In 2007, Liao finished his Ph.d. dissertation(Liao 2007) entitled “The Solutions on Multi-choice Games”. Chapter 1 of the dissertation mainly worked on two special cases of the H&R multi-choice Shapley value. One assumes that the weight function w(j) is a positive constant function for all j 6= 0 with w(0) = 0 and the other one assumes that the weight function w(j) = j for all j. If w(j) ’s are equal for all j > 0 then the formula of H&R multi-choice Shapley value can be significantly simplified to the original formula of the traditional Shapley value for the traditional games. Therefore, as a matter of fact, Definitions 1 and 2 in Chapter 1 of the dissertation  are simply the traditional Shapley value. Hence, in most part of Chapter 1, Liao was just writing “new results” of traditional games in terms of the notations of multi-choice games. Furthermore, the dissertation  did not cited (1994), (1995a) and (1996) which held the original ideas of its main part of chapter 1.
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