strategy-proof and efficient fair scheduling · strategy-proof and e cient fair scheduling yuan...

Strategy-proof and Efficient Fair Scheduling

Yuan Tian

Department of Economics

University of Chicago

Yuan Tian (University of Chicago) Mechanism Design of Fair Scheduling 1 / 49

Dynamic or sequential division problems

In each period or batch, an interval of indivisible heterogeneous goods must

be allocated to a set of players;

Goods in early periods must be allocated before those in later periods or

some batches of goods must be divided before other batches;

Players have dichotomous preferences: preference types are represented by

subsets of the goods to be allocated;

Players’ preferences may be dynamically or sequentially revealed to them as

private information;

No monetary transfers allowed.

Applications

Production task assignment on an assembly line;

Allocation of central server computing time;

Plug-in time division of electric car charging stations;

Scheduling of conference rooms;

Circulation of library books.

Contributions and challenges

Construct a dynamically payoff consistent, Pareto efficient, envy-free, and

strategy-proof division mechanism;

Discover a novel comparative statics result for the egalitarian solutions for

monotone and concave cooperative games regarding inter-personal payoff

comparisons;

Key feature of design: periodically promoting egalitarianism among players’

cumulative payoffs achieves strategy-proof-ness;

Major challenge: how to account for the lasting effect of a single player’s

deviation from truth-telling in a single period on the overall distribution of

cumulative payoffs of all players.

A quick demonstration

A set of three players N = {1, 2, 3};

Time horizon: [0, 7]; Interval [t, t + 1] is up for division in period t;

Player 2’s demand is fixed at [1, 2] ∪ [3, 5] ∪ [6, 7];

Player 3’s demand is fixed at [2, 4] ∪ [5, 7];

Player 1’s demand can be either θ1 = [1, 2] ∪ [4, 6] or θ1 = [1, 2] ∪ [3, 6];

How do player 1’s payoffs compare when her type is θ1 versus θ1

according to the mechanism designed in this work?

Cumulative payoff difference

Figure: Exponentiated difference in cumulative payoffs for player 1 with θ1 and θ1

3 3.5 4 4.5 5 5.5 61.3951

1.3952

1.3953

1.3954

1.3955

1.3956

1.3957

ated c

tive p

Student Version of MATLAB

Selected literature

Egalitarian solutions: Dutta and Ray (1989), Chen et al. (2013);

Assignment problems: Hylland and Zeckhauser (1979), Zhou (1990),Bogomolnaia and Moulin (2001), Bogomolnaia and Moulin (2004),Hashimoto et al. (2014);

Fair divisions: Neyman (1946), Steinhaus (1948), Weller (1985), Aziz and Ye(2014), Brams et al. (2006), Brams et al. (2008), Mossel and Tamuz (2010),Procaccia (2013);

Income distribution: Dasgupta et al. (1973), Rothschild and Stiglitz (1973),Shorrocks (1983);

Monotone comparative statics: Quah (2007), Topkis (1978), Topkis (2001),Milgrom and Shannon (1994), Vives (1990);

Dynamic mechanism design: Athey and Segal (2013), Bergemann andValimaki (2010), Gallien (2006), Pai and Vohra (2013), Pavan et al. (2013).

Overview

1 Introduction

2 Monotone and Concave Cooperative Games and Egalitarian Solutions

3 Sequential or Dynamic Division Problems

4 Sequential or Dynamic Division Mechanisms

5 Discussions

6 Conclusions and Future Research

Overview

1 Introduction

5 Discussions

Cooperative games in characteristic function form

A game Γ = (N, v(·)) with a set of players N where |N| = n < +∞:

The characteristic function v : 2N → R+ such that v(∅) = 0;

Γ is monotone if and only if for all S ⊆ T ⊆ N, v(S) ≤ v(T );

Γ is concave if and only if for all S ⊆ N and T ⊆ N,

v(S ∪ T ) + v(S ∩ T ) ≤ v(S) + v(T ).

For monotone and concave Γ and a ∈ Rn+ and b > 0, Γ ≡ (N, v(·))

with v(S) = b · v(S) +∑

ai ,∀S ⊆ N

is also a monotone and concave game.

Lorenz dominance

Given any z ∈ Rn+, let z be the vector generated by rearranging the entries of

z in a weakly increasing order;

Given two vectors y ∈ Rn+ and y ∈ Rn

+ such thatn∑

yi =n∑

y Lorenz dominates y iff ∀1 ≤ m ≤ n,m∑

yi ≥m∑

Lorenz dominance is an antisymmetric partial order of y and y;

Mutual Lorenz dominance between y and y ⇒ y is a permutation of y .

