A reference point approach to coalition games

One of the basic concepts of the theory of coalition games is the core of the game, together with its various extensions. However, the core is usually a set, a subset of Pareto set. Therefore, all problems of vector optimization and multiple criteria decision support arise when selecting points within a core. One of interactive decision support approaches is the use of reference points and the maximization of corresponding achievement functions. The paper proposes some ways of defining and using reference points that result in equitable allocations. One way of defining such reference points is to use known solutions of coalition games, such as Shapley value or Banzhaf value. Another way of defining such reference points might be an extension of Raiffa–Kalai–Smorodinski solution to coalition games, proposed in the paper. The properties of resulting equitable allocations in the core of the game are examined. The possibility of empty core and allocations in an extended core is also examined and the concept of maximal robustness point is introduced; this point might also be used as a reference point. Possible applications of this extension of the theory of coalition games concern business negotiations. An illustrative example of negotiating a cooperative merger of three or four high-tech firms is given. The basic conclusion from the analysis and the examples presented in the paper is that axiomatic solutions of coalition games, e.g. Shapley and Banzhaf values, should be considered as reference points for vector optimization and multicriteria decision support rather than as normative solutions. Moreover, beside these axiomatic solutions there are several other possible definitions of reference points, such as the Raiffa solution or the maximal robustness point, that might be preferred by negotiators when solving practical coalition games. This has been confirmed by simulated gaming exercises performed by students under a strong motivation to treat simulated game seriously.