The Escalator’s Dilemma

This article is part of a series entitled Everyday Game Theory. See also:
1. The Escalator’s Dilemma; 2. Electoral College; 3. Passing Curiosity; 4. Lesson Time

The key to surviving the Moscow subway system is what I might call the gruff grunt.

The underground subway car is packed; in ten seconds, the doors will close, and by that I mean close. Simply plant your shoulder squarely into that of the stranger in front of you, and emit a low noise of affirmation. Venture a nod of the head if this stranger deigns to crane his neck backwards. He’ll then grunt back, and plow his way further into the crowd. “Осторожно: двери закрываются.” (“Caution: doors are closing.”) We’ve made it this time.

One also must navigate. Ride the escalator upwards, away from the platform, and proceed through a long marble hallway – past that row of inexplicable shops (why are they camped in the subway?) selling “goods” ranging from cigarettes to bras – and then take the stairs downward, precisely two levels, to finally transfer from the orange line to the light-blue line. Just don’t continue further to the dark-blue line. (The simultaneous presence of the two blues might seem confusing to us English speakers, but Russian uses distinct words for the two colors – chalk up a victory for Sapir-Whorf – and, in any case, the subway lines’ Russian names are streets, and have nothing to do with the colors that the informational signs, and we Americans, use to represent them.)

But to survive the subway’s escalators? To survive the Moscow subway system’s escalators requires much, much more.

It’s rush hour, and our evening class – held on Thursdays in Moscow’s southwestern Vavilova district – begins in 10 minutes. I’m waiting for the escalator. The choke point awaits ten meters ahead, but a massive crowd has ballooned beneath it, filling the marble plaza.

But something’s strange. The escalator is half-empty! Despite the swelling crowd, the escalator’s riders have dutifully lined up, standing uniformly along the escalator’s right side. Half of the escalator’s resources are being left unused. They’re allowing others – those who wish to walk – to pass them on the left. Once in a while, a hasty walker streams up the left side.

Then, something stranger happens. A single rider mounts on the left, but stands. Suddenly, as if a floodgate has been released, riders pour onto the escalator at double the pace, filling both sides of the escalator. Before a few minutes have passed, the thick crowd has nearly cleared.

What we’ve observed could only possibly have occurred as an emergent result of an intricate system of interaction between members of a collective: a system of micromotives and macrobehavior. [1]

But what exactly is going on?

First theory: politeness. The standers right are cooperators.

“What grand unity,” I once thought to myself. Everyone’s impatient, packed in, and in a hurry. This is the Moscow subway. But, in a spectacular display of collective organization, – we hold strong together – everyone has filed up on the right side. An unbroken column stretches from bottom to top. Those who wish to walk are free to do so. If even one person had mounted on the left, then the entire left-hand column would have been broken. Walking would have become impossible.

We can view this game-theoretically. The game is as follows. Suppose that there is a very long line below the escalator. The game has two overall states: left lane empty and left lane full. As each player mounts the escalator, he makes his choice: to stand right or to stand left. Within the game state in which the left lane is empty, each player universally individually prefers to stand left than to stand right. Indeed, by standing left, that player can mount the escalator exactly one position ahead of where he otherwise would have mounted had he stood right. Thus by standing left, an individual player earns higher utility. On the other hand, standing left is damaging. If a single player stands left, then all walkers become forced to stand instead. (We assume that those who prefer to walk are external to the game.) Finally, if a single player does stand left, then standing right no longer acts to preserve the option of walking – the column has already been broken, after all – and standing left ceases to be harmful.

Thus, the game state left lane empty is an unstable equilibrium, characterized by mass cooperation. Each person foregoes individual utility so that walkers may walk. But if even a single person defects – or stands left – then the game state shifts into the stable equilibrium left lane full. Players no longer have reason to stand right, so it’s indeed an equilibrium, and standing left is not likely to be foregone, so the equilibrium is stable. In this state, although the crowd moves faster, walkers are unable to walk, and this stable equilibrium is a Kaldor-Hicks (please see this) suboptimum. Thus the flood of people we observed was a “disaster”.

Though this analysis seems tempting – it was my preliminary hypothesis, and indeed it does explain our earlier observations – a few things don’t seem quite right. For one, very few people actually prefer to walk, and meanwhile many, many people gain utility by a faster-moving line. Thus, it would seem that left lane full is actually a Kaldor-Hicks optimum. But then why do people forego it? Indeed, now it seems that, by standing right, players harm both themselves and all others. There must be another reason that people stand right.

This system will prove much more complicated than we first thought.

Second theory: culpability aversion. The standers right are defectors.

