Ομιλία Theory Group AUEB - 12/10/2012

   Fair pision of multiple stochastic pies
    within the Nash bargaining solution:
     Modeling and real-life application

            Athanasios C. Karmperis, PhD, c.

Contact Details

Address: NTUA, School of Mechanical Engineering, Sector of Industrial Management &
    Operational Research, Iroon Polytechniou 9, 15780, Zografos, Greece
Tel:  +30 210 7721066
Fax:   +30 210 7723571
Email: athkarmp@mail.ntua.gr


      1
Our research Team:

Ilias P. Tatsiopoulos (Professor), Konstantinos Aravossis (As.
Professor), Anastasios Sotirchos (LLM, Dr. Mech. Engineer),
and Athanasios C. Karmperis (PhD c.)

The model presented here is included in our recent paper:

“On the fair pision of multiple stochastic pies to multiple agents
within the Nash bargaining solution”, PloS ONE, Vol. 7, Issue 9,
art. No. e44535

This paper as well as the Proofs of Propositions and Theorems
presented here can be found:

  http://dx.plos.org/10.1371/journal.pone.0044535


2
Topics:

Introduction

Problem Description and Mathematic Formulation

Computing Solutions for Fair Surplus Division

Basic Features

Computation Algorithm

A Real-life Application

Discussion and Future Research
3
                                  Introduction (1)

   Cooperative game theory examines situations where at least two
   players benefit by working together or sharing some cost.

   Their objectives are partially cooperative, as they aim at reaching
   an agreement and partially conflicting, because each player has
   its own utility function regarding the negotiation outcome.

   Each player is faced with a set S of feasible bargaining outcomes,
   any of which presents the result if all participants agree upon.

A cooperative game for the coalition N = {1, 2}, is:

   Either a pair (N, p) with a characteristic function p : 2N →    and
   p(Ø) = 0, which represents the collective payoff,

   Or a pair (N, c) and a characteristic function c : 2N → and
   c(Ø) = 0 that describes the cost for agents who cooperate in
   accomplishing a specific task.

The solution of the game is a vector x Є N representing the allocation
   of the overall profit p(N) or cost c(N) to each agent.
    4
                           Introduction (2)

The Basic Problem: Two players (1 and 2) negotiate over a
   surplus that is yielded through cooperation.
-  Each player has a utility function (von Neumann and
   Morgenstern, 1944).
-  S is as a sub-set of 2 (for n-players of N) and is assumed
   to be closed, convex, non-empty and bounded.




5
                              Introduction (3)
-    Some natural axioms of the solution
-    “One states as axioms several properties that it would be
    natural for the solution to have and then one discovers that
    the axioms actually determine the solution uniquely” (Nash,
    1953)

    Axiom 1. Feasibility




  6
                 Introduction (4)




Axiom 2. Inpidual rationality




7
               Introduction (5)




Axiom 3. Pareto Optimality




8
          Introduction (6)




Axiom 4. Symmetry




9
                         Introduction (7)




Axiom 5. Independent of Irrelevant Alternatives




10
                               Introduction (8)




Axiom 6. Scale Invariance

This implies that the solution x is independent of the scale that is
used in measuring the players’ utilities, i.e. if we multiply the
utilities of all players i = 1,2,3,…,n by constants (a1, a2, a3, … , an),
then we have the feasible set S’ and we get the relative solution
x’ through the multiplication of the players’ coordinates by these
constants (Jain and Mahdian, 2007).




11
                                 Introduction (9)



 The Nash bargaining solution (NBS):
 Nash J F, (1950). The bargaining problem, Econometrica 18, pp.
 155–162, available at:
     http://www.math.mcgill.ca/vetta/CS764.dir/nashbarg.pdf

 There is a unique solution in satisfying Axioms 1 to 6, i.e.
 feasibility, inpidual rationality, Pareto optimality, symmetry,
 independence of irrelevant alternatives and independence of
 equivalent utility representations.

 Roth, A.E., (1979): Axiomatic models of bargaining, presents is
 detail all axioms used by Nash, available at:
http://kuznets.fas.harvard.edu/~aroth/Axiomatic_Models_of_Bargaining.pdf

 The NBS is a function that maximizes the geometric average of
 the players’ gains through the agreement, instead of settling for
 the disagreement point d.

  12
                             Introduction (10)

In other words, the NBS for a two-person bargaining game, is the
solution maximizing the product of the excesses: (x1 – d1) (x2 – d2),
subject to constraints: x1 ≥ d1, and x2 ≥ d2.




13
                             Introduction (11)




Harsanyi (1959), (1963) showed that the NBS can be easily
expanded in a finite set of n > 2 players: N = {1,2, .. , n}:


            Max. Π(xi – di) ,
             s.t. xi ≥ di
              iЄN

The solution of the game is a vector x Є N representing the
allocation of the overall profit p(N) to each agent.

The finite set N is called the grand-coalition, every subset in which
this set can be pided is called coalition and a coalition with just
one player is called singleton




14
                                Introduction (12)




   In cases with identical players (equal disagreement outcomes and
   symmetric utility functions), then the solution is the symmetric NBS
   and the surplus is pided equally: xi = xk ,∀ i, k ∈ N
   Kalai (1977) introduced the asymmetric NBS, as the solution
   maximizing the product of the excesses:

              (x1 – d1) p (x2 – d2) q

   where p, q are positive numbers such that: p + q = 1
Available at:
   http://www.datalaundering.com/download/IntJGT-6-3-129-133.pdf


   The symmetric NBS is the special case where: p = q = 1/2

   That is, in cases with non-identical players (unequal disagreement
   outcomes or/and and asymmetric utility functions), then the surplus is
   pided unequally for at least two players i, k Є N :
               xi ≠ xk
   15
                              Introduction (13)



Criticisms in the NBS about its real-life applications arise from the
assumptions used in the model (in real-life these assumptions can
be rarely found):

There is full information among players regarding their utility
function and their disagreement outcomes.

