Opening Day

It's the most won-der-ful tiiiimmmeee of the year.....

Baseball gets underway today. I love the start of baseball season, and I love the start of the fantasy season. I'll be watching as many games as possible this weekend without completely turning my back on real life. Of course, I can always dissertate and grade papers in front of the television.

My 2011 in Fantasy Baseball: An Introduction

As part of my effort to become more knowledgable about the in-and-outs of fantasy baseball (and to better procrastinate), I've joined a handful of different league types this season. Last year was my first year playing fantasy baseball, and I played in a non-standard 8x7 roto league (the usual ten plus K/OBP/Errors for hitters and CG/Shutouts for pitchers).

This year, I'm in four main leagues: a standard 5x5 roto league, a non-standard 7x7 (the usual plus doubles/Ks for hitters and QS/HR allowed for pitchers) keeper league, a head-to-head points league with weekly lineups, and a head-to-head (each) category league with daily lineups. This allows me to try out strategies and analysis on a more-than-theoretical level, and see how the leagues differ (the better to write about them).

So a few times a month, I'll post updates on how the leagues are going. To start, though, I'll post my teams and preseason strategies for each league. Here we go:

Head-to-head points league

League Name: Bernie's Bombers, 10 teams

My Team: Waveland and Clark

The details: I detailed the scoring elsewhere, but I'll repeat it here: +1 point per hit, walk, total bases (a single is then 2pts, a triple 4pts, etc), steal, run or RBI, -1 point for a strikeout or error; 3 points per inning pitched (or 1pt per out), 6 for a win, 5 for a save, -3 for a loss, 1 per strikeout, -1 per hit or walk issued, 4 per complete game, 2 per shutout, 10 points for a no-hitter. No start limits, 11 matchup acquisitions per week, no other transaction limits, weekly lineup changes.

Roster Composition: C, 1B, 2B, 3B, SS, 4OF, Util, 5SP, 2RP, 6 Bench, 1 DL.

My Roster: Victor Martinez, Billy Butler, Robinson Cano, Ryan Zimmerman, Stephen Drew, Andrew McCutchen, Shane Victorino, Nick Markakis, Denard Span, Adrian Beltre (starting hitters); Jered Weaver, Francisco Liriano, Daniel Hudson, Tim Hudson, Jhoulys Chacin, Jonathan Paplebon, Chris Perez (starting pitchers); Brian Roberts, Logan Morrison, Jason Bay, Miguel Tejada, Jake Peavy, Carlos Zambrano (bench).

The Plan: I targeted hitters early, since every week they'll play five to seven games as opposed to one or two for a starter. By maximizing my edge there, I can get just enough pitching between who I currently have and whatever good looking two-start pitcher is available to win most weeks. I made sure to target players who are better in points leagues than most roto leauges, especially Butler and Markakis. I also made sure that I had the ability to exploit some of the roster quirks, like Zambrano's RP eligibility (since he'll get more points than a closer if he starts twice in a matchup).

I drafted Garret Jones as a backup 1B, but the news that he'll platoon led me to drop him. I picked up Tejada because a) if he has any kind of a season he'll be OK for points, and b) with the uncertainty around Drew's opening-week availability I needed someone I could slide in at SS, and the 3B eligibility is just gravy. Instead of dropping Peavy, I'm hanging onto him so I can stash him when he becomes DL-eligible. I don't need the space this week, and there's no sense in dropping him until he becomes a burden on my roster (or proves worthless).

Standard(5x5) roto league

League Name: cp, 10 teams

Team Name: Cubbie Blues

The Details: Standard categories: Runs, RBI, Home Runs, Steals, Average for hitters; Wins, Saves, Ks, ERA, WHIP for pitchers. No acquisition/transaction limits, no hitter start limits, 200 start season limit for pitchers.

Roster Composition: C, 1B, 2B, 3B, SS, CI, MI, 5OF, Util, 9 pitchers (undifferentiated), 3 Bench, 1 DL.

My Roster: Joe Mauer, Adrian Gonzales, Dan Uggla, Martin Prado, Jimmy Rollins, Neil Walker, Billy Butler, Ryan Braun, Andrew McCutchen, Jay Bruce, Brett Gardner, Chris Coghlan, Jose Tabata (starting hitters); Justin Verlander, Max Scherzer, Tim Hudson, Jonathan Sanchez, Clay Buchholz, Daniel Hudson, Francisco Cordero, Leo Nunez, Brandon Lyong (starting pitchers), Ryan Dempster, Josh Beckett, Tsuyoshi Nishioka (bench).

The Plan: There isn't one. By the second round of the draft, it looked like amateur hour (one team took Stephen Strasburg and Ike Davis first and second). I've got plenty in each category- Mauer/Gonzales/Prado/Braun are .300+ hitters, I have eight 90+ run hitters, six 20+ home run hitters, three 100+ RBI hitters, and a projected 135 stolen bases from five guys. None of my pitchers are necessarily anchors in ERA/WHIP (as in under 3.00/1.20), but they're all solid for the most part. I can certainly afford to swing a trade, especially since I have an excess of above-average pitching and a stellar lineup. Saves may be a problem, but they can be found. If I don't win this league, I either had rotten luck or screwed something up royally. Just for fun, I may try and never make a (non-DL-related) transaction during the season.

I find the trick with a starts cap is to have two or three pitchers who you can throw out every week regardless of matchup (last year I never had fewer than two of Wainwright, Cain, Verlander, and Josh Johnson on my roster at once). Right now I only have Verlander, but many of the remaining starters all have to potential to be that guy as well. I'll possibly have to trade for one more such pitcher, but it shouldn't be a problem to pull off.

Head-to-Head Each Category

League Name: Cubs Fans Fantasy Baseball, 10 teams

Team Name: Cubs Expatriot

The Details: Standard 5x5 Categories, weekly matchups, 11 matchup acquisitions, no transaction or start limits, daily lineup changes, 10 IP/week minimum.

Roster Composition: C, 1B, 2B, 3B, SS, CI, MI, 5OF, Util, 9 pitchers (undifferentiated), 3 bench, 1DL

My Roster: Victor Martinez, Joey Votto, Dustin Pedroia, Martin Prado, Rafael Furcal, Erick Aybar, Kevin Youkilis, Ryan Braun, Andre Ethier, BJ Upton, Colby Rasmus, Juan Pierre, Jose Tabata (starting hitters); Jered Weaver, Francisco Liriano, Daniel Hudson, Brandon Morrow, Jonathan Paplebon, JJ Putz, Huston Street, Ryan Franklin, Rafael Soriano (starting pitchers); Jordan Zimmerman, Michael Pineda, Chris Coghlan (bench).

The Plan: I drafted to dominate pitching in this one. I stocked up on average-to-good relievers so I can win saves and anchor my ERA/WHIP each week. This allows me to throw out starters based on K/Win potential as I see fit, especially with daily lineup changes. I should win no fewer than three and quite possibly up to five pitching categories per week this way, in addition to having trade chips. For hitters, I went with roster flexibility. I have 2 eligible at 1B, 2 at 2B, 2 at SS, 2 at 3B (once Youkilis gains eligibility), 3 at CI, and 4 at MI. This allows me to put basically any infielder on the trading block without having to scramble too hard for a replacement if I need to make a move. In general, I should win average and runs most weeks, be ultra-competitive for home runs, and do OK in steals/RBI. Essentially, my roster is tailored to win at least five categories most weeks, with the ability to win eight pretty easily and ten never out of the question.

Non-standard (7x7) Roto League

League Name: Fantasy All-Stars, 10 teams

Team Name: Cubbie Blues (again).

The Details: Standard 5x5 categories plus doubles and Ks for hitters, QS and HR allowed for pitchers. No acquisition/transaction limits, no hitter start limits, 250 starts cap, daily lineup changes. Keeper league, can keep five players for an unlimited time with each keeper taking a pick off the top of the draft (e.g. 1 keeper loses a first round, 5 loses your first five picks).

Roster Composition: C, 1B, 2B, 3B, SS, CI, MI, 4OF, 2Util, 4SP, 2RP, 2 Pitcher (undifferentiated), 4 Bench, 2 DL.

My Roster: Brian McCann, Billy Butler, Brandon Phillips, Jose Bautista, Jose Reyes, Aaron Hill, Gaby Sanchez, Matt Kemp, Josh Hamilton, Brett Gardner, Nick Markakis, Denard Span, Luke Scott (starting hitters); Matt Cain, Francisco Liriano, Shaun Marcum, Jordan Zimmerman, Jonathan Paplebon, Andrew Baily, Wandy Rodriguez, Joel Hanrahan (starting pitchers); Anibal Sanchez, Mitch MOreland, Michael Cuddyer, Chris Coghlan (bench).

The Plan: Since I took over this team for a vacated owner, I was already at a disadvantage. I ended up keeping McCann, Bautista, Kemp, Hamilton, and Butler from the prior year. Butler isn't great, but this is about as ideal a roto team for him as exists since he hits doubles and doesn't strike out much. For the most part, I was left to plug holes; it's hard to draft well when one team can keep Albert Pujols, Ryan Zimmerman, Hanley Ramirez, Justin Upton and Felix Hernandez (or for that matter Ryan Braun, Joe Mauer, Robinson Cano, CarGo, and Joey Votto).

Saves were going to be an issue for me even before Bailey got hurt, and there isn't much to choose from when I can move him to the DL. I need to hope Cain/Liriano/Marcum can be those "every start" guys you need when there's a starts cap. I also managed to limit the number of fly-ball pitchers to account for HR allowed, though not as much as I might hope. I did manage to get some value plays given the extra categories- on top of Butler, Markakis, Gaby Sanchez, McCann, Phillips and Reyes all gain value from the league settings. What I really need to do are pick up some high-end relievers to put in the undifferentiated pitcher spots to help anchor WHIP/ERA, since pretty much everyone is 3.20+/1.20+ in those categories; at this point though that might be antithetical to picking up saves.