Few people prefer to walk. Yet when a passenger does wish to walk, he may be quite resolute. If he’s blocked from doing so by a stander left, he may become irritated. He might approach close behind this person above him. He could explicitly ask him to move. He may even try to push past.

As before, our game has two states: left lane empty and left lane full. Again, a long line awaits below, and walkers are external to the game. Each player, as before, makes the choice: stand left or stand right. And as before, players gain a small amount of individual utility by standing left.

But this utility – that gained by advancing one position – is, really, almost negligible. Meanwhile, a stander left experiences a few powerful disutilities that we haven’t yet recognized. For one, he risks enduring the social discomfort of being tailgated by an impatient walker. And by being the one who broke the column – in, moreover, a highly visible sense – he risks inducing additional social animosity from all prospective walkers, and even from standers. Thus, on the net, standing left involves a strong individual disutility.

But each act of standing left reduces this prevailing disutility. Indeed, when people begin to break the column – risking social shame – they absorb the burden facing prospective standers left. After all, if a stander left resides below other standers left, then the one lower down cannot be blamed for obstructing the column. The disutility diminishes. Eventually, standing left begins to carry only the slight utility that we’ve mentioned before. Both files become filled. If enough people stand left, the game state shifts.

It shifts into a strong Kaldor-Hicks optimum. Indeed, though now a few pesky walkers are forced to stand, the entire crowd moves twice as fast, producing massive, widespread utility gains. Furthermore, this optimum is also an equilibrium, because players no longer avoid standing left.

This is a critical mass game with a “very stable” and Kaldor-Hicks-suboptimal equilibrium.

Significantly, the utility gains associated with this optimum fall exclusively upon those lower down in line. The individuals who cause the state shift – who first stand left, that is – do not partake in the benefit triggered by this shift.

Thus the standers right are defectors. Though they know (do they?) that, by standing left, they might save time for many of people below them – hurting only a few walkers in the process – they refuse to, because they prefer not to singlehandedly bear the burden of risking angering walkers. Instead, they elect to endure the much-smaller individual disutility of standing right and foregoing one position.

The standers left on the other hand? Heroes. By standing left, and shouldering the burden of risk of social reprisal, they induce a tectonic shift in the game state, which permits waves of other people below them to board the escalator twice as fast. They are the cooperators.

The game, to be precise, actually involves a family of possible utility assignment schemes. The larger the crowd is, the longer wait times become, and the more insistent walkers get. This, in turn, threatens a heavier burden on prospective standers left, and makes the associated potential disutility higher. This, finally, makes the suboptimum harder to defeat. Thus as the crowd fluctuates, so do the values of the function connecting any given number of cooperators to its associated prevailing disutility, and, of course by consequence, the number of cooperators required to enact a state change. The larger crowds get, the more awe-inspiring the state shift.

Ironically, even those most insistent on walking – those most in a hurry – would pass through the escalator must faster in the left lane full state than they ever could by first waiting much longer in line and then walking up the empty left lane. Thus fascinatingly, though earlier I claimed that the left lane full equilibrium was a Kaldor-Hicks optimum – benefiting all standers, and hurting only would-be walkers – it’s indeed, though perhaps unwittingly to these would-be walkers, a Pareto optimum! Indeed, nearly all prospective walkers — all of them, that is, except for possibly a few near the very front — stand to gain by a state shift. Thus this is a true “tragic” collective action problem: a self-aggravating Pareto pessimum, we could say.

The dynamics of norms, and self-propagating conventions

The very existence of this problem rests on a set of poor social norms.

When the crowds are sufficiently small, left lane empty is a Kaldor-Hicks optimum. Standers wait only for a short time, and those truly in a hurry can sprint up the stairs if they like. When crowds are sufficiently large, left lane full is a Kaldor-Hicks, and indeed a “nearly Pareto”, optimum, for reasons outlined above. If play were perfect (by the Kaldor-Hicks standard), players would shift to the left lane full state as soon as crowds became so large that the harm imposed by general wait times surpassed the benefits accorded by an empty left lane to impatient walkers.

In practice, we see the game fall into the left lane empty state inappropriately often. This happens because of poor set of social norms which dictate that walkers should always be yielded to. The above-described collective action problem exists partly because of the presence of these norms.

We want the game as often as possible in its optimal state. It’s not clear, right away, that rules or norms at all, of any form, are a good way to achieve this. Perhaps we’d be better off if passengers simply chose what seemed best at any given moment.

It’s also not clear why, or how, the existing prevailing norm happened to arise. It’s possible, for example, that yield to walkers first developed in situations featuring small crowds — situations in which left lane empty really reigned as a Kaldor-Hicks optimum. The convention persisted. It’s proven hard to break.

Another set of norms could be better.

Yield to walkers except when crowds begin to collect below the escalator. Then fill both lanes.