Players are rational seeking to maximize their utility

The disagreement outcomes are fixed

The pie over which players negotiate is pisible and its size is fixed
(deterministic model)

A Query:

What happens if we relax one assumption that is the cooperation’s
overall return (including the surplus) consists of multiple pies with
uncertain sizes.
(stochastic model)
16
                 Problem description and mathematic formulation (1)

Assumptions

A finite number of agents indexed by i. Let N = {1,2,3,…,n} denote
the grand-coalition.

Assumption 1. There is complete information among agents, who
examine to cooperate having fixed disagreement payoffs.
Payoff vector: (C1,C2, … ,Cn).
Objectives: partially cooperative – conflicting.

Assumption 2. The different gains and losses that are yielded
through cooperation form a finite set of pies J = {1,2,3,…,m} that is
called pie-set. These pies are pisible, and are assumed to be
stochastic variables. Specifically, all pies indexed by j follow
normal probability distribution functions:
                           ( P j -µ j ) 2
                         -
             2
               (
       Π j (µ j , σ j ) = 1 / 2π σ j e)     2σ j
                               2




where σj > 0 for all pies, µj > 0 for gains and µj < 0 for losses.


17
                    Problem description and mathematic formulation (2)


Assumption 3. Agents are rational, i.e. each agent should get at
least as much as it could obtain through the non-cooperative
option. Clearly, the cooperation yields a nonnegative surplus S.

                    ∑Πj
                    m

That is, the N’s overall return:

                            ∑ Ci
                    j =1
                            n

equals the sum of disagreement payoffs:
                            i =1



           ∑ Π = ∑ Ci + S ⇔ S = ∑ Π - ∑ Ci
           m      n             m      n
               j                    j
plus the surplus S:
           j =1    i =1             j =1    i =1



Assumption 4. All agents are risk-neutrals, i.e. they are indifferent
between the m pies, since they consider only the overall expected
return when making investment decisions. In particular, the
bargaining outcome is the NBS, which is a vector:
           U = (U 1 , U 2 ,..., U n )
representing the expected (or mean) inpidual surplus shares
(pidends) which are allocated to players.
18
                     Problem description and mathematic formulation (3)
Axioms

Axiom 1. Coalitional rationality:

             µ N = ∑ µ - ∑ Ci > 0
                 m       n
                      j

                 j =1     i =1
                                         (1)

Since all agents are risk-neutrals, they negotiate over the S’s
mean value considering their expected pidends (µi). The
bargaining outcome is an allocation, i.e. a vector U Є N
representing the ratio of the µN that is pided in agent i Є N:

                 µ i = µ N (U i )                (2)

Axiom 2. Inpidually rational:
                     Ui > 0                 (3)
Axioms 3 and 4. Linear Invariance and Independence of
Irrelevant Alternatives:
Let F be the feasible set of allocations. If F ’ is obtained from F by
multiplying all agents’ utilities by αi then the solution of the new
game is obtained by multiplying each agent’s coordinate in the
first solution by αi. Moreover, the solution remains the same for
each subset of F that includes the specific solution.
19
                   Problem description and mathematic formulation (4)

Axioms 5 and 6. Feasibility and Pareto-Optimality: Efficient
allocation of the overall surplus:

               ∑ µi = µ N ⇔∑ Ui = 1
               n         n

                                       (4)
               i =1      i =1

    j
Let p i denote the ratio of pie j Є J allocated to i Є N.

                   µ i = ∑ (µ j p ij ) - C i
                          m

Player’s i expected pidend:
                        j =1

Efficient allocation of all pies j Є J:
                            ∑ p ij = 1
                             n


                             i =1
                                       (5)

Axiom 7. Symmetry (or asymmetry): All players identical, then:
                     (2)
       U 1 = U 2 = U 3 = ... = U n ⇔µ 1 = µ 2 = µ 3 = ... = µ n    (6)

If players non-identical, then asymmetric NBS, i.e. the surplus
can be pided either equally (Eq. (6)), or unequally for at least
two agents i, k Є N.

20
                            xi ≠ xk
                    Problem description and mathematic formulation (5)




Πi : the inpidual stochastic surplus share allocated to i.

                    Π i = ∑ (Π j p ij ) - C i
                          m


                          j =1
                                          (7)

A solution is a [P]nXm matrix, in which the 1,2,..,n rows denote the
players and the 1,2,..,m columns denote the pies, i.e. each
element represents the ratio of pie j which is allocated to agent i:

                          p1
                          1
                             p 12  ...  p 1m
                          p 12  p2   ....  pm
           [P ]n× m = [p ij ]n× m = [      2       2
                                        ]  (8)
                          ...  ...  ...  ...
                          p 1n  p2
                              n
                                 ....  pm
                                     n



The characteristic function:

             [P ]n×m × [Π j ]m×1 - [Ci ]n×1 = [Πi ]n×1 (9)

21
                      Problem description and mathematic formulation (6)




   Axiom 8. Fair (proportional) pision:

   Aristotle in “Nicomachean Ethics”:

“Equals should be treated equally and unequals unequally, in
   proportion to the relevant inequality”
“What is just, is proportional”

Following this maxim, the surplus is pided with fairness, when the
   stochastic pidends are distributed in proportion to the NBS, i.e.:

               µ i = ∑ (µ j p ij ) - C i = µ N ( U i )
                      m

   Expected (mean) values:    j =1
                                       (10)


                   µ1 µ 2 µ 3    µ
   Standard deviations:       =  =  = ... = n          (11)
                   σ1 σ 2 σ 3    σn

                         , ∀i,k∈ N
                   µi  σi  Ui
That is:                =  =                (12)
                   µk σk Uk
   22
                     Problem description and mathematic formulation (7)


   This implies that in the 3 Confidence Intervals:

   1st:  probability : (µ i - σ i ≤ Πi ≤ µ i + σ i ) ≈ 0.6827
   2nd:  probability : (µ i - 2σ i ≤ Πi ≤ µ i + 2σ i ) ≈ 0.9545
   3rd:  probability : (µ i - 3σ i ≤ Πi ≤ µ i + 3σ i ) ≈ 0.9973

For example:

If:   U = (U1 , U 2 , U 3 ) = (0.3,0.5,0.2), µ N = 100,

Then the sharing mechanism should satisfy: µ1 = 30, µ 2 = 50, µ 3 = 20,

and: σ1 = 3, σ 2 = 5, σ 3 = 2, or: σ1 = 0.6, σ 2 = 1.0, σ 3 = 0.4,

where σi are unknowns (depend on the normal probability distributions
  of the pies).
                        , ∀ i , k = 1,2,3
                 µi  σi  Ui
        In other words:   =  =
   23             µk σk Uk
         Computing solutions for fair surplus pision (1)




General Method.