In general, I should be good in average, doubles, and maybe Ks for hitters, with the chance to be decent in HR/R/Steals. I don't have anyone who's great in any of those (except Bautista for HR and Gardner for Steals), but I have plenty of above-average guys. RBIs will likely be a big hole. Pitching could get ugly, but I should be good in HR allowed and Ks, decent in QS/Wins, and I can readily improve my roster for ERA/WHIP. I'm not ready to give up on the season before it starts, but I think I might have to start thinking about next year by the end of June. If I look out of it, I'll punt saves entirely to try and get some good keepers out of the trades. Of the four leagues, this is the one I have the least certainty about in terms of future performance. The only way this season will be a bust, though, is if I'm essentially stuck with the same five keepers as I had going into the season.

Category Drains

Consider this the flipside of my last post. These are players who in the top-15 at each position (as determined by aggregate VUM) who represent the biggest drain on an individual statistic for a 5x5 roto league. I'm not saying don't roster them, just be aware of how they might affect your season stats.

First Base

Runs- Carlos Lee. HomeRuns- Carlos Lee. RBI- Carlos Lee/Kendrys Morales/Carlos Pena. Steals- Justin Morneau/Adrian Gonzales/Adam Dunn. Average- Carlos Pena.

Second Base

Runs- Juan Uribe. Home Runs- Brian Roberts. RBI- Brian Roberts. Steals- Martin Prado/Kelly Johnson/Juan Uribe. Average- Rickie Weeks.

Third Base

Runs- Pablo Sandoval. Home Runs- Michael Young/Chase Headly. RBI- Pablo Sandoval/Chase Headly. Steals- Casey McGehee. Average- Mark Reynolds.

Shortstop

Runs- Ian Desmond. Home Runs- Elvis Andrus. RBI- Elvis Andrus. Steals- JJ Hardy. Average- JJ Hardy/Jimmy Rollins.

Catcher

Runs- Yadier Molina. Home Runs- Yadier Molina. RBI- John Jaso. Steals- Matt Wieters/Miguel Montero/Geovany Soto/JP Arencibia. Average- JP Arencibia

Outfield

Runs- Mike Stanton. Home Runs- Juan Pierre/Brett Gardner/Michael Bourn. RBI- Michael Bourn. Steals- Nick Swisher/Andre Ethier/Mike Stanton. Average- BJ Upton

Since most leagues start 40-50 outfielders, the list of outfielders is drawn from the top-40 rather than the top 15.

These are all players who, relative to their position, can be expected to be a drain in the given category. Some of these are obvious, and some of these are trivial (are you really expecting to get steals when you draft Adam Dunn?). But, when you draft one of these players, you have to keep the statistic listed in mind as something that needs to be addressed later if it hasn't been already.

I'll update this later today with the exact numbers.

Value Under Maximum: Position by Statistic Targets

Value Under Maximum is a nice compound measure. At it's core, it is the percentage of best possible production a player gets across all categories slightly scaled for position. In a 5x5 league, a player with a VUM of 5 is leading the league in all five standard categories (and is therefore the best possible contributor in all 5 categories), while a VUM of 0 means the player contributes literally nothing.

But VUM is aggregated by adding the VUM at each statistic/category; and these individual VUMs give you a good idea of who to target given a position and statistic of need. You can do this just as easily by looking at statistic totals, to be honest. But you can set a specific aggregate VUM cutoff first and eliminate players with serious liabilities.

If I may put on my statistics wonk hat for a moment, you can assess "significantly worse than average" by using a 1-tailed student's t-test. For the top 250 batters, that means any player who is worse than the mean by roughly 1.66 standard deviations is, in a statistical sense, worse than average with a probability of 95 percent. What that means is that the projected performance is worse than average by an amount more than what could be accounted for by normal year-to-year differences in player performance.

In a standard 5x5 league, the mean aggregate VUM is 2.52 and the standard deviation is 0.654, meaning a significantly below-average player has an aggregate VUM of roughly 1.44. Using that as a cutoff, we'd actually only be left with the bottom 20 fantasy players. So let's be more lax. Let's put the probability at just better than 50-50, which leaves us with a cutoff value of 2.08. Any player with a VUM of less than 2.08 is therefore probably more trouble than they're worth to roster. But anyone with a VUM between 2.08 and 2.52 is worse than average, but not by so much that they hurt more than they help.

By position, here are the best players to target for a specific statistic who are flawed but won't have a significant downside (parentheses are overall VUM, stat-specific VUM):

First Base:

Runs- Justin Smoak (2.49, 0.61). Home Runs- Justin Smoak (2.49, 0.47). RBI- Freddie Freeman (2.40, 0.62). Steals- Daric Barton (2.21, 0.174). Average- Xavier Nady (2.17, 0.89)

First Base is incredibly deep, and the criteria established eliminate the top-20 1B. In a 10-team league, you don't need to dig this deep.

Second Base:

Runs- Brian Roberts (2.47, 0.80). Home Runs- Bill Hall (2.23, 0.53). RBI- Juan Uribe (2.39, 0.64). Steals- Chone Figgins (2.17, 0.85). Average- Howie Kendrick (2.39, 0.93)

Third Base:

Runs- Placido Polanco (2.45, 0.73). Home Runs- Kevin Kouzmanoff (2.40, 0.43). RBI- Kevin Kouzmanoff (2.40, 0.62). Steals- Placido Polanco (2.45, 0.16). Average- Placido Polanco (2.45, 0.92).

Shortstop:

Runs- Yunel Escobar (2.38, 0.68). Home Runs- Alex Gonzales (2.26, 0.39). RBI- Yuniesky Betancourt (2.31, 0.61). Steals- Alexei Casilla (2.50, 0.51). Average- Starlin Castro (2.39, 0.89).

Catcher:

Runs- John Jaso (1.82, 0.62). Home Runs- Chris Ianetta (1.86, 0.6). RBI- Matt Wieters (1.99, 0.65). Steals- Russel Martin (1.72, 0.28). Average- Yadier Molina (1.77, 0.86)

For catchers, only Mauer has a VUM above the mean and the only catchers that meet the original qualifications are the rest of the consensus top-5 (McCann, V-Mart, Posey, Santana) plus Kurt Suzuki. As such, I simply picked the best catcher that wasn't one of the top-5 for each category.

Outfield:

Runs- Logan Morrison (2.50, 0.72). Home Runs- Tyler Colvin (2.50, 0.52). RBI- Marlon Byrd (2.49, 0.57). Steals- Julio Borbon (2.41, 0.48) or Michael Brantley (2.35, 0.41) if you hate Borbon as much as I do. Average- Logan Morrison (2.50, 0.87).

Consider all these guys mid- to late-round stat boosters. They are the best at the statistic listed at their position who meet the "below-average-but-not-horrendous" qualifications.

Know Your League, or Why I Hate Mark Reynolds

It's standard advice, but it bears repeating: know your league.

KNOW. YOUR. LEAGUE.

As I've said before, and will probably continue to say at least once per week, most fantasy baseball rankings are based on a 5x5 rotisserie scoring system. But many leagues, and quite easily your league if you play seriously (or as seriously as fantasy baseball warrants) do not use the standard five categories; some aren't even rotisserie. If you tally more batter categories than the standard runs/RBI/HR/steals/batting average, then the value of batters change based on what those categories are. If you're in a points league, then points are accrued by methods above and beyond the standard five categories. In non-standard leagues, basing your strategy exclusively on 5x5 rankings and expecting to win is folly. It's like wondering why you're getting pulled over driving through Canada when the sign reads "speed limit 100." You're looking at the wrong damn thing. Canada's on the metric system, and you're only supposed to be going 60 miles per hour.

This is what brings me to Mark Reynolds. Reynolds is just fine for a 5x5 league. His 162-game average for his career is a respectable 90 runs/35 home runs/100 RBI/12 steals/0.242 batting average. The MLB-average batter during that same period of time is roughly 82/15/80/10/0.263 per Baseball Reference (if my math is correct). That makes Reynolds a roughly average contributor in runs and steals, well above average in homers and RBI, and a significant (but not insurmountable) negative in batting average.

[Last year's .198 BA was pretty abysmal, and the signals on that are mixed: his GB/FB ratio dipped and his line-drive percent plummeted along with his BaBip, but he was as good/bad as ever in terms of pitchers per plate appearance, strike percentages, etc.]

Even still, the remaining fantasy numbers are usually fine (if slightly depressed due to the BA). Basically, he's a flawed player worth a mid-round pick and top-10 potential every year at his position for fantasy purposes. If you need power, why not take him?

But that's for standard leagues. True story: I've never played in a league that doesn't count strikeouts as a hitter statistic. True statistic: Reynolds strikes out 221 times per 162 games as opposed to 192 for Ryan Howard, 183 times for Adam Dunn, 171 for Carlos Pena, 168 for BJ Upton, 159 for Dan Uggla, 150 for Matt Kemp, and 130 for Prince Fielder; all of these players are notorious for striking out. Reynolds whiffs far more often than any of them.

If you're playing in a league where you have to manage batter strikeouts, all of a sudden Reynolds isn't just a liability in batting average, he's an albatross for strikeouts too. You have to build a roster that explicitly minimizes strikeouts if you draft Reynolds, or else punt the category.