Generalized Lorenz dominance (Shorrocks (1983))

Given two vectors y ∈ Rn+ and y ∈ Rn

y generalized Lorenz dominates y iff ∀1 ≤ m ≤ n,m∑

yi ≥m∑

Generalized Lorenz dominance does not requiren∑

yi =n∑

y generalized Lorenz dominating y implies thatn∑

yi ≥n∑

Generalized Lorenz dominance implies Lorenz dominance;

Egalitarian solutions to concave cooperative games

Definition (Egalitarian solution)

The egalitarian solution of a monotone and concave game Γ = (N, v(·)), where

n = |N|, is a vector y∗ ∈ Rn+ such that

y∗ is not generalized Lorenz dominated by any y ∈ Rn+ such that

y ∈{z ∈ Rn

+ :∑

zi ≤ v(S),∀S ⊆ N

}≡ V .

From now on, let e(T − S) represent the average incremental from S to T :

∀S ⊂ T ⊆ N, e(T − S) ≡ v(T )− v(S)

|T | − |S | .

Algorithm E Example of Algorithm E

Definition (Algorithm E)

Given any monotone and concave game Γ = (N, v(·)),

1 Let S0 = ∅ and ∅ ⊂ S1 ⊆ N be one of the smallest candidates that

e(S1 − ∅) ≤ e(S − ∅),∀∅ ⊂ S ⊆ N;

2 For all k ≥ 1, let Sk+1 ⊃ Sk be one of the smallest candidates that

e (Sk+1 − Sk) ≤ e(S − Sk),∀S ⊃ Sk ;

3 Terminate in step K if the only candidate for SK is N;

4 Set y∗i = e (Sk+1 − Sk) if i ∈ (Sk+1\Sk) for some 0 ≤ k ≤ (K − 1);

5 For all i ∈ N, the clique of i at y∗: Qi (y∗) ≡ (Sk+1\Sk) if i ∈ (Sk+1\Sk).

Algorithm E : feasibility and dominance

Lemma (Feasibility and dominance)

For any monotone and concave game Γ = (N, v(·)),

1 Algorithm E is well-defined and e(Sk+1 − Sk) ≥ e(Sk − Sk−1); Appendix: proof

2 The y∗ generated by Algorithm E is an element of the feasible set V ;

3 y∗ generalized Lorenz dominates any y ∈ V .

1 Show that there is a smallest candidate in each step of Algorithm E ;

2 Induction on k for subsets of Sk ;

3 By contradiction:∑m

i=1 yi >∑m

i=1 yi implies that y /∈ V .

“Duality” of egalitarian solutions

Lemma (“Duality”)

Given a monotone and concave game Γ = (N, v(·)), let {Sk}Kk=1 be any sequence

of subsets generated by Algorithm E . Then

∀1 ≤ k ≤ K , e(Sk − Sk−1) = maxS⊂Sk

e(Sk − S).

Proof plan: induction on k for subsets of Sk . Step 1 in Algorithm E :

∀∅ 6= S ⊂ S1, e(S1 − ∅) = e(S1 − S)|S1| − |S ||S1|

+ e(S − ∅)︸︷︷︸≥e(S1−∅)

|S ||S1|

Suppose true for k > 1: ∀S ⊂ Sk , e (Sk − S) ≤ e (Sk − Sk−1);

Want to show that ∀S ⊂ Sk+1, e (Sk+1 − S) ≤ e (Sk+1 − Sk).

Proof of “Duality”

S ⊂ Sk or Sk ⊂ S ⊂ Sk+1. Otherwise, (Sk ∩ S) ⊂ Sk and (Sk ∪ S) ⊃ Sk .

≤e(Sk+1−Sk )︷︸︸︷e(Sk+1 − S) =

|Sk+1| − |Sk ||Sk+1| − |S |

≥e(Sk−Sk−1)︷︸︸︷e(Sk+1 − Sk) +

|Sk | − |S ||Sk+1| − |S |

≤e(Sk−Sk−1)︷︸︸︷e(Sk − S) ,

e(Sk+1 − Sk)︸︷︷︸≥e(Sk+1−S)

=|Sk+1| − |S ||Sk+1| − |Sk |

e(Sk+1 − S) +|S | − |Sk ||Sk+1| − |Sk |

e(S − Sk)︸︷︷︸≥e(Sk+1−Sk )

≤e(Sk+1−Sk )︷︸︸︷e (Sk+1 − S) ≤

≤e(Sk+1−Sk )︷︸︸︷e (Sk+1 − (Sk ∪ S))

|Sk+1| − |Sk ∪ S ||Sk+1| − |S |

+ e (Sk − (Sk ∩ S))︸︷︷︸≤e(Sk−Sk−1)

|Sk | − |Sk ∩ S ||Sk+1| − |S |

A thought experiment of changing a game

Given a game Γ = (N, v(·)), what if a player i enlarges the feasible set?