24
                                    Computing solutions for fair surplus pision (2)


Stage 1. Partition the pie-set J into two nonempty subsets: JA, JB,
  and the grand-coalition N into two nonempty coalitions: NA NB.


Partition of the pie-set J = {1,2,..,m} into two subsets: JA = {1,…,g},
and JB = {g+1,…,m}, with 1 ≤ g < m, the characteristic function is
presented in Eq (13):

                              ∑Π j
                              g



     [  p i{1,.., g }  p i{g+1 ,..,m }  ] [    j =1
                                      ] - [C ]        = [Π i ]n×1  (13)
                              ∑Πj
                        n× 2     m        2 ×1    i n ×1


                             j = g +1


Partition of the grand-coalition N into two coalitions: NA = {1,…, h},
NB = {h+1,…,n}, with 1 ≤ h < n. We have 4 unknowns, which are
the ratios of each subset: JA = {1,…,g} and JB = {g+1,…,m}
allocated to each coalition: NA = {1,…, h} and NB = {h+1,…,n}:


                            p {1,.., h }}
                                 g
                                      p {1g+1h,..m }
                [P ]2×2 = [        {1,..,
                             {1,.., g }
                                       { ,.., }
                                        { g+1 ,..m }  ]  2× 2
                                                         (14)
                            p {h+1 ,..n }    p {h+1 ,..n }
25
                                    Computing solutions for fair surplus pision (3)




Characteristic function:


             [P ]n×m × [Π j ]m×1 - [Ci ]n×1 = [Πi ]n×1                    (9)




                     ∑Π          ∑ Ci
                     g           h
                          j
  p{{1,..,h }}
       g
          p{{1g+1h,..m}                       Π{1,..,h}
[   1,..,      ,.., }
                  ] [  j =1
                            ] -[  i =1
                                   ]2×1 = [ Π         ]    (15)
                     ∑Π j         ∑ Ci
   {1,..,g }    {g+1,..m}   2×2   m      2×1    n               2×1
  p{h+1,..n} p{h+1,..n}                           {h +1,..,n}
                     j = g +1       i = h +1

26
                                 Computing solutions for fair surplus pision (4)



Theorem 1. For each pair of two coalitions and two subsets that can
  arise from the partition of the grand-coalition N = {1,..,n} into:

NA = {1,..,h} and NB = {h+1,..,n}, (NA U NB = N; NA ∩ NB = Ø)

   and the partition of the pie-set J = {1,..,m} into:

JA ={1,..,g} and JB ={g+1,..,m}, (JA U JB = J; JA ∩ JB = Ø),

   there is a unique [P] 2 x 2 matrix:

              p {1,.., h }}
                   g
                        p {1g+1h,..m }
           [     {1,..,       { ,.., }
                                 ]
                ,.., }
             p {{1h+1g,..n }    p {{h+1 ,..n }}
                          g+1 ,..m    2× 2




, which ensures fairness within the NBS satisfying:

                                  ∑ Ui
                                    h

        µ {1,.., h }        σ {1,.., h }       i =1
                 =             =
                                   ∑ Ui
                                     n
        µ { h +1,.., n }     σ { h +1,.., n }
                                  i = h +1
27
                                                  Computing solutions for fair surplus pision (5)


  Stage 2. Continuous partitions of the coalitions for n - 1 times
Partition of NA = {1,..,h} into coalitions {1,.., f } ,{f +1,…,h}, 1≤f<h:

                                p{1,..,h} ∑ Π                  ∑Ci
                                       g                  f
                                 {1,..,g }       j
          p{{1,..,gf }} p{{1g+1f,..m}
                    ,.., }
                                                                       Π{1,.., f }         (16)
      [      1,..,
                          ] [            j =1
                                                 ] -[      i =1
                                                              ]  =[              ]
          p{{1f+1g,..h} p{{g+1,..h}}       p{1,..,h} ∑ Π                    ∑Ci
             ,.., }      1,..m   2×2             m        2×1       h     2×1                2×1
                   f+
                                {g+1,..m}           j                      Π{ f +1,..,h}
                                       j = g +1             i = f +1


                                          p {1,.., gf }}
                                           {1,..,        p {1g+1f,..m }
                                                       { ,.., }
Which has a unique matrix:                        [                           ]
                                            ,.., }
                                         p {1f+1g,..h }
                                          {           p {g+1,..h }}
                                                       { f+
                                                         1 ,..m    2× 2



Partition of NB = {h+1,…,n} into {h+1,…,k},{k+1,…,n}, h+1≤k <n:
                           p {h +1,.., n } ∑ Π                   ∑ Ci
                                     g                    k
                             {1,.., g }         j
    {1,.., g }       {g+1,..m }
  p {h +1,.., k }    p {h +1,.., k }                                                  Π{ h +1,.., k }
[                      ] [            j =1
                                                 ] -[     i = h +1
                                                              ]    =[               ]   (17)
                           p {{h +11,.., nm}} ∑ Π j                 ∑ Ci
    {1,.., g }       { g+1,..m }  2× 2             m           2×1       n     2×1                  2×1
  p {k+1,..n }      p {k+1,..n }         g + ,..,                                      Π{ k +1,.., n}
                                    j = g +1                i = k +1


                                                        }
                                                 p {1h,..,1g,.., k }   p {h +1,.., k }}
                                                                g+1 ,..m

Which also has a unique matrix:                               [   { +           {
                                                                        ]
                                                    ,.., }
                                                 p {{1k+1g,..n }     p {{k+1 ,..n }}
                                                                g+1 ,..m      2× 2



  28
                   Computing solutions for fair surplus pision (6)


In particular, the partitioning of coalitions into two nonempty
coalitions is continued, until eventually all agents form singletons:
{1}, {2},…, {n}, i.e. for n-1 times.




