Towards fairer measures of player performance in one-day cricket

Traditionally, evaluation of individual player performances in one-day cricket has been based on measures such as batting and bowling averages, and strike and economy rates. It is recognized, within the game of cricket, that such measures have severe limitations in assessing the true performances and abilities of players. Whenever a player's performance statistics are quoted, there is nearly always some verbal qualification that is necessary in order to place the statistics into context. Such comments include the stage of an innings a player usually bats or bowls, as this significantly affects a player's opportunity for scoring runs or taking wickets. Further, traditional measures do not allow the comparison of the skills of batting and bowling as they are based on incompatible scales. This paper uses the well-established Duckworth/Lewis methodology to create alternative measures of player performance. These measures, it is argued, have the potential to reflect better the performances of players having regard for the stages of innings that runs are scored or conceded and wickets taken or lost. The proposed measures could impact in evaluating player performance for a particular match, a series of matches, or in the longer term over players' careers. Further, the methodology enables an objective measure to be created that compares and combines the performances of batsmen and bowlers and of a player's all-round ability. Data from international matches are used to illustrate the methodology and how it compares with existing measures.     

To bat or not to bat: An examination of match outcomes in day-night limited overs cricket

The tradition of tossing a coin to decide who bats first in a cricket match introduces a randomly assigned advantage to one team that is unique in sporting contests. The potential importance of the toss rule in determining cricket match results has been the subject of some investigation, which is further advanced in this paper that utilizes a data set relating to the increasingly popular, but contentious, day-night form of limited overs cricket as played at international level. We employ logit regression models to examine the effects of winning the toss and choice of batting order on the likelihood of a match victory, while controlling for home advantage and (relative) team quality. Our findings suggest that winning the toss and batting first increases the probability of winning whereas winning the toss and bowling first does not.     

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.     

Modelling a baseball game to optimise pitcher substitution strategies incorporating handedness of players

^** Email: hirotsu.nobuyoshi{at}jiss.naash.go.jp^*** Email: m.wright{at}lancaster.ac.uk This paper proposes a method for identifying the optimal strategyfor substituting players in a baseball game, taking into considerationthe handedness of players, which is one of the main factorsin terms of managerial decision-making for substitution. Usinga Markov chain model, we incorporate the effect of the handednessof players by introducing the concept of the defensive earnedrun average as a measure of the defensive ability of pitchersand calibrating the batting probabilities of players dependingon their handedness. We then develop a dynamic programming formulationincluding the effect of the handedness of players. This methodis illustrated using a match based on the real line-ups of theColorado Rockies and the San Francisco Giants in the NationalLeague of Major League Baseball, especially focusing on theintroduction of a relief pitcher in consideration with his handedness.     

Modelling match outcomes and decision support for setting a final innings target in test cricket

^** E-mail: p.a.scarf{at}salford.ac.uk This paper considers the problem of timing the declaration ofthe third innings in test cricket. Data on the outcomes of recenttest matches are analysed in order to develop simple decisionsupport tools. The first of these tools presents match outcomeprobabilities given the position of the match at a potentialdeclaration point. These probabilities are determined usinga multinomial logistic regression model that is fitted to thetest match data. This idea is then extended to consider progresstowards a declaration—match outcome probabilities areconsidered as a function of target aimed for and run-rate. Thedecision tools have been implemented on a spreadsheet and examplecalculations are presented. The modelling described has thepotential for practical use in test matches.     

Power Measures and Solutions for Games Under Precedence Constraints

Games under precedence constraints model situations, where players in a cooperative transferable utility game belong to some hierarchical structure, which is represented by an acyclic digraph (partial order). In this paper, we introduce the class of precedence power solutions for games under precedence constraints. These solutions are obtained by allocating the dividends in the game proportional to some power measure for acyclic digraphs. We show that all these solutions satisfy the desirable axiom of irrelevant player independence, which establishes that the payoffs assigned to relevant players are not affected by the presence of irrelevant players. We axiomatize these precedence power solutions using irrelevant player independence and an axiom that uses a digraph power measure. We give special attention to the hierarchical solution, which applies the hierarchical measure. We argue how this solution is related to the known precedence Shapley value, which does not satisfy irrelevant player independence, and thus is not a precedence power solution. We also axiomatize the hierarchical measure as a digraph power measure.     

