Strategy-proof and Efficient Scheduling
Yuan Tian
Department of Economics
University of Chicago
August 30, 2013
Yuan Tian (University of Chicago) Scheduling August 30, 2013 1 / 57
Introduction
The problem of fair division (cake-cutting)
Division of a continuum of heterogeneous goods among players;
Why continuum?
Is convenient when modeling demand as time intervals;
Subsumes random assignments/matching of discrete goods;
Allows for continuous characteristics:
e.g. same good with different prices are treated as different goods;
Restricted preference domain:
Goods are either acceptable (needed) or unacceptable (unneeded);
No transfers allowed.
Yuan Tian (University of Chicago) Scheduling August 30, 2013 2 / 57
Introduction
Examples
Objects:
Player Children Students Employees State governments
Goods Cake Schools Tasks Federal funded projects
Time:
Player Airlines/Committees E-cars Readers
Goods Runway/Room occupancy Power outlet Reading time
Yuan Tian (University of Chicago) Scheduling August 30, 2013 3 / 57
Introduction
Contribution
Characterized a class of strategy-proof and Pareto efficient
mechanisms that accommodates fairness objectives such as
envy-freeness and arbitrary guaranteed shares of demand.
Yuan Tian (University of Chicago) Scheduling August 30, 2013 4 / 57
Introduction
Plan/approach and technique
1 Propose a social welfare function G (·) that is potentially truthfully
implementable in weakly dominant strategies;
2 Verify a non-inferiority condition that ensures the truthful
implementability of G (·);
3 Verify the existence of a feasible schedule (hence a direct mechanism)
supporting the allocation that maximizes G (·);
Main technique: monotone comparative statics of constrainedoptimization problems with lattice programming Literature review
Yuan Tian (University of Chicago) Scheduling August 30, 2013 5 / 57
Introduction
A quick demonstration
Player 1: (0, 1) always; Player 2: (0.5, l);
Choose x1 and x2 to maximize (2 ln x1 + ln x2) subject to
x1 ≤ 1, x2 ≤ l − 0.5, and x1 + x2 ≤ max l , 1 ;
The solutions (allocations, not payoffs):
x∗1 = 1.5− l , x∗2 = l − 0.5 l ≤ 5
6,
x∗1 =2
3· (max l , 1) , x∗2 =
1
3· (max l , 1) 5
6< l ≤ 1.5,
x∗1 = 1, x∗2 = l − 1 l > 1.5.
Yuan Tian (University of Chicago) Scheduling August 30, 2013 6 / 57
Introduction
Players’ payoffs
Suppose 2’s true type l is l∗ = 1.2. What is her payoff as a function of l?
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
0.25
0.5
0.75
1
1.25
Player 2's payoff
Player 1's payoff
Sum of players' payoffs
Yuan Tian (University of Chicago) Scheduling August 30, 2013 7 / 57
Introduction
Literature of cake-cutting
Continuous and heterogenous cake [0, 1] with utility 1 to any player.
Proportionality : among n players, each obtains utility of at least 1/n;
Envy-freeness: no player prefers any other player’s piece to her own;
Strategy-proofness: all players report preferences truthfully;
Piecewise uniformity/dichotomy : the cake can be divided into
acceptable and unacceptable pieces for any player.
Yuan Tian (University of Chicago) Scheduling August 30, 2013 8 / 57
Introduction
Literature of cake-cutting-continued
Mechanism Players Prop. EF SP Unif.
Cut-and-Choose 2 Yes Yes No No
Moving Knife n Yes No No No
Selfridge-Conway 3 Yes Yes No No
Chen et al. (2013) n Yes Yes Yes Yes
Brams and Taylor (1996), Robertson and Webb (1998), Procaccia (2013).
