Posts tagged ‘expected value’

What is the Essence of a Two By Four?

Decades ago, common carpentry practice (later set in stone by written regulations) specified that certain applications needed a 2 inch by 4 inch board.   The reason this board was chosen was not due to its size per se (in most cases, for cost and space issues, I am sure folks would love to have gotten away with something smaller).  This size board was chosen for a specific application by its load-carrying ability.   For example, two inch by four inch boards spaced every 16 inches apart created acceptably strong framing for a wall.

Anyway, after many years of making lumber, the timber and lumber industry found ways to make the 2 inch by 4 inch board much stronger.  Well, not always stronger, but more uniform in strength such that the weakest board in a batch was much closer to the average than before.  But for standards, this has about the same effect -- 2 inch by 4 inch boards could be considered to be much stronger since the expected value had to be set at the minimum that might be encountered.

So now, all the standard applications are over-designed.  We can get away with a smaller, cheaper board than a 2x4.  Or, for those of you less focused on capitalism and more focused on environmentalism, we can use fewer trees to build the same house.  But how do we switch an entire industry that is steeped from birth as to what a 2x4 should be used for?  How do we rewrite a myriad of regulations that all call for a 2x4?

Well, in the lumber industry, they redefined the 2x4 to actually be something like 1.5 x 3.5 actual inches, a board which under new production processes has the same predicted strength as the old 2" by 4" boards.  In effect, they decided that the essence of a 2x4 was not its dimensions, but its load-carrying ability.  Almost any engineer can understand this immediately.  This means we still frame walls with 2x4's spaced every 16 inches, but the lumber is smaller and less expensive than it was before.  Standards and training don't have to change.  Architects maybe had to adjust a bit because their wall widths changed slightly, but a 3.5 inch board width actually is a nice number because with sheets of 3/4 inch drywall on both sides it makes for a nice round number 6" thick wall.

All of this is background to this absurd story, is using this history to try to commit legal blackmail against a couple large home store chains (via Overlawyered):

Two home improvements stores are accused of deceiving the buyers of four-by-four boards, the big brother to the ubiquitous two-by-four.

The alleged deception: Menards and Home Depot (HD) market and sell the hefty lumber as four-by-fours without specifying that the boards actually measure 3½ inches by 3½ inches.

The lawsuits against the retailers would-be class actions, filed within five days of each other in federal court for the Northern District of Illinois. Attorneys from the same Chicago law firm represent the plaintiffs in both cases. Each suit seeks more than $5 million.

“Defendant has received significant profits from its false marketing and sale of its dimensional lumber products,” the action against Menards contends.

“Defendant’s representations as to the dimension of these products were false and misleading,” the suit against Home Depot alleges.

The retailers say the allegations are bogus. It is common knowledge and longstanding industry practice, they say, that names such as two-by-four or four-by-four do not describe the width and thickness of those pieces of lumber.


I Am Not Sure the State of Florida Understands This Whole Internet Thingie

Every three years I have to endure a sales tax audit from the State of Florida.  This year they actually sent me well in advance a list of all the paperwork they needed.  I sent everything to them electronically weeks ago.  So why do they have their auditor fly to Phoenix, stay in a hotel, and do her analysis of this paperwork on her laptop in my office?  In the hour since she has been here she has not asked me for one thing.  It is just bizarre.  Given that I have been audited by them twice in the past and never owed more than forty or fifty bucks in back taxes from computation errors, I am pretty sure her flight cost way more than the expected value of her trip, particularly since she had done nothing so far she could not have done (better probably) in her own office.

The Meaning of Health "Insurance"

Megan McArdle has a column I am going to excerpt at great length (sorry Ms. McArdle).  This is great article on a topic I have tried to explain many times here

After all, the insurance company has to make money.  That has to mean that the expected value of the claims they pay out is lower than the expected value of the premiums their customers pay in.  In some sense, then, the expected value of your insurance premium is negative.

