Imagine 120,000 people in Boston, beginning to march in a parade all the way across the country to LA. Also imagine that this is not the actual America, but an alternate America where Boston and LA are connected by a long winding mesa, that snakes across the country in a southwesterly direction. It's about 2 to 3 kilometers wide and flat on top. There are sharp drop-offs to the northwest and southeast of this mesa.
The parade initially moves off to the southwest, trying to stay toward the center of the mesa, and avoid the sharp drop-offs to the left and the right. Unfortunately, the land is flat, the visibility is poor, and the mesa does not go southwest in a straight line. So it's not easy to avoid the edges.
After a few miles, the parade appoints a man named Alan as their leader. He is seen as having an uncanny ability to lead the parade in the right direction. In fact, he simply has a good ear, and hears murmurs from people in the parade who think they see signs of one of the edges of the mesa, and he adjusts the parade's path accordingly.
Alan also adds more structure to the entire process. Now people are lined up 12 abreast, in a total of 10,000 rows. The entire surface of the mesa is painted with parallel lines running east-west and north-south, ¼ kilometer apart, which forms a grid like city blocks. Alan says the parade must follow the lines, even though the land is completely flat on top of the mesa. The problem here is that the mesa runs to the southwest, and thus if the parade goes for any extended distance straight to the south or straight to the west, it eventually reaches a sharp drop-off. Thus at each corner Alan makes a decision---turn to the west or turn to the south.
By the time they reach Pennsylvania, Alan has grown old and retires from being the parade leader. He was so successful at keeping the parade away from the edges that some of the younger marchers have never even seen a cliff edge. They begin to believe that the entire world is flat, and that the parade could follow a straight line indefinitely, without even going over the edge. These "flat earthers" formed a group called "Modern Managers of Transport" and argue that Alan never had the power that people assumed, and indeed had no ability to lead them to Los Angeles. And that there are no cliffs.
The next leader decides to make decisions by committee. At each ¼ kilometer intersection, he consults with the 12 people in the front row, who vote on which direction to take---south or west. But in a sense not much has changed, as each of those 12 behave pretty much like Alan, listening to chatter in the vast parade behind them for signs that people have seen a sharp drop-off to the left or the right.
In Ohio there was a close call, as visibility was very poor and the mesa became quite narrow. Some marchers actually slid off one edge, and had to climb back up. A few never made it.
In southern Illinois the first female parade leader took over. By this time there was great interest in the characteristics of the leader. Some parade leaders were seen as having a "south bias", while others leaders were seen as having a "west bias" depending on which direction they seemed to prefer. However, at the back of the parade an old man named Milton suggested that the public greatly exaggerated the influence of parade leaders, that the leader actually had very little discretion over direction except in the very short run. Rather the terrain determined the path and the parade leader needed to follow that terrain. He pointed out that the mesa was only a few kilometers wide, and that if they really wanted to end up in LA, then leaders actually had no real control over the total number of left turns and right turns. More right turns now led to more left turns later.
The new female leader did a fine job of keeping the parade along a steady path, far from the steep drop-off. Although she had a west bias, she actually pushed the parade slightly to the south of the optimal line at the center of the mesa. And although her performance was excellent, she was replaced after a short period, soon after entering Missouri. Some said that having a female leader bothered the leading men in the parade, and that it was male chauvinism that caused her premature replacement.
The next leader was nicknamed JP, and he decided that the entire process needed reform. Improvements in technology allowed each parade participant to be given an electronic clicker that communicated with JP's handheld computer, with buttons for south and west. At each intersection, 120,000 pairs of eyes would search for clues of a drop-off, and click one of the two buttons. In essence, the parade would vote on which direction to take at each intersection. JP would then tabulate the votes on his laptop, and lead the parade in the new direction.
This was widely seen as a new system, where in a sense the parade was now the leader, and the putative leader followed the wishes of the 120,000 marchers. But followers of Milton insisted that it really wasn't all that much different, that even as far back as Connecticut the leader had mostly been getting out in front of a parade that had already collectively made up its mind on which direct it wanted to take. Even then, any errors by Alan, any moves inconsistent with the wishes of the marchers, had to be quickly corrected to avoid the cliff edge.
JP's next reform was to dispense with the grid. He suggested that since the mesa top was flat, it would be best to simply march to LA as directly as possible; no reason to go straight for arbitrary ¼ kilometer increments, before each turn. So at each step of the way the marchers now voted, and the path of the parade no longer looked like a step function; it looked more random, and somewhat smoothed out.
By Kansas there were enough improvements in artificial intelligence that it was decided that the next leader would be a robot. The marchers would continue to vote, and the votes would be sent to the robot at the front of the parade, which would adjust its path accordingly.
By eastern Colorado, the parade had achieved success in terms of finding the smoothest, most efficient path to LA. This led some creative thinkers to work on new ideas for future parades. Work began on how future parades might go to alternative cities. The debate was no longer about how to get to LA, now it was a debate about which city was the best destination. And at this point there were some important theoretical breakthroughs.
Researchers called the line along the center of the mesa from Boston to LA the "natural route of interest". This was the most efficient way of getting from Boston to LA. Sound parade management meant keeping the parade as close to the natural route as possible. It was also discovered that alternate mesa paths occasionally forked off from the Boston-LA line. Each alternative destination, whether it be San Francisco or Houston, had its own natural route of interest. People with the south bias suggested that Houston was a better destination for the parade, whereas those with a west bias preferred San Francisco as an alternative.
There was also a dispute about how to exit onto alternative mesa paths. The majority, called the NKers, thought that exits off of the left side looped around under the mesa before heading to the right, and vice versa. A smaller group called the NFs suggested that if you wanted to go right, then you should simply exit to the right.
This parable shows that it's not always clear whether a leader is actually leading a parade, or just getting out in front. The same point is made in another post over at TheMoneyIllusion, in a different way.
PS. It's not easy to find physical analogues for economic concepts. How do you model the zero bound? How do you model the fact that the natural route of interest may itself change as a result of policy. As you walk, the shape of the Earth actually changes? This is all too much for me.