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ParaTuGames.wl
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ParaTuGames.wl
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(* ::Package:: *)
(* :Title: ParaTuGames.m
: Release Date : 02.05.2023
*)
Off[Needs::nocont]
(* :Context: TUG`ParaTuGames` *)
(* :Summary:
This package provides some extensions to the package TuGame for
modeling and calculating solutions and properties for cooperative games with
transferable utilities in parallel.
*)
(* :Author:
Holger Ingmar Meinhardt
Department of Economics
University of Karlsruhe (KIT)
holger.meinhardt@wiwi.uni-karlsruhe.de
*)
(* :Package Version: 1.0.4 *)
(*
:Mathematica Version: 12.x, 13.x
*)
(*:Keywords:
Dual Game, Superadditive Game, Convex Game, Strong Convex Game, Average-Convex Game,
Kernel, balancing Maximum Excesses.
*)
(* :Sources:
Theo Driessen, Cooperative Games, Solutions and Applications, Kluwer Academic
Publishers, Dordrecht, 1988.
E. Inarra and J. Usategui, The Shapley value and average convex games,
IJGT, 22, 13-29, 1993.
M. Maschler, The Bargaining Set, Kernel and Nucleolus, Handbook of Game
Theory, Chapter 18, 591-647, 1992.
M. Maschler, B. Peleg and L.S. Shapley, Geometric Properties of Kernel,
Nucleolus and related Concepts, in Mathematics of Operations Research,
Vol4 Nov. 1979, p. 303-338.
J-E. Martinez-Legaz, Dual Representation of Cooperative Games based on
Fenchel-Moreau Conjugation, Optimization, pp. 291-319, Vol. 36, 1996.
H. I. Meinhardt, An LP approach to compute the pre-kernel for cooperative games,
Computers and Operation Research, Vol 33/2 pp. 535-557,2006.
H. I. Meinhardt, The Pre-Kernel as a Tractable Solution for Cooperative Games:
An Exercise in Algorithmic Game Theory, forthcoming in: Theory and Decision Library C,
Springer Publisher, Heidelberg. pp. 1-247, 2013.
A. Meseguer-Artola, Using the Indirect Function to characterize the Kernel of a TU-Game,
Departament d'Economia i d'Historia Economica, 1997.
R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970.
R.E. Stearns, Convergent Transfer Schemes for N-Person Games,
Transaction American Mathematical Society, 449-459, 1968.
Hal Varian (Ed.). Economics and Financial Modeling with Mathematica,
Springer, 1992.
*)
(*
:History:
See ChangeLog.
*)
Needs["TUG`coop`CooperativeGames`"];
Needs["TUG`TuGames`"];
Needs["TUG`TuGamesAux`"];
Which[$OperatingSystem === "Unix", Needs["TUG`vertex`VertexEnum`"],
$OperatingSystem === "Windows", Needs["TUG`vertex`VertexEnum`"],
True, Needs["VertexEnum`"]
];
ParaAntiPreKernel::usage =
"ParaAntiPreKernel[game,payoff,options] computes an anti-pre-kernel solution by relying on
the anti of Algorithm 7.2.1 of Meinhardt (2013).";
ParaAntiPreKernelQ::usage =
"ParaAntiPreKernelQ[game,payoff,opts] checks whether the vector 'payoff' is an element of the anti-pre-kernel.
ParaAntiPreKernelQ checks also the efficiency condition in contrast to the function MinExcessBalanced.";
ParaAvConvexQ::usage =
"ParaAvConvexQ[game] checks the average-convexity of the game.
It returns 'True' or 'False'. Calling the function with the option will return
the sum of the marginal contributions for each coalition S w.r.t. to each
superset S union {j}. These values must be non-negative.";
ParaConvexQ::usage =
"ParaConvexQ[game] checks if the Tu-game is convex. It returns the value 'True' or 'False'.";
ParaSuperAdditiveQ::usage =
"ParaSuperAdditiveQ[game] checks if a game is superadditive.";
ParaPreKernel::usage =
"ParaPreKernel[game,payoff,options] computes a pre-kernel element by iteratively solving
a system of linear equations in parallel mode. (cf. Algorithm 8.2.1 of Meinhardt (2013))";
ParaPreKernelQ::usage =
"ParaPreKernelQ[game,payoff,options] checks whether the (pre-)imputation 'payoff' is an element of the pre-kernel.
ParaPreKernelQ checks also the efficiency condition in contrast to the function MaxExcessBalanced.";
ParaPreKernelElement::usage =
"ParaPreKernelElement[game,payoff,options] computes a pre-kernel element by iteratively
determining a direction of improvement in parallel mode. The iteration process stops whenever the
direction of improvement is equal to the null vector. (cf. Algorithm 8.3.1 of Meinhardt (2013)).";
ParaModiclus::usage =
"Modiclus[game,opts] computes the modiclus as the projection of the pre-nucleolus from the
dual cover game onto the player set T of the original game. Do not confound this command
with the function ModifiedNucleolus[]. The algorithm is based on a method by Peleg to translate
the definition of the Nucleolus into a sequence of linear programs on the pre-imputation set.
A simplex method is now used to increase its computational reliability. For its default value
'False' the function Kernel[game] will be invoked to avoid infinite loops. To increases the
computational reliability in cases of numerical issues the following methods can be used:
RevisedSimplex, CLP, GUROBI, MOSEK, or Automatic. Default setting is Automatic. This option
must be used in connection with CallMaximize->False. For getting more precise results one
can even set Method->{InteriorPoint, Tolerance->10^-10}.";
ParaIsModiclusQ::usage =
"IsModiclusQ[game,payoff,opts] checks whether the provided payoff vector is the modiclus of the game.
For its default value 'False' the function Kernel[game] will be invoked to avoid infinite loops.
To increases the computational reliability in cases of numerical issues the following methods can be used:
RevisedSimplex, CLP, GUROBI, MOSEK, or Automatic. Default setting is Automatic. This option
must be used in connection with CallMaximize->False. For getting more precise results one
can even set Method->{InteriorPoint, Tolerance->10^-10}.";
ParaModPreKernel::usage =
"ParaModPreKernel[game] computes a modified pre-kernel element as the solution
of the pre-kernel from the excess comparability cover game.
Do not confound this command with the function ModifiedKernel[].";
ParaIsModPreKernelQ::usage =
"ParaIsModPreKernelQ[game,payoff] checks whether the provided payoff vector is a modified
pre-kernel element of the game.";
ParaProperModPreKernel::usage =
"ParaProperModPreKernel[game] computes a proper modified pre-kernel element as the projection
of the pre-kernel from the dual cover game onto the player set T of the original game.
