The Blogg

After the first mini-crash of the Philadelphia Eagles season, I wrote a bit about the problems I saw with the team. After the latest debacles against Cincinnati and Baltimore I was ready to launch into a tirade about how the team needs to jettison all of its aging talent and start building to be competitive again in 5-6 years. With the perspective of a few days, I now think it is possible to make this a good team in only a year or two if the management plays it’s cards correctly.

First, keep starting your starters through the rest of the year. We will be keeping most of them, and there is no reason to have them even more rusty than necessary at the beginning of next season. Some of the ones that should not be on this team next year may have trade value if they keep playing and show themselves worthy. Second, rotate everybody in throughout the games. With no realistic chance of making the playoffs this year, there is no need to have people like an injured Brian Westbrook in there pounding the ball all day long and risking further injury. More importantly, there are some current backups that will need to contribute next year, they need the practice, and the coaching staff needs to evaluate their talent. In particular, if they are even considering keeping Lorenzo Booker on the team next year, we need to see a lot of him in the next 5 games.

As for next year, let’s start with the coaching staff. I won’t be disappointed if Andy Reid goes after this year, but I think it can work with him staying. However, the only way he should be allowed to remain the head coach is if he agrees to completely turn over the job of general manager to someone else and allow his offensive coordinator to call the offensive plays. Reid is a good motivator and organizational guy, but he has made it quite clear that he cannot be trusted to do those two things well. If, as I have heard, Marty Mornhinweg was calling the plays against Baltimore, then he probably needs to go as well. Jim Johnson should stay.

It is not McNabb’s fault that the Eagles refuse to run the ball (unless he is frequently audible-ing away from running plays), or that his receivers drop the ball more often than they should. It is, however, his fault that he has been wildly inaccurate throwing the ball for much of this year. I suspect that he has a lot of good football left in him, but if he does not get things figured out quickly, I think it is time to move on.

The wide receivers, other than Jackson, are not great. They are, however, good enough that replacing them should not be a high priority. At running back, we mostly just need to use what we have. Nearly every time Buckhalter touches the ball, good things happen. Starting next season, he must get at least 10 carries per game, including most of the short-yardage work. Westbrook needs to get an additional 15 carries each game in addition to catches. Until we commit to running the ball early and often, we will not have success running when we need to later in the game and defenses will be able to sell out against the pass. We do need a third guy here as a backup in case of injury, and from what little I have seen of Booker, he is not competent.

I think Dan Klecko has done an admirable job at fullback, and with a season at that position behind him, I would expect him to be better next year. Still, there definitely needs to be a serious contender to replace him in training camp. Tight end is a huge issue. L. J. Smith needs to go, and while Celek is a nice player, I don’t think he is good enough. We need a strong blocking tight end to hold the edge on stretch plays, where Westbrook has been most effective historically. The tight end also needs to be a good pass-catcher to make the West Coast offense work properly. Unfortunately, the best tight end that will be a free agent this year is Desmond Clark, who is not quite the star I would hope for. Thus, I would be targeting this position with one of the early-round draft picks.

The offensive line is going to be quite interesting. First, the management needs to have a very long, very serious talk with Shaun Andrews in the off-season and make sure he is committed to playing. If he is, move him to one of the tackle positions. If making a serious attempt at being competitive next year, we cannot let both Runyan and Thomas go. Unfortunately, holding on to either one is bad long term. I say keep Runyan and let Thomas walk. Forget about Justice, and start training King Dunlap to be your next right tackle. Put Herremans and Jean-Gilles at guard and keep Jackson at center, with Nick Cole as the main backup for all three spots. Draft another guard/center if you can get one.

On the defensive side things have been pretty good this year. The defensive line has had some issues against the run a few times, but the only disappointment has been Chris Clemons. I would keep them all. The linebackers are ok, but we absolutely must find someone who can cover tight ends. I don’t know who that might be, but it is not Chris Gocong.

