**Ludonomy**
[Ashwin Reddy](https://ashwinreddy.github.io/)
The ancients took games seriously, perhaps more seriously than us, giving them a venerated place in their stories and rituals. After Ragnarök, the Nordics say, new gods will find artifacts in the grass -- pieces for the chesslike game Tafl, the remainders of an extinguished world. Or, in the inciting incident of the Indian epic poem _Mahabharata_, the Pandavas lose their kingdom and all their possessions in a rigged game of dice. Even today, the Olympics serve as a reminder that the Greeks once celebrated the name Zeus with an _agon_, an athletic contest.
Suppose that in hoping to recover old ideals, we believe games still matter. But assuming life is a game, so what? Is it a cause for joy or despair? More to the point, what kind of game is it? A rat race? Or more like Theseus navigating the labyrinth and grabbing the Minotaur by his horns? A cold and calculated chess match? Perhaps the opposite -- a game of snakes and ladders?
A useful metaphor must ground its tenor in a more concrete vehicle. Yet games are hardly easier to understand than life itself, so the comparison wins us nothing. _Life is a game_ is an enervated aphorism which needs an essay to investigate and invigorate it.
# Define
> "That's enough about lessons," the Gryphon interrupted in a very decided tone: "tell her something about the games now."
> -- _Alice in Wonderland_
Without a definition of games, we are liable to talk past one another, but I do not claim an indisputable definition. The point is to bind the word to an agreed meaning. After that, I may use the word liberally and with impunity.
Game
: A game $G$ consists of player(s) $P_1, \dots, P_n$ in a space $S$, each attempting to reach their designated goal space $\Omega_{P_i}(G) \subset S$, while following rules which determine the shape of $S$; and they do this all for fun.
The definition splits into three components. The Ablation column below lists what a game would be without that component.
|Component|Key Phrase|Ablation|
|---------|----------|---------|
|Agency|_players ... each attempting to reach_|Event|
|Goal|_designated goal space_ $\Omega_{P_i}(G)$|Toy/Sandbox|
|Restrictions/Playfulness|_following rules which determine the shape of $S$; and they do this all for fun_|Activity|
[Table [components]: The ingredients of a game]
A game needs players empowered to make choices that dramatically shift the game's course. A player who lacks agency is not a player but someone experiencing an event, like a graduation ceremony or a wedding or an acid trip. In such events, one lets perception and feeling take over.
To direct their agency, players need a goal $\Omega(G)$, a privileged state in the space $S$. The $\Omega(G)$ for chess is checkmate, but the goal need not be a static arrangement of objects in $S$. It might be a mental state as in _Clue_, where the $\Omega(G)$ is knowing the details of the murder. It can be a physical space, like sumo wrestlers trying to stay inside the dohyo as long as possible. In last-man-standing or so-called infinite games, the $\Omega(G)$ is keeping the game going as long as possible. If the player decides on his or her own goals, then the game is a toy or a sandbox like Legos.
Finally, a game needs restrictions, rules or constraints, which impose limits on $S$ but in turn open up the possibility of play. Notably, players have to affirm these rules on their own terms. Moving a bulky sofa into a new apartment is an activity that affords agency and has a goal, but it is not a game. The constraints are defined purely by the circumstances. You can't drag the sofa through the door because the door is not wide enough, not because there are external rules preventing it. In contrast, the chess ruleset prevents its players from moving pieces however they like, and players subscribe to these rules for no other reason than to play chess.
Thus, we cover chess, crossword puzzles, ultimate frisbee, video games, and the rest.
# Instinctution
Definitions can't explain, and this one says nothing about the reasonable but misguided question _why do people play games?_
Here, it's helpful to lean on Gilles Deleuze's "Instincts and Institutions." An instinct directly satisfies some tendency or need of an organism. The form of the instinct depends on that particular organism (if it's hungry, it has one instinct; if it's sad, it has another) as well as its species (a bat might instinctively use echolocation, but a cow can't). An institution is a socially sanctioned method for satisfying a tendency or need, but it always introduces new, effectively arbitrary, undertakings:
> So money will liberate you from hunger, provided you have money; and marriage will spare you from searching out a partner, though it subjects you to other tasks.
>
In Deleuze's view, humans characteristically replace their animals instincts with institutions that structure their perception.