29
                   Computing solutions for fair surplus pision (7)

Through this process, if we compute the unique [P] 2 x 2 matrix for
each coalition, we can compute the ratio of each subset {1,..,g} and
{g+1,..,m} which is allocated to each agent.
Let π i denote the set of coalitions including agent i, which arise from
a specific set of partitions of the grand-coalition N into two nonempty
coalitions for n-1 times.
For instance: Partitions of the N = {1,2,3,4,5} into: {1} and {2,3,4,5},
and further into {3} and {2,4,5}, and further into {2} and {4,5}, and
further into {4} and {5}. The sets of coalitions including agents are:



                 π1 = {{1}}
                 π 2 = {{2,3,4,5}, {2,4,5}, {2}}
                 π 3 = {{2,3,4,5}, {3}}
                 π 4 = {{2,3,4,5}, {2,4,5}, {4,5}, {4}}
                 π 5 = {{2,3,4,5}, {2,4,5}, {4,5}, {5}}

30
                                        Computing solutions for fair surplus pision (8)

That is, the ratio of each subset of pies ({1,..,g} and {g+1,..,m}), which
is allocated to agent i, equals the product of ratios of the coalitions, in
which i is included, e.g. for agent 5:

  p 51,.., g }= ∏ p {k1,.., g } = ( p {12,.., ,g4}, 5 } )( p {{12,.., ,g5}} )( p {{14,.., }g } )( p {{15,.., g } )
   {
                    { ,3            ,4         ,5         }
          k π5
           ∈


 p {5g +1,.., m }= ∏ p {k g +1,.., m } = ( p {2 ,+31,,..,, 5m}} )( p {2 ,+41,,..,}m } )( p {4 ,+51},.., m } )( p {{5 }+1,.., m } )
                        g
                       { 4
                                    g
                                   { 5          {
                                               g           g

           k π5
           ∈


However, the ratio for agent i in each subset of pies {1,…,g} and
{g+1,…,m}, equals her ratios in all pies of the specific subset:

   p {51,.., g }=p1=p 5=...= p 5
           5
             2    g




   p {5g +1,.., m }=p 5 +1=p 5 + 2=...= p 5
             g   g      m




In other words, we compute the ratios of all pies j = 1,2,..,m which are
allocated to all agents i = 1,2,..,n. That is, we can compute a specific
matrix [P] n X m, which ensures that the agents’ pidends are
distributed in proportion to the NBS:
                                               , ∀i,k∈ N
                                         µi  σi  Ui
31                                         =  =
                                         µk σk Uk
                             Basic Features (1)




Number of possible partitions of the pie-set

Proposition 1. The number of possible partitions of a pie-set J into
  two nonempty subsets is given from the piecewise Eq (18):
           m- 1


            ∑
         m!  2   m!
   g ( m )=    +         , m = περιτος
       ( m-1)! l= 2 ( m-l )! l!
           m
                                  (18)
           +∑
            -1
         m!  2    m!     m! 1
   g ( m )=             +    , m = αρτιος
       ( m-1)! l= 2 ( m-l )! l!  m 2 2
                     ( !)
                     2




32
                                     Basic Features (2)

   Finite possible [P] n X m matrices for fair surplus pision
   Theorem 2. The number of possible [P] n X m matrices that ensure
     fairness for the surplus pision within a NBS, is finite and
     equals the product of the possible partitions of the pie-set into
     two subsets with the possible partitions of all coalitions into
     two nonempty coalitions for n-1 times:

      possible [P] n X m     =  f (n) g (m)               (19)
             n-1


       f (n -1)+∑
     n!        2  n!
f (n)=               f (n - k ) f (k ), n = περιτος
    (n-1)!     k=2 (n-k)!k!

             n                            (19.1)
       f (n -1)+∑
              -1
     n!       2  n!             n!  n 21
f (n)=               f (n - k ) f (k )+   f ( !) , n = αρτιος
    (n-1)!     k=2 (n-k)!k!           n 2 2 2
                            ( !)
                            2
                m- 1


              +∑
            m!    2 m!
      g ( m )=             , m = περιτος
          ( m-1)! l= 2 ( m-l )! l!
                m                         (19.2)
               +∑
                 -1
             m!   2   m!     m! 1
      g ( m )=              +    , m = αρτιος
           ( m-1)! l= 2 ( m-l )! l!  m 2 2
   33                     ( !)
                         2
                                              Basic Features (3)

        Computation of possible [P] n X m matrices for fair surplus
         pision with the Wolfram Research Mathematica:



                      n −1
              n!        2    n!            n −1
f@n_D := PiecewiseB::       f@n − 1D + ‚         f@n − kD f@kD ,   ∈ Integers && n ≥ 3>,
             Hn − 1L !      k=2 Hn − kL ! k !          2
             n −1
     n!       2     n!             n!   n 2 1 n
  :       f@n − 1D + ‚         f@n − kD f@kD +     fB F , ∈ Integers && n ≥ 4>>, 1F
    Hn − 1L !      k=2 Hn − kL ! k !         J n !N
                                  2  2 2 2
                                2
                  m −1
               m!   2    m!     m −1
g@ m_D := PiecewiseB::       +‚         ,   ∈ Integers && m ≥ 3>,
             H m − 1L ! l=2 H m − lL ! l !   2
         m −1
      m!  2     m!      m! 1 m
  :       + ‚         +     , ∈ Integers && m ≥ 4>>, 1F
    H m − 1L ! l=2 H m − lL ! l !  J !N
                    m 2 2 2
                    2

MatrixForm @Table@Table@f@nD g@ m D, 8 m, 2, 10<D, 8n, 2, 10<DD




        34
                          Basic Features (4)

Number of possible [P] n X m matrices for fair surplus pision




35
                                                                 Computation Algorithm (1)


Step 1. Randomly partition the pie-set J into two subsets: {1,…,g} and
{g+1,…,m} with 1 ≤ g < m.
Step 2. Randomly partition the grand-coalition N into two coalitions:
{1,…,h} and {h+1,…,n} with 1 ≤ h < n.
Step 3. Develop a Monte Carlo simulation model, in which the Ci , Π j
are inputs and the Π{1,..,h} , Π {h+1,..,n} the outputs, according to:

