Operations Research Situations and Games

6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011

Cooperative Game Theory. Operations Research
Games. Applications to Interval Games
Lecture 2: Operations Research Situations and Games

Sırma Zeynep Alparslan G¨k
o
S¨leyman Demirel University
u
Faculty of Arts and Sciences
Department of Mathematics
Isparta, Turkey
email:zeynepalparslan@yahoo.com

August 13-16, 2011


Outline
Introduction
Cooperative games
Economic and Operations Research situations and games
Market situations and big boss games
Bankruptcy situations and games
Sequencing situations and games
Airport situations and games
Final Remark
References

Introduction

Introduction
Operations research (OR) is an interdisciplinary mathematical
science that studies the eﬀective use of technology by
organizations.
Operations research use techniques of other sciences such as
statistics, optimization, probability theory, game theory,
mathematical modeling and simulation to solve the problems.
We pay much attention to the modelling part; that is how to
go from an economic or Operations Research situation to
game theory.
Economic and OR games: Cooperative games associated with
several types of economic and OR problems in which various
decision makers (players) are involved, who face a joint
optimization problem (in trying to minimize/maximize total
joint costs/rewards) and an allocation problem in how to
distribute the joint costs/rewards among them.

Cooperative games

Preliminaries on cooperative games
A cooperative game (Transferable utility (TU-game)) is a pair
< N, v >, where
N = {1, 2, ..., n} is the set of players
v : 2N → R is the characteristic function of the game with
v (∅) = 0.
v (S) - value of the coalition, S ⊂ N.
G N - the family of coalitional games with player set N.
We denote the size of a coalition S ⊂ N by |S|.
G N is a (2|N| − 1) dimensional linear space for which unanimity
games form a basis.
Let S ∈ 2N {∅}. The unanimity game based on S, uS : 2N → R
is deﬁned by
1, S ⊂ T
uS (T ) =
0, otherwise.

Cooperative games

Core

How to distribute the profit generated by the cooperating players?
An important role is played by the allocations in the core of the
game.
The core (Gillies (1959)) is defined by

C (v ) = x ∈ RN | xi = v (N), xi ≥ v (S)for each S ∈ 2N ,
i∈N i∈S

for each v ∈ G N .
i∈N xi = v (N): Efficiency condition
i∈S xi ≥ v (S), S ⊂ N: Coalitional rationality condition

Cooperative games

The Shapley value
Π(N): the set of all permutations σ : N → N of N.
P σ (i) := r ∈ N|σ −1 (r ) < σ −1 (i) : the set of all predecessors of i
with respect to the permutation σ.
Let v ∈ G N and σ ∈ Π(N).
The marginal contribution vector mσ (v ) ∈ Rn with respect to σ
and v has the i-th coordinate the value
miσ (v ) := v (P σ (i) ∪ {i}) − v (P σ (i)) for each i ∈ N.
The Shapley value (Shapley (1953)) φ(v ) of a game v ∈ G N is the
average of the marginal vectors of the game:
1
φ(v ) := mσ (v ).
n!
σ∈Π(N)

Cooperative games

Preliminaries on cooperative games
A game < N, v > is called
superadditive if v (S ∪ T ) ≥ v (S) + v (T ) for all S, T ⊂ N with
S ∩ T = ∅;
subadditive if v (S ∪ T ) ≤ v (S) + v (T ) for all S, T ⊂ N with
S ∩ T = ∅;
convex (or supermodular) if v (S ∪ T )+v (S ∩ T ) ≥ v (S) + v (T )
for all S, T ⊂ N;
concave (or submodular) if v (S ∪ T )+v (S ∩ T ) ≤ v (S) + v (T ) for
all S, T ⊂ N.
Each convex (concave) game is also superadditive (subadditive).
For a game v ∈ G N and a coalition T ∈ 2N {∅}, the subgame
with player set T , (T , vT ), is the game vT deﬁned by
vT (S) := v (S) for all S ∈ 2T .
For details see Branzei, Dimitrov and Tijs (2008).



We study some types of Economic and Operations Research
situations and their relation with game theory.


