The Infinite Card Game

  • 02.28.2018
  • etc

Kira and I were in Australia most of last month, and near where we were staying in Melbourne was a game shop. We had a free Friday night so I stopped by for my first Magic: The Gathering (MTG) draft event, and it got me thinking about designing card game systems.

MTG is a collectible card game with a great deal of strategic depth. Games with large state spaces like Chess and more recently Go have been more-or-less "solved"1; The state space of MTG is certainly orders of magnitude larger than Chess and Go, given the massive back catalog of cards (going back to 1993!)2 and the ever-growing number of interactions between them. Though the state space of Starcraft is likely larger (and people are working on "solving" it), to my knowledge MTG has not yet been solved in this sense.

For those unfamiliar with MTG, it's played between two or more players and involved constructing a deck of cards around a particular strategy. Some strategies may emphasize fast, aggressive plays ("aggro") which, if failing to win quickly, lose steam in longer matches. Others may focus on slowing opponents down by stopping plays short or making actions more expensive ("control"). And there are other strategies still.

MTG has a variety of game formats which govern how decks are constructed and can affect other game rules. These formats are broadly divided into Constructed and Limited formats. Constructed formats are where players carefully design and assemble their decks in advance. This gives plenty of space for creative, expressive strategies since players have a large pool of cards to select from. In contrast, Limited formats mean that players are given a small amount of random cards drawn from a set of cards and need to assemble a deck on-the-spot (a process called "drafting").

I've mostly played Constructed formats, but now that I've tried Limited a bit more I'm coming to prefer the randomness and uncertainty of Limited formats. In Limited you have to think on your feet more, design your deck more delicately (you aren't sure what to expect from your opponents), and work within a tighter set of constraints. It makes for more challenging and exciting games.

The problem with Limited is that each set has roughly 200-300 cards. After a few games you'll be familiar with every card and players have learned the strategies that work best within that set. Games start to get formulaic and stale. It loses that sense of uncertainty that makes Limited exciting in the first place. It isn't until the next set is released, with new cards and abilities, that things are interesting again.

These sets are carefully designed such that the cards have enough variation to keep things interesting, but not so much that they're totally incoherent (Mark Rosewater, the lead designer of MTG, has a great podcast delving into this design process). And they are meticulously balanced so no strategy is strictly better than any others.

That being said...I wonder if there's a way to design an infinite set, i.e. a dynamic and self-adjusting process which outputs a stream of cards to draw from for a never-ending Limited format. Such a system would need some rule scaffolding or framework (doesn't have to be MTG's) from which it can derive new mechanics and costs (some quantifier of their power), and then generate a balance of cards over some probability distribution.

For example, a core mechanic in MTG is that you have creatures that can attack opponents to damage them. Players can use their own creatures to "block" opponents' attacking creatures. These creatures have some cost to play ("cast") them; generally stronger creatures have the drawback of costing more to cast. Sometimes they may have other abilities which make them more or less versatile, which is compensated by a respective increase or decrease in casting cost. Sometimes you have creatures which are disproportionately cheap in casting cost for their strength, but these are rare.

Let's say the game for this infinite draft system has just this simple attacking-creature mechanic. Our creatures have only a strength and a casting cost. Generally, the stronger the creature, the higher it's casting cost. But not always -- on rare occasions we might have a strong creature that's a bit cheaper than normal. Finally, we add the additional constraint that weaker creatures are more likely, so we emphasize strong creatures as a more notable event.

What we're essentially saying is that the casting cost of a creature is dependent on its strength (and vice versa), which we can represent as a simple Bayes net:

G strength strength casting_cost casting_cost strength->casting_cost

When we want to create a new card, we can first sample its strength, then sampling a casting cost depending on the value we sampled for its strength.

We want a creature's strength to be a positive integer, say in the range $[1, 12]$. So we want a discrete probability distribution with finite support. We could use a Beta-binomial distribution, e.g. $\text{BetaBinomial}(\alpha=2, \beta=6, n=12)$, which has the properties we want:

$\text{BetaBinomial}(\alpha=2, \beta=6, n=12)$
$\text{BetaBinomial}(\alpha=2, \beta=6, n=12)$

Here creatures will tend to have a strength somewhere in $[1, 4]$ and very rarely above 6. Then we can do something similar with casting cost, except that it's dependent on the strength.

This is an extremely simple game and so not a very interesting one. We'd want to add in additional abilities that interact in interesting ways. For example, in MTG the "flying" ability makes a creature blockable only by other creatures with flying. So we can add in some small probability of a creature gaining flying, and have that also affect the casting cost's distribution.

MTG's Flying mechanic

A really nice version of this system is one where you can pass in an arbitrary network relating costs and abilities (a more complex example of the one above), and it would output card descriptions in some interchange form (e.g. JSON), and you can use that to print cards with whatever design you wanted.

A few years ago I prototyped a similar system of cost-based card generation, which incorporated different card types beyond creatures (which I called "units"), additional abilities, and procedurally-generated names.

An example generated card
An example generated card

This prototype doesn't incorporate intra-card balance beyond what falls out of cost-balanced cards. The relative effectiveness of various abilities are really hard to objectively quantify, since their costs are really relative to what the dominant strategies are, i.e. the metagame. So ideally this infinite draft system not only generates balanced cards but also tracks how people are playing them to learn which strategies seem over or underpowered, and correspondingly tweaks the costs of abilities related to those strategies.

The generation features I just described are more about balance but another interesting feature would be introducing a balance-drift so that gameplay never stagnates in an equilibrium of strategies. Perhaps once balance is achieved the system can gradually and temporarily bias the game towards different strategies to encourage different kinds of gameplay. That way there'd be an ebb-and-flow that keeps things interesting in the aggregate and subtly changing the overall feeling of the game.

For example, if the system sees that players almost exclusively play low-strength, cheap creatures, and almost no larger creatures, maybe it will start to slightly cheapen the larger creatures so they see more play. That in turn may cause a new strategy to dominate, maybe a slower gameplay with larger creatures, and eventually a different change would be introduced to shock the system to a different strategy equilibrium.

I've given a very hand-wavy outline of this system here and as described it by no means would match MTG's hand-designed depth and complexity. But I do like the idea of a general system where you input some mechanics and it outputs a series of cards to play a game with. You could, perhaps, model many different systems via a card game format this way.

  1. Not solved in the proper sense, but human players are reliably bested by computer players. 

  2. Based on's data, there are 18,191 unique cards as of Jan 21, 2018. 

Simulation & Understanding

  • 01.26.2018
  • etc

This is the written reference for a talk at the Digital Humanities + Data Journalism Symposium (the slides are here). A class based on the topics presented here is now at the New School.

