From 2000 to 2009, A-Rod was hitting approximately .064 home runs per plate appearance. In 2008 he hit .059 and in 2009 he hit .056, so it’s probably much fairer to use a slightly lower rate. I’m going to make the assumption that Rodriguez’s true production is about .055 home runs per plate appearance, since he exhibited a downward trend and his 2010 production has been very low. (It also cuts him some additional slack in the Choke Index.)

Simply, we should assume that A-Rod’s failure to produce is merely the result of chance, and not due to choking or media distraction or even Rodriguez’s discomfort with the special chipped baseballs. (A better man than I would call this the Numbered Ball Effect.) Then, we should see how likely that is.

At .055 home runs per plate appearance, the likelihood of going 42 plate appearances without a home run is or approximately .093. The Choke Index is simply or, in this case, .907. As it becomes progressively less likely that Rodriguez will go another plate appearance without hitting a home run, the Choke Index number rises. A theoretical Choke Index of 1 would indicate that the player’s lack of home run hitting is nearly impossible to describe by chance alone.

A-Rod’s Choke Index between #499 and #500 was about .877. This is a man who doesn’t handle milestones well.

Another example was Gary Sheffield in 2009, when he was attempting to hit his 500th home run. In the previous two years, he hit approximately .041 home runs per plate appearance. Much was made of Sheffield’s trouble hitting #500, but since he was hitting almost exclusively as a pinch hitter, he simply didn’t have many opportunities. Between his final plate appearance on September 26 of 2008 and his only plate appearance on April 17 of 2009, Sheffield went 21 plate appearances without hitting a homer. That gives him a choke index of .556.

Barry Bonds, meanwhile, was hitting .065 home runs per plate appearance in the seasons prior to his record-breaking home run #756. #755 was hit in Bonds’ first plate appearance on August 4, 2007. Bonds made 3 more plate appearances, all walks, in that game. He hit #756 in his third plate appearance only three days later on August 7.  He had August 5 off and made 4 plate appearances on August 6, meaning that Bonds went 9 plate appearances between home runs, giving him a choke index of .454.

Rodriguez will hit his 600th home run eventually, but it’s getting painful to watch.

Age: There are six players to have hit #600. Sammy Sosa was the oldest at 39 years old in 2007. Ken Griffey, Jr. was 38 in 2007, as were Willie Mays in 1969 and Barry Bonds in 2002. Hank Aaron was 37. Babe Ruth was the youngest at 36 in 1931. Alex Rodriguez, who is 35 as of July 27, will almost certainly be the youngest player to reach 600 home runs. If both Manny Ramirez and Jim Thome hang on to hit #600 over the next two to three seasons, Thome (who was born in August of 1970) will probably be 42 in 2012; Ramirez (born in May of 1972) will be 41 in 2013. (In an earlier post that’s when I estimated each player would hit #600.) If Thome holds on, then, he’ll be the oldest player to hit his 600th home run.

Productivity: Since 2000 (which encompasses Rodriguez, Ramirez, and Thome in their primes), the average league rate of home runs per plate appearances has been about .028. That is, a home run was hit in about 2.8% of plate appearances. Over the same time period, Rodriguez’ rate was .064 – more than double the league average. Ramirez hit .059 – again, over double the league rate. Thome, for his part, hit at a rate of .065 home runs per plate appearance. From 2000 to 2009, Thome was more productive than Rodriguez.

Standing Out: Obviously it’s unusual for them to be that far above the curve. There were 1,877,363 plate appearances (trials) from 2000 to 2009. The margin of error for a proportion like the rate of home runs per plate appearance is

Ordinarily, we expect a random individual chosen from the population to land within the space of 95% of the time. That means our interval is

That means that all three of the players are well without that confidence interval. (However, it’s likely that home run hitting is highly correlated with other factors that make this test less useful than it is in other situations.)

Alex’s Drought: Finally, just how likely is it that Alex Rodriguez will go this long without a home run? He hit his last home run in his fourth plate appearance on July 22. He had a fifth plate appearance in which he doubled. Since then, he’s played in five games totalling 22 plate appearances, so he’s gone 23 plate appearances without a home run. Assuming his rate of .064 home runs per plate appearance, how likely is that? We’d expect (.064*23) = about 1.5 home runs in that time, but how unlikely is this drought?

The binomial distribution is used to model strings of successes and failures in tests where we can say clearly whether each trial ended in a “yes” or “no.” We don’t need to break out that tool here, though – if the probability of a home run is .064, the probability of anything else is .936. The likelihood of a string of 23 non-home runs is

It’s only about 22% likely that this drought happened only by chance. The better guess is that, as Rodriguez has said, he’s distracted by the switching to marked baseballs and media pressure to finally hit #600.

Sheffield had a Batting Average, On-Base Percentage, Slugging Percentage and On-Base Plus Slugging of .225/.326/.400/.725 in 2008; Bonds was last active in 2007 and hit .276/.480/.565/1.045 (with the .480 OBP leading the National League). Clearly, something’s wrong. Collusion?