Some restrictions for how i changes the game Γ:

1 i does not affect any group excluding i ;

2 i only increases the values of groups including i ;

3 i affects smaller groups including i more than larger groups.

How will such changes affect the egalitarian solution y∗ of Γ?

Comparative statics of egalitarian solutions

Theorem (Maximal increase)

Let Γ = (N, v(·)) and Γ = (N, v(·)) be two monotone and concave games such

that there exists a unique i ∈ N such that, for all S ⊆ N,

1 i /∈ S ⇒ v(S) = v(S);

2 i ∈ S ⇒ v(S) ≤ v(S);

3 i ∈ S ⊂ T ⇒ v(T )− v(T ) ≤ v(S)− v(S).

Let y∗ and y∗ be the egalitarian solutions of Γ and Γ, respectively. ∀j ∈ N,

y∗i − y∗i ≥ y∗j − y∗j . (1)

Proof of Maximal increase I

From an earlier paper Tian (2013): y∗i ≥ y∗i ;

For any j ∈ N such that y∗j ≤ y∗i and j /∈ Qi (y∗), y∗j = y∗j by Algorithm E ;

For any j ∈ N such that y∗j ≥ y∗i and y∗j ≤ y∗i , y∗i − y∗i ≥ y∗j − y∗j ;

Hence, take any j such that y∗j ≥ y∗i and y∗j > y∗i ; Want to show y∗j ≤ y∗j ;

Let Sl+1 be the first entry in the sequence{Sl

l=1of subsets of N

generated by Algorithm E for Γ such that (Sk+1\Sk) = Qj(y∗) ⊆ Sl+1,

where{Sk}Kk=1

is the sequence generated by Algorithm E for Γ;

Keep in mind that this implies y∗j = e (Sk+1 − Sk);

Proof of Maximal increase II

Thus,(Sk+1 ∪ Sl

)⊃ Sl and

(Sk+1 ∩ Sl

)⊂ Sk+1.

Notice that y∗j > y∗i implies that i ∈ Sl by the definition of Algorithm E ;

y∗j ≤v(Sl+1

)− v

∣∣∣Sl+1

∣∣∣−∣∣∣Sl∣∣∣

Definition of Algorithm E︷︸︸︷

≤v(Sl ∪ Sk+1

)− v

∣∣∣Sl ∪ Sk+1

∣∣∣−∣∣∣Sl∣∣∣

≤v(Sl ∪ Sk+1

)− v

∣∣∣Sl ∪ Sk+1

∣∣∣−∣∣∣Sl∣∣∣

︸︷︷︸i affects smaller sets more

≤v (Sk+1)− v

(Sl ∩ Sk+1

|Sk+1| −∣∣∣Sk+1 ∩ Sl

∣∣∣︸︷︷︸

by concavity of Γ

≤ e (Sk+1 − Sk)︸︷︷︸by “Duality”

Unilateral disturbance

Lemma (Unilateral disturbance)

Let Γ = (N, v(·)) and Γ = (N, v(·)) be two monotone and concave games such

that there exists a unique i ∈ N such that, for all S ⊆ N,

1 i /∈ S ⇒ v(S) = v(S);

2 i ∈ S ⇒ v(S) ≤ v(S);

3 i ∈ S ⊂ T ⇒ v(T )− v(T ) ≤ v(S)− v(S).

Let y∗ and y∗ be the egalitarian solutions for Γ and Γ, respectively. Then

y∗i = y∗i ⇒ y∗ = y∗. Otherwise,[y∗j ≥ y∗i and j /∈ Qi (y∗)

]⇒ y∗j ≤ y∗j .

Overview

1 Introduction

5 Discussions

Players and types

A finite set N = {1, 2, 3, · · · , n} of players; The planner is player 0;

Time periods, finite or infinite, T ≡ {t}T−1t=0 ; |T | = T ≤ +∞;

In period t, a unit interval [t, t + 1] ⊂ R+ is up for division;

Player i ’s type in period t: θit ⊆ [t, t + 1]; Player i ’s type: θi ≡⋃

t∈Tθit ;

θit ∈ Θt is a finite collection of mutually disconnected closed intervals;

Profile of all players’ types in period t is represented by ~θt ≡ (θit)i∈N ;

Profile of all players’ types: θ ≡ (θi )i∈N =(θi , (θj)j∈(N\{i})

)≡ (θi , θ−i ).

Feasible divisions and preferences

A feasible division is a function C : R+ → N ∪ {0};

Set of all feasible divisions is represented by C;

The indicator function for player i given C : 1Ci (r) =

{1 if C (r) = i ,

0 otherwise;

i ’s discount factor for period t: βit > 0, ∀(i , t) ∈ N × T ;

i ’s period-t canonical utility of C given type θi : uit (C |θi ) ≡∫

1Ci (r) dr ;

i ’s canonical utility of C given type θi : ui (C |θi ) ≡T−1∑

βituit (C |θi );

Profile of all players’ period-t canonical utilities of C with θ: ~ut (C |θ).