Zero-sum constrained stochastic games with independent state processes

We consider a zero-sum stochastic game with side constraints for both players with a special structure. There are two independent controlled Markov chains, one for each player. The transition probabilities of the chain associated with a player as well as the related side constraints depend only on the actions of the corresponding player; the side constraints also depend on the player’s controlled chain. The global cost that player 1 wishes to minimize and that player 2 wishes to maximize, depend however on the actions and Markov chains of both players. We obtain a linear programming formulations that allows to compute the value and saddle point policies for this problem. We illustrate the theoretical results through a zero-sum stochastic game in wireless networks in which each player has power constraints     

A remark on the Harsanyi-Selten theory of equilibrium selection

We consider a class of 3-person games in normal form with two pure strategies for each player and two strict equilibrium points. To select one of these two strict equilibrium points as the solution, the equilibrium selection theory of Harsanyi and Selten is applied. The games are constructed in such a way that the a priori probabilities reflect somewhat poorly the risk situation of the players. It is argued and illustrated by examples that this might yield unreasonable results. The a priori probabilities would describe the risk situation of the players more completely if their definition were not based on the expectation of correlated decision behavior.     

Axiomatizations of Banzhaf permission values for games with a permission structure

In games with a permission structure it is assumed that players in a cooperative transferable utility game are hierarchically ordered in the sense that there are players that need permission from other players before they are allowed to cooperate. We provide axiomatic characterizations of Banzhaf permission values being solutions that are obtained by applying the Banzhaf value to modified TU-games. In these characterizations we use power- and player split neutrality properties. These properties state that splitting a player’s authority and/or contribution over two players does not change the sum of their payoffs.     

Sharing beliefs about actions

We consider a strategic situation in which each player may not know the probability distribution governing the information structures of his opponents, and consequently his beliefs about opponents' action choices are represented by a set of probability measures. Suppose that beliefs of all the players are common knowledge. Then for any subset of players, the marginal beliefs of those players (about the action choices of their common opponents) must share at least one probability measure.     

Optimal batting orders in cricket

Cricket teams are usually listed on the scoreboard in expected batting order, andalthough captains can choose to send in any of the remaining batsmen when awicket falls, they rarely depart from the usual batting order. By optimizing asimplified model using dynamic programming, this paper shows that in all formsof cricket significant increases in expected score result if captains allow avariable batting order and base their decision on the state of the game, ratherthan using a set batting order.     

Optimal challenges in tennis

The use of technology in sport to assist umpires has been gradually introduced into several sports. This has now been extended to allow players to call upon technology to arbitrate when they disagree with the umpire's decision. Both tennis and cricket now allow the players to challenge a doubtful decision, which is reversed if the evidence shows it to be incorrect. However, the number of challenges is limited, and players must balance any possible immediate gain with the loss of a future right to challenge. With similar challenge rules expected to be introduced in other sports, this situation has been a motivation to consider challenges more widely. We use Dynamic Programming to investigate the optimal challenge strategy and obtain some general rules. In a traditional set of tennis, players should be more aggressive in challenging in the latter stages of the games and sets, and when their opponent is ahead. Optimal challenge strategy can increase a player's chance of winning an otherwise even five-set match to 59%.     

Generalized Gambler’s Ruin Problem: Explicit Formulas via Siegmund Duality

We give explicit formulas for ruin probabilities in a multidimensional Generalized Gambler’s ruin problem. The generalization is best interpreted as a game of one player against d other players, allowing arbitrary winning and losing probabilities (including ties) depending on the current fortune with particular player. It includes many previous other generalizations as special cases. Instead of usually utilized first-step-like analysis we involve dualities between Markov chains. We give general procedure for solving ruin-like problems utilizing Siegmund duality in Markov chains for partially ordered state spaces studied recently in context of Möbius monotonicity.     

