# More on selfish merge and jams

I wrote earlier this week about selfish merging and traffic jams and this prompted some to ask if the selfish merge is really selfish. Update: There is more and new thinking in this later post on selfish merge being not so selfish.

There are two forms in which it is selfish. At its most basic, it is barging into line. A series of cars is traveling the road, and one car, who is behind all the others, waits for them to merge out of the vanishing lane, then zooms ahead of all of them, and get somebody up front to let them in where the merge has made things stop-and-go. 100 people behind the merger are delayed 5 seconds each, and he gains 500 seconds compared to joining the back of the line. That’s if you presume it’s a zero sum situation.

However, I believe it is worse than zero-sum, for a couple of reasons. A typical highway lane can handle 2,000 cars/hour, but only about 1,000 if traffic slows to a crawl. Cars that merge while traffic is still flowing are less likely to cause the collapse than those who attempt to merge from a stopped position at the end of the vanishing lane. It starts when somebody slows to let them in, or they barge in forcing somebody to brake.

Now if two lanes able to carry 2,000 cars/hour merge to one, we can only have smooth flow if there are in fact only 1,000 cars per hour (or fewer, since heavy merging reduces capacity to about 1,500 cars/hour) in each lane. If input is within the output capacity of the continuing lane, we can do fine. However, if slowing to stop and go reduces the chokepoint to 1,000 cars per hour, we can only handle 500/cars/hour/lane or the jam backs up for a long distance. Once input exceeds the output capacity we must take more dramatic steps to stop a long traffic jam.

This is the theory that supports metering lights on highways. As long as the highway flows at good speed, its capacity is high and sufficient for the traffic. If it gets a burst of high-demand, it collapses into a traffic jam. Thus, for people waiting at metering lights, while they are annoyed at waiting, in fact because everybody is being metered they will get there faster than if they don’t wait. For the car at the “tipping point” it can be the case that if they wait, they will join a smooth traffic flow, but if they rush into traffic, they will be the straw that breaks the camel’s back and slows everybody, including themselves.

My proposal is similar to metering lights, except for a merge. Merging reduces lane capacity as cars must increase spacing to allow safe merging. Or they must stop entirely in a jam. If demand starts to exceed capacity, my proposal is to prohibit merging well down the highway. The cars in the continuing lane zoom through without merging using the full capacity of the lane. However, from time to time they must stop (creating a waiting line) to let the cars in the vanishing lane through, also at full speed without merging. The volume of cars through the chokepoint is what matters here, and if we can increase that to 2,000 cars/hour instead of 1,000 cars/hour, we will have a far shorter jam when there is no choice but to have a jam (ie. more than 2,000 cars/hour coming in.) And by encouraging cars to merge early, we can avoid a jam when we have less than 1,500 cars/hour coming in. When we have something in between, we introduce a hopefully short single pause but maintain a little under 2,000 in output capacity. We would need experimentation to learn what the output capacity is with metered stopping.

### It's selfish no matter where you merge

Brad wrote:
>It starts when somebody slows to let them in,

And it doesn't matter how far back from the merge point that occurs. If there are plenty of gaps in the continuing lane, no one slows and you don't get a jam no matter how late the merging. If there aren't enough gaps, there will be a jam no matter how soon the merging starts, because someone has to slow up to create a space. In other words, merge jams don't happen until the input traffic exceeds the output capacity, and once that happens you are going to get a queue no matter what. The fastest way to clear the jam would be to use all available space and do a perfect zipper merge, but like a zipper the merging would only occur at exactly one spot - the end of the vanishing lane, using both lanes to the fullest. Note what happens to a zipper if you try to merge some of the teeth early: you get a jam. Of course, this can't happen in the real world. Note how a "selfish merge" is defined as anyone who tries to merge later than you did. Therein lies the problem.

### More notes

Actually, in theory merging should begin as far back as possible, ideally shortly after the last exit before the ending of the vanishing lane, because your bottleneck is the merge-point (and resulting single lane) and this and only this controls the capacity of the road over that stretch. The sooner people merge, the better, and the more time they have to merge the more capacity there is in the road.

As noted, a single lane of ordinary traffic carries 2,000 cars/hour, with only light lane-changing. When we try to do a lot more merging than normal (as in an upcoming lane vanishing) capacity drops as people need more space to perform their lane changing. However, the longer you give them to do the lane changing, the less we hurt the capacity of the highway. Give them 10 miles to complete the merge and (barring selfish actions) you could in fact perform the merge with only minimal reductions in capacity from a normal, non-merging single lane. It is only because people wait (because waiting to the last minute often benefits you slightly if traffic is slowing, and greatly — at the expense of others — if it is stopped) that we try to get a lot of the merging done in the last 1/4 mile, and thus we really reduce the capacity.