Do not confound this command with the function ModifiedKernel[].";
ParaIsProperModPreKernelQ::usage =
"ParaIsProperModPreKernelQ[game,payoff] checks whether the provided payoff vector is a proper modified
pre-kernel element of the game.";
ParaSMPreKernel::usage =
"ParaSMPreKernel[game] computes the simplified modified pre-kernel of the game.";
ParaIsSMPreKernelQ::usage =
"ParaIsSMPreKernelQ[game,payoff] checks if payoff is the simplified modified pre-kernel of the game.";
ParaBestCoalitions::usage =
"ParaBestCoalitions[game,payoff] computes the set of most effective coalitions that supports the claim of
player i against j, for all possible pair of players in parallel mode.";
ParaSetsToVec::usage =
"ParaSetsToVec[bestcoal,T,options] converts the set of most effective coalitions to a set of vectors
of length T in parallel mode. A vector reflects how the best arguments are distributed between a
bargaining pair (i,j) at a proposal. A plus sign indicates that the arguments are skewed in favor
of the player i, zero means that the arguments are balanced, and a minus sign indicates that the
arguments are skewed in favor of the player j. See also Meinhardt (2013).";
ParaDirectionOfImprovement::usage =
"ParaDirectionOfImprovement[game, payoff, options] determines a vector of improvement in order reduce
the maximum surpluses in parallel mode.";
ParaMaxSurplus::usage =
"ParaMaxSurplus[game,pi,pj,payoff] calculates the maximum surplus of player i over j with respect to the
imputation 'payoff' in parallel mode. Note that the efficiency condition will not be checked.";
ParaGameBasis::usage =
"ParaGameBasis[T] computes the basis of a |T|-person game in parallel.";
ParaCharacteristicValues::usage =
"ParaCharacteristicValues[unancrd_List,T,opts] computes the coalitional values from the vector of
unanimity coordinates in parallel.";
Options[ParaAntiPreKernel] = Sort[Options[PreKernel]];
Options[ParaAntiPreKernelQ] = Sort[Options[PreKernelQ]];
(* Options[ParaAvConvexQ] = Options[AverageConvexQ]; *)
Options[ParaBestCoalitions] = Sort[Options[BestCoalitions]];
Options[ParaSetsToVec] = Sort[Options[SetsToVec]];
Options[ParaPreKernelElement] = Sort[Options[PreKernelElement]];
Options[ParaDirectionOfImprovement] = Sort[Options[DirectionOfImprovement]];
Options[ParaPreKernel] = Sort[Options[PreKernel]];
Options[ParaPreKernelQ] = Sort[Options[PreKernelQ]];
Options[ParaMaxExcessBalanced] = Sort[Options[MaxExcessBalanced]];
Options[ParaMinExcessBalanced] = Sort[Options[MinExcessBalanced]];
Options[ParaExcessPayoff] = Sort[Options[ExcessPayoff]];
Options[ParaModPreKernel] = Sort[Options[ModPreKernel]];
Options[ParaProperModPreKernel] = Sort[Options[ProperModPreKernel]];
Options[ParaModiclus] = Sort[Options[Modiclus]];
Options[ParaIsModiclusQ] = Sort[Options[IsModiclusQ]];
DistributeDefinitions[Options[ParaPreKernel] = Sort[Options[PreKernel]]];
(* DistributeDefinitions[Options[ParaAvConvexQ] = Options[AverageConvexQ]]; *)
DistributeDefinitions[Options[ParaBestCoalitions] = Sort[Options[BestCoalitions]]];
DistributeDefinitions[Options[ParaPreKernelElement] = Sort[Options[PreKernelElement]]];
DistributeDefinitions[Options[ParaDirectionOfImprovement] = Sort[Options[DirectionOfImprovement]]];
DistributeDefinitions[Options[ParaPreKernel] = Sort[Options[PreKernel]]];
DistributeDefinitions[Options[ParaExcessPayoff] = Sort[Options[ExcessPayoff]]];
DistributeDefinitions[Options[ParaModPreKernel] = Sort[Options[ModPreKernel]]];
DistributeDefinitions[Options[ParaProperModPreKernel] = Sort[Options[ProperModPreKernel]]];
SetSharedFunction[ParaMaxSurplus];
SetSharedFunction[ParaAntiSurplus];
SetSharedFunction[ParaTIJsets];
SetSharedFunction[ParaW];
(* :Error Messages: *)
(* :One Argument: *)
ParaAntiPreKernel::argerr="One argument was expected.";
ParaAvConvexQ::argerr="One argument was expected.";
ParaConvexQ::argerr="One argument was expected.";
ParaGameBasis::argerr="One argument was expected.";
ParaModiclus::argerr="One argument was expected.";
ParaModPreKernel::argerr="One argument was expected.";
ParaPreKernelElement::argerr="One argument was expected.";
ParaPreKernel::argerr="One argument was expected.";
ParaProperModPreKernel::argerr="One argument was expected.";
SMPreKernel::argerr="One argument was expected.";
ParaSuperAdditiveQ::argerr="One argument was expected.";
(* :Two Arguments: *)
ParaAntiPreKernelQ::argerr="Two arguments were expected.";
ParaBestCoalitions::argerr="Two arguments were expected.";
ParaCharacteristicValues::argerr="Two arguments were expected.";
ParaDirectionOfImprovement::argerr="Two arguments were expected.";
ParaExcessPayoff::argerr="Two arguments were expected.";
ParaIsModiclusQ::argerr="Two arguments were expected.";
ParaIsModPreKernelQ::argerr="Two arguments were expected.";
