Near its northern terminus in Deadhorse, Alaska's Dalton highway is little more than a two lane dirt road barreling straight through what appears to be a massive frozen bog. But despite the remarkable paucity of truck stop diners, the equipment haulers who operate along this road have the constant company of the genuine source of their livelihood: the Trans-Alaska Pipeline. Unfortunately, both the highway and the pipeline are now being threatened by a slow-moving blob which could swamp the lifeblood of the state in fewer than ten years.
No, this isn't science fiction, nor is it unusual. Tearing down the mountain slopes of the Brooks Range is actually a creeping landslide, which has recently accelerated its pace to 1 cm per day. Beneath the lobe of sediment is waterlogged topsoil that moves and behaves like a liquid. And while this would be a trite incident in any other part of the state's uninhabited wilderness, geologists and policymakers are in a scramble to address the issue that happened to strike Alaska's most vulnerable chord.
We will use this problem to take a look at a branch of mathematical analysis called decision theory. As the name suggests, decision theory examines the factors that influence human decisions, with assumptions that can be derived from directly observable human behavior. This is essential, because to ensure that the theory is robust, we need players to make decisions in predictable, consistent ways. In the example that follows, we'll inspect the most important result of decision theory: the Expected Utility Maximization Theorem.
Six months ago, the Anchorage Daily News reported that the Alaska transportation department has considered four possible solutions: building a bridge over the slump, freezing the material in place, building a barrier, or removing the material at the toe (the face) of the lobe. Let's begin by examining the last option.
There are three terms we'll want to familiarize ourselves with here: states, prizes, and lotteries. For the sake of simplicity, let's assume that there are three states in our solution: the toe is unstable, the toe is not unstable, or the soil is frozen solid and difficult to remove.
Our prizes are contingent on the success or failure of our solution. If it succeeds, the prize will be (cost savings minus expenditure). If it fails, well, the prize is negative. It's just (-expenditure). The lottery tells us the probability of earning a certain prize if we are in a particular state, for every prize. As you might imagine, each of the potential options can be represented by lotteries.
Unfortunately, although the prizes are in monetary form, we can't just take them at face value. We need to transform them using something called a utility function, which spits out a number called utility (be careful not to confuse this with expected utility!). We can think of utility as a measure of satisfaction that discounts higher dollar amounts. To understand this, imagine that your employer gave you a $5,000 bonus today and then gave you a $5,000 bonus again tomorrow. If you're like most people, the second bonus would bring you less satisfaction than the first.
Although I won't calculate expected utility here, I'll describe how it's calculated. Suppose the ground is frozen and the operation fails. We first want to calculate the probability of obtaining the prize (-expenditure) given that the ground is frozen. We then multiply this number by the utility of (-expenditure) in the frozen ground state. Finally, we perform the above operation on each of the state-prize combinations and add them all up. This is the expected utility of the last option.
So how do we compare our options? The theorem states that our policymakers prefer one lottery over another if and only if the expected utility of the first lottery is greater than the expected utility of the second.
And believe it or not, this is exactly how the Alaska Department of Transportation will decide upon which option to pursue. Of course, they won't be performing all of these calculations, but it ultimately reduces to the same thing. Whenever we make a decision, we think about all the possible outcomes and the chance that each outcome will occur.
University of Alaska Fairbanks researcher Ronald Daanen has identified 150 lobes similar to this one in the Brooks Range, some of which threaten to pose similar issues. But for the North Slope oil rigs dependent on the road for supplies, these natural geological processes are lethal. The answer to this problem will be just another lesson in resolving the inveterate tension between man and nature.