Click the map to select a location

Results

At the chosen grid size, measure, and dimensions. Every space in use counts once.

Distribution of connectedness

Lorenz curve

Spaces ordered from least to most connected

Inequality and welfare

Inequality across grid sizes

The same city, by set of dimensions

Welfare across grid sizes

Sen’s welfare, finest grid = 100

The selected space across grid sizes

Masked spaces

Average from afar; inside, the least connected

What the selected location can reach

Access from the selected 125 m cell, each dimension scaled as in the rail; the tick is the median cell in use

1. Questions this game can answer

The game has no score and no target. These are questions it can help answer. None of them have one right answer, because the answer depends on the scale at which the city is read, the metric, and the dimensions counted.

Three from each group are drawn at random on every visit.

2. Background and context

The standard entry point to the study of inequality, of any form, is the question: inequality between whom? Asked of space, it becomes a question about those in (or using) one space against those in (or using) another—the incomes of one neighbourhood against another’s, say, or the gap between groups living on opposite sides of a city. Space in such a reading is a container. It records where people are, and the inequality belongs to them. In my doctoral research (Jana, 2025) I took a step back from this and turned the question around, to ask what role space itself plays in determining spatial inequality. How different are different spaces? Is one space more equal than another? Can spaces be altered to make them more equal? And, critically, what does equal mean in the context of space at all?

The standard economics reading of a city begins with production. Firms locate where they can minimise costs and maximise productivity, and workers then decide where to live based on what they can afford, so that those who earn less usually live further away from the sites of employment. The infrastructure, services, and amenities of a space tend to mirror this first hierarchy of wages and distance, in quality as much as in quantity, and what a space offers comes to reflect how productive it is. Soja (1980) and Harvey (2009), from different directions, challenge the notion of space as a passive backdrop to social processes: in their reading, the arrangement of a city produces inequality as well as reflecting it. And since the built city changes slowly, an arrangement can go on producing inequality long after the policies that made it have lapsed, as it has in cities of the global South whose planning was exclusionary.

Two further reasons are practical. First, space is what a city government regulates. Zoning, the allocation of land, the siting of a school or a clinic—these are all decisions about spaces, and regulating space is arguably the most powerful instrument the state has for changing a city. When locating a new housing project, for example, a planner asks whether the site is close to work, to schools and clinics, maybe even to public transport, which is in effect a question about how well a space will be served. Whether the new housing makes the place more or less unequal, or adds to the welfare of the city, is usually left to rules of thumb (and, many times, to anecdote). Second, the data. Household surveys are rarely representative below the scale of the city, and censuses are often anonymised below it too, while land-use and zoning records can go down to the plot. An inequality defined between spaces can be computed from records a city already keeps.

People enter this reading as the users of spaces, through Sen’s capability approach, which judges well-being by what people are able to do and to be (their capabilities) rather than by the resources they hold (Sen, 1979; Nussbaum, 2013). Resources become capabilities through conversion factors, i.e., the conditions that decide how much capability a given resource yields. These can be personal, social, or environmental, and the environmental ones—the built city among them—have received the least attention (Robeyns, 2005, 2017). Consider two households with identical incomes in different parts of a city. For one, a clinic, a school, and a job are a short trip away. For the other, much of the same income goes on reaching them, and less is left for anything else. A space’s access to what a life needs is, in this sense, a conversion factor for everyone who uses it: those who live there, those who work or study there, and those who come to its parks. Spatial welfare, as the thesis defines it, is the degree to which the arrangement of a city’s spaces lets their users turn resources into capabilities. Each space therefore counts once in the game, and the people who use it count through what it lets them do rather than as weights.

Taking spaces as the unit changes how inequality has to be measured, in two ways. First, a space has to be defined before it can be counted, and the definition changes the answer. The sensitivity of a result to the choice of spatial unit is known as the modifiable areal unit problem (MAUP), and it is conventionally treated as a bias to be minimised (Openshaw, 1984). When the object is inequality between spaces, however, how inequality moves as the spaces grow or shrink is evidence in its own right—about the scales at which a city sorts its opportunities. Second, access has many dimensions. A space near good jobs may well be near good schools and clinics too, and whether counting another dimension shows more inequality or less depends on how unevenly the new dimension is spread, and on whether it ranks spaces as the others do. Advantages that sit together compound, while advantages that sit apart partly offset one another.

