On top coalitions, common rankings, and semistrict core stability

The top coalition property of Banerjee et al. (2001) and the common ranking property of Farrell and Scotchmer (1988) are sufficient conditions for core stability in hedonic games. We introduce the semistrict core as a stronger stability concept than the core, and show that the top coalition property guarantees the existence of semistrictly core stable coalition structures. Moreover, for each game satisfying the common ranking property, the core and the semistrict core coincide.coalition formation, common ranking property, hedonic games, semistrict core, top coalition property

A network-based dynamical ranking system for competitive sports

From the viewpoint of networks, a ranking system for players or teams in sports is equivalent to a centrality measure for sports networks, whereby a directed link represents the result of a single game. Previously proposed network-based ranking systems are derived from static networks, i.e., aggregation of the results of games over time. However, the score of a player (or team) fluctuates over time. Defeating a renowned player in the peak performance is intuitively more rewarding than defeating the same player in other periods. To account for this factor, we propose a dynamic variant of such a network-based ranking system and apply it to professional men's tennis data. We derive a set of linear online update equations for the score of each player. The proposed ranking system predicts the outcome of the future games with a higher accuracy than the static counterparts.Comment: 6 figure

Reducing the Effects of Unequal Number of Games on Rankings

Ranking is an important mathematical process in a variety of contexts such as information retrieval, sports and business. Sports ranking methods can be applied both in and beyond the context of athletics. In both settings, once the concept of a game has been defined, teams (or individuals) accumulate wins, losses, and ties, which are then factored into the ranking computation. Many settings involve an unequal number of games between competitors. This paper demonstrates how to adapt two sports rankings methods, the Colley and Massey ranking methods, to settings where an unequal number of games are played between the teams. In such settings, the standard derivations of the methods can produce nonsensical rankings. This paper introduces the idea of including a super-user into the rankings and considers the effect of this fictitious player on the ratings. We apply such techniques to rank batters and pitchers in Major League baseball, professional tennis players, and participants in a free online social game. The ideas introduced in this paper can further the scope that such methods are applied and the depth of insight they offer

The game of go as a complex network

We study the game of go from a complex network perspective. We construct a directed network using a suitable definition of tactical moves including local patterns, and study this network for different datasets of professional tournaments and amateur games. The move distribution follows Zipf's law and the network is scale free, with statistical peculiarities different from other real directed networks, such as e. g. the World Wide Web. These specificities reflect in the outcome of ranking algorithms applied to it. The fine study of the eigenvalues and eigenvectors of matrices used by the ranking algorithms singles out certain strategic situations. Our results should pave the way to a better modelization of board games and other types of human strategic scheming.Comment: 6 pages, 9 figures, final versio

Ranking efficient DMUs using cooperative game theory

The problem of ranking Decision Making Units (DMUs) in Data Envelopment Analysis (DEA) has been widely studied in the literature. Some of the proposed approaches use cooperative game theory as a tool to perform the ranking. In this paper, we use the Shapley value of two different cooperative games in which the players are the efficient DMUs and the characteristic function represents the increase in the discriminant power of DEA contributed by each efficient DMU. The idea is that if the efficient DMUs are not included in the modified reference sample then the efficiency score of some inefficient DMUs would be higher. The characteristic function represents, therefore, the change in the efficiency scores of the inefficient DMUs that occurs when a given coalition of efficient units is dropped from the sample. Alternatively, the characteristic function of the cooperative game can be defined as the change in the efficiency scores of the inefficient DMUs that occurs when a given coalition of efficient DMUs are the only efficient DMUs that are included in the sample. Since the two cooperative games proposed are dual games, their corresponding Shapley value coincide and thus lead to the same ranking. The more an ef- ficient DMU impacts the shape of the efficient frontier, the higher the increase in the efficiency scores of the inefficient DMUs its removal brings about and, hence, the higher its contribution to the overall discriminant power of the method. The proposed approach is illustrated on a number of datasets from the literature and compared with existing methods

Top coalitions, common rankings, and semistrict core stability

The top coalition property of Banerjee et al. (2001) and the common ranking property of Farrell and Scotchmer (1988) are sufficient conditions for core stability in hedonic games. We introduce the semistrict core as a stronger stability concept than the core, and show that the top coalition property guarantees the existence of semistrictly core stable coalition structures. Moreover, for each game satisfying the common ranking property, the core and the semistrict core coincide.coalition formation

A network-based ranking system for American college football

American college football faces a conflict created by the desire to stage national championship games between the best teams of a season when there is no conventional playoff system to decide which those teams are. Instead, ranking of teams is based on their record of wins and losses during the season, but each team plays only a small fraction of eligible opponents, making the system underdetermined or contradictory or both. It is an interesting challenge to create a ranking system that at once is mathematically well-founded, gives results in general accord with received wisdom concerning the relative strengths of the teams, and is based upon intuitive principles, allowing it to be accepted readily by fans and experts alike. Here we introduce a one-parameter ranking method that satisfies all of these requirements and is based on a network representation of college football schedules.Comment: 15 pages, 3 figure