Thus, a game is an institution that tries to satisfy the need for play by giving us other tasks. However, play itself doesn't have an end. In "What's the Point If We Can't Have Fun?" David Graeber [suggests](https://thebaffler.com/salvos/whats-the-point-if-we-cant-have-fun) that our tendency to think in terms of capitalist economies and evolutionary fitness makes us confused about play for its own sake. If we're confused, we're looking at it backwards. Play must be important because species that have survived are expending resources on something oblique or orthogonal to their survival.
The question isn't _why do people play games_ but rather _where does play show up in games_?
We have created a definition that emphasizes the structure of games, but it impoverishes play by treating it as a mere matter of structure. It lacks the sense that players yearn for play. Unlike games, we won't and don't need a definition of play. [#Feezell10] gives us the sense that it is protean, at once a behavior or activity; a motive, attitude, or state of mind; a form or structure; and a meaningful experience. Rather than directly grappling with play, we will consider two prototypical examples of games. One fits our current definition perfectly. The other one defies it. In attempting to square the two, we will gain a more holistic sense for the sacred syllable of game.
# Solve
Optimization problems are a well-understood class of game in mathematics which offers insight into the structure of games. They are typically written in a symbolic form.
\begin{equation}
\label{eq:opt}
\min_{x \in X} f(x) \quad \text{s.t.} \quad \Big\{ g_i(x) \leq 0 \Big\}_{i=1}^n
\end{equation}
Let's apply our definition of game here. The player $P_1$ is the puzzle-solver. The space $S$ is given as $X$. The goal $\Omega(G)$ is that $P_1$ can express the value of the minimum $x$, and the rules are that the $x$ satisfies the $g_i$ constraints, and, additionally, $P_1$ must prove their correctness with valid mathematical logic.
At first, the naked form of Equation \ref{eq:opt} hardly looks like a game, but it perfectly describes, say, a race: minimum time = maximum speed, optimization at its purest. Behind every game is an Equation \ref{eq:opt}, but often the goals, constraints, and players change in an intricate dance which cannot ever be put into symbols.
In fact, symbols suppress the geometry of a game's landscape. Stephen Wolfram [shows](https://writings.stephenwolfram.com/2022/06/games-and-puzzles-as-multicomputational-systems/) that even simple games like Tic Tac Toe exist inside an intricately structured space:
![Figure [multiway]: Wolfram's visualizations of the state spaces of simple games, demonstrating that games are computational systems.](https://content.wolfram.com/uploads/sites/43/2022/06/multiway-hero-final-2.png)
Wolfram focuses in his computational essay on games with discrete state spaces and moves. For such games, the natural visualization is a graph. Each vertex is a state and the edges represent moves from one state to another. If the game has a continuous state space, however, the better description is a manifold.
We're all familiar with the manifold that is the Earth's surface,[^dimensions] and just as a physical atlas is a book of charts that allow me to navigate the earth, every manifold $M$ has a mathematical atlas $\mathscr{A}$ which contains charts $\varphi_\alpha$ that cover all of $M$.
[^dimensions]: The Earth's surface is an example of a 2-dimensional manifold because you need flat sheets of paper as the charts. A 1D manifold has number lines for charts, and a 3D manifold has cubes for charts, so on and so forth; you can have manifolds in $n$ dimensions for any integer $n \geqslant 1$.
$$
\mathscr{A} \triangleq \big\{ \underbrace{(U_\alpha, \varphi_\alpha)}_{\text{chart}}: \alpha \in I\big\} \quad \text{s.t.} \quad \bigcup_{\alpha \in I} U_\alpha = M
$$
A map $\varphi$ takes some portion of $M$, denoted $U$, and gives it coordinates in $\mathbf{R}^n$, which are easier to work with. For Earth, a $U$ could be the region of France, and the map $\varphi$ would assign latitude and longitude to all the points in France.
![Figure [earth]: Earth as a manifold, with one chart shown. Notice the flat geometry of the chart compared to the curved geometry on the surface. With enough charts, we could cover the whole surface.](https://upload.wikimedia.org/wikipedia/commons/a/a3/Triangle_on_globe.jpg)
Critically, players don't need to keep the whole manifold in their memory to play. As long as they know where they are and where the goal is, they can make progress, tracing out just one of many possible paths in the space of the manifold.
![Figure [path]: A path on the manifold of your screen.](https://upload.wikimedia.org/wikipedia/commons/d/d8/Path.svg)
Path
: A trajectory in the manifold, represented by a function $f: [0, 1] \to M$. If $M$ is a graph, then a sequence of edges connecting vertices on the graph.