That’s why they signal lane closings miles ahead when they can. When traffic is light, it doesn’t matter if we don’t obey quickly. Unfortunately, the “selfish merge” problem comes about because when traffic is heavy, the selfish merger gains by waiting, because they can pass other cars in the much thicker continuing lane, and if the continuing lane comes to a stop, they can zoom by stopped cars.

The ultimate selfish merger waits until just before the last spot. Unfortunately, the end of the vanishing lane is special, because you must stop there unless there is a large gap in the continuing lane. And this makes the problem even worse. Because you absolutely must stop, the only way nice people (suckers) can let you in is for them to stop. And now we’ve created stop-and-go traffic, and cut the capacity of the chokepoint way down.

There are two kinds of selfishness here. One is barging into a queue, which is selfish with only a marginal effect on any individual in a queue. It’s rude, but not the end of the world. The other is increasing the risk the queue will grind to a stop by delaying merging until near the chokepoint, thus concentrating in space the amount of merging and lowering the capacity of the chokepoint. That slows down everybody, including the selfish merger, and is thus the most unenlightened selfishness.

The problem is that the defectors do this as a group. It’s hard for any one defector to see how they are creating the jam. Indeed, if all the defectors can be convinced to cooperate, but you still defect, you win really big — there’s no jam and you pass everybody. Like in the prisoner’s dilemma, you need a way to make sure everybody sees the value to everybody, including themselves, of cooperating.

### Where it occurs

I believe there is a tipping point where the merge coverts to stop-and-go, and this immediately makes a serious reduction in the capacity of the chokepoint. As long as people are merging at speed, the chokepoint is able to handle about 1,500 cars/hour. As speed drops it gets less, and it is seriously less once you hit stop-and-go.

Now I have intuitions about the factors that convert it from merge-at-speed to stop-and-go, and this happens some time before we go over the capacity of the chokepoint.

I think at the very least you would agree that once it has become stop and go, we have a situation where we have fewer than 1,500 cars arriving per hour but we still have a big jam, because at stop-and-go we can only put out 1,000 cars. So we can be in a situation where we have 1,100 cars coming in (which could be easily handled if people were merging at speed) and 1,000 going out, resulting in a growing jam, that we would not be in if people in the jam and approaching the jam were to cooperate rather than defect.

If, in that situation, we got everybody to merge early (which the cooperators try to do) and we did not have defectors trying to zoom up the lane being emptied, then we would simply have a line of stopped cars ready to accelerate, the limiting factor their ability to accelerate and space themselves, which I would estimate is far greater than 1,100 cars per hour, and the jam would clear. (As soon as it does clear we are back to a single lane able to handle 2,000 cars/hour with only 1,100 coming in -- no jam risk at all.)

And yes, I would also contend that butting ahead in line to the detriment of others is selfish. We accept it in society when you see friends in line because we like the ability to join our friends in line. We react much more negatively if there's a long line and people just randomly walked up and entered the line near the front. Because people are polite, you can do that from time to time but do it enough and I think you'll see blows. However, my intuition here is that more is going on, that the person butting into the line is actually slowing down the whole line, and oddly, as a group, are slowing down themselves, the same way people who jumped metering lights would, as a group, are on occasion as an individual, slow down themselves as well as others.

### Waxing philosophic

And I had begun to believe that I was the only traffic prophet...

First (and as suggested in replies to your other posts on the subject), the selfish merger isn't strictly selfish. In fact, what the cooperative merger and the selfish merger have in common is that they are trying to reduce their individual blood pressure: the 'selfish' merger is frustrated by the extra pavement going unused, and the 'cooperative' merger is frustrated when people aren't following the merge early. Each driver tries to lower their blood pressure accordingly.

The fact is, however, that the 'selfish' merger is causing a problem - but that problem affects other people, not himself. The slowdown that is created by forcing a vehicle into the continuing lane at the merge point when inter-vehicle space is not available is self-amplifying, in that it increases the "emission probability" of future selfish mergers from the rear.

At the same time, the cooperative merger also increases this probability: both by slowing down the continuing lane early [i.e., at their chosen merge point], as well as by creating a "pressure differential" between the continuing lane and the merge lane [as a result of their lane transition].

It is quite reasonable to "resolve" a pressure differential by moving to the lane of reduced pressure - that is, to use the merge lane if it is more open. Not following this basic driving instinct eliminates any chance of a multi-lane highway achieving its maximum throughput.

The problem is this: drivers are informed of a merge early so that they will slow down and generate space. It is this space that will allow for a fluid merge at the actual merge point, because vehicles in the continuing lane will not need to slow down nearly as much to accommodate merges.

[Side note: a stop sends "shock waves" back through traffic, which produce a sort of A/C resistance that is independent of the D/C throughput; faster stops produce higher amplitude waves that cannot be damped down as quickly. This also contributes to the self-amplifying nature of traffic slowdowns.]

But drivers do not utilize the merge region to spread out and merge. They [we] have become polarized, into either the 'cooperative' camp or the 'selfish' camp, and the emergent behavior that results is the direct result of this polarization. Both camps' actions encourage inefficient merging.

