The first thing to know about the Keystone pipeline? It already exists. Here's a breakdown of the pipeline's various parts. (Gillian Brockell, Jhaan Elker and Kate M. Tobey/The Washington Post)

There's been a longstanding assumption in the minds of Americans that more access to oil from North America necessarily means lower gas prices. In 2008, with the "drill, baby, drill" mantra humming through national politics, Katie Couric (then at CBS) challenged vice presidential candidate Sarah Palin on the link between drilling and prices during her now-infamous interview. More recently, the argument has been made in favor of building the Keystone XL pipeline, set for a vote in the Senate on Tuesday.

That assumption, though, is mostly wrong. Or at least, the reality is far more complex. To wit:

* Keystone XL pipeline oil won't all stay in the United States. Oil from the pipeline, President Obama said last week, would be extracted from Canadian tar sands, processed and sent "through our land, down to the Gulf, where it will be sold everywhere else. That doesn't have an impact on U.S. gas prices."

The industry points out that not all tar sands oil heads overseas. There is shipping by rail and truck, as well as an existing (lower capacity) Keystone pipeline that's bringing the product into the United States already. Those shipments have been given some credit for helping to keep American gas prices flat. (We import more oil from Canada than any other country.)

The entire point of the pipeline, to Obama's point, is to open up the tar sands to international markets. The difficulty in getting the product to foreign customers has kept prices low — so low that it threatens the viability of extraction in the region.

* There's no direct link between more production in North America and national gas prices. Even if all of the oil from the tar sands came to the United States, the effects would be hard to measure. After all, we have seen a massive spike in oil production over the past few years thanks to the increase in hydrofracturing (fracking) in shale formations in North Dakota, Texas and Oklahoma. But the effect on gas prices isn't one-to-one.

The industry argues that having a larger domestic supply has meant that we see fewer fluctuations when there's turmoil in the Middle East. The price of gas at the pump (the red line on the graph above) has fluctuated a lot over the past three years, but hasn't continued the upward trajectory we saw from 2003 until the recession. But there's no clear mathematical correlation (or visual one) between domestic production and gas prices. (CNN, in 2011: " 'This drill drill drill thing is tired,' said Tom Kloza, chief oil analyst at the Oil Price Information Service.")

In fact, the Energy Information Administration pointed out this month that the price of gasoline is more closely linked to the price of a barrel of oil against an international benchmark than a domestic one. That international link exists even in areas with high production — and that's several years into the production boom. The reason, in brief: "Gasoline is a globally traded commodity, and prices are highly correlated across global spot markets."

When the global market sees an increased supply of oil — as it does with increased American extraction — that affects international prices. But the link is indirect.

* Keystone XL could make some regional prices higher. The indirect link mentioned above is largely an assessment of national prices. There's some indication, as CBS reported on Monday, that some local prices could increase.

There are two reasons why. The first is that less of the Albertan oil that is being purchased in the Midwest would end up going to the Midwest, instead shunting down to the Gulf Coast. The second is that opening up tar sands oil to the international market would (as mentioned) increase its price. So local refiners that are buying affordable tar sands oil at this point would see those prices increase. (As the EIA notes, however, even Midwest prices are pegged to international benchmarks.)

We tend to assume that the law of supply and demand is both rigid and omnipresent. In the case of oil — one of the more complicated commodity markets — the effects of more supply are much more subtle. The most direct beneficiaries of Keystone XL won't be consumers. But you probably already guessed that.