Yuan Tian (University of Chicago) Scheduling August 30, 2013 9 / 57
The Model
Routemap
1 Introduction
2 The Model
3 The ResultsA constrained optimization problemA non-inferiority conditionFeasibility and Pareto efficiency
4 Discussion and Future Research
Yuan Tian (University of Chicago) Scheduling August 30, 2013 10 / 57
The Model
Readers and schedules
One book in a public library; Continuous time;
Fixed and finite number of readers: N ≡ 1, 2, · · · , n;
A schedule µ ≡ (µ1(·), µ2(·), · · · , µn(·)), where
µi (·) : R+ → 0, 1 , ∀i ∈ N;
A feasible schedule µ:
∑
i∈Nµi (r) ≤ 1, ∀r ∈ R+.
The set of feasible schedules is F .
Yuan Tian (University of Chicago) Scheduling August 30, 2013 11 / 57
The Model
Preferences and Pareto efficiency
Reader i ’s type: finite union of open intervals on R+, θi ∈ Θ;
Reader i ’s utility from a schedule µ:
ui (µ; θi ) ≡∫
θi
µi (r) dr ;
A schedule µ Pareto dominates another schedule µ′ if and only if
ui (µ; θi ) ≥ ui
(µ′; θi
), ∀i ∈ N
and the inequality is strict for at least one i .
A feasible schedule µ is Pareto efficient if and only if there does notexist another feasible µ′ that Pareto dominates µ.
Yuan Tian (University of Chicago) Scheduling August 30, 2013 12 / 57
The Model
Circulation mechanisms
Reader i ’s reported type: ti ∈ Θ;
Profile of all readers’ types: t ≡ (t1, t2, · · · , tn) ∈ Θn; t ≡ (ti , t−i );
A (offline) circulation mechanism is a function C : Θn → F ;
C (t) = µ ≡ (µ1(·), µ2(·), · · · , µn(·)) ⇐⇒ (Ci (t)) (·) = µi (·),
for all (t, i) ∈ Θn × N.
Yuan Tian (University of Chicago) Scheduling August 30, 2013 13 / 57
The Model
Strategy-proofness and Pareto efficiency
A mechanism C is strategy-proof if and only if
ui (C (θi , t−i ); θi ) ≥ ui (C (ti , t−i ); θi ) ,
for all (ti , t−i , θi , i) ∈(Θ×Θn−1 ×Θ× N
);
A strategy-proof mechanism is Pareto efficient if and only if
C (θ)
is a Pareto efficient schedule for all θ ∈ Θn.
Notation: ∀x ∈ Rn, x ≡ (x1, x2, · · · , xn).
Yuan Tian (University of Chicago) Scheduling August 30, 2013 14 / 57
The Results
Routemap
1 Introduction
2 The Model
3 The ResultsA constrained optimization problemA non-inferiority conditionFeasibility and Pareto efficiency
4 Discussion and Future Research
Yuan Tian (University of Chicago) Scheduling August 30, 2013 15 / 57
The Results A constrained optimization problem
Routemap
1 Introduction
2 The Model
3 The ResultsA constrained optimization problemA non-inferiority conditionFeasibility and Pareto efficiency
4 Discussion and Future Research
Yuan Tian (University of Chicago) Scheduling August 30, 2013 16 / 57
The Results A constrained optimization problem
A constrained optimization problem: the components
An objective function G (x) ≡∑
i∈Ngi (xi ), where gi (·) : R+ → R is
continuous, strictly increasing, strictly concave for all i ∈ N;
∀t ∈ Θn and ∀S ⊆ N, let b(S ; t) ≡ λ(⋃
i∈S ti)—Lebesgue measure;
A budget:∑
i∈Sxi ≤ b(S ; t), referred to as binding when equal;
The budget set: B(t) ≡
x ∈ Rn+ :∑
i∈Sxi ≤ b(S ; t), ∀S ⊆ N
.
The optimum: x∗(t) ≡ arg maxx∈B(t)
G (x).