I played in a roto league that counted strikeouts last year, and among the top half of teams in the standings the fewest strikeouts was 1,058. Reynold's 211 strikeouts was 20% of that total by himself, never mind the other 9 people you had to play daily. The teams that rostered him at any point during the season finished last and sixth in strikeouts; they only played him for 48 and 21 games respectively. If instead, the former team rostered a 3B with similar production but half the strikeouts for just those 48 games, he would have won the league. Reynolds easily cost him 2 roto points in just strikeouts alone. He lost the league (to me) by 1.5 roto points. Lest you think that's impossible to find, half the strikeouts would have meant 106 over last season- still good for a 73rd-place tie and the company of multiple 20+ home run hitters.

OK, maybe you want some math to back it up. Using the Value Under Maximum metric, Reynolds is projected to be the #7 most valuable 3B for the upcoming season in a 5x5 league. I currently play in a 7x7 roto league, with Ks and doubles the extra batting stats. This is one statistic Reynolds is horrible in (Ks) and one he's slightly above-average in (doubles). With these categories added, Reynolds drops all the way down to the #25 third baseman. His VUM is 3.15 (on a 0-5 scale) in a standard roto league, while in my league it's 1.96 (on a -1 to 6 scale). The sheer number of strikeouts renders Reynolds unplayable in my roto league as a regular at even the corner infield spot- assuming the top 10 at 1B & 3B start at those positions (and not CI), he's still #28 on the list of CI-eligible players (#48 overall CI).

Drafting Reynolds in this league is like wearing a "kick me" sign. You're basically ceding first place, if not second and third as well. It's the fantasy equivalent of walking into a car dealership waving a wad of cash and declaring "I need something with wheels RIGHT NOW but I don't know what I want and I don't have the time to do any research or look anywhere else." You're just asking to be taken advantage of.

Incidentally, he went in the 12th round of my roto draft. The team that drafted him already dropped Wandy Rodriguez (his 10th round pick) for David Murphy (who's good, and I like him, but fighting for playing time), if you want to talk other indicators of knowledge. He's also got Adam Dunn (192), Chris Young (156), Jay Bruce (136), Travis Snyder (140), Ian Desmond (114) and Ian Stewart (153) starting; that's a projected 1,113 strikeouts from just under half the lineup. That's 6 of the top-30 strikeout hitters.

Reynolds isn't so hot in points leagues either. Under my BB league scoring system (detailed in the glossary), he's projected to get 394 points. That's a VORP at 3rd base of -0.244, meaning he'll produce at roughly 75.6% the capacity of a replacement-level 3B (521.2 points). That's good for the #25 3B, #181 overall hitter, and #244 overall player. He still went in the 12th round of that draft, at #111 overall. If you want an equivalency for that level of reach (using the method I detailed here), it's about the same cost as taking Ryan Zimmerman or Andrew McCutchen first overall.

In a points league, each strikeout counts against a player by a certain amount, say 1 point. If you know Reynolds is going to strike out 200-ish times, that means he starts off with a value of -200 points (give or take). Eighty walks and 35 homers are just about going to make that up. But what else are you expecting from Reynolds, other than strikeouts, home runs, and walks? He only averages 138 hits per 162 games, and only 65 extra-base hits, more than half of which are the home runs we've already factored out.

Or put it this way: he averages 221 Ks but 276 total bases per 162 games. For most points leagues, that's a net 55 points per 162 games, meaning any extra value comes from runs/RBI (which are significantly team dependent, and he plays for Baltimore), walks, and any quirks in your scoring system (e.g. counting hits on top of total bases).

Reynolds is ranked just outside the top-100 hitters in ESPN's preseason rankings, and with good reason. He's a solid contributor to a 5x5 team. But his flaws run deep enough that if your league isn't 5x5 roto, you have to seriously re-assess his worth. You have to know your league.

Value-Under-Maximum Roto Projections

Here's the 5x5 VUM-based rankings for each position. I thought you might like to see how they stack up. For hitters, the range is zero to five (since all categories are positive). For pitchers, the range is negative two to three (five categories, three positive but two negative). The VUM-values in parenthesis are added across all standard categories. In each category, VUM is a position-scaled percent of the best possible performance you could get in that category. That makes anything greater than 4 absolutely otherworldly, since that player is giving you more than 80% the best conceivable performance.

As a side note, pitchers will generally have low VUM rankings. Even a great ERA/WHIP is more than 50% the maximum value. That means the greatest starting pitcher, one who leads the league in Ks, Wins, WHIP and ERA, will likely only have a VUM of 1 or less (since Ks/Wins would each be one, but subtracting about 0.5 to 0.75 each for WHIP/ERA, leaving you with something between 0.5 and 1). As such, it's not a good idea to compare VUM between pitchers and hitters without some sort of standardization (e.g. dividing by the maximum score or creating a z-score). If that were the case, I'd recommend adding something between two and three to a pitcher's VUM to shift the range up to the same 0-5 range that hitters occupy. Without further ado (and as always, based on ESPN's 2011 projections):

First Base:

Albert Pujols (4.288), Adrian Gonzales (3.758), Miguel Cabrera (3.746), Joey Votto (3.732), Adam Dunn (3.558), Mark Teixeira (3.493), Ryan Howard (3.461), Kevin Youkilis (3.449), Prince Fielder (3.395), Paul Konerko (2.964)

Second Base:

Robinson Cano (3.451), Dan Uggla (3.349), Dustin Pedroia (2.955), Ben Zobrist (2.844), Ian Kinsler (2.829), Rickie Weeks (2.825), Martin Prado (2.786), Brandon Phillips (2.770), Aaron Hill (2.728), Kelly Johnson (2.714)

Third Base:

Evan Longoria (3.969), David Wright (3.777), Jose Bautista (3.745), Alex Rodriguez (3.597), Ryan Zimmerman (3.383), Adrian Beltre (3.352), Mark Reynolds (3.151), Pedro Alvarez (3.013), Ian Stewart (2.965), Michael Young (2.945)

Shortstop:

Hanley Ramirez (4.110), Troy Tulowitzki (3.838), Jose Reyes (3.182), Alexei Ramirez (3.014), Derek Jeter (3.001), Jimmy Rollins (2.996), Elvis Andrus (2.991), Rafael Furcal (2.911), Stephen Drew (2.879), Ian Desmond (2.643)

Catcher:

Joe Mauer (2.615), Victor Martinez (2.485), Brian McCann (2.283), Buster Posey (2.254), Carlos Santana (2.282), Kurt Suzuki (2.052), Miguel Montero (1.979), Matt Wieters (1.985), Geovany Soto (1.889), Carlos Ruiz (1.718)

Outfield:

Carl Crawford (3.987), Carlos Gonzales (3.8), Nelson Cruz (3.717), Ryan Braun (3.716), Matt Holliday (3.613), Justin Upton (3.583), Matt Kemp (3.52), Jayson Werth (3.426), BJ Upton (3.423), Curtis Granderson (3.364)

The next ten outfielders (in order) are: Shin-Soo Choo, Hunter Pence, Andrew McCutchen, Chris Young, Shane Victorino, Jay Bruce, Alex Rios, Jason Heyward, Josh Hamilton, Mike Stanton.

Starting Pitcher:

Roy Halladay (0.547), Tim Lincecum (0.359), Felix Hernandez (0.348), Jon Lester (0.334), Cliff Lee (0.308), CC Sabathia (0.299), Justin Verlander (0.225), Clayton Kershaw (0.151), Cole Hamels (0.127), Mat Latos (0.112).

The next ten: Roy Oswalt, Ubaldo Jimenez, Dan Haren, David Price, Tommy Hanson, Jered Weaver, Chris Carpenter, Yovani Gallardo, Francisco Liriano

Relief Pitcher:

Heath Bell (0.461), Neftali Feliz (0.397), Brian Wilson (0.338), Matt Thornton (0.264), JJ Putz (0.263), Mariano Rivera (0.208), Joakim Soria (0.202), Carlos Marmol (0.113), Jonathan Paplebon (0.104), Huston Street (0.097)

Not too bad, but we do see a few surprises. Ian Stewart and Pedro Alvarez cracking the top-10 at 3B is certainly one of them, but they may sniff the bottom of that at the end of the season. The whole order at 2B, if not necessarily the players involved, seems really odd. Dan Uggla a top-5 2B, yes. Dan Uggla the number two 2B? It could happen if he hits enough bombs. I don't quite buy Zobrist as top-4 at the position, as much as I might like him, but it's pretty closely bunched in the 4-6 (or even 4-10) range.

And lastly, catcher is horrible. The hitter rankings are directly comparable, which means the top catcher (Mauer) falls outside the top 10 at every other hitting position. And this in spite of the fact that Mauer is predicted to pace the league in batting average! He starts off with a guaranteed VUM of at least one, and still can't tack on enough to crack the top 50 (actually, I have him as the #108 hitter in a 5x5 league).

VUM is a ratio statistic within positions, meaning that if player A has a VUM twice as high as player B then player A is twice as valuable assuming they have the same eligibility. Kurt Suzuki, who you can probably pick up in round 20 or so, has a VUM that is 78% that of Mauer, who is off the board by round three. That means you can get 78% Mauer's production some seventeen rounds later. If ever you needed proof to wait on a catcher, that's it.

Value Under Maximum

Ok, so I mentioned in my last post just a little bit ago the flaws of pVORP. After tinkering with the idea of a sort of "value under maximum" measure, here's what I've got:

The best way to boost your standing in a given roto category is to draft the player who will (or is projected to) lead the league in that category. If I need steals, then the best thing I can do is draft someone who will swipe the most bags in a season. The corollary is that any player who will not pace that category is doing so by some fractional amount. That fraction is then a good indicator of his value in that category. That means, for any given player, their worth in a category is their projected production divided by the projected maximum in that category.

But, as always, rosters are built around certain constraints. No catcher will lead the league in steals, for example, and it's unreasonable to expect them to. Why should I devalue a catcher who won't steal any bases when no catcher will steal many, especially since I have to play a catcher? A catcher who steals five bases isn't necessarily worth less than an outfielder who steals ten. Five steals from a catcher is gravy, ten steals from an outfielder means he needs to be doing something else well.

