Modification of shapley value and its implementation in decision making

Zaremba, Leszek; Zaremba, Cezary S.; Suchenek, Marek Article Modification of shapley value and its implementation in decision making Foundations of Management Provided in Cooperation with: Faculty of Management, Warsaw University of Technology Suggested Citation: Zaremba, Leszek; Zaremba, Cezary S.; Suchenek, Marek (2017) : Modification of shapley value and its implementation in decision making, Foundations of Management, ISSN 2300-5661, De Gruyter, Warsaw, Vol. 9, Iss. 1, pp. 257-222, https://doi.org/10.1515/fman-2017-0020 This Version is available at: https://hdl.handle.net/10419/184629 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/4.0 Foundations of Management, Vol. 9 (2017), ISSN 2080-7279 DOI: 10.1515/fman-2017-0020 257 MODIFICATION OF SHAPLEY VALUE AND ITS IMPLEMENTATION IN DECISION MAKING Leszek ZAREMBA*, Cezary S. ZAREMBA**, Marek SUCHENEK*** *Academy of Finance and Business Vistula, Institute of Management, Warszawa, Poland e-mail: l.zarem[email protected] ** Peaceful Games Cezary Zaremba, Warszawa, Poland e-mail: peacefulgam[email protected] *** California State Univ. Dominquez Hills, Computer Science Department, Carson, Calif. USA e-mail: [email protected] Abstract: The article presents a solution of a problem that is critical from a practical point of view: how to share a higher than usual discount of $10 million among 5 importers. The discount is a result of forming a coalition by 5 current, formerly competing, importers. The use of Shapley value as a concept for co-operative games yielded a solution that was satisfactory for 4 lesser importers and not satisfactory for the biggest importer. Appropriate modification of Shapley value presented in this article allowed to identify appropriate distribution of the saved purchase amount, which according to each player accurately reflects their actual strength and position on the importer market. A computer program was used in order to make appropriate calculations for 325 permutations of all possible coalitions. In the last chapter of this paper, we recognize the lasting contributions of Lloyd Shapley to the cooperative game theory, commemorating his recent (March 12, 2016) descent from this world. Keywords: characteristic function, coalition, cooperative game, core of game, imputation, Shapley value. JEL: C71, F51, F12. 1 Introduction We consider the decision problem that the importers of raw materials face while minimizing their costs by means of cooperation with their competitors. It has been communicated to the first author by an employee of a Polish consulting company. The solution of the problem that we present here enables the importers to make substantial savings commensurate with their actual contributions to the cooperating group. It turns out that the proposed solution based on the original concept of the Shapley value for cooperative games has one weakness  it discriminates the biggest importer who did not accept that solution. Our approach eliminates this weakness. The problem we solve in this article has both practical and theoretical ramifications. In particular, it is satisfactory for all the grand coalition members, and not just to some or majority of them. Our improvement is based on appropriate modification of the Shapley value concept for cooperative games (see Comment 1); it was carried out by a computer program used to perform the necessary calculations. Politics of collective rationality, to which this article is related, has been discussed at several occasions in the economic literature of the subject (cf. Weirich, 2012) where cooperation and rationalism in the language of game theory has been broadly investigated. Some authors pointed out (e.g., Ott, 2016) the problems arising from conflict solving and competitor cooperation by means of strategic alliances and joint decision making. A popular mathematical model used in problem solving that incorporates cooperative games has been presented in Peleq (2007). Lloyd Shapley (1923 2016) played a major role in the development of those games. He was awarded a Nobel Prize in 2012 in Economics together with Alvin Roth. A fairly thorough account of