The world is very complex of course and seems inexorable in its acceleration of this complexity. Problems we face expand in difficulty in that they 1) grow to impossible scales but 2) simultaneously become increasingly sensitive to the smallest detail.

In 1973 Rittel & Webber described such problems as "wicked". Their paper Dilemmas in a general theory of planning attempts to enumerate the characteristics of such problems, which include but are not limited to:

  • difficult to agree on the problem (or whether there's a problem at all)
  • there is no comprehensive perspective
  • wicked problems are interconnected
  • no way to verify or experiment solutions
  • there may be (very) delayed feedback loops
  • every wicked problem is unique
  • are nonlinear (outputs are disproportionate to the inputs)
  • cannot be reduced to its parts (i.e. emergent)
  • they are massive & intimidating

For example, consider the issue of climate change:

  • Some people don't agree that climate change in general is an issue, some people think it's happening but it isn't caused by people, and so on.
  • No individual or organization really has a complete understanding of the issue. There isn't even a single unifying perspective since it affects different communities in very different ways.
  • It's deeply entangled with many other problems, e.g. displacement of populations, economic crises, ecological destabilization, globalization, etc
  • We can't really experiment with solutions because the problem is so complex it's unlikely we'll ever be able to establish any convincing causal relation between the intervention and the outcome.
  • Any intervention may change the nature of the problem such that we'd be dealing with a completely different problem, so any learnings from the intervention might be obsolete.
  • The feedback delay between an intervention and its outcome may not be felt until years or decades later.
  • Nonlinear dynamics like tipping points
  • Global scale, many moving parts, many different interests, varying degrees of cooperation, is, as Emily Eliza Scott puts it, "multiscalar, multitemporal, multidimensional, and multidisciplinary"

I was listening to an interview with the philosopher Frithjof Bergmann who's involved in a project called New Work. In the interview he put forward a claim that every problem can be traced back to our relationship with work. I was skeptical at first but after some thought there is a plausible causal chain between jobs and climate change. Employment and "job creation" are of course major political issues. Some part of this probably involves manufacturing (though perhaps in decreasing amounts in more developed economies) which requires stoking the flames of consumption to keep demand up; this kind of production drives energy usage and pollution and waste which end up contributing to climate change.

It seems that although technology is often responsible for the exacerbation of such problems and their complexity, there should also be some means of technologically mitigating them. Not solve them outright -- that takes much more than just new tech -- but at least better equip us to do so ourselves.

I think simulation has some potential to help here.

Simulation requires that we formalize our mental models; they are a "working memory" that support larger such models, the process of designing a simulation can lead to increase understanding, at the very least identify gaps in our knowledge, and they provide a canvas for us to sketch out possible alternatives.

What simulation?

There are many different forms simulation can take. Here I'm referring to both agent-based modeling and system dynamics (i.e. causal loop diagrams) and focusing on their applications to social systems ("social simulation").

Agent-based modeling (ABM) is a method which models systems via modeling its constituent elements (e.g. people) and describing how they interact. It's often contrasted with equation-based modeling (i.e. differential equations) which is far more common but limited by assumptions hard-coded into it:

  • usually assumes homogeneous population
  • usually continuous variables
  • cannot observe local detail, generally focuses only on aggregate phenomena
  • top-down (from global perspective as opposed to individual)
Agent-based modeling
Agent-based modeling

Agent-based modeling, on the other hand, supports heterogeneity, is flexible in the variables it supports (discrete or continuous), provides views at both the local and global level, allows for learning/adaptive agents, and more.

System Dynamics
System Dynamics

System dynamics models systems by stocks (quantities that change over time) and flows (rates of change relating stocks). System dynamics is like a higher-level abstraction over agent-based modeling, in which we don't represent individuals in a population explicitly but instead as aggregates.

An especially appealing aspect of both ABMs and system dynamics that they are much more intuitive than differential equations (I feel bad leveraging our education system's inability to make people comfortable with math, but oh well), so they are more immediately accessible to a wider audience.

Differential equations
Differential equations

They are also exceptionally flexible in what they can represent, ranging from ant colonies to forest fires to urban segregation to religious experiences.


Language Evolution Simulation (Fatih Erikli)
Language Evolution Simulation (Fatih Erikli)

This simulation shows how a language could evolve over time by exchanging vocabulary with random mutations.

Differentiation without Distancing. Explaining Bi-Polarization of Opinions without Negative Influence (Michael Mäs, Andreas Flache)
Differentiation without Distancing. Explaining Bi-Polarization of Opinions without Negative Influence (Michael Mäs, Andreas Flache)

This study looks at how opinions may become polarized without requiring negative influence ("individuals’ striving to amplify differences to disliked others"). In the image you see everyone starting from the same intermediate position and eventually becoming bi-polarized.

Learning To Be Thoughtless: Social Norms and Individual Computation (Joshua M. Epstein)
Learning To Be Thoughtless: Social Norms and Individual Computation (Joshua M. Epstein)

This paper models how individuals not only learn which social norms to adopt but also learn how much to think about adopting a social norm (e.g. if a social norm is prevalent enough we may adopt it without consciously deciding to). On the left, the blue and yellow colors indicate a binary norm, and the image depicts, from top to bottom, how the adoption of those norms ebb over time. There are always little pockets of resistance (e.g. yellow amongst majority blue) that eventually bloom into the new majority and eventually fall to the same fate.

From Private Attitude to Public Opinion: A Dynamic Theory of Social Impact (Andrzej Nowak, Jacek Szamrej, Bibb Latané)
From Private Attitude to Public Opinion: A Dynamic Theory of Social Impact (Andrzej Nowak, Jacek Szamrej, Bibb Latané)

This model describes how opinions shift according to social influence. I really like that this model was printed out like this.

A Distributed Platform for Global-Scale Agent-Based Models of Disease Transmission (Jon Parker, Joshua M. Epstein)
A Distributed Platform for Global-Scale Agent-Based Models of Disease Transmission (Jon Parker, Joshua M. Epstein)

This is a massive epidemiology model at a self-described "global-scale" of 6 billion agents, running in reasonable time. This particular study models the H1N1 (swine flu) virus.


The Simulados project models hunter-gatherer groups in India from the years 10000BC to 2000BC. It's part of a broader initiative, Simulpast, which "aims to develop an...interdisciplinary methodological framework to model and simulate ancient societies".

Cities: Skylines (Paradox Interactive)
Cities: Skylines (Paradox Interactive)

Paradox Interactive's Cities: Skylines (similar to SimCity) models their citizen's activities as an agent-based model.

MONIAC (Monetary National Income Analogue Computer)
MONIAC (Monetary National Income Analogue Computer)

A non-digital example, MONIAC is a hydraulic system dynamics model of the UK economy.