ParaIsProperModPreKernelQ::argerr="Two arguments were expected.";
ParaIsSMPreKernelQ::argerr="Two arguments were expected.";
ParaMaxExcessBalanced::argerr="Two arguments were expected.";
ParaMinExcessBalanced::argerr="Two arguments were expected.";
ParaPreKernelQ::argerr="Two arguments were expected.";
ParaSetsToVec::argerr="Two arguments were expected.";
(* :Four Arguments: *)
ParaMaxSurplus::argerr="Four arguments were expected.";
ParaAntiSurplus::argerr="Four arguments were expected.";
(* Based on Algorithm 8.2.1 of Meinhardt (2013) *)
(* User interface to compute a pre-kernel element. *)
ParaPreKernel[args___]:=(Message[ParaPreKernel::argerr];$Failed);
ParaPreKernel[game_,opts:OptionsPattern[ParaPreKernel]] :=
Block[{pay},
pay = Table[v[T],{Length[T]}]/Length[T];
ParaPreKernel[game,pay,opts]
];
ParaPreKernel[game_, payoff_List, opts:OptionsPattern[ParaPreKernel]] := Block[{dimpay,rclim},
dimpay = Dimensions[payoff];
rclim=If[Length[T] > 11,1024,256];
Which[Length[dimpay]===2,
Which[(Last[dimpay]===Length[T] && Depth[payoff] ===3), Block[{$RecursionLimit = rclim}, ParaPreKernelAlg2[game,#, opts]&/@ payoff //Union],
True, ParaPrintRemark[payoff]],
Length[dimpay]===1,
Which[(First[dimpay]===Length[T] && Depth[payoff] === 2 ), Block[{$RecursionLimit = rclim},ParaPreKernelAlg2[game, payoff, opts]],
True, ParaPrintRemark[payoff]],
True, ParaPrintRemark[payoff]
]
];
(* Main Functions *)
ParaPreKernelAlg2[game_, payoff_List, opts:OptionsPattern[ParaPreKernel]] :=
Block[{rattol,sil, smc, meff, matE, vlis, alpv},
sil = OptionValue[Silent];
smc = OptionValue[SmallestCardinality];
rattol=OptionValue[RationalTol];
meff = ParaBestCoalitions[game, payoff, MaximumSurpluses -> False, SmallestCardinality -> smc];
matE = -ParaSetsToVec[meff, T, EffVector -> True];
vlis = ParallelMap[MapThread[v[#1] &, {#}] &, meff, Method -> "CoarsestGrained",DistributedContexts -> None];
alpv = ParallelMap[ReplaceAll[#, List -> Subtract] &, vlis, Method -> "CoarsestGrained",DistributedContexts -> None];
PrependTo[alpv,v[T]];
err=Norm[matE.payoff+alpv]^2;
If[LessEqual[err,1.5*rattol],Return[payoff],
xvec=-PseudoInverse[matE].alpv;
ParaPreKernelAlg2[game,xvec,opts]]
];
(* Based on Algorithm 8.3.1 of Meinhardt (2013) *)
(* User interface to compute a pre-kernel element. *)
ParaPreKernelElement[args___]:=(Message[ParaPreKernelElement::argerr];$Failed);
ParaPreKernelElement[game_,opts:OptionsPattern[ParaPreKernelElement]] :=
Block[{pay},
pay = Table[v[T],{Length[T]}]/Length[T];
ParaPreKernelElement[game,pay,opts]
];
ParaPreKernelElement[game_, payoff_List, opts:OptionsPattern[ParaPreKernelElement]] := Block[{dimpay,rclim},
dimpay = Dimensions[payoff];
rclim=If[Length[T] > 11,1024,256];
Which[Length[dimpay]===2,
Which[(Last[dimpay]===Length[T] && Depth[payoff] ===3), Block[{$RecursionLimit = rclim}, ParaPreKernelAlg3[game,#, opts]&/@ payoff //Union],
True, ParaPrintRemark[payoff]],
Length[dimpay]===1,
Which[(First[dimpay]===Length[T] && Depth[payoff] === 2 ), Block[{$RecursionLimit = rclim},ParaPreKernelAlg3[game, payoff, opts]],
True, ParaPrintRemark[payoff]],
True, ParaPrintRemark[payoff]
]
];
(* Main Functions *)
ParaPreKernelAlg3[game_, payoff_List, opts:OptionsPattern[ParaPreKernelElement]] :=
Block[{sil, smc, optst, doi, optstep, itpay,tol,brc,pinv},
sil = OptionValue[Silent];
smc = OptionValue[SmallestCardinality];
optst = OptionValue[CalcStepSize];
pinv = OptionValue[PseudoInv];
rattol=OptionValue[RationalTol];
{optstep, doi} = ParaDirectionOfImprovement[game, payoff, MaximumSurpluses -> False, CalcStepSize -> optst, PseudoInv->pinv,Silent -> sil, SmallestCardinality -> smc];
itpay = payoff + optstep*doi;
If[SameQ[sil,False],
Print["doi=", doi];
Print["optstep=", optstep];
Print["itpay=", itpay];,
True];
If[Depth[itpay]!=2,Return[payoff],True];
tol=Table[1.5*rattol,{Length[T]}];
brc=Apply[And,MapThread[LessEqual[#1,#2] &,{Abs[doi],tol}]];
If[SameQ[brc,True], Rationalize[itpay,rattol], ParaPreKernelAlg3[game, Rationalize[itpay,rattol], CalcStepSize -> optst, Silent -> sil, SmallestCardinality -> smc]]
];
ParaDirectionOfImprovement[args___]:=(Message[ParaDirectionOfImprovement::argerr];$Failed);
ParaDirectionOfImprovement[game_, payoff_List, opts:OptionsPattern[ParaDirectionOfImprovement]] :=
Module[{sil, smc, optst, meff, matE, mopt, varpay, mex, submex, setpay, grmex, doi, optstep,pinv},
sil = OptionValue[Silent];
smc = OptionValue[SmallestCardinality];
optst = OptionValue[CalcStepSize];
pinv = OptionValue[PseudoInv];
mopt= OptionValue[MaximumSurpluses];
{meff, mex} = ParaBestCoalitions[game, payoff, AntiPreKernel -> False, MaximumSurpluses -> True, SmallestCardinality -> smc];
matE = -ParaSetsToVec[meff, T, EffVector -> True];
submex = ParallelMap[{1, -1}.# &, mex];
varpay = x[#] & /@ T;
setpay = MapThread[Rule, {varpay, payoff}];
