Wednesday, December 26, 2012

A Chicken in Every Pot, A Car in Every Garage (And an Armed Police Officer in Every School [and three eyes on every fish]))

If you've been paying attention to the news in the last week or so (or been in casual vocal contact with any other human being), you're probably familiar enough with the most recent tragic school shooting.  If you're reading this post in the future, you may well be (here's hoping against it, though) familiar enough with a new more recent tragic school shooting.

Well, not if the executive vice president of the NRA Wayne LaPierre has anything to say about it.  You see, the NRA was originally created by man.  But they rebelled.  They evolved.  There are many copies(?)  And they have a plan.

Hard to verify Hoover quote, check.  Simpsons reference, check.  BSG reference, check.

Now that I have those bases covered I think we can safely take a look at Wayne LaPierre's "plan to keep schools safe."  If you're feeling like you could kill some more time on the internet today you could easily Google that phrase, spend some time with the results, then come back to this post.  I'll still be here.

Before we start, I want to say that I want to try to stay fair on this one.  I really do.  My views on guns are far from liberal, but they're also far from crazy.  I do think that there are very important places for guns in a free society.  At the same time, I think there are bad places for guns in a free society.  Unlike Antonin Scalia, I'm somewhat of a constitutional originalist when it comes to the second amendment. 

Regardless, my views on guns are unimportant here.  What I want to evaluate is the feasibility of Wayne LaPierre's plan, not whether or not we should have guns.  You can argue that one on your own time, but the bottom line is that the government is never going to take away all guns, nor are they going to allow all guns.  It's just the way of the world.  If you're fighting for either of those extremes I want to introduce you to my friend Sisyphus.

I've seen a lot of people just throw this plan under the bus without any fair analysis, so I figured I'd give some fair analysis a try.  We'll see how that goes.

Back to the plan.  If you haven't read about it by now, Wayne LaPierre has suggested that the best and easiest way to keep our schools safe is to simply place an armed guard in every school that wants them.

That would cost a lot of money, though, right?  Apparently not, because he just wants to just use a volunteer force of "local volunteers", like retired police officers, retired fire fighters, retired emergency responders, and military veterans.  Why not just cover all our bases and add Wal-Mart greeters to that list?

Frankly, that plan seems a little weak compared to a funded plan, so in a minute we'll get to the bulk of the numbers on what the actual funded plan would look like.  For now though you might be wondering why the 'volunteer' plan might not be the best solution.

I have a lot of respect for civil servants, and I applaud their service.  Frankly - unlike most in government at the moment - I think we should pay them quite a bit better than we do, and we shouldn't be raiding their pensions, etc, etc.  I do believe there would also be a number of them who would be willing to volunteer at a school to help out things, as well.  Since this is totally volunteer, though, it seems like the retired police officers would be the only ones to maybe be able to provide their own guns, though maybe you have to turn in your gun when you retire?  Not totally sure on that one. 

So, these civil servants would not only be volunteering their time, but also be providing their own weapons?  Okay, so to be fair to them I don't think beggars can be choosers, and we'd have to let anyone willing to do this bring whatever gun they wanted.

Again, I want to say that I honestly have the utmost respect for those that serve our country, and I always try to thank our military veterans (and actives!) when I get a chance.  Like most groups, though, there are always outliers.  Just like there are extremes in the civilian population there are also extremes in the veteran population.   

Accuse me of cherry-picking if you will, but do you need me to list some potential military veterans that I wouldn't like having guard a school full of kids with the weapon of their choice?  How about the Oklahoma City Bomber?  Served in Desert Storm, received a Bronze Star for service.  Had to spend some time and effort (and the time and effort of two co-conspirators) to put together the single deadliest act of domestic or international terrorism on US soil pre-9/11.

Now, imagine if instead of having to come up with a pretty detailed plan he was invited to guard a school with an assault rifle?  Actually, don't imagine that.  That's horrible.  This is the situation you don't want to end up encountering.

So, let's just consider that Wayne LaPierre will eventually come to the same conclusions that a totally free plan has some flaws, and the way to go is a paid plan.  It can't be that expensive, right?  Anyway, this is about protecting kids, so why not spend some money on it?

Now, let's make this clear here.  I was in high school when Columbine happened.  I remember what it was like to be in any high school in the country in the aftermath of that.  My high school was halfway across the country but it was one of those things that could have clearly happened anywhere.  It was pretty crazy, and it dominated the news cycle for a while.