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The Results A constrained optimization problem
Interpretations
G (x) is a social welfare function: Nash collective utility;
(Bogomolnaia and Moulin (2004));
G (x) is Paretian: increase in any xi , ceteris paribus, increases G (x);
(Mas-Colell et al. (1995));
The budget set B(t) describes necessary conditions for the
existence of a feasible and Pareto efficient schedule;
Sufficiency will be established later. Feasibility
Yuan Tian (University of Chicago) Scheduling August 30, 2013 18 / 57
The Results A constrained optimization problem
An example
Let t1 = (0, 12) ∪ (23, 35), t2 = (7, 17), and t3 = (13, 25);
(13 25)
(7 17)
(0 12) (23 35)
out of scale
Yuan Tian (University of Chicago) Scheduling August 30, 2013 19 / 57
The Results A constrained optimization problem
An example-the optimum
g1(x1) = ln(x1), g2(x2) = 3 ln(x2), and g3(x3) = 2 ln(x3);
⇒ G (x) = ln(x1) + 3 ln(x2) + 2 ln(x3);
B(t) is given by
0 ≤ x1 ≤ 24, 0 ≤ x2 ≤ 10, 0 ≤ x3 ≤ 12; x1 + x2 + x3 ≤ 35;
x1 + x2 ≤ 29, x2 + x3 ≤ 18, x1 + x3 ≤ 34.
The optimum: x∗1 = 17, x∗2 = 10, and x∗3 = 8; Calculations
Yuan Tian (University of Chicago) Scheduling August 30, 2013 20 / 57
The Results A non-inferiority condition
Routemap
1 Introduction
2 The Model
3 The ResultsA constrained optimization problemA non-inferiority conditionFeasibility and Pareto efficiency
4 Discussion and Future Research
Yuan Tian (University of Chicago) Scheduling August 30, 2013 21 / 57
The Results A non-inferiority condition
A non-inferiority condition
Lemma
For any profile of reported types t, S ′ ⊆ N, and ∆S ′ > 0, let
B (t,∆S ′) ≡
x ∈ Rn+
∣∣∣∣∣∑
i∈S ′
xi ≤ b(S ′; t) + ∆S ′ ;
∑
i∈Sxi ≤ b(S ; t), ∀S ⊆ N and S 6= S ′
and x∗ ≡ arg maxx∈B(t,∆S′ )
G (x) and x∗ = arg maxx∈B(t)
G (x), then
x∗i ≥ x∗i ,∀i ∈ S ′.
Yuan Tian (University of Chicago) Scheduling August 30, 2013 22 / 57
The Results A non-inferiority condition
Comments
A thought experiment on arbitrarily increasing the budget of a subset
of readers and see how their optimal reading times change;
The proof is based on the following observation: fix a vector x , if the
budgets for S1 ⊆ N and S2 ⊆ N are binding, then the budget for
S1 ∩ S2 is also binding. Calculations
Yuan Tian (University of Chicago) Scheduling August 30, 2013 23 / 57
The Results A non-inferiority condition
Plan of proof of non-inferiority
Suppose, on the contrary, there is some i such that x∗i < x∗i ;
Name this group S2; Name the group S1 such that x∗i > x∗i ;
Look for a pair of vectors y∗, y∗ ∈ Rn+ such that y∗ + y∗ = x∗ + x∗,
and min x∗i , x∗i ≤ y∗i , y∗i ≤ max x∗i , x∗i , ∀i ∈ N
and x∗ 6= y∗ ∈ B and x∗ 6= y∗ ∈ B;
Yuan Tian (University of Chicago) Scheduling August 30, 2013 24 / 57
The Results A non-inferiority condition
Plan of proof of non-inferiority: the goal
Concavity of G (x) implies that
G (x∗) ≥ G (y∗)⇒ G (y∗) ≥ G (x∗) .
Equivalently, look for two subsets S1 ⊆ S1 and S2 ⊆ S2 such that
∑
i∈Sx∗i = b(S)⇒
∣∣S ∩ S1
∣∣ =∣∣S ∩ S2
∣∣ ,∀S ⊂ S ′;
∑
i∈S
x∗i = b(S)⇒∣∣∣S ∩ S1
∣∣∣ =∣∣∣S ∩ S2
∣∣∣ ,∀S ⊂ S ′.