Shapley’s achievements may be found in Roth (2005), where lasting contribution of Shapley to cooperative game theory was recognized in a form of 20 papers written by 24 authors, who either reviewed or continued the research 258 Leszek Zaremba, Cezary S. Zaremba, Marek Suchenek initiated by Shapley’s five remarkable 1953 papers listed in the bibliography. A shorter presentation of Shapley’s role in the development of cooperative game theory is given in the last chapter of this article. Another type of antagonistic games constitutes 2-person, zero-sum games. They may be either static in their nature or dynamic. The last category includes 2-person, zero-sum games where the actions of players are governed by differential equations; see Zaremba (1979,1980, 1982, 1984a, 1984b, 1986, 1989). In such games, there is no place for cooperation, however. Coming back to our main topic, we assume that the amount of discount offered to each importer by the exporter depends on the total value of the import according to the Table 1. Table 1. Discount correlation to the import value in millions of PLN Company size Bottom limit of the import value Percentage of discount very large 700 4 large 300 3 medium 125 2 small 50 1 very small 0 0 The information contained in Table 1 can also be presented graphically, as shown in Fig.1. Figure 1. Discount function in millions of PLN The solution method presented below does not depend on numeric parameters shown in the table above; it may be straightforwardly implemented for any other table presenting the correlation between the discount and the amount of import amount. We assume that there are a number of importers of the raw materials operating on the market, including the existing competitors A, B, C, D and E, whose annual expenditure on purchasing the raw material are shown in the first version in the Table 2a. Table 2a. The amounts of import for particular companies in millions of PLN A B C D E 310 150 130 100 60 We also consider a version Table 2b of the above scenario in order to illustrate the influence of slightly different import quotas on final solution. Modification of Shapley Value and its Implementation in Decision Making 259 Table 2b. The amounts of import for particular companies in millions of PLN A B C D E 290 150 130 100 60 A consulting company that provided services to A, B, C, D and E importers recommended that they combine their forces and act as one large importer. This should enable them to obtain the biggest 4% discount as they are going to import raw material totaling 750 million PLN (Table 2a) or 730 million PLN (Table 2b). However, this recommendation gave rise to a complicated problem: how to divide the 4% discount as each of the 5 importers had a different idea of how to share the saved amount of purchase. The said problem may be formulated as follows: Problem 1: Propose a formula, or a methodology, for the division of saved money by a coalition of 5 importers resulting from obtaining a 4% discount among the coalition members, so that each of them perceives the share he cashes is reflecting adequately their contribution to the coalition and position on the importer market. Definition 1: The characteristic function of an nperson cooperative game with a set C of players (referred to as a grand coalition in this paper) assigns to each subset S of C the maximum value v(S) that coalition S can guarantee itself by coordinating the strategies of its members, no matter what the other players do (Thomas 1986, p.86). Cooperation Games (cf. Malawski, Wieczorek, Sosnowska, 2006, pp.127-150) provide a mathematical model for solving this kind of decision problems. Our approach will benefit from this formalism with discounts playing the role of payouts v(S) for all possible coalitions S. Discounts that each coalition is entitled to are presented in Tables 3a and 3b in accordance with the data presented in Table 1 and Tables 2a, 2b, respectively. In this paper, we are also going to use two classic concepts that are useful while solving the cooperative games, that is, a core, first defined explicitly by (Gillies, 1959), and the Shapley value, which always exists, is unique and belongs to the core of the game; see