Characterizing Mechanism Design
Over
Discrete Domains
Ahuva Mu’alem and Michael Schapira
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”?
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 | i=1..n}, Vi = R+, f (v) = argmax(vi)
– [Nisan, Ronen]’s scheduling problem: find a partition of the tasks
T1..Tn to the machines that minimizes maxi costi (Ti ).
3
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 : v ( f(v , v )) – p (v , v ) ≥ v ( f(w , v )) – p (w , v ).
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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 : v ( f(v , v )) – p (v , v ) ≥ v ( f(w , v )) – p (w , v ).
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- 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.
5
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 : v ( f(v , v )) – p (v , v ) ≥ v ( f(w , v )) – p (w , v ).
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Main Problem: Which social choice functions are truthful?
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Truthfulness and Monotonicity
Truthfulness vs. Monotonicity
Example: 1 item Auction with 2 bidders [Myerson]
1 wins
v1
●
p2
2 wins
v1
●
v2
p2
●
v'2 v2
Mon.  Truthfulness

Mon.  Truthfulness
player 2 wins and pays p2.
the curve is not monotone player 2 might untruthfully bid
v’2 ≤ 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.
9
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.
“Simple”-Monotonicity

Weak-Monotonicity

Cyclic-Monotonicity
Dfn : V is called WM-domain if any social choice function on V
satisfying Weak-Monotonicity is truthful implementable.
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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.
“Simple”-Monotonicity

Weak-Monotonicity

Cyclic-Monotonicity
Dfn: V is called WM-domain if any social choice function on V
satisfying Weak-Monotonicity is truthful implementable.
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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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Combinatorial
Auctions with
single minded
bidders [LOS]

1
Convex Domains
[Saks+Yu]

item Auctions
CyclicMonotonicity
Truthfulness
[Rochet]
Essentially
Convex Domains
[Monderer]

[Myerson]

WM-Domains
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Combinatorial
Auctions with
single minded
bidders [LOS]

1
Convex Domains
[Saks+Yu]

item Auctions
CyclicMonotonicity
Truthfulness
[Rochet]
Essentially
Convex Domains
[Monderer]

[Myerson]

WM-Domains
Discrete Domains??
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
Monge Domains
Integer
Grid Domains



WM-Domains
0/1 Domains
 Strong-Monotonicity
Truthfulness
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Weak / Strong / Cyclic –
Monotonicity
Weak-Monotonicity

Cyclic-Monotonicity
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) > 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) > 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) > vi (b) + ui (a).
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Example: A single player,
2 alternatives a, and b, and
2 possible valuations v1, and v2.
a
b
v1 v2
1 0
0 1
a
b
v1 v2
1 0
0 1
 Majority satisfies Weak-Mon.
f(v1) = a, f(v2) = b.
 Minority doesn’t.
f(v1) = b, f(v2) = a.
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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) > 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) > 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) > 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.
a
b
c
v1
0
-2
1
v2
1
0
-2
v3
-2
1
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.
a
b
c
v1
0
-2
1
v2
1
0
-2
v3
-2
1
0
f satisfies Weak-Monotonicity , but not Cyclic-Monotonicity:
v1 v2
a
0
1
b
-2 0
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Discrete Domains:
Integer Grids and Monge
Integer Grid Domains are SM-Domains
but not WM-Domains
Prop[Yu 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.
Monge Matrices
Dfn:
B=[br,c] is a Monge Matrix
if for every r < r’ and c < c’:
br, c + br’, c’ > br’, c+ br, c’.
1
0
-2
-1
2
1
0
1
2
5
8
9
0
4
8
9
0
4
8
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Example: 4X5 Monge Matrix
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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 Bi =[br , c ]
in which br , c = vi c( ar )
vi 1 vi 2 vi 3 vi 4 vi 5
is a Monge matrix.
a1
Examples:
• Single Peaked Preferences
• Public Project(s)
1
a2 0
a3 -2
a4 -1
2
1
0
1
2
5
8
9
0 0
4 4
8 8
9 10
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) > vi (b) + ui (a).
There are two cases to consider:
…
vi 1 vi 2 vi 3 vi 4 vi 5
a1
1
a2 0
a3 -2
a4 -1
2
1
0
1
2
5
8
9
0 0
4 4
8 8
9 10
A simplified Congestion Control Example:
•
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•
Consider a single communication link with capacity C > 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’= d’1 , d’2 , . . . , d’n , the
capacity of the link is shared between the players in the
following recursive manner (known as fair queuing [Demers,
Keshav, and Shenker]): If d’i > C / n then allocate a
capacity of C / n to each player.
Otherwise, perform the following steps: Let d’k be the lowest
declared value. Allocate a capacity of d’k to player k. Apply
fair queuing to share the remaining capacity of C - d’k
between the remaining players.
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A simplified Congestion Control Example (cont.):
Assume the capacity C=5, 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
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”.
a1
1
1
1
1
1
1
2
2
2
2
1
2
3
3
3
a4
1
2
3
4
4
a5
1
2
3
4
5
Claim: Every social choice function
that is aligned with a Monge
a2
Domain is truthful implementable.
a3
Thm:
Monge Domains are WM-Domains.
Proof: …
vi 1 vi 2 vi 3 vi 4 vi 5
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 Tardos]’s 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 (it’s a special case of
[Nisan and Ronen] setting). It’s a discrete, but not a Monge
Domain. They use Cyclic-monotonicity to show truthfulness.
• Find more applications of Monge Domains (Single vs. Multiparameter problems).
• Relaxing the requirements of Monge Domains: a partial order
on the alternatives/valuations instead of a complete order.
Thank you