This game offers a way to work through these ideas on cities that do not exist. Each city is generated from a few rules about where housing, jobs, schools, clinics, parks, and public amenities go, and any part of it can be repainted by hand. In Reading the City the user can change three things—the size of the spaces the city is read at, the measures of access and of inequality, and the dimensions counted—and watch how the distribution of access between spaces responds. Spatial Interventions adds a school, a clinic, a park, a jobs centre, or another use, and shows what it does to inequality between spaces and to welfare at every size of space, and how the credit divides when several are built. The numbers do not describe a real city. What they show is how the ideas behave when the same city is read in different ways.

3. The city

What is a city in the game, and what is a space in it? Every city here is a square 20 km on a side, held as a lattice of 125 m cells, 160 by 160. Each cell holds one specific land use—housing, sites of employment, education, health and community, open space, public amenities, or mixed use—or is vacant, and access is computed cell by cell (§4). The page opens on a city drawn at random, and Generate a new city draws another; each city has a number, shown under the button, that reproduces it. The numbers quoted in this Guide are all for one example city, laid out in rings around one centre, which can be opened in the simulator.

The city can be read at five grid sizes: 125 m, 250 m, 500 m, 1 km, and 2 km. A different grid size changes what counts as a space. A space counts only if some of its land is in use. In the example city, for instance, 11,264 of the 25,600 cells are in use, and the number of spaces falls to 2,884 at 250 m, 759 at 500 m, 209 at 1 km, and 60 at 2 km.

The five grid sizes show one side of what geographers call the MAUP, i.e., the dependence of a result on the size of the units used. The tessellation switch shows the other side of the MAUP, its dependence on where the lines are drawn. It replaces the squares with hexagons of the same area at each size from 250 m upwards, each made of the 62.5 m quarter-cells whose centres fall inside it.

A new city starts from one of five layouts: rings around one centre (Burgess, 1925); sectors along transport lines (Hoyt, 1939); several nuclei (Harris & Ullman, 1945); two residential districts separated by a buffer of chosen width; and a city laid out at random. The layout sets the city’s footprint—176 km² of the 400 km² square in the example city—and every cell inside the footprint is in use. Land outside it is vacant, as is the buffer’s band. How much land each use takes is set by the land-use shares, whose default is inspired by the URDPFI guidelines (Ministry of Urban Development, Government of India, 2015): housing 54 percent, sites of employment (i.e., commercial and industrial areas) 16, open space (i.e., recreational and green areas) 16, education 5, health and community 2, public amenities 2, and mixed use 5. The help text beside the shares says how the guidelines were adapted, and any of the predefined shares can be edited. The guidelines do not provide for mixed use explicitly, so the game gives it 5 percent, taken from the other uses in proportion to their shares. Mixed use therefore has land in every city, but it is not among the dimensions counted by default; ticking it under Dimensions adds it.

Where each use goes is decided by a propensity score for every cell. Housing fills the footprint first, and every other use then takes its share from the housing cells with the highest score, employment first, then open space, then the rest. The score adds up four terms along with a little noise. Each term answers a question a planner would ask about a facility, and each has its own slider under How amenities are placed:

  • Clustering: do the facilities come in groups, a campus or a market, say, or one at a time?
  • Centrality: are they drawn to the centre, or pushed out to the edge?
  • Co-location with housing: are they placed where residential density is highest, or kept away from it?
  • Hierarchy: is provision nested, a large centre with smaller ones inside each part of the city, and smaller ones again inside those?

The random layout ignores the score altogether and scatters every use. It is the benchmark every other layout is read against. In a random city every cell has the same expected access, except near the edge, where a cell has fewer neighbours; whatever inequality a random city shows comes from its edge and from chance. In our example city, laid out at random instead, the Gini is 0.127 at 125 m, against 0.220 for the rings.

