Game Theory in the Decision Review System in the sport of Cricket.
Cricket is an historic game aging back to the early 1900s in England. It is referred to as the “gentleman’s game”. From its conception, cricket has come a long way with new formats, new rules and new playing styles. However, along with the rise of technology, cricket has introduced DRS (Decision Review System). Cricket has always relied on umpires to make close-call decisions whether a batsman is out or not. A few of these close-call decisions were LBW (Leg before wicket: if the ball hits the log and was eventually going to hit the wicket otherwise) and caught behind (the ball slightly touched the bat and was caught by the keeper). When an umpire makes a decision, there is always a chance he might get it wrong especially in such cases as he or she is human after all. Hence, by integrating modern technology into cricket, we have ways to find out and verify the decisions made by umpires to see if it the batsman is actually out or not.
However, the way and how this technology is used is a huge part of how modern day cricket is played. There are two teams, the batting team and the bowling team. Once the bowling team bowls a ball and they think the batsman is out through the two ways mentioned above (LBW or caught behind), they appeal to the umpire and the umpire makes his decision. Now, if the decision is out, the batsman is given 15 seconds to challenge and review the decision by the umpire if he firmly believes he is not out and vice versa if the decision is not out, the bowling team has 15 seconds to consult and challenge the decision if they feel like the umpire got it wrong. After reviewing, the umpires use technology like ultra-edge and ball tracking to confirm if the batsman was out or not.
If this is the case, then why don’t the teams review every decision that goes against them? This is because the number of reviews for each team is limited to 2 in the longer formats 0f the game and 1 in the shorter format. If a team challenges the umpires decision and uses a review, if the original decision given by the umpire was correct, they would lose their review and if their review is successful and the decision is overturned they keep their review.
This is where game theory comes into play where a team weighs the payoffs for using a review to overturn an incorrect decision by the umpire. However, this is influenced by many other factors how confident the players are that the umpire’s decision is incorrect, the stage of the game whether they feel like losing a review could harm them in the long run as they might need it later, how desperate they are to get a certain batsman out if he’s performing well. By keeping all these factors in mind, my knowledge of the sport and knowledge about game theory, I came up with the following pay-off matrix. Keep in mind, this pay-off matrix is based under a scenario which is from the perspective of the bowling team who think that the batsman is out and the umpire gives a decision of not-out and the players are considering using up one of their reviews. The first number is the payoff for the fielding team, and the second number is the payoff for the batting team.
As we can see, a Nash pure equilibrium cannot be set up here as the bowling team can never fully know whether the batsman is out or not.
Now if we take external factors like the stage of play into account too like mentioned above, we can come up with probabilities p for whether the team takes a review and 1-p for if they do not and that way they can calculate their best possible decision according to the scenario.
Citation: https://www.economist.com/game-theory/2012/02/06/up-for-review