Yuan Tian (University of Chicago) Scheduling August 30, 2013 25 / 57
The Results A non-inferiority condition
Sketch of proof of non-inferiority: notations
Define P ⊆ S1
P ≡
p ∈ S1
∣∣∣∣∣ ∃S ⊂ S ′ such that p ∈ S and∑
i∈Sx∗i = b(S)
;
S(p) ≡
S ⊂ S ′
∣∣∣∣∣ p ∈ S and∑
i∈Sx∗i = b(S)
;
The core of p at x∗: M(p) ≡⋂
S∈S(p)
S ,∀p ∈ P; Notice M(p) ∈ S(p);
Similarly for Q ⊆ S2, S(q), and M(q), ∀q ∈ Q.
Yuan Tian (University of Chicago) Scheduling August 30, 2013 26 / 57
The Results A non-inferiority condition
Sketch of proof of non-inferiority: observations
O(p) ≡ M(p) ∩ S2 6= ∅, O(q) ≡ M(q) ∩ S1 6= ∅, ∀(p, q) ∈ P × Q;
S1 and S2: Black , P and Q: Green ;
O(p) and O(q): Blue , S1 and S2: Red ;
Figure : Groupings of readers
STRATEGY-PROOF AND EFFICIENT SCHEDULING 23
although S and S are only restricted to be proper subsets of S0, (29) holds trivially for S0 since S1 and S2621
have the same cardinality by (28). 622
Clearly, yi and y
i satisfy the conditions in (24) for all i and the inequalities are strict for at least one i if623
S1 6= ; 6= S2. To construct the the two subsets in (28) hence Lemma 4, I first show the following result.624
Lemma 5. If the budget for S is binding at x, then S \ S2 6= ;. Similarly, if the budget for S is binding625
at x, S \ S1 6= ;.626
Proof of Lemma 5. I will prove the first statement—the second follows identically. S \ S2 = ; implies that627
S \ S0 S1 [ Se 6= S0.
By Lemma 3, the budget for S \ S0 must also be binding at x. However, this implies628
b (S \ S0; t) =X
i2S\S0
xi <
X
i2S\S0
xi ) x /2 B(t,S0).
As Lemma 4 suggests, let629
P (
p 2 S1 : 9S S0 such that p 2 S andX
i2S
xi = b(S; t)
);
630
Q
8<:q 2 S2 : 9S S0 such that q 2 S and
X
i2S
xi = b(S; t)
9=; .
That is, P and Q are the set of readers in S1 and S2 such that there exists a group including these readers631
whose budgets are binding at their corresponding points. Furthermore, let632
S(p) (
S S0 : p 2 S andX
i2S
xi = b(S; t)
)and M(p)
\
S2S(p)
S, 8p 2 P ;
633
S(q)
8<:S S0 : q 2 S and
X
i2S
xi = b(S; t)
9=; and M(q)
\
S2S(q)
S, 8q 2 Q.
Let634
O(p) M(p) \ S2 6= ; if p 2 P 6= ;.635
O(q) M(q) \ S1 6= ; if q 2 Q 6= ;.M(p) (M(q)) is the intersection of all binding budgets at x (x) that include p (q). I shall refer to M(p)636
(M(q)) as the core of p (q) at x (x). The point reference may be dropped later since the hat-symbol637
should clarify when I am referring to x. The notion of the core is well-defined and nonempty (p 2 M(p),638
for example) as long as P and/or Q is nonempty. Two readers can share the same core. However, for each639
reader at each optimum, the core is uniquely defined since the intersection of any two distinct candidate640
cores is a proper subset of both, which denies either of the candidates of being a core.641
The following illustration presents an example of the various subgroups of readers.642
x1 · · · x
p · · · x|S1| x
|S1|+1 · · · x|S1|+|SE | x
|S1|+|SE |+1 · · · xq · · · x
|S0|
^ ^ ^ ^ ^ k k k _ _ _ _ _x
1 · · · xp · · · x
|S1| x|S1|+1 · · · x
|S1|+|SE | x|S1|+|SE |+1 · · · x
q · · · x|S0|
0BBBB@
1CCCCA
Yuan Tian (University of Chicago) Scheduling August 30, 2013 27 / 57
The Results A non-inferiority condition
Algorithm O: initiation
S1 Without loss of generality, take i1 ∈ P in the first step and proceed to
the second step;
S2 In step two, take any i2 ∈ O(i1). If i2 /∈ Q, set S2 = i2 and
S1 = i1 and terminate. Otherwise, proceed to the next step;
Yuan Tian (University of Chicago) Scheduling August 30, 2013 28 / 57
The Results A non-inferiority condition
Algorithm O: odd-numbered steps
In any odd-numbered step k ≥ 3, choose ik ∈ O(ik−1).