A class of cooperative games suitable to model market situations
with two corners regarding the outcome of cooperation is big boss
games.
In one corner there is a powerful player called the big boss; the
other corner contains players that need the big boss to beneﬁt
from cooperation.
In a big boss game, the big boss has veto power (i.e. the worth of
each coalition which does not include the big boss is zero) and the
characteristic function of the game has a monotonicity property
(i.e. joining the big boss is more beneﬁcial when the size of
coalitions grows larger) and a union property (expressed in terms of
marginal contributions to the grand coalition of coalitions and
individuals).


Example (Find the treasure)
There is a hotel with 6 rooms. In one of the rooms a treasure is
hidden. The value of the treasure is 600 units. Each room’s
1
probability of having the treasure is same, i.e. 6 .
Agent O is allowed to look only in one room and to keep the
treasure if found.
Agents A and B know something about the position of the
treasure:
1. If the treasure is in room 3 or 6, then agent A knows it.
2. If the treasure is in room 4,5 or 6 then agent B knows it.
The big boss game with O as a big boss. v ({O}) = 1 600 = 100,
6
v ({A}) = v ({B}) = 0, v ({O, A}) = 3 600 = 300,
6
v ({O, B}) = 4 600 = 400, v ({A, B}) = v (∅) = 0,
6
v ({O, A, B}) = 5 600 = 500.
6


Big boss games (Muto et al. (1988), Tijs (1990))
< N, v > is a big boss game with n as big boss if
(i) v ∈ G N is monotonic, i.e.,
v (S) ≤ v (T ) if for each S, T ∈ 2N with S ⊂ T ;
(ii) v (S) = 0 if n ∈ S;
/
(iii) v (N) − v (S) ≥ i∈NS (v (N) − v (N {i}))
for all S, T with n ∈ S ⊂ N.
Property (ii) expresses the veto power of the big boss: the
worth of each coalition which does not include the big boss is
zero.
Property (i) says that joining the big boss is more beneﬁcial
when the size of coalitions grows larger, while (iii) expresses
the fact that the force of non big boss players is in their union.
We denote the set of all big boss games with n as a big boss by
BBG N .


Big boss games
Let v ∈ G N . For each i ∈ N, the marginal contribution of player i
to the grand coalition N is Mi (v ) := v (N) − v (N {i}).
The core C (v ) of a big boss game is always nonempty and equal to

{x ∈ I (v )|0 ≤ xi ≤ Mi (v ) for each i ∈ N {n}} .

For a big boss game < N, v > (with n as a big boss) of v ∈ G N
two particular elements of its core are the big boss point B(v )
deﬁned by
0, if j ∈ N {n}
Bj (v ) :=
v (N), if j = n,
and the union point U(v ) deﬁned by

Mj (N, v ), if j ∈ T {n}
Uj (v ) :=
v (N) − i∈N{n} Mi (N, v ), if j = n.


τ -value of a big boss game

For big boss games τ -value, introduced by Tijs (1981), is the
element in the center of the core.

τ -value of a big boss game < N, v > is deﬁned by

B(v ) + U(v )
τ (v ) = .
2
Note that if the game is a convex big boss game then
τ (v ) = φ(v ).

For details see Tijs (1981, 1990).


Example
Let < N, v > be a three-person game with v (N) = 5,
v ({1, 3}) = 3, v ({2, 3}) = 4, v (S) = 0 otherwise.
This game is a big boss game, with 3 as a big boss (from
deﬁnition). Now,
M1 (v ) = v (N) − v (N {1}) = 5 − 4 = 1,
M2 (v ) = v (N) − v (N {2}) = 5 − 3 = 2.
C (v ) = x ∈ R3 : x(N) = v (N), 0 ≤ x1 ≤ M1 (v ), 0 ≤ x2 ≤ M2 (v )
3
3
C (v ) = x ∈R : xi = 5, 0 ≤ x1 ≤ 1, 0 ≤ x2 ≤ 2
i=1
C (v ) = conv {(0, 0, 5), (1, 0, 4), (0, 2, 3), (1, 2, 2)}, which is a
paralellogram.
B(v ) = (0, 0, 5), U(v ) = (1, 2, 2) and τ (v ) = ( 1 , 1, 7 ) (core
2 2
elements).