Properties of feasible divisions

A feasible division C is Pareto efficient if and only if

there does not exist a feasible division C ′ 6= C such that

ui (C ′|θi ) ≥ ui (C |θi ) , ∀i ∈ N,

with the inequality being strict for at least one i ;

A feasible division C is envy-free if and only if for all (i , j) ∈ N × N,

ui (C |θi ) ≥T−1∑

1Cj (r) dr .

Three assumptions

Assumption (Common discount factor)

There exists a sequence {βt}T−1t=0 such that βit = βt , for all (i , t) ∈ N × T .

Assumption (Finite maximum utility)

T−1∑

βt < +∞.

Assumption (Finite type space)

∀(i , t,

(θit , θit

))∈(N × T ×Θ2

), θit ∩ θit ∈ Θt , θit ∪ θit ∈ Θt , and |Θt | < +∞.

Information structure and direct mechanisms

{βt}T−1t=0 and {Θt}T−1

t=0 are common knowledge among N ∪ {0};

For any player i , θi is player i ’s private information;

Players’ types can be dynamically revealed to the players:

However, θit must be revealed to i before t;

The space of all possible types is represented by Θ ≡T−1∏

A direct mechanism (or “mechanism”) is a function M : Θn → C.

Properties of mechanisms

M is strategy-proof if and only if ∀(i ,(θi , θi

), θ−i

)∈(N ×Θ2 ×Θn−1

ui (M(θi , θ−i )|θi ) ≥ ui(M(θi , θ−i

)|θi)

M is payoff consistent (or “consistent”) if and only if

∀(θ, θ, t

)∈ (Θn ×Θn × T ) , ~θt =

~θt for all 0 ≤ t ≤ t implies that

uit (M (θ) |θi ) = uit(M(θ)|θi), ∀i ∈ N and 0 ≤ t ≤ t;

M is Pareto efficient (envy-free) if and only if

M(θ) is a Pareto efficient (envy-free) feasible division for all θ ∈ Θn.

Overview

1 Introduction

5 Discussions

Notations

For all 1 ≤ t ≤ T , let χt ≡t−1∑

For all (θ,S , t) ∈ Θn × 2N × T , let v(S |~θt

)≡∫⋃

i∈S θit

1 dr ;

For all S ⊂ T ⊆ N, let e(T − S |~θt

v(T |~θt

)− v

(S |~θt

|T | − |S | ;

Let V(~θt

)≡{y ∈ Rn

+ :∑

yi ≤ v(S |~θt

), ∀S ⊆ N

Let Et ≡{e(T − S |~θt

):(S ,T , ~θt

)∈(2N , 2N ,Θn

),S ⊂ T ⊆ N

}∪ {0};

Let δt ≡ min{βt · |z1 − z2| : (z1, z2) ∈ E 2

t and (z1 − z2) 6= 0}⇒ δt > 0.

Two sequences

Definition (Two Sequences)

Recursively define two sequences {ρt}T−1t=1 and {∆t}Tt=1 as follows:

1 Set ∆1 = δ0; Set ρ1 so that 0 < ρ1 < min

χ1 + ∆1,δ1

Set ∆2 = min {(1− ρ1) ∆1, δ1} − ρ1χ1 > 0;

2 Given ∆t , set ρt so that 0 < ρt < min

χt + ∆t,δt

Set ∆t+1 = min {(1− ρt) ∆t , δt} − ρtχt > 0.

Mechanism R Formal definition

Let W : R+ → R be strictly increasing and strictly concave and continuous on R+;

Given any type profile θ, R(θ) is recursively constructed as follows:

1 In period 0, select C1 ∈ C to maximize∑

W (ui0 (C1|θi )) within V(~θ0

2 Given Ct ∈ C, let xit (Ct |θi ) ≡t−1∑

βsuis (Ct |θi );

3 In period t, select Ct+1 ∈ C to replicate Ct up to t

and to maximize∑

W (ρtxit(Ct |θi ) + βtuit (Ct+1|θi )) within V(~θt

4 In any period s, only select feasible divisions so that r /∈ θis ⇒ Cs+1(r) 6= i .

The result

Proposition (C-PE-EF-SP)

Mechanism R is consistent, Pareto efficient, envy-free, and strategy-proof.

Consistency: by definition;

The maximizer of∑

W (ρtxit(Ct |θi ) + βtuit (Ct+1|θi )) in V(~θt

)is unique;

Pareto efficiency: by definition;

W (ρtxit(Ct |θi ) + βtuit (Ct+1|θi )) is strictly increasing in uit(·|·);

Hence, focus on envy-free-ness and strategy-proof-ness;

When clear, notations regarding θ may be dropped for brevity.

