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Quelle tu_game.pvs Sprache: PVS

%---------------------------------------------------------------------
% A formal theory of cooperative TU-games in PVS 4.2
%
% Author: Erik Martin-Dorel,
% University Montpellier 2 & University of Perpignan
%
%   THEORY tu_game[U]
%
%   Version 1.0    2009/05/06    Initial version
%   Version 1.1    2009/05/10    Identifiers renamed
%   Version 1.2    2009/05/14    COV_core proved
%   Version 1.3    2009/10/03    Universal quantifiers merged
%
% This work has been done at the University of Perpignan
% within the Master's Thesis, under the supervision of
% Marc Daumas and Annick Truffert, with the help of Michel Ventou
%
% This library is available under GNU Lesser General Public License,
% either version 3 of the Licence, or any later version  (cf. infra)
%---------------------------------------------------------------------

tu_game[U: TYPE+]: THEORY

BEGIN

  IMPORTING players_set[U],
            coalition_fun,
            imputations

  % cooperative Transferable-Utility game
  tu_game: TYPE = [N: players_set, v: coalition_fun[U,N]]
  % which is a dependent 2-ary tuple type

  N: VAR players_set
  powset(N): TYPE = powset[U,N]
  set_vect(N): TYPE = set_vect[U,N]

  g: VAR tu_game
  setFP(g): set_vect(g`1) = setFP[U,g`1,g`2]
  setPI(g): set_vect(g`1) = setPI[U,g`1,g`2]
  setI(g):  set_vect(g`1) =  setI[U,g`1,g`2]
  core(g):  set_vect(g`1) =  core[U,g`1,g`2]

  ss: VAR [g:tu_game -> set_vect(g`1)]
  solution_concept?(ss): bool = FORALL (g:tu_game): subset?(ss(g), setFP(g))
  solution_concept: TYPE+ = (solution_concept?) CONTAINING core
  % core_type: JUDGEMENT core HAS_TYPE solution_concept

  s: VAR solution_concept
  a_test: LEMMA subset?(s(g), setFP(g)) % and this FOR ALL game g.

  % Thanks to the equivalence between
  % [[t_1,...,t_n] -> t] and [t_1,...,t_n -> t],
  % our definition of "tu_game" as a tuple type makes
  % the functions defined on it very handy to use.
  test_param: LEMMA s(g) = s(g`1, g`2)

  %--------------------------------
  % Predicates on solution_concept:
  %--------------------------------

  % 1) Pareto optimal (PO)
  PO(s): bool = FORALL g: subset?(s(g), setPI(g))

  % 2) anonymous (AN)
  NN: VAR players_set
  tau_type(N,NN): TYPE = (bijective?[(N),(NN)])
  tau_v(g:tu_game,NN:players_set,tau:tau_type(g`1,NN)): coalition_fun[U,NN] =
    g`2 o inverse_image(tau)
  tau_x(g:tu_game,NN:players_set,tau:tau_type(g`1,NN))(
    x:[(g`1)->real]): [(NN)->real] =
    LAMBDA (i:(NN)): x(inverse(tau)(i))
  % AN(s): bool = FORALL g,NN:
  %               FORALL (tau:tau_type(g`1,NN)):
  AN(s): bool = FORALL (g:tu_game, NN:players_set,
                        tau:tau_type(g`1,NN)):
  s(NN,tau_v(g,NN,tau)) = image(tau_x(g,NN,tau),s(g))

  % 3) equal treatment property (ETP)
  interchangeable?(g)(i,j:(g`1)): bool =
    FORALL (S:setof[(g`1)]): (NOT member(i,S)) AND (NOT member(j,S)) =>
    g`2(add(i,S)) = g`2(add(j,S))
  % ETP(s): bool = FORALL g: FORALL (x:(s(g))):
  %   FORALL (i,j:(g`1)): interchangeable?(g)(i,j) =>
  %   x(i) = x(j)
  ETP(s): bool = FORALL (g:tu_game, x:(s(g)), i,j:(g`1)):
    interchangeable?(g)(i,j) => x(i) = x(j)

  % 4) desirability (DES)
  more_desirable?(g)(i,j:(g`1)): bool =
    FORALL (S:setof[(g`1)]): (NOT member(i,S)) AND (NOT member(j,S)) =>
    g`2(add(i,S)) >= g`2(add(j,S))
  % DES(s): bool = FORALL g: FORALL (x:(s(g))):
  %   FORALL (i,j:(g`1)): more_desirable?(g)(i,j) =>
  %   x(i) >= x(j)
  DES(s): bool = FORALL (g:tu_game, x:(s(g)), i,j:(g`1)):
    more_desirable?(g)(i,j) => x(i) >= x(j)

  % 5) nullplayer property (NPP)
  nullplayer?(g)(i:(g`1)): bool =
    FORALL (S:setof[(g`1)]): (NOT member(i,S)) =>
    g`2(add(i,S)) = g`2(S) + g`2(singleton(i))
  % nullplayer(g): TYPE = (nullplayer?(g)) % if needed
  NPP(s): bool = FORALL (g:tu_game, x:(s(g)), i:(g`1)):
  % FORALL (i:nullplayer(g)): x(i) = g`2(singleton(i))
  nullplayer?(g)(i) => x(i) = g`2(singleton(i))

