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# Weighted Matching in Chess Tournaments

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## Abstract

In many chess tournaments, e.g. when the Swiss system is used, the number of players is much larger than the number of rounds to be played. In such tournaments the pairing for a round depends on the results in earlier rounds, and the pairing process can be very complicated. In these pairing systems the main goals are to let players with equal scores play together, and that each player should alternately play white and black, with the restriction that no player may face the same opponent more than once. The paper describes how a weighted matching algorithm is used to find 'the best pairing' by converting the pairing rules into penalty points.
... A pairing algorithm which is widely used for chess tournaments with many participants (e.g., Chess Olympiad, large Open tournaments) is the Swiss pairing system (see [25,26,27]). This was first used in 1903 in the Swiss national tournament [58] although there are claims [35] that it was used in Zurich as early as in 1895. It was first formalized as an algorithm and programmed by Olafsson [58]. ...
... This was first used in 1903 in the Swiss national tournament [58] although there are claims [35] that it was used in Zurich as early as in 1895. It was first formalized as an algorithm and programmed by Olafsson [58]. By construction, the Swiss pairing system has three main goals: ...
... Alternatively, we may start from the bottom score group and float up. In formally defining the Swiss pairing algorithm, Olafsson [58] argues for a bi-directional score group order: in the top half of the tournament, the odd player floats down, in the bottom half, they float up and conflicts are iteratively resolved around the middle. ...
Article
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Let n and k be integers such that n ≥ 2 and 1 ≤ k ≤ n. In this paper, we consider the problem of finding an ordered list of the k best players out of n participants by organizing a tournament of rounds of pairwise matches (comparisons). Assuming that (i) in each match there is a winner (no ties) (ii) the relative strength of the players is constant throughout the tournament and (iii) the players' strengths are transitive, the problem is equivalent to partially sorting n different, comparable objects , allowing parallelization in rounds. The rounds are restricted as one player can only play one match in each round. We propose concrete pair-ing algorithms and make conjectures about their performance in terms of the worst case number of rounds and matches required. The research article was started by professor Antal Iványi who sadly passed away during the work and was completed in his honor by the co-author. He hopes, in this modest way, to reflect his deep admiration for professor Iványi's many contributions to the theory, practice and appreciation of algorithm design and analysis.
... As will be shown in Section 3, the EOMM framework converts the problem of determining optimal match assignment to the problem of seeking MWPM on a weighted graph. MWM/MWPM have broad applications in other fields, including creating pairs following specific rules in chess tournaments [34], schdeduling training sessions among NASA shuttle cockpit simulators [4] and transmitting images over networks [36]. In a similar spirit, Ólafsson [34] leverages MWPM algorithm to determine opponents. ...
... MWM/MWPM have broad applications in other fields, including creating pairs following specific rules in chess tournaments [34], schdeduling training sessions among NASA shuttle cockpit simulators [4] and transmitting images over networks [36]. In a similar spirit, Ólafsson [34] leverages MWPM algorithm to determine opponents. Their goal, however, was to create matches maximally adhering specific rules of chess tournament, which is different than ours to optimize for player engagement. ...
Conference Paper
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Matchmaking connects multiple players to participate in online player-versus-player games. Current matchmaking systems depend on a single core strategy: create fair games at all times. These systems pair similarly skilled players on the assumption that a fair game is best player experience. We will demonstrate, however, that this intuitive assumption sometimes fails and that matchmaking based on fairness is not optimal for engagement. In this paper, we propose an Engagement Optimized Matchmaking (EOMM) framework that maximizes overall player engagement. We prove that equal-skill based matchmaking is a special case of EOMM on a highly simplified assumption that rarely holds in reality. Our simulation on real data from a popular game made by Electronic Arts, Inc. (EA) supports our theoretical results, showing significant improvement in enhancing player engagement compared to existing matchmaking methods.
... As will be shown in Section 3, the EOMM framework converts the problem of determining optimal match assignment to the problem of seeking MWPM on a weighted graph. MWM/MWPM have broad applications in other fields, including creating pairs following specific rules in chess tournaments [34], schdeduling training sessions among NASA shuttle cockpit simulators [4] and transmitting images over networks [36]. In a similar spirit, Ólafsson [34] leverages MWPM algorithm to determine opponents. ...
... MWM/MWPM have broad applications in other fields, including creating pairs following specific rules in chess tournaments [34], schdeduling training sessions among NASA shuttle cockpit simulators [4] and transmitting images over networks [36]. In a similar spirit, Ólafsson [34] leverages MWPM algorithm to determine opponents. Their goal, however, was to create matches maximally adhering specific rules of chess tournament, which is different than ours to optimize for player engagement. ...