The hierarchy slider draws on a cascade, essentially the p-model of Meneveau and Sreenivasan (1987) laid on a map. The square is divided into a five by five array of districts, each 4 km across. Each district is split into four quadrants, weighted in a random order; each quadrant is split into four again, with a fresh order; and so on, down to the 125 m cells. A cell’s cascade weight is the product of the weights along its path, and the slider sets how strongly a use follows it. By default it is at zero for every use, and §6 discusses what a city built entirely by the cascade does across the grid sizes.

Any cell can then be repainted. With Edit zoning set to Paint, a use picked from the palette is painted by clicking or dragging on the map, with a brush of 125 m, 250 m, or 500 m. Painting with Vacant clears a cell’s use, Undo steps back one stroke, and Reset returns to the generated city. Painting changes the city itself, whatever grid size is being shown, and the results are recomputed when the stroke ends.

4. Access

Every cell has a value of access to each dimension, measuring how much of the city’s sites of employment, say, or its open space that cell can get to. Distance, for now, is measured in a straight line between cell centres, and a cell’s distance to itself is taken as 0.5214 × 125 m, the mean distance between two random points in a square of that side. Street networks and travel speeds are left for a later version.

The default access measure is gravity, in the tradition of Hansen (1959). The access of cell i to dimension d adds up the area occupied by that dimension across the city, each site counted for less the further away it is,

Aid=∑jℓjdf(dij),

where ℓjd is the area occupied by dimension d in cell j, in m², dij is the distance between the two cells, and f is a decay. The default decay is exponential, with a half-life h set by the Reach slider,

f(d)=2−d/h,

so that at the default of 1.5 km a site 1.5 km away counts half as much as one next door, and a site 3 km away a quarter. Gaussian, step, and power forms are offered as alternatives. The step counts everything within the reach, while the power form, d−α, ignores the reach altogether and has its own exponent.

The two extremes of the reach slider denote the following: first, at zero reach a cell counts only its own land, i.e., access becomes the land-use map itself. A housing cell then scores zero on every dimension, more than half the spaces at 125 m score zero, and the Gini of the example city is 0.590. Second, at long reach every cell gets to almost everything, and access flattens: at 8 km the same Gini is 0.062, against 0.220 at the default. Under the step decay, which stops at the reach, a cell beyond the reach of every site of a dimension scores zero on it—what the thesis calls a spatial fragment—and the map hatches such cells. With a step of 1 km, for example, 6,683 of the 11,264 cells in use get to nothing on at least one dimension.

Gravity is only one type of access measure, and the user can switch to others using the drop-down menu:

MeasureAccess of cell i to dimension dWhat it assumes
Gravity (default)∑jℓjdf(dij)Everything counts, discounted by distance.
Cumulative opportunities∑jℓjd1[dij≤t]Everything within a threshold t, the reach, counts in full. It is gravity with the step decay, under the name planners know it by.
EndowmentℓidOnly the cell’s own land: gravity at zero reach.
Nearest facilityf(minj{dij:ℓjd>0})Only the nearest site counts.
Competition (Shen, 1998; Luo & Wang, 2003)∑jRjf(dij), with Rj=ℓjd/∑kPkf(dkj)Access net of the people who share it, the natural measure for jobs and school places. Pk is the population of housing cell k, whose density falls with distance from the centre.
Utility (Ben-Akiva & Lerman, 1985)1βln(1+∑jℓjde−βdij), with β=ln2/hThe expected value of the best of a choice among destinations.

For a single dimension, utility ranks the cells as exponential gravity does, since it is an increasing function of the same sum, and in the example city every one of the 11,264 cells in use keeps its percentile for employment. Yet the Gini of employment access at 125 m is 0.284 under gravity, and 0.021 under utility. This illustrates how, for the same layout of a city, the measured inequality can vary by the choice of access measure.

Before the dimensions are added, each is scaled—by default, by its largest value over the cells in use:

Ki(S)=∑d∈SAidmaxkAkd,

where S is the set of dimensions ticked and Ki is the cell’s connectedness. Each scaled dimension then lies between zero and one, and connectedness between zero and |S|, the number of dimensions. The alternative divides by the mean instead, so that each dimension averages one. For a dimension read alone the choice changes nothing, since the indices of §5 do not depend on units; in the sum, however, it is a weighting. The Gini of the composite is the sum, over the dimensions, of each one’s Gini times its Gini correlation with the composite (the correlation of the dimension with the composite’s ranks, relative to its correlation with its own), weighted by its share of the mean (Lerman & Yitzhaki, 1985). Scaling by the maximum therefore gives less weight to a dimension whose access is piled into a few places, while scaling by the mean weights every dimension alike. In the example city the Gini at 125 m is 0.220 under the maximum, and 0.227 under the mean.

