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 ).

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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

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.

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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

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.

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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-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

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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 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

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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

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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

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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.

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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 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: …

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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. Multi- parameter problems).

• Relaxing the requirements of Monge Domains: a partial order on the alternatives/valuations instead of a complete order.

Thank you