Decision Making using Game Theory: An introduction for managers

85Games with saddle pointsPlayer 1abcdefgPlayer 2a b c d e f g6 7 5 11 8 12 1014 12 12 11 10 12 1310 7 8 13 4 10 98 4 2 13 8 14 1013 7 8 14 5 14 1012 9 8 13 9 12 810 7 3 14 7 14 10Figure 5.9A large matrix with a saddle point.Strategy Column 1 Column 2Row 1abRow 2cdFigure 5.10Dominance and admissability.Figure 5.9 is a seven-by-seven matrix and closer inspection reveals asaddle point at row b, column e, where the value of the game is 10. In azero-sum game, since the pay-oV for one player is the negative of thepay-oV for the other, a game is said to be fair if the value of the game iszero. None of the games we have considered so far has been fair in thatsense. In the case of the surgeon charged with incompetence, the gameis unfair because the surgeon cannot ignore the allegation. In the caseof the new Toronto Airport bus timetable, the game is unfair becausenot changing the timetable is not an option.Dominance and inadmissibility in large saddle point gamesSometimes one of a player’s pure strategies dominates the other ones inthat it yields an outcome at least as good against any strategy that theopposing player may select. It cannot make sense for a rational playerto select a dominated strategy and it is thus said to be inadmissible.More formally, one row dominates another when elements in thedominant row are larger and as large (smaller in the case of dominantcolumns) as the corresponding elements in the inadmissible row. Forexample, in Figure 5.10, row 1 dominates row 2 if a c and b d, forall a, b, c, d R.