Contents:
1. Introduction
2. Getting Started
3. Custom Installation
4. Requirements
5. Acknowledgment
6. License
7. Citation
8. MATLAB File Exchange
The game theoretical Matlab toolbox MatTuGames provides about 500 functions for modeling, and calculating some solutions as well as properties of cooperative games with transferable utilities. In contrast to existing Matlab toolboxes to investigate TU-games, which are written in a C/C programming style with the consequence that these functions are executed relatively slowly, we heavily relied on vectorized constructs in order to write more efficient Matlab functions. In particular, the toolbox provides functions to compute the (pre-)kernel, (pre-)nucleolus, anti (pre-)kernel, and modiclus as well as game values like the Banzhaf, Myerson, Owen, position, Shapley, solidarity, and coalition solidarity value and much more. In addition, we will discuss how one can use Matlab's Parallel Computing Toolbox in connection with this toolbox to benefit from a gain in performance by launching supplementary Matlab workers. Some information are provided how to call our Mathematica package TuGames within a running Matlab session.
In order to get some insight how to analyze a cooperative game, a so-called transferable utility game, with the Game Theory Toolbox MatTuGames, we discuss a small example to demonstrate of how one can compute some game properties or solution concepts, like convexity, the Shapley value, the (pre-)nucleolus or a pre-kernel element.
For this purpose, consider a situation where an estate is insufficient to meet simultaneously all of the debts/claims of a set of claimants, such a situation is known in game theory as a bankruptcy problem. The problem is now to find a fair/stable distribution in the sense that no claimant/creditor can find an argument to obstruct the proposed division to satisfy at least partly the mutual inconsistent claims of the creditors. In a first step, we define a bankruptcy situation while specifying the debts vector and the estate that can be distributed to the creditors. We restrict our example to a six-person bankruptcy problem with a debts vector given by
>> d = [40.0000 32.0000 11.0000 73.3000 54.9500 81.1000];
and an estate value which is equal to
>> E = 176;
We immediately observe that the estate E is insufficient to meet all
of the claims simultaneously. It should be obvious that with these values
we do not have defined a cooperative game, however, these information
are enough to compute a proposal how to divide the estate between the
creditors. A fair division rule which is proposed by the Babylonian Talmud,
is given by
>> tlm_rl=Talmudic_Rule(E,d)
>>
tlm_rl =
20.0000 16.0000 5.5000 48.3500 30.0000 56.1500
However, this distribution rule does not incorporate the coalition formation process. Thus, we might get a different outcome when we consider the possibility that agents can form coalitions to better enforce their claims. This means, we have to study the corresponding cooperative game. This can be constructed while calling the following function
>> bv=bankruptcy_game(E,d);
Having generated a game, we can check some game properties like convexity
>> cvQ=convex_gameQ(bv)
cvQ =
logical
1
The returned logical value indicates that this game is indeed convex. This must be the case for bankruptcy games. In addition, we can also verify if the core of the game is non-empty or empty. To see this one needs just to invoke
>> crQ=coreQ(bv)
crQ =
logical
1
which is answered by affirmation. This result confirms our expectation, since each convex game has a non-empty core.
After this short introduction of game properties, we turn our attention now to some well known solution concepts from game theory. We start with the Shapley value, which can be computed by
>> sh_v=ShapleyValue(bv)
sh_v =
23.5175 18.7483 6.4950 44.3008 33.3317 49.6067
A pre-kernel element can be computed with the function
>> prk_v=PreKernel(bv)
prk_v =
20.0000 16.0000 5.5000 48.3500 30.0000 56.1500
which must be identical to the distributional law of justice proposed by the Talmudic rule. Moreover, it must also coincides with the nucleolus due to the convexity of the game. To see this, let us compute first the nucleolus and in the next step the pre-nucleolus
>> nc_bv=nucl(bv)
nc_bv =
20.0000 16.0000 5.5000 48.3500 30.0000 56.1500
>> pn_bv=PreNucl(bv)
pn_bv =
20.0000 16.0000 5.5000 48.3500 30.0000 56.1500
We observe that both solutions coincide, which must be the case for zero-monotonic games. To check that these solutions are indeed the pre-nucleolus can be verified by Kohlberg's criterion
>> balancedCollectionQ(bv,pn_bv)
ans =
logical
1
>> balancedCollectionQ(bv,nc_bv)
ans =
logical
1
In order to verify that the solution found is really a pre-kernel element can be done while typing
>> prkQ=PrekernelQ(bv,prk_v)
prkQ =
logical
1
Furthermore, with the toolbox we can also compute the modiclus of the game, which takes apart from the primal power also the preventive power of coalitions into account.