Shapley (1953a, 1953d) and for example, Malawski, Wieczorek, Sosnowska (2006, pp.134-136). In the next paragraph, we are going to present a solution to the problem raised by the consulting company by calculating the Shapley value for this cooperative game. Table 3a. Characteristic function v(S) in case of Table 2a in millions of PLN coalition discount v(S) coalition S discount v(S) coalition S discount v(S) A, B, C, D, E 30 A, B, E 15.6 B, C 5.6 A, B, C, D 20.7 A, C, E 15 B, D 5 A, B, C, E 19.5 B, C, E 10.2 B, E 4.2 A, B, D, E 18.6 A, D, E 14.1 C, D 4.6 A, C, D, E 18 B, D, E 9.3 C, E 3.8 B, C, D, E 13.2 C, D, E 5.8 D, E 3.2 A, B, C 17.7 A, B 13.8 A 9.3 A, B, D 16.8 A, C 13.2 B 3 A, C, D 16.2 A, D 12.3 C 2.6 B, C, D 11.4 A, E 11.1 D 1 E 0.6 260 Leszek Zaremba, Cezary S. Zaremba, Marek Suchenek Table 3b. Characteristic function v(S) in case of Table 2b in millions of PLN coalition discount v(S) coalition discount v(S) coalition discount v(S) A, B, C, D, E 29.2 A, B, E 15 B, C 5.6 A, B, C, D 20.1 A, C, E 14.4 B, D 5 A, B, C, E 18.9 B, C, E 10.2 B, E 4.2 A, B, D, E 18 A, D, E 13.5 C, D 4.6 A, C, D, E 17.4 B, D, E 9.3 C, E 3.8 B, C, D, E 13.2 C, D, E 5.8 D, E 3.2 A, B, C 17.1 A, B 13.2 A 5.8 A, B, D 16.2 A, C 12.6 B 3 A, C, D 15.6 A, D 11.7 C 2.6 B, C, D 11.4 A, E 10.5 D 1 E 0.6 However, the resulting solution was not accepted by the biggest Polish importer who argued that their biggest input in the 5-person coalition had been underestimated in the Shapley value. So, we will modify the Shapley value concept for cooperative games in the 3rd chapter (Commentary 1) with the aim to develop a new discount distribution (pay-outs to individual players), which – we claim  will not be questioned by any of the players. The obtained solution is unique and always exists. We will demonstrate that it also belongs to the core (see, Definition 3) of the studied game. 2 Solution of the Problem by means of the Shapley Value In each n-person game, the following natural question arises: When a coalition is formed, how does it share its payout (reward) between its own individual members? Definition 2a: (Thomas, 1986, p.90) An imputation in an n-person game with characteristic function v(S) defined for all coalitions S is a vector: ),...,,( 21 n xxxx  satisfying: (i) 54321 xxxxx  = v(N), where N is the set of all players; (ii) )i(vxi for i = 1, 2,…, n. In the context studied here, the conditions (i)  (ii) can be formulated as follows: (a) EDCBA xxxxx     = v{A, B, C, D, E}; (b) )A(vxA; )B(vxB; )C(vxC; )D(vxD; )E(vxE. Definition 2b: (Thomas, 1986, p.92) We say that an imputation )x,...,x,x(x n21  is rational for coalition S if, and only if the sum of payouts it generates for all members of S is greater or equal to v(S). Let’s note that if a certain imputation: )x,...,x,x(x n21  does not satisfy the condition stated in Definition 2b for some coalition S, then S will have no monetary incentive to participate in such share of rewards (discounts in our case). Therefore, we restrict our analysis, without loss of generality, to imputations that are rational for all coalitions S. Definition 3: (Thomas, 1986, p.92) The set of all imputations which are rational for all coalitions S is called the core of game. The natural candidate for the solution of problem 1 is the Shapley value (see Shapley 1953a and for example Thomas 1986, pp.101-103). It is a concept that in some rational way takes into account what (how much) each of player “contributes” to the largest coalition’s reward. In the studied case with 5 players, the Shapley value: x* = *)x*,x*,x*,x*,x(x EDCBA  Modification of Shapley Value and its Implementation in Decision Making 261 is such an imputation whose coordinates represent an “average” contribution (savings) each of these 5 players brings when it joins the coalition {A, B, C, D, E} at all possible stages of its creation. Let us explain it. The idea is that the players arrive at game in random order. When any player, for example D, arrives at some already existing coalition S, he brings (contributes) an extra amount to it, namely: v(DS ) – v(S). Note 1: We should use {D} rather than D, but for simplicity we will not be doing it, hoping it will not cause any misunderstanding. Suppose