A game can now be compactly described as a manifold on which players aim to trace out solutions, simply paths terminating in $\Omega(G)$.
Solution
: A path that ends at $\Omega(G)$ while obeying all the constraints.
Formalization makes solving a game simple.
1. Enumerate or represent all possible paths that end at $\Omega(G)$.
2. Select the one which is optimal by some algorithm or heuristic.
Problem solving is all about thinking in this way. Working backwards is the technique of finding a path by traversing from $\Omega(G)$. A more powerful example is Lagrangian mechanics, where we model a physical system as a game. The player is Mother Nature,[^gt] and the space is a manifold which represents a physical system (e.g. mass on a spring, blocks on a ramp, etc.). We impose the following condition on Nature: at time $t_0$ the system must be in the configuration $\mathbf{q}_0$ and at time $t_1$ it must be in configuration $\mathbf{q}_1$.
[^gt]: This is a standard practice in game theory. We say that we're modeling Nature as a pseudo-player, and the actions are called moves by Nature.
![Figure [solutions]: Various paths $\mathbf{q}(t)$ that start at $\mathbf{q}_1$ on the left and end at $\Omega(G)$ on the right. The player, i.e. Nature, needs to select the optimal one.](https://upload.wikimedia.org/wikipedia/commons/0/00/Homotopy_between_two_paths.svg)
The laws of physics make for many possible paths, but Nature can select only one. It can be shown that Nature selects the optimal one as defined by
$$
\frac{\delta S}{\delta \mathbf{q}(t)} = 0, \tag{Hamilton's Principle}
$$
where this equation is analogous to finding an optimum for a function $f: \mathbf{R} \to \mathbf{R}$ by solving $f'(x)=0$. The quantity $S$ is called the action and it tracks the total excess energy $\mathcal{L}$, known as the Lagrangian, over the time interval.
$$
S[\mathbf{q}] \triangleq \int_{t_0}^{t_1} \mathcal{L}\big(t, \mathbf{q}(t), \dot{\mathbf{q}}(t)\big) \, \mathrm{d}t. \tag{Action}
$$
A standard technique in the toolkit of physicists, Lagrangian mechanics is an application of Equation \ref{eq:opt}. The upshot is that thinking in terms of optimization and search problems in a space is an incredibly practical tool in the real world.
Another example goes back to Wolfram's discrete setting. In the Tower of Hanoi problem, there are three rods with a stack of disks on the rightmost rod organized from smallest to largest going down. Your job is to transfer the disks to the leftmost rod, one disk at a time, but without ever putting a larger disk on top of a smaller one.
![Figure [Hanoi]: The Tower of Hanoi, a classic problem from introductory computer science classes. ](https://content.wolfram.com/uploads/sites/43/2022/06/sw060722hanoiimg3.png)
We can generate the solution using recursion to avoid creating the whole graph.
```Python
def printMove(fr, to):
print(f'move from {fr} to {to}')
def towers(n, fr, to, spare):
if n == 1:
printMove(fr, to)
else:
towers(n - 1, fr, spare, to)
towers(1, fr, to, spare)
towers(n - 1, spare, to, fr)
```
[Listing [Recursion]: Python program to compute Hanoi solutions recursively.]
In all these examples, we clearly specified a game, then computed ready-to-go solutions using problem solving techniques and reasoning. The paths are guaranteed to end at $\Omega(G)$. In the best case, we can solve a game completely.
Solved game
: A game for which we can produce solutions from any position.
Only after formalizing a game can we submit it to our will like this. What I want is a word to refer to a game's mathematical structure so that we can identify this possibility.
Ludonomy[^ety-ludonomy]
: The game as it appears when expressed through logical propositions, symbols in algebra, or as a geometric object, enabling it to be represented and solved mathematically.
Ludonomers are logicians, computer scientists, mathematicians, philosophers, and lawyers, fluent in the rules (and the implications) of a game.
Remember this about ludonomy, if nothing else: it is where we take up the mantle of Shiva and by solving games, put them to rest. This is remarkable but also bittersweet. People stop playing solved games. Adults know how to force a draw in Tic Tac Toe, hence only children play it. But new games emerge where old ones are buried. We go on to construct games for which we must become better, faster, stronger.