The Coarse grids switch offers two ways of denoting a space that extends beyond one cell. Average, the default, gives each space the average connectedness of the nu cells in use inside it,

Ku=1nu∑c∈uKc,

so the measure stays fixed and only the space changes. Recompute, on the other hand, adds up the area each dimension occupies in a space first and computes access afresh between the centres of the spaces, so the measure changes with the grid size as well as the space. The two differ most when the reach is short against the grid size: in the example city the Gini at 2 km is 0.246 under Average and 0.275 under Recompute. Recompute is offered for squares only.

5. Inequality and welfare

Every space in use counts once: a space at the edge with a little housing weighs as much as a dense one at the centre. With N spaces in use and mean connectedness μ, the Gini index is half the mean absolute difference between two spaces, relative to the mean:

G=12N2μ∑u∑v|Ku−Kv|.

The other indices on offer are Theil’s index, the mean log deviation, GE(2) (i.e., half the squared coefficient of variation), and Atkinson’s (1970) index with an inequality aversion of one, along with two ratios, the Palma ratio and the ratio of the 90th to the 10th percentile. All of them except the two ratios respect the transfer principle: moving connectedness from a better-connected space to a worse-connected one never raises them.

Spatial welfare is read with Sen’s (1976) index,

WS=μ(1−G).

Draw two spaces at random, with replacement, and WS is the expected connectedness of the worse of the two; it rises when connectedness rises on average, and falls when the spaces grow apart. Under scaling by the maximum it has a ceiling, the number of dimensions, which would be reached only if every space were as well served as the best on every dimension, and the readout gives the share of the ceiling reached: 1.81 of 5 at 1 km in the example city, or 36 percent. Under scaling by the mean there is no ceiling, and the readout gives the level and the Gini instead.

The results come in three rows. The first is the city at the chosen size: the distribution of connectedness across its spaces, the Lorenz curve, and the indices. The second reads the same city at all five sizes. Here the inequality chart draws three nested sets of dimensions (the first ticked, the first three, and all of them) and, for the Gini, a faint fourth line that counts each space by its land in use instead of once—the benchmark of §6. The welfare chart sets each set’s welfare at 125 m to 100, because adding a dimension raises welfare by construction and levels are not comparable across sets; a switch shows the levels anyway.

Selecting a location on the map follows the space that contains it across the five sizes. Both the Gini and Sen’s index split exactly across the spaces,

G=1N∑uΔu2μ,WS=1N∑u2(1−Fu)Ku,

where Δu is the space’s mean absolute gap to the spaces of the city and Fu is its percentile. The panel for the selected space gives its share of each, relative to an average space, and its percentile at each size: the multiscalar reading of Olteanu, Randon-Furling, and Clark (2019), applied here to connectedness. A space at either end of the distribution holds a large share of the Gini and one in the middle a small share, while a space’s share of welfare weighs its connectedness by how far below the top it sits.

The third row has two cards. The first, Masked spaces, shows what an average hides. At the chosen size a space is masked when its average puts it in the middle fifth of spaces, while one of its 125 m cells is in the bottom fifth of all cells. The card draws every space as a line from its least connected cell to its best connected, placed at its percentile among the spaces, with the middle fifth shaded and the city’s bottom fifth of cells below a dashed line, so that a masked space is an orange line in the band that crosses below it. The map’s Masked spaces view marks the same bottom fifth of cells and outlines the same spaces. At 125 m a space is a single cell, and nothing can be masked. In the example city no space is masked at 250 m or 500 m either; one is at 1 km, and at 2 km eleven of the twelve spaces in the middle fifth are. The second card, What the selected location can reach, gives the selected cell’s access on each dimension, against the median cell in use.

