A study on the evolution of pitching staff usage.

The major trends in pitching staff usage have been:

Conclusions related to why most general trends makes sense:

Some graphs, part 1: Historical trends in usage

Average delay in start of pitchers (rotation length)
This graph was constructed as follows: for each pitcher's start, count the number of days and his team's games since his previous start. If this is his first start of year, first start after a relief appearance, or first start in 9 days, ignore the data point. Average these counts of delays each year.
The graph shows 1-man "rotations" were normal when the major leagues (as National Association) began in 1871, but rapidly expanded to 4 by 1897.
In fact, the most common delay of games (not average) (you could say most common rotation length) was:
1 in 1871-1881
2 in 1882-1890
3 in 1891-1897
4 in 1898-1925, 1927, 1929, 1934-1936, 1938, 1959, 1969-1970, 1972-1974
5 in 1926, 1928, 1930-1933, 1937, 1939-1958, 1960-1968, 1971, 1975-2005
So the decision to go to a 5th starter was apparently difficult. But now there is overwhelming consensus, 5-game delay being about 5x as common as 4-game in recent years, and about 10x as common as 6-game.

Fraction of innings pitched by team's top N pitchers
By Top N pitchers I mean the N pitchers with most innings pitched for particular team each year, for N = 1, 2, ... 10. The graph shows 1 pitcher did most of a typical teams pitching till early 1880's, now it takes about 4 pitchers to do 50%, and top 10 do about 82%.

Average duration of an appearance
Average duration of an outing for starters and relievers, strong downward trend is not slowing. (This implies number of relievers per game is rising, in fact it is strongly, now about 2.7, this isn't shown here).
The longer lines (labeled "ST") are from season-total data and only uses pitchers who always relieved or always started, since I don't have the information to allocate their total innings between starts and reief appearances. The shorter lines (labeled "PBP") are from www.retrosheet.org play-by-play data, and includes all appearances. As you may expect, pitchers who were only starters averaged longer starts and pitchers who were only relievers (for a year) had shorter relief outings.

Fraction of pitcher removals which are at start of inning
The interruption in the trend in 1973 is of course due to the DH and pitchers nto being removed for pinch hitters. The significance of the trend I think is that managers believe being used at predictable times benefits pitchers (as setup pitchers in 8th, closers in 9th), and that stress is minimized when say 1-inning outings are just one inning and the pitcher only needs to warm up once.

Last batter faced retired before mid-inning removal
It used to be pitchers were "knocked out of the box", i.e. removed for ineffectiveness, that is not nearly so true anymore. 25% of mid-inning removals are after last batter was retired; this figure was only 5% 50 years ago and has risen steadily.

How often is starter in at start of 9th with 1-run lead?
That fraction of times that the starting pitcher is in among games with the given score (with lead). This is more evidence that being effective won't keep a starter in: even with a 1-0 lead the starter was out over 80% of the time in 2005.

Avg standard deviation of reliever entry
Define entry point as number of outs recorded by pitching team in game (e.g. 24 for entry at start of 9th), for each relief pitcher compute the standard deviation of this number (each year) and average over all pitchers (each year).
The graph shows entry time for each reliever is becomming more consistent.

The emergence of the closer, graph of quantity of them
With a little thought about how to define a closer, I say a pitcher is a closer in a particular season if all of this are true: The graph shows how many closers there are each year (in 1957-2005 period). Who, you may wonder, were the early (pre-1975) closers:

Some graphs, part 2: Studies of effectiveness

First, an important conclusion:

Pitchers are more effective when used as relievers than when used as starters.
I come to this conclusion from the 1957-2005 data, doing self-comparisions of pitchers' performances when they were used both as starters and relievers in the same season. For each pitcher-season with such a dual role, I compute:

[his batting average against as a reliever] - [his batting average against as a starter]

And then I average all these pitcher-season results together, each weighted by the lesser of his at-bats against as reliever or at-bats against as starter, and
the grand result is:

Total weight: 830704 at-bats.
Relieving compared to starting: -.00540 batting average
+.02799 strikeout rate

Batting average against starters, relievers by year
I'd say most likely the intrensic skill of relievers was lower than starters until later 1970s, and lately they have apparently been better -- over-performing by more than the reliever advantage of .0054 BA calculated above.

Batting average against starters, relievers by month
I made this graph to see if either starters or relievers wear down more than the other through the season, apparently they do not.
Note batting averages are highest in the hottest months.