[^ety-ludonomy]: Formed from Latin _ludus_ meaning game and Greek _nomos_ meaning law, custom, or rule.
# Language
> "Our language can be seen as an ancient city: a maze of little streets and squares, of old and new houses, and of houses with additions from various periods; and this surrounded by a multitude of new boroughs with straight regular streets and uniform houses."
> -- Wittgenstein
Perhaps no mathematician loved games and play as much as Lewis Carroll, author of _Alice in Wonderland_ (1865) and its sequel _Through the Looking-Glass_ (1871). When he wasn't writing about Euclidean geometry and logic or finding recursive ways to compute determinants, he was writing poetry and novels and designing puzzles.
In _Through the Looking-Glass_, Humpty Dumpty unknowingly reveals to Alice that conversations are turn-based games.
> "In that case we start fresh,” said Humpty Dumpty,
> “and it’s my turn to choose a subject—” (“He talks about it just as if it was a game!” thought Alice.)
>
Alice stops short of calling the conversation a game, but we can take her premise to its logical conclusion. A conversation is a game where two or more players are playing in the space of ideas. They have agency insofar as they can say something which will advance the conversation, either along the line it was already moving or turning it somewhere else. The goal is to keep the conversation engaging for all players as long as they can, like a tennis rally. There must be rules because you cannot just say anything in a conversation, and Paul Grice tells us what they are.
|Maxim|Directive|Meaning|
|----|---------|--------|
|Quantity|Be informative|Say as much as you need but no more than that.|
|Quality|Be truthful|Don't say what is false or lacking in reason.|
|Relation|Be relevant|Stay on topic.|
|Manner|Be clear|Be brief and orderly; avoid ambiguity and obscurity.|
[Table [Grice]: Grice's Maxims tell us four principles that (good) conversationalists follow.[^grice]]
[^grice]: It's easier to see the rules in the transgressions. When someone says only _yes_ or _no_ to your questions in a conversation, or tells you an obvious lie without irony, or goes on a long digression without explanation, or speaks in a complicated way, we feel they don't know how to play the game of conversation.
We have shown that a conversation has ludonomy, but it shares almost no similarities with the examples in section [Solve]. The path of a conversation is completely abstract and can't reasonably be represented in a mathematical structure. Consequently, I can't point to an $\Omega(G)$ in that space, nor can I predict what you would say or where the conversation will go, nor does a conversation have a "solution." To play a conversation is to experience it. Therefore, any conception of games relying solely on ludonomy is insufficient. There must be another side to games which can capture the gamelike aspect of conversation.
Ludwig Wittgenstein coined the term _Sprachspiel_ (German for language-game) to talk about such games, which include
>
> - Giving order ands obeying them
> - Describing the appearance of an object, or giving its measurements
> - Constructing an object from a description (a drawing)
> - Reporting an event
> - Speculating about an event
> - Forming and testing a hypothesis
> - Presenting the results of an experiment in tables and diagrams
> - Making up a story; and reading it
> - Play-acting
> - Singing catches
> - Guessing riddles
> - Making a joke; telling it
> - Solving a problem in practical arithmetic
> - Translating from one language to another
> - Asking, thanking, cursing, greeting, praying.
>
Both ludonomy and _Sprachspiel_ tell us something important about games, so we'd like a common basis to talk about both. Since _Sprachspiele_ resist a simple mathematical structure, we reverse the approach: force mathematics into language.
That's not a surprising claim because mathematics has its own symbols and language for talking about its objects of study. If I take a ludonomous game like chess, then I see that it too has a language called [algebraic notation](https://en.wikipedia.org/wiki/Algebraic_notation_(chess). Since a game is an institution, people have to talk to one another about the game they are planning, about valid and invalid moves, better and worse strategies. That fact tells us that every game of ludonomy has a _Sprachspiel_ to it. Therefore, despite the rigorous structure of chess, different players have different styles, just as different people have their own way of talking. This is what it means to have _style_. The best players of any game show their agency by finding a unique way of expressing themselves in the game's _Sprachspiel_.
# Academics
> "Regarding truths, the artist has a weaker morality than the thinker. He definitely does not want to be deprived of the splendid and profound interpretations of life, and he resists sober, simple methods and results."