I could, instead, divide a players expected production by the projected leader at that position. However, that has the problem of all positions not being equal in production. Since a catcher isn't going to steal many bases, why should the best base-stealing catcher (who will get maybe 10 steals) have the same value as the best base-stealing outfielder (who will get 50ish steals)?

The simplest solution, if not necessarily the best, is to account for both of these facts with a straight average. A catcher may not steal many bases, nor should I overvalue the best base-stealing catcher, but a catcher who does steal bases is a bonus nonetheless. What you then get for a given statistic is:

x/[(max(p)+max(l))/2]

where x is a player's production in a category, max(p) is the best projection at that position, and max(l) is the best projection for the league (AL, NL, or MLB). Rewritten, you get:

2x/(max(p)+max(l))

You can do this for any statistic the league uses. It also has the following advantages:

1) It's additive. You can add together a player's value under maximum for all statistics and get a rough idea of the player's overall value. The score has no units and so you're not, for example, adding runs to steals. Since all statistics count equally, there's no reason to weight it or average it.

2) It works for positive and negative statistics. If you have a statistic that is counted negatively (such as batter Ks, where the fewest Ks means the most roto points) you calculate it the same way but simply subtract it when aggregating or look for lower values instead of higher ones within a position.

3) It's simple. You don't have to adjust for league size, roster composition, etc. The maximum is always the maximum. This is true no matter how many teams are in the league. The position- and league-best stats are constant no matter how many catchers, infielders, outfielders, utility, etc. you roster.

4) It's generative. You can use it for any statistic in the league with ease to come up with league-specific rankings.

5) It works for counting stats and rate stats. Because the number is essentially a percent value placed on production, the nature of the statistic doesn't matter.

6) It has a circumscribed range. The value under maximum can only be between 0 and 1. For the league leader in a statistic, his production is x, the maximum league value is x, and the maximum position value is also x. That gives 2x/(x+x) = 2x/2x = 1. So a value of 1 is the best possible production for the statistic. Similarly, if a player gets none of something (e.g. zero steals), then the numerator is zero and therefore the value under maximum is zero. By the same token, the range for the aggregate score is zero to the number of statistics counted (e.g. 0-5 in a 5x5 league).

7) It properly devalues pitchers. The conventional wisdom in roto leagues is that pitchers will not contribute in all categories, as closers usually don't get wins and starters don't get saves. This inherently accounts for that, since (for example) starters will have a zero value under maximum for saves. This means when ranking all players (pitchers and batters), the pitchers will generally range from 0-4 while hitters from 0-5 (in a 5x5 league).

8) It accounts for the decreased impact of any one category as statistics expand. As the categories increase, the weight of a player being poor in one category decreases. For example, if I have a player with a low batting average, that's 20% of his value in a 5x5 league. In a 7x7 league, that's only 14.3% of his value. Since the range of the aggregate increases as the number of categories increases, the amount that the low batting average factors in decreases as well.

9) Best of all, VUM is more or less a ratio statistic. Within positions, the VUMs are divisible, and between positions a VUM ratio works OK (though not perfectly). That means if Player A has a VUM that is 85% that of Player B, then Player A is (in the aggregate) about 85% as good as Player B for fantasy purposes (especially if they play the same position).

So there you have it. Instead of using pVORP, I'm going to calculate my roto rankings using Value Under Maximum (VUM).

Further analysis of pVORP

Earlier I detailed some of the difficulties in calculating value in a roto league. It is certainly more difficult than calculating value in a points league, to be sure. Each category has its own value, and so categories are not interchangeable. Furthermore, though the position of a player doesn't matter insofar as a direct contribution is concerned (as in, you should generally get power from 1B and speed from a middle infielder, but vice-versa yields no net difference), but getting less-than-expected contributions from a position does matter (it's harder to get power if you're not getting it from first base).

This led me to the idea of partial value over replacement player, or pVORP. You take a player's specific contribution to a category (x), and subtract from it replacement production for that statistic at that position (r), and divide by the total production in that statistic from an entire roster of replacement players (R). This yields (x-r)/R, which is essentially excess production given a position and a category/statistic. If you're in a 5x5 league, then every batter has five separate pVORP values, one each for runs, RBI, HR, SB, and batting average. You can then add them together to get an aggregate value for the player.

It's important to note, though, that a team of replacement players would yield one point in every roto category. Moreover, a team of replacement players would be in last place in every category by a significant margin. Positive pVORP values aren't "bonus" production, it's wholly necessary.

Here's a perfect example. Let's say you draft an OF who should steal about 30 bases. A team of replacement players might steal 130 bases, and a replacement outfielder will steal about eight. That makes this players pVORP for steals (30-8)/130 = 0.169. That's a good value, but even if it's the best value in the league all it means is that this particular OF will contribute about 16.9% more steals than expected relative to replacement level. If the remainder of your roster has little or no stolen base value, you'll still end up in last place in the category.

Moreover, imagine you have a 1B who steals 20 bases. Since you expect fewer steals at 1B, replacement level may be around 3. This particular player's pVORP for steals is then (20-3)/130= 0.131. This value is only slightly less than the pVORP for the above OF, but the OF is contributing 50% more steals. It's just that you expect more steals from an OF. The two do not make a comparable absolute contribution, even if they make similar relative contributions.

What this means is that pVORP is best used in comparing players within a position. It gives you a way to quantify the overall value of two different types of players, like Dan Uggla and Chone Figgins. When comparing pVORP between positions, then, some qualifications are necessary. If a 1B has a higher pVORP than a 2B, what it means is that the 1B is more valuable as a first baseman than the 2B is as a second baseman. It does not necessarily tell you which is more valuable overall.

The other thing to note is that pVORP is going to skew a little in favor of players who steal bases. Steals has a top-heavy distribution; replacement level at a position is often in the single-digits and always less than 20, and so the total replacement level for stolen bases is small. However, the best base-stealers can swipe upwards of 40 bags a season. For example, there are multiple outfielders who can steal 40+ bases, but because so many OF are rostered (and steals drops off so quickly), replacement level is only ten bases, give or take a few for roster composition. This skew means that a base-stealing OF will have a high pVORP in steals. This will give them a relatively high aggregate pVORP, whatever their other contributions may be.

There are possible statistics to use instead that would get rid of that, such as a value-under-maximum statistic. You would, essentially, take an average maximum for each statistic at each position over a given period of time instead of predicted replacement level and do essentially the same calculations. For example, if at 1B the average maximum number of HR is 40, then first baseman x has a production under maximum of x-40. You then calculate this maximum value for every position. You then add up these values (say, to 429 home runs). That gives you the number of HR you would get if that were the only stat you focused on. That 1B then has a value under maximum of (x-40)/429. Every player would have a negative value, with less negative values being better. This has flaws of its own, though.

pVORP is a fine stat to use, but like all measures it has its limits. In this case, that limit is that it is best used to compare within, as opposed to between, positions.

Calculating value in a roto league

Calculating value in a points league is easy. Because each statistic counted has a specific value, there is an equivalency between statistics, and all that matters is points accrued over a given period of time (game/week/matchup/season). This is not the case in a rotisserie (or for that matter H2H categories) league. Statistics are not interchangeable, since the number you accumulate in each contributes to your team points in a different way. In a points league, for example, runs are one point each and RBI are one point each. If I get 1,800 combined runs and RBI, the proportions don't matter since all possible allocations (from 0 runs and 1,800 RBI to 1,800 runs and 0 RBI) come to the same point total. In a roto league, the distribution matters.

In 2009, the last year for which I could find the relevant information (here), it took 1,189.2 runs to win the category in a 10-team league (with 13 starting hitters) and 1,168.2 RBI to win that category. That means if all 1,800 are in one category or the other, you net 11 points (first place in one and last in the other). An even split (900 and 900) will only net you 2 points, since neither 900 runs nor 900 RBI break the 2nd place average in those categories. You can (approximately) calculate the optimal distribution, but that's beside the point. The idea is that the distribution of those statistics matters in roto but not in points.

Since categories are not interchangeable, you have to calculate value in each category separately. You can do this using a simple VORP formula, which is expected production (x) minus replacement production (r) divided by replacement production (r again). This yields the typical (x-r)/r formula. But you can't simply calculate replacement value in each category for each position, either. Much as points are points no matter how they are accumulated in a points league, stats are stats in a roto league no matter what position they come from. Calculating value by using only typical position production leads to wildly asymmetrical results.

As an example, let's focus on home runs and steals. At position A, replacement level is 30 home runs and 2 steals. At position B, replacement level is 15 home runs and 20 steals. If I get 30 home runs and two steals from position A and 15 home runs and 20 steals from position B, the aggregate VORP is zero for each stat at both positions. However, if I get 15 home runs and 20 steals from position A and 30 home runs and 2 steals from position B, the VORP values are this:

Home Runs A: (15-30)/30= -0.5

Steals A: (20-2)/2 = 9

Home Runs B: (30-15)/15= 2

Steals B: (2-20)/20= -0.9

Adding the VORP values together in this case gets you 9.4, which is obviously very different from zero. Nor are the VORP values multiplicative, since in each case you get -1.

What I would do instead is calculate a partial value over replacement (pVORP). Essentially, you take a roster full of replacement players at each position and calculate what the total replacement value is in a category. In the above example, total replacement home runs is 45. You then divide straight production above replacement (x-r) by total replacement home runs. So at positions A and B, typical production still gives you a pVORP of zero (production above replacement is 0, and 0 multiplied by anything is 0). Atypical production gets you a pVORP of (15-30)/45= -15/45= -1/3 at position A. At position B the atypical production gives you (30-15)/45= 15/45 = 1/3. Add them together, and you get zero again. Since you're getting no value over replacement, this is what you want.