System dynamics applied to religious experiences
System dynamics applied to religious experiences

Arlen Wolpert was an engineer and a scholar of religion who used system dynamics to describe a powerful religious experience he had. I like this example because it demonstrates the method's breadth of applicability, and it's really curious to see how he framed the different components of the experience.

Agent-based modeling in more detail

Most of my interest is in agent-based modeling rather than system dynamics, so I'll go into a little more depth here by walking through Schelling's model of segregation. This model was developed to demonstrate that even when individuals have "tolerant" preferences about who they live near, neighborhoods may in aggregate still end up segregated.

Every agent-based model has two core components: the agents and their environment.

In Schelling's model, agents are typically modeled as households, placed in a 2D grid environment. This is similar to cellular automata, like the Game of Life; CAs can be thought of as a subset of ABMs.

Each household has a "type" (typically representing race) and a tolerance threshold $t$. This threshold defines the minimum amount (percent) of similar neighbors required for the household to be "satisfied". An unsatisfied household will move to a random unoccupied spot in the grid.

Schelling basics
Schelling basics

Here we consider Von Neumann neighbors; that is, the spaces to the north, east, west, and south of a cell. This is in contrast to a Moore neighborhood, which includes all eight surrounding cells.


Here's how an Schelling agent would be defined in System Designer (still under heavy development! The API will change!), an agent-based modeling framework Fei Liu and I are currently developing:

import syd
import numpy as np

class SchellingAgent(syd.Agent):
    state_vars = ['type', 'position', 'satisfied', 'threshold']

    async def decide(self):
        neighbors = await
        same = np.where(neighbors == self.state.type)[0]
        satisfied = same.size/neighbors.size >= self.state.threshold
        self.submit_var_update('satisfied', satisfied)
        if not satisfied:
            await, self.state.type, self.addr)

And here's how the Schelling environment (world) would be defined:

import random

class SchellingWorld(
    state_vars = ['grid', 'vacancies']

    async def decide(self):

    async def queue_random_move(self, position, agent_type, agent_addr):
        if self.state.vacancies:
            new_pos = self.state.vacancies.pop(random.randrange(len(self.state.vacancies)))
            self.submit_update(self.move_position, position, new_pos, agent_type)
            agent_proxy = await self.container.connect(agent_addr)
            await agent_proxy.submit_var_update('position', new_pos)

    def update_vacancies(self, state):
        state.vacancies = list(self.vacancies(empty_val=0))
        return state

Then, to run the simulation:

from itertools import product

n_types = 2
n_agents = 50
width = 10
height = 10
grid = np.zeros((width, height))
positions = list(product(range(width), range(height)))

# value of 0 is reserved for empty positions
types = [i+1 for i in range(n_types)]

sim = syd.Simulation(node)
world, world_addr = sim.spawn(SchellingWorld, state={'grid': grid, 'vacancies': []})

for _ in range(n_agents):
    position = positions.pop(0)
    type = random.choice(types)
    agent, addr = sim.spawn(SchellingAgent, state={
        'type': type,
        'position': position,
        'satisfied': 0.,
        'threshold': threshold
    }, world_addr=world_addr), position))

    for report in sim.irun(n_steps, {
        'n_satisfied': (lambda ss: sum(s.satisfied for s in ss if hasattr(s, 'satisfied')), 1)
        print('mean satisfied:', report['n_satisfied']/n_agents)
Schelling's model of segregation
Schelling's model of segregation

One key question when developing an ABM is: what constitutes an agent? Is it an individual, a family, a firm, a city, a nation, a cell, a grain of sand, etc? That is, what scale is relevant for the simulation? Of course, this depends on the exact problem at hand, but one idea which is particularly interesting is multiscale simulations.

Level-of-Detail in video games
Level-of-Detail in video games

In video games there is this notion of "level-of-detail" (LOD). When you are very close up to an object in a game (say, a rock) the engine loads a very high-resolution texture or more complex mesh so you can see all its detail. If however you're looking at the rock from far away, you won't be able to see the fine details and so it would be a waste to load such a high-resolution texture. Instead, a low-resolution texture or lower-poly mesh can be loaded and you won't be able to tell the difference.

Maybe the same idea can be applied to agent-based modeling. If you are "zoomed out" you don't see the actions of individuals but instead, for example, cities. It's not as straightforward as it sounds but it has a lot of appeal.

ABMs and Machine Learning

How does machine learning factor into agent-based models? Many agent-based models are fairly simple, relying on hand-crafted rules, but they can get very sophisticated too.

For example, in our project Humans of Simulated New York, Fei and I wanted to generate "plausible" simulated citizens. To do so we used ACS PUMS data (individual-level Census data) to learn a Bayes net (a generative model) which would capture correlations present in the data (this generation code is available on GitHub).

For example we could generate a citizen "from scratch":

>>> from people import generate
>>> year = 2005
>>> generate(year)
    'age': 36,
    'education': <Education.grade_12: 6>,
    'employed': <Employed.non_labor: 3>,
    'wage_income': 3236,
    'wage_income_bracket': '(1000, 5000]',
    'industry': 'Independent artists, performing arts, spectator sports, and related industries',
    'industry_code': 8560,
    'neighborhood': 'Greenwich Village',
    'occupation': 'Designer',
    'occupation_code': 2630,
    'puma': 3810,
    'race': <Race.white: 1>,
    'rent': 1155.6864868468731,
    'sex': <Sex.female: 2>,
    'year': 2005

Or specify some value, e.g. generate me a person from a particular neighborhood. It's not perfect, but it worked well enough for our purposes.

Similarly, we used a logistic regression model trained to predict the probability of two people being confidants (from Social Distance in the United States: Sex, Race, Religion, Age, and Education Homophily among Confidants, 1985 to 2004, Jeffrey A. Smith, Miller McPherson, Lynn Smith-Lovin. University of Nebraska - Lincoln. 2014), using that to bootstrap the simulation's social network.

HOSNY also included businesses which produced and sold goods (including consumer good firms, which produced food, capital equipment firms, and raw material firms). They needed to decide how much of a good to produce and what to price it at. It's quite difficult to come up with a set of rules that will work well here, so we implemented them with Q-learning (as described in "An agent-based model of a minimal economy, a basic reinforcement learning algorithm (reinforcement learning algorithms are the class of techniques used by AlphaGo, for instance).

Similarly, we implemented the government as a reinforcement learning agent, using Q-learning to decide on tax rates.

These are just a few examples of where the fields of simulation and machine learning overlap.

A systems language

In The Image of the City, urban planner Kevin Lynch interviews people in multiple cities to understand how they form "mental maps" of their cities.