6. What the game shows

Three results say what to expect of the charts in Reading the City; a fourth, on the welfare value of a site, is in §7.

Counting spaces lets the reading move. Suppose the grids nest, each coarse space takes the average of the cells inside it, and each space counts in proportion to its land in use. The coarse distribution is then a mean-preserving contraction of the fine one—each coarse value is an average of fine values—and no index that respects the transfer principle can be higher at the coarser size (Dasgupta, Sen, & Starrett, 1973). Coarsening, in other words, could only average inequality away. The faint line on the inequality chart is this benchmark. Counting each space once drops its last premise. At a coarser size a space is a larger unit that still counts once, and the set of spaces is a different set; inequality between spaces can therefore rise or fall with the grid size. In the example city it rises. The Gini goes from 0.220 at 125 m to 0.246 at 2 km with each space counted once, while the benchmark falls from 0.220 to 0.210. The rise comes from how the city’s spaces are defined.

What a dimension adds depends on how it ranks the spaces. Write the inequality cost in Sen’s index as C=μG, i.e., half the mean absolute difference between two spaces. For two dimensions x and y, since |a+b|≤|a|+|b|,

C(x+y)≤C(x)+C(y),soWS(x+y)≥WS(x)+WS(y),

with equality exactly when x and y rank every pair of spaces the same way (Lerman & Yitzhaki, 1985). Adding a dimension therefore raises welfare by at least its own welfare, and by more when it ranks the spaces differently from the dimensions already counted, since its unevenness then partly cancels theirs. At 1 km in the example city, employment alone gives a welfare of 0.283 and open space alone 0.487; together they give 0.788, more than the sum of the two. The Gini of the composite, being a relative index, can move either way: 0.303 for employment, 0.174 for open space, and 0.208 for the two together. The co-location slider is the dial for this effect. With open space placed beside housing rather than away from it (and no pull to the centre), welfare at 125 m is 2.295 against 1.956, and the Gini 0.186 against 0.199.

A cascade city loses the same inequality at every step. Suppose the total area occupied by a dimension is spread over the cells in proportion to the cascade weights of §3, with weights pk=(1+aek)/4 on the four quadrants at each split and e=(1,13,−13,−1), and suppose it is read at zero reach. Theil’s index then falls by the same amount at every doubling of the grid size,

T(g)−T(2g)=ln4−H(p),H(p)=−∑k=14pklnpk,

which at a=0.6 is 0.106 at every step from 125 m to 2 km. It follows that equalising the top j of k levels of the cascade (equalising budgets between districts, say) removes exactly j/k of the index, and leaves the rest inside the districts. On a logarithmic axis of grid size a cascade city draws a straight line, and a city with a characteristic scale, such as a buffer of fixed width or a centre of fixed size, bends it at that scale. The game gives each cell a single use, so a city built with the hierarchy slider only approximates the cascade. The straight line is what a bend is read against; what a bend would look like in a real city is left for later.

7. Spatial interventions

In the second mode the user makes a planning decision and sees what it does to inequality between spaces and to spatial welfare, and whether the answer depends on how the city is read. There is no budget: the cost of a decision is its consequence for inequality and welfare, read at every grid size under the levers set in Reading the City.

An intervention is a school, a clinic, a park, a jobs centre, a public amenity, or mixed use, placed by clicking the map on a square block that runs from one 125 m cell (15,625 m²) to 1 km². By default it is built on vacant land only, and a block that is partly in use is built on its vacant part, with the ledger saying how much of the block was left as it was. In a generated city the vacant land lies beyond the footprint and in the buffer layout’s band, so an intervention on vacant land extends the city at its edge; painting cells vacant in Reading the City makes sites inside it. The Existing uses switch lets an intervention replace a use instead. The land given up then enters the accounts like any other change, the converted cells are hatched on the map, and the ledger lists what each intervention took.

Under the gravity measure the change an intervention makes is exact and local. Adding Δℓ of a dimension at cell j raises the access of every cell i by Δℓf(dij); the changes from several interventions add up; and every scaled value moves by that amount over the dimension’s maximum, unless the intervention raises the maximum itself, which scales every cell down. Land brought into use also adds spaces. A space that was wholly vacant enters the count once it holds any land in use, and a new space at the edge enters poorly connected. An intervention that raises the access of every existing space can therefore still lower welfare between spaces, and whether it does can depend on the grid size.