(As an aside, the reason it's important for pVORP to be additive is that the statistics themselves are additive. Even the rate stats are additive, insofar as they are calculated by adding up the individual values for the team in both the numerator and the denominator. For example, the batting average for two players is the sum of all hits divided by the sum of all at-bats).

You then essentially do this for every statistic, hitter or pitcher, that your league uses. In a later post, I'll give you the values for 5x5 statistics plus several other common ones (e.g. quality starts, holds) calculated for a 10-team league. It gets complicated, especially at pitcher where many leagues use undifferentiated spots, but I'll see if I can't make some simplifying assumptions.

More on Adjusted PAR

I've been thinking about how I've been using adjusted PAR (and to a lesser extent, projected points) in draft prep. To date, I've been using a top-to-bottom qualitative ranking for both measures as the basis for a draft order, rather than treating each quantitatively. That begs the question: what does happen if you use xPts and adj PAR as quantitative values, rather than using qualitative rankings?

The short answer is, it depends. Namely, it depends how you take both into account. A straight average of the two is .5*(xpts+adj PAR), or:

This in turn simplifies into:

The quantities (xPts-rPts) cancel out, and by turning ".5" into "1/2" we get:

This is as good a way as any to do it, though you can weight the average if you wish. Weighting it is only going to modify the base ratio (xPts^2)/rPts.

This retains the same basic advantages of adj PAR, in that it squares xPts (putting more value on extra points accrued at the top while bunching players in the middle and bottom), scales for replacement value at a position, and preserves positional rankings.

Using a straight average, though, what we see is a greater weight put on total points. Players at deep positions (large rPts value) move up in the draft. For example, Miguel Cabrera, Adrian Gonzales, Joey Votto, Mark Teixeira, Prince Fielder, and Billy Butler all move up four to ten spots; in some cases it's not more only because they cannot go higher (e.g. Gonzales moving up to #2; he can't leapfrog Pujols since they play the same position but Pujols will score more). This becomes even more pronounced beyond "replacement range," which we'll call anything outside the top-15 at a position.

The reason for this is that within that range xPts rank greatly outpaces adj PAR rank; the latter is negative and so it cannot rank above the total quantity of starting players (for Bernie's Bombers, that would be no higher than #171). The further outside "replacement range" you get, the more pronounced this effect becomes. As an example, take Adam LaRoche. He projects to 469 points, which is #20 among first basemen. This is going to give him a very negative adj PAR (-81.06, to be exact), which is 353rd among ranked players. However, his 469 points is 136th-most overall. Players with discrepancies this large between adj PAR and xPts rankings are going to benefit the most from this new calculation.

The opposite is going to happen at shallow positions, such as catcher. Every single one of the top-20 catchers moves down in the draft order, because (generally) they will have relatively poor xPts rankings but high adj PAR rankings. The same holds true for pitchers, both starters and relievers.

Essentially, at positions where xPts rankings are higher than adj PAR, the players will (on the whole) move up. Where xPts rankings are lower than adj PAR, the players will (on the whole) move down.

This doesn't affect the top of the draft so much, since those players have high xPts values anyway. In fact, the top-10 in my rankings changes in order but not in composition, and while the order changes a lot only three new players move into the top-50 (first five rounds) and four new players move into the top-100 (though the order is shuffled more the more the list expands).

This greater weight on total points is, perhaps counter-intuitively, more valuable in deep leagues (more spots to fill, more teams). In shallow leagues such as mine, you have plenty of opportunities to roster high-point players, so you need to maximize your edge at shallow positions. At first base, the difference between the #4 first baseman and the #8 first baseman is only 50 points, less than a 10% difference. As such, who I get in that range doesn't matter so much. At catcher, the difference between #4 and #8 is 90 points, which is a 21% difference. In that case, it matters a great deal if I miss out on a top catcher. Since everyone can get a good first baseman, it's relatively more important to get a good catcher.

In deeper leagues with more teams and/or more positions (such as corner or middle infield), not everyone can get a good player at a deep position, much less a shallow one. There's also more flexibility if the added positions are more open. In a 10-team league with corner infield, middle infield, and two utility spots (these do exist) one person can play up to four players from any infield position (or six outfielders, or three catchers). The increased flexibility means a set position is a smaller part of your overall roster composition, so the scarcity at that position matters less. It's better for me to draft an extra first baseman (whom I can still start) who will score more points than whatever catcher I have to reach for.

In larger leagues, the percentage of teams in an equivalent position to you if you miss a shallow position is greater. If you miss a top-4 catcher in a 10-team league, then 60% of the teams are scrambling among flawed starters. If you miss a top-4 catcher in a 16-team league, then 75% of the teams are in the same position. How much you're "marginally screwed" by a missed position decreases, so there's little or no point in making sure you get the best of the dregs. It's better to concentrate on where you can actually get points instead, which is usually at a deeper position.

The moral of the story is that there's no right way to use a metric or create pre-draft rankings, but the manner in which use a metric or create rankings has to be carefully thought out. You need to account for roster size, roster flexibility, number of teams, etc. You can create a mathematical/statistical reason for a certain ranking system, but just as often it's as easy or easier to try out a couple of systems and eyeball it. No draft is going to proceed perfectly efficiently, so the general rankings (e.g. how to weight scarcity, the rough area a player occupies) matter more than specific ranking positions.

When should you draft an injured player?

Baseball players get hurt. It happens to every player at some point. They'll miss time due to the injury, and even after they return they won't likely be performing at their full capacity right away, if they even get back there at all during the season.

When a good player gets hurt before the season, a player you may want on your fantasy team, the question then becomes when they should be drafted. Obviously, they shouldn't be drafted as if they were healthy, because they will not give you production equal to the value of that draft spot. You have to assess the value they will give you during the season, and draft accordingly.

There's an inherent uncertainty to this, since information on a) when the player will return, and b) how the player will produce when he does return is wholly imperfect. There may be a timetable for return, but the player may race ahead of or fall behind that projection. Similarly, a player is likely to return before he is 100% healthy, but how much that affects production is an open question.

Take, for example, Chase Utley. Before his knee issues cropped up, he was generally considered one of the top three second basemen available. Now that he'll be out for an indeterminate period of time, and his production when he does return is in question, his value has dropped. So how far should he fall in the draft before he is worth a pick?

The first determination you have to make when assessing an injured player is when he'll be healthy enough to start for your fantasy team. A player may get back into his actual team's lineup when he's "70% healthy," but this may not be enough production to start him on your team. Assuming that a phrase such as "70% healthy" roughly translates to "will perform at 70% of his expected pre-injury production," then he isn't worth starting until "X% healthy" has a value of X= 1/(1+VORP). At that point, he'll be performing at exactly the level of a replacement player. In this case, since a player is missing time, VORP cannot be assessed in terms of total season points; instead it should be assessed in terms of production rate. The best measure for this would be either points per game or points per PA.

Using Bernie's Bombers scoring, Utley's xPts/PA is 0.9524, which gives him an xPts/PA VORP of 0.101. This means he shouldn't be in your lineup until he's "90% healthy," since 1/1.101 = 0.908.

Next, we have to make an assumption as to when Utley will be 90% healthy. The outlook on his knee isn't so good, so let's assume he won't be back in action and worth starting on a fantasy team until right after the All-Star Break. Further, let's assume he increases production by 1% per week. That would mean his first week back on your roster he's at 90%, then 91% the following week, 92% the week after that, etc. And further, let's assume he hits a plateau at 95% capacity, since the injury is a) irreversible damage to knee cartilage, b) knee injuries affect all aspects of a player's game, and c) actually playing won't help the knee heal.

As you can see, that's a lot of assumptions. That's what makes drafting injured players so difficult, and why most people try to stay away as much as possible. But let's press on.

Up until the All-Star Break, you have to play a different second baseman. Chances are, that second baseman will be at or around replacement level. A full-season projection for a replacement-level second baseman is 523.25 points. For the first 14 weeks of the season, that's your expected prorated production. The season is 25 weeks long, so for the first 14 weeks your 2B production will be 14*(523.25/25) = 293 points, or about 21 points per week. During Utley's first week back (week 15) he'll also be at replacement, so he'll net you 21 points. The next week (week 16) he'll be at 91%, or 21.04 points, then 21.27 during week 17, 21.5 during week 18, 21.74 during week 19, and then 22 points per week for weeks 20-25.

That puts your expected 2B production over the course of the season at the sum of your expected pre-All-Star Break production from a replacement player plus Utley's expected weekly post-return production, or 531.55 points. Essentially, drafting Utley as your primary 2B is equivalent to drafting a 2B with an xPts value of 531.55 points.

Using this value, we can then calculate adj PAR for your 2B spot. Using the adj PAR formula of ((xPts^2)-(xPts*rPts))/rPts, you have ((531.55^2)-(531.55*523.25))/523.25, which is 8.432. That yields rankings for your expected 2B production at #80 for xPts and #146 for adj PAR. The average of those two (113) puts Utley at #100 overall, or roughly the end of the 11th round.

So, under these assumptions, Chase Utley should fall from the 2nd round to somewhere in the range of the 11th-12th round due to lost production. That makes him roughly the #9 second baseman, below Ian Kinsler and Brian Roberts but above Kelly Johnson and Aaron Hill. Obviously, Roberts has injury concerns of his own, so perhaps Utley is the #8 second baseman.

Now, these calculations are made under a number of assumptions about playing time, post-injury production, etc, any or all of which may (actually, will) be wrong to varying degrees. Utley may miss more or less than 14 weeks, and (as a player who is known for toughness) he may produce at up to 100% production at some point after he gets back, and perhaps fairly soon after he returns. It also assumes you draft essentially a replacement-level player and not a player whose production is significantly below replacement.