If ik /∈ P, terminate (“early termination” at k);
Set S1 = ik and S2 = ik−1.
Otherwise, if O(ik) ∩ i1, i2, · · · , ik−1 6= ∅,
⇒ l ≡ max l : il ∈ O(ik) ∩ i1, i2, · · · , ik−1 .
Terminate and save the ordered sequence(ik , il , il+1, · · · , ik−1
).
Otherwise, proceed to the next step.
Yuan Tian (University of Chicago) Scheduling August 30, 2013 29 / 57
The Results A non-inferiority condition
Algorithm O: even-numbered steps
In any even-numbered step l ≥ 4, choose il ∈ O(il−1).
If il /∈ Q, terminate (“early termination” at l);
Set S1 = il−1 and S2 = il.
Otherwise, if O(il) ∩ i1, i2, · · · , il−1 6= ∅,
⇒ k ≡ max
l : il ∈ O(il) ∩ i1, i2, · · · , il−1.
Terminate and save the ordered sequence(ik , ik+1, · · · , il−1, il
).
Otherwise, proceed to the next step.
Yuan Tian (University of Chicago) Scheduling August 30, 2013 30 / 57
The Results A non-inferiority condition
Algorithm O: after termination
After terminations that are not early,
Set S1 to be the set of all the odd-numbered terms in the saved
ordered sequence;
Set S2 the set of all the even-numbered terms, preserving the order.
Yuan Tian (University of Chicago) Scheduling August 30, 2013 31 / 57
The Results A non-inferiority condition
Algorithm O: observations
It terminates in finitely many steps;
When it terminates, the saved ordered sequence contains equal
number of readers from S1 and S2;
Adjacent terms in the saved ordered sequence always appear in
binding constraints together;
For small enough ξ > 0 and S1 and S2 from Algorithm O,
y∗ = x∗ + ξ ·(1S1− 1S2
)∈ B;
y∗ = x∗ − ξ ·(1S1− 1S2
)∈ B.
Yuan Tian (University of Chicago) Scheduling August 30, 2013 32 / 57
The Results Feasibility and Pareto efficiency
Routemap
1 Introduction
2 The Model
3 The ResultsA constrained optimization problemA non-inferiority conditionFeasibility and Pareto efficiency
4 Discussion and Future Research
Yuan Tian (University of Chicago) Scheduling August 30, 2013 33 / 57
The Results Feasibility and Pareto efficiency
Feasibility
Proposition (Existence of a feasible schedule)
Given any t ∈ Θn, for all x ∈ B(t), there exists a feasible schedule µ ∈ F
such that∫
ti
µi (r) dr = xi , (1)
where, recall, x ≡ (x1, x2, · · · , xn) and µ ≡ (µ1, µ2, · · · , µn).
Let F(x ; t) be the set of all such feasible schedules.
Yuan Tian (University of Chicago) Scheduling August 30, 2013 34 / 57
The Results Feasibility and Pareto efficiency
Proof of feasibility: the components
Given any t ≡ (t1, t2, · · · , tn),
Let Ti (ti ) be the set of all end points of ti ;
Let T (t) ≡⋃
i∈NTi (ti ) and m(t) ≡ |T (t)|; Let ε ∈ T (t);
Arrange the elements of T (t), indexed by j , in the increasing order;
T (t) =ε1(t), ε2(t), · · · , εm(t)(t)
.