that the grand coalition of all 5 importers has been formed in such a way that at first, player D joined player E, next C joined the coalition {E, D}, next B arrived to the coalition {E, D, C}, and finally A joined the coalition {E, D, C, B}. Therefore, when importer E started the creation of coalition {E, D, C, B, A}, his contribution to this largest coalition’s reward was just v({E}) = v(E) = 0.6, according to Table 2a. Note 2: We should remember that coalition {E, D, C, B, A} is the same as coalition {A, B, C, D, E}. When importer D joined E, he brought an additional award (discount) of v{E, D} – v({E}) = 3.2 – 0.6 = 2.6 million PLN. Similarly, when C joined the coalition {E, D}, he brought an extra discount of v({E, D, C}) – v({E, D}) = 5.8 – 3.2 = 2.6 million according to Table 2a. When B arrived at coalition {E, D, C}, he brought additional discount of: v({E, D, C, B}) – v({E, D, C}) = 13.2 – 5.8 = 7.4 million. Finally, when A joined coalition {E, D, C, B}, he brought the largest discount of: v({E, D, C, B, A}) – v({E, D, C, B}} = 30 – 13.2 = 16.8 million. It is easy to notice that the grand coalition can also be formed in 119 = 5! – 1 different ways than the one {E, D, C, B, A}. Note 3: The foregoing is a sequence and not a set of players that we have examined; for example, {C, B, D, A, E} is another sequence of the same grand coalition. Performing analogous calculations as above for {C, B, D, A, E}, one can summarize the results obtained above in the Table 4. Table 4. Contributions of players to greater discount order of creation of grand coalition contribution of player A contribution of player B contribution of player C contribution of player D contribution of player E total {E, D, C, B, A} 16.8 7.4 2.6 2.6 0.6 30 {C, B, D, A, E} 9.3 3 2.6 5.8 9.3 30 Following this procedure of Shapley for all 118 = 5! – 2 remaining permutations, and next averaging the results obtained in each column (for each importer), we arrive at the Shapley imputation: *)x*,x*,x*,x*,x(x EDCBA . Proposition 1: The Shapley value for the cooperative game with characteristic function presented in Table 3a is the imputation *,x*,x*,x*,x(x DCBA  *)xE = (11,68; 5,89; 5,14; 4,11; 3,18). Dividing the coordinates of this vector by the amounts of import for the companies A, B, C, D, E, one obtains the Shapley value in percentage terms (3.77%; 3.93%; 3.95%; 4.11%; 5.29%). One could understand why the biggest importer A felt disappointed and discriminated when he was told that the Shapley imputation for him was considerably less than 4.12%4310   million he was almost certain to receive, hoping for even more because he had the largest contribution to the import value of 750 million. It is known (see, e.g., Malawski, Wieczorek, Sosnowska, 2006, pp.134-136) that the proposition below is true. In order to gain a greater insight, we will however demonstrate a proof in this particular case. Proposition 2: The Shapley value for the cooperative game with characteristic function presented in Table 3a belongs to the core of this game. Proof. It is enough to verify (see Table 5a below) that the discounts (11.68; 5.89; 5.14; 4.11; 3.18) offered by Shapley solution: *)x*,x*,x*,x*,x(x EDCBA  262 Leszek Zaremba, Cezary S. Zaremba, Marek Suchenek to each importer sum up for all 31 = 125 coalitions to greater amounts than those specified by the characteristic function (Table 3a). We leave to the reader a verification of this claim just for one coalition {A, C, E}. In fact, Shapley’s solution offers for {A, C, E} the total discount of: 20 million = 11.68 + 5.14 + 3.18 while the characteristic function is not so much generous. Indeed, according to assumptions made in Table 1 and Table 2a, we have v{A, C, E} = 15 million = %3)60130310(    . Let us recall that v(S) is the maximum value that coalition S can guarantee itself by coordinating the strategies of its members, no matter what the other players will do. Table 5a. Discounts attributable to each coalition according to Shapley value and Table 2a coalition discount v(S) coalition discount v(S) coalition discount v(S) A, B, C, D, E 30 A, B, E 20.75 B, C 11.03 A, B, C, D 26.82 A, C, E 20 B, D 10 A, B, C, E 25.89 B, C, E 14.21 B, E 9.07 