If I'm remembering things right I think that was the point that my high school hired on some local police to - you know - just kind of be around.  They added some extra security in terms of only having certain doors be open at certain times (which made sense anyway), and having people at or around those doors.  Realistically the biggest things that helped were those sorts of small things that would come out of a security consultation, like don't have two dozen doors to all corners of the school open at all hours with no one guarding or accountable for those doors.

Some of you might be a little young, but this was some pre-9/11 world going on.  We didn't have to take our shoes off at airports or submit to radiation treatments every time we wanted to fly.  The TSA was still several years off.  Things just felt safer.  Columbine shattered that, and cops at the doors made things just feel a little better.  They certainly helped piece of mind, if nothing else, but looking back at it as an adult instead of a high-schooler they certainly weren't free.  To be fair, this was the 90s, though, when we actually still cared about spending money where it was needed.

Anyway, let's take a look at the numbers.

The US Department of Education has numbers on the number of public and private primary and secondary educational institutions in the US.  There's also information on colleges, but I think the tone of the plan is really just to protect kids until they're not kids any more.  Anyway, protecting colleges with individual good guys with guns would get pretty complex and messy for big schools, so let's assume they can handle themselves. 

Anyway, for the most recent year I could find data (2009-2010), there were 98,817 public primary and secondary schools and 33,366 private public and secondary schools in the US.

Oh, 132,183 is more than you were expecting?  Yeah, there are a lot of kids in the country that need protecting.  That's why this is so important - kids are our most valuable natural resource.  Or is 'that kids are our most valuable natural resource' thing also from the 90s?  I get the feeling that oil might be winning these days.

Now, Wayne LaPierre has said that this wouldn't be forced on schools, so there's probably plenty that would opt out.  I have spent a decent amount of time working with numbers from the last presidential election to try to come up with some reasonable way to speculate on what percent of schools might refuse this service, but I've run into dead end after dead end.

I can easily find numbers on the popular and electoral votes, and what way seats in Congress that were on the table went, but exit poll information on gun ownership and/or gun control unfortunately looks to have been dropped from 2012 questions.  In 2008 it was reported that about 40% of homes owned guns, with that number being a bit higher for voters who identified Republican (~60%) and a bit lower for voters who identified Democratic or Independent (~25%).

That still doesn't help much, as you might not want to own a gun but still be okay putting an armed guard in a school.  I'm going to err on the side of even splits here and say that for whatever multitude of reasons half of the schools in the US might refuse this plan.  We could really dig in and figure out this number, but I think it's not time well spent.  I can think of a lot of noise that's going to drive this one way or the other, and in the end I'm imagining that most of it will wash out to the center.    

The annual salary of police officers seems to be fairly normal when examined by any one source, at least for the most current sets of data that I could find.  The means do seem to jump around, though, anywhere from around $40-50K.  The average of averages (oh that's probably going to piss more people off than talking about guns- just spend some time reading Twitter archives the week before the last election or search for something like 'Nate Silver wrong' [looks like he wasn't though, eh?]) seems to be in the low 40s, which seems like a safe estimate.  We wouldn't want to be hiring anyone but the best, right?  For simplicity's sake let's just call their annual salary $40,000 - if they're working in a school they need to be underpaid at least a little bit.

Keep in mind that this is only during the 9 month school year, and not a full annual job, so we get to cut that pay to account for three months off in the summer.  So, if half of the schools get one police officer at the door, and he brings his own gun from his job as a police officer (is that cool?), then the manpower cost we're talking about (annually), is:

(132,183/2)*(3/4)*$40,000 = $1,982,745,000

Okay, 1.9 billion dollars.  That's not horrible.  

Having one officer at each of these schools is also the kind of thinking that might have worked in the 90s.  Looking back at it, I think it really was just about peace of mind and a low threshold to stop people whose hearts weren't really in it (or out of it, rather).

It may be the case that a lot of you didn't live through the cold war.  If you've done the math on when I was in high school you know that I only caught the tail end of it.

That said, there's a reason why we had enough nuclear ordinance to blow up every city on the planet a few times over - it's because the USSR also had that much ordinance.  We kept building because they kept building, and they kept building because we kept building.  Are you familiar with the term 'arms race'?

