The order of actions in contests may generate different psychological effects which, in turn, may influence contestants' probabilities to win. The Prouhet-Thue-Morse sequence in which the first "n" moves is the exact mirror image of the next "n" moves should theoretically terminate any advantage to any of the contestants in a sequential pair-wise contest. The tennis tiebreak sequence of serves is the closest to the Prouhet-Thue-Morse sequence that one can find in real tournament settings. In a tiebreak between two players, A and B, the order of the first two serves (AB) is a mirror image of the next two serves (BA), such that the sequence of the first four serves is ABBA. Then, this sequence is repeated until one player wins the tiebreak. This sequence has been used not only in tennis, but also recently in the US TV presidential debates. In this study we analyse 1,701 men's and 920 women's tiebreak games from top-tier tournaments between the years 2012 to 2015. Using several different strategies to disentangle the effect of serving first from the effect of selection, we find that, for both genders, serving first does not have any significant effect on the winning probabilities of the two players, implying that the ABBA sequence is fair. We thus argue that it might be useful for other sports and contests in general to consider adopting the ABBA sequence in order to improve fairness.
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