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Characterizing Mechanism Design Over
Discrete Domains
Ahuva Mu’alem and Michael Schapira
Mechanisms: elections, auctions (1st / 2nd price, double, combinatorial, …), resource allocations … social goal vs. individuals’ strategic behavior.
Main Problem: Which social goals can be “achieved”?
Motivation
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 : vi ( f(vi , v-i)) – pi(vi , v-i) ≥ vi ( f(wi , v-i)) – pi(wi , v-i).
4
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.
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 : 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?
6
Truthfulness and Monotonicity
Truthfulness vs. Monotonicity Example: 1 item Auction with 2 bidders [Myerson]
v1
v2
Mon. Truthfulnessplayer 2 wins and pays p2.
p2
2 wins
1 wins
v1
v2v'2p2
●●
●
Mon. Truthfulnessthe 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
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.
Truthfulness vs. Monotonicity
10
Cyclic-MonotonicityWeak-Monotonicity“Simple”-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.
Truthfulness vs. Monotonicity
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Cyclic-MonotonicityWeak-Monotonicity“Simple”-Monotonicity
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.
12
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.
13
14
Cyclic-MonotonicityTruthfulness[Rochet] Convex Domains
[Saks+Yu] Combinatorial Auctions with single minded bidders [LOS] Essentially
Convex Domains [Monderer]
WM-Domains
1 item Auctions[Myerson]
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Cyclic-MonotonicityTruthfulness[Rochet] Convex Domains
[Saks+Yu] Combinatorial Auctions with single minded bidders [LOS] Essentially
Convex Domains [Monderer]
Discrete Domains?? WM-Domains
1 item Auctions[Myerson]
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Monge Domains
Integer Grid Domains
WM-Domains
0/1 Domains
Strong-Monotonicity Truthfulness
Weak / Strong / Cyclic – Monotonicity
Cyclic-MonotonicityWeak-Monotonicity
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).
Monotonicity Conditions
18
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 doesn’t. f(v1) = b, f(v2) = a.
v1 v2
a 1 0
b 0 1
v1 v2
a 1 0
b 0 119
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).
Monotonicity Conditions
20
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
21
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
22
v1 v2
a 0 1
b -2 0
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.
Dfn:
B=[br,c] is a Monge Matrix
if for every r < r’ and c < c’:
br, c + br’, c’ > 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
25
Monge Matrices
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)
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 0a2 0 1 5 4 4a3 -2 0 8 8 8a4 -1 1 9 9 10
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: …
Monotonicity on Monge Domains
vi 1 vi
2 vi 3 vi
4 vi 5
a1 1 2 2 0 0a2 0 1 5 4 4a3 -2 0 8 8 8a4 -1 1 9 9 10
A simplified Congestion Control Example:
• 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.
28
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 smallercapacity share by reporting a higher vi
j.And so, The Fair queuing rule dictates an “alignment”.
Claim: Every social choice functionthat 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
Monge Domains
Claim: Every social choice function that is aligned with a Monge Domain is truthful implementable.
Thm: Monge Domains are WM-Domains.
Proof: …
31
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. Multi- parameter problems).
• Relaxing the requirements of Monge Domains: a partial order on the alternatives/valuations instead of a complete order.
Thank you