Lito Sheppard will not be coming back, which is unfortunate. Sheldon Brown and Asante Samuel are quite capable as the starters (although Samuel has not played as befits his contract), but the depth is shaky. Thankfully we should be getting this year’s draft pick Jack Ikegwuonu off of IR, but it remains to be seen whether or not he will actually be good. If Dawkins is willing to come back at a reduced salary and play situationally as the hybrid linebacker/safety role the team was looking at a few years ago, fantastic. If he is too proud to not be the starter, then we have to let him go. If he does not make any team, seriously consider hiring him as a defensive backs coach or for the “team spokesman” role that was create for Hugh Douglas a few years ago. Quintin Mikell is good, and Considine is good enough, but hopefully Demps will push for the free safety spot.

While Rocca was been inconsistent, his average is fine. It is probably time to get rid of Akers unless he shows in the rest of the year that he can consistently hit in the 40-yard to 55-yard range as he once could. Although Jackson has done a good job as the punt returner, he is the kind of player that you want on the field for every offensive snap. Thus, it would be wise to look at someone else who might fill the special teams position.

Do I think this is going to happen and the Eagles are going to return to the glory days of the early 2000s? No, but all I can do is speculate and hope. I fully anticipate getting blown out by the Cardinals on Thursday, but I’ll be watching and living and dieing with every play. It’s the burden of fan-hood.

Note: If you are only interested in the mathematics but not how they are being applied, jump down a few paragraphs.

I am co-running a game of Dungeons & Dragons for a group of my friends, and one of my responsibilities during the last week was to come up with some reasonable rules for simulating large-scale combat. By reasonable, I mean that an outcome based on it should be close to what would be achieved by somehow running the normal rules on thousands of characters, that calculating results should not be arduous, and that there should still be a way for lucky die rolls to impact the outcome.

I came up with the following system, which I think does a fair job of meeting each of those goals: If a group of x characters of type A attacks a group of y characters f type B, we determine the probability p that a given member of A would hit a member of B using the standard rules. (That is, taking into account all bonuses and penalties, what percentage of 1d20 rolls, when added to an A‘s attack bonus, will meet or exceed a B‘s armor class?) We then assume that each character scores on average p hits, for a total number of hits h = x * p.

To include some randomness into this process, we then multiply h by ( 0.8 + 0.05 * 1d8 ). Next, we determine the number of average hits from a member of A that will be necessary to knock a member of B out of the fight, k, which is simply a matter of dividing the number of hit points a member of B has by the average damage of an attack from a member of A. Divide the number of hits by the number required for a kill to find the number of casualties c = h / k. If c > y, then all y units die. Otherwise, c units are killed. It is important to keep track of partial deaths for future rounds. Because one group having all of its members act before all of the members of another group would be a huge advantage, we presume that all participants act simultaneously. That is, the size of each group is fixed during the round, and casualties are removed from each group at the end of the round.

This is not a perfect system, because it does take into account critical hits and other parts of the standard rules. However, it does seem to work well. I ran a short simulation between 100 type As with an attack bonus of +8, damage of 1d8+4, armor class 17, and 36 hit points and 50 type Bs with attack bonus +10/+5, damage 2d4+5, armor class 20, and 45 hit points. After 12 rounds, 34.57 As and 10.60 Bs remained.

However, this system still requires that a fair number of calculations be made for each round of what will probably be a (relatively) quite long battle. Thus, I tried to find a way to handle some arbitrary number of rounds all at once with very few calculations. To do so, I substituted a single die roll and thus a single number of kills per attacker for the group of rounds. Then the number of members of opposing groups A and B left alive at the beginning of round n can be calculated from the following simultaneous equations:

Unfortunately, those equations are recursively defined. I learned at some point a few strategies for solving recurrence relations, but not sets of simultaneous recurrence relations with an unfixed number of terms. Knowing no analytic way to find a closed-form solution for these equations, I decided to unroll the first few values and look for a pattern. To make it easier to read, I have used a for the initial number of type A, b for the initial number of type B, c for the number of kills per A, and d for the number of kills per B. The results are below.

There are some interesting patterns here. Every result has an a term with coefficient 1. Every result has a bd term with coefficient -n. Every result has an acd term with a coefficient in the series 0, 0, 1, 3, 6, 10. I wrote a recurrence relation for this as well, x(n) = x(n-1) + n – 1, but could not find a closed-form solution for it either. The Online Encyclopedia of Integer Sequences has some interesting entries about it, but nothing that was leading me in the direction I wanted to go. Finally, there are other higher-order terms added in a pattern that was not clear with so few entries.