If you haven't noticed, people who have been shooting up places in recent years have been engaging in an arms race to get ahead of just these sorts of patches like putting a good guy in every school.  The two most recent shootings (Aurora, Colorado and Newtown, Connecticut) had shooters using some pretty heavy gear including bulletproof vests.  A simple cop who tried to stop them with nothing but a handgun would have likely been gunned down as well.

So I don't want to spend much (any) time googling the kind of things that I need to google in order to figure out what it would cost to deck out a police officer in bulletproof gear, but it's not going to be cheap.  Even if we could work it down to less than $1,000 per officer to get them set up with guns and gear, we're then looking at another $65K or so initial costs.  It's a small drop in the bucket when you're adding it to 1.9 billion, but it would be money that would need to come from somewhere.

Let's revisit the fact that this plan did originally call for volunteers, and that some police officers might be willing to donate some of their time.  Even if all of them donated all of the time needed, we're still talking about $1,982,745,000 being removed from the workforce.  Those officers could be earning that amount of money a year, so by making them all volunteers you've removed that income from the system.  This included income tax, and frequent readers of this blog know that we can figure out the income tax paid for any salary pretty quick, excluding things like deductions, etc.

If each of these officers could be making $40,000 a year, they'd potentially be paying $6,125 each simply in federal income taxes.  Over all the officers we'd be looking at:


Well, so if you just let these police officers work their normal jobs, you'd have a decent amount of money to outfit them.  This tax money would also be coming back if you were paying them, though that also requires a $1.9 billion cost annually.

Let's imagine by some stroke of fancy that someone comes up with $1.9 billion dollars to implement this plan.  Can we feel safe after that?

Well - like I said - this kind of feels like 90s thinking.  By the 90s most of us had forgotten about the largest single school killing in US history.  Here's some good trivia - anyone think they know what the single largest school massacre in US history is?

If you guessed anything in the last 80 years you're unfortunately incorrect.  If you've never heard of it I would strongly suggest taking a few minutes to read the wikipedia page about the Bath School disaster:

It was actually pretty crazy, and took place way back in 1927.  It is a testament to what one really pissed off person can (could?) do with a little know-how, a little credibility, and a whole lot of planning.

If you want something a little more recent, look no further than the Oklahoma City bombing of 1995, or - I don't know - maybe 9/11.  An armed guard at either of those events would have been a bit too little too late.

We have a department of homeland security to protect us from some of these big picture things these days, and they seem to be doing a pretty okay job.  Their budget?  A bit more than our $1.9 billion - just under $50 billion.

That's still a lot of money, but it's also been pretty effective.  Keep in mind that our $1.9 billion cost is only labor, and doesn't involve any of the infrastructure and administration that would be needed to employ 60,000+ individuals.  Now, all of that would still be shy of $50 billion, but we might start getting into the general area of having a department of school safety.

Sorry to keep tossing out 'keep in minds', but let's keep in mind that aside from the Bath School disaster the second most deadly school killing was not at one of the schools we've covered with our $1.9 billion, but at a college - Virginia Tech.

The reason that I didn't look at colleges at the beginning is because they make things drastically more complicated.  If you've ever been around a large state school you know that there's not just one classroom building that we can toss a police officer at and limit access to.  A college like Virginia Tech is going to have dozens of classroom buildings, which means dozens of police officers in this plan.  You might start getting into the ballpark of a solid million dollars to cover a single college with police officers.  The short answer is that costs are going to start building as we dig deeper and deeper to cover every gap.

I also just realized that we've mentioned the Aurora shooting, but it's not covered under our $1.9 billion either.  Once the kids realize that they're only safe at school because of the armed police officer at the front door they're going to start wondering why they're not being protected to that degree in other places, like movie theaters.

Looks like we should start tossing some more money at funding police academies because at this rate we're going to need a whole lot of police officers.  Maybe we should start with pushing Police Academy 8 out the door faster to get people interested in the police life.  Hopefully we'll have an armed police officer at all the midnight screenings of what will likely be the best of the last 6 Police Academy movies.

I'm having some trouble nailing down an exact number, so I'm going to err on the low side here.  It looks like that estimate would put the number of movie theaters in the US at around 5,000.  If we put an armed police officer in each of those, we'd be looking at another $200 million dollars, partly because these officers would have year round jobs and not just 9 months of the year.

Sure, some movie theaters would also refuse this service, and since we're getting close to a big number let's say we ballpark a cost in there for the total plan that comes right to around $2 billion dollars.

