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
Cost Allocation Problems
Alan and Bob
Several agents
want to rent a car
have to carry out a common
project
they can rent the car for one day
each (Saturday for Alan at 70 €,
Sunday for Bob at 40 €)
they can perform it separately
or they can rent the car for the
weekend (at 90 €)
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
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 )
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
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)
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
1
ri  c (i )  mi
ECAi  m i  g(N )
n
M i  min { c (S ) - 
m | i  S ,S  N }
k S \ i k g(S )  c (S ) mi
ri

i

S
ACAi  m i g(
N
)
s.t.
i  c (N )
i

N
i N ri g i  min  g(S ) | i  S 
CGAi  m i  g(N )
gi
i N g i
   M  (1 - )m
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?
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
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 = {4};
c1
Sh(1) = c1 /4
Sh(2) = Sh(3) = c1 /4 + c2 /3
Sh(4) = c1 /4 + c2 /3 + c3
c2
c3
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 = k=i,j ij , where ii is the cost for a player
of kind i to repair the facility to the level i and ik (k > i ) is the cost
for a player of kind i to repair the facility from the level
k - 1 to the level k
c (S ) 
 S  gi
i 1, j (S )
Aij (S )
Infrastructure Cost Games
Maintenance Cost Game
N = {1, 2, 3, 4}
11
12
222
223
33
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
13
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
Thank you for your attention!
Bankruptcy Games
A set of agents N = {1, ..., n} has claims d = (d1, ..., dn) over an
estate E with:
i=1,n di > E
A feasible division x has to satisfy:
• 0  xi  di
• i=1,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 }
back
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 }
back
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}
2
s
3
t
1
2
2
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
back
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}
3
4
2
3
1
5
2
4
1
0
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
back
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
A single
The two agents
are agent
v (I) = v (II) = v (III) = 0
cannot switch
not adjacent
v (I, II) = 30
= (5x9 – 3x5)+
v (I, III) = 0
=
(5x9
– 3x5)
+ (5x8 – 4x5)+
+prefers
:
Agent
II
v (II, III) = 0
= (3x8 –
9x4)
+
the
switch
v (I, II, III) = 50
back