Applications of TU Games

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Applications of TU Games

• Vito Fragnelli
• University of
• Eastern Piedmont

Outline Cost Allocation Problems Infrastructure
Cost Games Short Survey
Politecnico di Torino 24 April 2002
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Cost Allocation Problems
Alan and Bob want to rent a car they can rent
the car for one day each (Saturday for Alan at 70
, Sunday for Bob at 40 ) or they can rent the
car for the weekend (at 90 )
Several agents have to carry out a common
project they can perform it separately or they
can do it jointly (if the total cost decreases)
Allocate among the agents the total cost of the
joint project
The equal sharing (45 each) will be rejected by
Bob
Allocation should be fair
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Cost Allocation Problems
It is a pair ?N, c ?, where N 1, ..., n set
of agents c cost function c (S ) is the cost
of the project if it is jointly performed by the
agents in S, satisfying their necessities
N Alan, Bob c (Alan) 70, c (Bob) 40, c
(Alan, Bob) 90
The agents not included in S cannot benefit from
this project, so that they could have to carry
out their own separate ones
The solution is a vector x in Rn x should be
efficient ?i?N xi c (N )
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Cost Allocation Problems
The Tennessee Valley Authority Problem (30)
Some interventions (dams and reservoirs) were
planned in the Tennessee River valley in order to
encourage the economy

• provide water for energy
• control flooding
• improve navigation

The fair allocations should satisfy

• Stand-alone cost test no subset of agents S
  should be charged more than the cost of an
  alternative project specialized for S
• ?i?S xi ? c (S )
• Incremental cost test no subset of agents S
  should be charged less than the additional cost
  including them in the project
• ?i?S xi ? c (N ) - c (N \ S )

By efficiency the two properties are equivalent
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Cost Allocation Problems
The Tennessee Valley Authority Problem (30)

• Does there exists a vector that satisfies the
  previous conditions?
• How do select a single vector?

Point solutions Separable costs methods

• ECA Equal Charge Allocation
• ACA Alternate Costs Avoided
• CGA Cost Gap Allocation (?-value)

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Cost Allocation Problems
Separable costs methods

• Separable cost of player i (marginal cost)
• mi c (N ) - c (N \i )
• Non separable cost
• g(N ) c (N ) - ?i?N mi

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Cost Allocation Problems
Cost Allocations Problems and Game Theory
A cost allocation problem ?N, c ? induces a
cooperative cost game ?N, c ?

• The cost game ?N, c ? has non-empty core?
• How do select a single core allocation?

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Infrastructure Cost Games
Some users need an infrastructure They are
grouped according to different requests,
corresponding to increasing levels of the
infrastructure or of the facilities that build up
the infrastructure, as g1, ..., gk An
infrastructure has costs independent from the
number of users (Building costs) and costs
depending on the number of users (Maintenance
costs)
Infrastructure cost game Building cost game
Maintenance cost game
The Shapley Value is additive
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Infrastructure Cost Games
Building Cost Game
It is precisely an Airport Game (Littlechild
Thompson, 1973) assign to a subset of planes the
cost of the larger strip required c (S ) Cj(S)
where j (S ) max j S ? gj ? ?
N g1 ? g2 ? g3 g1 1 g2 2, 3 g3
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Sh(1) c1 /4 Sh(2) Sh(3) c1 /4 c2 /3
Sh(4) c1 /4 c2 /3 c3
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Infrastructure Cost Games
Maintenance Cost Game
If a player of kind i uses (and damages) a
facility of level j he has to refund an amount
Aij ?ki,j ?ij , where ?ii is the cost for a
player of kind i to repair the facility to the
level i and ?ik (k gt i ) is the cost for a player
of kind i to repair the facility from the level
k - 1 to the level k
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Infrastructure Cost Games
Maintenance Cost Game
N 1, 2, 3, 4
Sh(1) ?11 3/4 ?12 1/2 ?13 Sh(2) Sh(3)
?22 1/2 ?23 1/4 1/3 ?12 Sh(4) ?33 1/4
1/3 ?12 1/2 ?13 1/2 2 ?23
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Other Applications
Short survey

• Voting games (power indices)
• Bankruptcy games
• Operations Research games

• Production Linear Transformation games
• Network games
• - Flow games
• - Connection games (Spanning tree games, Forest
  games, Extension games, Fixed tree
  games)
• - Shortest Path games
• Sequencing, Assignment Permutation games
• Inventory games
• Location games
• Transportation games

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Thank you for your attention!
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Bankruptcy Games

• A set of agents N 1, ..., n has claims d
  (d1, ..., dn) over an estate E with
• ?i1,n di gt E
• A feasible division x has to satisfy
• 0 ? xi ? di
• ?i1,n di E
• It is possible to define a (pessimistic) game ?N,
  v ? whose core coincides with the set of feasible
  divisions
• v (S ) max 0, E - ?i?S di

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Linear Production Games
A set of agents N 1, ..., n has a set of
resources b (b1, ..., bn) they can produce a
set of m goods according to a (fixed) technology
represented by the matrix A each unit of good j
produce an income cj , j 1, ..., m A group of
agents S ? N can use their resources bS ?i?S
bi to maximize their income It is possible to
define a TU game as v (S ) max c Tx s.t. Ax
? bS x ? 0
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Flow Games
A set of agents N 1, ..., n operates on a
transportation network each agent control a set
of arcs A group of agents can use only the arcs
they control in order to maximize the flow from
the source s to the sink t
N 1, 2, 3
v (1) 0 v (2) 1 v (3) 0 v (1, 2) 3 v
(1, 3) 0 v (2, 3) 2 v (1, 2, 3) 4
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Minimum Cost Spanning Tree Games
A set of agents N 1, ..., n wants to be
connected to a common source 0 building
suitable connections at minimum cost
N 1, 2, 3
Non monotonic game v (1) 5 v (2) 1 v (3)
4 v (1, 2) 4 v (1, 3) 8 v (2, 3) 3 v (1,
2, 3) 6
Monotonic game v (1) 4 v (2) 1 v (3) 3 v
(1, 2) 4 v (1, 3) 6 v (2, 3) 3 v (1, 2,
3) 6
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Sequencing Games
A set of agents N 1, ..., n waits for a
service each agent i has a fixed service time
ti at a fixed cost per unit of time ci Adjacent
agents can reorder themselves for improving their
global service time
N I, II, III t (5, 3, 4) c (5, 9, 8) u
(1, 3, 2) ? optimal order (Smith) II, III, I
v (I) v (II) v (III) 0 v (I, II) 30 v (I,
III) 0 v (II, III) 0 v (I, II, III) 50
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