We rely on similar mental maps when navigating non-corporeal landscapes. In Postmodernism Fredric Jameson references The Image of the City when calling for improved versions of this "cognitive mapping". As he describes it:

a situational representation on the part of the individual subject to that vaster and properly unrepresentable totality which is the ensemble of society's structures as a whole

Jeff Kinkle quotes Jameson's discussion of alienation and mobility in particular:

Lynch taught us that the alienated city is above all a space in which people are unable to map (in their minds) either their own positions or the urban totality in which they find themselves. […] Disalienation in the traditional city, then, involves the practical reconquest of a sense of place and the construction or reconstruction of an articulated ensemble which can be retained in memory, and which the individual subject can map and remap along the moments of mobile, alternative trajectories.

Kinkle further notes:

an inability to cognitively map the mechanisms and contours of the world system is as debilitating politically as being unable to mentally map a city would be for a city dweller.

Can we, again quoting Kinkle quoting Jameson, develop a way of thinking about systems "so vast that [they] cannot be encompassed by the natural and historically developed categories of perception with which human beings normally orient themselves"? Is there any way to better represent these complex systems in which we are submerged, along with their incredibly nuanced causal chains, in a way that makes them more manageable and navigable?

How do we do so without relying on the crutch of conspiracy theories, inappropriately reducing the complexity of a situation in order to make it more comprehensible?

For cities we now have Google Maps. What about for these social systems?

This seemingly intractable problem has sometimes been called the "crisis of representation" or "representational breakdown". Speaking of climate change, but equally applicable to any similarly overwhelming problem:

What kinds of representation (visual and otherwise) are adequate to the task of conveying climate change, and perhaps most importantly, to stemming dysphoric paralysis while triggering critical thinking and action? (Archives of the Present-Future: On Climate Change and Representational Breakdown, Emily Eliza Scott)

There are many different ways of representing something like climate change. There isn't really a right one; they all have different effects.

Maldives Underwater Cabinet Meeting protest
Maldives Underwater Cabinet Meeting protest

For example: depicted here is a protest in the Maldives (which is at a very high risk of being submerged in the near future) where President Nasheed and his cabinet sent an SOS message to the UN climate change summit from underwater.

But what if we want to better comprehend the "total" system, with its interconnections, insofar as it even can be represented?

The Network of Global Corporate Control (Stefania Vitali, James B. Glattfelder, Stefano Battiston)
The Network of Global Corporate Control (Stefania Vitali, James B. Glattfelder, Stefano Battiston)

When it comes to systems the network is perhaps the most common type of representation. Networks are a fantastic abstract form but perhaps ineffective in communicating a system's less technical aspects. It is capable of describing the full system (in terms of its relational structure) but even this may serve only to intimidate. Such intimidation may be the only affective effect of such a representation, failing to depict the human experiences that emerge from and compel or resist the system.

For example, Bureau d'Etudes' An Atlas of Agendas is a meticulously researched mapping of organizations throughout the world, so detailed that a magnifying glass accompanies the book (they were originally produced at a larger scale on walls). It is undeniably impressive but induces vertigo and helplessness in its unmanageable intricacy.

_An Atlas of Agendas_
An Atlas of Agendas

This kind of network mapping also doesn't capture the dynamism of such a system.

In contrast to network mapping, Cartographies of the Absolute (the title is a direct reference to the aforementioned Jameson book), Jeff Kinkle and Alberto Toscano examine how narrative fiction provides a more experiential representation, i.e. an aesthetics, discussing for example The Wire and cinema from the peak of New York City's "planned shrinkage" (such as Wolfen). Kinkle describes the Cartographies of the Absolute project as such:

The project is an investigation into various attempts in the visuals arts to cognitively map the contours of the current world system and its influence on various local settings...focus[ing] on the techniques artists, filmmakers, and cartographers use to portray global forces in all their complexity without being merely didactic or reverting to antiquated aesthetic models.

For example, the chapter on The Wire, "Baltimore as World and Representation":

[The Wire] address[es] the drug trade, de-industrialization, city hall, the school system and the media...mapped both vertically (making internal hierarchies explicit) and horizontally (tracking their entanglements and conflicts with other 'worlds' spread throughout the city...we to see how each world affects the ones around it...

This type of representation is more visceral and emotional but may obscure the totality of the underlying system (although The Wire seems to do a good job).

Is there some kind of representational "language" that falls in between? In Iain M. Bank's Culture novels he describes a language called "Marain" which "appeal[s] to poets, pedants, engineers and programmers alike". Is there an equivalent here, one that can similarly bridge the technical and the emotional representations, allowing the expressiveness of both?

The Culture's Marain
The Culture's Marain

In Ted Chiang's The Story of Your Life (which is being made into a movie now), a researcher studies an alien language which has two components - spoken (Heptapod A) and graphic (Heptapod B). The aliens tend to prefer the graphic one because the spoken language is limited by the arrow of time - it goes in one way and you have to "wait" to hear all or most of the sentence to grasp the meaning. With the graphic language, which I picture as large mandala-like ideograms, the whole sentence is consumed at once, there in front of you, which, for these aliens at least, lead to an instant and more total understanding of what was being communicated. Is there an analog of that for us?

Heptapod B as depicted in _Arrival_
Heptapod B as depicted in Arrival
Heptapod B more as I imagined it (depicted: Bismillah)
Heptapod B more as I imagined it (depicted: Bismillah)
Heptapod B more as I imagined it (depicted: a mandala)
Heptapod B more as I imagined it (depicted: a mandala)

In Thinking in Systems, Donella H. Meadows makes a similar observation:

there is a problem in discussing systems only with words. Words and sentences must, by necessity, come only one at a time in linear, logical order. Systems happen all at once.

... Pictures work for this language better than words, because you can see all the parts of a picture at once.

The possible impact of such a language in affecting our way of thinking can be seen in the effect of transitioning from Roman numerals to Hindu-Arabic numerals. Wilensky & Papert refer to these numeral systems as an example "representational infrastructure", calling this shift a "restructuration" which ultimately led to arithmetic being accessible to essentially everyone. Roman numerals were not amenable to - they did not afford, to use a design term - certain mathematical operations that are now second-nature to many; with Roman numerals only a specially-trained group of people could do such arithmetic. Wilensky is the creator of NetLogo, a very popular framework for agent-based modeling, so he seems to be positioning ABMs as a restructuration.

So what kind of language - what kind of representational infrastructure - is needed for everyone to be comfortable with complex systems?

Social simulation itself may not be this language, but it seems inevitable that it will at least be the engine for it. Perhaps we can build richer experiences on top of this engine.

A canvas for alternatives

Simulation can build an appreciation of the arbitrariness of the present, insofar as things could have played out different and still can play out differently ("another world is possible").

