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I. Preston and J. Thomas, 2000, "Batting Strategy in Limited Overs Cricket" The Statistician (Journal of the Royal Statistical Society: Series D) 49, 95-106.

Cricket is a game with many interesting strategic aspects. This paper attempts to understand batting strategies employed in limited overs cricket games. We investigate both those strategies that are most likely to lead to victory and those that are actually played. There are several reasons why these issues are of interest. An understanding of optimum strategies is of value in an obvious practical sense to those who play the game and want to win. Strategies played also determine the way in which probabilities of victory evolve over the course of games. An understanding of this may help in judging how teams stand at any point in a game and may therefore also be relevant to the design of rules for adjusting scores in rain-interrupted games. In principle, it could also be useful in the evaluation of player performance. Our results suggest that optimum strategy depends on the way in which the likelihood of dismissal depends upon attempted scoring rate. Our empirical analysis of actual batting behaviour allows us to estimate the parameters of this relationshi.

In this paper we consider the question of optimum batting strategy in a simplified dynamic programming representation of a limited overs cricket game. Clarke has previously investigated optimum strategy in limited overs cricket using dynamic programming methods and has simulated solutions numerically. We extend this by imposing a greater theoretical structure on the problem. This allows us to derive conclusions which can be generalised beyond his particular numerical specifications and which hopefully illuminate the source of his results. For example, his conclusion that the popular strategy of building up runs slowly has little justification, at least in the second innings, is mirrored in our findings. We suggest reasons however why such a strategy may have more justification in the first innings.

We also use data on English county-level limited overs games to estimate a model of cricketers' behaviour. Bairam, Howells and Turner have investigated the empirical relationship between measures of performance, including batting and bowling inputs, and both victory probabilities and points accumulation. This is a rather different exercise however from investigating the aspects of within-match strategy considered here. Our statistical framework takes the form of an interesting variant on conventional survival analysis models. Kimber and Hansford have previously applied survival analysis to data on individual innings. Our work goes further in its attempt to relate estimates to the effects of covariates on hazards within the context of a stuctural model of batting strategy.

Duckworth and Lewis have developed and estimated a target reduction rule based on ideas of neutrality between teams and balance in run-scoring resources. This rule, adopted by several cricketing authorities, is undoubtedly a valuable improvement on crude proportional reduction rules and we hope that the sort of analysis in this paper can illuminate its possible foundations in the context of a model of cricketing behaviour.

I. Preston and J. Thomas, 2002, "Rain Rules for Limited Overs Cricket," paper presented to The Statistician (Journal of the Royal Statistical Society: Series D), 51, 189-202.

The game of cricket suffers from a requirement for clement weather of a type rarely guaranteed by the climates in which the game is played. Interruptions due to poor weather have proven a particular difficulty for design of rules for the limited overs version of the game, since such interruptions affect the fortunes of the two teams very differently depending upon the point in the game at which they occur and, if prolonged and ignored, would destroy the contest and therefore much of the interest of the game. Many rules have been suggested for adjusting the terms of the game in such a way as to maintain fairness between the two teams in cases of weather affected matches. However none have proven wholly satisfactory. The methods, their strengths and their weaknesses are summarised by Duckworth and Lewis, whose own rule has won attention and increasingly widespread adoption.

The Duckworth-Lewis rule is based on the notion of correcting for loss of "run-scoring resources.'' In this it can be seen as a development of the idea underlying simpler rules based on run rates. We are sympathetic to the general notion of run-scoring resources but there are several dimensions to resources - primarily balls available and wickets standing - and the question is how to value them appropriately against each other. Duckworth and Lewis appear implicitly to interpret the value of run-scoring resources as ``average total score'' given match circumstances. We explore an alternative approach in which they are valued according to their contribution to probability of victory. That is to say we use the shadow values in the dynamic programming problem that the teams are attempting to solve.

This approach has several advantages. For one thing a rain rule which adjusts runs so as to preserve teams' advantage judged according to their own objectives appears to us to capture well the notion of fairness. Secondly, while application would require complex calculations, the underlying concept is easily understood. Thirdly, rules such as this which recognise the team's own objectives are tactically neutral in the sense that they will give teams no incentive to vary strategy in anticipation of rain. Unless probabilities of victory after rain rule adjustment are monotonically increasing in the probabilities before the interruption then teams expecting rain may adjust behaviour to seek to maximise the probabilities after the anticipated application of the rule. The main disadvantage of the rule is probably the difficulty, explained below, in applying it consistently to cases where play has to be abandoned.