grmex = v[T] - Total[x[#] & /@ T] /. setpay;
PrependTo[submex, grmex];
If[SameQ[sil,False], Print["submex=", submex],True];
If[SameQ[pinv,False],
doi = LeastSquares[matE,-submex,Tolerance -> 10^(-10)];,
doi = -PseudoInverse[matE].submex];
optstep = If[SameQ[optst,True], ParaDelStar[doi, matE, submex], 1];
If[SameQ[mopt,False],{optstep,doi},{optstep,doi,mex}]
];
ParaDelStar[doi_List, matE_List, smex_List]:=
Block[{edvec,nrsq,tol},
edvec = matE.doi;
nrsq =Norm[edvec]^2;
tol=1.5*10^(-12);
If[LessEqual[Abs[nrsq],tol], 0, - smex.edvec/nrsq]
];
(* Computing the anti pre-kernel *)
ParaAntiPreKernel[args___]:=(Message[ParaAntiPreKernel::argerr];$Failed);
ParaAntiPreKernel[game_,opts:OptionsPattern[ParaPreKernel]] :=
Block[{pay},
pay = ParallelTable[v[T],{Length[T]}]/Length[T];
ParaAntiPreKernel[game,pay,opts]
];
ParaAntiPreKernel[game_, payoff_List, opts:OptionsPattern[ParaPreKernel]] := Block[{dimpay,rclim},
dimpay = Dimensions[payoff];
rclim=If[Length[T] > 11,1024,256];
Which[Length[dimpay]===2,
Which[(Last[dimpay]===Length[T] && Depth[payoff] ===3), Block[{$RecursionLimit = rclim}, ParaAntiPreKernelAlg2[game,#, opts]&/@ payoff //Union],
True, ParaPrintRemark[payoff]],
Length[dimpay]===1,
Which[(First[dimpay]===Length[T] && Depth[payoff] === 2 ), Block[{$RecursionLimit = rclim},ParaAntiPreKernelAlg2[game, payoff, opts]],
True, ParaPrintRemark[payoff]],
True, ParaPrintRemark[payoff]
]
];
ParaAntiPreKernelAlg2[game_, payoff_List, opts:OptionsPattern[ParaPreKernel]] :=
Block[{sil, smc, meff, matE, vlis, alpv,err},
sil = OptionValue[Silent];
smc = OptionValue[SmallestCardinality];
meff = ParaBestCoalitions[game, payoff, AntiPreKernel -> True, MaximumSurpluses -> False, SmallestCardinality -> smc];
matE = -ParaSetsToVec[meff, T, EffVector -> True];
Parallelize[vlis = MapThread[v[#1] &, {#}] &/@ meff;
alpv = ReplaceAll[#, List -> Subtract] & /@ vlis, Method -> "CoarsestGrained",DistributedContexts -> None];
PrependTo[alpv,v[T]];
err=Norm[matE.payoff+alpv]^2;
If[LessEqual[err,1.5*10^(-12)],Return[payoff],
xvec=-PseudoInverse[matE].alpv;
ParaAntiPreKernelAlg2[game,xvec,opts]]
];
(* Section Modiclus, Modified and Proper Modified Pre-Kernel *)
ParaModiclus[args___]:=(Message[ParaModiclus::argerr];$Failed);
ParaModiclus[game_,opts:OptionsPattern[ParaModiclus]] :=
Block[{mthd,ovls, dcvals, lt, t0, t1, DCGame, mdnc},
mthd=OptionValue[Method];
ovls = v[#] & /@ Coalitions; (* Storing original game values. *)
t0 = T; (* Storing original game values. *)
dcvals = ParaDualCover[game];
lt = Length[T];
t1 = Range[2*lt];
DCGame = DefineGame[t1, dcvals];
mdnc = PreNucleolus[DCGame,Method->mthd];
DefineGame[t0, ovls]; (* Redefine the original game. *)
Take[mdnc, lt]
];
ParaIsModiclusQ[args___]:=(Message[ParaIsModiclusQ::argerr];$Failed);
ParaIsModiclusQ[game_,payoff_List,opts:OptionsPattern[ParaIsModiclusQ]] :=
Block[{ovls, dcvals, lt, t0, t1, dpay, DCGame, bcQ},
mthd=OptionValue[Method];
If[SameQ[Total[payoff] - v[T], 0] && Apply[And,NumericQ[#] &/@ payoff], True, Return[False]];
ovls = v[#] & /@ Coalitions; (* Storing original game values. *)
t0 = T; (* Storing original game values. *)
dcvals = ParaDualCover[game];
lt = Length[T];
t1 = Range[2*lt];
DCGame = DefineGame[t1, dcvals];
dpay= Flatten[{payoff,payoff}];
bcQ = BalancedCollectionQ[DCGame,dpay,Method->mthd];
DefineGame[t0, ovls]; (* Redefine the original game. *)
Return[bcQ]
]
ParaModPreKernel[args___]:=(Message[ParaModPreKernel::argerr];$Failed);
ParaModPreKernel[game_,opts:OptionsPattern[ParaModPreKernel]] :=
Module[{pay},
pay = Table[v[T],{Length[T]}]/Length[T];
ParaModPreKernel[game,pay,opts]
];
ParaModPreKernel[game_, payoff_List, opts:OptionsPattern[ParaModPreKernel]] := Module[{dimpay,rclim},
dimpay = Dimensions[payoff];
rclim=If[Length[T] > 11,1024,512];
Which[Length[dimpay]===2,
Which[(Last[dimpay]===Length[T] && Depth[payoff] ===3), Block[{$RecursionLimit = rclim}, ParaFuncModPreKernel[game,#, opts]&/@ payoff //Union],
True, PrintRemark[payoff]],
Length[dimpay]===1,
Which[(First[dimpay]===Length[T] && Depth[payoff] === 2 ), Block[{$RecursionLimit = rclim},ParaFuncModPreKernel[game, payoff, opts]],
True, PrintRemark[payoff]],
True, PrintRemark[payoff]
]
];
(* Main Functions *)
ParaFuncModPreKernel[game_, payoff_List, opts:OptionsPattern[ParaModPreKernel]] :=
Module[{sil, smc, optst, pinv, ovls, t0, dcvals, dcgame, doi, optstep, itpay,tol,brc},
sil = OptionValue[Silent];
smc = OptionValue[SmallestCardinality];
optst = OptionValue[CalcStepSize];
pinv = OptionValue[PseudoInv];
ovls = v[#] & /@ Coalitions; (* Storing original game values. *)
t0 = T; (* Storing original game values. *)
dcvals = ParaECCoverGame[game,payoff];
dcgame = DefineGame[t0, dcvals];
{optstep, doi} = ParaDirectionOfImprovement[dcgame, payoff, MaximumSurpluses -> False, CalcStepSize -> optst, PseudoInv->pinv,Silent -> sil, SmallestCardinality -> smc];