Bi-monotonic allocation schemes (bi-mas)
We denote by Pn the set {S ⊂ N|n ∈ S} of all coalitions
containing the big boss.
Let v ∈ G N be a big boss game with n as a big boss. We call a
scheme a = (aiS )i∈S,S∈Pn a bi-monotonic allocation scheme
(bi-mas) (Branzei, Tijs and Timmer (2001)) if
(i) (aiS )i∈S is a core element of the subgame < S, v > for each
coalition S ∈ Pn ,
(ii) aiS ≥ aiT for all i ∈ S {n} with S ⊂ T and anS ≤ anT for all
S, T ∈ Pn with S ⊂ T .
Interpretation: In a bi-mas the big boss is weakly better oﬀ in
larger coalitions, while the other players are weakly worse oﬀ.
For (total) big boss games the τ -value generates a bi-mas.
Further, each core element generates a bi-mas for these
games.


Example

Let < N, v > be a big boss game (with 3 as big boss) and
v ({i}) = 0 for i ∈ N, v ({1, 2}) = 0,
v ({1, 3}) = 6, v ({2, 3}) = 5, v (N) = 9.
The (total) τ -value is a bi-monotonic allocation scheme:
 
1 2 3
(1, 2, 3)  2 1 1 5 2 
 2
1

(1, 3)  3 ∗ 3 .
 
(2, 3)  ∗ 21 21 
2 2
(3) ∗ ∗ 0

In larger coalitions player 3 is better oﬀ, other players worse oﬀ.



Bankruptcy situations and bankruptcy games have been intensively
studied in literature (O’Neill (1982), Aumann and Maschler
(1985)).
The story is based on a certain amount of money (estate) which
has to be divided among some people (claimants) who have
individual claims on the estate, and the total claim is weakly larger
than the estate.
A bankruptcy situation with set of claimants N is a pair (E , d),
where E ≥ 0 is the estate to be divided and d ∈ RN is the vector
+
of claims such that i∈N di ≥ E .



We assume that 0 ≤ d1 ≤ d2 ≤ . . . ≤ dn and denote by BR N the
set of bankruptcy situations with player set N.
The total claim is denoted by D = i∈N di .
A bankruptcy rule is a function f : BR N → RN which assigns to
each bankruptcy situation (E , d) ∈ BR N a payoﬀ vector
f (E , d) ∈ RN such that 0 ≤ f (E , d) ≤ d (reasonability) and
i∈N fi (E , d) = E (eﬃciency).



The proportional rule (PROP) is one of the most often used in real
life, deﬁned by
di
PROPi (E , d) = E
j∈N dj

for each bankruptcy problem (E , d) and all i ∈ N.

Another interesting rule the rights-egalitarian (f RE ) rule is deﬁned
1
by fi RE (E , d) = di + n (E − i∈N di ), for each division problem
(E , d) and all i ∈ N.



To each bankruptcy situation (E , d) ∈ BR N one can associate a
bankruptcy game vE ,d deﬁned by vE ,d (S) = (E − i∈NS di )+ for
each S ∈ 2N , where x+ = max {0, x}.
The game vE ,d is convex and the bankruptcy rules PROP and f RE
provide allocations in the core of the game (Aumann and Maschler
(1985)).


Example
A bankruptcy situation (E , d) is given by E = 500 and
d = (100, 200, 300). The associated bankruptcy game (by
deﬁnition) vE ,d is as follows

S ∅ {1} {2} {3} {1, 2} {1, 3} {2, 3} N
.
vE ,d (S) 0 0 100 200 200 300 400 500

For example vE ,d ({1, 2}) = max 0, E − i∈N{1,2} di =
max {0, 500 − 300} = 200.

I (vE ,d ) = conv {(0, 100, 400), (0, 300, 200), (200, 100, 200)} ,

C (vE ,d ) = conv {(0, 200, 300), (100, 200, 200), (100, 100, 300)} .


Example continues...
Marginal vectors of the bankruptcy game can be observed from the
following table.
 
σ σ σ σ
 m1 (vE ,d ) m2 (vE ,d ) m3 (vE ,d ) 
 
(123) 
 0 200 300 

(132) 
 0 200 300 
.
(213) 
 100 100 300 

(231) 
 100 100 300 

(312)  100 200 200 
(321) 100 200 200

200 500 800
φ(vE ,d ) = (
, , ) ∈ C (vE ,d )
3 3 3
(bankruptcy games are convex).