If re-training drivers were possible, this would be easy to fix.

I think the simplest solution is road re-design, as suggested by one of your readers' observations overseas: make both lanes "merge lanes" somehow: maybe the lane divider, rather than being dotted, should also 'merge' into the continuing lane, implying that "yes, your lane, too, is going away". This might force the 'cooperative' merger to utilize the "actual" merge lane and thus reduce the pressure differential.

A gradual merge, rather than a sharp merge, also makes sense, so that both lanes feel a 'squeeze' that urges them to cooperate over time before the merge point.

Maybe the dividing line could be wavy: if each lane is alternately 'squeezed', the instinct to slow down would be triggered alternately, naturally generating the "zipper" action that most efficiently executes the merge.

Whatever the solution, drivers, consider this: the other guy has a good idea, and the solution must involve everyone.

### No 'extra pavement'

The mistake the selfish merger is making is the belief there is "extra pavement" to be used. This is an illusion. All that matters is the chokepoint and the volume of cars per hour that can go through it. It doesn't actually matter from a flow standpoint where we imagine the lane to end. The flow is limited by the flow of the checkpoint. (The one difference it makes is that cars will travel faster on a 2 lane road getting to the checkpoint than on the one lane road after it, but the number of cars that gets through per hour is the same whether you put the chokepoint right after the last interchange (and just cone off the "extra" space) or do it at the very end.

As I see it, all that matters is the flow rate of the chokepoint. If we drop to stop-and-go, that rate is a fair bit lower than if we have a smooth zipper merge over a longer stretch of road while all cars maintain a decent speed. Attempting to merge where there is no room to do it at speed forces the chokepoint to stop and go, and makes it slower for everybody. In many cases, in fact, it makes it slower even for the selfish merger, because they usually have to endure a little stop and go traffic before they can get to the point where the vanishing lane is empty enough to zoom up to the end. In those cases, like crashing through metering lights, it is not just selfish but stupidly selfish.

There are other instances where it is more simply selfish, and the selfish merger can immediately move into the emptying lane, and wait only a short time for somebody to let them in at the chokepoint. However, there is an argument that this still should take longer than the trip would take in a zipper merge where you never slow all that much.

### No mistakes

You're absolutely right - the "extra" pavement that is there does not mean that throughput can be increased by using it like a normal lane. That pavement is there to facilitate the merge - to give drivers the time they need to negotiate the merging details with one another. The selfish merger is, I also agree, foolishly ignoring this fact.

Yet the motivation of the selfish merger - that of resolving a local, apparent "lane pressure differential" - is quite valid in any other circumstance. And because we're not going to curb that instinct (pardon the pun), any traffic control solution must presume its existence, and work within that framework; I am personally a cooperative merger, but I will often drive between the lanes to 'encourage' cooperative merging by specifically by reducing the local pressure differential behind me.

I think, going off on a tangent here, every merge generates a local 'eddy' in traffic, which whorls backwards with respect to the merging driver. In normal traffic flow, the "from" lane speeds slightly to fill in the extra space, and the "to" lane slows slightly; thus the eddy is 'absorbed'. In heavier traffic flow, the pressure differential created behind the eddy encourages drivers to move out the slowed "to" lane and into the faster "from" lane. This should cancel the eddy out (from that point back). But the extra spacing generated by the lane change negotiations themselves the cause of the reduction in throughput of the lanes involved.

A lane without any allowed merging [such as a carpool lane] eliminates eddies, but at the expense of reducing the flow rate to that of the slowest driver in the lane [on average]. Multi-lane highways exist to circumvent this problem.

Whatever the solution, I think it has to take into account the 'selfish' merger into account from the start. If you want to reduce merging events when input exceeds output, why not just have lane-warning bumps that go higher and lower based on highway volume?

### The stops

The problem actually comes when the selfish merger finally does their lane change. This happens one of two ways. Before the traffic collapses to stop and go, but when it's thick, they will merge into a spot that is smaller than the safety margin of the drivers behind, causing them to slow or even stop. The first one who stops triggers the collapse to stop and go.

At this point, curiously, the problem is now caused by cooperative mergers who stop to let the selfish merger in. While many of us refuse to do this, there are always those who do. And in some cases selfish mergers "force" such a stop by barging in and relying on the fact that you won't dent your bumper just to be angry at a selfish merger. This regular need for traffic in the main lane to stop prevents the stop-and-go collapse from being corrected.

It is however, very hard to correct the instincts that cause cooperating drivers to let people in, and even harder to make them actively block barged entry, which is difficult and can inspire road rage. To generate a new ethic of merging, I think metering lights are required, as we are starting to understand them, and we instinctively obey red/green. You add the rule "Lane change prohibited (breakdowns excepted) while merge metering is in effect" and perhaps something happens.

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