share by reporting a higher vi
j. And so, The Fair queuing rule dictates an
alignment. Claim Every social choice
function that is aligned with a Monge Domain is
truthful implementable. Thm Monge Domains are
WM-Domains. Proof
vi 1 vi 2 vi 3 vi 4 vi 5
a1 1 1 1 1 1
a2 1 2 2 2 2
a3 1 2 3 3 3
a4 1 2 3 4 4
a5 1 2 3 4 5
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Monge Domains
Claim Every social choice function that is
aligned with a Monge Domain is truthful
implementable. Thm Monge Domains are
WM-Domains. Proof
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Related and Future Work

• Archer and Tardoss setting scheduling jobs
  on related parallel machines to minimize makespan
  is a Monge Domain.
• Lavi and Swamy unrelated parallel machine,
  where each job has two possible values High and
  Low (its a special case of Nisan and Ronen
  setting). Its a discrete, but not a Monge
  Domain. They use Cyclic-monotonicity to show
  truthfulness.
• Find more applications of Monge Domains (Single
  vs. Multi- parameter problems).
• Relaxing the requirements of Monge Domains a
  partial order on the alternatives/valuations
  instead of a complete order.

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• Thank you