Article
Full-text available
Matchmaking connects multiple players to participate in online player-versus-player games. Current matchmaking systems depend on a single core strategy: create fair games at all times. These systems pair similarly skilled players on the assumption that a fair game is best player experience. We will demonstrate, however, that this intuitive assumption sometimes fails and that matchmaking based on fairness is not optimal for engagement. In this paper, we propose an Engagement Optimized Matchmaking (EOMM) framework that maximizes overall player engagement. We prove that equal-skill based matchmaking is a special case of EOMM on a highly simplified assumption that rarely holds in reality. Our simulation on real data from a popular game made by Electronic Arts, Inc. (EA) supports our theoretical results, showing significant improvement in enhancing player engagement compared to existing matchmaking methods.
... The two papers closest to ours focus on modeling the exact FIDE pairing criteria and computing the prescribed pairings. Olafsson (1990) pairs players using a maximum weight matching algorithm on a graph, where players and possible matches are represented by vertices and edges. Edge weights are set so that they model the 1985 FIDE pairing criteria. ...
Preprint
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The International Chess Federation (FIDE) imposes a voluminous and complex set of player pairing criteria in Swiss-system chess tournaments and endorses computer programs that are able to calculate the prescribed pairings. The purpose of these formalities is to ensure that players are paired fairly during the tournament and that the final ranking corresponds to the players' true strength order. We contest the official FIDE player pairing routine by presenting alternative pairing rules. These can be enforced by computing maximum weight matchings in a carefully designed graph. We demonstrate by extensive experiments that a tournament format using our mechanism 1) yields fairer pairings in the rounds of the tournament and 2) produces a final ranking that reflects the players' true strengths better than the state-of-the-art FIDE pairing system.
... The work by Edmonds [8] on this problem greatly influenced the role of polyhedral theory on algorithm design [23]. On the other hand, the problem found applications in several domains [26,1,22,6,2]. In particular routing problems are an important area of application, and its procedures often appeared as subroutines of other important algorithms, the most notable being Christofides' algorithm for the traveling salesperson problem [5]. ...
Preprint
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The min-cost matching problem suffers from being very sensitive to small changes of the input. Even in a simple setting, e.g., when the costs come from the metric on the line, adding two nodes to the input might change the optimal solution completely. On the other hand, one expects that small changes in the input should incur only small changes on the constructed solutions, measured as the number of modified edges. We introduce a two-stage model where we study the trade-off between quality and robustness of solutions. In the first stage we are given a set of nodes in a metric space and we must compute a perfect matching. In the second stage $2k$ new nodes appear and we must adapt the solution to a perfect matching for the new instance. We say that an algorithm is $(\alpha,\beta)$-robust if the solutions constructed in both stages are $\alpha$-approximate with respect to min-cost perfect matchings, and if the number of edges deleted from the first stage matching is at most $\beta k$. Hence, $\alpha$ measures the quality of the algorithm and $\beta$ its robustness. In this setting we aim to balance both measures by deriving algorithms for constant $\alpha$ and $\beta$. We show that there exists an algorithm that is $(3,1)$-robust for any metric if one knows the number $2k$ of arriving nodes in advance. For the case that $k$ is unknown the situation is significantly more involved. We study this setting under the metric on the line and devise a $(19,2)$-robust algorithm that constructs a solution with a recursive structure that carefully balances cost and redundancy.
... This paper also permits lower quotas and project closures, but our focus is on cardinal utilities rather than ordinal preferences. Cardinal utilities can facilitate a more flexible representation of preferences and they occur naturally in various matching problems, such as the solution of symmetric indefinite systems [31], the organization of Chess tournaments [28] or car sharing [15]. Moreover, various methods for converting ordinal preferences into cardinal utilities are known, such as the normalized Borda score [30]. ...
Article
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We study a natural generalization of the maximum weight many-to-one matching problem. We are given an undirected bipartite graph $$G= (A\, \dot{\cup }\, P, E)$$ with weights on the edges in E, and with lower and upper quotas on the vertices in P. We seek a maximum weight many-to-one matching satisfying two sets of constraints: vertices in A are incident to at most one matching edge, while vertices in P are either unmatched or they are incident to a number of matching edges between their lower and upper quota. This problem, which we call maximum weight many-to-one matching with lower and upper quotas (WMLQ), has applications to the assignment of students to projects within university courses, where there are constraints on the minimum and maximum numbers of students that must be assigned to each project. In this paper, we provide a comprehensive analysis of the complexity of WMLQ from the viewpoints of classical polynomial time algorithms, fixed-parameter tractability, as well as approximability. We draw the line between $$\textsf {NP}$$-hard and polynomially tractable instances in terms of degree and quota constraints and provide efficient algorithms to solve the tractable ones. We further show that the problem can be solved in polynomial time for instances with bounded treewidth; however, the corresponding runtime is exponential in the treewidth with the maximum upper quota $$u_{\max }$$ as basis, and we prove that this dependence is necessary unless $$\textsf {FPT}= \textsf {W}[1]$$. The approximability of WMLQ is also discussed: we present an approximation algorithm for the general case with performance guarantee $$u_{\max }+1$$, which is asymptotically best possible unless $$\textsf {P}= \textsf {NP}$$. Finally, we elaborate on how most of our positive results carry over to matchings in arbitrary graphs with lower quotas.
... Probably the most distributed parallel ranking method is the Swiss System, which has several good description [15,16,19,20,21,22,23,29,33,39,50,59,60,63 ...
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Let n, k and p be positive integers (2 ≤ n and 1 ≤ k ≤ n). We rank the k largest numbers of n different numbers using at most p processors. We organize a tournament consisting of pairwise matches ending always with the win of the better player. Let R(n, k, p) denote the minimal number of rounds, necessary to rank the largest k given numbers (in worst case). We use the parallel mathematical model proposed in 1883 by Lewis Carroll, propose a new ranking algorithm Combined and using it we characterize R(n, k, p) and its expected values.
... Hence, SE is suitable field where we can introduce ML algorithms to develop many more process, product, people and project. Many topics in the Software Engineering [50] [62] can be a fertile soil where many problems and questions can be achieved and solved by ML. We provide comparative study of some software development tasks using learning algorithms which had been using by many authors and researchers. ...
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Software Project Management (SPM) is the most important and toughest job in software engineering organizations. For successful project management, it needs the proper planning and schedul- ing of each software development umbrella activity. Software plan- ning needs proper communication, requirement elicitation, suitable patterns and phase wise testing of software for fulfilling the clients requirements and completing the umbrella activities. All these can be effectively achieved and predicted by accurate estimates of the bud-get and schedule. Here, literature review is illustrated by the most com- monly used Machine Learning (ML) techniques such as neural net- works, case based reasoning, classification and regression trees, rule induction, genetic algorithm and genetic programming for the sub- domain of SPM. In current era, ML imparts consistently promising accuracy in some SPM fields.
Article
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Pairwise comparisons have become popular in the theory and practice of preference modelling and quantification. In case of incomplete data, the arrangements of known comparisons are crucial for the quality of results. We focus on decision problems where the set of pairwise comparisons can be chosen and it is designed completely before the decision making process, without any further prior information. The objective of this paper is to provide recommendations for filling patterns of incomplete pairwise comparison matrices based on their graph representation. The proposed graphs are regular and quasi-regular ones with minimal diameter (longest shortest path). Regularity means that each item is compared to others for the same number of times, resulting in a kind of symmetry. A graph on an odd number of vertices is called quasi-regular, if the degree of every vertex is the same odd number, except for one vertex whose degree is larger by one. We draw attention to the diameter, which is missing from the relevant literature, in order to remain the closest to direct comparisons. If the diameter of the graph of comparisons is as low as possible (among the graphs of the same number of edges), we can decrease the cumulated errors that are caused by the intermediate comparisons of a long path between two items. Contributions of this paper include a list containing (quasi-)regular graphs with diameter 2 and 3 up until 24 vertices. Extensive numerical tests show that the recommended graphs indeed lead to better weight vectors compared to various other graphs with the same number of edges. It is also revealed by examples that neither regularity nor small diameter is sufficient on its own, both properties are needed. Both theorists and practitioners can utilize the results, given in several formats in the appendix: plotted graph, adjacency matrix, list of edges, Graph6’ code.
Chapter
Ranking rules are a crucial component of every tournament design. This chapter argues for a better consideration of theoretical results in the construction of ranking criteria. Our axiomatic approach provides insight into three fields: points scoring systems applied to determine the outcome of championships including multiple competitions, ranking in Swiss-system tournaments where the strength of opponents has to be taken into account, and tie-breaking criteria in round-robin tournaments. A method is proposed to decide the final ranking in a round-robin league if the season is stopped and cannot resume. The previous FIFA World Ranking, used between 2006 and 2018, illustrates how many ways a rating formula can go awry.
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