> -- Nietzsche
|Ludonomy|_Sprachspiel_|
|--------|--------------|
|Ludonomy is the enclosed institution of the game with a clear goal. It requires concentration and careful analysis of rules. It sees the game as a puzzle and wants us to use our mind and learning to uncover the structure and come to an objective answer. It wants everything laid out in a hierarchical manner so that we, the players, can see what its purpose is and then exploit its advantages. Once we see all things in a stable coherence, we can find the optimal trajectory and simply follow it.|_Sprachspiel_ is about the outward-looking instinct and language of play. It wants to feel how the world is laid out over time and space, not to comprehend it in one gulp. It wanders around the space, hoping to find something curious, something that defies expectation. _Sprachspiel_ orients itself towards the aesthetic. It asks whether a thing has a use which hasn't been considered yet. It likes intuition and creating new things out of old ones. The _Spieler_ digs for Pluto's sparkling gems in the earthy soil.|
Every game has both sides because it must have a ludonomy to capture the rules as well as a _Sprachspiel_ that makes the game an institution, not just the fancy of an individual. Of course, different games are comprised of different proportions of the two. In this diagram, we consider the major disciplines of academics.
****************************************************************
* Math Science Art
* +----+ ^ ^ ^ \
* | +-+-----+----------------+--------------+----+---+
* +----+ v /
* Ludonomy Philosophy Sprachspiel
*
****************************************************************
We draw a box for Ludonomy because it can swallow games whole and an arrow for _Sprachspiel_ because it can extend indefinitely. We'll never exhaust, for example, all the ways of using writing to communicate.
Each discipline is a game that provides its own frame for seeing and understanding its subject matter. Taken alone, we can doubt what a single discipline has to tell us. A typical philosophy paper can't and won't bring in a mathematical perspective. A work of art usually doesn't do the science it references justice. But the principle of consilience tells us that if fields with different operating assumptions and techniques converge on the same conclusions, we should believe that conclusion more. For this reason, it's important to speculate just a little bit on what these foundational games do.
_**Math**_. Technical papers in pure mathematics are a sequence of definitions, lemmas, theorems, and proofs. They leave you to develop the intuition; the paper's purpose is to lay out the facts. Heavy on Ludonomy and light on _Sprachspiel_, math still needs language to communicate. Grice's Maxims are always in play: the paper needs to say something interesting, relevant, and informative in a clear way. And one needs a sense of adventure and curiosity in mathematics as well.
_**Science**_. Science pursues explanations in the natural world, or what amounts to the same, discovering the rules of Nature's game. Given observations of how the game has turned out, can we say what the ludonomy of the game is? Richard Feynman found games a natural metaphor for science:
> There are a number of special techniques associated with the game of making observations,
> and much of what is called the philosophy of science is concerned with a discussion of these techniques.
> The interpretation of a result is an example. To take a trivial instance, there is a famous joke about a man
> who complains to a friend of a mysterious phenomenon. The white horses on his farm eat more than the black horses.
> He worries about this and cannot understand it, until his friend suggests that maybe he has more white horses than black ones.
>
On the one hand, Wittgenstein says speculating about an event and dealing with hypotheses are _Sprachspiel_. The farmer is also clearly communicating with his friend to interpret the results. On the other hand, the basic inference (more white horses -> the white horses eat more in total) is a trivial example of ludonomy at play.
Explanations in science ought to follow Grice's maxims as well: they should be interesting, relevant, informative, and clear. [#Wojtowicz20] tries to tether the Ludonomy and _Sprachspiel_ of explanations in this direction using Bayesian probability.
_**Philosophy**_. A serious discussion about the purpose of philosophy could take up a book of its own. All that must be said is that philosophy's tendency toward logic and uncovering the structure of things pulls it toward ludonomy while its creative and discursive nature pushes it toward _Sprachspiel_. If we want to prevent ourselves from minimizing its value, we need to stop ourselves from allowing it to slip all the way to ludonomy. The goal of philosophy is not to create a totalizing framework in which everything is given a narrow purpose. At least, such a stance would hardly hold up in today's age. Rather, philosophy has to invent concepts and tell us what games are worth playing as well as how to play them.
_**Art**_. Art includes creating, understanding, and critiquing it, whether writing, drawing, painting, sculpting, or singing. Of course, we see a great sense of play in Picasso and Dali. We love artists who revolutionize the game around their own style. Undoubtedly, artists have the agency to put elements together on the canvas, in the marble, whatever the medium may be. The goal of a piece of art is never clear, but the artist must bind themselves to _some_ rules to create something meaningful and sensible.
