Service sports include two-player contests such as volleyball, badminton, and squash. We analyze four rules, including the Standard Rule (SR), in which a player continues to serve until he or she loses. The Catch-Up Rule (CR) gives the serve to the player who has lost the previous point—as opposed to the player who won the previous point, as under SR. We also consider two Trailing Rules that make the server the player who trails in total score. Surprisingly, compared with SR, only CR gives the players the same probability of winning a game while increasing its expected length, thereby making it more competitive and exciting to watch. Unlike one of the Trailing Rules, CR is strategy-proof. By contrast, the rules of tennis fix who serves and when; its tiebreaker, however, keeps play competitive by being fair—not favoring either the player who serves first or who serves second.

It is well known that Nash equilibria may not be Pareto-optimal; worse, a unique Nash equilibrium may be Pareto-dominated, as in Prisoners’ Dilemma. By contrast, we prove a previously conjectured result: every finite normal-form game of complete information and common knowledge has at least one Pareto-optimal nonmyopic equilibrium (NME) in pure strategies, which we define and illustrate. The outcome it gives, which depends on where play starts, may or may not coincide with that given by a Nash equilibrium. We use some simple examples to illustrate properties of NMEs—for instance, that NME outcomes are usually, though not always, maximin—and seem likely to foster cooperation in many games. Other approaches for analyzing farsighted strategic behavior in games are compared with the NME analysis.

Recently, Artificial Intelligence (AI) technology use has been rising in sports to reach decisions of various complexity. At a relatively low complexity level, for example, major tennis tournaments replaced human line judges with Hawk-Eye Live technology to reduce staff during the COVID-19 pandemic. AI is now ready to move beyond such mundane tasks, however. A case in point and a perfect application ground is chess. To reduce the growing incidence of ties, many elite tournaments have resorted to fast chess tiebreakers. However, these tiebreakers significantly reduce the quality of games. To address this issue, we propose a novel AI-driven method for an objective tiebreaking mechanism. This method evaluates the quality of players’ moves by comparing them to the optimal moves suggested by powerful chess engines. If there is a tie, the player with the higher quality measure wins the tiebreak. This approach not only enhances the fairness and integrity of the competition but also maintains the game’s high standards. To show the effectiveness of our method, we apply it to a dataset comprising approximately 25,000 grandmaster moves from World Chess Championship matches spanning from 1910 to 2018, using Stockfish 16, a leading chess AI, for analysis.

This paper contributes to the literature on pure equilibria in symmetric zero-sum games in two main ways. First, we introduce new sufficient conditions, including interchangeability and weak quasiconcavity, for the existence of such equilibria. Second, we uncover relationships between these newly introduced conditions and existing ones. For instance, we demonstrate that the class of weakly quasiconcave games generalizes the class of quasiconcave games and ordinal potential games. Additionally, we show that exact potential games satisfy the interchangeability condition. However, no logical relationship exists between interchangeability and (weak) quasiconcavity.