Friday, June 26, 2009

Flipping Bias

Here's something that should make perfect sense, but most people don't believe. There's no such thing as a biased coin - only a biased toss, if there's a catch. No, not "but there's a catch," literally "if" there's a catch preventing the coin from bouncing or rolling and there is no rotational spin on the flat axis of the coin. The summary of the explanation can be found in Teaching Statisitics by Gelman and Nolan.

"But," you might say, "what if the coin's center of gravity is closer to the head side?" or "what if one edge is heavier?" or "what if it's slightly concave like a frisbee?" Still, the coin is not the source of any bias that results. As long as the coin does not bounce or roll after it lands, it has an equal chance of heads and tails (I've never observed a coin landing vertically on its edge but I won't completely rule it out!). Just think, a coin tossed with only "flipping" momentum spends 50% of it's time heads up, and 50% of it's time heads down. No bounce, no roll, no spin, no bias.

E.T. Jaynes is Professor of Physics (and, as it turns out teaches statistics to physics students) at Washington University in St. Louis. To illustrate, he used a pickle jar (to view this you need GSView for free or some other postscript file, *.ps, viewer), which is top-heavy and concave. Tossing it 100 times without spin, bounce or roll, he got results (p=0.54) that were not statistically different from 0.5 (z = 0.04/sqrt(0.5^2/100) = 0.8 --> P(z>0.8) = 0.424). He also tossed it to favor tails (allowing it to roll) and to favor heads (by appling rotational spin to it). When it was tossed in a way that made it roll, heads came up zero out of 100 trials, and when it was tossed with rotational spin, it came up heads 99 times.

So, the moral of the story is this. I've discussed before the notion of bias in flipping one versus two discs to determine the pull preceding an ultimate game. This tells us that the disc (uneven as it might be) is not biased unless the tosser applies bias. If the bias in the tosses is the same, then even is a dominant choice, even if the direction of the bias is unknown (but it should be known if you see the toss - will it spin or roll?). This can be proven as long as you know that the area of a square is more than the area of a rectangle with equal perimeter, i.e. it can be shown that p^2 + (1-p)^2 >= 2p(1-p), with equality only at p = 0.5. However, that point is now moot. If the tosser is not manipulating the toss, any single flip (heads or tails) and any double flip (odd or even) has "fair odds."

I'm convinced. Hope you are too.

Tuesday, June 23, 2009

Government's Net Worth?

This got me thinking:

US To Trade Gold Reserves For Cash Through

What exactly is the net worth of the US Government? Yes, the deficit/debt are large, and should be reduced, but how much is the stuff lying around that belongs to the public (land, fighter jets, buildings, etc.) worth? Anyone know of a place to try to get a ballpark for this information?

Monday, June 22, 2009

The Income Distribution Argument

I'd like to propose that the main hubub about health care is about distribution of income, i.e. a shift in the burden of health care from a "benefits received" principle to a "ability to pay" principle. To make any sort of shift like this you have to accept the principle that health care has certain characteristics attributable to a public good (something whose provision is non-excludable and benefits us all) or a merit good (something worth making more universially accessible). These premises are debatable, but I won't deal with them. Let's just suppose that we shift from fully individual, private-paid to fully public-paid (this is, of course not the change that would really be taking place because our current system is not fully private-pay, and no proposal on the table is fully public-pay).

There's not much reason to believe that the financial costs will be too much higher in public-pay as they are in a private-pay system, at least not per-person. But a private, free-market system tells me that if I want 10,000 dollars of benefit, I pay 10,000 (actually, since these contracts are not usually "actuarially fair" it would be more like paying 10000 and getting 8 or 9 thousand). Same goes for everyone else. This is a very "fair" stystem in terms of "benefits received," i.e. what you put in is what you get out.

Unfortunately, there are a lot of people who don't have the dough for 10,000 of health insurance, and there might not be a lower-priced option they can afford. Besides, if there is, it might not cover a lot of things or have a really high deductible (in fact the deductible might be so high that if they ever got so sick that their costs ever required insurance, it might put those same folks into bankruptcy - 68 percent of all individual bankruptcies are heath care related - anyway, so why bother!). This adds to the dimensions of adverse selection and moral hazard typical to the insurance market. Another "fairness" metric is a person's "ability to pay." If a good has "merit" qualities, and some folks cannot afford even a minimal level of it then what you pay is could be indexed to what you earn. Everyone gets 10000 in benefits, but a rich person pays 15000 or more, and a poor person pays a 5-6000. But is it "fair?" Yes, in a manner of speaking, because if we agree that everyone "should" have 10000 in basic coverage, then we wouldn't expect a person earning just 15000 a year to pay 10000 of it to for health insurance. (N.B. At moments like this I almost wish I remembered the game-theory formal notational definitions of "fair" and "envy-free," as well as the proofs for why Walrasian (free-market) equilibria do not generally exhibit this property....almost.)

Here's where I think the income distribution question slips in. Basically, the fight is this: Republicans represent rich folks, and Democrats represent poor folks. Making health care universal will necessitate a higher burden on richer folksto provide a benefit for poorer folks who cannot afford it. The poorest already get medicaid. Median wages have fallen over the last 9 years. Private health care costs are rising. Households view health care as a necessity. Somthing's gotta give. I suspect that if the Bush economic record were more favorable to the middle quintile of the income distribution then the "get costs under control" rhetoric would have been more effective, but rising health costs coupled with falling middle-class incomes means most people are wise enough to know that cutting costs around the edges won't do much. So, let's look at a couple of the proposals:

1. Sort of the two-tiered "keep what you want if you like it" option. People happy with their stuff now get to keep it but you still pay a percentage of your income into the public system. Of course, they will have to help pay for the publicly funded option, so their costs will either go up, or their coverage will change (by moving into the public pool). Poor people could get covered even if they cannot pay for it themselves, or can only afford a small portion of the burden. This would be a little like the education system, which, of course, is not perfect either.
2. Fully public "single-payer" system. This abandons the private option, and everyone receives the same basic plan. However, payment is based on a percentage of your income, thus shifting the payment burden up the income distribution even farther.
3. "Tax Credits." Rich folks love this option. Being able to deduct health care from their taxable income is great for them for two reasons: one, they all purchase health care anyway; and two, they pay a higher marginal tax on the income they get to deduct. This option would actually be MORE regressive than the current system (and if you think the revenues to make up for the "credit" are growing on trees you're crazy!). Good God, if you believed the republicans you'd think that tax cuts are could discover a cure for AIDS and build moon colonies, but guess what? They won't insure more people.
4. Stay with what we have and try to curb costs. Either way cost-efficiency needs to be addressed. Pulling more people into the pool for normal coverage might cut some costs per-person, but might increase costs in the aggregate, so this might not be such a bad thing.

Unless you believe that health care is a merit good, there will probably continue to be 12-15% of the population uninsured, and a growing percentage who are underinsured.