        Π {1,.., h } = p {1,.., h } ∑ Π + p {1,.., h }                  ∑ Π - ∑ Ci
                      g                           m                h
                {1,.., g }          j          { g+1 ,..m }           j
                                                                           (20)
                      j =1                        j = g +1             i =1


        Π{h+1,..n } = p{{1h+1g,..n } ∑ Π j + p{{h+1,..n }} ∑ Π j - ∑ Ci
                        g                         m                n
                  ,.., }        g+1 ,..m
                                                                           (21)
                        j =1                      j = g +1          i = h +1

                                      {      }         {      }
Step 4. Select a specific value of p {1,.., h } = 1 - p {1h,..,1g,.., n }, and estimate the
                            1,..,
                               g
                                +
 {g +1,.., m }   {g +1,.., m }
p {1,.., h } = 1 - p {h +1,.., n } in order to fulfill:

        µ {1,..,h } = µ N ∑ U i = p {1,..,h } ∑ µ + p {1,..,h } ∑ µ - ∑ Ci
                h                  g                     m           h
                            {1,.., g }        j      {g+1,..m }          j

                i =1                 j =1                 j = g +1         i =1
                                                                           (22)

        µ{h+1,..n} = µ N ∑Ui = p{h+1,..n} ∑µ + p{h+1,..n} ∑µ - ∑Ci
                 n                   g                   m             n
                            {1,..,g }          j    {g+1,..m}            j

                i =h+1                  j =1               j = g +1         i =h+1
                                                                           (23)
 36
                                        Computation Algorithm (2)


For the scenario that fulfils Eqs (22), (23) run the MCS and estimate
σ{1,..,h}, σ{h+1,..,n}. If the following Eq (24) is fulfilled then go the next
Step, otherwise examine alternative values (simply by increasing /
decreasing its initial value), until you find the unique [P] 2 X 2 matrix:
                              ∑ Ui
                               h

               µ {1,.., h }  σ {1,.., h }
                     =       = i =n1                     (24)
               µ {h+1 ,..n } σ {h+1 ,..n }  ∑ Ui
                              i = h +1

           {1,.., g } {1,.., g }     { g +1,.., m } { g +1,.., m }
Step 5. Use the (p {1,.., h } , p {h +1,.., n } , p {1,.., h } , p {h +1,.., n } ) , and return to Step 2, i.e.
randomly partition both the {1,…,h} and {h+1,…,n} coalitions into two
pairs of nonempty coalitions: {1,…,f}, {f+1,…,h} and {h+1,…,k},
{k+1,…,n}, respectively, and compute the unique ratios. Specifically,
the 2 to 5 Steps should be followed for n-1 times.
Step 6. Compute the ratio of each subset {1,…,g} and {g+1,…,m}
allocated to each agent, through the product of the ratios of the agent-
coalitions in which the agent is included. Moreover, each agent’s ratios
are equal in all pies of each subset, so illustrate the [P] n X m matrix.

 37
                                 Computation Algorithm (3)


Step 7. Develop another MCS model, in which the computed [P] n X m ,
the [Π j] m X 1 , and the [Ci] n X 1 matrices are inputs, and the [Π i] n X 1 is
the output according to Eq (9):



      [P ]n×m × [Π j ]m×1 - [Ci ]n×1 = [Πi ]n×1                 (9)

Run the simulation and estimate:

µ 1 , µ 2 , µ 3 ,..., µ n , and  σ 1 , σ 2 , σ 3 ,..., σ n ,

in order to verify that the pidends are allocated to all agents with
fairness:

              , ∀i,k∈ N
       µi  σi  Ui
         =  =
       µk σk Uk


38
                             A real-life application (1)


   A consortium of scholars for a specific public project.
   The project concerns a set of technical studies for a construction
   project on behalf of the MoD in 2011. According to the tender Call,
   there are 8 different studies to be developed by a consortium of
   studiers, each with specific qualification requirements:

   Category / Class                Pre-estimation of
No          Type of Study
     Degree                  Remunerations ( €)
1        -     Tender               63.927,73
2        -     Safety                8.427,99
3     6 / Β’ or Γ’  Architecture            43.980,64
4     8 / Γ’ or ∆’  Static               46.674,04
5       9 / Ε’   Electro mechanic         564.722,18
6      10 / Α’   Transportation            3.706,14
7      13 / Α’   Hydraulic             10.969,22
8      21 / Γ’   Geotechnical            86.168,52
   39
                      Total      828.576,46
                            A real-life application (2)


  Following this Call, a specific consortium of 6 scholars was formed,
  and after negotiation, they have agreed in the following:



               Allocation of Remunerations
       Class
Player
      Degree
              Study       Tender + Safety
              (100%)        (Surplus)
 1     06 Β    Architecture        5,82%
 2      08 Γ     Static         6,17%
 3      09 Ε  Electro mechanic       74,68%
 4     10 Α   Transportation        0,49%
 5     13 Α     Hydraulic         1,45%
 6      21 Γ   Geotechnical        11,39%
                   Total     100,00%

   40
                                          A real-life application (3)


 Notations.
 Grand-coalition (studiers): N = {1,2,3,4,5,6}
 Pie-set (studies):     J = {1,2,3,4,5,6,7,8}
 Bargaining outcome for surplus pision:
U = (U1 , U 2 , U 3 , U 4 , U 5 , U 6 ,) = (0.0582,0.0617,0.7468,0.0049,0.0145,0.1139)
 Profit-sharing mechanism:
                p1
                1  p12  3
                      p1  p 14  5
                            p1    6
                                p1  7
                                   p1  8
                                     p1
                p12  p2
                   2  p3
                      2  p4
                         2   p5
                            2    p6
                                 2  p7
                                   2  p8
                                      2

                p1   2
                   p3  p3  4
                        p3   p5    6
                                p3  7
                                   p3  p8
  [P ]6×8 = [p ij ]6×8 = [  3      3      3         3
                                        ]
                p14  p2
                   4  p3
                      4  p4
                         4   p5
                            4    p6
                                 4  p7
                                   4  p8
                                      4

                p1
                5
                   2
                   p5  p3
                      5
                         4
                        p5   p5
                            5
                                 6
                                p5  7
                                   p5  p8
                                      5

                p16  2
                   p6  p3
                      6
                         4
                        p6   p5
                            6    p6
                                 6  p7
                                   6  p8
                                      6




