Abstract. The rules of many sports are not fair -they do not ensure that equally skilled 3 competitors have the same probability of winning. As an example, the penalty shootout in soccer, 4 wherein a coin toss determines which team kicks first on all five penalty kicks, gives a substantial 5 advantage to the first-kicking team, both in theory and practice. We show that a so-called Catch-6Up Rule for determining the order of kicking would not only make the shootout fairer but also 7 is essentially strategyproof. By contrast, the so-called Standard Rule now used for the tiebreaker 8 in tennis is fair. We briefly consider several other sports, all of which involve scoring a sufficient 9 number of points to win, and show how they could benefit from certain rule changes, which would 10 be straightforward to implement. 11

Abstract. The rules of many sports are not fair -they do not ensure that equally skilled competitors have the same probability of winning. As an example, the penalty shootout in soccer, wherein a coin toss determines which team kicks first on all five penalty kicks, gives a substantial advantage to the first-kicking team, both in theory and in practice. We show that a so-called Catch-Up Rule for determining the order of kicking would not only make the shootout fairer but is also essentially strategyproof. By contrast, the so-called Standard Rule now used for the tiebreaker in tennis is fair. We briefly consider several other sports, all of which involve scoring a sufficient number of points to win, and show how they could benefit from certain rule changes which would be straightforward to implement.

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We analyze the 78 2 × 2 distinct strict ordinal games, 57 of which are conflict games that contain no mutually best outcome. In 19 of the 57 games (33%), including Prisoners' Dilemma and Chicken, a cooperative outcome-one that is at least next-best for each player-is not a Nash equilibrium (NE). But this outcome is a nonmyopic equilibrium (NME) in 16 of the 19 games (84%) when the players start at this outcome and make farsighted calculations, based on backward induction; in the other three games, credible threats can induce cooperation. In two of the latter games, the NMEs are "boomerang NMEs," whereby players have an incentive to move back and forth between two diagonally opposite NMEs, one of which is cooperative. In Prisoners' Dilemma, the NE and one NME are not Pareto-optimal, but we conjecture that in all two-person games with strict preferences, there is at least one Pareto-optimal NME. As examples of NMEs that are not NEs, we analyze two games that plausibly model the choices of players in international relations: (i) no first use of nuclear weapons, a policy that has been adopted by some nuclear powers; and (ii) the 2015 agreement between Iran, and a coalition of the United States and other countries, that has forestalled Iran's possible development of nuclear weapons.