Yuan Tian (University of Chicago) Scheduling August 30, 2013 35 / 57
The Results Feasibility and Pareto efficiency
Proof of feasibility: the network Γ(x ; t) Construction
Recall the example: t1 = (0, 12) ∪ (23, 35), t2 = (7, 17), and t3 = (13, 25);
⇒ T (t) = 0, 7, 12, 13, 17, 23, 25, 35 . Let x = (16, 5, 7)⇒ x ∈ B(t).
0∞
,,7 ∞ //
∞
))
i
16
&&
12∞
,,Ψ
7
==
5
88
1
44
4 //
6**
2
&&10
!!
13 ∞ //
∞
))
ii 5 // Ω
17∞
,,23 ∞ //
∞
<<
iii
7
88
25
∞
??
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The Results Feasibility and Pareto efficiency
Max-flow-min-cut
Lemma
The maximum flow (and the minimum cut) of the network Γ(x ; t) is
∑
i∈Nxi .
A simple proof by contradiction since x ∈ B(t).
Yuan Tian (University of Chicago) Scheduling August 30, 2013 37 / 57
The Results Feasibility and Pareto efficiency
Proof of the max-flow by graph: a non-minimum cut
0∞
++7 ∞ //
∞
''
i
16
$$
12∞
++Ψ
7
@@
5
::
1
55
4 //
6))
2
$$10
13 ∞ //
∞
''
ii 5 // Ω.
17∞
++23 ∞ //
∞
>>
iii
7
::
25
∞
BB
Yuan Tian (University of Chicago) Scheduling August 30, 2013 38 / 57
The Results Feasibility and Pareto efficiency
Feasibility-the example
07
,,7
5
((
i
16
%%
12
Ψ
7
>>
5
99
1
44
4 //
6**
2
%%10
13
4
((
ii 5 // Ω
173
,,23
2
==
iii
7
99
25
7
@@
µ: i : (0, 7) ∪ (23, 32) ii : (7, 12) iii : (13, 20)
Yuan Tian (University of Chicago) Scheduling August 30, 2013 39 / 57
The Results Feasibility and Pareto efficiency
Mechanism G
Definition
For any G (x) satisfying the regularity conditions, let Mechanism G be
defined by
G(t) ∈ F (x∗(t); t) , ∀t ∈ Θn,
where, recall,
x∗(t) ≡ arg maxx∈B(t)
G (x).
F (x∗(t); t) is non-empty by the feasibility—mechanism G is well-defined.
Yuan Tian (University of Chicago) Scheduling August 30, 2013 40 / 57
The Results Feasibility and Pareto efficiency
Strategy-proofness and Pareto efficiency
Proposition
Mechanism G is strategy-proof and Pareto efficient.
Pareto efficiency is straight-forward to show: G (x) is a Paretian;
For strategy-proofness, suffices to show deviations to supersets and
subsets cannot be profitable;
Take ti 6⊂ θi and ti 6⊃ θi and θ′i ≡ ti ∩ θi 6= ∅. Take any t−i ,
ui (G (t) ; θi ) = ui
(G (t) ; θ′i
)≤ ui
(G(θ′i , t−i
); θ′i)
= ui
(G(θ′i , t−i
); θi)≤ ui (G (θi , t−i ) ; θi ) .
Yuan Tian (University of Chicago) Scheduling August 30, 2013 41 / 57
The Results Feasibility and Pareto efficiency
Strategy-proofness: deviations to subsets
Let B0 = B(θ). When ti ⊂ θi , let x(k)∗ ≡ arg maxx∈Bk
G (x), where
Bk ≡
x ∈ Rn
+
∣∣∣∣∣∣∑
i∈Sk′
xi ≤ b (Sk ′ ; t) , 1 ≤ k ′ ≤ k
and∑
i∈Sk′
xi ≤ b (Sk ′ ; θ) , k < k ′ ≤ 2n − 1
;
Let x∗ be the optimum when i reports truthfully. Clearly, x(0)∗ = x∗.
x(k)∗i ≤ x(k − 1)∗i ⇒ ui (G(t); θi ) ≤ ui (G(θ); θi ).