Where are we ever going to get $2 billion dollars annually to put an armed guard in half of our schools and most of our movie theaters?  I had a few guesses, and it looks like my first one paid off, so to speak.  Want to guess how much money is spent lobbying in D.C. annually?

Well, every year from 2008-2011 it's been well over $3 billion each year.  There's no reason to believe 2012 will be any different when it's all said and done.  So, if the lobbying arm of our government decided to step back and protect our kids they'd be able to cover the labor costs and probably pick up a big chunk of the administration costs.  They could also throw a few pizza parties and maybe a pretty sweet holiday party.

Let's say they don't want to, though.  Where else can we find big piles of money?  Well, numbers are still hard to come by, but a number of estimates have the donations collected by candidates for the 2012 presidential election well into the billions of dollars - some as high as $6 billion dollars.  That would only cover 3 out of every 4 years, but we'd hopefully find another way to supplement that.  The NRA has a budget of $250-300 million annually, so that would actually make a pretty reasonable dent if they wanted to hire on some good guys.  If they didn't feel like it, I"m sure we could find some other ways to come up with money.

Like, maybe raise taxes?  Oh, wait, that's where I'll get in trouble.  No new taxes.

Realistically, though, the most effective way to throw money at this would likely be to create a department of the government to handle things like this.  Or throw money at existing departments, like homeland security and health and human services, or maybe increase the budget for the department of education.

Overall, I'm not sure what the bottom line is here.  $2 billion is in the ballpark of something that's not super easy to come up with, but not completely outlandish.  It's a stopgap, to be sure, and I'm not even sure it would work.  It would be like heavy and unwarranted use of antibiotics - the longer we used a plan like this the more likely we'd come across someone willing to compete in the arms race and go one level better to break the system.

So, I'm not sure I have any good solutions on this, unfortunately.  This is one of those things where I was hoping that something would fall out in the middle, but nothing really did.  My apologies.  Maybe someone will have some better solutions in the comments?

Wednesday, December 19, 2012

On Fevered Temperature Measurements

Let me apologize in advance for the brevity of today's post.  You'll see very shortly that I've been a bit under the weather the last few days.  It's the very fact that I've been on and off bedridden with fever that got me to thinking about temperature and thermometers.

Specifically, I've been wondering how accurate any readings of my temperature have been.

Over the course of an hour or so (I've been a bit bored), I took thirty measurements of my temperature to see just how much any one of those measurements got it right.

Let's cut to the chase, shall we?

To go back to last week, the mean and median are both 100.8, and the mode is 100.7.  The standard deviation, or the average amount that any given measurement deviates from the mean, is around .6. 

That means that the 95% confidence interval around my presumed actual temperature of somewhere around 100.8 is actually 99.6 to 102.0!

So, I guess my thermometer could be a bit more accurate.  In fact, it's really quite poor.  

Wednesday, December 12, 2012

Means, Medians, & Modes: Come on Down! (Games of The Price is Right)

You are the first three contestants on The Price is Right!

Hopefully you can hear The Price is Right theme in your head the moment you see those words.  If you can't, Google it.  Or call in sick tomorrow and watch some daytime TV.

Today we're talking about the holding pit of The Price is Right - Contestants' Row.

After my last post about The Price is Right a friend called me out on the fact that I could just watch a whole bunch of The Price is Right episodes and code them to pick up on the human behavior side.  I tried to dodge that idea a bit by explaining that I felt that there was a lot to learn - through simulation - about the situations the humans on the show find themselves in.

All said I knew that he was right - at some point I was going to have to sit down and code a bunch of The Price is Right Episodes.  I set my DVR to record every new episode in the series and before I knew it I had plenty of episodes to pull data from.

There are a lot of games on The Price is Right that have things that are difficult to code, and there are also games that are so infrequent as to be very difficult to code in any reasonable quantity.  For example, I've at the moment coded 15 episodes.  With six games on every show that means I've seen 90 pricing games.  Plinko has come up once.

I plan to keep coding episodes here and there to come back to some of the things that I simply need more data for.  After 15 episodes there should be something that I should be able to examine, though, right?

There are a few things that are constant on every show.  Every show has the Showcase Showdown at the end of the show.  Two individuals make bids on two showcases, which means that I've seen 30 showcase bids.