If[SameQ[sil,False], Print["doi=", doi], True];
If[SameQ[sil,False], Print["optstep=", optstep], True];
itpay = payoff + optstep*doi;
If[SameQ[sil,False], Print["itpay=", itpay], True];
If[Depth[itpay]!=2,Return[payoff],True];
tol=Table[1.5*10^(-6),{Length[T]}];
brc=Apply[And,MapThread[LessEqual[#1,#2] &,{Abs[doi],tol}]];
DefineGame[t0, ovls];
If[SameQ[brc,True], Rationalize[N[itpay],10^(-6)], ParaFuncModPreKernel[game, itpay, CalcStepSize -> optst, Silent -> sil, SmallestCardinality -> smc]]
];
ParaECCoverGame[game_, payoff_] :=
Module[{exc, mexc, dv, sx, df, assg, dmexc, pvals, dvals, vals},
exc = ParaExcessPayoff[game, payoff];
mexc = Max[exc];
dv = (v[T] - v[#]) & /@ Reverse[Coalitions];
assg = MapThread[Rule, {Map[x, T], payoff}];
sx = x[#] & /@ Coalitions /. assg;
df = dv - sx;
dmexc = Max[df];
pvals = v[#] + mexc + 2*dmexc & /@ Coalitions;
dvals = dv + dmexc + 2*mexc;
vals = MapThread[Max[#1, #2] &, {pvals, dvals}];
vals = Flatten[{0,Drop[vals,1]}];
vals = Drop[vals, -1];
Flatten[{vals, v[T]}]
];
ParaProperModPreKernel[args___]:=(Message[ParaProperModPreKernel::argerr];$Failed);
ParaProperModPreKernel[game_,opts:OptionsPattern[ParaProperModPreKernel]] := Block[{ovls, dcvals, lt, t0, t1, DCGame, mdnc},
ovls = v[#] & /@ Coalitions; (* Storing original game values. *)
t0 = T; (* Storing original game values. *)
dcvals = DualCover[game];
lt = Length[T];
t1 = Range[2*lt];
DCGame = DefineGame[t1, dcvals];
mdnc = ParaPreKernel[DCGame,opts];
(*Print["mdnc0=",mdnc];*)
DefineGame[t0, ovls]; (* Redefine the original game. *)
Take[mdnc, lt]
];
ParaProperModPreKernel[game_,payoff_List,opts:OptionsPattern[ParaProperModPreKernel]] := Block[{ovls, dcvals, lt, t0, t1, DCGame, mdnc,dcpay},
ovls = v[#] & /@ Coalitions; (* Storing original game values. *)
t0 = T; (* Storing original game values. *)
dcvals = ParaDualCover[game];
lt = Length[T];
t1 = Range[2*lt];
dcpay=Flatten[{payoff,payoff}];
DCGame = DefineGame[t1, dcvals];
mdnc = ParaPreKernel[DCGame,dcpay,opts];
(* Print["mdnc=",mdnc];*)
DefineGame[t0, ovls]; (* Redefine the original game. *)
Take[mdnc, lt]
];
ParaIsModPreKernelQ[args___]:=(Message[ParaIsModPreKernelQ::argerr];$Failed);
ParaIsModPreKernelQ[game_, payoff_List] :=
Block[{ovls, dcvals, lt, t0, t1, DCGame, pmpkQ},
ovls = v[#] & /@ Coalitions;(*Storing original game values.*)
t0 = T;(*Storing original game values.*)
dcvals = ParaECCoverGame[game, payoff];
DCGame = DefineGame[t0, dcvals];
pmpkQ = ParaPreKernelQ[DCGame, payoff];
DefineGame[t0, ovls];(*Redefine the original game.*)
Return[pmpkQ]];
ParaIsProperModPreKernelQ[args___]:=(Message[ParaIsProperModPreKernelQ::argerr];$Failed);
ParaIsProperModPreKernelQ[game_,payoff_List] := Block[{ovls, dcvals, lt, t0, t1, dpay, DCGame, pmpkQ},
ovls = v[#] & /@ Coalitions; (* Storing original game values. *)
t0 = T; (* Storing original game values. *)
dcvals = ParaDualCover[game];
lt = Length[T];
t1 = Range[2*lt];
DCGame = DefineGame[t1, dcvals];
dpay= Flatten[{payoff,payoff}];
pmpkQ = ParaPreKernelQ[DCGame,dpay];
DefineGame[t0, ovls]; (* Redefine the original game. *)
Return[pmpkQ]
];
(* Dual Extension of the primal game *)
ParaDualExtension[game_] := Block[{lt, T1, cls, cl1, clset, vlset, dlext,vals},
lt = Length[T];
cls=Subsets[T];
T1 = Range[lt + 1, 2*lt];
cl1 = Subsets[T1];
Parallelize[
clset = Table[Join[cls[[i]], #] & /@ cl1, {i, 1, Length[cl1]}];
vlset = Table[v[cls[[i]]] + v[T] - v[Complement[T, #]] & /@ Coalitions, {i, 1, Length[cl1]}];
dlext = Flatten[MapThread[MapThread[List[#1, #2] &, {#1, #2}] &, {clset, vlset}], 1] // Sort,
Method -> "CoarsestGrained",DistributedContexts -> Automatic];
vals = Last[#] & /@ dlext;
{vals, dlext}
];
(* Primal Extension of the dual game *)
ParaPrimalExtension[game_] :=
Block[{lt, T1, cl1, cls,clset, vlset, plext,vals},
cls=Subsets[T];
lt = Length[T];
T1 = Range[lt + 1, 2*lt];
cl1 = Subsets[T1];
Parallelize[
clset = Table[Join[cls[[i]], #] & /@ cl1, {i, 1, Length[cl1]}];
vlset = Table[v[#] + v[T] - v[Complement[T, cls[[i]]]] & /@ Coalitions, {i, 1, Length[cl1]}];
plext = Flatten[MapThread[MapThread[List[#1, #2] &, {#1, #2}] &, {clset, vlset}], 1] // Sort,
Method -> "CoarsestGrained",DistributedContexts -> Automatic];
vals = Last[#] & /@ plext;
{vals, plext}
];
ParaDualCover[game_] := Block[{dvals, dexts, pvals, pexts},
{dvals, dexts} = ParaDualExtension[game];
{pvals, pexts} = ParaPrimalExtension[game];
MapThread[Max[#1, #2] &, {dvals, pvals}]
];
(* Start of the section related to the simplified modified pre-kernel/nucleolus of a game. *)
ParaSMPreKernel[args___]:=(Message[ParaSMPreKernel::argerr];$Failed);
ParaSMPreKernel[game_] := Block[{ovls, dv, av, AVGame, smpk},
ovls = v[#] & /@ Coalitions;(*Storing original game values.*)
dv = DualGame[game];
av = (ovls + dv)/2;