>> mnc_bv=Modiclus(bv)
mnc_bv =
22.5067 17.7567 7.4533 41.8600 37.1100 49.3133
Checking this solution can be established while invoking a modified Kohlberg criterion.
>> modiclusQ(bv,mnc_bv)
ans =
logical
1
The return value is a logical one, hence the solution is the modiclus. For bankruptcy game we can rely on the computation of the anti pre-nucleolus as a simple cross-check to figure out that the solution is correct (cf. Meinhardt 2018c).
>> apn_bv=Anti_PreNucl(bv)
apn_bv =
22.5067 17.7567 7.4533 41.8600 37.1100 49.3133
We observe that both solutions coincide, hence this gives additional evidence that the computation of the modiclus was correct. Moreover, for the class of convex games the modiclus must belong to the core, which can be examined through
>> belongToCoreQ(bv,mnc_bv)
ans =
logical
1
However, if this should still not be enough evidence, then we can referring to the axiomatization of the modiclus, which is characterized by SIVA, COV, EC, LEDCONS, and DCP, whereas DCP can also be replaced by DRP (cf. Meinhardt 2018c).
Apart of SIVA (Single Valuedness), the toolbox can examine COV
>> COV_mnc_bv=COV_propertyQ(bv,mnc_bv,'','','MODIC')
COV_mnc_bv =
struct with fields:
covQ: 1
sol_v2: [23.5067 18.7567 8.4533 42.8600 38.1100 50.3133]
sgm: [23.5067 18.7567 8.4533 42.8600 38.1100 50.3133]
v2: [1x63 double]
x: [22.5067 17.7567 7.4533 41.8600 37.1100 49.3133]
as well as EC
>> ECQ_mnc_bv=EC_propertyQ(bv,mnc_bv,'MODIC')
ECQ_mnc_bv =
struct with fields:
propQ: 1
y: [22.5067 17.7567 7.4533 41.8600 37.1100 49.3133]
x: [22.5067 17.7567 7.4533 41.8600 37.1100 49.3133]
and LEDCONS
>> [LEDC_mnc_bv, LEDCGPQ_mnc_bv]=Ledcons_propertyQ(bv,mnc_bv,'MODIC')
LEDC_mnc_bv =
struct with fields:
ledconsQ: 1
rgpq: [1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]
ledpropQ: [1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]
LEDCGPQ_mnc_bv =
1x4 cell array
{'vS'} {2x62 cell} {'impVec'} {1x63 cell}
to finally check DCP
>> DCP_mnc_bv=DCP_propertyQ(bv,mnc_bv,'MODIC')
DCP_mnc_bv =
struct with fields:
propQ: 1
xQ: 1
y: [22.5067 17.7567 7.4533 41.8600 37.1100 49.3133 22.5067 17.7567 7.4533 41.8600 37.1100 49.3133]
x: [22.5067 17.7567 7.4533 41.8600 37.1100 49.3133]
and DRP
>> DRP_mnc_bv=DRP_propertyQ(bv,mnc_bv,'MODIC')
DRP_mnc_bv =
struct with fields:
propQ: 1
xQ: 1
y: [22.5067 17.7567 7.4533 41.8600 37.1100 49.3133 22.5067 17.7567 7.4533 41.8600 37.1100 49.3133]
x: [22.5067 17.7567 7.4533 41.8600 37.1100 49.3133]
By this example, we observed that the axiomatization of the modiclus was satisfied, from we which can conclude that the modiclus of the game was found by this evaluation. Of course, the toolbox offers in addition routines to examine the axiomatization of the pre-nucleolus, pre-kernel, anti pre-nucleolus, anti pre-kernel, modified as well as proper modified pre-kernel, and Shapley value.
Moreover, the toolbox offers to the user the possibility to create several game class objects to perform several computations for retrieving and modifying game data with the intention to ensure a consistent computation environment. Hence, these classes should avoid that some data from a different game are used or that game data are unintentionally changed, which allow the user to concentrate on the crucial aspects of analyzing the game instead of dealing with the issue of supplying the correct game data. Such a class is, for instance TuSol, which executes several computations in serial for retrieving and storing game solutions. A class object, let us call it sclv, is created by calling TuSol with at least one argument, that is, the values of the characteristic function. The other two input arguments can be left out. However, if they are supplied, then the second specifies the game type, for instance cv for the class of convex games. Whereas the last argument specifies the game format, which is for the discussed example mattug to indicate that the coalitions are ordered in accordance with their unique integer representation to carry out some computation under MatTuGames.
