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The Strategic Games Foundation offers funding to complete a PhD with a topic related to the mathematics of sport, gambling and conflicts. Some suggestions for projects are outlined below.
Methods to reduce the draw probability in test cricket
The percentage of test matches resulting in a draw in 2015 is about 15% (although in 2003 it was around 25%). The draw probability can be reduced in test cricket by increasing the number of allowable overs, where the current system has a maximum of about 450 overs (90 overs over 5 days). Given that ODI cricket plays a maximum of 100 overs in a day, it could then appear ‘practical’ to extend the number of overs in test cricket from 90 to 100 overs per day. Also, an additional 6th day could also appear to be a ‘practical’ strategy to reduce the draw probability. As above, the percentage of tests that resulted in a draw decreased from about 25% in 2003 to about 15% in 2015. These statistics indicate that players are playing more aggressively to score runs to increase their chances of winning the match due to the limited number of overs available to bowl the opposing side out twice to reduce the draw probability, and this strategy inadvertently increases the chances of the opposing side winning since by scoring runs faster there may be an increased chance of losing wickets. Thus, it appears that by extending the number of allowable overs in test cricket by possibly 100 overs in a day or 6 days (or both), players may resort back to the style of play prior to 2015 which could prevent increasing the chances of the opposing side winning but still maintaining their win probability. This appears to be in the interest of improving the quality of the game of cricket. There is also the situation where a team batting second scored fewer runs compared to the other team in the 1st innings and is unable to win the test in the 2nd innings due to not having enough overs remaining and is thus playing defensively to play for a draw. It would again seem in the interest of cricket particularly for spectators to extend the length of allowable overs in test cricket, which would then give this team the opportunity to have a ‘realistic’ chance of winning even with say a collapse of the 1st innings. Another method to reduce the draw probability in test cricket is by playing only one innings for each side (compared to the standard two innings) and a further method to reduce the draw probability is then to introduce a Duckworth/Lewis/Stern rule on when to reduce a standard two innings match to one innings based on rain delays amongst other factors. This project will obtain quantitative results on the methods outlined above using analytic and simulation models which could then be used by regulators to make informed decisions on test cricket scoring systems.
Methods to reduce the draw probability in soccer
The percentage of soccer matches resulting in a draw after the standard 90 minutes of play is about 25%. Often extra time (typically 30 minutes) is used to determine a winner in soccer and if teams are still level after extra time, then a penalty shootout is used to determine a winner. This project will consider alternate methods of both to reduce the draw probability and determine a winner. More extra time to the typical 30 minutes and the golden point rule are analyzed in this project. A more innovative method to reduce the draw probability is by implementing a countback mechanism based on the number of scoring shots blocked by the goalie that would have scored a goal given that teams are level after the final siren. This project will obtain quantitative results on the methods outlined above using analytic and simulation models which could then be used by regulators to make informed decisions on soccer scoring systems.
Game theory with risk
The Minimax Theorem is the most recognized theorem for determining strategies in a two-person zero-sum game. Other common strategies exist such as the maximax principle and minimize the maximum regret principle. All these strategies follow the Von Neumann and Morgenstern linearity axiom which states that numbers in the game matrix must be cardinal utilities and can be transformed by any positive linear function f(x)=ax+b, a>0 without changing the information they convey. This project defines risk-averse strategies for a two-person zero-sum game where the linearity axiom may not hold, and subsequently an equilibrium is obtained where the ‘value’ of the game for the favourable player is less than the ‘value’ under the Nash Equilibrium expectation. With connections to gambling theory, there is evidence to show why it can be optimal for the favourable player to adopt risk-averse strategies. Game theory with risk is also applied to two-person nonzero-sum games.
Game theory strategies in tennis
Game theoretic solutions in tennis are undertaken to determine how often a player should take more risk on the second serve, how often to serve-and-volley and how often to chip-and-charge to the net off a second serve amongst other problems. Similar game theory strategies are also applied to other sports.
Mathematical modelling in volleyball
A Markov Chain model is applied to volleyball to calculate win probabilities and mean lengths with the associated variances, conditional on both the scoreboard and the server. A feature of this model is that it predicts outcomes conditional on both the scoreboard and the server. The inclusion of the server in the event space is an essential requirement of this model, and arises from the rule in volleyball that the winner of each point must serve on the following point. The average probability of a team winning a point on serve is less than 0.5, and so rotation of serve is commonplace. The key to the analysis of an evenly contested set is the observation that, from the situation the scores are level (after at least 46 points have been played), the team that wins the set must eventually win two successive points. If the two points are shared then the score is level once again, although a rotation of server may have occurred. This scoring structure when the scores are level after 46 points have been played combined with the method of rotating the serve combined, distinguishes volleyball from other racket sports such as tennis, squash, badminton and table tennis.
Caribbean Stud Poker when collusion exists
Analysis is carried out on Caribbean Stud Poker (CSP) by colluding non-verbal information to demonstrate that advantage play is possible, and with some simple strategies CSP can yield a 1.1% player advantage.