  %   NPP(s): bool = FORALL g: FORALL (x:(s(g))):
  %   FORALL (i:(g`1)): nullplayer?(g)(i) =>
  %     x(i) = g`2(singleton(i))

  % 6) covariant under strategic equivalence (COV)
  lin(N: players_set, a:real, x: [(N) -> real])(i:(N)): real
    = a*x(i)
  affine(N: players_set, a:real, b: [(N) -> real])
    (x: [(N) -> real]): [(N) -> real] =
    LAMBDA (i:(N)): lin(N,a,x)(i) + b(i)
  dif(N: players_set, y, b: [(N) -> real])(i:(N)): real
    = y(i) - b(i)
  affineinv(N: players_set, a:nzreal, b: [(N) -> real])
    (y: [(N) -> real]): [(N) -> real] =
    LAMBDA (i:(N)): dif(N,y,b)(i) / a
  bstar(N: players_set, b: [(N) -> real]): coalition_fun[U,N] =
    LAMBDA (S:powset(N)): tot(S, b)
  affinestar(N: players_set, a:real, b: [(N) -> real])
    (v: coalition_fun[U,N]): coalition_fun[U,N] =
    LAMBDA (S:powset(N)): a*v(S) + bstar(N,b)(S)
  % COV(s): bool = FORALL (N:players_set):
  %   FORALL (a:posreal, b:[(N) -> real]):
  %   FORALL (v: coalition_fun[U,N]):
  %   s(N,affinestar(N,a,b)(v)) = image(affine(N,a,b), s(N,v))
  COV(s): bool = FORALL (N:players_set, a: posreal, b: [(N) -> real],
                         v:coalition_fun[U,N]):
    s(N,affinestar(N,a,b)(v)) = image(affine(N,a,b), s(N,v))

  % 7) single valued (SIVA)
  SIVA(s): bool = FORALL g: is_finite(s(g)) AND card(s(g)) = 1

  % 8) nonemptiness (NE)
  NE(s): bool =   FORALL g: nonempty?(s(g))

  % 9) Reasonableness, on both side (REAS)
  % REAS(s): bool = FORALL (N:players_set, v:coalition_fun[U,N]):
  %   FORALL (x:(s(N,v))): FORALL (i:(N)):
  % (EXISTS (S1:powset(N)): (NOT member(i,S1)) AND x(i) >= v(add(i,S1))-v(S1))
  % AND
  % (EXISTS (S2:powset(N)): (NOT member(i,S2)) AND x(i) <= v(add(i,S2))-v(S2))
  REAS(s): bool = FORALL (N:players_set, v:coalition_fun[U,N],
                          x:(s(N,v)), i:(N)):
    (EXISTS (S1:powset(N)): (NOT member(i,S1)) AND x(i) >= v(add(i,S1))-v(S1))
    AND
    (EXISTS (S2:powset(N)): (NOT member(i,S2)) AND x(i) <= v(add(i,S2))-v(S2))

  %--------------------------------

  % DES_ETP_lemma1: LEMMA FORALL g: FORALL (i,j:(g`1)):
  %   interchangeable?(g)(i,j) => more_desirable?(g)(i,j)
  interch_impl_desir: LEMMA FORALL (g:tu_game, i,j:(g`1)):
    interchangeable?(g)(i,j) => more_desirable?(g)(i,j)

  % DES_ETP_lemma2: LEMMA FORALL g: FORALL (i,j:(g`1)):
  %   interchangeable?(g)(i,j) => interchangeable?(g)(j,i)
  interch_is_sym: LEMMA FORALL (g:tu_game, i,j:(g`1)):
    interchangeable?(g)(i,j) => interchangeable?(g)(j,i)

  DES_implies_ETP: THEOREM DES(s) => ETP(s)

  SIVA_implies_NE: THEOREM SIVA(s) => NE(s)

  PO_core: THEOREM PO(core)

  affine_bij: LEMMA FORALL (N:players_set,
    a: posreal, b,x: [(N) -> real]):
    affine(N,a,b)(affineinv(N,a,b)(x)) = x
  COV_core: THEOREM COV(core)

END tu_game

%---------------------------------------------------------------------
% License for use and distribution
%
% Copyright (C) 2009 Erik Martin-Dorel.
%
% GNU LGPL information:
% ---------------------
%
% This library is free software: you can redistribute it and/or
% modify it under the terms of the GNU Lesser General Public License
% as published by the Free Software Foundation, either version 3 of
% the License, or (at your option) any later version.
%
% This library is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
% GNU General Public License for more details.
%
% You should have received a copy of the GNU Lesser General Public
% License along with this library.
% If not, see <http://www.gnu.org/licenses/>.
%---------------------------------------------------------------------

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