"Another world is possible"
"Another world is possible"

So many terrible systems and practices are maintained under the pretence of being "natural" (e.g. the infuriating "human nature" argument), so there we often see rhetoric justifying such arrangements along those lines; they are framed as spontaneous and organically-formed. They usually aren't, and even if they are, that's not a very convincing justification for their continuation.

This speculative aspect of simulation is really appealing; it carves out this space for experimentation and imagining utopias.

Humans of Simulated New York
Humans of Simulated New York

Fei Liu and I approached our project Humans of Simulated New York in this way. HOSNY was implemented as a simple economic simulation primarily based on New York City Census data, but we allowed people to play with the parameters of the world along three axes: food/agriculture, health, and technology/labor and see how the lives of citizens change in each alternate world.

HOSNY world specification
HOSNY world specification

There is a parallel here to cartography - in the late 1960's and early 1970's The Detroit Geographical Expedition and Institute (DGEI) focused on producing "oughtness maps": "maps of how things are and maps of how things ought to be." (The Detroit Geographic Expedition and Institute: A Case Study in Civic Mapping, Catherine D'Ignazio). William Bunge, one of the co-founders of the DGEI (the other was Gwendolyn Warren), noted (probably referencing Marx):

Afterall, it is not the function of geographers to merely map the earth, but to change it.

I feel a similar sentiment can be said about simulation. Perhaps we can see the emergence of simulation equivalents of the practices of "countermapping" and "radical cartography".

Detroit money transfers (Fitzgerald: Geography of a Revolution, William Bunge)
Detroit money transfers (Fitzgerald: Geography of a Revolution, William Bunge)
Detroit Recommended Redistricting to place black children under "sympathetic authority"
Detroit Recommended Redistricting to place black children under "sympathetic authority"

Exploring policy-space

This kind of systems mapping and modeling has more "practical" applications. For example, identifying points of influence/leverage points within a network.

One area of interest is in public policy modeling. The way policy is handled now seems totally nonsensical, and of course there is more than just ignorance about consequences to blame (i.e. politics), but I wonder if this sort of modeling wouldn't have at least some positive impact.

How to defeat ISIS
How to defeat ISIS
The Cycle of International Stupidity
The Cycle of International Stupidity

A comprehensive textbook on the topic, Modeling Complex Systems for Public Policies, was published last year so it seems like a fairly active area of research. In the text researchers discuss modeling transit systems, education systems, economies, cities,

_Modeling Complex Systems for Public Policies_
Modeling Complex Systems for Public Policies

One active and ambitious initiative in this area is the Eurace@Unibi model. It's a joint venture involving multiple European research institutes and seeks "to provide a micro-founded macroeconomic model that can be used as a unified framework for policy analysis in different economic policy areas and for the examination of generic macroeconomic research questions." (The Eurace@Unibi Model: An Agent-Based Macroeconomic Model for Economic Policy Analysis) The goal is to develop an economic model that has more representative power than existing methods (i.e. dynamic stochastic general equilibrium, or DSGE models, which, as the name suggests, are focused on equilibriums) and so would be a better tool for determining economic policy. I don't have enough experience with the model or economics itself to really comment on its relative effectiveness, but it's interesting nonetheless.


In urban planning a similar initiative exists, UrbanSim, an "open source simulation platform for supporting planning and analysis of urban development, incorporating the interactions between land use, transportation, the economy, and the environment."


There is an important distinction that needs to be brought up here: speculation vs prediction. Such an application of social simulation is not about predicting the outcome of a policy but rather about understanding what the possibilities are.

I try to mention this point a lot because social simulation is often misconstrued as something like Asimov's "psychohistory", which is described as a means of predicting the trajectories of civilizations.

Bret Victor, in What can a technologist do about climate change?, puts it this way:

Modeling leads naturally from the particular to the general. Instead of seeing an individual proposal as “right or wrong”, “bad or good”, people can see it as one point in a large space of possibilities. By exploring the model, they come to understand the landscape of that space, and are in a position to invent better ideas for all the proposals to come. Model-driven material can serve as a kind of enhanced imagination.

Scott E. Page makes a similar caveat in the preface to Modeling Complex Systems for Public Policies:

Complex systems do not represent a silver bullet, but another arrow in the policy maker's quiver. More accurately, all of these tools put together can be thought of as multiple imperfect arrows that provide insight into what is likely to happen, what could happen, and how what happens might spill into other domains.

That is, simulation is more about counterfactuals (what-ifs), laying out the possibility space and shining a light on more of it than we would without such tools. It's not about predicting anything, but about exploring and speculating and moving towards understanding.

In any case, this application of simulation induces anxiety...there are too many examples of technology amplifying the power of those who already have too much of it. It's a common topic in science fiction: in the miniseries World on a Wire (which shares the source material behind The Thirteenth Floor but is quite a different experience), the IKZ (Institute for Cybernetics and Future Science) is pressured to use their simulated world to predict steel prices for the steel industry.

_World on a Wire_
World on a Wire

This furthers the urgency of popularizing this way of thinking. As we're seeing with the proliferation of machine learning, such technological vectors of control are especially potent when the public don't have opportunities to learn how they work.

A means for discourse

One side effect of translating mental models into simulation code is that you generally need to render your assumptions explicit. One assumptions are made unambiguous in code, it is easier to point to them, discuss them, and perhaps modify them. You can fork the model and assert your own perspective.

This becomes clear when you look at games which rely on simulation, like SimCity. As Ava Kofman points out in Les Simerables:

To succeed even within the game’s fairly broad definition of success (building a habitable city), you must enact certain government policies. An increase in the number of police stations, for instance, always correlates to a decrease in criminal activity; the game’s code directly relates crime to land value, population density, and police stations. Adding police stations isn’t optional, it’s the law.

SimCity 5 design document
SimCity 5 design document

Chris Franklin (Errant Signal) similarly notes many problematic assumptions in Civilization 5, such as framing only certain societies as "civilizations" (the rest are "barbarians", which are literally under the same banner as wild animals), and further asserting that the only valid form of social organization and actor in history is the nation-state. As such it precludes the possibility of alternative forms of society; the game simulation leaves no space for such things.

These two instances could be framed as examples of what Ian Bogost calls "procedural rhetoric", defined as:

an argument made by means of a computer model. A procedural rhetoric makes a claim about how something works by modeling its processes in the process-native environment of the computer rather than using description (writing) or depiction (images). (Persuasive Games: The Proceduralist Style, Ian Bogost)

Another example includes molleindustria's Nova Alea, a game about speculative real estate investment and gentrification in which you pump money into a city block to (if all goes well) see it grow and later reap the profits.

molleindustria's Nova Alea
molleindustria's Nova Alea

I'm also developing a video game, The Founder: A Dystopian Business Simulator, which argues that the logic that compels Silicon Valley is inherently destructive and destabilizing. In it you found and manage a startup for as long as you can satisfy the required growth targets. Maintaining growth requires hiring cheap labor, replacing it where possible, commingling with the government, pacifying employees, and more.