A, B, D, E 24.86 A, D, E 18.97 C, D 9.25 A, C, D, E 24.11 B, D, E 13.18 C, E 8.32 B, C, D, E 18.32 C, D, E 12.43 D, E 7.29 A, B, C 22.71 A, B 17.57 A 11.68 A, B, D 21.68 A, C 16.82 B 5.89 A, C, D 20.93 A, D 15.79 C 5.14 B, C, D 15.14 A, E 14.86 D 4.11 E 3.18 This way we proved the proposition with the help of a computer program which performed all the calculations. One can carry out a similar reasoning for the data presented in Table 2b, thus arriving at the following proposition. Proposition 3: The Shapley value for the cooperative game with characteristic function presented in Table 3b is the imputation = (10.47; 6; 5.24; 4.22; 3.27). Dividing the coordinates of this vector by the amounts of import for particular companies A, B, C, D and E, one obtains the Shapley value in percentage terms (3.61%; 4%; 4.03%; 4.22%; 5.47%). In this case, the disappointment of importer A was smaller than previously because according to Table 2b, he himself was not entitled to receive a 3% discount. However, he still questioned why the smallest importers got the highest discounts. Let’s note here that the 5 payoffs (discounts) to the players A, B, C, D and E do not sum up this time to 30 million PLN, but to 29.2 million PLN because 2.29%4730  . Proposition 4: The Shapley value for the cooperative game with characteristic function presented in Table 3b belongs to the core of this game. Once again, knowing that the fact above is true, we will demonstrate it in the particular example studied. Clearly, the proof will again follow the same lines as in the case of Proposition 2 with Table 3a and Table 5a replaced by Table 3b and Table 5b (below), respectively. Commentary 1: In Chapter 3, we modify the Shapley idea, by taking into account not only all possible ways (permutations) by which the full coalition can be created (as Shapley did), but also we take into account all possible ways by which the remaining (smaller) coalitions could have been created. Modification of Shapley Value and its Implementation in Decision Making 263 Table 5b. Discounts attributable to each coalition according to Shapley value and Table 2b coalition discount v(S) coalition discount v(S) coalition discount v(S) A, B, C, D, E 29.20 A, B, E 19.75 B, C 11.24 A, B, C, D 25.93 A, C, E 18.99 B, D 10.22 A, B, C, E 24.99 B, C, E 14.52 B, E 9.28 A, B, D, E 23.97 A, D, E 17.97 C, D 9.46 A, C, D, E 23.21 B, D, E 13.5 C, E 8.52 B, C, D, E 18.74 C, D, E 12.74 D, E 7.5 A, B, C 21.71 A, B 16.47 A 10.47 A, B, D 20.69 A, C 15.71 B 6.00 A, C, D 19.93 A, D 14.69 C 5.24 B, C, D 15.46 A, E 13.75 D 4.22 E 3.28 3 Solution of Problem 1 by means of market strength (MS) method Now we will propose another solution concept to Problem 1 by means of the so-called market strength method. This method will produce such imputation )x ˆ ,x ˆ ,x ˆ ,x ˆ ,x ˆ (EDCBA of discounts which fully takes into account the real strength of each of importers A, B, C, D and E on the raw material market. It turns out that the solution )x ˆ ,x ˆ ,x ˆ ,x ˆ ,x ˆ (EDCBA assigns higher discount to the largest importer than the Shapley solution. Similarly, as in the Shapley’s approach, the MS method requires the computation of average marginal discount (contribution) that each importer brings to the existing coalition when he joins it. But unlike as in Chapter 2, this time we take into account all 325 coalitions (of all possible sizes), not just the biggest one, as was the case with the methods based on the Shapley value. The information gained this way therefore yields more information about the strength of each importer on the raw material market than in the Shapley’s approach. That information (knowledge) is next utilized in the construction of a new imputation: )x ˆ ,x ˆ ,x ˆ ,x ˆ ,x ˆ (EDCBA we propose as a solution. Consider thus: 1 =         0 5 5-member coalition, 5 =         1 5 4 - member coalitions, 10 =         2 5 3-member coalitions, 10 =         3 5 2-member coalitions and finally 5 =         4 5 importers A, B, C, D and E, which formally may be viewed as 1-member coalitions. The number of all sizes’ coalitions is: 3251521061024512011 4 5 !2 3 5 !3 2 5 !4 1 5 !5 0 5                                             Our computer program thus analyzed 325 permutations, averaging all marginal contributions the players bring when they join an already existing coalition. We demonstrate this by examining just 10 listed below permutations representing 10 ways to create a coalition: ACEBD (9.3; 4.5; 3.9; 10.5; 1.8); ACEDB (9.3; 12; 3.9; 3; 1.8); EBCA (9.3; 3.6; 6; -; 0.6); EBCD (-; 3.6; 6; 3; 0.6); EBD (-; 3.6; -; 5.1; 0.6); ECA (11.2; -; 3.2; -; 0.6); CA (10.6; -; 2.6; -; -); DB (-; 4; -; 1; -); B (-; 3; -; -; -); D (-; -; -; 1; -). 