Envy-free-ness: what if i envies j in period τ

Suppose R is not envy-free. Then there exists a triple (i , j , τ) ∈ N × N × T :

0 ≤ uiτ (R(θ)|θi ) <∫

1R(θ)j (r) dr ≤ ujτ (R(θ)|θj) ; (2)

⇒ ρτxiτ + βτuiτ ≥ ρτxjτ + βτujτ

ρτ<1===⇒ xiτ − xjτ ≥

βτ (ujτ − uiτ )

ρτ> βτ (ujτ − uiτ );

Since, otherwise, a transfer from j to i in τ will increase∑

W (xiτ , uiτ );

(2) and (3) together imply xi(τ+1) > xj(τ+1): i does not envy j by (τ + 1).

For strategy-proof-ness: observations of Mechanism R

Fix a profile θ and period t; For any S ⊆ N, let xSt ≡∑

i∈S xit

|S | ;

Define period game Γt ≡(N, v

(·|~θt))

and let u∗t be its egalitarian solution;

When [t, t + 1] is divided by Mechanism R, uit = u∗it −ρt

(xit − x

Qi (u∗t )

After [t, t + 1] is divided, xi(t+1) = βtu∗it + (1− ρt)xit + ρt x

Qi (u∗t )

Strategy-proof-ness: plan of the proof

“One-shot deviation principle” by Blackwell (1965) applies;

Suppose only player i deviates only in period τ ;

Suffices to just consider deviations to supersets or subsets (Tian (2013));

Focus on deviations to supersets of the true type: let θiτ ⊃ θiτ ;

Suffices to show: uiτ ≥ uiτ ;

And ∀t > 1, xi(τ+t) − xi(τ+t) ≤ xi(τ+1) − xi(τ+1);

And ∀t > 1, xi(τ+t) − xi(τ+t) ≥ 0. Details

Strategy-proof-ness: sketch of the proof

When u∗iτ = u∗iτ , Mechanism R proceeds identically for θiτ and θiτ ; Unilateral

Instead, βt ·>0︷︸︸︷

(uiτ − uiτ ) =

>0︷︸︸︷βt · (u∗iτ − u∗iτ ) −

<βt ·(u∗iτ−u∗iτ )︷︸︸︷

(xQi (u

∗t )

t − xQi (u

∗t )

); Sequences

Moreover, for all j 6= i ,(xi(τ+1) − xi(τ+1)

)≥(xj(τ+1) − xj(τ+1)

); Details

For all t ≥ 1, Details

xi(τ+t+1) − xi(τ+t+1) ≤maxj∈N

{xj(τ+t+1) − xj(τ+t+1)

≤maxj∈N

{xj(τ+t) − xj(τ+t)

}≤ · · · ≤ xi(τ+1) − xi(τ+1).

Overview

1 Introduction

5 Discussions

Why are the ρt ’s so small?

Example (ρt = 1)

N = {1, 2, 3}, T = 25, Θt = {∅, [t, t + 1]};

Set ρt = 1 for all 1 ≤ t ≤ 24—call this mechanism R1;

Fix θ1 = [0, 11] ∪ [20, 25] and θ2 = [7, 17];

Deviating agent 3: θ3 = [5, 25] and θ3 = [9, 25];

u3 (R1 (θ3, θ−3) |θ3) = 9.5 < 10 = u3

(θ3, θ−3

Period and cumulative allocations for ρt = 1Table 5: Period and cumulative payoffs by rt = 1 for q and q

i\t 5 6 7 8 9 10 11 12 13 14 15 16... ...20 21 22 23 24 u

1y 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 .5 NA

1x 5 5 5 5 5 5 5 5 5 5 5 5 5 6 7 8 9 9.5

2y 0 0 1 1 .5 .5 .5 .5 .5 .5 .5 .5 0 0 0 0 0 NA

2x 0 0 0 1 2 2.5 3 3.5 4 4.5 5 5.5 6 6 6 6 6 6

3y 1 1 0 0 .5 .5 .5 .5 .5 .5 .5 .5 0 0 0 0 .5 NA

3x 0 1 2 2 2 2.5 3 3.5 4 4.5 5 5.5 9 9 9 9 9 9.5

1y 1 1 0 0 0 0 0 0 0 0 0 0 1 .5 .5 .5 .5 NA

1x 5 6 7 7 7 7 7 7 7 7 7 7 7 8 8.5 9 9.5 10

2y 0 0 1 1 0 0 .5 .5 .5 .5 .5 .5 0 0 0 0 0 NA

2x 0 0 0 1 2 2 2 2.5 3 3.5 4 4.5 5 5 5 5 5 5

3y 0 0 0 0 1 1 .5 .5 .5 .5 .5 .5 0 .5 .5 .5 .5 NA

3x 0 0 0 0 0 1 2 2.5 3 3.5 4 4.5 8 8 8.5 9 9.5 10

Periods 1 through 4 and 17 through 19 are ignored since there is only one player (player 1 and 3, respectively) whose

type includes these periods.

