We continue with our very low key World Cup theme. It’s so low key that you’d be hardpressed to call it a theme. But we don’t march to the beat of any one.
Cricket fans are usually pretty nerdy. Here is a fan who is exceptionally so. Shehan sends this one in to once and for all explain what the fuck went down in Mumbai before the final started.
Many criticised Sri Lanka’s tactics in the final of the recently concluded cricket world cup from the bowling changes and field placements to their decision to bat first after winning the toss. Least amongst these criticisms was the team combination Sri Lanka went in with which included four changes from the semi final.
As in most sports, the decisions made both prior and during the game by one team in cricket have a direct impact on the welfare of the opposing team and vice-versa and this is mutually recognised. This is known as strategic interdependence. Game theory is a mathematical analysis of strategy where a game is a model of strategic interdependence which attempts to predict behaviour and advice strategy.
In this report I will attempt, using game theory to study the choices Sri Lankan management would have considered in team selection for this crucial game. In section 2.0 of this report I will address the decision problems faced by team management whilst in section 3.0 I will construct a model to study Sri Lanka’s possible actions. In section 4.0 I will analyse the model developed in the previous section to identify Sri Lanka did indeed get their team wrong. I will emphasise the limitations of this study in section 5.0 of this report whilst offering my conclusions in section 6.0.
2.0 SRI LANKA’S DECISION PROBLEM
Most decisions are difficult as sometimes they involve an element of risk when nature interferes, they could be strategic when they are dependent on the actions of others and they could be complex when the information available can’t be comprehended. Some decisions are a combination of these three factors.
2.1 The strategic impact on team selection
Going in to a cricket match with the best possible mix of specialist batsmen and bowlers is vital given the opposition and the information you have and don’t have about them. For this final, Sri Lanka who was clear underdogs knew that India has an extremely strong batting line up, most of who were in top form during the tournament. Their main objective would be to select the best team combination that would give them the best chance of countering India’s strengths and with a bit of luck (as opposed to a lot of luck) even win the game. So should they too pack their team with batsman to try and out-bat India or should they go in with more bowlers and try contain India’s batsman and bank on the selected few batsman doing the job?
India was also aware that the Sri Lankan bowlers had performed the best out of all the teams in the tournament, so what team combination would India go in with? Would they bank on their top six batsman to score the runs and play five bowlers to contain the Sri Lankan batsmen’s score or go in with the extra specialist batsman to counter for Sri Lanka’s strong bowling attack? As the teams have to be announced together minutes prior to the toss there is no way of making decisions based on the actions of your opponent. The team announcements need to be made simultaneously.
2.2 The impact of risk on team selection
2.2.1 The pitch
Apart from your opposition’s team characteristics how the pitch will play would also impact on one’s team combination. As the final was a day/night game held in India it is well known that the pitch will be predictable and play flat during the first half of the game. In the second half under lights, pitches tend to slow up and take spin if conditions don’t change. However if moisture in the air is high it tends to form dew which then makes life harder for bowlers to grip the ball.
2.2.2 The toss
How will Sri Lanka get to use the pitch? Will they get to bat first on a flat pitch and have their blowers’ defend the target under lights or will they have to bowl first and keep India’s score to a minimum and chase in the night? This will depend on the outcome of the toss which offers a 50-50 chance to each captain. Sri Lanka needs to factor this event of chance into their decision making process.
3.0 BUILDING A GAME TO STUDY SRI LANKA’S ACTIONS
3.1 Identifying the rules of the game
3.1.1 The players
These are the decision makers involved. This game is a two player game in which the players are India and Sri Lanka.
Actions involve all the possible alternatives between which a player can decide. In this game actions are the possible team combinations. Whilst there are quite a few possible combinations a cricket team could opt for, in this game I will consider only two, the first being the combination of seven batsmen with four bowlers (7,4) and the second six batsmen with five bowlers (6,5). As mentioned previously, actions would occur simultaneously.
The combination of all players possible actions with payoffs. The payoffs in this game are rank outcomes for each player with 4 being the best and 1 being the worst.
This involves what the players know about their opponents, actions and payoffs. Here, the two players know the general strengths and weaknesses of each other however do not have information about the action the other would take.
Games in which players can communicate are termed as corporative whilst games such as this where the two teams will not discuss their team combination with each other are termed as non corporative.
3.2 Representing the simultaneous game
A game tree can be used represent this game where nodes represent the outcomes and decisions available to both teams with branches representing alternative actions. The information set represents information imperfection where Sri Lanka would not be certain of the team combination India would have gone in with.
The tree diagram on the right indicates how the risks of both the outcome of the toss as well as the nature of the pitch explained in section 2.2 impact the four possible outcomes of the game. Whilst the toss has been assigned probabilities of 50-50 it has been assumed the pitch is assured to play flat during the day and a 40-60 probability between the pitch taking spin or dew playing a part in the night. This is as Mumbai, the venue of the final had a reputation of dew in early April.