AVGame = DefineGame[T, av];
smpk = ParaPreKernelElement[AVGame];
DefineGame[T, ovls];(* Redefine the original game.*)
Return[smpk];
];
ParaIsSMPreKernelQ[args___]:=(Message[ParaIsSMPreKernelQ::argerr];$Failed);
ParaIsSMPreKernelQ[game_, payoff_] := Block[{ovls, dv, av, AVGame, smpkQ},
ovls = v[#] & /@ Coalitions;(*Storing original game values.*)
dv = DualGame[game];
av = (ovls + dv)/2;
AVGame = DefineGame[T, av];
smpkQ = ParaPreKernelQ[AVGame, payoff];
DefineGame[T, ovls];(* Redefine the original game.*)
Return[smpkQ];
];
(* End of the section related to the simplified modified pre-kernel of a game. *)
(* Selecting the set of lexicographically smallest coalitions. *)
ParaBestCoalitions[args___]:=(Message[ParaBestCoalitions::argerr];$Failed);
ParaBestCoalitions[game_,payoff_List,opts:OptionsPattern[ParaBestCoalitions]]:=
Block[{dimpay},
dimpay = Dimensions[payoff];
Which[Length[dimpay]===2,
Which[(Last[dimpay]===Length[T] && Depth[payoff] ===3), Map[ParaBestcoalij01[game,#,opts]&, payoff],
True, ParaWrongDimension],
Length[dimpay]===1,
Which[(First[dimpay]===Length[T] && Depth[payoff] === 2 ),ParaBestcoalij01[game,payoff,opts],
True, ParaWrongDimension],
True, ParaWrongDimension
]
];
paralistIJ[T_List]:=Flatten[ParallelTable[Table[{i, j}, {j, i + 1, Length[T]}], {i, 1, Length[T]}]];
ParaBestcoalij01[game_, payoff_List,opts:OptionsPattern[ParaBestCoalitions]] :=
Block[{anti, maxsurp, allc, plvec,sij,sji,plj,pli,payass,amax,ramax,exc,exvec,intcoal,selcij,selcji,sigcoal},
anti = OptionValue[AntiPreKernel];
allc = OptionValue[AllCoalitions];
maxsurp = OptionValue[MaximumSurpluses];
plvec = Partition[paralistIJ[T],2];
Parallelize[pli = Map[First[#] &, plvec];
plj = Map[#[[2]] &,plvec];
sij=MapThread[ParaTIJsets[#1,#2] &,{pli,plj}];
sji=MapThread[ParaTIJsets[#1,#2] &,{plj,pli}];
payass = MapThread[Rule,{x /@ T,payoff}],Method -> "CoarsestGrained",DistributedContexts -> True];
If[SameQ[anti,False],
amax = ParallelTable[ParaMaxSijSurpluses[game,sij[[i]],sji[[i]],payass],{i,Length[sij]},Method -> "CoarsestGrained",DistributedContexts -> True],
amax = ParallelTable[ParaAntiSijSurpluses[game,sij[[i]],sji[[i]],payass],{i,Length[sij]},Method -> "CoarsestGrained",DistributedContexts -> True]
];
exc = First[ParaExcessPayoff[game, payoff]];
exvec = Drop[Drop[exc, 1], -1];
intcoal = Drop[Drop[Subsets[T], 1], -1];
ramax = Map[Reverse[#] &, amax];
selcij = ParallelTable[ParaSelCoal[sij[[i]], intcoal, exvec, amax[[i]]],{i,Length[amax]},Method -> "CoarsestGrained"];
selcji = ParallelTable[ParaSelCoal[sji[[i]], intcoal, exvec, ramax[[i]]],{i,Length[ramax]},Method -> "CoarsestGrained"];
sigcoal = MapThread[{Flatten[#1], Flatten[#2]} &,{selcij, selcji}];
If[SameQ[maxsurp,False], sigcoal,{sigcoal,amax}]
];
ParaSelCoal[setsij_List, coal_List, redexc_List, maxexc_List, opts:OptionsPattern[ParaBestCoalitions]] :=
Block[{allc, smc, detpos, extval, poscoal, extcoal},
allc = OptionValue[AllCoalitions];
smc = OptionValue[SmallestCardinality];
detpos=MapThread[List,{coal,redexc}];
extval = Last[#] &/@ Cases[detpos,{#,___}] &/@ setsij // Flatten;
poscoal = Position[extval, First[maxexc]];
extcoal = Extract[setsij, poscoal];
(* Taking the coalition with smallest/largest (First/Last) cardinality if extcoal > 1 *)
If[Length[extcoal] === 1, extcoal,
Which[allc === True, extcoal, True,
Which[smc === True, First[extcoal],
smc === False, Last[extcoal],
True, First[extcoal]]]]
];
ParaSetsToVec[args___]:=(Message[ParaSetsToVec::argerr];$Failed);
ParaSetsToVec[mg_List, T_List, opts:OptionsPattern[ParaSetsToVec]] :=
Block[{effvec, zrv, pscoal, replzr,coasts, onesoft},
effvec = OptionValue[EffVector];
zrv = Table[0, {i, Length[T]}];
pscoal = Map[Outer[List, #] &, mg];
replzr = Parallelize[Map[ReplacePart[zrv, 1, #] &, #] &/@ pscoal];
coasts = Parallelize[MapThread[Subtract[#1, #2] &, #] &/@ replzr];
onesoft = Table[1,{i,Length[T]}];
If[effvec==True, Prepend[coasts,onesoft], coasts]
];
ParaMaxSurpluses[game_, payoff_List,dir_List] :=
Block[{pli,plj,maxpi,maxpj,res},
Parallelize[pli = First[#] &/@ dir;
plj = #[[2]] &/@ dir;
maxpi = MapThread[ParaMaxSurplus[game,#1,#2,payoff]&,{pli,plj}];
maxpj = MapThread[ParaMaxSurplus[game,#1,#2,payoff]&,{plj,pli}];
MapThread[List,{maxpi,maxpj}],Method -> "CoarsestGrained",DistributedContexts -> None]
];
(* We refrain from overloading due to its negative effect on the performance of the (Anti-)Pre-Kernel computation!!! *)
ParaMaxSijSurpluses[game_, sij_List, sji_List,payoff_List] :=
Block[{maxpi,maxpj},
maxpi = ParaMaxSijSurplus[game,sij,payoff];
maxpj = ParaMaxSijSurplus[game,sji,payoff];
Return[{maxpi,maxpj}]
];
ParaAntiSurpluses[game_, payoff_List,dir_List] :=
Block[{pli,plj,minpi,minpj,res},
Parallelize[pli = First[#] &/@ dir;
plj = #[[2]] &/@ dir;
minpi = MapThread[ParaAntiSurplus[game,#1,#2,payoff]&,{pli,plj}];