The point isn't the specific numbers on Utley, the point is that this is how you go about discounting players who have a preseason injury. You make assumptions about return date and post-return efficacy, and follow these steps.

Glossary

Given the proliferation of statistical and blog-specific terms, I thought a glossary would be a good idea. Some of these terms you're likely to know cold if you're reading a fantasy baseball website. Some terms are more obscure, and some are ones I invented. I will keep this continually updated each time I use a new term.

Adj(usted) PAR: Adjusted Points Above Replacement. A method of scaling value by positional scarcity in points leagues. Calculated by multiplying expected points by VORP, yielding the formula ((xPts^2)-(xPts*rPts))/rPts

aPk: Actual Draft Pick. Where a player is taken in a draft. Used to calculate scalar inefficiency.

BB (or Bernie's Bombers) Roster: The roster composition for my head-to-head points leagues. Starts one each at catcher, first base, second base, third base, and shortstop, four outfielders, five starting pitchers, two relief pitchers.

BB (or Bernie's Bombers) Scoring: The scoring system specific to my head-to-head points league. For hitters, one point per hit, walk, run, RBI, or stolen base, one point per single (on top of the one for a hit, two total), two per double (three total), three per triple (four total), and four per home run (five total), -1 point per strikeout and error. For pitchers, 3 points per inning pitched, one point per strikeout, -1 point per hit or walk given up, -2 points per earned run, 4 points per complete game, 2 points per shutout, 10 points per no-hitter, 6 points per win, -3 points per loss, 5 points per save.

H2H: Head-to-head league, in which two teams match up in a given week. Can be either points or categories.

PAR: Points Above Replacement. The number of points a player is expected to get over a player of the same position who should be available at any given time and/or is just barely good enough to start. Calculated by xPts-rPts.

Points league: A league in which each outcome has a pre-determined point value.

Position scarcity: An informal term denoting how many good players are at a position and/or how poor a position is at replacement level.

pVORP: Partial Value Over Replacement Player. Used to calculate value in a roto league. Is equal to a player's production minus replacement production at position divided by the expected production from a team of replacement players for a single statistic.

Replacement/Replacement Player: The expected best player available at a position on the waiver wire, or the player you should expect to get if you fill a starting position last among your league.

Rotisserie (or Roto) league: A league in which each category is calculated constantly over the course of a season. Teams are awarded points equal to the inverse of their standing in a category. In a league of x teams, the 1st place team in a given category gets x points, second gets x-1, third gets x-2, and so on down until the last place team gets one point. Counting stats are summed throughout the season, rate categories are taken using a true average (i.e. the batting average is calculated by taking the sum of all hits by active players and dividing by all at-bats by active players). Generally, a league is described as "a x b", where a is the number of categories used for hitters and b is the number of categories used for pitchers. For the vast majority of leagues, a=b.

rPts: Replacement-level points. Calculated, generally, by averaging the two or three lowest-point-total draftable players and the two or three highest-point-total waiver wire players in a points league. For a league of x teams starting y players, this means averaging the expected point totals of players (x*y)-1, (x*y), (x*y)+1, and (x*y)+2 when players are ranked in descending order of expected point totals. For example, in a ten-team league that starts one second baseman per team, rPts is the average of the 9th, 10th, 11th, and 12th highest point-total second basemen.

Scalar (or Draft) Inefficiency: A formula that assigns a number to every draft selection based on where the selected player is ranked and the placement of the draft pick. Calculated by the formula 10*[(aPk-xPk)/(10+aPk)]. The formula yields a negative value when a player is taken earlier than expected and a positive value if a player is taken later than expected. Allows for equivalencies between separate draft actions. Generally speaking, for an acceptable pick this value is not less than -1 (the value of reaching for the #51 player with a median pick in the fifth round).

Standard Points League: Scoring system used for a standard points league. For hitters, 1 point per total base, run scored, RBI, walk and stolen base, -1 point per strikeout. For pitchers, 3 points per inning pitched, 10 points per win, -5 points per loss, 5 points per save, 1 point per strikeout, -1 point per hit or walk issued, -2 points per earned run.

Standard (or 5x5) Roto League: The most common rotisserie format. The categories are batting average, home runs, runs, RBI, and stolen bases for hitters. For pitchers, the categories are strikeouts, wins, ERA, WHIP, and saves.

VORP: Value Over Replacement Player. Calculated using (xPts-rPts)/rPts. Yields a decimal which is the percentage of points player X scores that are above replacement for a points league.

VUM: Value Under Maximum. A position-scaled percentage of best possible performance for a statistic (e.g. steals, RBI) in a roto/category league. Can be calculated for an individual stat with the formula (2x)/(max(p)+max(l)), where x is the projected performance, max(p) is the highest projection at the position, and max(l) is the best projected performance in the league. An aggregate can be created for a player by adding the VUMs for each individual statistic.

xPk: Expected pick. Where a given player is expected to be chosen in the draft. For my purposes, calculated by ranking players in ascending order by the average of their rank in xPts and their rank in adj PAR, with ties broken in favor of a higher xPts value.

xPts: The expected number of points a player is projected to accrue over the course of a season based on preseason projections. For the most part, this blog uses ESPN's 2011 player projections unless otherwise noted.

Is it possible to draft aggressively?

Drafting in a snake format is, by and large, a passive affair. The only true action is the first pick in the draft, since every pick after that is at least partially conditional on the pick(s) made before. Who you choose with the third pick depends on who is there after the first two picks, and who is there at the second pick depends on who is chosen first. You may think you've acted in taking Hanley Ramirez second overall among the alternatives, aggressively filling a need at shortstop. But, if Albert Pujols were still there with the second pick, would you still draft Hanley Ramirez even if he were still available? I would guess not.

But, within these constraints, is it possible to draft aggressively? That is, can you make a draft decision that forces other people to change their strategy and/or alter the value of their picks?

To that end, one possibility easily comes to mind: drafting your utility player early. As I said inmy post on calculating value, in a 10-team league there are ten starting players each at first, second, third, short, and catcher (since everyone plays multiple outfielders, it's difficult to accomplish this task using the outfield position). Drafting an extra player from the top ten at one of these positions to put in your utility spot changes this equation, though. When such an action occurs, a player who was originally below replacement now has to be drafted into a starting spot.

This clearly puts the person who gets stuck with this player at a disadvantage. But, because this player must be drafted and started, it also changes replacement value. If someone drafts two of the top-10 second basemen, the #11 second baseman is now a starting-caliber player instead of a top reserve. This means replacement value, which is calculated using the last two rostered players and the first two reserves, is now the average of the #10-13 players instead of #9-12. This will lower replacement value by some amount a. Lowering replacement value raises adj PAR. The change in adj PAR is calculated thus (x is projected points, r is pre-draft replacement value, and a is the change in replacement value from drafting a utility player):

This simplifies into the following formula:

If adj PAR rises enough then his adj PAR ranking (relative to the remaining players available) will rise. If the adj PAR ranking rises, then the expected draft position (calculated by averaging adj PAR ranking and total points ranking) rises as well. Because you've increased the scarcity of this position, you've changed the dynamics of the draft. It may be by a little or a lot, but the dynamics have changed.

One more thing to note- the change doesn't just happen within the position. Overall draft rankings are created by averaging adj PAR rankings and total points rankings. If you change the adj PAR for one position, you change the adj PAR rankings of every remaining player. This creates a ripple effect that changes the averages, and therefore the order, of the entire rest of the draft.

You can maximize the change in rankings the following ways. First, choose a position where many of the remaining players are further down in the rankings of players left. If a player is near the top of the rankings of players remaining, then there's nowhere for him to rise to. More valuable players will see a greater increase in adj PAR, since the amount of change is directly proportional to the square of expected points. But these players are more likely to be near the top of the rankings of players left.

Second, the shallower the position, the greater the increase in adj PAR. Adj PAR has an inverse relationship with the square of original replacement value [r(r-a) = (r^2)-(a*r)]. Therefore, shallower positions will see a greater increase in relative value among the remaining players. This has to be balanced with the fact that you may be getting fewer total points from a player by drafting out of a shallow position.

Last, adj PAR increases more the larger the difference in replacement value you create. The difference is a positive multiplier in the numerator and subtracted out in the denominator, so as a increases the numerator increases and the denominator decreases.

Here are the a-values for the first utility pick at each position (using ESPN's 2011 projections):

Catcher: 5.75

First Base: 4.5

Second Base: 3

Third Base: 5.6

Shortstop: 3

Now, every way to maximize the advantage of this move has a counterpoint that could hurt you. You are likely to get fewer total points drafting a second player out of a shallow position as opposed to a deep one. Players in deep positions, being more likely to have more total points, will see a greater increase in adj PAR, but they are already likely near the top of the remaining players. So the best position with which to do this has a) low replacement value, and b) a top-heavy skew in the distribution.

This sounds an awful lot like catcher.

Replacement value for a catcher, pre-draft, is 398.5 points, the worst among hitters. It's top-heavy, with the top two catchers ranked #23 (Mauer) and #30 (Victor Martinez) overall in total points, and gets very bad in a hurry. In fact, the rest of the position looks like this (overall point ranking in parenthesis): McCann (96), Posey (97), Santana (111), Suzuki (121), Wieters (181), Montero (188), Yadier Molina (195), Jaso (207), Soto (235) and Ruiz (241). The a-value for taking an extra catcher is 5.75, highest among the five relevant positions.

Well mock this possibility up in a subsequent post, but first I'd like to roll out another example. This actually happened twice in the first five rounds in my draft. One owner took first basemen with back-to-back picks in the first and second round, and then I took a second third basemen in the fourth round after taking one in the second round.