Yuan Tian (University of Chicago) Scheduling August 30, 2013 42 / 57
The Results Feasibility and Pareto efficiency
Strategy-proofness: deviations to supersets-1
Let x∗ be the optimum when i reports truthfully;
Arrange all subsets of N in a sequence, indexed by k , as follows:
1 Place all subsets that do not include i or subsets including i with
unbinding budgets at x∗ in the front of the sequence. Suppose there
are k such subsets;
2 The remaining subsets all have binding budgets at x∗ and include i ;
Among these, always place supersets before subsets.
Yuan Tian (University of Chicago) Scheduling August 30, 2013 43 / 57
The Results Feasibility and Pareto efficiency
Strategy-proofness: deviations to supersets-2
Let B0 = B(θ) and x(0)∗ = x∗. Let M(i) be the core of i at x∗;
By construction, M(i) ⊆ Sk , ∀k > k;
Construct Bk by replacing only the budget b (Sk ; θ) in Bk−1 with
b (Sk ; t) for the k-th subset in the sequence constructed above;
Let x(k)∗ ≡ arg maxx∈Bk
G (x). Notice x(k)∗ = x(0)∗ = x∗, ∀k ≤ k ;
Yuan Tian (University of Chicago) Scheduling August 30, 2013 44 / 57
The Results Feasibility and Pareto efficiency
Strategy-proofness: deviations to supersets-3
Let M ′(i) ≡ M(i)\ i.By non-inferiority,
x(k + 1)∗i ′ ≥ x(k)∗i ′ , ∀i ′ ∈ M ′(i) and k ≥ k .
ui (G(t); θi ) = λ
θi ∪
⋃
i ′∈M′(i)
ti ′
−
∑
i ′∈M′(i)
x(2n − 1)∗i ′
≤ λ
θi ∪
⋃
i ′∈M′(i)
ti ′
−
∑
i ′∈M′(i)
x∗i ′ = ui (G(θ); θi ).
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Discussion and Future Research
Routemap
1 Introduction
2 The Model
3 The ResultsA constrained optimization problemA non-inferiority conditionFeasibility and Pareto efficiency
4 Discussion and Future Research
Yuan Tian (University of Chicago) Scheduling August 30, 2013 46 / 57
Discussion and Future Research
Relations to Chen et al. (2013)
Corollary (Envy-freeness)
When gi (·) = g(·) for all i ∈ N, Mechanism G reduces to Mechanism 1 in
Chen et al. (2013), hence is proportional and envy-free.
Mechanism 1 in Chen et al. (2013)
Find and allocate ∪i∈S ti to S such that
S = arg minS⊆N
λ
(⋃
i∈Sti
)/|S |
;
Repeat with the remaining reading time and readers until all gone.
Yuan Tian (University of Chicago) Scheduling August 30, 2013 47 / 57
Discussion and Future Research
Guaranteed shares of demand
Corollary
Let G (x) =∑
i∈N αi ln(xi ) where αi > 0, ∀i and∑
i∈N αi = 1. Then
x∗i ≥ αi · λ (ti ) .
Modified Mechanism 1
Find and allocate ∪i∈S ti to S such that
S = arg minS⊆N
λ
(⋃
i∈Sti
)/(∑
i∈Sαi
);
Repeat with the remaining reading time and readers until all gone.
Yuan Tian (University of Chicago) Scheduling August 30, 2013 48 / 57
Discussion and Future Research
Future research
Online interval scheduling mechanisms;
Capacities and quotas;
Piecewise constant value densities.
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Monotone comparative statics of constrained optimization
Paper Obj. fn. Budget set
Milgrom and Shannon (1994) Super-modular Strong set order
Antoniadou (2004) N.A. Direct value order
Quah (2007) S.M. & concave C-flexible order
Topkis (2001), Athey (2002), Vives (2001).
