Decision Making using Game Theory: An introduction for managers

95Interval and ordinal scales for pay-offsFigure 5.19SurgeonStrategyAcceptincompetencyassessmentRejectincompetencyassessmentA reconfigured matrix with ordinal pay-offs.Hospital boardDismiss withretirementpackagegvgRetain&retrainbvbalthough they preserve the relative ranking of preferences. So pay-oV‘3’ is less favourable than ‘4’ and more favourable than ‘2’, although ‘4’is not twice as favourable as ‘2’. Pay-oV matrices in mixed-motivegames (Chapter 6) are usually ordinal, while cooperative and zero-sumgames tend to have interval scales. In theory, ordinal pay-oVs can beadjusted, like interval pay-oVs, by linear transformations, but in practiceit makes no sense to do so.Games with ordinal pay-offs and saddle pointsGames with ordinal pay-oVs can be solved easily if they have saddlepoints. Figure 5.19 is Figure 5.7 reconWgured as an ordinal pay-oVmatrix, and vb, b, g and vg represent ‘very bad’, ‘bad’, ‘good’ and ‘verygood’, respectively.It is still clear that b is a saddle point, since b is the minimum in itsrow and the maximum in its column. In other words,vb b g, for all values of vb, b, g RSo, it is not necessary to have exact quantitative measurements of therelative desirability of the pay-oVs in a zero-sum game in order to Wnda solution.Games with ordinal pay-offs, but without saddle pointsConsider Figure 5.20, an ordinal reconWguration of Figure 5.15, inwhich w, b and e represent the worst, bad and excellent pay-oVs for thesenior deputy. p and 1 p are the senior deputy’s mixed-strategyprobabilities and q and 1 q and those for the junior deputy.