 In this reduced game, Fiona is certain to gain a strictly higher payoff by choosing \(D^1\) than if she chooses \(C^1\), so \(D^1\) is her unique best choice. Of course, when Fiona chooses \(D^1\), Alan, being rational, responds by choosing \(D^2\). If Fiona and Alan know: (i) that they are both rational, (ii) that they both know the payoff structure of the game, and (iii) that they both know (i) and (ii), then they both can predict what the other will do at every node of the Figure 1.1.a game, and conclude that they can rule out the \(D^1,C^2\)-branch of the Figure 1.1.b game and analyze just the reduced game of the following figure: