Duckworth–Lewis and Twenty20 cricket

Originally designed for 1-day cricket, this paper considers the use of the Duckworth–Lewis method as an approach to resetting targets in interrupted Twenty20 cricket matches. The Duckworth–Lewis table is reviewed and an alternative resource table is presented. The alternative table is constructed using observed scoring rates from international Twenty20 matches. A desideratum of a resource table is monotonicity in both the rows and columns corresponding to wickets and overs respectively. Consequently, a Gibbs sampling scheme related to isotonic regression is applied to the observed scoring rates to provide a non-parametric resource table. Taking into account the more aggressive batting style of Twenty20 compared to 1-day cricket, the resultant resource table is seen to possess sensible features. A discussion is provided concerning the use of the Duckworth–Lewis method applied to Twenty20.     

The Duckworth-Lewis-Stern method: extending the Duckworth-Lewis methodology to deal with modern scoring rates

The famous (and occasionally infamous) Duckworth-Lewis methodology for dealing with interruptions in limited overs cricket matches made its international debut in early 1997. For nearly 20 years, it has set the standard for target adjustment at nearly all levels of the game. In that time, though, it has not been static. In 2003, the Professional Edition of the method was introduced to handle changes to scoring patterns that were becoming apparent in modern cricket. We here introduce the Duckworth-Lewis-Stern (DLS) method, designed to deal with the now common extreme scoring rates seen in limited overs matches, particularly Twenty20. In addition, we outline key principles governing the structure and properties of target adjustment methods and compare DLS with other proposed procedures.     

A fair method for resetting the target in interrupted one-day cricket matches

A method is described for setting revised target scores for the team batting second when a limited-overs cricket match has been forcibly shortened after it has commenced. It is designed so that neither team benefits or suffers from the shortening of the game and so is totally fair to both. It is easy to apply, requring nothing more than a single table of numbers and a pocket calculator, and is capable of dealing with any number of interruptions at any stage of either or both innings.The method is based on a simple model involving a two-factor relationship giving the number of runs which can be scored on average in the remainder of an innings as a function of the number of overs remaining and the number of wickets fallen. It is shown how the relationship enables the target score in an interrupted match to be recalculated to reflect the relative run scoring resources available to the two teams, that is overs and wickets in combination. The method was used in several international and domestic one-day competitions and tournaments in 1997.     

A constructive solution to a problem of ranking tournaments

A tournament is an oriented complete graph. The problem of ranking tournaments was firstly investigated by P. Erd?s and J. W. Moon. By probabilistic methods, the existence of ?? ?? unrankable” tournaments was proved. On the other hand, they also mentioned the problem of explicit constructions. However, there seems to be only a few of explicit constructions of such tournaments. In this note, we give a construction of many such tournaments by using skew Hadamard difference sets which have been investigated in combinatorial design theory.     

The shifted Turán sieve method on tournaments II

In a previous work [5], we developed the shifted Turán sieve method on a bipartite graph and applied it to problems on cycles in tournaments. More precisely, we obtained upper bounds for the number of tournaments which contain a small number of r-cycles. In this paper, we improve our sieve inequality and apply it to obtain an upper bound for the number of bipartite tournaments which contain a number of 2r-cycles far from the average. We also provide the exact bound for the number of tournaments which contain few 3-cycles, using other combinatorial arguments.     

A criterion for comparing and selecting batsmen in limited overs cricket

The batting average statistic has been used almost exclusively to assess the worth of a batsman. It reveals a great deal about the potential performance of batsmen in cricket played at the first class level. However, in the one-day game, strict limits on the number of balls bowled have introduced a very important additional dimension to performance. In the one-day game, it is clearly not good enough for a batsman to achieve a high batting average with a low strike rate. Runs scored slowly, even without the loss of wickets, will generally result in defeat rather than victory in the one-day game. Assessing batting performance in the one-day game, therefore, requires the application of at least a two-dimensional measurement approach because of the time dimension imposed on limited overs cricket. In this paper, we use a new graphical representation with Strike rate on one axis and the Probability of getting out on the other, akin to the risk–return framework used in portfolio analysis, to obtain useful, direct and comparative insights into batting performance, particularly in the context of the one-day game. Within this two-dimensional framework we develop a selection criterion for batsmen, which combines the average and the strike rate. As an example of the application, we apply this criterion to the batting performances of the 2003 World Cup. We demonstrate the strong and consistent performances of the Australian and Indian batsmen as well as provide a ranking of batting prowess for the top 20 run scorers in the tournament.     

Strategy selection and outcome prediction in sport using dynamic learning for stochastic processes

Stochastic processes are natural models for the progression of many individual and team sports. Such models have been applied successfully to select strategies and to predict outcomes in the context of games, tournaments and leagues. This information is useful to participants and gamblers, who often need to make decisions while the sports are in progress. In order to apply these models, much of the published research uses parameters estimated from historical data, thereby ignoring the uncertainty of the parameter values and the most relevant information that arises during competition. In this paper, we investigate candidate stochastic processes for familiar sporting applications that include cricket, football and badminton, reviewing existing models and offering some new suggestions. We then consider how to model parameter uncertainty with prior and posterior distributions, how to update these distributions dynamically during competition and how to use these results to make optimal decisions. Finally, we combine these ideas in a case study aimed at predicting the winners of next year’s University Boat Race.     

On local characterizations of Hamiltonian tournaments

We characterize the hamiltonian tournaments that admits exactly one spanning cycle and the hamiltonian tournaments with the least number of 3-cycles by their hamiltonian subtournaments which have the same properties respectively. AMS(MOS) Subject Classification (1980)-05C20. Work performed under the auspices of the Gruppo Nazionale di Topologia (Fondi MURST 40%). This paper is in its final form and no version of it will be submitted for publication elsewhere     

Rating players in test match cricket

In general, the evaluation of player performance in test cricket is based on measures such as batting and bowling averages. These measures have a number of limitations, among which is that they fail to take into account the context in which runs are made or conceded and wickets are taken or given away. Furthermore, batting and bowling averages do not allow the comparison of performances in these two disciplines; this is because batting and bowling performances are measured using different metrics. With these issues in mind, we develop a new player rating system for test cricket. We use multinomial logistic regression to model match outcome probabilities session by session. We then use these probabilities to measure the overall contribution of players to the match outcome based on their individual batting, bowling and fielding contributions during each session. Our measure of contribution has the potential for rating players over time and for determining the ‘best’ player in a match, a series or a calendar year. We use results from 104 matches (2010–2012) to illustrate the method.     

A Contribution to the Second Neighborhood Problem

Seymour’s Second Neighborhood Conjecture asserts that every oriented graph (without digons) has a vertex whose first out-neighborhood is at most as large as its second out-neighborhood. It is proved for tournaments, tournaments missing a matching and tournaments missing a generalized star. We prove this conjecture for classes of oriented graphs whose missing graph is a comb, a complete graph minus two independent edges, or a cycle of length 5.     

On Stockmeyer's non-reconstructible tournaments

Stockmeyer (The falsity of the reconstruction conjecture for tournaments. J. Graph Theory, 1 (1977) 19–25) constructed an infinite family of non-reconstructible tournaments. It has been known for some time that there is a flaw in the proof given there. We provide an alternative constructive proof which is much simpler than the original proof.     

Problems and conjectures concerning connectivity, paths, trees and cycles in tournament-like digraphs

In this paper we collect a substantial number of challenging open problems and conjectures on connectivity, paths, trees and cycles in tournaments and classes of digraphs which contain tournaments as a subclass. The list is by no means exhaustive but is meant to show that the area has a large number of interesting open problems. We also mention problems for general digraphs when they are relevant in the context.     

Hamiltonian-connected tournaments

We characterize weakly Hamiltonian-connected tournaments and weakly panconnected tournaments completely and we apply these results to cycles and bypasses in tournaments with given irregularity, in particular, in regular and almost regular tournaments. We give a sufficient condition in terms of local and global connectivity for a Hamiltonian path with prescribed initial and terminal vertex. From this result we deduce that every 4-connected tournament is strongly Hamiltonian-connected and that every edge of a 3-connected tournament is contained in a Hamiltonian cycle of the tournament and we describe infinite families of tournaments demonstrating that these results are best possible.     