If you read my last post on The Wheel (or if you've ever seen The Price is Right ever) you know that twice a show three individuals have the chance to make two spins each, for 24 potential spins an episode.  That's 360 potential spins, though a lot of those spins are highly interrelated.  It's more fair to simply consider each set of three people as an event, leaving us with 30 wheel events.

The most abundant source of information across The Price is Right episodes is the information from Contestant's Row.  Six times a show, four contestants each make a bid.  Again, these sets of bids are fairly interrelated, but that still leaves us with 90 sets of bids.

There's a lot of information in these bids, and there's a lot of potential things to look at.  After only a bit of thinking I realized that these bids would potentially make a great discussion about means, medians, and modes.  There's more that I'll get to, but if you've always struggled to remember these sorts of stats (or are teaching a stats class and can't get it across to your students), hopefully this sort of practical example might help.

Now, before we get into it we should clarify exactly what happens during Contestants' Row.  Four contestants are pulled from the audience and shown a prize.  The goal is to guess the price of the price, WITHOUT GOING OVER.  You can think of it like an auction.  You want to get the item for a deal, but you want to beat the other contestants that are also trying to get a deal.  The highest bid that still got the item for a deal (e.g. didn't overpay) is the winner.

For example, if the bids on an item are 600, 700, 800, and 1000, and the price of the prize is 799, the contestant who bid 700 is the winner.  The contestants who bid 800 and 1000 were willing to pay too much, and the contestant who bid 700 was closer to the price than the contestant who bid 600.

If everyone bids over the price of the prize (e.g. if the price in the above example was 599), the bids reset and contestants start again.

Because of this, if a contestant thinks that all the other contestants have overbid they will frequently bid $1 - if they're right in their assumption everyone else will be disqualified by being over the price and they will win.

When a contestant wins they leave Contestants' Row and play a pricing game.  For the next bid their spot is filled with a new contestant, but the others remain the same.  The new contestant always get first bid, and bids move to the right (from the stage).

One of the main questions I had was about what the modal bid would be across all bids.  For those reading this for the stats review, the mode of a distribution is the number that is used the most frequently.  While a lot of numbers are used there is one that is used differently from all others.  That number is 1, the loneliest number.  So, is $1 the modal bid?  Let's see.

[By the way, can anyone that works for Google and works on Google Docs get on adding histogram functionality to your spreadsheets?  It really can't be that hard, right?  I love using Google Spreadsheets to make graphics, but not being able to make one of the most basic is a huge bummer.  They should be able to look like the graph below]

Well, you can see by that large spike at $1 that $1 bids are in fact the mode.  Of all 360 bids, 24 are $1 bids.  That might not seem like much (less than 10%), but the next most frequent bid is $800, with a frequency of 17.  After that it drops off pretty quick.

This is across all contestants, though, and there's good reason to believe that dollar bids are more likely to come later in the chain of bidding.  If the first contestant bids $1 they run the risk of the second contestant bidding $2, effectively nullifying their bid.  That second contestant would run the risk of the third contestant bidding $3, who would also most certainly lose when the final contestant (hopefully!) bid $4.  So how does this look if we break it down by bid order?

As expected, almost all of the $1 bids are put in by the last contestant to bid.  A handful are placed by the third contestant to bid, but only 1 was placed by the second or first to bid.  You might be able to see that there are some $2 and $3 bids by the fourth contestant - these are in reaction to second and third contestants taking the dollar option away from them.

The mode of the fourth graph is very much $1, but the mode shifts out into the regular distribution for all earlier bidders.  The strength of the mode is much smaller in effect for the first three bidders - the mode for the first bidder is $1200, the mode for the third bidder is $600, and the mode for the second bidder is...multi-modal!

I feel like the second contestant spot just won $500 for a perfect bid.

What is multi-modal, you ask?  Well, when there's no single mode, but several.  The most common case is bimodality, where two modes exist.  That actually happens to be the case here - $800 and $850 both occurred 5 times, and are each modes.  You can see though that a 5-frequency value being the mode is much less powerful than the 20:4 cut that occurs from $1 to the next best number for the fourth contestant.

So, not shockingly, $1 bids are used a lot - more than any other singular bid.  Those $1 bids are much more likely to occur later in the bidding process, and are non-existent in this sample for the first bidder.

Modes, got it?  Good.

If you're familiar with making the above graphs you've likely noticed that I've removed the means from the sides - no use giving those away before it's time.