>> scl_bv=TuSol(bv,'cv','mattug');
Having created the class object scl_bv, one can invoke a computation of getting results for some selected solution concepts while executing
>> scl_bv.setAllSolutions
ans =
TuSol with properties:
tu_prk: [20.0000 16.0000 5.5000 48.3500 30.0000 56.1500]
tu_prn: [20.0000 16.0000 5.5000 48.3500 30.0000 56.1500]
tu_prk2: []
tu_sh: [23.5175 18.7483 6.4950 44.3008 33.3317 49.6067]
tu_tauv: [24.0807 19.2646 6.6222 44.1279 33.0809 48.8237]
tu_bzf: []
tu_aprk: [22.6550 17.9050 7.3050 41.8600 37.1100 49.1650]
prk_valid: 1
prn_valid: 1
prk2_valid: 0
aprk_valid: 1
tuvalues: [1x63 double]
tusize: 63
tuplayers: 6
tutype: 'cv'
tuessQ: 1
tuformat: 'mattug'
tumv: 176.0000
tumnQ: 0
tuSi: [62 61 59 55 47 31]
tuvi: [0 0 0 0 0 0]
tustpt: []
which stores apart of the solution concepts also some important data of the game. This class object can then be used, for instance, to determine the modified pre-kernel of the underlying game
>> mpk_bv=scl_bv.ModPreKernel
mpk_bv =
22.6550 17.9050 7.3050 41.8600 37.1100 49.1650
>> mpkQ_bv=scl_bv.ModPrekernelQ(mpk_bv)
mpkQ_bv =
logical
1
or the proper modified pre-kernel through
>> pmpk_bv=scl_bv.PModPreKernel
pmpk_bv =
22.2100 17.4600 7.7500 41.8600 37.1100 49.6100
>> pmpkQ_bv=scl_bv.PModPrekernelQ(pmpk_bv)
pmpkQ_bv =
logical
1
or much more. For a deeper discussion of the function set provided by the toolbox consult the Manual or type help mat_tug to get a short overview.
To install the toolbox, we recommend a custom installation. Having downloaded the .mltbx file, navigate to it within the Matlab file explorer, double click on the mltbx file mat_tugV1d9.mltbx and click "install". Alternatively, right click on the .mltbx, and click "Install."
Additional instructions can be found at the URL:
The mltbx file mat_tugV1d9.mltbx is provided at
This release of MatTuGames was developed and tested using Matlab R2021a and earlier releases. A set of functions use the Optimization Toolbox and the cdd-library by Komei Fukuda, which can be found at the URL:
as well as the Matlab interface to the cdd solver CDDMEX:
Alternatively, in order to get even full scope of operation of the graphical features, one can also install the MPT3 toolbox that can be downloaded from
which ships with CDDMEX. We strongly recommend the user to apply the MPT3 toolbox, in particular of using the graphical features of our toolbox.
For the computation of the pre-kernel and related solutions the SuiteSparse for Matlab is recommend that can be got from the URL
If you do not want to use SuiteSparse, then replace the function qr_dec by pinv in all functions
for the pre-kernel and related solutions. The same argument applies for the function qrginv. It should be noted that this may cause accuracy issues with the consequence that the result is incorrect.
To run the toolbox even in parallel mode, Matlab's Parallel Computing Toolbox is needed.
For connecting the Mathematica Package TuGames, the Mathematica Symbolic Toolbox is required, which can be found under the URL:
whereas TuGames Version 2.6.1 can be downloaded from the URL:
We recommend a custom installation with paclet, which can be found at
The MatTuGames toolbox should work with all platforms.
Moreover, the toolbox works also with the game theory toolbox written by Jean Derks, which can be requested from:
We added some adjusted files that fix a problem with closed loops under certain game classes.
This toolbox can be used to compute the pre-nucleolus up to 10-persons, if one has no license of Matlab's optimization toolbox.
Finally, the toolbox MatTuGames offers interfaces to access the solvers of CVX, CPLEX, GLPK, GUROBI, HSL, IPOPT, MOSEK, and OASES. The CPLEX interfaces are compatible with version 12.10.
To summarize, apart of the mentioned software, the toolbox requires the following MATLAB toolboxes:
MATLAB Parallel Server, Optimization Toolbox, Parallel Computing Toolbox, Signal Processing Toolbox, Statistics and Machine Learning Toolbox, Symbolic Math Toolbox
to get full functionality in serial as well as in parallel.
The author acknowledges support by the state of Baden-Württemberg through bwHPC.
Of course, the usual disclaimer applies.
This project is licensed under the FreeBSD License - see the LICENSE file.
For citation consult the URL:
For additional comments see also the retro site
For additional information of the current version consult
Holger I. Meinhardt Institute of Operations Research University of Karlsruhe (KIT) E-mail: Holger.Meinhardt ät wiwi.uni-karlsruhe.de holger.meinhardt ät partner.kit.edu