Let us calculate PROP and f RE
PROP1 (E , d) = d1 dj E = 100 500 =
600
250
3 ,
j∈N
d2 200 500
PROP2 (E , d) = dj E = 600 500 = 3 ,
j∈N
d3 300
PROP3 (E , d) = dj E = 600 500 = 250.
j∈N
f1RE (E , d) = d1 + 3 (E − i∈N di ) = 100 + 3 (500 − 600) = 200 ,
1 1
3
RE (E , d) = d + 1 (E − 1 500
f2 2 3 i∈N di ) = 200 + 3 (500 − 600) = 3
f3RE (E , d) = d3 + 3 (E − i∈N di ) = 300 + 3 (500 − 600) = 800
1 1
3
250 500
Note that PROP(E , d) = ( 3 , 3 , 250) and
f RE (E , d) = ( 200 , 500 , 800 ) are also core elements, because the
3 3 3
game is convex.
Note that f RE (E , d) = φ(vE ,d ).



Waiting lines are part of everyday life (people standing in line,
service jobs, manufacturing jobs, machines to be repaired,
telecommunication transmissions etc..).

We consider the sequencing situations with one queue of players,
each with one job, in front of a machine order.

Each player must have his/her job processed on this machine, and
for each player there is a cost according to the time he/she spents
in the system.


First problem is to find an optimal order of the jobs taking into
account the individual processing times and the costs per unit of
time because it is useful for reducing the cost connected with the
time spent in the system.
After an optimal order is found, second problem is to motivate the
agents to switch their positions according to the new order since
agents are basically interested in their individual benefit .
First problem: An optimal order may be obtained simply
reordering the jobs for weakly decreasing urgency indices
(Smith (1956)).
Second problem: Cooperative games arising from such
sequencing situations is useful to solve this problem which is
done by offering fair shares of the gain generated by
cooperation(Curiel, Pederzoli and Tijs (1989)).



Formally, a one-machine sequencing situation is a 4-tuple
(N, σ0 , α, p) where:
N = {1, ..., n} is the set of jobs;
σ0 : N → {1, ..., n} is a permutation that deﬁnes the initial
order of the jobs;
α = (αi )i∈N ∈ Rn is a non-negative real vector, where αi is
+
the cost per unit of time of job i;
p = (pi )i∈N ∈ Rn is a positive real vector, where pi is the
+
processing time of job i.


Given a sequencing situation and an ordering σ of the jobs, for
each i ∈ N we denote by P(σ, i) the set of jobs preceding i,
according to the order σ.
The time spent in the system by job i is the sum between the
waiting time that jobs in P(σ, i) need to be processed and the
processing time of job i yielding the related cost

αi j∈P(σ,i) pj + pi .

The (total) cost associated with σ:

Cσ , is given by Cσ = i∈N αi j∈P(σ,i) pj + pi .



The optimal order of the jobs σ ∗ produces the minimum cost
Cσ∗ = i∈N αi j∈P(σ ∗ ,i) pj + pi or the maximum cost saving
Cσ0 − Cσ∗ .
Smith (1956) proved that an optimal order can be obtained
reordering the jobs according to decreasing urgency indices, where
α
the urgency index of job i ∈ N is deﬁned as ui = p i .
i
The following question arises: Is it possible to share this cost
savings Cσ0 − Cσ∗ among the agents in such a way that no agent
will protest? This question can be answered by using cooperative
games.


Example

Let us ﬁnd the optimal order of the three agents, where
α1 = 20, α2 = 60, α3 = 100 and p1 = 2, p2 = 3, p3 = 4.

The urgency indices are
u1 = α1 = 20 = 10, u2 =
p
1
2
α2
p2 = 60
3 = 20 and u3 = α3
p3 = 100
4 = 25.

Optimal order of service is σ ∗ = (3, 2, 1).