minpj = MapThread[ParaAntiSurplus[game,#1,#2,payoff]&,{plj,pli}];
MapThread[List,{minpi,minpj}],Method -> "CoarsestGrained",DistributedContexts -> None]
];
ParaAntiSijSurpluses[game_, sij_List, sji_List,payoff_List] :=
Block[{minpi,minpj},
minpi = ParaAntiSijSurplus[game,sij,payoff];
minpj = ParaAntiSijSurplus[game,sji,payoff];
Return[{minpi,minpj}]
];
ParaMaxSurplus[args___]:=(Message[ParaMaxSurplus::argerr];$Failed);
ParaMaxSurplus[game_, pi_, pj_, payoff_List] :=
Block[{payass},
payass = Which[Depth[payoff]==3, MapThread[Rule,{x /@ T,#}]& /@ payoff,
Depth[payoff]==2, MapThread[Rule,{x /@ T,payoff}],
True, Print["The input 'payoff' is not a list."];Return[]];
Which[Depth[payass] == 5,Max[ReplaceAll[(v[#] - x[#]) & /@ ParaTIJsets[pi,pj],#]] &/@ payass,
Depth[payass] == 4,Max[ReplaceAll[(v[#] - x[#]) & /@ ParaTIJsets[pi,pj],payass]],
True, Print["Wrong data format."];Return[]]
];
ParaMaxSijSurplus[game_,sij_List, payass_List] :=
Block[{z0},
Max[ReplaceAll[Map[v[#] - x[#] &, sij],payass]]
];
ParaAntiSurplus[args___]:=(Message[ParaMaxSurplus::argerr];$Failed);
ParaAntiSurplus[game_, pi_, pj_, payoff_List] :=
Block[{payass},
payass = Which[Depth[payoff]==3, MapThread[Rule,{x /@ T,#}]& /@ payoff,
Depth[payoff]==2, MapThread[Rule,{x /@ T,payoff}],
True, Print["The input 'payoff' is not a list."];Return[]];
Which[Depth[payass] == 5,Min[ReplaceAll[(v[#] - x[#]) & /@ ParaTIJsets[pi,pj],#]] &/@ payass,
Depth[payass] == 4,Min[ReplaceAll[(v[#] - x[#]) & /@ ParaTIJsets[pi,pj],payass]],
True, Print["Wrong data format."];Return[]]
];
ParaAntiSijSurplus[game_,sij_List, payass_List] :=
Block[{z0},
Min[ReplaceAll[Map[v[#] - x[#] &, sij],payass]]
];
ParaTIJsets[i_Integer, j_Integer]:=DeleteCases[Cases[ProperCoalitions,{___,i,___}],{___,j,___}];
ParaExcessPayoff[args___]:=(Message[ParaExcessPayoff::argerr];$Failed);
ParaExcessPayoff[game_,payoff_List, opts:OptionsPattern[ParaExcessPayoff]]:= Block[{dispmat,assg,li,res},
dispmat = OptionValue[DisplayMatrixForm];
If[Depth[payoff] == 2 || Depth[payoff] == 3,
li = If[Length[Dimensions[payoff]]==1,{payoff},payoff];
Parallelize[assg = MapThread[Rule,{Map[x,T],#}] & /@ li;
res = (v[#]-x[#])& /@ Coalitions;
res = ReplaceAll[res,#]&/@ assg,Method -> "CoarsestGrained",DistributedContexts -> Automatic];
Which[dispmat == False, res, True, ParaDisplayErgb[res]],
ParaPrintRemark[payoff]
]
];
ParaDisplayErgb[payoff_List]:= Block[{exc,coal,mpc},
exc = payoff;
coal = Subsets[T];
mpc = Map[Global`Co,coal];
MatrixForm[PrependTo[exc,mpc]]
];
(* User interface to check for (anti) pre-kernel element. *)
ParaPreKernelQ[args___]:=(Message[ParaPreKernelQ::argerr];$Failed);
ParaPreKernelQ[game_, payoff_List,opts:OptionsPattern[ParaPreKernelQ]] :=Block[{rattol,tolv,graval,dimpay},
rattol = OptionValue[RationalTol];
graval = v[T];
dimpay = Dimensions[payoff];
tolv=1.5*rattol;
Which[Length[dimpay] === 2,
Which[ (Last[dimpay]===Length[T] && Depth[payoff] ===3),MapThread[And,{(Abs[Total[#] - graval]<=tolv) & /@ payoff,ParaMaxExcessBalanced[game, payoff,RationalTol->rattol]}],
True, ParaPrintRemark[payoff]],
Length[dimpay] === 1,
Which[(First[dimpay]===Length[T] && Depth[payoff] === 2 ), MapThread[And,{{Abs[Total[payoff] - graval]<=tolv},{ParaMaxExcessBalanced[game, payoff,RationalTol->rattol]}}],
True, ParaPrintRemark[payoff]],
True, ParaPrintRemark[payoff]
]
];
ParaAntiPreKernelQ[args___]:=(Message[ParaAntiPreKernelQ::argerr];$Failed);
ParaAntiPreKernelQ[game_, payoff_List,opts:OptionsPattern[ParaAntiPreKernelQ]] :=Block[{rattol,tolv,graval,dimpay},
rattol = OptionValue[RationalTol];
graval = v[T];
dimpay = Dimensions[payoff];
tolv=1.5*rattol;
Which[Length[dimpay] === 2,
Which[ (Last[dimpay]===Length[T] && Depth[payoff] ===3), MapThread[And,{(Abs[Total[#] - graval]<= tolv) & /@ payoff,ParaMinExcessBalanced[game, payoff,RationalTol->rattol]}],
True, ParaPrintRemark[payoff]],
Length[dimpay] === 1,
Which[(First[dimpay]===Length[T] && Depth[payoff] === 2 ), MapThread[And,{{Abs[Total[payoff] - graval]<=tolv},{ParaMinExcessBalanced[game, payoff,RationalTol->rattol]}}],
True, ParaPrintRemark[payoff]],
True, ParaPrintRemark[payoff]
]
];
ParaMaxExcessBalanced[args___]:=(Message[ParaMaxExcessBalanced::argerr];$Failed);
ParaMaxExcessBalanced[game_, payoff_List,opts:OptionsPattern[ParaMaxExcessBalanced]]:= Block[{rattol,dimpay},
rattol = OptionValue[RationalTol];
dimpay = Dimensions[payoff];
Which[Length[dimpay] === 2,
Which[(Last[dimpay]===Length[T] && Depth[payoff] ===3), ParaMaxExcessBalCheck[game, #,RationalTol->rattol] & /@ payoff,
True, ParaPrintRemark[payoff]],
Length[dimpay] === 1,
Which[(First[dimpay]===Length[T] && Depth[payoff] === 2 ), ParaMaxExcessBalCheck[game, payoff,RationalTol->rattol],
True, ParaPrintRemark[payoff]],
True, ParaPrintRemark[payoff]]
];