The Founder
The Founder

With simulation the mechanics of a system as you see it must be laid bare. They become components which others can add to or take from or rearrange and demonstrate their own perspective. This is no doubt utopian of me to say, but I would love to see a culture of argumentation via simulation!

Exploration & education

Simulations can be used to argue, but they can also be used for education, falling under the rubric of "explorable explanations". That is, when imbued with interactivity, simulations can essentially function as powerful learning tools where the process of exploration is similar to or indistinguishable from play.

Nicky Case is doing a lot of amazing work here. Their projects are great at communicating the pedagogical appeal of simulation. Parable of the Polygons (developed with Vi Hart) is a wonderful presentation of Schelling's segregation model. Their emoji simulation prototype is also awesome. (I highly recommend their talk How to Simulate the Universe in 134 Easy Steps.

Parable of the Polygons
Parable of the Polygons
emoji simulator
emoji simulator

Nicky also maintains a collection of explorable explanations, framing them as such:

What if a book didn't just give you old facts, but gave you the tools to discover those ideas for yourself, and invent new ideas? What if, while reading a blog post, you could insert your own knowledge, challenge the author's assumptions, and build things the author never even thought of... all inside the blog post itself?

What if, in a world where we're asked to constantly consume knowledge, we construct knowledge?

Working off a similar strand of thought is Buckminster Fuller's proposed "World Game":

In the 1960's Buckminster Fuller proposed a “great logistics game” and “world peace game” (later shortened to simply, the “World Game”) that was intended to be a tool that would facilitate a comprehensive, anticipatory, design science approach to the problems of the world. The use of “world” in the title obviously refers to Fuller's global perspective and his contention that we now need a systems approach that deals with the world as a whole, and not a piece meal approach that tackles our problems in what he called a “local focus hocus pocus” manner. (Buckminster Fuller Institute)

Fuller's World Game
Fuller's World Game

I loved this idea of a shared game through which people learn to think on a global scale, about the interrelatedness of all things, imbuing an acknowledgement of our connectedness as a species from a young age.

It reminds me of the fictional game Azad, from The Player of Games, which is played annually by the Empire of Azad and literally dictates how the civilization is run and what it values as a culture. The game has enough depth and complexity that a player can reflect their entire philosophy and perspective about the world through how they play. It's a beautiful capacity for a game to aspire to.


As far as I know Fuller's World Game never materialized, but a game which seems to trace its intellectual lineage to this proposal is Strange Loop Games' ECO, where players must collaborate in a simulated ecosystem to build a lasting society, involving both an player-run economy and a government, and culminating in cooperating to prevent an extinction event (a meteor impact).

Strange Loop Games' ECO
Strange Loop Games' ECO

Looking forward into the future, the speculative paper Optimists’ Creed: Brave New Cyberlearning, Evolving Utopias (Circa 2041) (Winslow Burleson & Armanda Lewis) postulates an educational method called "cyberlearning" which integrates VR, AI, and simulation:

Cyberlearning provides us (1) access to information and (2) the capacity to experience this information’s implications in diverse and visceral ways. It helps us understand, communicate, and engage productively with multiple perspectives, promoting inclusivity, collaborative decision-making, domain and transdisciplinary expertise, self actualization, creativity, and innovation (Burleson 2005) that has transformative societal impact.


This is the holy grail of simulation to me - such an educational experience could have an incredible effect.

MIT's CityScope project is a step in this direction, providing an interactive "Tactile Matrix" - a city in the form of Legos - that is connected to an urban simulation.

CityScope's Tactile Matrix (MIT)
CityScope's Tactile Matrix (MIT)

Experience generation

Last but not least: experience generation. Earlier I asked how we imbue simulation with the narrative richness other media provides. That is, how do we more accurately communicate the experiences that a simulation might (ostensibly) represent?

Humans of Simulated New York tries to tackle this question. We were initially curious about "breathing life" into data. Data about human phenomenon are meant to translate experiences into numbers which may reveal something about the underlying structure of these phenomenon. Perhaps if we take this structure, parameterized with these numbers, we can use simulation to generate some experience which better approximates the experience that initially generated the data.

Many "experience generators" already exist. Video games are probably the most ubiquitous example. Games like Kentucky Route Zero and Gone Home are powerful ways of telling the stories of individuals.

Kentucky Route Zero
Kentucky Route Zero
Gone Home
Gone Home

These particular examples at least are highly-scripted narratives; ultimately they are very guided and linear.

Some games go beyond this by not writing a particular story but instead defining the world in which many stories can happen. Games of this category include Cities: Skylines, Stellaris (similar to a galactic Civilization game), and famously Dwarf Fortress. These games describe rules for how a world (or universe) works and provides players the space to create their own stories.

Dwarf Fortress
Dwarf Fortress

These games, however, are not derived from real-world data and, perhaps as a result of this, lack the same narrative power that games like Gone Home have. It seems like if we can create game from a simulation meant to approximate our society (at least the aspects relevant to the stories we wish to tell) we can create a deeply engaging way to learn about each other. Sort of like Fuller's World Game, but one that doesn't privilege the global perspective and equally appreciates the individual one.

We certainly did not achieve this with HOSNY but this is part of what motivated the project. We were interested in creating a simulation that modeled a typical day in the life of a New Yorker of a particular demographic, as described by Census data (in particular, New York City American Community Survey data spanning 2005-2014). The following sketch gives an idea:

Early HOSNY form
Early HOSNY form

The simulation agents ("simulants") had very simple behaviors. They needed to buy food for them and their family and, unless they were independently wealthy or a business owner, had to find employment to get the money to do so. They also had to pay rent, could get sick, could visit the doctor if they could afford it, and could start a business if they could afford it. They also varied a bit in personality - this part was totally separate from the dataset - which influenced their decision making.

HOSNY simulant design
HOSNY simulant design

A social simulation needs a social network so we used a model from Social Distance in the United States: Sex, Race, Religion, Age, and Education Homophily among Confidants, 1985 to 2004 (Jeffrey A. Smith, Miller McPherson, Lynn Smith-Lovin. University of Nebraska - Lincoln. 2014) which looked at what factors correlated with two people being confidants. Given two simulants, this model output a probability that they would be friends.

HOSNY social network formation
HOSNY social network formation

The full simulation design document is available here.