Next, I though I might try to do some substitutions to get a recurrence relation solely in terms of A(i), without any references to B(i) other than the constant B(0). From this, perhaps I could do something. I was able to transform it using the following sequence, but it has not gotten me closer to a clear solution.

At this point I am unsure what else I might try. I am simultaneously disappointed in my inability to solve this, and pleased that I have found an instance of a difficult problem in my everyday, non-academic life. If any more serious mathematicians read this, I would be quite interested to see your thoughts, or perhaps even a solution. While my application is fairly specific, it seems like the general case of a problem with two populations that decrease over time by a factor of the other’s size would be common enough.

I am currently sitting in on a course (this is the first time it has been offered, and I have already completed my credit requirement). Because I am not actually registered, I try to be a silent observer rather than participating in the classroom. This morning, however, after listening to the class fumble for the answer to what I thought was a simple question for 5 minutes I had to interject. We moved on, but I was thinking about how pathetic this situation was. After all, I am the only person in the room with no real motivation to study the topic, had missed 2 out of the last 3 lectures and done none of the reading, and seemed to understand the material better than anyone else.

As my annoyance grew, I interrupt the professor to make a point and one of the most inane things I have ever heard came out of my mouth. Before I had even finished, I was apologizing and asking him to move on and ignore me. Almost exactly the same thing happened several weeks ago, and apparently I did not learn to keep my mouth shut. The first instance was bad enough, but crow always tastes worse the second time around.

I’m thinking about working on a small project. I’d prefer not to say what it is, but feel free to speculate. As a precursor to this, I need to gather a list of the types of things that shaped the psyches of members of my generation — world events, new products, pop culture memes, etc. I am more interested in things from my late childhood (the 1990s), but of course it is much easier to remember more recent things. So far, I have the following list:

• East and West Germany are reunified and the Iron Curtain falls
• Operations Desert Shield / Storm / Strike / Cleanup
• Mahmoud Ahmadinejad and Kim Jong Il
• Hurricanes Katrina and Rita
• The Oklahoma City bombing, 9/11, and terrorism in general
• School shootings at Jonesboro, Columbine, Virginia Tech, and elsewhere
• The standoff between Branch Davidians and ATF at Waco
• The suicides of Heaven’s Gate cultists
• Worldcom, Enron, and other accounting scandals
• The stock market crash of 2008
• The Valerie Plame affair
• The Monica Lewinsky scandal
• The War on Terror
• Genocide in Rwanda, Kosovo, and Darfur
• The death of Princess Diana and conspiracy theories
• The Rodney King beating, trial, and riots
• The O. J. Simpson trial
• Jack Kevorkian’s actions and trial
• Grunge, Rap, Electronic, and Emo music
• Cell phones and other gadgets
• Google, Facebook, Myspace, Ebay, and other popular web sites
• Napster, Kazaa, and other filesharing programs
• Video games
• Huffing, crystal methamphetamine, and recreational use of prescription drugs
• The Simpsons, Seinfeld, Friends, and other popular television shows
• American Idol, Survivor, and other reality television shows
• Who Wants To Be A Millionaire? and other game shows
• The overhyped Y2K bug
• Computer viruses, worms, and malware
• Diagnosis of ADD and ADHD, and treatment with Ritalin
• Ever-present advertising for Viagra, Cialis, and other treatments for erectile dysfunction
• The Exxon Valdez oil spill
• The Tiananmen Square protests
• Bush I breaks “Read my lips: no new taxes” promise
• Sexual abuse by Catholic priests
• Ross Perot and Ralph Nader make attempts to break out of the two-party system
• Tonya Harding has Nancy Kerrigan attacked
• Lorena Bobbitt dismembers her husband John
• The “Don’t Ask, Don’t Tell” policy is enacted in the military
• The European Union is created by the Treaty of Maastricht
• The Million Man March
• Mad Cow Disease (Bovine Spongiform Encephalopathy)
• The murder of Jean-Benet Ramsey
• Barry Bonds and the baseball steroids scandal
• Murder of Matthew Shepard
• The DMCA and USA-PATRIOT Act
• The International Space Station
• The murder by Andrea Yates of her children
• The debate over the removal of Terri Schiavo’s feeding tube

Am I missing anything obvious?