Understanding and critiquing art is also a game. Reading a literary book or watching a cinematic movie is a game because you have to choose how to take in what you're seeing. If you skip to the end of the book or movie, you didn't "win." The point is to follow a path in the space of imagination. In some sense, this path is fixed because the concrete materials of the artwork don't change. But the experience of the path _is_ different for different viewers. There are better and worst readers and viewers depending on how well they make sense of and use that experience and can convey it to others.
|Discipline|Ludonomy|_Sprachspiel_|
|----------|--------|-------------|
|**Math**|Pure math|Applied math|
|**Science**|Theoretical|Experimental|
|**Philosophy**|Analytical|Continental|
|**Art**|Art Criticism/History|Avant garde|
# Chaos Emerges
Mathematicians, scientists, philosophers, and artists are not working on the same problems they were a hundred years ago. If they are, the problems changed when it reached their hands. Some games have turned into cadavers through ludonomy. Others stopped being interesting. Point being, games evolve to stay relevant. To get precise about what keeps a game interesting, we'll need the language of emergence, chaos, and entropy.
According to Assad and Packard, emergence is when a system like a game shows behavior that could not have been predicted from the rules. The bigger the gap between the formal rules (ludonomy) and the behavior of the system in the wild (_Sprachspiel_), the more "emergent." For example, biological organisms are emergent because it's hard to deduce that amino acids could give rise to life.
|Category|Meaning|
|---------|-------|
|Non-emergent|Behavior is immediately deducible upon inspection of the specification or rules generating it.|
|Weakly emergent|Behavior is deducible in hindsight from the specification after observing the behavior.|
|Strongly emergent|Behavior is deducible in theory, but its elucidation is prohibitively difficult.|
|Maximally emergent|Behavior is impossible to deduce from the specification.|
[Table [Emergence]: Scales of emergence.]
If you've ever played a board game with complicated rules but began to see and enjoy the strategy after playing for a few rounds, you have experienced emergence. The rules were abstract, so you couldn't see that in practice they led you to make compelling choices.
Emergence focuses on the structure of the game, but chaos and entropy can help us see the behavior inside the game space. Chaos theory is a technical field examining deterministic systems which exhibit local predictability with long-term unpredictability. Take a double pendulum.
It moves smoothly, as you would expect. But the path it traces out is unpredictable, even at a mathematical level. Despite having the equations -- you can even write them out using a Lagrangian as in sec. [Solve] -- there's no way to compute the real motion without simulating it step by step. In this way, chaos theorists have found that math, the ultimate ludonomy, undermines its own capabilities in some instances.
Finally, we have entropy, in the sense of information theory. The concept is subtle and needs its own [exposition](https://ashwinreddy.github.io/writeups/info-theory.html), but briefly, entropy captures a way of thinking about information as surprise. It says that if the receiver of a message didn't think that message was likely, then they receive a lot of information when they see it in reality. The higher the entropy of a random variable $X$, the more uncertain one is about which realization $x$ they will observe. The concept of information entropy has spread into a number of areas, including statistics, where the principle of maximum entropy was discovered.
Principle of maximum entropy
: If you have many potential models for a system, select the one which has the highest entropy, the most uncertainty, admitting the widest range of possibilities (all three are equivalent).
To use a model which has lower entropy is to _pretend_ you know what's surprising and what's not.
For a concrete example, look at stories. A story with minimal entropy is one you can predict from beginning to end, thereby conveying little information. But a high entropy story where, dreamlike, characters and events pop in and out of the narrative at random, isn't good either. These extremes are the simplest and most devastating criticisms of a novel or a movie: it was boring, or it was nonsensical. A good narrative strikes a balance, where the audience sees it can go in multiple ways or has multiple meanings.
A good story, following Equation \ref{eq:opt}, needs constraints. They don't need to be realistic, but the story needs to be consistent in following them. A _deus ex machina_ is a type of inconsistency because, by definition, the story never signalled such an intervention. Thus, a story aims for maximum entropy, given its own narrative constraints. The information flows, now smoothly, now abruptly, but it must end up somewhere interesting, remarkable, _unexpected_.