A sequencing game is a pair < N, v > where N is the set of
players, that coincides with the set of jobs, and the characteristic
function v assigns to each coalition S the maximal cost savings
that the members of S can obtain by reordering only their jobs.
We say that a set of jobs T is connected according to an order σ if
for all i, j ∈ T and k ∈ N, σ(i) < σ(k) < σ(j) implies k ∈ T .
A switch of two connected jobs i and j, where i precedes j,
generates a change in cost equal to αj pi − αi pj .
We denote the gain of the switch as

gij = (αj pi − αi pj )+ = max{0, αj pi − αi pj }.


For a connected coalition T according to an order σ is
v (T ) = gij .
j∈T i∈P(σ,j)∩T
If S is not a connected coalition, the order σ induces a partition in
connected components, denoted by S/σ. For a nonconnected
coalition S, v (S) = v (T ) for each S ⊂ N.
T ∈S/σ
The characteristic function v of the sequencing game can be
deﬁned as v = gij u[i,j] , where u[i,j] is the unanimity game
i,j∈N:i<j
deﬁned as:
1 if {i, i + 1, ..., j − 1, j} ⊂ S
u[i,j] (S) = .
0 otherwise



Curiel, Pederzoli and Tijs (1989) show that sequencing games are
convex games and, consequently, their core is nonempty.
Moreover, it is possible to determine a core allocation without
computing the characteristic function of the game.
They propose to share equally between the players i, j the gain gij
produced by the switch and call this rule the Equal Gain Splitting
rule (EGS-rule).
EGSi (N, σ0 , α, p) = 2 k∈P(σ,i) gki + 1 j:i∈P(σ,j) gij for each
1
2
i ∈ N.


Example

Let us ﬁnd the optimal order of the three agents where
α1 = 20, α2 = 10, α3 = 30 and p1 = 1, p2 = 1, p3 = 1.
The urgency indices are
u1 = α1 = 20, u2 = α2 = 10 and u3 = α3 = 30.
p
1
p
2
p
3

Optimal order of service is σ ∗ = (3, 1, 2).

Solution: Go from σ0 to optimal order by neigbour switches and
share equally the gain. This can be done by two neigbour services.
First 2 and 3 switch, gain is g23 = α3 p2 − α2 p3 = 30 − 10 = 20.
Second 1 and 3 switch, gain is g13 = α3 p1 − α1 p3 = 30 − 20 = 10.
EGS(N, σ0 , α, p) = (∗, 10, 10) + (5, ∗, 5) = (5, 10, 15) ∈ C (v ).


The sequencing game < N, v > (convex) with N = {1, 2, 3} can
be constructed as follows: v ({1}) = v ({2}) = v ({3}) = v (∅) = 0
v ({1, 2}) = 0 (1 is more urgent than 2), v ({1, 3}) = 0 (switch is
not allowed because 2 is in between), v ({2, 3}) = 20,
v (N) = 20 + 10 = 30.
Note that v = 20u[2,3] + 10u[1,3] .
For example
v ({2, 3}) = 20u[2,3] ({2, 3}) + 10u[1,3] ({2, 3}) = 20 + 0 = 20,
v (N) = 20u[2,3] ({1, 2, 3}) + 10u[1,3] ({1, 2, 3}) = 20 + 10 = 30.

I (v ) = conv {(0, 0, 30), (0, 30, 0), (30, 0, 0)} ,

W (v ) = C (v ) = conv {(0, 0, 30), (0, 30, 0), (10, 0, 20), (10, 20, 0)} .



Marginal vectors of the sequencing game can be observed from the
following table.
 
σ σ σ σ
 m1 (v ) m2 (v ) m3 (v ) 
 
(123) 
 0 0 30 

(132) 
 0 30 0 
.
(213) 
 0 0 30 

(231)  10
 0 20 

(312)  0 30 0 
(321) 10 20 0

φ(v ) = ( 10 , 40 , 40 ) ∈ C (v ) (sequencing games are convex).
3 3 3



In airport situations costs of the coalitions are considered (Driessen
(1988)).
Airport situations lead to concave games and the Shapley value
belongs to the core of the game.
Baker(1965)-Thompson(1971): only users of a piece of the runway
pay for that piece and they share the cost of it equally.
Littlechild and Owen (1973) showed that the Baker-Thompson rule
corresponds to the Shapley value.



