Virat Kohli averages 38.6 over 186 matches in the IPL (as of October 20, 2020), striking at 131, to go with a mountainous 50.8 average in T20 internationals. This frequently earns him a mention among the best T20 batsmen, according to many watchers of the game. It is no surprise, then, that a recent article examining Kohli's anchor role, and questioning whether an anchor is needed at all in modern T20, was met with widespread uproar across the internet.

Kohli does have almost unparalleled conventional stats, resulting from his strategy of knocking singles and doubles around before a final phase of risky hitting. This gets him high scores in some instances, but what of the times he cannot transition to that big-hitting stage of the knock? His slow approach leaves his team behind the ideal run rate. Kohli spends the first half of his innings minimising risk. If he gets out early, he has consumed deliveries and scored slowly, but he has not survived often enough to make up for it adequately. His net worth is a tug of war between how frequently he crosses the barrier, beyond which he starts scoring faster, and the slowness of his early innings.

How can we gauge such a player's gross impact over a whole season? Do his half-centuries and centuries cancel out the detriment he causes an innings when he gets out in his early phase? We can answer this using a contextual measure of runs, called the Runs Above Average (RAA). This quantifies how much the output of a batsman is better or worse compared to the average score in a given "situation", which is defined by the innings phase, wickets lost before the ball is bowled, and ground.

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Considering all IPL matches played since 2015, an average runs-per-ball score is calculated for each situation. For instance, the average runs scored per ball at the Feroz Shah Kotla in Delhi in the powerplay with one wicket down in this period is 1.25. If a batsman hits a four in this situation, his RAA for that ball is 4 - 1.25 = 2.75. We can compute RAA for one innings, or a whole season, and it tells us the player's contribution in comparison to an average player, had he played in the same situations.

Looking at data from the IPL from 2015 onwards (until the KXIP vs DC match on the 20th of October), Kohli's average RAA per innings by phase puts him in the "average" category before the death overs: -0.14 RAA in the powerplay, -0.41 in the middle overs, and 2.85 in the final four overs. He scores below the average batsman's output in two out of three phases. He has a positive RAA at the death; how often that nullifies his below-par scoring early on can be quantified by his innings RAA. If this number is positive, he has made up for whatever he does early on, by making a net-positive contribution to the innings.

Kohli's mean innings RAA is 0.42, and he has a positive RAA in 44% of his innings. So despite scoring faster than most in the death overs, he does not enter that phase often enough to have a very high net-positive effect compared to the average batsman. In comparison, AB de Villiers has a positive contribution in 60% of his innings, and his mean RAA per innings is 5.8. The following plot shows Kohli's runs against his RAA for all innings in the IPL since 2015. Many long innings have a negative RAA, which means that he wastes a large number of balls staying under the "average" expected scoring rate.

His season-wise RAA totals show his contributions over a whole season. Even in 2016, his bumper year, he scored only 40 runs over the average player in total. In the last two seasons, 2020 included, he has been a shade under average. His long innings do not make up for his sedate beginnings.

In comparison, here is the plot of runs vs RAA for Rishabh Pant. He too is slow at the start, but only two 20-plus run innings of his have a negative RAA. He does not cost his team runs if he gets a start, and most of his innings yield positive returns, accounting for context.

The average innings RAA (IRAA) tells us the typical innings-wise contribution of a batsman, and the average accumulated RAA value after facing ten balls in the innings (10RAA) tells us how quickly the batsman starts at the crease. Here is a plot of the two values for the 50 highest scorers in the IPL since 2015. Players with a 10RAA value of less than zero start slow: Kane Williamson, Manish Pandey, Virat Kohli and Shubman Gill are the "anchors" who make up this category. Shane Watson and Chris Gayle are notorious for starting slow (even if by powerplay standards), and MS Dhoni's weakness against spin when he walks in is on display in his very low 10RAA score.

Players who have a higher IRAA than 10RAA make up for their slow starts, but even among those, Dhoni, Gill and Pandey have a negative IRAA. Andre Russell, Pant, David Warner, de Villiers and Jos Buttler make up an elite cluster: they start in the positive and improve upon their contribution as the innings progresses. Interestingly, Sunil Narine loses relative value if he stays at the crease for longer: his 10RAA is very high, but his IRAA is lower.

The same metric can be used to tell how many runs a bowler concedes relative to the average bowler in a given situation. In a bowler's case, a negative RAA is better, because it means he has saved runs compared to the expected rate. The following table lists the 15 players with the best average RAA values for bowlers, among those who have bowled 600 or more legal balls in the IPL since 2015. In Jofra Archer's case, it effectively means that he concedes 5.39 runs fewer per innings than the "average" bowler would, bowling in the same situations.