But insurance does make everyone better off, because it covers very large costs that most people would have trouble paying.  Even most really good savers would have a hard time replacing the value of their house, or paying off a $250,000 judgement for an auto accident.  The expected value of those incidencts is very, very negative--more than just the value of the cash, you have to factor in the horror of being homeless or bankrupt.  When you factor in the homelessness, the bankruptcy, and so forth, the slighly negative expected financial value is more than outweighed by the positive value of being protected against personal catastrophe.  Not to mention the peace of mind one gets from not having to worry about homelessness, etc.

This is the magic of risk pooling.  But notice that it's the catastrophe which makes insurance a good deal.  You wouldn't get much value from buying "grocery insurance".  At best, you'd be paying an extra administrative fee to route your routine expenses through an insurer, rather than paying them directly.  At worst, you'll end up with bills skyrocketing as all sorts of perverse incentives appear.  After all, if the insurer is paying all your grocery claims, why not load up on filet mignon instead of ground turkey?

But insurers try very hard never to sell insurance for less than the cost of your expected claims.  If you expect to buy $10,000 worth of groceries next year, it will not charge you less than that for a "grocery policy".  And if we all drive up the costs of grocery insurance by consuming more, the insurer can do one of two things: raise everyone's "insurance premiums" to cover a filet mignon budget, or create a list of "approved groceries" that it will cover, and start hassling anyone who tries to file an excessively expensive claim.

Sound familiar?

This is why you should always have liability insurance, but should think twice about collision damage coverage.  It's why high deductibles are a good idea--for small expenses, it's better to self insure.  And it's why "catastrophic" health plans, which only cover the sort of extremely expensive events that most people would have difficulty financing, are a much better deal than the soup-to-nuts plans that most people get through their employers.  Those plans are expensive, both because they're paying for a higher percentage of your expenses, and because they drive up utilization--which means that they drive up next year's premiums even more.  Imagine what your car insurance would cost if it covered gasoline, routine maintenance, and those little air freshener trees you hang from the rearview mirror.  Then stop asking why health insurance costs so much.

But Kathleen Sebelius, the Secretary of HHS, thinks that catastrophic insurance isn't really insurance at all.

At a White House briefing Tuesday, Health and Human Services Secretary Kathleen Sebelius said some of what passes for health insurance today is so skimpy it can't be compared to the comprehensive coverage available under the law. "Some of these folks have very high catastrophic plans that don't pay for anything unless you get hit by a bus," she said. "They're really mortgage protection, not health insurance."

She said this in response to a report from the American Society of Actuaries arguing that premiums are going to rise by 32% when Obamacare kicks in, as coverage gets more generous and more sick people join the insurance market.  Sebelius' response is apparently that catastrophic insurance isn't really insurance at all--which is exactly backwards. Catastrophic coverage is "true insurance".  Coverage of routine, predictable services is not insurance at all; it's a spectacularly inefficient prepayment plan.

The last two lines are why I knew from the very beginning that the promise I would get to keep my health insurance was a lie.  Because I have true insurance, rather than a pre-payment plan for incidental health-related expenses, and the folks who wrote Obamacare think of insurance as pre-paid medical care (in fact, I believe they think of private insurance as a Trojan Horse for all-inclusive single payer government health care).

Hosed At Any Price -- An Update on Geithner Plan Analysis

I had someone ask me whether the results in this post on the economics of Geithner's latest brainstorm were an artifact of the selected purchase price for the distressed asset of 150.  The answer is no.  Investors are willing to buy this asset on these terms at any price under 175, and banks are willing to sell for any price over 100.  Here is the graph of expected values as a function of the purchase price


Note the taxpayer gets hosed at any price  (kind of the Obama-Geithner update on "unsafe at any speed")  Two things I had not realized before:

  • Without competition among investors to drive up the price, a very large percentage of the taxpayer subsidy goes to the investors rather than the banks.
  • There is an interesting incentive to collude here between banks and investors.  The best outcome for both is for investors to pay a high price to banks and then have the bank kick back some portion to the investor.