Consider the aircraft fee problem of an airport with one runway.
Suppose that the planes which are to land are classiﬁed into
m types.
For each 1 ≤ j ≤ m, denote the set of landings of planes of
type j by Nj and its cardinality by nj .
Then N = ∪m Nj represents the set of all landings.
j=1
Let cj represent the cost of a runway adequate for planes of
type j. We assume that the types are ordered such that
0 = c0 < c1 < . . . < cm .



We consider the runway divided into m consecutive pieces Pj ,
1 ≤ j ≤ m, where P1 is adequate for landings of planes of
type 1; P1 and P2 together for landings of planes of type 2,
and so on. The cost of piece Pj , 1 ≤ j ≤ m, is the marginal
cost cj − cj−1 .
The Baker-Thompson rule is given by
βi = j [ m nr ]−1 (ck − ck−1 ) whenever i ∈ Nj . That is,
k=1 r =k
every landing of planes of type j contributes to the cost of the
pieces Pk , 1 ≤ k ≤ j, equally allocated among its users
∪m Nr .
r =k
The airport TU game < N, c > is given by
c(S) = max {ck |1 ≤ k ≤ m, S ∩ Nk = ∅} for all S ⊂ N.


Example
The players 1, 2 and 3 own planes which need landing strip of
length |AD|. The strip |AD| is divided into 3 pieces |AB|, |BC |
and |CD|. Player 1 needs to use the strip of length |AB| with the
cost k1 , player 2 needs to use the strip of length |AC | with the cost
k1 + k2 , player 3 needs to use the strip of length |AD| with the
cost k1 + k2 + k3 .
In cooperation they can share strips leading to the cost game
< N, c > with N = {1, 2, 3},
c(∅) = 0, c({1}) = k1 , c({2}) = c({1, 2}) = k1 + k2 ,
c(N) = c({1, 3}) = c({2, 3}) = c({3}) = k1 + k2 + k3 .
The Baker Thompson rule is
β = (β1 , β2 , β3 ) = ( k1 , k1 + k2 , k1 + k2 + k3 ).
3 3 2 3 2
We know that Baker-Thompson rule is equal to the Shapley value.


Marginal vectors of the airport game can be observed from the
following table.
 
σ σ σ
 m1 (c) m2 (c) σ
m3 (c) 
 
(123)  k1
 k2 k3 

(132)  k1
 0 k2 + k3 
.
(213) 
 0 k1 + k2 k3 

(231) 
 0 k1 + k2 k3 

(312)  0 0 k1 + k2 + k3 
(321) 0 0 k1 + k2 + k3

k1 1 1
φ(c) = (
, (2k1 + 3k2 ), (6k3 + 3k2 + 2k1 )).
3 6 6
φ(c) ∈ C (c) (airport games are concave).

Final Remark

Final Remark

For other interesting Operations research games see

Operations Research Games: A Survey

by Borm, Hamers and Hendrickx published in TOP (the
Operational Research journal of SEIO (Spanish Statistics and
Operations Research Society)).

References

References
[1]Aumann R. and Maschler M., “Game theoretic analysis of a
bankruptcy problem from the Talmud”, Journal of Economic
Theory 36 (1985) 195-213.
[2]Baker J.Jr., “Airport runway cost impact study”, Report
submitted to the Association of Local Transport Airlines, Jackson,
Mississippi (1965).
[3]Borm P., Hamers H. Hendrickx R., (2001)Operations Research
Games: A Survey, TOP 9, 139-216.
[4]Branzei R., Dimitrov D. and Tijs S., “Models in Cooperative
Game Theory”, Game Theory and Mathematical Methods,
Springer (2008).
[5]Branzei R., Tijs S. and Timmer J., “Information collecting
situations and bi-monotonic allocation schemes”, Mathematical
Methods of Operations Research 54 (2001) 303-313.

References

References
[6] Curiel I., Pederzoli G. and Tijs S., “Sequencing games”,
European Journal of Operational Research 40 (1989) 344-351.
[7] Driessen T., “Cooperative Games, Solutions and Applications”,
Kluwer Academic Publishers (1988).
[8] Gillies D. B., “Solutions to general non-zero-sum games”. In:
Tucker, A.W. and Luce, R.D. (Eds.), Contributions to the theory
of games IV, Annals of Mathematical Studies 40. Princeton
University Press, Princeton (1959) pp. 47-85.
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