Quasi-random tournaments

We introduce a large class of tournament properties, all of which are shared by almost all random tournaments. These properties, which we term “quasi-random,” have the property that tournaments possessing any one of the properties must of necessity possess them all. In contrast to random tournaments, however, it is often very easy to verify that a particular family of tournaments satisfies one of the quasi-random properties, thereby giving explicit tournaments with “random-like” behavior. This paper continues an approach initiated in several earlier papers of the authors where analogous results for graphs (with R.M. Wilson) and hypergraphs are proved.     

Neighborhood monotonicity, the extended Zermelo model, and symmetric knockout tournaments

In this paper, neighborhood monotonicity is presented as a natural property for methods of ranking generalized tournaments (directed graphs with weighted edges). An extension of Zermelo’s classical method of ranking tournaments is shown to have this property. An estimate is made of the proportion of ordered pairs that all neighborhood-monotonic rankings of symmetric knockout tournaments have in common. Finally, numerical evidence for the asymptotic behavior of the extended Zermelo ranking of symmetric knockout tournaments is presented.     

Obstructions for acyclic local tournament orientation completions

The orientation completion problem for a fixed class of oriented graphs asks whether a given partially oriented graph can be completed to an oriented graph in the class. Orientation completion problems have been studied recently for several classes of oriented graphs, including local tournaments. Local tournaments are intimately related to proper circular-arc graphs and proper interval graphs. In particular, proper interval graphs are precisely those which can be oriented as acyclic local tournaments. In this paper we determine all obstructions for acyclic local tournament orientation completions. These are in a sense minimal partially oriented graphs that cannot be completed to acyclic local tournaments. Our results imply that a polynomial time certifying algorithm exists for the acyclic local tournament orientation completion problem.     

政府支出能规避对外贸易风险吗？——来自“灵猫六国”的经验证据

以"灵猫六国"等新兴经济体为样本,采用边界检验和自回归分布滞后模型等方法,从时间序列视角研究了一国政府支出与对外开放程度之间的长期影响关系,并验证了Rodrik(1998)提出的政府支出规避对外贸易风险假说的存在性.研究结果显示:除越南以外,其他五国的政府支出与对外开放程度之间均存在着显著的长期影响关系,但政府支出规避对外贸易风险假说仅在中国、印尼和南非三个国家成立.继Islam(2004)发现该假说在OECD国家不成立之后,本文的研究结论又表明政府支出规避对外贸易风险假说在"灵猫六国"等新兴经济体中同样也不具有普遍性.     

(1, 2)-Symplectic Metrics, Flag Manifolds and Tournaments

It has recently been shown by Mo and Negreiros that a necessarycondition for an invariant almost complex structure on the complexfull flag manifold F(n) to admit a (1, 2)-symplectic invariantmetric is that its associated tournament be cone-free. In thispaper, a canonical stair-shaped form is given for such tournaments,and this is applied to show that the condition is also sufficient;in the process, all the associated (1, 2)-symplectic metricsare described. 2000 Mathematics Subject Classification 53C55,05C20, 58E20.     

Constructions Over Tournaments

We investigate tournaments that are projective in the variety that they generate, and free algebras over partial tournaments in that variety. We prove that the variety determined by three-variable equations of tournaments is not locally finite. We also construct infinitely many finite, pairwise incomparable simple tournaments.     

Resistance and conductance in structured Zermelo tournaments

The mathematical model that Zermelo developed for ranking by paired comparisons and that was later popularized by Bradley and Terry has several attractive theoretical properties, but computation of the associated ratings may involve solution of a system of several high-degree polynomial equations in several variables. This paper describes how to define quantities analogous to electrical resistance and conductance for certain generalized tournaments in such a way that these quantities are well-behaved with respect to certain types of decomposition of tournaments and permit comparison of the ratings of pairs of nodes. Application of this theory is illustrated through consideration of specific examples.