Academics emphasizes the necessary work of ludonomy, but we must also be humble about what we know and acknowledge that a healthy amount of uncertainty enables play. In _Black Swan_ and _Antifragile_, Nassim Taleb makes space for the family of disorder we're talking about, which includes
> uncertainty; variability; imperfect, incomplete knowledge; chance; chaos; volatility; disorder; entropy; time; the unknown; randomness; turmoil; stressor; error; dispersion of outcomes; unknowledge
>
# Autonomous Intelligence
Play is at the core of the academic disciplines, or so I argue. Then perhaps the kernel of intelligence is play. Where do Ludonomy and _Sprachspiel_ come in?
In recent years, machine learning has become the most promising candidate in creating software systems that are intelligent. However, nobody can say with certainty what the principles of intelligence are. The machine learning position paper [#Ma22] focuses on two values that systems of autonomous intelligence -- both humans and computers -- must express: self-consistency and parsimony. These two more or less correspond to the binary we've been harping on here.
||Self-Consistency|Parsimony|
|---|----------------|---------|
|Motive|How to learn|What to learn|
|Question|How to compute the solution to a problem?|What's the right way to frame and capture the problem in simpler terms?|
|Technical Field|Game Theory|Information Theory|
|Game Type|Ludonomy|_Sprachspiel_|
[Table [Learning-Values]: Two principles for learning.]
The basic challenge of AI now revolves around the question of models understanding their world. While deep neural networks can pick up on complex patterns, the fundamental question is how to ensure they're picking up on the right ways of solving their given task. Self-consistency is the value of having a model that's correctly predicting what will happen next. The paper "advocate[s] the need of an _internal_ game-like mechanism for any intelligent agent to be able to conduct self-learning via self-critique!" In other words, a ludonomy for the game of making improvements to oneself is necessary to intelligence.
Parsimony is a more difficult word to grasp. Luckily, Jorges Luis Borges' short story "On Exactitude in Science" is only one paragraph long.
> In that Empire, the Art of Cartography attained such Perfection that the map of a
> single Province occupied the entirety of a City, and the map of the Empire, the entirety
> of a Province. In time, those Unconscionable Maps no longer satisfied, and the
> Cartographers Guilds struck a Map of the Empire whose size was that of the Empire, and
> which coincided point for point with it. The following Generations, who were not so
> fond of the Study of Cartography as their Forebears had been, saw that that vast Map
> was Useless, and not without some Pitilessness was it, that they delivered it up to the
> Inclemencies of Sun and Winters. In the Deserts of the West, still today, there are
> Tattered Ruins of that Map, inhabited by Animals and Beggars; in all the Land there is
> no other Relic of the Disciplines of Geography.
>
The Cartographers Guild are ludonomers hoping to create a complete manifold. But, as Borges points out, such a manifold is useless because it doesn't focus our attention. The job of a map is to simplify the world so we can navigate it. That's parsimony. How do we create systems that know what the right representations of the world are? There cannot be a general theory of such representations because the right representation depends on the situation. We have to design AI systems that can do this reliably.
It seems, as per [#Eysenbach22], that constraining these systems to also follow the principle of maximum entropy from section [Chaos Emerges] makes them able to solve challenging variations on the same problem. I'm hopeful that AI systems will do better when they incorporate language-based explanations and other _Sprachspiele_ into their learning algorithms (see [#Lampinen21] and [#Liu22]).
In [Awakening from the Meaning Crisis](https://www.youtube.com/playlist?list=PLND1JCRq8Vuh3f0P5qjrSdb5eC1ZfZwWJ), John Vervaeke discusses the Ludonomy/_Sprachspiel_ dichotomy in the language of cognitive science and philosophy. Vervaeke believes in a framework he calls Relevance Realization. While humans are amazing at Ludonomy, solving problems and seeing underlying structures in things, a lot of this happen unconsciously. It's possible for us to fixate on the wrong thing, frame the problem the wrong way, not exploring possibilities outside the stories we tell ourselves. We too need ludonomy to intellectualize and solve problems, to find discrepancies (self-consistency). But we also need _Sprachspiele_ to discover what to focus on (parsimony), to see what else might be relevant. We can't survive without both values.
# Life, a Game?
> "What I cannot understand," said Arthur, "is why you should
> take the trouble to think about man and his problems, or to sit
> in a committee on them, if the only respectable thing about him
> is the way he treats a few pets. Why not let him extinguish himself without fuss?"
> [...]
> Merlyn said: "It is because one likes to tinker with things, to play with possibilities."