 Such that the players’ profits: Π i (µ i , σ i ) satisfy the axiom of fair
                              2


 (proportional) pision: µ i
                         , ∀i,k∈ N
                 σi     Ui
                =    =
              µk σk Uk

  41
                                           A real-life application (4)
   6 out of 8 pies are inpisible, since the “Architect study (no 3) can be
   developed only by the Architect, the Static study (no 4) only by the
   Civil Engineering, etc): p1 p12 1 0 0 0 0 0
                  1

                       p 12  p2
                           2  0  1  0  0  0  0
                       p1   2
                          p3  0  0  1  0  0  0
         [P ]6×8 = [p ij ]6×8 = [   3
                                         ]
                       p 14  p2
                           4
                             0  0  0  1  0  0
                       p1
                       5
                           2
                          p5  0  0  0  0  1  0
                       p1
                       6
                           2
                          p6  0  0  0  0  0  1
   Since there is uncertainty over the size of the remunerations that
   will be finally achieved by the grand-coalition N, a normal
   probability distribution function is assigned in each pie
A/A  Type of Study              Mean value: µ j      Standard deviation: σ     j

j=1  Tender                    63.927,73          28.767,47
j=2  Safety                     8.427,99          3.792,59
j=3  Architecture                 43.980,64          2.199,03
j=4  Static                    46.674,04          2.333,70
j=5  Electro mechanic               564.722,18          28.236,11
j=6  Transportation                 3.706,14            185,31
j=7  Hydraulic
    42                      10.969,22            548,46
j=8  Geotechnical                 86.168,52          4.308,43
                                             A real-life application (5)


     Initially, we examine the case where each of the pisible pies (No 1:
     Tender, and No. 2: Safety), is allocated proportionally among agents:
                      0,0582  0,0582  1  0  0  0  0  0
                      0,0617  0,0617  0  1  0  0  0  0
                      0,7468  0,7468  0  0  1  0  0  0
        [P ]6×8 = [p ij ]6×8 = [                       ]
                      0,0049  0,0049  0  0  0  1  0  0
                      0,0145  0,0145  0  0  0  0  1  0
                      0,1139  0,1139  0  0  0  0  0  1

     That is we use the above profit-sharing mechanism as input in a Monte
     Carlo model within the characteristic function:

            [P ]6×8 [Π j ]8×1 = [Π i ]8×1
     The simulation gives the following results:
µ1 = 48191.6, µ 2 = 51138.1, µ 3 = 618756.7, µ 4 = 4060.6, µ 5 = 12018.3, µ 6 = 94409.5
σ1 = 2803.5, σ 2 = 2967 .3, σ 3 = 35365 .5, σ 4 = 233.6, σ 5 = 687.5, σ 6 = 5452 .4
      43
                              A real-life application (6)


   These results verify that this mechanism is unfair, because:

   µ1 U1 0.0582 µ1 U1 0.0582 µ1 U1 0.0582 µ1 U1 0.0582
     =  =   ,  =  =   , =   =   , =   =    ,
   µ 2 U 2 0.0617 µ 3 U 3 0.7468 µ 4 U 4 0.0049 µ 5 U 5 0.0145
   µ1 U1 0.0582 µ 2 U 2 0.0617 µ 2 U 2 0.0617     µ  U  0.0145
     =  =   , =   =   ,  =  =    ,....., 5 = 5 =
   µ 6 U 6 0.1139 µ 3 U 3 0.7468 µ 4 U 4 0.0049    µ 6 U 6 0.1139


   However, the profits’ standard deviations are not in proportion to
   the bargaining outcome U:

σ 1 µ 1 U 1 σ 1 µ 1 U1 σ 1 µ 1 U1 σ1 µ 1 U1 σ 1 µ 1 U1
  ≠  =  ,  ≠  = , ≠   = , ≠  = ,  ≠  =  ,
σ 2 µ 2 U 2 σ3 µ 3 U3 σ 4 µ 4 U 4 σ5 µ 5 U5 σ6 µ 6 U6
σ2 µ2 U2 σ2 µ2 U2        σ5 µ 5 U5
 ≠  = ,  ≠  =  ,.......,  ≠  =
σ3 µ 3 U3 σ 4 µ 4 U 4      σ6 µ 6 U6

   Further, we follow the computation algorithm
    44
                                                    A real-life application (7)

Due to the fact that some pies are inpisible, a partition of the pie-set is
followed into two sets: JA (pisible subset) and JB (inpisible subset):
           Pie-set
        J = {1, 2, … , 8}




    JA = {1, 2}     JB = {3, 4, 5, 6, 7, 8}             1  0   0   0    0  0
                                       0  1   0   0    0  0
Set of pisible pies    Set of inpisible pies             0  0   1   0    0  0
                              [P ]6j×6J B = [
                                ∈
                                                        ]
i.e. the set of inpisible pies is fixed:                  0  0   0   1    0  0
                                       0  0   0   0    1  0
                                       0  0   0   0    0  1

                                         p1
                                         1   p 12
                                         p 12  p2
                                             2

                                         p1   2
                                            p3
                              [P ]     =[             ]
                                j JA∈       3
and we look for a [P ]6j×2J A matrix:
                  ∈
                  ∈
                  ∈
                  ∈              6× 2
                                         p 14  p2
                                             4

                                         p1
                                         5
                                             2
                                            p5
                                         p1
                                         6
                                             2
                                            p6


Characteristic function:        [P ]6j× 2J A [Π j ]2j ×1J A + [P ]6j× 6J B [Π j ]6j×1J B = [Π i ]6×1
                       ∈      ∈            ∈            ∈