A Robust Heuristic for Batting Order Optimization Under Uncertainty

Baseball teams are faced with a difficult scheduling problem every day: given a set of nine players, find the optimal sequence in which they should bat. Effective optimization can increase a team's win total by up to 3 wins per season, and 10% of all Major League teams missed the playoffs by 3 or less wins in 1998. Considering the recent $252 million contract given to one player, it is obvious that baseball is a serious business in which making the playoffs has large financial benefits. Using the insights gleaned from a Markov chain model of baseball, we propose a batting order optimization heuristic that performs 1,000 times faster than the previous best heuristic for this problem. Our algorithm generates batting orders that (i) are optimal or near-optimal, and (ii) remain robust under uncertainty in skill measurement.     

A mathematical analysis of badminton scoring systems

The International Badminton Federation recently introduced rule changes to make the game faster and more entertaining, by influencing how players score points and win games. We assess the fairness of both systems by applying combinatorics, probability theory and simulation to extrapolate known probabilities of winning individual rallies into probabilities of winning games and matches. We also measure how effective the rule changes are by comparing the numbers of rallies per game and the scoring patterns within each game, using data from the 2006 Commonwealth Games to demonstrate our results. We then develop subjective Bayesian methods for specifying the probabilities of winning. Finally, we describe how to propagate this information with observed data to determine posterior predictive distributions that enable us to predict match outcomes before and during play.     

Emergence and nonemergence of alternating offers in bilateral bargaining

We study in what circumstance players alternate offers in bilateral bargaining. To examine this question, we suppose that players choose whether to take the initiative in each period. The player who tries to take the initiative is able to make an offer only when the other player does not. The probability that a player tries to take the initiative is referred to as the frequency of initiative taking. We assume that this is conditioned on mutually observable states and is, once chosen, unchangeable. When players make their frequency of initiative taking dependent on the identity of the latest proposer, the players alternate their offers (possibly with some stochastic delay). In contrast, when players always use the same frequency of initiative taking, or when players only distinguish odd-numbered from even-numbered periods for the frequency of initiative taking, both players constantly try to take the initiative. Consequently, an impasse arises.     

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.     

Repeated congestion games with bounded rationality

We consider a repeated congestion game with imperfect monitoring. At each stage, each player chooses to use some facilities and pays a cost that increases with the congestion. Two versions of the model are examined: a public monitoring setting where agents observe the cost of each available facility, and a private monitoring one where players observe only the cost of the facilities they use. A partial folk theorem holds: a Pareto-optimal outcome may result from selfish behavior and be sustained by a belief-free equilibrium of the repeated game. We prove this result assuming that players use strategies of bounded complexity and we estimate the strategic complexity needed to achieve efficiency. It is shown that, under some conditions on the number of players and the structure of the game, this complexity is very small even under private monitoring. The case of network routing games is examined in detail.     

De-synchronized clocks in preemption games with risky prospects

We study an optimal timing decision problem where an agent endowed with a risky investment opportunity trades the benefits of waiting for additional information against a potential loss in first-mover advantage. The players’ clocks are de-synchronized in that they learn of the investment opportunity at different times. Thus, the model captures situations where players are heterogeneous with respect to the amount of information that they possess at any instant. In this framework, previous literature has uncovered an inverted-U shaped relationship between a player’s equilibrium expected expenditures and the measure of his competitors. This result no longer holds when the increase in the measure of players leads to a decrease in the degree of clock synchronization in the game. We show that the result reemerges if information arrives only at discrete times, and thus, a player’s strategic beliefs are updated between decision times in a measurably meaningful way.     

Pyramidal values

We propose and analyze a new type of values for cooperative TU-games, which we call pyramidal values. Assuming that the grand coalition is sequentially formed, and all orderings are equally likely, we define a pyramidal value to be any expected payoff in which the entrant player receives a salary, and the rest of his marginal contribution to the just formed coalition is distributed among the incumbent players. We relate the pyramidal-type sharing scheme we propose with other sharing schemes, and we also obtain some known values by means of this kind of pyramidal procedures. In particular, we show that the Shapley value can be obtained by means of an interesting pyramidal procedure that distributes nonzero dividends among the incumbents. As a result, we obtain an alternative formulation of the Shapley value based on a measure of complementarity between two players. Finally, we introduce the family of proportional pyramidal values, in which an incumbent receives a dividend in proportion to his initial investment, measured by means of his marginal contribution.