Effectiveness by # of batters faced so far in game
The rising graphs show that both relievers and starters steadily lose effectiveness the longer they are in a game, except maybe the first 6 batters or so especially in the case of relievers. The vertical axis shows the difference between batting average against pitchers who have faced N batters (N on horizontal axis) and a predicted value for batting average in those circumstances, where that predicted value accounts for batter's and pitcher's season statistics. Specifically the predicted value is the average value of
predicted BA = [Batter's BA]*[Pitcher's BA against] / [Major League BA]
where each batting average is for current season.
So, we can see that pitchers go from having something like a 5-8 BA point advantage when they are fresh to a 10 or more point disadvantage after 25+ batters faced.

(relative) batting avg against relievers, based on batters faced prev N days
This shows the difference between an the observed batting average against relievers with particular rest levels and the predicted batting average (as described for previous graph)
Vertical axis is this relative BA and horizontal is batters faced in previous N days (different graphs for N=1, 2, 3).

(relative) strikeout rate against relievers, based on batters faced prev N days
As previous graphs, but for strikeouts (relative to predicted strikeouts).

Reliever Effectiveness By Usage

-3 -2 -1   WGT    diff BA     diff KA 
          707643 -0.001588  -0.000434 
       X  356953  0.000533   0.002121 
    X     362683  0.001171   0.000518 
    X  X   90228  0.000164   0.002082 
 X        208516  0.001909  -0.002056 
 X     X   83889  0.001694  -0.004051 
 X  X      53466  0.001124  -0.001963 
 X  X  X   12164  0.000580  -0.001067 

X in -N column means pitcher pitched N days before (X in -1 can mean first game of doubleheader the same day)
WGT: min of batters faced in that situation, batters faced all others.
average diff in BA
Conclusion: pitchers best if not used at all previous 3 days, but by small amount

Sarters by days of rest

Days TotWeight  Diff_ba    Diff_ka 
 0        28   0.083924    0.073666 
 1       376   0.006580   -0.045849 
 2      4486   0.001406    0.011964 
 3     26824  -0.002721    0.010433 
 4    654211  -0.000633    0.002648 
 5   1771294   0.000983   -0.000750 
 6    834691   0.000872   -0.001044 
 7    192325  -0.000308   -0.001046 
 8     80049  -0.000725   -0.000223 
 9    170044   0.001840   -0.003696 
>9    756357  -0.003962    0.011828 

Again this is a self-comparison of pitchers with themselves in the same season, showing performance for pitchers starting on the Nth day (first column) since their previous start. Each pitcher each season is a data point; these data points are averaged for each season with a weight which is the minimum of: (the pitcher's # of batters faced on the Nth day, the pitcher's # of batters faced in all other starts that season).
For example, pitchers on 5th day had batting average against .000983 higher than in their other starts in the same season (in a weighted average sense), on on 4th day .000633 lower.
Conclusion: pitchers have actually done better on 4th day than on 5th, so it seems we should go back again to a 4-man rotation.

Holding 9th inning leads

Something may be about to give

With more relief pitchers in each game, shorter outings for all pitchers, more frequent outings for relievers, more relievers per game, and better performances by pitchers used for short outings the trend is towards a situation where all pitchers are used in short frequent outings. Use of relievers tends to optimize effectiveness, the purpose of starters has largely become to eat innings.
I think the continuance of the starter role is now just from tradition. Not everyone can pitch in a relief role, because even 1 inning every 2, or even 2 innings every 3 games is considered too much pitching, and less than that would mean too many pitchers would be required on a staff.
In trying to maximize effectiveness of pitchers, we have developed a sort of di-morphic notion of pitchers: some make best contributions is about 6-inning outings every 5 games, and some with 1-inning outings in about half the games, but there may be an optimum use somewhere in-between.

A's 1993 "pitching-groups" experiment

What we are trending towards is something like what LaRussa tried with the A's July 19-24 1993. His recently successful team was getting hit hard and losing. He announced an experiment: a 3-group rotation: Groups were: Val Poppel/Darling/Campbell Mohler/Witt/Briscoe Welch/Downs/Gossage Each pitcher is limited to around 50 pitches, any might start (tho it was really a Van Popppel/Mohler/Downs 3-man rotation) and there were other pitchers with normal relief roles (e.g. Eckersley). I suspect that ones prejudices against this sort of thing are overcome (e.g. starters must go 5 for a win) it would be a better system.
The A's lost 5 of the 6 games, with ERA around 4.6, similar to how they'd been doing otherwise, and the experiment ended.

A note on the importance of closer

Wondering, how much importance do they have (e.g. per inning or per batter)
Construct table of win probabilties by game situation:
    Home-Vis Score
    Which Inning/Which Half
    Which of the 24 inning situations

The average of the absolute value of the change in a team's chance of winning a game per batting event is .025.
Averaged in categories accoring to what kind of pitcher is on the mound, starters and relievers nearly equal (about .025), since 1957. Typical value for closer is 0.045.