The first 11 picks of the draft went like this: Pujols, Hanley Ramirez, Crawford, Cano, Longoria, Braun, Tulowitzki, Carlos Gonzales, Cabrera, Fielder, Adrian Gonzales. So the same owner drafted Fielder and Adrian Gonzales, leaving (since Pujols and Cabrera were already drafted) seven owners competing for six above-replacement first basemen. Now, I think Votto or Teixeira would have been a better choice than Fielder, but a) the quantitative effect on whomever remains is the same so long as both players are above-replacement, and b) there's no dissuading a Brewers fan from drafting Prince Fielder.

Here's how the next handful of first basemen ranked overall before the draft as compared to after the pick:

Texeira (#22 pre-draft, #22 post-pick)

Votto (#25, #23)

Youkilis (#35, #31)

Butler (#41, 41)

Carlos Lee (#84, 82)

Dunn (#92, #88)

Gaby Sanchez (#100, #97)

Ryan Howard (#101, #98)

Paul Konerko (#110, #105)

Justin Morneau ( #118, #107)

James Loney (#126, #114)

There wasn't much change, but players did move up; this effect was greater as the players dipped below replacement value. This is, of course, is in part due to the fact that a) the remaining top-4 had little room to move, and b) many of them are closely clustered in value. And the optimal next ten picks, instead of looking like this:

Halladay, Holliday, Felix Hernandez, Mauer, Pedroia, Victor Martinez, Cliff Lee, Tim Lincecum, Andrew McCutchen, Ryan Zimmerman.

instead look like this:

Halladay, Holliday, Mauer, Felix Hernandez, Pedroia, Victor Martinez, Andrew McCutchen, Cliff Lee, Ryan Zimmerman, Tim Lincecum.

It's not a large change, but the ripple effects of selecting a utility player early are noticeable right away.

After I picked in the fourth round of the draft, there was a similar situation with third base. With Longoria, Wright, Zimmerman, Alex Rodriguez Bautista, and Beltre off the board, five teams were competing for four above-replacement third baseman. Here's how the next handful of third baseman stack up in overall rank pre-draft, after the Fielder/Gonzales duo, and after the Beltre pick:

Casey McGehee (#73 pre-draft, #63 after Fielder/Gonzales, #71 after Beltre)

Michael Young (#75, #65, #76)

Placido Polanco (#95, #85, #96)

Miguel Tejada (#96, #86, #98)

Pablo Sandoval (#99, #91, #101)

Jhonny Peralta (#163, #140, #150)

Aramis Ramirez (#166, # 151, #155)

Pedro Alvarez (#174, #156, #162)

**Update**

(Initially, I really screwed this part up. Below is the corrected version)

After (if not because of) the Beltre pick, the rankings of the remaining third basemen were re-depressed with respect to the beginning of round two. I would argue that this is due to the fact that, at pick #37, the draft had proceeded relatively inefficiently. That is, after 37 picks there remained a full seven players available who should have already been drafted. By contrast, after the Gonzales/Fielder picks, there were only three players out of eleven who should have been drafted. The less efficient a draft has been up to that point, the less the effect of a utility pick.

The simple reason for this is any player who remains but should already be off the board has disproportionately high value, and so should be picked next regardless of any change in positional scarcity. As an example, here (using pre-draft rankings) are the expected 38-50 picks (x indicates a player already drafted by that point):

Prado, Cruz(x), Reyes, Butler, Huff, Kershaw, Delmon Young, Haren, Alex Rodriguez(x), Tommy Hanson, Vernon Wells, Greinke, Werth.

Using the pre-draft rankings, here are the expected 38-50 picks after the draft started to unfold (y indicates should have been picked before):

Victor Martinez(y), Victorino(y), Etheir(y), Verlander(y), Markakis(y), Pence(y), Ichiro Suzuki(y), Prado, Reyes, Butler, Huff, Kershaw, Delmon Young

That's a full half of the picks. Those seven players will top the rankings of remaining players either way. Any ripple effects will only be apparent after the players who have fallen are off the board.

Put another way, let's look at the aggregate scalar inefficiency value of players left on the board who should have been taken. We'll assume aPk is the next pick after the utility pick, which is #12 for the Fielder/Gonzales picks and #38 for the Beltre pick. At pick #12, the players left who should have been taken are (scalar inefficiency value in parentheses):

Roy Halladay (6.667), Matt Holliday (4.667), and Felix Hernandez (0.455).

The players at this point who should not have been taken are (scalar inefficiency calculated at #12 again) are:

Prince Fielder (-7.73), Carlos Gonzales (-2.73), and Tulowitzki (-0.455)

Aggregating the values (sum of players who fell minus sum of players who were picked early), we get a value of about 22.704.

By contrast, after the Beltre pick (#38), here's the values of the fallen players left:

Victor Martinez (6.05), Victorino (3.68), Ethier(2.11), Verlander(1.84), Markakis (1.32), Pence (0.53), Ichiro Suzuki (0.26)

And here's the value of the players who should have been left:

Chase Utley (-40.833; that's what ignoring an injury gets you), Alex Rodriguez (-1.667), Josh Hamilton(-5.0), Matt Kemp(-3.125), Ryan Howard(-13.125), Justin Upton(-2.92), Nelson Cruz (-0.208).

Aggregating the same way as we did before, we get a whopping 82.665 for our scalar inefficiency value. The way other teams used their picks was so glaringly inefficient that it dwarfed any effects of changing replacement value.

So is it possible to draft aggressively using the utility spot? Yes, but only to the extent that people are drafting efficiently. If people are letting players fall relative to their rankings, then the benefit of drafting those players greatly outweighs any possible benefit of drafting a utility player. The more inefficient the draft, the less the relative value of changing replacement level at a position.

It's also worth noting that the other league members have to respond to this aggressive pick. If they don't then the only effect is someone drafts a below-replacement player at that position. The benefit, however, is that if they take note of it (in a naive, intuitive way since they're not doing these actual calculations) you create a different set of rankings for the other members, while you can follow the original rankings. This inefficiency is something that can then be exploited to your benefit in the draft.

Given all of these conditions, it is likely best to draft a utility player early only if he is among the top players left in your rankings and none of the other players around him fill a need or you have concerns about them. The potential benefit is relatively small, and the draft is likely to have plenty of inefficiencies due to players falling, so the change in the rankings is more of a side benefit of being able to draft a utility player rather than a sound strategy in and of itself. You may still leave yourself with a valuable trade chip, especially at a shallow position, so it's not necessarily a bad strategy. It's just that the strategy doesn't have a big enough impact to seriously consider in most cases.

Later, I'll mock up what is likely the best-case scenario for this: drafting both Mauer and Victor Martinez early in a fairly efficient draft.

A quick link: SWIP

Baseball Guys has a few posts up on SWIP, a new metric that attempts to measure control of the strike zone. The basic stat is (K-BB)/IP. It rewards guys who get strikeouts by setting up hitters and controlling location rather than with extreme, uncontrolled movement. I'd like to see how it stacks up to K:BB ratio in predicting performance, which perhaps I'll do at a later date. In the meantime, here's part I and part II. Part III will be up later.

**Update**

Part III right here.

Overvalued/Undervalued players in points leagues

Most fantasy analysis websites base their rankings on 5x5 rotisserie leagues. Some of them have separate rankings for points leagues, but those for the most part are tucked away and/or not given much extra analysis. Moreover, the rotisserie rankings are given privilege; ESPN for example puts players in their rotisserie rank-order in their draft rooms no matter the league format.

Rotisserie leagues vary in the statistics they use, but the standard 5x5 format (which is how the rankings are created) use Runs, Batting Average, RBIs, Home Runs, and Stolen Bases for hitters and Wins, Saves, Strikeouts, ERA, and WHIP for pitchers. You then get points based on the inverse rankings in a statistic (e.g. the team with the most RBIs gets 10 points, 2nd place gets 9 points, etc).

This scoring system bears only a tangential relationship to points leagues. Runs, RBIs, Home Runs, Stolen Bases, Wins, Saves, and Strikeouts all count in a points league, but they have different values (as opposed to equal weight in a roto league). To wit, in a roto league you need to balance accumulating home runs with accumulating stolen bases to maximize your standing in both categories. In a points league, home runs accrue points faster than stolen bases and have more value; in fact, you can theoretically win each matchup without ever having anyone on your team steal a base. In a roto league, you can't forfeit those points and expect to win. Moreover, batting average/ERA/WHIP bear only a tangential relationship to scoring in a points league. These are rate stats, and better rates mean better point totals in a points league. But in a points league, there may be a theoretical matchup in which two teams each get, say, 60 hits (all singles). Team A gets those 60 hits in 200 at-bats, team b gets 60 hits in 230 at-bats. In a roto league, team A had a much better week (because he has a higher batting average). In a points league, the two are equal. Better rate stats don't mean more points, they simply mean a greater likelihood of more points.

Because of these disconnects, certain players will be ranked higher or lower in a roto league than they would be in a points league. Since many team owners, even in points leagues, will be influenced by roto league rankings it is important to get a handle on where value is improperly assessed. Using ESPN's top-300 preseason rankings and my points league rankings (the explanation of which is detailed here), here are some notable undervalued/overvalued players.

Overvalued Players (better in roto than points)

1) Carlos Gonzales, Colorado OF (#9 ESPN, #18 Points): You're getting a good player here either way. The issue is that, early on, reaching for a player is disproportionately costly. So reaching nearly a full round for an outfielder is a really bad play this early in the draft.

2) David Wright, NYM 3B/Alex Rodriguez, NYY 3B (#11/23, #26/46): See above, only more so.