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Let S0 = S1 ∩ S2 and S3 = S1 ∪ S2. First notice the simple fact that
⋃
i∈S1
ti
∩
⋃
i∈S2
ti
⊇
⋃
i∈S0
ti ,
b (S3; t) =b (S1; t) + b (S2; t)− λ
⋃
i∈S1
ti
∩
⋃
i∈S2
ti
≤b (S1; t) + b (S2; t)− b (S0; t)
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∑
i∈S1
xi +∑
i∈S2
xi =∑
i∈S3
xi +∑
i∈S0
xi ≤ b (S3; t) +∑
i∈S0
xi
≤ b (S1; t) + b (S2; t)− b (S0; t) +∑
i∈S0
xi
≤ b (S1; t) + b (S2; t) ,
∑
i∈S1
xi +∑
i∈S2
xi = b (S1; t) + b (S2; t)
⇒∑
i∈S0
xi = b(S0; t).
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Calculation of the example
Let η(S) be the Lagrange multiplier of S ⊆ N. The Lagrangian is
L = G (x) +∑
S⊆Nη(S) ·
(b(S ; t)−
∑
i∈Sxi
).
In the example, the optimal values of the Lagrange multipliers are
η∗(2) =1
20, η∗ (2, 3) =
13
68, and η∗ (1, 2, 3) =
1
17.
And η∗(S) = 0 for all other S , by the KKT conditions.
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Proof of feasibility: the network
Definition
Define the following directed network Γ(x ; t):
create a source Ψ,
a node (called a bin node) for each bin (εj(t), 1 ≤ j ≤ m(t − 1)),
a node (called a reader node) for each reader (represented in
lowercase Roman numerals),
and a sink Ω.
Create the arcs as follows.
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Definition of the network-continued
Connect the source to all bin nodes with arcs pointing to the binnodes—I will refer to such arcs as source-bin arcs;
Assign capacity (εj+1(t)− εj(t)) to the arc connecting the source tothe node corresponding to the bin with end points εj(t) and εj+1(t);
Connect the reader nodes to the sink with arcs pointing to the sink—Iwill refer to such arcs as reader-sink arcs;
Assign capacity xi to the arc connecting reader i ’s node to the sink;
Connect a bin node to a reader node with an arc pointing to thereader node—referred to as a bin-reader arc—if and only if the bin isa subset of the reader’s reported demand;
Assign infinite capacities to all bin-reader arcs.
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References I
Antoniadou, E. (2004, June). Lattice programming and consumer theorypart ii: Comparative statics with many goods.http://ssrn.com/abstract=1370352.
Athey, S. (2002). Monotone comparative statics under uncertainty. TheQuarterly Journal of Economics 117(1), 187–223.
Bogomolnaia, A. and H. Moulin (2004). Random matching underdichotomous preferences. Econometrica 72(1), 257–279.
Brams, S. and A. Taylor (1996). Fair Division: From Cake-Cutting toDispute Resolution. Cambridge University Press.
Chen, Y., J. K. Lai, D. C. Parkes, and A. D. Procaccia (2013). Truth,justice, and cake cutting. Games and Economic Behavior 77(1), 284 –297.
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References II
Mas-Colell, A., M. D. Whinston, J. R. Green, et al. (1995).Microeconomic theory, Volume 1. Oxford University Press New York.
Milgrom, P. and C. Shannon (1994). Monotone comparative statics.Econometrica 62(1), 157–180.
Procaccia, A. D. (2013). Cake cutting algorithms. Chapter 13 inHandbook of Computational Social Choice.
Quah, J. K.-H. (2007). The comparative statics of constrainedoptimization problems. Econometrica 75(2), 401–431.
Robertson, J. and W. Webb (1998). Cake-Cutting Algorithms: Be Fair IfYou Can. AK Peters Natick.
Topkis, D. M. (2001). Supermodularity and Complementarity. PrincetonUniversity Press.
Vives, X. (2001). Oligopoly Pricing: Old Ideas and New Tools. The MITPress.
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