ParaMaxExcessBalCheck[game_,payoff_List,opts:OptionsPattern[ParaMaxExcessBalanced]]:=
Block[{rattol,plpr,rvpr,asspay,sij,sji,msrplij,msrplji,msrij,msrji,lthij,tolvec,sysij,sysji,eqQ},
rattol = OptionValue[RationalTol];
plpr = Partition[paralistIJ[T],2];
rvpr = Map[Reverse[#] &, plpr];
asspay = ParaAssgPay[payoff];
Parallelize[
sij = ParaTIJsets[#[[1]], #[[2]]] & /@ plpr;
sji = ParaTIJsets[#[[1]], #[[2]]] & /@ rvpr,
Method -> "CoarsestGrained",DistributedContexts -> Automatic];
{msrplij,msrplji}= {ParaMaxExcess[sij, asspay],ParaMaxExcess[sji, asspay]};
{msrij,msrji} = {msrplij - msrplji,msrplji - msrplij};
lthij = Binomial[Length[T],2];
tolvec = Table[1.5*rattol, {i, lthij}];
sysij = Union[MapThread[LessEqual, {Abs[msrij], tolvec}]];
sysji = Union[MapThread[LessEqual, {Abs[msrji], tolvec}]];
eqQ = Apply[Join, {sysij, sysji}];
Apply[And, eqQ]
];
ParaMinExcessBalanced[args___]:=(Message[ParaMinExcessBalanced::argerr];$Failed);
ParaMinExcessBalanced[game_, payoff_List,opts:OptionsPattern[ParaMinExcessBalanced]]:= Block[{rattol,dimpay},
rattol = OptionValue[RationalTol];
dimpay = Dimensions[payoff];
Which[Length[dimpay] === 2,
Which[(Last[dimpay]===Length[T] && Depth[payoff] ===3), ParaMinExcessBalCheck[game, #,RationalTol->rattol] & /@ payoff,
True, ParaPrintRemark[payoff]],
Length[dimpay] === 1,
Which[(First[dimpay]===Length[T] && Depth[payoff] === 2 ), ParaMinExcessBalCheck[game, payoff,RationalTol->rattol],
True, ParaPrintRemark[payoff]],
True, ParaPrintRemark[payoff]]
];
ParaMinExcessBalCheck[game_,payoff_List,opts:OptionsPattern[ParaMaxExcessBalanced]]:=
Block[{rattol,plpr, rvpr, asspay,sij,sji,msrplij,msrplji,msrij, msrji,lthij,tolvec,sysij,sysji,eqQ},
rattol = OptionValue[RationalTol];
plpr = Partition[paralistIJ[T],2];
rvpr = Map[Reverse[#] &,plpr];
asspay = ParaAssgPay[payoff];
Parallelize[
sij = ParaTIJsets[#[[1]], #[[2]]] & /@ plpr;
sji = ParaTIJsets[#[[1]], #[[2]]] & /@ rvpr,
Method -> "CoarsestGrained",DistributedContexts -> Automatic];
{msrplij,msrplji} = {ParaMinExcess[sij, asspay],ParaMinExcess[sji, asspay]};
{msrij,msrji} = {msrplij - msrplji,msrplji - msrplij};
lthij = Binomial[Length[T],2];
tolvec = ParallelTable[1.5*rattol, {i, lthij}];
sysij = Union[MapThread[LessEqual, {Abs[msrij], tolvec}]];
sysji = Union[MapThread[LessEqual, {Abs[msrji], tolvec}]];
eqQ = Apply[Join, {sysij, sysji}];
Apply[And, eqQ]
];
ParaAssgPay[payoff_List] := Block[{vars},
vars = x[#] & /@ T;
MapThread[Rule, {vars, payoff}]
];
ParaMaxExcess[mgij_List, asspay_List] := ParallelMap[ParaMaxExc[#, asspay] &, mgij,Method -> "CoarsestGrained",DistributedContexts -> Automatic];
ParaMinExcess[mgij_List, asspay_List] := ParallelMap[ParaMinExc[#, asspay] &, mgij,Method -> "CoarsestGrained",DistributedContexts -> Automatic];
ParaMaxExc[mg_List, asspay_List] := Max[ReplaceAll[(v[#] - x[#]) & /@ mg, asspay]];
ParaMinExc[mg_List, asspay_List] := Min[ReplaceAll[(v[#] - x[#]) & /@ mg, asspay]];
(* Deriving a game from unanimity coordinates. *)
ParaCharacteristicValues[args___]:=(Message[ParaCharacteristicValues::argerr];$Failed);
ParaCharacteristicValues[coord_List,T_,opts:OptionsPattern[]]:= Block[{z0},
Which[ Length[coord] === 2^Length[T] , ParaDetWorth[coord,T] ,
True, ParaWrongCoordDimension[coord, T]]
];
ParaDetWorth[coord_List, T_, opts:OptionsPattern[]]:=Block[{tugb, cval},
tugb = ParaGameBasis[T];
cval = tugb.Drop[coord,1];
Prepend[cval,0]
];
ParaGameBasis[args___]:=(Message[ParaGameBasis::argerr];$Failed);
ParaGameBasis[T_] := Block[{mgsys,gb},
mgsys = Drop[Subsets[T], 1];
gb = ParallelTable[If[SubsetQ[#,mgsys[[i]]], 1, 0] & /@ mgsys, {i, Length[mgsys]}];
Transpose[gb]
];
(* Checking convexity and average-convexity *)
ParaConvexQ[args___]:=(Message[ParaConvexQ::argerr];$Failed);
ParaConvexQ[game_]:=
Block[{liste},
liste = Flatten[ParaIncreasingMargContributions[game,#] & /@ T,1];
Apply[And,Apply[And,liste,1]]
];
ParaIncreasingMargContributions[game_,i_Integer]:=
ParallelTable[(v[#]-v[DeleteCases[#,i]] <=
v[Union[Flatten[{#,j}]]] - v[DeleteCases[
Union[Flatten[{#,j}]],i]])& /@ Take[ParaW[i],Length[ParaW[i]]-1],
{j,1,Length[T]},Method -> "CoarsestGrained",DistributedContexts -> Automatic];
ParaW[i_Integer]:= Cases[Coalitions,{___,i,___}];
ParaAvConvexQ[args___]:=(Message[ParaAvConvexQ::argerr];$Failed);
ParaAvConvexQ[game_] := Block[{dispre, pwset, chsum,delmp},
pwset = Drop[Subsets[T],1];
DistributeDefinitions[pwset];
chsum = ParallelMap[ParaCheckSumQ[#, T] &, pwset,Method -> "CoarsestGrained"];
delmp = DeleteCases[chsum, {{}, {}}];
Apply[And, Union[chsum]]
];
ParaCheckSumQ[teilmg_List, T_] :=
Block[{supset, smarg, dispres},
supset = ParaOberMenge[teilmg, T];
smarg = ParaSumMargContribution[#, teilmg] & /@ supset;
Apply[And, Union[smarg]]
];
ParaOberMenge[teilmg_List, T_List] := Block[{compl,subc,prop},
compl = Complement[T, teilmg];