By Martin Gardner
A suite of brainteasers with illustrations by way of Jeff Sinclair
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Additional resources for Classic brainteasers
3. ’s method  may obtain negative interval-valued Shapley-like values ∗ À 00 Á of players, which are not rational. 2, ϕ1 υ is not a positive interval even if all coalitions’ values are positive intervals. However, the quadratic programming methods always assures that the interval-valued least square solutions are positive if all coalitions’ values are positive intervals. 4. 2, according to the method proposed by Han et al. , the interval lengths of the interval-valued Shapley-like values of players are identical.
40)) can be directly applied to compute the intervalvalued least square solutions of interval-valued cooperative games. ’s method  is respectively used to compute the interval-valued Shapley-like value of each player. 2. In the quadratic programming methods, the distance is used to measure the differences between interval-valued payoffs and interval-type values of coalitions. Thus, we can effectively avoid the magnification of uncertainty resulted from the subtraction of intervals. ’s method  may not overcome this disadvantage.
Cent Eur J Oper Res. 2011;19:523–32. 22. Alparslan G€ ok SZ, Branzei O, Branzei R, Tijs S. Set-valued solution concepts using intervaltype payoffs for interval games. J Math Econ. 2011;47:621–6. 23. Peleg B, Sudh€ olter P. Introduction to the theory of cooperative games, Series C: game theory, mathematical programming and operations research, vol. 34. 2nd ed. Berlin: Springer; 2007. 24. Branzei R, Dimitrov D, Tijs S. Models in cooperative game theory: crisp, fuzzy, and multichoice games, Lecture notes in economics and mathematical systems, 556.