The verdict table reports, at every grid size, the change in welfare, in the Gini, and in Theil’s index from everything placed, against the city before it, along with the number of spaces before and after, and it calls the verdict better on both counts, worse on both, or mixed. One of the Guide’s questions opens on a park at the northern edge of the example city, on a block of 250,000 m² of which 218,750 m² is vacant and built. Read at 250 m and 500 m, the park makes the city worse on both counts, and read at 125 m it lowers welfare while barely moving the Gini; read at 1 km and 2 km, it makes the city better on both. On the spaces that were in use before it, though, welfare rises by 0.09 to 0.12 percent and the Gini falls at every grid size. The flip comes entirely from the spaces the park adds—14 at 125 m, 6 at 250 m, and 2 at 500 m, each at the edge and poorly connected—while at 1 km and 2 km the park falls inside spaces already in use and adds none.

The Where next view shades vacant land by the welfare value of placing the chosen use there, and Best next site places it where that value is highest. The value is a first-order approximation. The derivative of Sen’s index with respect to one space’s connectedness is 2(1−Fi)/N, so to first order the value of a site j is

Vj=2N∑i(1−Fi)ΔKi,

i.e., the gain the site gives each space, weighted by how far below the top that space already is. The least connected space’s gain counts nearly twice the average, and the best connected’s nearly nothing; these are the rank weights of Sen’s welfare function (Weymark, 1981). The same formula explains why maximising total access, with every space weighted one, and maximising welfare choose different sites: a site where access is already high adds to the mean, and little to welfare. The page adds two terms to the sum, one for the new land joining the space it sits in and one for the dimension’s maximum rising, and it leaves out spaces changing rank and new spaces entering the count, of which the second is the larger error. Where the block adds no new space (for a school of 62,500 m², at 500 m and above), the value is within 1 to 6 percent of the exact change on three test cities. At 125 m and 250 m, where the school’s own land becomes new spaces, it overstates the exact change by 18 to 51 percent. The shading, then, is a guide to where to look—close at the coarser sizes and rough at the finest—while the verdict table is exact. It is computed for the gravity family, in the Average mode. In the example city the best site for such a school is the same corner of vacant land read at 125 m and at 1 km, and a different one read at 2 km.

Where several interventions are placed, two more questions arise: how does the credit divide among them, and does the order of building matter? The end point is the same in every order, but the path to it is not. Each intervention’s share is its Shapley value (Shapley, 1953; Shorrocks, 2013), i.e., its contribution to welfare averaged over every order in which the interventions could be built,

ϕk=1m!∑π[WS(Bkπ∪{k})−WS(Bkπ)],

where m is the number of interventions and Bkπ the set built before k in order π. The shares add up to the total change, but the effects of each intervention alone do not, because each changes what the others add. Under the gravity measure the changes in access themselves add up, so the shares depart from the effects alone only where interventions re-rank spaces, add spaces or raise a dimension’s maximum. The chart of welfare along the programme draws the path in every order, and marks the order as placed, the greedy order (which at each step builds whichever remaining intervention raises welfare most), and the best and worst orders by average welfare along the way. Shares and orders are computed for up to six interventions, or 720 orders; beyond six, the verdict is still given.

A second question opens on three projects in the example city: the same park, a school at the eastern edge, and a jobs centre of 1 km² at the north-eastern edge. Together they make the city better on both counts read at 125 m and 2 km, and mixed at 250 m, 500 m, and 1 km, while on the spaces in use before them welfare rises by 0.83 to 0.92 percent and the Gini falls at every size. The greedy order differs from the order placed, and it changes with the grid size: the school, the jobs centre, and then the park at 125 m, 250 m, and 2 km; the jobs centre, the school, and the park at 500 m; the park, the jobs centre, and the school at 1 km. The school on its own adds 0.15 percent to welfare read at 125 m, and takes away 0.47 percent read at 1 km. Whether the same arithmetic should credit welfare to the dimensions themselves, rather than to interventions, is a question left open.

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