What’s wrong, in my estimation, is still that Bonds represents a negative externality on his team’s production, reputation, and revenue; Sheffield, meanwhile, is less of a threat to ticket sales. Despite being unpopular and saying bizarre things, Sheffield has not yet to my knowledge irritated fans to the extent that Bonds has, nor is he quite the clubhouse menace Bonds is said to be.

Of course, time will tell whether Sheffield produces $400,000 worth of runs for the ailing Mets.

Barry Bonds is, without a doubt, one of the most controversial figures in baseball. He’s currently trying, again, what he tried last year – shopping himself around for the league’s minimum salary. (Thanks to the Sports Law Blog for the link.) Inside, I’d like to briefly discuss collusion and look at the incentives involved with this situation.

Collusion in the economic sense involves an agreement to act inefficiently. To give an oversimplified stock example, suppose a competitive market for chili peppers. That is, there are many people growing chili peppers and many people who want to buy chili peppers. Each bushel of peppers costs about $1 to produce and process from seed to sale. The market is big enough that no one producer or consumer can affect the overall price of the peppers. If one person raises his prices, the others can easily undercut him, and if he wants to sell his peppers he’ll have to meet them. As a result, the sale price approaches the (marginal) cost of production ($1).

Now suppose that only one person grows chilies. They still cost $1 to produce, but the chili monopolist can maximize his profits however he chooses. (The standard assumption is that he’ll set the price equal to marginal revenue, not marginal cost.) If I had made up a demand function and were in the mood to do calculus, I could figure out exactly how many bushels he would produce, what their price would be, and how many fewer bushels that would be than the perfectly competitive market would yield, but the important thing is that monopolies derive more profit than competitive firms, at a cost to the consumer. (Let’s leave aside the idea of a natural monopoly, where high production costs make it more efficient for only one firm to produce.)

So, what’s collusion? Simply, it’s an agreement by two or more firms to act like a monopoly and split the much higher monopoly profits. This is good for the firms and (generally) bad for the consumers because there’s a loss of welfare – some people want the product but at the higher monopoly price they can’t afford it and the monopoly produces fewer goods than the competitive market.

And what does this have to do with Bonds? Consider that Bonds is a producer of runs, and production can be roughly measured by the OPS stat. (There are other methods, but I’ll follow SI’s lead.) Assume that there’s a positive relation between previous OPS and salary (that is, that when negotiating contracts you can put OPS into a formula and that formula will spit out a salary, and that most salaries more or less line up with it), and that Bonds would play either right field or DH, so defense wouldn’t impose an additional cost. A rational team, when offered Bonds’ OPS number for a significantly smaller salary than it would otherwise have to shell out for the same number, should sign Bonds for $400,000. It would be fairly easy to make the leap from this statement of rational behavior to declaring that the teams are behaving irrationally and therefore must be colluding.

SI make a good counterpoint – in economic terms, Bonds’ clubhouse demeanor and notoriety would cause high amounts of negative utility in terms of unhappy and distracted teammates in the first case and customers unwilling to buy tickets to see Bonds in the second. Both of these can affect the profit of the firms and would have to be accounted for in any analysis of the value the team can expect to derive from Bonds. Suppose the most extreme case – that all fans are so disgusted by Bonds that they refuse to attend games he plays in, and that his teammates are so distracted by his attitude that they produce zero runs. In that case Bonds will still be a high producer, but he would still have a decidedly negative effect on profit and on his team’s record.

SI also makes two suggestions I don’t agree with – that Bonds’ felony charges might provide a disincentive in that the team has a high probability of losing him, and that Bonds’ age makes him unattractive. The first assertion ignores the fact that Bonds will still probably produce more than $400,000 worth of runs in a partial season, so in a strict runs-to-salary relation he still represents a net profit. The second doesn’t take into account that at age 42 in 2007 Bonds was worth $15.5 million dollars and that he would have to have declined at an almost impossibly high rate to fail to produce $400,000 worth of runs. Bonds will be out of shape and fragile, but the comparison need not be Bonds now to Bonds in 2007. I’d much rather have an out-of-shape and fragile Barry Bonds as my designated hitter than a $400,000 wet-behind-the-ears rookie. Hell, considering how fragile David Ortiz is, I’d rather take Bonds at $400,000 than Ortiz at his market salary. (The Red Sox could spend the difference on a new fifth starter now that Curt Schilling retired.)

Collusion thus doesn’t make much sense to me in this case, since the incentive to break collusion would be so strong. Stable collusion requires some method for the colluders to punish a fellow colluder who cheats. In a market this would be by simply returning to competitive prices, depriving the undercutter of his excess profits. Here, unless Selig is directly involved (unlikely), there’s no way for the teams to punish anyone for signing Bonds. (Yes, I suppose they could start plunking the batters every time, but that’s not sustainable.) The alternative explanation that makes the most sense to me is the disutility argument – that production and ticket sales will suffer for any team that employs Bonds. If, however, the SI article is correct in its idle suggestion that Bud Selig has ordered teams not to sign Bonds, Selig should be ashamed of himself.

Inexpensive runs
will not placate angry fans.
“Cheap offense” indeed.