> -- _The Book of Merlyn_
No, at least not at first. We don't always have agency. Our goals are typically nested in a hazy and shifting stack. As in, right now I should be studying so I can do well on the test so I can get a good grade so I can graduate, so what should I think of my not studying right now? In addition, the rules are never quite fully spelled out. Most unfortunate of all, we may lose the _spiritus ludi_.
But a metaphor isn't an equation. That life comes with its own responsibilities, duties, and labor doesn't invalidate the sentiment. We just have to ask whether games are a useful lens of looking at life. I say they are.
For they describe even the negative aspect of living, which we feel when we compare life to a labyrinth or a maze. It's there when someone says they're _disillusioned_ with life. _dis + in + ludere_. Not-being-in-it-to-play. Albert Camus found his metaphor for life in the gamelike punishment of Sisyphus pushing a boulder up a hill for it to come rolling back down. And surely it's not a coincidence that some people think we might be living in a simulation.
In ages past, when religion and culture could not be decoupled so easily, one could have believed in God's ludonomous plan, where all things came together in a hierarchical unity. The sidewalk nihilist now looks at that plan with beady eyes and weepy ones. Not for nothing does he do this, for there is a dark side to games. Just take addiction. It's when someone engages in an activity for its own sake, gaining a short-term predictable reward, not knowing -- whether willfully or not -- where they will end up in the long-term. That's play and chaos in the darkest possible terms.
We're so acquainted with this negative sense of the game that the positive sense sounds like a fairy tale. Here's what we can say, in our own vocabulary, on our own terms.
If life is a game, then it has a ludonomy and a _Sprachspiel_. The ludonomy is the orienting mechanism, providing a goal $\Omega(G)$ and some basic rules and grammar to form paths that can take us there. The _Sprachspiel_ is what makes us understand each other a little better and start to articulate ideas that are in tantalizingly close reach. If you look closely, you'll see the push and pull between the two at many levels. It occurs at the level of evolution, a theory whose grammar would suggest that only the most adaptated will survive, but, nonetheless, is compatible with the emergence of peacocks whose display of brilliant feathers makes them eminently vulnerable. You'll see it in a poem with an intricately crafted structure of words that still sings sweetly into your ears. You'll see it in yourself because human brains need to be capable of both values.
Find a ludonomy that makes sense, and pay attention to the _Sprachspiel_ which stretches it to its limit. With any luck, these two should carve out a real manifold for you. However small your sampling of the vast ocean of knowledge, ask of it what it makes possible, more than what it shuts down and denies. Because you don't want to know how it ends, and you can't know how it ends. Stepping back from _disillusion_ to _illusion_ is not a kind of deception, rather a greater clarity of what is in and out of your control.
I don't particularly care whether life is a game or not. I'm interested in a simpler, more challenging question: how do we play with it?
# Appendix
## Players
We assumed good faith players throughout the essay, but it's worth reflecting on deviant player types and behavior.
|Type|Rule-abiding|Goal-aiming|
|-----|------------------|-----------------|
|Players|Yes|Yes|
|Triflers|Yes|No|
|Cheats|No|Yes|
|Spoilsports|No|No|
[Table [Players]: Types of players]
- Players follow the rules and are aiming for the goal.
- A trifler follows the rules but isn't interested in the goal given. They're either delaying or interested in some other goal. This is not necessarily a negative.
- A cheater wants to reach the goal but breaks rules to get there. Cheaters threaten the game.
- A spoilsport doesn't want to play. They neither follow the rules nor aim for the given goal. Spoilsports indeed spoil the sport.
## Bibliography
Images on this page are not mine. Click on them to go to the source.
[#Eysenbach22]: [Maximum Entropy RL (Provably) Solves Some Robust RL Problems](https://arxiv.org/abs/2103.06257)
[#Feezell10]: [A Pluralist Conception of Play](https://philpapers.org/rec/FEEAPC)
[#Lampinen21]: [Tell me why! Explanations support learning relational and causal structure](https://arxiv.org/abs/2112.03753)
[#Liu22]: [Mind's Eye: Grounded Language Model Reasoning through Simulation](https://arxiv.org/abs/2210.05359)
[#Ma22]: [On the Principles of Parsimony and Self-Consistency for the Emergence of Intelligence](https://arxiv.org/abs/2207.04630)
[#Wojtowicz20]: [From Probability to Consilience: How Explanatory Values Implement Bayesian Reasoning](https://arxiv.org/abs/2006.02359)
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Published 24 October 2022