The RAA does not consider the effects of preserving or losing wickets, because it seeks to quantify the player's individual output in the context of the average player, and the ultimate currency of wins and losses is runs. Wickets affect team scoring in highly complex, non-linear ways that are beyond the scope of simple models.

The RAA is useful to analyse a player's career, or season, or a single innings. Can it be used to gauge a player's contribution to a team's winnings over a season? Can we calculate how much a player is worth in a team, relative to the average player? The Wins Above Average (WAA) metric does this, by effectively translating a team's aggregate runs scored and conceded into win probability.

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Using data of team performances over a season, a relationship can be formulated between the runs scored and conceded by a team and its win percentage. It must be noted that teams that chase and win score limited runs: they don't need to score more than the target. To rectify this, the runs conceded and scored are adjusted in the case of chasing wins, where a margin of wickets and balls remaining is converted into the extra expected runs the winning team would have scored had they completed their innings. This ensures that all win margins are homogenised into units of runs scored, and a run tally that rewards teams for chasing wins can be calculated for every team

The win percentage of a team in a season follows a "logistic" relationship with the ratio of the total runs scored to the total runs conceded*, which is represented by a curve shown in the following graph. Each dot represents one team in one year of the IPL since 2015. (As a sanity check, the curve shows us the logical result that a team that concedes as much as it scores should win 50% of the time, on average.)

How is this useful in judging a player's worth in terms of wins? We can replace a chosen player by an "average" player, by subtracting the chosen player's RAA from his team's run tally for bowling or batting. This replacement changes the team's win percentage, in accordance with the relationship above. This difference in win percentage over a typical 14-match season is defined as the player's WAA. Positive RAA values for batsmen, and negative ones for bowlers, correspond to positive WAA - these contributions are beneficial for their team results. This way, a player's output in each season can be expressed in terms of their WAA, while accounting for the situations they have batted or bowled in. How many wins has a player contributed to in a 14-match season, over and above the average player's contribution? The WAA depends not only on the player's individual output, but also the quality of his team. A player in a weaker side has a higher WAA compared to another in a stronger team with the same RAA, because the former relatively contributes more to each win.

Excluding the 2020 season, the best individual season for a batsman in the IPL is Pant's 2018. He scored 684 runs, but his RAA was 168, which corresponded to 1.4 wins for his team. Pant, Warner and de Villiers feature repeatedly on the list of top batting seasons, all having at least one season with a WAA of 1 or more.

Among bowlers, Jasprit Bumrah has two campaigns at the top, with a WAA of 0.92 in his best season. He is followed by his Mumbai Indians team-mate Lasith Malinga, which shows why Mumbai have been generally successful over the past few editions of the league. Mustafizur Rahman comes fourth with his 2016 season, and it is no surprise that the Sunrisers Hyderabad emerged champions that year, with Warner's season ranking fourth in the batting RAA table and Bhuvneshwar Kumar also featuring among the best efforts by bowling RAA. Narine, Archer and Rashid Khan make up the rest of the top ten.

Here is the leader board for the current season, until October 20. Nicholas Pooran has been the most valuable batsman in a mostly lacklustre Kings XI campaign, his contribution being worth 1.37 games over the average player. Shikhar Dhawan's rich vein of attacking form puts him in second place. KL Rahul, despite being the top run scorer, has a season RAA of just 5.8 because of his largely sedate approach, while Virat Kohli is at -2.9. Aaron Finch (-0.37), Dhoni (-0.26) and Kedar Jadhav (-0.25) have the worst season WAAs.

Among the bowlers, Rashid Khan and Archer are on top. Delhi's Axar Patel, along with pace duo Anrich Nortje and Rabada are vital instruments in their roaring campaign, which is reflected in their places on the table. Chris Morris, the go-to death bowler for RCB this season has saved 40 runs over the season compared to an average bowler, given that he bowls in the toughest phase. Bumrah, after a cold start, has come back to his dependable ways. Surprisingly, Russell finds a place in the top ten with his crucial death bowling for KKR.

More complex methods that incorporate wicket-taking and wicket-saving ability are natural extensions to this model, but this serves as a first stepping stone to comparing player performances with the typical player in the league, contextualising their run outputs, and reading them in the language of wins for their team.

* The log odds of victory taking all non-tied matches with results were fit to the ratio of the adjusted runs scored and conceded. The model parameters were significant at the 99% significance level, with an R2 of 0.59