We can view the state of social norms as itself the product of a dynamic process. For some reason, always yield to walkers has become the norm, and it persists, as a stubbornly stable equilibrium. Yield to walkers may be one of those norms – like driving on one particular side of the road [1] – which endures because deviations are very difficult to introduce. Yield to walkers is a self-propagating convention.

The state comprising our alternative norm is, it’s tempting to believe, a stable equilibrium. If, by some miracle, the prevailing norm could be reversed, then the years of relentless inefficiency could finally come to an end. This is unlikely to occur.

In my imagination – in the wild, improbable sequel to this story – a government emerges. “Please fill both lanes, both lanes, please” drones the weary subway attendant, speaking in Russian, through a loudspeaker, to the shuffling, grey masses, wrapped in overcoats and scarves. One passenger, looking up, steps into the left row, delicately, but confidently, and looks up and smiles. Another follows, and then another. And the flow of the great river of people is diverted towards good.

References:

  1. Thomas C. Schelling, Nobel Laureate in Economics: Micromotives and Macrobehavior
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3 comments on “The Escalator’s Dilemma

  1. Ben says:

    I have used language loosely, in a certain sense, and I’ll make that sense clear here.

    Measures of efficiency like Pareto or Kaldor-Hicks are used to compare economic states, or outcomes. We can say things like this economic outcome is more efficient than that one or among this family of economic outcomes, this one is optimal. These efficiency measures can only be invoked with a clear set of states in mind.

    I’ve freely said here things like “both lanes full is a (Pareto or Kaldor-Hicks) optimum”. This statement is ill-defined. Optimum among which class of economic states?

    One plausible way to make this precise would be something like the following. We can first imagine a vast family of possible economic states in the subway station — each specifies a full arrangement of standers, walkers, players and utility states. Then we conclude that, among all of those, some particular one is an optimum. Though the statement understood in this way now makes sense, this is not actually what we have done.

    What we’ve done is something more like the following. In any one of these comprehensive economic states, the left lane is either empty or full. If we fix any one of those economic states, and then reverse the game state — the state of the escalator — then we can compare the two outcomes. We might observe that switching from left lane empty to left lane full tends to produce (Pareto or Kaldor-Hicks) improvements. This latter idea is really what I mean when I say, again loosely, that “both lanes full is a (Pareto or Kaldor-Hicks) optimum”.

    This new understanding of the language can be “substituted” into the above piece seamlessly.

  2. Richard says:

    Interesting read. I’ve experienced this on the London underground. People there are quite happy to (vocally) remind you that standers stand on the left. (Or was it the right?)

    “Within the game state in which the left lane is empty, each player universally individually prefers to stand left than to stand right. Indeed, by standing left, that player can mount the escalator exactly one position ahead of where he otherwise would have mounted had he stood right.”

    Is that so? Aren’t you overlooking the possibility that encroaching upon another person by standing side by side, rather than by being behind them, is less desirable, ceteris paribus?

    When you say: “It shifts into a strong Kaldor-Hicks optimum. Indeed, though now a few pesky walkers are forced to stand, the entire crowd moves twice as fast, producing massive, widespread utility gains.”

    You almost sound like you’re drawing what Derek Parfit called “the repugnant conclusion” and praising it as a desirable result. The repugnant conclusion, you might recall, is when we are forced to say (among other things) that very tiny albeit massively widespread increases in utility are worth (among other things) a few localized though very intense cases of disutility – as long as this doesn’t amount to a zero sum game or worse (i.e. as long as the net return on utilities is positive overall).

    “The individuals who cause the state shift – who first stand left, that is – do not partake in the benefit triggered by this shift.”

    Unless they are game-theoretically trained and note their own heroism!

    “Ironically, even those most insistent on walking – those most in a hurry – would pass through the escalator must faster in the left lane full state than they ever could by first waiting much longer in line and then walking up the empty left lane. Thus fascinatingly, though earlier I claimed that the left lane full equilibrium was a Kaldor-Hicks optimum – benefiting all standers, and hurting only would-be walkers – it’s indeed, though perhaps unwittingly to these would-be walkers, a Pareto optimum!”

    This depends on the relative lengths of the queue and the escalator, I assume. For some ratio of escalator length to line length to walker fitness, a walker might still benefit from freedom to walk, surely?

    Avoid the repugnant conclusion, Ben. We keep the lane free because it’s not acceptable for us to live in a society in which the intense struggles of arbitrary individuals can be callously swept away by a seething tide of petty utilities, whose individual comings and goings are barely noticed by each of us anyway. If people care enough to walk or, in rare circumstances, are ever desperate enough to run, then we should not impose a crippling dearth of utility upon them just because it involves a greater surge of utility in the masses, a surge so distributed that each of us barely notices it on an individual level. You might as well praise the killing of one man because we can distribute his blood, bone marrow and vital organs among ten, at least the benefits of that cruel act would be felt by the beneficiaries.