Hindsight and Risk-based Decision Making

Last weekend I was watching an NFL game (I forget which one) and the team, which already had a solid lead, was considering going for a TD rather than a field goal at fourth and goal.  The announcer was going "Bad idea, bad decision.  Take the field goal and the sure points.  You don't want to risk getting the other team back in the game with the emotional prop of stopping you at fourth and goal."  Well, the team went for it and made the touchdown, after which the announcer said "I guess it was a good decision after all."

But was it?  If you choose to hit a nineteen in blackjack, and pull a deuce, was it a good decision?  If you  placed a 50-50 bet that a normal die roll will come up with a "6", and it does, was that a good decision?  I would say no.  I would argue that both decisions were bad decisions, despite the fact they happened to yield positive results for the decision-maker.  The reason is that, given the information the decision-maker had at the time of the decision, both moves have an expected value less than zero.

I won't bore my audience with a digression too far into expected value and decision trees.  Suffice it to say that the standard approach for making decisions in uncertainty is to list the possible outcomes of the decision, assign values and probabilities to each outcome, and then total up the sums.  The decision that yields the highest value times probability is the is the one that you would expect, on average, to yield the highest value.   Take the example of the bet on the die roll above.  If you bet a dollar, you would win a dollar on a roll of "6", which is a 16.7% probability.  You would lose a dollar on a roll of 1-5, which is a 83.3% probability.   The value of the "don't bet" decision is zero.  The value of the "bet" decision is 16.7% x $1 plus 83.3% x -$1 equals -$0.67.  So the "no bet" decision is best, since at zero it is higher than the negative outcome of the "bet" decision.  Here is a more complete discussion of the decision tree process.

A couple of provisos:

  • When the situation is more complex, the trick of course is to assign the right values and probabilities.  We can assign these exactly for cards and dice, but it's a little harder for something in the business world, like say Enron's decision to enter the broadband business.  But managers are paid the big bucks to do their best.  And managers have tools at their disposal to manage their lack of information.  For example, once you build a base-case, you can ask questions like  "OK, I am not sure about the size of the broadband market, but how large does it have to potentially be to offset the risk involved."
  • Like many real-world processes as the approach the asymptotes,  things get a bit squirrelly for really small probability events, particularly when they have very large financial values (positive or negative) attached.  Small probability positive events are essentially a lottery, and many people buy lottery tickets, even though we know the expected value is less than the price.  I play blackjack too, despite a negative expected value, because I get non-monetary benefits from the play.  Small probability negative events are called disasters, and are things we insure for.  Many times the decision to buy insurance has a negative expected value, but we do it anyway because we would sleep better at night knowing that we may be throwing away a little expected value, but we have pre-empted an event that would bankrupt us.  Here we get into interesting topics of risk profiles and risk tolerance, which I will avoid.

Unfortunately, in evaluating historical decisions, we often ignore the state of facts and risks the decision-maker faced at the time of the decision.  We argue Mead should have pursued Lee harder after Gettysburg, because we know now Lee's army got trapped behind a swollen river. The Chargers shouldn't have traded half their assets** to move up one spot in the draft to get Ryan Leaf.  And Enron should not have entered the broadband business.   We treat the decision makers in each of these as boneheads today (we even threw Skilling in jail, as much for his failed business decision as for any fraud).  But all of these evaluations are based on the outcomes, not on what the decision-makers were facing at the time.  Mead had been in charge of the army for less than a week, had driven Lee from a battlefield for the first time ever, and had a primary charge of defending Washington.  It is hard to believe today, but the Peyton Manning and Ryan Leaf were considered nearly equivalent in quality in the '98 draft, and the Chargers trade might have been perfectly appropriate if they had actually gotten a Manning-quality quarterback.  Enron's vision of broadband looked like it would become an enormous business, which in fact it did, just five years too late for them.

** The Chargers traded an inventory of picks and players to the Arizona Cardinals, who, true to form, did nothing with this goldmine.  The Cowboys, by contrast, arguably built a whole dynasty in the 90's off the slew of picks they got in the Herschal Walker trade with Minnesota.