Definition 3. Fix the types of players 1 and 2 and consider two different preferences for player 3:946

8>><>>:

q1 = [0,11][ [20,25], q2 = [7,17], q3 = [5,25] [q ] ;

q1 = [0,11][ [20,25], q2 = [7,17], q3 = [9,25]⇥q⇤.

The outcome of this alternative mechanism is presented in Table 5. Notice that a player with type947

q3 will receive a strictly higher payoff by misrepresenting as q3. Observe that player 3 actually948

deviates in more than one period in this example.949

The spirit of this alternative mechanism is actually identical to that of Mechanism R—both950

build in a reward and punishment mechanism when dividing in period t that is based on players’951

cumulative payoffs up until bt . The difference lies in the seemingly insignificantly small reward952

or punishment given by Mechanism R that has an ample effect on incentives. Take a deviation953

Table: Columns represent periods; Clear rows represent period allocations in therespective period; Shaded rows represent cumulative allocations before the corresponding

periods; Top (bottom) half represents when 3 reports θ3

Why are the ρt ’s not zero then?

A pure strategy for player i : σi : Θ→ Θ;

Given a mechanism M, a profile of pure strategies σ ≡ (σi (·))i∈N comprises a

pure strategy Nash equilibrium (PSNE) if and only if, for all i , there does

not exist a σ′i 6= σi such that

ui (M (σ′i (θi ), θ−i ) |θi ) ≥ ui (M (σi (θi ), θ−i ) |θi ) ,∀θ ∈ Θn

with the inequality being strict for at least one θ;

Let Σ(M) represent the set of all PSNE’s of mechanism M.

Efficiency measure: price of anarchy

Given any mechanism M, define the price of anarchy (POA) of M as

POA (M) ≡ supθ∈Θn

T−1∑

i∈N θi)∩[t,t+1]

infσ∈Σ(M)

ui (M (σ(θ)) |θi )

Order of computation: fix a type profile θ, compute the ratio between

the maximum achievable total payoff and

the total payoff from the worst performing PSNE;

Calculate the highest aforementioned ratio among all type profiles.

Infinite refined division problems and ρt = 0

An infinite refined problem is one where Θt = {∅, [t, t + 1]}

T = +∞ and θi is an infinite union of the intervals in {[t, t + 1]}+∞t=0 ;

Let R0 be defined by setting ρt = 0, for all t ≥ 0;

R0 equally divides [t, t + 1] among all players demanding [t, t + 1];

Reporting demand for R+ is a weakly dominant strategy for all players;

Truth-telling is the strictly dominant strategy for all players for R. Details

Price of anarchy comparison

Proposition

POA (R0) ≥ n = |N| while POA (R) = 1.

Focus on POA (R0):

Suffices to construct an example where the ratio in the definition of POA is n;

Consider θi =⋃

k∈N∪{0}

[i + kn, i + kn + 1] but θi = R+, ∀i ;

Any interval is demanded by only one player but equally divided among all.

Overview

1 Introduction

5 Discussions

Devised a strategy to construct a consistent, Pareto efficient, envy-free, andstrategy-proof dynamic division mechanism R for players with potentiallydynamically revealing dichotomous preferences;

Accommodated infinite time horizon with a common discount factor;

Design of Mechanism R can be easily extended to division problems withtime varying length of intervals;

Mechanism R possesses strict incentive properties and stronger efficiencyperformance than repetition of equal divisions for infinite refined problems byat least a factor of the size of the set of players;

Discovered a novel comparative statics result on the egalitarian solutions ofmonotone and concave cooperative games involving inter-player comparisonof changes in payoffs.

Future research

Axiomatization of dynamic division mechanisms;

Development of fairness and incentive measures and improvement ofefficiency measures;

Characterization of potential “efficiency-fairness-incentive” frontiers fordivision problems;

Characterization of maximal domain for compatibility of PE, EF, and SP inexchange economies and renewed examination of price formation mechanisms;

(Wild) conjecture: for any domain where there exists a division mechanismthat accommodates PE, EF, and SP, the mechanism must be thepseudo-market equilibrium mechanism in Hylland and Zeckhauser (1979).

Thank You!