  45
                                      A real-life application (8)


Step 1. Partition the pie-set J into: {1} and {2}.
Step 2. Random partition of the grand-coalition N into two coalitions:
{2, 3, 5, 6} and {1, 4}.
Step 3. Develop a Monte Carlo simulation model, in which the
[P ]6j×2J A , [Π j ]2j×1J A , [P ]6j×6J B , [Π j ]6j×1J B are inputs and the Π{2356} , Π {14} the
  ∈     ∈     ∈     ∈

outputs, according to:
             }
     Π{2356 } = p{12356 } Π 1 + p{ 2} } Π 2 + Π 4 + Π 5 + Π 7 + Π 8
           {        { 2356                          (20)
          {1}     {2
     Π{14} = p{14} Π 1 + p{14}} Π 2 + Π 3 + Π 6
                                                (21)

                     p{ 2} } and we compute the
Step 4. We examine alternative values of { 2356
         { 2356
             (
unique elements p{2} } = 0.95, p{14}} = 0.05, p{1} } = 0.8375, p{14} = 0.1624
                {
                 2
                        { 2356
                                {1}
                                                 )
which fulfill:    µ {2356} σ {2356} U 2 + U 3 + U 5 + U 6 0.9369
               =    =           =
           µ {14}  σ {14}    U1 + U 4     0.0631

 46
                           A real-life application (9)

Step 5. We return to step 2 and we follow the 2 to 5 steps for n - 1 =
6 - 1 = 5 times, following specific partitions:




47
                                                  A real-life application (10)

    Step 6. We compute the ratios for each agent in each subset (1}, (2}:

           p1 = (p {{1} }p {{1}} ) = (0.1642)(0.7877) = 0.1279
    Agent 1:   1    14   1



           p12 = (p {{14}}p {{12}} ) = (0.05)(0.98) = 0.049
                2




    Agent 2:
                  }
            p1 = (p {{12356 } )(p {{12}} ) = (0.8375)(0.2883) = 0.2414
            2

  ,         p 2 = (p {{ 2 } } )(p {{2 }} ) = (0.95)(0.04) = 0.038
             2     2356    2



               }
         p1 = (p {{12356 } )(p {{1} } )(p {{1}} ) = (0.8375)(0.7117 )(0.9959) = 0.5936
    Agent 3: 2 3
             {2 }
                     356
                     {2 }
                          3
                          {2 }
         p 3 = (p {2356 } )(p {356 } )(p {3} ) = (0.95)(0.96)(0.841) = 0.7669

,
         p14 = (p {{1} }p {{14}} ) = (0.1624)(0.2122) = 0.0344
    Agent 4: 2     14
              {2 } {2 }
         p 4 = ( p {14 }p {4 } ) = (0.05)(0.02) = 0.001


         1    {1}     {1}     {1}     {1}
    Agent 5: p 5 = ( p{{2356 } )(p {{356 } )( p {{56}} )( p {52 }) = (0.8375)(0.7117)(0.04)(0.5339) = 0.1295
          2    2}      2}      2     {
                                  }
           p 5 = (p {2356 } )(p {356 } )(p {56 } )(p {5 } ) = (0.95)(0.96)(0.159)(0.112) = 0.0162

                 }
           p1 = (p {{12356 } )(p {{1} } )(p {{1} } )(p {{1}} ) = (0.8375)(0.7117 )(0.04)(0.466) = 0.0113
    Agent 6:   6           356    56     6


     48
                       2}
           p 6 = ( p {{2 } } )( p {{356 } )( p {{56} } )( p {{6 }} ) = (0.95)(0.96)(0.159)(0.888) = 0.1287
            2
                 2356
                              2      2
                                                   A real-life application (11)
                             p1 1
                                 p 12      0.1279       0.049
      And we illustrate the matrix:         p 12  p2       0.2414       0.038
                                  2

                             p1   p3 2
                                        0.5936      0.7669
                    [P ]6j× 2J A = [
                      ∈
                              1
                               3
                                    ]= [                    ]
                             p4   p2 4
                                         0 .034      0 .001
                             p1
                              5
                                 2
                                 p5       0.1295       0.016
                              1   2
                             p6   p6       0.011      0.1287
  ,
      Step 7. We use it as well as the [Π ]2×1 , [P ]6 × 6 B , [Π ]6 ×1
                        j  A         jj J
                                    ∈   B     j J
                                            ∈         j J
                                                      ∈
                                                            as inputs in
      another Monte Carlo simulation model: [Π ] = [P ] [Π ]       i 8 ×1
                                               j JA
                                               ∈
                                               6× 2
                                                    j  j JA
                                                        ∈
                                                      2 ×1  + [P ]6j × 6J B [Π
                                                              ∈      j
                                                                      ]
                                                                      j JB
                                                                      6 ×1
                                                                        ∈



      And the simulation gives:
      µ1 = 48191.6, µ 2 = 51138.1, µ 3 = 618755.9, µ 4 = 4060.6, µ 5 = 12018.3, µ 6 = 94409.8
      σ1 = 2706.1, σ 2 = 2734 .2, σ 3 = 36420 .4, σ 4 = 234 .4, σ 5 = 722.64, σ 6 = 5681 .1
,
             µ1 σ1 U1 0.0582 µ1 σ1 U1 0.0582 µ1 σ1 U1 0.0582
               =  =  =   ,  =  =  =   ,  =  =  =    ,
             µ 2 σ 2 U 2 0.0617 µ 3 σ 3 U 3 0.7468 µ 4 σ 4 U 4 0.0049
             µ1 σ1 U1 0.0582 µ1 σ1 U1 0.0582 µ 2 σ 2 U 2 0.0617
               =  =  =   , =   =  =   , =   =  =    ,
    Verifying the   µ 5 σ 5 U 5 0.0145 µ 6 σ 6 U 6 0.1139 µ 3 σ 3 U 3 0.7468
    Fair pision   µ 2 σ 2 U 2 0.0617 µ 2 σ 2 U 2 0.0617 µ 2 σ 2 U 2 0.0617
               =  =  =   , =   =  =    , =   =  =    ,
              µ 4 σ 4 U 4 0.0049 µ 5 σ 5 U 5  0.0145 µ 6 σ 6 U 6  0.1139
    Among agents:
             µ 3 σ 3 U 3 0.7468 µ 3 σ 3 U 3  0.7468 µ 3 σ 3 U 3 0.7468
               =  =  =   , =   =  =    , =   =  =    ,
             µ 4 σ 4 U 4 0.0049 µ 5 σ 5 U 5  0.0145 µ 6 σ 6 U 6 0.1139
       49     µ 4 σ4 U4  0.0049 µ 4 σ 4 U 4  0.0049 µ 5 σ 5 U 5  0.0145
               = =  =    , =   =  =    , =   =  =
             µ 5 σ5 U5  0.0145 µ 6 σ 6 U 6 0.1139 µ 6 σ 6 U 6 0.1139
                                                                     A real-life application (12)