The end result was a participatory economic simulation which players, after joining via their phone, are assigned a random simulant. It's their duty to ensure the well-being of their simulant by voting on and proposing legislation they believe will help improve their life.

HOSNY player screen
HOSNY player screen

I don't think we quite achieved generating a compelling narrative with this particular project, but I do think the general methods used will eventually support a successful one.

System Designer (SyD)

Fei and I are working on a framework, System Designer (SyD for short), which tries to tie together and realize these ideas I've discussed here. The goal is to create an accessible tool for constructing, running, and playing with social simulations and develop an accompanying educational program.

Early sketch of System Designer
Early sketch of System Designer

Our current blurb for describing it:

An open-source simulation toolkit and conceptual framework to make complex systems thinking (how did we get here/where are we now?) and the simulation of alternative worlds (where can we go?) more accessible to designers, activists, urban planners, educators, artists....

The framework is still in its early stages, but the plan is to include:

  • an engine for running simulations (Python)
    • multi-node support
    • extensibility
    • common data format for sharing/forking models
  • a protocol for communicating with the engine
    • to build arbitrary interfaces to build & tweak simulations
    • to build arbitrary visualizations on top
  • a GUI editor for building/tweaking simulations
  • workshops to teach the method & tool
Game of Life built with SyD
Game of Life built with SyD

Wrap up and come down

Clearly I'm very excited about the possibilities here, but I also want to add as a caveat: I'm not advocating this as some monolithic silver bullet (even though it might sound that way). I think simulation (in particular, agent-based modeling and system dynamics) is a very interesting methodology that has an inherent appeal and provides powerful new ways of looking at the world. But I also have reservations and uncertainties which I would be happy to discuss afterwards. There is one main concern I want to address, and it basically amounts to: "how well can we actually model people?"

This is a difficult question that has many different phrasings and different answers for some and no answers for others.

As a general and perhaps unsatisfying answer, I don't, at my current level of understanding of the field, think we can ever model people "perfectly" or even at an extremely high-fidelity (perhaps due to a combination of limits of computation and of what we can understand about human behavior generally), but I do think we can model people at a convincing and useful level. To quote George Box (this quote had to be in here somewhere): "all models are wrong, but some are useful".

The promise/spectacle of big data has revived a notion of "social physics", the idea that we can describe immutable laws which govern human behavior. In fact, Bruno Latour seems to be arguing for the abandonment of methods like agent-based modeling in favor of big-data-driven social science. Most of us are familiar with how flawed the idea of "big data == truth" is (Nathan Jurgenson goes into more detail about the shortcomings here), but to put it succinctly: processes of data collection and analysis are both subject to bias, and the "infallible big-data" rhetoric completely blinds practitioners to it.

In 2014, Joshua Epstein published Agent_Zero: Toward Neurocognitive Foundations for Generative Social Science. It is an ambitious attempt at defining a "first principles" agent that models what he describes as the three core components to human behavior: the cognitive (rational), the affective (emotional), and the social. The book is fascinating and I think there is a lot to be learned from it, but it also needs to be considered with great care because it too seems subjectable to a biased understanding of what constitutes human behavior.

Of course, there is no methodology that is immune to potential bias, and as Jurgenson notes in View from Nowhere, the main safeguard is training practitioners to recognize and mitigate such bias as much as possible. The same caveat needs to be applied to the methods of social simulation, and even so, I still believe them to hold a great deal of promise that needs to be further explored.

n. the awareness of the smallness of your perspective, by which you couldn't possibly draw any meaningful conclusions at all, about the world or the past or the complexities of culture, because although your life is an epic and unrepeatable anecdote, it still only has a sample size of one, and may end up being the control for a much wilder experiment happening in the next room.

from the dictionary of obscure sorrows

Rendering the invisible visible

  • 09.30.2016
  • etc
Friedman's "Lesson of the Pencil"
Friedman's "Lesson of the Pencil"

In Free to Choose, Milton Friedman presents a pencil (the idea is borrowed from a poem) in order to sort of marvel at the wonder of the free market, how all these disparate people and resources were coordinated to produce something as simple as a pencil, and he's sort of coming from the angle of it's great because you don't have to think about that production process at all, it's totally abstracted away from you, and he doesn't seem to be interested in acknowledging how fraught the politics of production is (or maybe he just doesn't care).

Is there a way to do the opposite? To make the production process of say your smartphone - conflict mining and poor working conditions and publicly-funded research and so on - undeniably obvious in the object itself? To really make that whole production network felt ("real") to the end user? And going a step further, what happens after the consumer is done with it? For example, e-waste dumps in Ghana and China.

Smartphone mineral mining (Congo; Marcus Bleasdale/National Geographic)
Smartphone mineral mining (Congo; Marcus Bleasdale/National Geographic)
Foxconn production (Bloomberg)
Foxconn production (Bloomberg)
E-waste (Agbogbloshie, Ghana; Wikipedia)
E-waste (Agbogbloshie, Ghana; Wikipedia)

Sourcemap maps out the supply chains of various products and I wonder what can be built on top of it.


Learning about learning

  • 01.08.2016
  • etc

As part of my OpenNews fellowship I have been catching up on the theory behind machine learning and artificial intelligence. My fellowship is ending soon, so today I'm releasing the notes from the past year study to bookend my experience.

To get a good grasp of a topic, I usually went through one or two courses on it (and a lot of supplemental explanations and overviews). Some courses were fantastic, some were real slogs to get through - teaching styles were significantly different.

The best courses has a few things in common:

  1. introduce intuition before formalism
  2. provide convincing motivation before each concept
  3. aren't shy about providing concrete examples
  4. provide an explicit structure of how different concepts are hierarchically related (e.g. "these techniques belong to the same family of techniques, because they all conceptualize things this way")
  5. provide clear roadmaps through each unit, almost in a narrative-like way

The worst courses, on the other hand:

  1. lean too much (or exclusively) on formalism
  2. seem allergic to concrete examples
  3. sometimes take inexplicit/unannounced detours into separate topics
  4. just march through concepts without explaining how they are connected or categorized

Intuition before formalism

In machine learning, how you represent a problem or data is hugely important for successfully teaching the machine what you want it to learn. The same is, of course, for people. How we encode concepts into language, images, or other sensory modalities is critical to effective learning. Likewise, how we represent problems - that is, how we frame or describe them - can make all the difference in how easily we can solve them. It can even change whether or not we can solve them.

One of the courses I enjoyed the most was Patrick Winston's MIT 6.034 (Fall 2010): Artificial Intelligence. He structures the entire course around this idea of the importance of the right representation and for people, the right representation is often stories (he has a short series of videos, How to Speak, where he explains his lecturing methods - worth checking out).

