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The Tennis Problem |
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| The hierarchical scoring structure of tennis
(game, set, match) results in not all points
having equal value to each side, and thus
seeming to call for different strategies
in order to try to proceed through each point
expending the effort that the player specific
value of the point justifies. There are papers that have investigated some of the issues in tennis (e.g. [1], [2]). It can be shown numerically that there is some advantage in selectively reducing effort expended to win some points in the greater goal of seeking to win the match. Attached is a paper draft [3] that investigates some of the surprising effects on strategy/tactics arising from the hierarchical scoring structure and its effects mentioned above. The numerical method has been implemented in Excel using a simple probability based, energy attrition model. The assumption is that one of the players is playing to maximal effort at all times and the other can vary effort in response to the situation. The problem then reduces to one of optimisation for the player who can control their effort. The numerical model suggests a strategic advantage effected through the strategic/tactical process of establishing criteria to put certain levels of effort into the play for a particular point. The more general problem, and one that would be more useful to solve, relates to the situation where each player can choose to vary point-wise strategy in response to the match situation. A solution to this problem would seem to be useful in developing an understanding of the sensitivity of outcomes in military situations to choice of strategic options during the various hierarchical levels of a conflict. For example, commanders in planning a course of action are often faced with the dilemma of sending in a reserve force behind a leading force. The dilemma is that an adversary expecting a reserve may either reinforce their defence or leave the battlefield altogether, either way the effect of surprise may be nullified and resources wasted. This situation is conceptually identical to the dilemma faced by a tennis player who must win a difficult point, but is on a second serve. Should the player serve safely, the opponent may be waiting and take advantage of an easy shot, but if the server serves with maximum force the opponent may be put in a difficult position, but there is a greater risk of a double fault. Such situations arise in many other conflict situations, for example in economics and trade negotiations. The issue here is take into account the effect of the hierarchical nature of the problem. An interesting solution would enable evaluation of the effect of variation of the rules of the game, such as variation of the number of hierarchical levels in the game, or the scoring rules. This would be beneficial because it would develop means to enable investigation of the sensitivity of the target situation to the wide range of variations of situation that may be encountered. Desired Outcomes Our interest is to obtain insights in relation to the following issues: a. The non-equivalence of value of the points depending on the current score in the game, set and match. b. The effect on the probability of winning the match arising from depleting available capability through the effort to win the point. c. The ability to generalise from tennis to a more complex game structure (i.e. where there is not the convenience of discrete play events between just the two equivalent players or teams that are present in tennis.) d. The definition of a model of match outcome into which the effect of morale or other psychological effects can be incorporated. References [1] M. Walker & J. Wooders, Equilibrium play in matches: Binary Markov games, 7 July 2000, www.u.arizona.edu/~mwalker/BMG.pdf [2] M. Walker & J. Wooders, Minimax play at Wimbledon, 7 November 1998, American Economic Review, www.u.arizona.edu/~mwalker/ WimbldnScan990910.pdf [3] Ferris, submitted, Emergence: An Illustration of the Concept for Education of Young Students, INCOSE 2003, Washington, July. |
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