Rain rules based on the idea of keeping probabilities of victory constant across interruptions offer a feasible, intelligible and fair alternative to other rules. Furthermore even if the case for their use is not regarded as persuasive it is still interesting to compare them to alternative rules to judge how the application of those other rules is likely to change the balance of advantage within a game. For example, simulations demonstrate that under the widely used Duckworth-Lewis rule, probabilities of victory after an interruption are not in a one-to-one relationship with probabilities before, teams already doing well at the point of interruption are favoured and anticipation of rain may lead teams towards cautious batting.

I. Preston, S. F. Ross and S. Szymanski, 2000, "Seizing the Moment: A Blueprint for Reform of World Cricket," paper presented to ESRC Study Group, Economics of Sport, Arts and Leisure, 13 December 2000.

The crisis in cricket threatens the future of the oldest organised team sport . The revelations of the King Commission in South Africa included the admission by the former South African captain, Hansie Cronje, that he had accepted bribes for fixing matches and suborned the corruption of other team members. The Qayyum Report in Pakistan found the former Pakistani captain, Salim Malik, to have fixed matches and reported a failure to cooperate with its enquiry by another former captain, Wasim Akram. Most recently, the investigation by the Central Bureau of Investigation of the Indian Police Force pointed to extensive corruption involving among others the former Indian captain, Mohammed Azharuddin, and also alleged the improper involvement of other international players, including former England, West Indies, Sri Lanka and New Zealand captains and players of Australia, with bookmakers. All of this has seriously undermined the credibility of the international game. Moreover, while the investigation commissioned by the governing body of world cricket, the International Cricket Council (ICC), and led by the former UK Metropolitan Police Commissioner, Sir Paul Condon, may draw a line under the current scandals, there must be a real risk that it will uncover even more evidence of wrongdoing.

In the short term it is clear that cricket will have to deal with the cheats. Given the relatively small number of genuine stars in the game, this process will inevitably prove difficult. The reluctance of national cricket boards to take decisive and immediate action against alleged culprits even where they have admitted talking to bookmakers, highlights the dilemma of the authorities. Wedded to a not unnatural concern to retain those stars that draw the crowds, there is also genuine sympathy for individuals whose careers may have been distinguished and whose lapses may by comparison have been minor.

In the longer term, there is need to reform the structure of the international game to ensure that the conditions that encouraged cheating to flourish are eliminated. In this paper we argue that the fundamental problem is the meagre financial rewards received by the top international cricketers. This view has already been accepted by some senior figures in the cricket hierarchy. The new chairman of the MCC, Lord Alexander of Weedon, has reportedly called for cricketers to be paid more so that they are less likely to be tempted by match-fixing offers (BBC website, 17 November 2000). However the response of the England and Wales Cricket Board shows that acceptance is not widespread: "The England and Wales Cricket Board has noted the remarks reportedly made by MCC President, Lord Alexander of Weedon, that international cricketers should be paid more so that they are less likely to be tempted by offers to be involved in match-fixing. The ECB Management Board takes the view that the pay of many international cricketers has increased significantly in recent times and they are now well rewarded for their skill and expertise and that, in any event, there is no justification for any international cricketer to succumb to corruption. (ECB Press Release 18 November 2000)" We believe high salaries would be a significant disincentive to cheating, as we believe it is in most other major sports. The incentive to cheat is increased when there is an imbalance between the financial status of the players and the financial returns to the sport as whole. In cricket the top players earn a much smaller fraction of the total revenues generated by their efforts than is the case in other major international sports.

Low pay for international cricketers is the consequence of an organisational structure that uses the revenue from popular international matches to subsidise domestic competitions and that undermines the bargaining position of top players over pay. Removing the cross-subsidy might provide a temporary respite, but since competitions are not capable of generating enough income to cover costs they would have to be severely curtailed. This in turn would limit the supply of new talent coming into the game and hence undermine its long term future.

We believe that there is an alternative way to deal with the underpayment of professional cricket players. In most sports outside of North America income and thereby professional salaries are boosted by the existence of international club competitions that complement the international representative game. The introduction of international club cricket would in our view add a significant dimension to international cricket that would command spectator and media attention. Competition among international clubs for the best players would be a sure way to bid up the salaries of international cricketers.

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Ian Preston
Department of Economics
University College London
Gower Street, London WC1E 6BT