Comparison of theories for payoff disbursement of coalition values

Summary and Conclusion

The present paper addresses the problem of comparison of models ofpayoff disbursement in coalition formation studies which make point,line, or area predictions. A satisfactory solution to this problem is criticalfor model comparison, which has been the major focus of research oncoalition forming behavior during the last decade. The goal of this paperis to devise and subsequently apply a test procedure which, in comparingthe models to each other, offsets the advantage that the less specific modelhas over its competitor. In addition, the test procedure should employmeasures of error which yield intuitive results and are consistent with theprinciples underlying present coalition theories.It was contented that both the error measure of Bonacich and the netrate of success of Selten and Krishker suffer from serious deficiencies.Bonacich's approach allows for degrees of confirmation of a model butemploys an index of error which yields counterintuitive results. Theapproach of Seken and Krischker also defies intuition and commonpractice by treating all payoff vectors that do not fall in the model'sprediction set in exactly the same manner. The test procedure proposedin the present paper allows the prediction set of a model to expanduniformly in all the directions (dimensions) of the outcome space untilit encompasses all the observed payoff vectors which lie in this space. Indoing so it generates a function, called a support function, which relatesthe cumulative proportion of observed payoff vectors within the expand-ed set of predictions against the relative size of this set. By comparing toeach other the cumulative proportions for two different models when therelative sizes of their expanded predictions sets are held equal, theprocedure offsets the advantage possessed by the less specific modelwhich initially prescribes a larger or more dispersed prediction set.Like the index of error E devised by Bonacich, the procedure proposedin the present paper incorporates the intuitive idea that differentoutcomes diffentially confirm a theory if they are not contained in itsprediction set. Error is allowed to be continuous even if the theory underconsideration is algebraic. Statistical tests of algebraic theories in otherareas of psychology are almost always based on this assumption. Theprocedure also incorporates the shortest rather than the mean squareddistance between a payoff vector and a set of predicted payoff vectorsas the appropriate measure of error. The shortest distance is appropriatebecause coalition theories are mute with respect to the degree of ‘importance’, ‘representativeness’, of 'typicality' of the predictions they make.The procedure seems to yield satisfactory results. When applied to thetwo studies by Rapoport and Kahan (1976) and Kahan and Rapoport(1980) it has not favored models making line predictions over modelsmaking point predictions. It has established either strong or weak domi-nance relations between all pairs of models tested in these two studies.And it has confirmed the major conclusions of the two studies, which hadbeen originally reached by less rigourous tests of a smaller number ofmodels.