3) Josh Hamilton, Tex OF (#17, #62): The issue here seems to be playing time (ESPN projects about 550 plate appearances). In a roto league, players who miss time still do well for you in any rate categories and produce well while they're in the lineup in the counting categories. Moreover, since most roto leagues have daily lineups you can move him in and out of the lineup as he gets injured or takes a day off. In a points league, someone who doesn't come to the plate doesn't get you points, and the rates don't mean anything. The weekly lineups in a points league also means if he gets injured or rested you have a dead spot in the lineup. If the Rangers rest him frequently to try and keep him healthy, he's not getting enough plate appearances per week to justify one of your first five picks.

4) BJ Upton, TB OF (#65, #116): There are a whole slew of players in this category (Chris Young, Drew Stubbs, Colby Rasmus and Jimmy Rollins come to mind) who get plenty of home runs and steals with a moderate-to-low batting average and often (but not always) strikeouts. Because they are a plus in at least two (and often three) roto categories, they get ranked high for roto leagues. This is why you'll read the phrase "power/speed combo" in just about any draft article. But they don't get enough hits and/or too many strikeouts to justify a high ranking in points leagues, especially since steals are not on equal footing with home runs in a points league (as detailed here, home runs have a much higher correlation to points output than steals). Again, these are fine players to have but only if they are drafted appropriately.

5) Mark Reynolds, Bal 3B (#158, #313): Every year, someone thinks Mark Reynolds is worth the home runs. That statement could apply to fantasy baseball or actual major-league GMs. He never is; he strikes out too much to overcome everything else he may do. Strikeouts are costly. That doesn't matter as much in a 5x5 roto league (though I watched him nearly torpedo a team last year in a roto league that counted batter Ks), he is simply a player with a big positive in one category (home runs) and a big negative in one (batting average). But in a points league, someone who strikes out 200 times (as Reynolds does) starts off with negative value that has to be overcome. Assuming an 8-point average value for home runs (+1 for hit, +4 for home run, +1 run, +2 for 2 RBI), it would take 28 home runs to overcome his projected 222 strikeouts for this season. This is worth putting in italics: striking out 222 times requires 28 two-run home runs for Reynolds just to have non-negative value. If he's "only" going to hit 35 home runs, then he needs to get more non-home-run hits and plenty of walks (probably 80+) to add real positive value. That's just not going to happen. This goes for any high-power, high-strikeout player like Adam Dunn (#37, #92) or Carlos Pena (#149, #280); Reynolds just happens to be the most extreme example.

6) Any player who only contributes stolen bases: And I mean really only contributes stolen bases; we're talking no power, mediocre batting average, etc. Fantasy experts (and "experts," and amateurs) will tell you that cheap steals are available later in the draft. That's valuable for a roto league, but steals aren't worth enough in points leagues to justify drafting many of these players quite where they are ranked. A prime example is Brett Gardner (#103, #165), but Michael Bourn (#90, #152), Jacoby Ellsbury (#52, #89), Rajai Davis (#136, #201) and Elvis Andrus (#80, #93) qualify as well.

Undervalued Players (worth more in points)

1) Billy Butler, KC 1B/Nick Markakis, Bal OF (#88/139 ESPN, #41/33 points): Nick Markakis has to be the most extreme example of this phenomenon, but Billy Butler also qualifies: the player who just hits. Neither of these players hit many home runs or steal bases, they don't get many RBI, and maybe they score runs. The one thing they really do is hit for a high average. This makes them one-category (maybe two) players in roto leagues, but contained in that high average are plenty of doubles (and triples). These don't count in roto leagues, but add valuable extra points in a points league. Both Butler and Markakis are projected to hit above .300 (180+ hits) with 40 or so doubles but 20 or fewer home runs. In a roto league, this makes them just .300 hitters contributing in average. In points leagues, their non-home-run hits are projected to be worth 379 points (for Butler) and 371 points (for Markakis), which would be enough to place them in the top-250 overall. This doesn't even account for home runs, walks, steals, runs, or RBI. They are draftable even without these stats, so it doesn't matter that they don't get enough of them to justify a high pick in a roto league. Each additional tick in those categories just pushes them farther up the list of draftable players, in their cases enough to make them both worth one of your first five picks.

2) Matt Holliday, StL OF (#14, #5): Everyone loves Matt Holliday. He hits for average and hits for power, he'll steal a few bases, and he hits behind Albert Pujols. No one will let him slip past the second round. The point is that he's the top-rated outfielder, and you shouldn't let him slip out of the first round.

3) Joe Mauer, Min C/Victor Martinez, Det C (#30/#44, #12/#15): In roto leagues, the strategy is to wait on a catcher, simply seeing what's available at your draft spot. That's especially true if you don't get one of these two (or McCann/Posey later on). The idea is that, as scarce as catcher may be, you can get more valuable fantasy contributors than these early in the draft. The more valuable catchers aren't worth enough as overall contributors to take too high. But in a points league, that scarcity comes into play more. If a catcher contributes enough points overall then he's worth a high pick (Mauer is #23 for overall points and Martinez #30, more than Cliff Lee, Tim Lincecum, Jose Reyes, or CC Sabathia). The scarcity of points at catcher then adds extra value to those already-good point totals, justifying a high pick.

4) Kurt Suzuki, Oak C (#208, #86): This case is essentially a combination of the reasons detailed in #1 and #3. He's projected to hit for an OK average (about .270) with a few home runs (14). But he's also projected to get 25 doubles, and OK run/RBI numbers that increase his value in a way that doesn't apply to roto leagues. That he does this from a scarce position increases that value immensely. Moreover, catcher falls off a cliff after Suzuki; the drop from him to the next-best catcher will cost about 55 points. Those 55 points at catcher are worth about 50 spots in the rankings.

5) Logan Morrison, Fla OF/Jose Tabata, Pit OF (#268/257, #78/113): These are two outfielders who only do one thing well (runs for Morrison, steals for Tabata), but don't do anything poorly. That makes them fairly unsexy late-round picks in roto leagues, since they're only taken if you specifically need their one good category. But they do just enough of everything else to generate plenty of points for a points league format.

6) Any under-discussed bat who projects to hit in one of the top-2 spots in the order: In points leagues, it's important to get players who come up to bat a lot. More plate appearances means more opportunities, and more opportunities means more points. Even if such a player doesn't do any one thing in outstanding fashion, if he hits at the top of the order he'll be solid enough often enough to contribute. Each spot down in the batting order costs a player somewhere on the order of 15-20 plate appearances projected over the course of a full season; that means the difference between batting 1st and batting 9th is about 150 plate appearances. That's fewer hits, fewer walks, runs, RBI, etc. Take two players with identical OBP, one batting first and one batting ninth. The league-average on-base percentage is .325, and so the player batting first will have 150*0.325 = 48.75, or about 49 more times on base. That means the minimum difference between the two players is 49 points, easily 10% of total points in the middle-to-late rounds. That doesn't take into account hits, extra base hits, runs, steals, etc., which will push the difference upward. Moreover, because a) this is fantasy baseball (where the top hitters are drafted), and b) these are players hitting high in the order (where OBP is valued), the OBP in question is likely greater than 0.325; you could well gain 80-100 points just by targeting a leadoff hitter over someone hitting in the bottom half of the lineup.

So, to recap: In a points league, overvalued players are injury/rest prone, strike out a lot, hit for power with low average (and likely strikeouts), power/speed combos, and one-category stolen base contributors. Undervalued players are hit for a high average, hit plenty of doubles, bat high in the order, and come from scarce positions.

Addendum to post on drafting inefficiencies

I detailed here a way to calculate the penalty involved in reaching for a player given the difference between his expected pick based on rankings (xPk) and where he's actually taken (aPk). I said in that post you want to draft the best player available in at least the first five rounds. Using that as a rule of thumb, it is possible to derive how far you should reach for a player.

Assuming you want to draft the best player available in the first five rounds, the smallest possible penalty is then taking a player who sits just outside that value in your league. In a ten team league, that means taking the #51 ranked player with pick #50 (the last pick in the fifth round). Using the formula (10*(aPk-xPk))/(10+aPk), that gives you a value of negative one-sixth (-.16666). That is a good starting point for the maximum penalty you would want to accrue; anything larger than that has similar negative value to reaching in the first five rounds. So by setting the formula in question equal to -0.16666, you end up with the following two equations:

Equation 1: aPk= (60*xPk-10)/61

Equation 2: xPk = (10+61*aPk)/60

Equation 1 is how early you can take the player ranked at xPk without accruing a significant penalty, while Equation 2 is how far you can reach with your current pick without getting more than that same penalty.

If you have the fifth pick in the tenth round (pick 95), then the farthest you want to reach is only down to pick (10+(61*95))/60= 96.75, or roughly pick 97. Similarly, if you really want the 95th ranked player, then you should only take him as early as ((60*95)-10)/61= 93.27, or pick 93. This is obviously quite stringent, and you can set the value more liberally. It does jive with the notion that you should draft best overall as much as possible in the first ten rounds, though.

The most I would set the value to is -1. That is approximately the average cost of reaching for the #51 player with any given pick in the fifth round. That yields

Equation 1: aPk= ((10*xPk)-10)/11

Equation 2: xPk= (10+(11*aPk))/10

Using these standards, the farthest you want to reach with the 95th pick is (10+(11*95))/10= 105.5, or player #106 (about one round). If you want the 95th ranked player, then the earliest you should take him is ((10*95)-10)/11= 85.45, or about pick 85 (again, one round).

**Update**

Using the -1 scalar value standard (detailed immediately above), here's a good rule of thumb for number of picks to reach: take the round, and add 0.55 (that is, in round 1 do not reach more than 1.55 picks, 7.55 picks in round 7, etc). The calculations assume a 10-team league and a median pick (pick 5.5 in each round). The number should be higher for a lower pick in the round, and smaller for a higher pick.