Decision Making using Game Theory: An introduction for managers

26Games of skillExample 2.4 Examination success and time given to tutoringKPMG (UK), the British subsidiary of the worldwide accountancy and businessservices Wrm, has been analysing the examination results (for chartered status) ofits trainees over a number of years in relation to national trends and has foundthat the number of hours of direct tutoring is one of the determinants of howwell students do. Up to a certain point, overall results (as measured by thenumber of students achieving distinction grades) improve as more timetabledinstruction is given, but after that, results decline as the students’ practicalexperience diminishes.The relationship between the number of hours timetabled per week for directinstruction (h) and the percentage by which the results are below the company’sinternational benchmark (r), which is complicated by other variables such as age(a) and the percentage of students without a prior qualiWcation in a numeratediscipline (n), is found to be given by the equation:r n[(h 4)1/2 h]aThe company wishes to determine the optimal number of teaching hours for itstrainees so as to maximise overall results, subject to a minimum requirement of4 hours per week imposed by the Institute of Chartered Accountants code ofpractice.Holding n and a constant, the derivative of r with respect to h is:r'(h) n/a[ 1 2 (h 4)1/2 1]Solving the equation r'(h) 0 gives:12 (h 4)1/2 1(h 4) 1/2 1 2h 4.25Putting this result back into the equation for r gives:r(h) n/a[(0.25) 1/2 4.25]3.75n/aIn other words, the company’s results are maximised at [3.75n/a]