  3. Richard says:

    Oof. We’ll that’s me told off(!). I haven’t been so thoroughly scolded since I was a young’un.

    But seriously though, I don’t see much of a tone in what I was writing, and it certainly wasn’t written in the spirit in which it has been read. But apologies for any confusion. I used the phrase ‘repugnant conclusion’ only because it is the name of a well known philosophical issue in utilitarian theories of normativity. I do not find anything *you* have written repugnant.

    Anyway… my first remark, like my others, *was* written after reading your entire piece. The second theory only mentions the following:

    “a stander left experiences a few powerful disutilities that we haven’t yet recognized. For one, he risks enduring the social discomfort of being tailgated by an impatient walker. And by being the one who broke the column – in, moreover, a highly visible sense – he risks inducing additional social animosity from all prospective walkers, and even from standers. Thus, on the net, standing left involves a strong individual disutility.”

    Here you mention, explicitly, the disutility from being judged by standers but do not mention explicitly the discomfort and disutility that could be associated *specifically* with standing closer to a complete stranger than is necessary. I was just fleshing out the possible disutilities. Disutility from breaking the trend is one thing but disutility from personal discomfort at the unnecessary proximity to a stranger (who may want his space) is another.

    Your second comment seems to miss the point a bit. The repugnant conclusion is precisely that everyone’s benefiting is taken as justification for someone’s suffering, *even when* the benefit felt by each individual in the benefitted majority is barely noticed first-personally and yet the suffering felt by each individual in the minority is very great first-personally. If the utilities add up, then the conclusion follows. You may think this rarely arises, but it is a consequence of the view you seem to develop. It’s an old problem that afflicts all such utility based moralizing, I’m afraid.

    What I’ve offered *is* an attack, sure, but only on a theory which draws the ‘repugnant conclusion’. (I sense you do not like the word ‘repugnant’, which is fine. But take that up with Parfit not me.)

    Just consider the following: why does the convention to keep the left clear arise at all? You conclude that it is because people are merely bullied into it by irate walkers and that, in a world of heroes, fewer people would avoid the social discomfort in order to benefit the masses and thus produce a greater quantity of utility. But this presupposes a partition of the population into walkers and non-walkers, neither cell of which partition is empty.

    But why aren’t all people walkers in the first place? The answer is presumably because the majority aren’t walkers. If the majority *were* walkers, then they would, by the social pressure you mention, have everyone walking. So the majority must be such that they do not stand to benefit more from walking than from standing (otherwise they’d walk). But the losses they suffer from having to wait a bit longer (due to the existence of left lines reserved for walkers) must not then be sufficient to prompt them to walk at the escalator rather than stand, otherwise (again) the majority would be walking.

    My conclusion contains a rather more congenial view of people, namely that each person does not see the small, perhaps tiny, amount of benefit they’d gain if all were standing as sufficient to warrant imposing great disutility on the those who need to travel with some urgency. Each person is free to think to himself: if my need is great enough I have the option of walking. This freedom, this possibility, bestows great widespread comfort upon each of us, even though we may never require it. Space for walkers allows the passage of more types of people than your uncharitably caricatured tyrant walkers, it also allows those whose need is very great and very worthy to pass unobstructed.

    Think of it this way: a thousand people, most of whose needs are insufficient to warrant their walking in the first place, could arrive thirty seconds or a minute earlier at their destinations for a measly (if even noticeable) gain in utility, while one person is ten or twenty minutes too late to the hospital bed of a loved one, or to see his child’s school play etc. incurring a devastating loss of utility and damage to his personal well-being. Call me an optimist, but I think the standing masses allow for the swift passage of those with urgent needs because the standing masses do not necessarily see their own lesser needs – even collectively – as out-weighing those of the urgent walkers. Indeed, one day they too might need to walk rather than stand, but it’s not today. Hence they stand.

    Here’s another intuition pumping analogy. Distributing a lifetime’s supply of nourishment among a million who are otherwise already well-fed gives each of the million a small increase in utility. But what heartless majority would be indifferent between this option and one where the same quantity of utility is achieved by giving the nourishment to some individual who would otherwise have insufficient nourishment to support his life? The example is a dramatic outlier at the fringes of a theory of norms with principles like yours. But if the principles you seem to favour are adopted, we must prepare to be indifferent between giving the fed million one more morsel while leaving the individual to starve, and giving the individual enough food to live on for a lifetime but denying the fed million its extra chomp. But clearly we should choose the latter option. So the principles must be rejected as they stand.

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