Provide $360,000 of total funding equally spilt amongst 6 PhD projects ($60,000 per project) on the mathematics of sport, gambling and conflicts. The 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. 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 over in a day or 6 days (or both), players may resort back to the style of play prior to 2015 which would 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 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 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 is about 25%. By building a Markov Chain model this project will develop alternate scoring systems to reduce the draw probability based on the amount of time played and what lead is required to win at a particular time in the game. Countback mechanisms are also investigated to determine a winner if scores are level after a fixed amount of playing time. The results 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. By modelling lawsuits as a casino game, an arbitration value is obtained in a litigation game, where the amount awarded to the victim is less than expectation and shown to be “fairer” when compared to the amount obtained using the Von Neumann and Morgenstern game theory framework. Game theory with risk is then applied to two-person nonzero-sum games.

**Game theory strategies in sport with a particular focus on 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 then applied to other sports.

**Mathematical modelling out in sport**

Recursion modelling in Excel is used to obtain chances of winning and parameters of distribution of points in a match from any score line for tennis, volleyball and pickleball.

**Caribbean stud poker when collusion exists**

Analysis is carried out on Caribbean Stud Poker by colluding non-verbal information to demonstrate that advantage play is possible, and with some simple strategies can yield a 1.1% advantage.