Proof of Feasibility and dominance I

Let Γ = (N, v(·)) be a monotone and concave game,

1 ∀R ⊂ S ⊂ T ⊆ N,

e(T − R) = e(S − R)⇒ e(T − R) = e(S − R) = e(T − S);

2 ∀R ⊂ T ⊆ N and R ⊂ S ⊆ N, where R ⊂ (T ∩ S) and T 6⊆ S and S 6⊆ T ,

such that e(S − R) = e(T − R), then

min {e [(T ∪ S)− R] , e [(T ∩ S)− R]} ≤ e(S − R) = e(T − R);

3 ∀R ⊂ T ⊆ N and R ⊂ S ⊆ N, where R = (T ∩ S) and T 6⊆ S and S 6⊆ T ,

such that e(S − R) = e(T − R), then

e[(T ∪ S)− R] ≤ e(S − R) = e(T − R).

Proof of Feasibility and dominance II

Proof.

1 e(T − R) = e(T − S) · |T | − |S ||T | − |R| + e(S − R) · |S | − |R||T | − |R| ;

2 By the concavity of Γ, Back

e [(T ∩ S)− R] · (|T ∩ S | − |R|) + e [(T ∪ S)− R] · (|T ∪ S | − |R|)

≤ e(S − R) · (|T ∪ S |+ |T ∩ S | − 2|R|) ;

3 e[(T ∪ S)− R](|T ∪ S | − |R|) ≤ e(T − R) · (|T |+ |S | − 2|R|).

e (Sk+1 − Sk−1) =e (Sk+1 − Sk) · |Sk+1| − |Sk ||Sk+1| − |Sk−1|

+ e (Sk − Sk−1) · |Sk | − |Sk−1||Sk+1| − |Sk−1|

Formal definition of Mechanism R Back to main text

Definition (Mechanism R)

Let {ρt}T−1t=1 be any sequence selected in Two Sequences. Given any type profile θ,

1 Select any feasible division C1 such that

~u0 (C1|θ) = arg maxy∈V (~θ0)

W1(yi ) and r /∈ θi ⇒ C1(r) 6= i ,∀r ∈ R+

for a strictly increasing and strictly concave W1 : R+ → R continuous on R+;

2 Given Ct , select any Ct+1 such that Ct+1(r) = Ct(r) for all r ≤ t and

~ut(Ct+1|θ) = arg maxy∈V (~θt)

Wt+1 (ρt · xit(Ct |θi ) + βtyi ) ≡ ~yt+1

and r /∈ θi ⇒ Ct+1(r) 6= i ,∀r ∈ R+, for any strictly increasing and strictlyconcave function Wt+1 : R+ → R continuous on R+;

3 Define R(θ) to be the pointwise limit of the sequence {Ct}Tt=1.

Details for Proof for SP I

1 j /∈ Qi (u∗τ )⇒ ≤ 0⇒ ® < ¬; Sequences

2 j ∈ Qi (u∗τ ) and j ∈ Qi (u∗τ ), ®= 0⇒ ¬ ≥ ; Maximal increase

3 j /∈ Qi (u∗τ )⇒ ≤ 0⇒ ® < ¬. Sequences Back

(xi(τ+1) − xi(τ+1)

)−(xj(τ+1) − xj(τ+1)

¬︷︸︸︷βτ (u∗iτ − u∗iτ )−βτ

︷︸︸︷(u∗jτ − u∗jτ

+ ρτ

[(xQi (u

∗τ )

τ − xQi (u

∗τ )

)−(xQj (u

∗τ )

τ − xQj (u

∗τ )

︸︷︷︸®

Details for Proof for SP II

By the definition {∆t}Tt=1, xi(τ+1)− xi(τ+1) ≥ ∆τ+1. By the definition of {ρt}T−1t=1 ,

xi(τ+t)−xi(τ+t) ≥ ∆τ+t ⇒ xi(τ+t+1) − xi(τ+t+1)

= (1− ρτ+t)(xi(τ+t) − xi(τ+t)

)+ ρτ+t

(xQj(u∗τ+t)τ+t − x

Qj(u∗τ+t)τ+t

≥(1− ρτ+t)∆τ+t − ρτ+tχτ+t ≥ ∆τ+t+1 > 0;

xj(τ+t+1) − xj(τ+t+1) = (1− ρτ+t)

¬︷︸︸︷(xj(τ+t) − xj(τ+t)

+ρτ+t

(xQj(u∗τ+t)τ+t − x

Qj(u∗τ+t)τ+t

︸︷︷︸

Both ¬ and are less than or equal to maxj∈N

{xj(τ+t) − xj(τ+t)

}. Back Back

Details for price of anarchy

Consider only i deviates in period τ ;

For any player j ,

xj(τ+1) − xj(τ+1) = βτ(u∗jτ − u∗jτ

)+ ρt

(xQj (u

∗τ )

τ − xQj (u

∗τ )

⇒[u∗jτ > u∗jτ ⇐⇒ xj(τ+1) > xj(τ+1)

Also, because of equal division, u∗iτ > u∗iτ ⇐⇒ u∗jτ < u∗jτ ;

Hence, u∗iτ > u∗iτ ⇒ xi(τ+2) − xi(τ+2) < xi(τ+1) − xi(τ+1);

While, u∗iτ > u∗iτ ⇒ xi(τ+t) − xi(τ+t) > 0, ∀t ≥ 1. Back

References I

Athey, S. and I. Segal (2013). An efficient dynamic mechanism.Econometrica 81(6), 2463–2485.