    This matrix is one out of 945 matrices [P ]6× 2 A that can be computed.
                                                        j J
                                                        ∈

    For example, if we follow a different set of partitions:
       Set of pisible pies                         Grand-coalition
          J Α = {1, 2}                           N = {1, 2, 3, 4, 5, 6}




     subset         subset                 Coalition           Coalition
      {1}          {2}                  {1, 3, 4}          {2, 5, 6}
                                   P{134}{2} = 78.490%       P{256}{2} = 21.510%
  ,                                 P{134}{1} = 99.953%       P{256}{1}= 0.047%




                          Coalition        Singleton         Singleton        Coalition
                          {4, 1}          {3}            {6}          {2, 5}
                         P{41}{2}= 8.700%    P{3}{2}= 91.300%     P{6}{2}= 59.920%      P{25}{2}= 40.080%
                         P{41}{1}= 2.377%    P{3}{1}= 97.623%     P{6}{1}= 45.477%      P{25}{1}= 54.523%



,
                   Singleton         Singleton                       Singleton       Singleton
                     {4}            {1}                          {5}          {2}
                   P{4}{2}= 8.100%     P{1}{2}= 91.900%                    P{5}{2}= 19.000%  P{2}{2}= 81.000%
                   P{4}{1}= 0.473%     P{1}{1}= 99.527%                    P{5}{1}= 92.584%  P{2}{1}= 7.416%



                                       p1
                                       1
                                             p 12        0.0236     0.0627
                                       p 12    p2
                                             2
                                                       0.0001     0.0698
    Then we compute:                           p1      2
                                             p3         0.9757     0.7166
                            [P ]6j× 2J A = [
                               ∈         3
                                                ]= [                     ]
                                       p 14    p2
                                             4
                                                       0 .0001     0.0055
                                       p1
                                       5
                                             2
                                             p5         0.0002     0.0163
     50                                  1      2
                                       p6     p6         0.0002     0.1288
                                                                        A real-life application (13)

    … or a different set of partitions:
      Set of pisible pies                           Grand-coalition
        J Α = {1, 2}                            N = {1, 2, 3, 4, 5, 6}




    subset          subset                   Coalition          Singleton
     {1}            {2}                  {1, 2, 4, 5, 6}         {3}
                                   P{12456}{2} = 27.350%      P{3}{2}= 72.650%
                                   P{12456}{1} = 9.922%      P{3}{1}= 90.078%

  ,

                             Coalition         Singleton
                             {1, 2, 4, 5}         {6}

                           P{1245}{2}= 57.100%     P{6}{2}= 42.900%
                           P{1245}{1}= 11.439%     P{6}{1}= 88.561%




,                      Coalition        Singleton                         Then we compute:
                       {2, 4, 5}         {1}

                   P{245}{2}= 58.600%     P{1}{2}= 41.400%
                   P{245}{1}= 18.524%     P{1}{1}= 81.476%

                                                            p1
                                                             1    p 12     0.0092    0.0646
                                                             1
             Coalition         Singleton
                                                            p2    p2 2
                                                                        0.0001    0.0698
              {4, 5}           {2}
                                                            p1    p3 2
                                                                        0.9008    0.7265
            P{45}{2}= 23.691%
            P{45}{1}= 99.980%
                         P{2}{2}= 76.309%
                         P{2}{1}= 0.020%              [P ]6j× 2J A = [
                                                   ∈          3
                                                             1    2
                                                                    ]= [              ]
                                                            p 4   p 4
                                                                        0 .0009   0 .0054
      Singleton          Singleton
                                                            p1 5   p52
                                                                        0.0011    0.0162
        {4}             {5}

     51
     P  {2}
      {4} =
        {1}
           25.000%     P{5}{2}= 75.000%                                 p 16   p62
                                                                        0.0878    0.1173
     P {4} =  45.421%     P{5}{1}= 54.579%
                                 Discussion and Future Research (1)

      Limitations of the model:

      This stochastic model is a static model, rather than a dynamic one, as
      it does not considers the effect of time.

      Fairness can be achieved if the bargaining outcome is not fixed, as for
  ,
      example in the case study:

    U = (U1 , U 2 , U 3 , U 4 , U 5 , U 6 ,) = (0.0582,0.0617,0.7468,0.0049,0.0145,0.1139)

      In particular, this outcome should include the proportional pision rule
      of the final outcomes (Remunerations of the different studies), i.e.
,
      those that will be realized. However, the contract among studiers
      should include a mathematic formula:

                           ,∀ i ∈ N
                  1   2
                        Φi
                  p =p =
                       ∑ Φi
                  i   i   n


                        i =1




      52
                           Discussion and Future Research (2)

    Future research issues:

    Development of a code, e.g. in MatLab for the computation algorithm.

    The application of the proposed method in other real-life applications,
    such as water-sharing problems, power distribution case, tax-paying
  ,
    problems, etc.

    The application of other solution concepts, e.g. Shapley value,
    nucleolus in stochastic environments, in which the size of the pie over
    which agents negotiate is uncertain.
,


    The development of another model, in which one (or more than one)
    natural axioms will be used, in order to define a unique solution.

    To examine situations by relaxing the assumptions used in our model:
    i.e. what happens if pies are inpisible, or/and follow asymmetric
    probability distributions (e.g. lognormal), if players are risk-averse
    or/and risk-seeking and are not indifferent between the m pies, etc.
     53
              THANK YOU!

           Athanasios C. Karmperis




………. And some marketing for our
   recently published Book :




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