264 Leszek Zaremba, Cezary S. Zaremba, Marek Suchenek Let us now explain what do we mean by writing EBD (-; 3.6; -; 5.1; 0.6). Dash on the place of the first coordinate means that the first importer (A) does not participate in this coalition. Similarly, dash on the place of third coordinate means that the third importer (C) does not participate in coalition EBD either, which is obviously true. The second coordinate (equal to 3.6) indicates that the second importer (B) contributes (brings marginal discount) at average of 3.6 million when joining an existing coalition of any size (4 or 3 or 2 or 1 or 0). Computing the average strength of player A (the same remark applies also to any other importer) on the raw materials market, the computer program adds up the marginal contributions (extra discounts) that importer A brought to the already existing coalitions when he joined them. The sum of all the marginal contributions divided by the number of such instances yields the average strength of importer A on the market. To clarify this point even further, let us assume that there are only 10 (listed above) ways the coalitions can be formed; in reality, there are 325 of them. Then the average contribution of player A would be equal to (9.3+9.3+9.3+11.2+10.6)/5 = (49.7)/5 = 9.94. Indeed, we divide by 5 (not by 10) the sum of all contributions of player A because A joined existing coalitions only 5 times. We are now about to define this new solution determined by the MS method. In case of data displayed in Table 2a, our computer program calculated the average strengths for importers A, B, C, D and E on the raw material market obtaining the following imputation (10.99; 4.99; 4.29; 3.18; 2.32) of discounts. Since the sum of such calculated contributions (extra discounts) equals 25.77, which is less than 30 = )60130150310(%4  , we should proportionally increase all coordinates of the vector (10.99; 4.99; 4.29; 3.18; 2.32) to receive another imputation for which the sum of its coordinates will be equal to 30. As a result, we will obtain the imputation (12.795; 5.81; 4.995; 3.70; 2.70). Proposition 5: The solution )x ˆ ,x ˆ ,x ˆ ,x ˆ ,x ˆ (EDCBA of the cooperative game problem 1 with characteristic function given in Table 3a, determined according to the MS method proposed in this article, is the following imputation of discounts (12.795; 5.81; 4.995; 3.70; 2.70). Dividing the coordinates of this vector by the amounts of import for companies A, B, C, D and E, one arrives at the solution (4.13%; 3.87%; 3.84%; 3.70%; 4.50%). Comparing this new imputation (12.795; 5.81; 4.995; 3.70; 2.70) with the Shapley value (11.68; 5.89; 5.14; 4.11; 3.18), one can see that importer A became the key beneficiary of MS method. Clearly, the same conclusion holds when we compare these two solutions in percentage terms, that is, (4.13%; 3.87%; 3.84%; 3.70%; 4.50%) with the Shapley solution (3.77%; 3.93%; 3.95%; 4.11%; 5.29%). Note 4: One might argue that it is not altogether clear that A should be the main beneficiary. Assuming so seems similar to the Marxian economy that distributes rewards according to the investment effort and resources. However, if E with its “small” contribution increased everybody else's reward by a large margin, he would be in a good negotiating position to command a lion’s share of these rewards. If, however, he has to compete with other “small” importers, then his rewards should be equal to the minimum acceptable by any of these “small” importers. Perhaps, the assumption that coalitions split their rewards regardless of what other actors on the market are doing is a root cause of this flaw. In this way, we have arrived at a new result. Proposition 6: The solution )x ˆ ,x ˆ ,x