Aziz, H. and C. Ye (2014, December). Cake cutting algorithms for piecewiseconstant and piecewise uniform valuations. In T.-Y. Liu and Y. Ye (Eds.), Weband Internet Economics - 10th International Conference, WINE 2014, Beijing,China, December 14-17, 2014, Proceedings, Volume 8877 of Lecture Notes inComputer Science, pp. 1–14. Springer.

Bergemann, D. and J. Valimaki (2010). The dynamic pivot mechanism.Econometrica 78(2), 771–789.

Blackwell, D. (1965, 02). Discounted dynamic programming. The Annals ofMathematical Statistics 36(1), 226–235.

Bogomolnaia, A. and H. Moulin (2001). A new solution to the randomassignment problem. Journal of Economic Theory 100(2), 295–328.

Bogomolnaia, A. and H. Moulin (2004). Random matching under dichotomouspreferences. Econometrica 72(1), 257–279.

References II

Brams, S. J., M. A. Jones, and C. Klamler (2008). Proportional pie-cutting.International Journal of Game Theory 36(3-4), 353–367.

Brams, S. J., M. A. Jones, C. Klamler, et al. (2006). Better ways to cut a cake.Notices of the AMS 53(11), 1314–1321.

Chen, Y., J. K. Lai, D. C. Parkes, and A. D. Procaccia (2013). Truth, justice, andcake cutting. Games and Economic Behavior 77(1), 284 – 297.

Dasgupta, P., A. Sen, and D. Starrett (1973). Notes on the measurement ofinequality. Journal of Economic Theory 6(2), 180 – 187.

Dutta, B. and D. Ray (1989). A concept of egalitarianism under participationconstraints. Econometrica: Journal of the Econometric Society 57(3), 615–635.

Gallien, J. (2006). Dynamic mechanism design for online commerce. OperationsResearch 54(2), 291–310.

Hashimoto, T., D. Hirata, O. Kesten, M. Kurino, and M. U. Unver (2014). Twoaxiomatic approaches to the probabilistic serial mechanism. TheoreticalEconomics 9(1), 253–277.

Hylland, A. and R. Zeckhauser (1979). The efficient allocation of individuals topositions. The Journal of Political Economy 87(2), 293–314.

References III

Milgrom, P. and C. Shannon (1994). Monotone comparative statics.Econometrica 62(1), 157–180.

Mossel, E. and O. Tamuz (2010). Truthful fair division. In Algorithmic GameTheory, pp. 288–299. Springer.

Neyman, J. (1946). Un theoreme d’existence. CR Acad. Sci. Paris 222, 843–845.

Pai, M. M. and R. Vohra (2013). Optimal dynamic auctions and simple indexrules. Mathematics of Operations Research 38(4), 682–697.

Pavan, A., I. Segal, and J. Toikka (2013). Dynamic mechanism design: Amyersonian approach.

Procaccia, A. D. (2013). Cake cutting algorithms. Chapter 13 in Handbook ofComputational Social Choice.

Quah, J. K.-H. (2007). The comparative statics of constrained optimizationproblems. Econometrica 75(2), 401–431.

Rothschild, M. and J. E. Stiglitz (1973). Some further results on themeasurement of inequality. Journal of Economic Theory 6(2), 188 – 204.

Shorrocks, A. F. (1983). Ranking income distributions. Economica 50(197), 3–17.

References IV

Steinhaus, H. (1948). The problem of fair division. Econometrica 16(1), 101–104.

Tian, Y. (2013). Strategy-proof and efficient offline interval scheduling and cakecutting. In Web and Internet Economics, pp. 436–437. Springer.

Topkis, D. M. (1978). Minimizing a submodular function on a lattice. OperationsResearch 26(2), 305–321.

Topkis, D. M. (2001). Supermodularity and Complementarity. PrincetonUniversity Press.

Vives, X. (1990). Nash equilibrium with strategic complementarities. Journal ofMathematical Economics 19(3), 305–321.

Weller, D. (1985). Fair division of a measurable space. Journal of MathematicalEconomics 14(1), 5–17.

Zhou, L. (1990). On a conjecture by gale about one-sided matching problems.Journal of Economic Theory 52(1), 123–135.

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