Many technical fields benefit from presenting difficult concepts as stories and analogies - for example, many concepts in cryptography are taught with "Alice and Bob" stories, and introductory economics courses involve a lot of hypothetical beer and pizza. For some reason concepts are easier to grasp if we anthropomorphize them or otherwise put them into "human" terms, anointing them with volition and motivations (though not without its shortcomings). Presenting concepts as stories is especially useful because the concept's motivation is often clearly presented as part of the story itself, and they function as concrete examples.

Introducing concepts as stories also allows us to leverage existing intuitions and experiences, in some sense "bootstrapping" our understanding of these concepts. George Polya, in his Mathematics and Plausible Reasoning, argues that mathematical breakthroughs begin outside of formalism, in the fuzzy space of intuition and what he calls "plausible reasoning", and I would say that understanding in general also follows this same process.

On the other hand, when concepts are introduced as a dense mathematical formalism, it might make internal sense (e.g. "$a$ has this property so naturally it follows that $b$ has this property"), but it makes no sense in the grander scheme of life and the problems I care about. Then I start to lose interest.

Concept hierarchies

Leaning on existing intuition is much easier when a clear hierarchy of concepts is presented. Many classes felt like, to paraphrase, "one fucking concept after another", and it was hard to know where understanding from another subject or idea could be applied. However, once it's made clear, I can build of existing understanding. For instance: if I'm told that this method and this method belongs to the family of Bayesian learning methods, then I know that I can apply Bayesian model selection techniques and can build off of my existing understandings around Bayesian statistics. Then I know to look at the method or problem in Bayesian terms, i.e. with the assumptions made in that framework, and suddenly things start to make sense.


Some learning tips

Aside from patience and persistence (which are really important), two techniques I found helpful for better grasping the material were explaining things to other and reading many sources on the same concept.

Explain things to other people

When trying to explain something, you often reveal your own conceptual leaps and gaps and assumptions and can then work on filling them in. Sometimes it takes a layperson to ask "But why is that?" for you to realize that you don't know either.

Most ideas are intuitive if they are explained well. They do not necessarily need to be convoluted and dense (though they are often taught this way!). If you can explain a concept well, you understand it well.

Read a lot

People conceptualize ideas in different ways, relating them to their own knowledge and experiences. A concept really clicks if someone with a similar background to yours explains it because the connections will be more intuitive.

The same concepts are often present in many different fields. They may be referred to by different names or just have different preferred explanations. Seeing a concept in different contexts and introduced in different ways can provide a really rich understanding of it. If you can explain a concept in many different terms, you probably have a strong multi-perspective understanding of it.

Just add water

  • 11.14.2015
  • etc
No Man's Sky (Rock Paper Shotgun)
No Man's Sky (Rock Paper Shotgun)

The go-to rebuttal to increasing automation tends to be something around how creativity could not be emulated by computers, at least, not for awhile. There's some truth to that statement depending on how you define "creativity", I suppose. The least charitable definition might be synonymous with "content generation", a domain typically exclusive to humans - artists, musicians, writers, and so on - but computers have made some claim to this territory.

The poster child of the automated (i.e. procedural) content generation beachhead is No Man's Sky, which generates a literal universe, complete with stars and beaches and creatures, using interweaving mathematical equations. The universe it generates will reportedly take 5 billion years to explore, and that's just one of those universes. In theory, an infinite amount of universes can be generated by setting different seed values (the value which is used to determine all other values in the game). This interview with Sean Murray goes a bit more into depth on their approach.

No Man's Sky procedural creatures (via)
No Man's Sky procedural creatures (via)

These procedurally-generated universes aren't random, though they are designed to give that appearance. They are completely deterministic depending on the seed value - this is a critical property since there needs to be consistency. If you leave a planet, you should not only be able to come back to that planet, but it should be in a similar state to when you left it.

A big challenge in procedural content generation is figuring out a way of creating things which feel organic and natural - for No Man's Sky, the solution is the impressive-sounding Superformula, developed by John Gielis in 2003. In polar coordinates, with radius $r$ and angle $\varphi$, parameterized by $a, b, m, n_1, n_2, n_3$, the Superformula is:

$$ r\left(\varphi\right) = \left( \left| \frac{\cos\left(\frac{m\varphi}{4}\right)}{a} \right| ^{n_2} + \left| \frac{\sin\left(\frac{m\varphi}{4}\right)}{b} \right| ^{n_3} \right) ^{-\frac{1}{n_{1}}}. $$

The above is the formula for two dimensions, but it is easily generalized to higher dimensions as well by using spherical products of superformulas (cf. Wikipedia).

Some forms resulting from the 2D formula:

2D Superformula-generated shapes (Wikipedia)
2D Superformula-generated shapes (Wikipedia)

Procedural generation is a really beautiful approach to game design. It's not so much about creating a specific experience but rather about defining the conditions for limitless experiences. No Man's Land is far from the first to do this, but it is the first (as far as I know) to have all of its content procedurally generated. The game Elite (from 1984!) also had procedurally-generated universe. Elite's universe was much simpler of course, but used a clever approach using the Fibonacci sequence to simulate randomness:

$$ \begin{aligned} x_0 &= \text{seed} \\ x_n &= x_{n-1} + x_{n-2} \end{aligned} $$

And then taking the last few digits of values generated from this sequence.

For example, take the Fibonacci sequence:

$$ 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, \dots $$

Let's jump at head in the sequence:

$$ 1597, 2584, 4181, 6765, 10946, 17711, 28657, \dots $$

If we just look at the last three digits, the resulting sequence looks quite random:

$$ 597, 584, 181, 765, 946, 711, 657, \dots $$

The great thing about this is that these universes can be shared by sharing seed values.

Elite's procedurally-generated universe (Gamasutra)
Elite's procedurally-generated universe (Gamasutra)

Procedural content generation is interesting, but other games focus on "procedural" stories. These games have (some) manually generated content - creatures, buildings, etc - but focus on designing complex systems which form the bedrock of the game's ability to spawn wild and fantastic stories. Dwarf Fortress and RimWorld are two great examples, which are essentially fantasy and sci-fi world-simulators (respectively) which model things like each individual's mental health, weather patterns, crop growth, and so on. No one writes the stories ahead of time, no one has a premeditated experience for you put in place - it's all off-the-cuff based on the dynamics of the game's rules.

RimWorld (via)
RimWorld (via)

The stories that come out of these games are amazing. Dwarf Fortress has an especially vibrant community based around the often absurd things that happen in game (for example, see r/dwarffortress, or for a more academic treatment, see Josh Diaz's thesis).

With the independent game industry continually expanding, I think we'll see more of these kinds of games. They can be developed with relatively small teams but have the richness and depth (and often to a better degree) of a massively-built studio game.