ˆ ,x ˆ ,x ˆ (EDCBA of the cooperative game problem 1 with characteristic function given in Table 2a, determined according to the MS method, belongs to the core of the game. Proof. It is enough to make sure that the discounts )x ˆ ,x ˆ ,x ˆ ,x ˆ ,x ˆ (EDCBA = (12.795; 5.81; 4.995; 3.70; 2.70) sum up for all 31= 125 coalitions to greater amounts (Table 6a) than those featured in Table 3a. We leave to the reader justification of this claim for coalition {A, C, E} only. In fact, by summing up the coordinates A x ˆ, C x ˆ, E x ˆ, we obtain 12.79 + 4.99 + 2.70 = 20.48, which is more than the discount v{A, C, E} = 15 million = %3)60130310(  listed in Table 3a. Modification of Shapley Value and its Implementation in Decision Making 271 sions). Warszawa-Wrocław: Wydawnictwo Naukowe PWN. [43] Mercik, J., 2013. Index of Power for Cabinet. In: Holler and Owen, eds., Power Indices and Coalition Formation, pp.371-380. New York: Springer Science+ Business Media. [44] Mertens, J-F., Neyman, A., 1981. Stochastic Games. International Journal of game Theory, 10, pp.53-66. [45] Mertens, J-F., 2002. Stochastic Games. In: R. Aumann and S. Hart, eds., 2002. Handbook of Game Theory with Economic Applications, Vol.3. North-Holland: Elsevier, pp.1809-1827. [46] Milnor, J., Shapley, L., 1978. Values of Large Games II: Oceanic Games. Mathematics of Operations Research, 3, pp. 290-307; initially in: Rand Corporation, RM-2649, Santa Monica,1961. [47] Monderer, D., Samet, D., and Shapely, L., 1992. Weighted Values and the Core. Journal of Political Economy, 21, pp.27-39. [48] Monderer, D., Samet, D., 2002. Variations on the Shapley Value. In: R. Aumann and S. Hart, eds., 2002. Handbook of Game Theory with Economic Applications, Vol. 3. North-Holland: Elsevier, pp.2055-2075. [49] Moriariti, S. ed., 1983. Joint Cost Allocation. Tulsa: Univ. of Oklahoma Press. [50] Nowak, A., Radzik, T., 1994. A Solidarity Value for n-Person Transferable Utility Games. International Journal of Game Theory, 23(1), pp.4348. [51] Ott, U., 2006. International Joint Ventures: An Interplay of Cooperative and Noncooperative Games under Incomplete Information. Houndmills Basingstoke: Palgrave Macmillan. [52] Owen, G., 1977. Values of Games with a Priori Unions. In: R. Henn and O. Moeschlin. Mathematical Economics and Game Theory. Berlin: Springer Verlag, pp.76-88. [53] Peleq, B., 2005. Introduction to the Theory of Cooperative Games, 2 ed. Berlin: Springer. [54] Raghavan, T., Non-sero-sum Two-Person Games. In: R. Aumann and S. Hart, eds., 2002. Handbook of Game Theory with Economic Applications, Vol. 3. North-Holland: Elsevier, pp.1687-1718. [55] Roth, A., 1984. The Evolution of the Labor Market for Medical Interns and Residents: A Case Study in Game Theory, Journal of Political Economy, 92, pp.991-1016. [56] Roth, A., Sotomayor, M., 1990. Two-Sided Matching: A Study in Game-Theoretic Modelling and Analysis. Econometric Society Monograph. Cambridge: Cambridge University Press. [57] Roth, A., 2005. The Shapley Value: Essays in Honor of Lloyd S. Shapley. Cambridge: Cambridge Univ. Press. [58] Shapley, L., 1953a. A Value for n-Person Games. In: eds., H.W. Kuhn and A.W. Tucker. Contributions to the Theory of Games, Vol. 2. Ann. Math. Studies 28. Princeton: Princeton Univ. Press, pp.307-317. [59] Shapley, L., 1953b. Adaptive and Non-Adaptive Set Functions. Ph.D. thesis. Department of Mathematics, Princeton Univ. [60] Shapley, L., 1953c. Open Questions. In: ed., H.W. Kuhn. Report of an Informal Conference on the Theory of n-Person Games. Princeton University. [61] Shapley, L., 1953d. Quota Solutions of n-Person Games. In: eds., H.W. Kuhn and A.W. Tucker. Contributions to the theory of Games, Vol. 2, Ann. Math. Studies 28, Princeton Univ. Press, pp.343-359. [62] Shapley, L., 1953. Stochastic Games. Proceedings of the National Academy of Sciences, 39, pp.1095-1100. [63] Shapley, L., Shubik M., 1954. A Method for Evaluating a Distribution of Power in a Committee System. American Political Science Review, 48, pp.787-792. [64] Shapley, L., 1959. The Solution of a Symmetric Market Game. Annals of Mathematics Studies, 40, pp.145-162. [65] Shapley, L., 1961a. Values of Large Games III: A Cooperation with 2 Large Stockholders. RM26-50 Rand Corporation, Santa Monica. [66] Shapley, L., 1961b. Values of Large Games IV: An 18-Person Market Game. RM-2860 Rand Corporation, Santa Monica. [67] Shapley, L., 1962a. Compound Simple Games, I: Solutions of Sums and Products. RM -3192-PR, The Rand Corporation, Santa Monica. 