Let me now consider generating numbers for each outcome in order to carry out a mathematical analysis of this model. The basis of the numbers I generate would be to determine the “expected combined team ability” given the team combination each player opts for and the effects of nature on this team combination. The numbers and reasoning behind them are as follows.
- Batsmen rating will be multiplied by 1.2 if opponent goes in with 4 bowlers as it assumes batsmen would be advantaged by facing less quality bowlers.
- Bowler rating will be multiplied by 1.2 if opponent goes in with only six batsmen as they have less batsmen to dismiss.
- Assumes the total strength of Sri Lanka’s bowling unit would be at an advantage or disadvantage given the conditions and the same goes for Indian batsmen.
- There would be no double counting as I am working out individual team strengths.
I have assumed a rating of 15 points for each Indian batsman as opposed to 10 for each Sri Lankan as on paper and recent form, the Indian batsmen were clearly superior. I however have assumed a rating of 12 points for each Sri Lankan bowler as opposed to 10 points per Indian bowler as the Sri Lankan’s even though marginally, had a better attack. This final was viewed as a battle between India’s batsmen and Sri Lanka’s bowlers.
The game tree can now be represented numerically with the expected team strength for each team under the four possible outcomes. On the extreme right lie the individual expected team strengths of Sri Lanka bowling first and bowling second. These two values are then multiplied by the expected probabilities of bowling first or second to generate expected values for the outcome shown on the left.
Example: Calculating expected team strengths with India and Sri Lanka both going with a 7, 4 combination and Sri Lanka bowling second (second box, extreme right).
Rough wicket (40%)
Batting: (7 x 15 x 1.2 x 0.7) = 88
Batting: (7 x 15 x 1.2 x 1.3) = 164
Expected team strength of Sri Lanka bowling second:
India: (40% x 128) + (60% x 204) = 174
Sri Lanka: (40% x 146) + (60% x 118) = 129
The combination of all four possible expected team strengths given the selected team combination and impacts of nature can be represented in a 2 x 2 matrix as follows.
This table justifies why India went into the game as favourites. Whatever the combination either team went in with, India’s expected team strength would always be higher than Sri Lanka’s. But does this mean Sri Lanka need not bother strategising? Not at all. As in most sports, some games begin with one player or team being the favourite and the other the underdog just as in this case. Still both players/teams would strategise, with India seeking to maximise its advantage whilst Sri Lanka would seek to minimise this gap so that proper execution of game plans and a little bit of good fortune could see them with the title. The bigger this gap the harder it would be for Sri Lanka.
The table below on the left indicates the difference in expected team strengths between these two teams whilst the table on the right indicates the outcomes based on ranked payoffs with 4= best to 1= worst. So for India the bigger the gap the better, whilst for Sri Lanka it would be the opposite.
4.0 ANALYSIS OF THE MODEL
An outcome is a Nash Equilibrium (NE) if every player is doing the best they can given what the other player is doing. The outcome 3, 2 is a NE as at this outcome both India and Sri Lanka are playing their best reply. A Dominant Strategy Equilibrium (DSE) is an outcome where both players have a dominant strategy. In this game 3, 2 is also a DSE as Sri Lanka has a dominant strategy in fielding a 6, 5 team combination and India has a dominant strategy in fielding a 7, 4 combination.
An outcome is Pareto Efficient if both players can’t agree to go to a better outcome. If the above four outcomes were in a game where both players could gain/lose by moving to another outcome, every outcome would be Pareto Efficient. However as cricket is a zero sum game, the concept of Pareto Efficiency becomes irrelevant here.
India went into the final with a 7, 4 combination and Sri Lanka with a 6, 5 combination. Whilst it is unlikely they used game theory as a strategic tool, from the analysis conducted on the above described model it could be seen that both teams did indeed start the game with the best possible combination.
5.0 LIMITATIONS OF THE MODEL
This model is a simplification of the real pre-game context. The assumptions made in this model are general enough to be true and aid analysis. However in reality things do not always work as assumed and cannot be captured in a model. Some of these limitations are as follows;
- Only two combinations considered between batsmen and bowler split. Teams have many more possible combinations they could consider which includes the mix of fast bowlers and spin bowlers within the decided number of bowlers. Factoring these choices would have made the model more complex but may have generated a different analysis.
- The assumptions consider a constant average value for each team’s batsmen and bowlers in calculating team strength. However this assumption maybe too generic. If India decided to replace a batsman of the calibre of Tendulkar with another batsman does not make it a like for like replacement (so it’s not 15 replaced with another 15).
- The model assumes the team that won the toss would have chosen to bat first. This assumption need to always hold true. Had India won the toss they may have still opted to bat second.
- The probabilities assigned to pitch conditions in the night are subjective and changing these probabilities will offer different expected team strengths.
In cricket, a team’s outcome depends not only on its decisions but also on the behaviour of their opponents as well as chance. Good leaders would think strategically and anticipate the behaviour of their opponents and the environment and make decisions to improve/enhance the team’s chances.
Game theory can help teams to understand and analyse available actions and predict the behaviour of opponents and design strategy and tactics.