Well, it's time.  Means.  You can think of means as averages, because that's what they are.  A mean is the sum of all values divided by the number of values.  If you had two values (please don't use the mean of two values) that were 10 and 20, you can find the mean to be 15 by summing 10 and 20 (30), and then dividing by the number of values (2), which gives you 30/2 or 15.

So what's the mean bid by contestant placement?

You can see that the mean drops quite a bit by the fourth contestant.  That drop is real, though it might not be meaningful.  Since the mean uses a sum of all numbers, having a whole bunch of numbers far outside the rest of the numbers will pull that mean - down, in this case.  How would the means look if we just took out all the $1 bids?

You can see pretty clearly that the mean is being pulled down by those $1 bids where those bids are more frequent - the most for the 4th contestant.

If means are being influenced, how about a different metric?

Medians are somewhat like means, except that instead of giving a straight average they give the value that is...well, medium?

If you have four friends and find yourself in the same room (or maybe in line at midnight to see a movie about Hobbits) with not much to talk about, line yourselves up by height.  The height of the person in the middle - the third person from either the top or bottom - is your median height.   

It doesn't matter if you line up from shortest to tallest or tallest to shortest, and it doesn't matter which side you count from.  It doesn't matter what the height of the other people are - just the person in the middle.  You might have two friends that are an inch shorter than you and two friends and inch taller, or you might have two short friends and two NBA centers - it doesn't matter.  All that matters is the height of the person in the middle.

Here's what the median bids look like:

The medians are still being impacted (because 20 values are being removed from the forth contestant), but to a smaller degree.

All this is well and good, but these statistics are all pretty unimportant without context.  That context is the actual cost of the prizes being bid on.  The actual prices of prizes is a somewhat more interesting chart:

What is really interesting to notice is that there are no prizes - in the shows that I've watched - that are valued less than $500 (the lowest is $538).  There are also no prizes valued higher than $3000 (the highest is $2880).

Take note of that, contestants who bid $4500 on a regular basis.

The mean prize value is $1283.
The median prize value is $1195.
The modal prize value is...well, less important (it's multi-modal and in the same range, though).

All said, it seems that $1 bids are somewhat unnecessary - a bit of $400 or $500 seems to serve the same purpose (though perhaps without the showmanship).  

It raises an interesting point, though.  Given the information so far, what values would you need to stay under to be confident to different levels that you're not going to overbid?

Well, we can look at percentiles to get a feel for this.  Given the bids I've collected, a bid of $538 has an exceptionally low chance of being higher than the price of the prize.  What bid would be a bit higher but have a 95% chance of not going over?


Not a bad bid, it would seem.  You're not going to go over, but you're also likely to be a bit far from the actual price.

If you want to bump up your odds of winning at the cost of busting, you can be 75% confident in not going over with a bid of $811.  $800 is actually a pretty popular bid, so people might be picking up on this a bit.  There's not a lot of situations where everyone goes over (it's been once in the shows I've watched).

If you want to have a 50-50 shot at staying safe or going over your bid would be...well, I've already told you that.  Think about it for a minute before you read on, if you can't come up with it.

The value of the 50th percentile is the median - $1195.

$1200 is also a pretty popular bid - especially for the first to bid.  

Want to run on the wild side a bit more?  A bid of $1761 gives you a 75% chance of being over the value of the prize.  A bid of $2276 gives you a 95% chance.

A bid of $4500?  Look, don't do that.  Don't do that, people.

It does look like position had an impact on the sorts of bids that a contestant will make, so how does position impact your odds of winning?  Well, it's actually pretty interesting.

The forth position seems to have a bit of an advantage.  More than that, the third position seems to have a bit of a disadvantage.  From a purely speculatory standpoint it does seem that the contestant first to bid often puts forth a pretty good bid fairly close to the actual price.  The second has the opportunity to bid fairly well, especially if the first hasn't.  Good bids may be in short supply by the time it gets to the third contestant, and the third contestant also doesn't yet have the advantage of the last bid.  If they bid $1 the forth position can easily bid $2.  If the third position bids one dollar more than the highest bid (also a good strategy), the forth still has the option of going one dollar higher.

So what's the best bid?  Good question.  To a large degree it seems to be dependent pretty heavily on your position in the bidding order.  

If you do find yourself in Contestants' Row the best advice might be to make sure to not bid $4500 without good cause.  Other than that, just play it smart and hope to find yourself in that forth spot.  Happy bidding!