272 Leszek Zaremba, Cezary S. Zaremba, Marek Suchenek [68] Shapley, L., 1963). Compound Simple Games, II: Some General Composition Theorems. RM - 3643-PR, The Rand Corporation, Santa Monica. [69] Shapley, L., 1964a. Solutions of Compound Simple Games, I: Solutions of Sums and Products. Annals of Mathematics Studies, 52, pp.267305. [70] Shapley, L. 1964b. Values of Large Games VII: A General Exchange Economy with Money. RM-4248, Rand Corporation, Santa Monica. [71] Shapley, L., 1967a. On Balanced Sets and Cores. Naval Research Logistics Quarterly, 14, pp.453460. [72] Shapley, L., 1967b. On Committees. In: eds., F. Zwicky and A. Wilson. New Methods of Thought and Procedure. New York: Springer, pp.246270. [73] Shapley, L., Shubik, M., 1969. On Market Games. Journal of Economic Theory, 1, pp.9-25. [74] Shapley, L., 1971. Cores of Convex Games. International Journal of Game Theory, 1, pp.1126. [75] Shapley, L., Shubik, M., 1972. The assignment Game I: The Core. International Journal of Game Theory, 1, pp.111-130. [76] Shapley, L., 1977. A Comparison of Power Indices and a Nonsymmetric Generalization. Paper P-5872. The Rand Corporation, Santa Monica. [77] Shapley, L., Shapiro, N., 1978. Values of Large Games I: A Limit Theorem. Mathematics of Operations Research, 3, pp.1-9; initially in Rand Corporation, RM-2648, Santa Monica (1960). [78] Shubik, M., 1962. Incentives, Decentralized Control, the Assignment of Joint Costs and Internal Pricing. Management Science, 8(3), pp.325-343. [79] Shubik, M., 1982. Games Theory in the Social Science. Cambridge: MIT Press. [80] Sosnowska, H., 1997. Coalitions and Rationality, Control & Cybernetics 26(4), pp.623-624. [81] Sosnowska, H., 2014. Banzhaf Value for Games Analyzing Voting with Rotation. Operations Research & Decisions, 24(4), pp.75-88. [82] Sosnowska, H., 2016. Approval Voting as a Method of Prediction in Political Votings. Case of Polish Elections. LNCS Transactions on Computational Collective Intelligence, 23, pp.17-28. [83] Steunenberg, B., Schmidtchen, D., and Kobolt, C., 1999. Strategic Power in the European Union: Evaluating the Distribution of Power in Policy Games. Journal of Theoretical Politics, 11(3), pp.339-366. [84] Thomas L., 1986. Games, Theory & Applications. Chichester: Ellis Horwood. [85] Weirich, P., 2012. Collective Rationality: Equilibrium in Cooperative Games. Oxford: Oxford Univ. Press. [86] Young, H., 1994. Cost Allocation. In: Aumann and Hart, eds., Handbook of Game Theory with Economic Applications, Vol.2. North-Holland: Elsevier, pp.1194-1230. [87] Zaremba, L., 1979. On the Existence of Value in the Varaiya-Lin Sense in Differential Games of Pursuit. Journal of Optimization Theory & Applications, 29, pp.135-145. [88] Zaremba, L., 1980. On the Existence of Value in Pursuit-Evasion Games with Restricted Coordinates, Journal of Optimization Theory & Applications, Vol. 30, pp.451-470. [89] Zaremba, L., 1982. Existence of Value in Differential Games with Fixed Time Duration. Journal of Optimization Theory & Applications, Vol. 38, pp.581-598. [90] Zaremba, L., 1984a. Existence of Value in Games Governed by Generalized Differential Equations. Systems & Control Letters, Vol. 4, pp.237-341 [91] Zaremba, L., 1984b. Existence of Value in Generalized Pursuit-Evasion Games. SIAM Journal on Control & Optimization Vol. 22, pp.894-901. [92] Zaremba, L., 1986. Existence Theorems for Games of Survival. Journal of Optimization Theory & Applications, Vol. 48, pp.431-446. [93] Zaremba, L., 1989. Optimality Principles of Dynamic Programming in Differential Games. Journal of Mathematical Analysis and Applications, Vol. 138, pp.43-51. [94] Zaremba, L., Zaremba, C., 2010. Rozwiązanie uogólnionego problemu optymalnej alokacji zasobów (Solution to Generalized Optimal Resource Allocation Problem). Zarządzanie Zmianami – Biuletyn Naukowy i Informacyjny Wyższej Szkoły Zarządzanie – the Polish Open University, 6, pp.1-19.