Category Archives: philosophy

Goals, Anti-Goals and Multi-player Games

In this article I will try to address Keith Burgun‘s assertion that games should have a single goal and his analysis of certain kinds of goals as trivial or pathological. I will try to demonstrate that multi-player games either reduce to single player games or necessitate multiple goals, some of which are necessarily the sorts of goals which Burgun dismisses as trivial. I’ll try to make the case that such goals are useful ideas for game designers as well as being necessary components of non-trivial multi-player games.

(Note: I find Keith Burgun’s game design work very useful. If you are interested in game design and have the money, I suggest subscribing to his Patreon.)

Notes on Burgun’s Analytical Frame

The Forms

Keith Burgun is a game design philosopher focused on strategy games, which he calls simply games. He divides the world of interactive systems into four useful forms:

  1. toys – an interactive system without goals. Discovery is the primary value of toys.
  2. puzzle – bare interactive system plus a goal. Solving is the primary value of the puzzle.
  3. contests – a toy plus a goal all meant to measure performance.
  4. games – a toy, plus a goal, plus obfuscation of game state. The primary value is in synthesizing decision making heuristics to account for the obfuscation of the game state.

A good, brief, video introduction to the forms is available here. Burgun believes a good way to construct a game is to identify a core mechanism, which is a combination of a core action, a core purpose, and a goal. The action and purpose point together towards the goal. The goal, in turn, gives meaning to the actions the player can take and the states of the interactive system.

On Goals

More should be said on goals, which appear in many of the above definitions. Burgun has a podcast which serves as a good long form explication of many of his ideas. There is an entire episode on goals here. The discussion of goals begins around the fifteen minute mark.

Here Burgun provides a related definition of games: contests of decision making. Goals are prominent in this discussion: the goal gives meaning to actions in the game state.

Burgun raises a critique of games which feature notions of second place. He groups such goals into a category of non-binary goals and gives us an example to clarify the discussion: goals of the form “get the highest score.”

His analysis of the poorness of this goal is that it seems to imply a few strange things:

  1. The player always gets the highest score they are capable of because the universe is deterministic.
  2. These goals imply that the game becomes vague after the previous high score is beaten, since the goal is met and yet the game continues.

The first applies to any interactive system at all, so isn’t a very powerful argument, as I understand it. Take a game with the rules of Tetris except that the board is initialized with a set of blocks already on the board. The player receives a deterministic sequence of blocks and must clear the already present blocks, at which point the game ends. This goal is not of the form “find the highest score” or “survive the longest” but the game’s outcome is already determined by the state of the universe at the beginning of the game. From this analysis we can conclude that if (1) constitutes a downside to the construction of a goal, it doesn’t apply uniquely to “high score” style goals.

(2) is more subtle. While it is true that in the form suggested, these rules leave the player without guidelines after the goal is met, I believe that in many cases a simple rephrasing of the goal in question resolves this problem. Take the goal:

G: Given the rules of Tetris, play for the highest score.

Since Tetris rewards you for clearing more lines at once and since Tetris ends when a block becomes fixed to the board but touches the top of the screen, we can rephrase this goal as:

G': Do not let the blocks reach the top of the screen.

This goal is augmented by secondary goals which enhance play: certain ways of moving away from the negative goal G' are more rewarding than others. Call this secondary goal g: clear lines in the largest groups possible. Call G' and goals like it “anti-goals.”

This terminology implies the definition.

If a goal is a particular game state towards which the player tries to move, an anti-goal is a particular state which the player is trying to avoid. Usually anti-goals are of the form “Do not allow X to occur” Where X is related to a (potentially open ended) goal.

Goals of the “high score” or “survive” variety are (or may be) anti-goals in disguise. Rephrased properly, they can be conceived of in anti-goal language. Of course there are good anti-goals and bad ones, just as there are good goals and bad goals. However, I would argue that the same criteria applies to both types of goals: a good (anti) goal is just one which gives meaning to the actions a person is presented with over an interactive system.

Multi-Player Games and Anti-Goals

I believe anti-goals can be useful game design, even in the single player case. In another essay I may try to make the argument that anti-goals must be augmented with mechanics which tend to move the player towards the anti-goal against which players must do all the sorts of complex decision making which produces value for players.

However, there is a more direct way of demonstrating that anti-goals are unavoidable aspects of games, at least when games are multi-player. This argument also demonstrates that games with multiple goals are in a sense inevitable, at least in the case of multi-player games. First let me describe what I conceive of as a multi-player game.

multi-player game: A game where players interact via an interactive system in order to reach a goal which can only be attained by a single player.

The critical distinction I want to make is that a multi-player game is not just two or more people engaged in separate contests of decision making. If there are not actions mediating the interaction of players via the game state then what is really going on is many players are playing many distinct games. A true multi-player game must allow players to interact (via actions).

In a multi-player game, players are working towards a win state we can call G. However, in the context of the mechanics which allow interaction they are also playing against a (set of) anti-goals {A}, one for each player besides themselves. These goals are of the form “Prevent player X from reaching goal G“. Hence, anti-goals are critical ingredients to successful multi-player game design and are therefore useful ideas for game designers. Therefore, for a game to really be multi-player then there must be actions associated with each anti-goal {A}.

An argument we might make at this point is that if players are playing for {A} and not explicitly for G then our game is not well designed (for instance, it isn’t elegant or minimal). But I believe any multi-player game where a player can pursue G and not concern herself with {A}, even in the presence of game actions which allow interaction, is a set of single player games in disguise. If we follow our urge to make G the true goal for all players at the expense of {A} then we may as well remove the actions which intermediate between players and then we may as well be designing a single player game whose goal is G.

So, if we admit that multi-player games are worth designing, then we also admit that at least a family of anti-goals are worth considering. Note that we must explicitly design the actions which allow the pursuit of {A} in order to design the game. Ideally these will be related and work in accord with the actions which facilitate G but they cannot be identical to those mechanics without our game collapsing to the single player case. We must consider {A} actions as a separate (though ideally related) design space.

Summary

I’ve tried to demonstrate that in multi-player games especially, anti-goals, which are goals of the for “Avoid some game state”, are necessary, distinct goal forms worth considering by game designers. The argument depends on demonstrating that a multi-player game must contain such anti-goals or collapse to a single player game played by multiple people but otherwise disconnected.

In a broader context, the idea here is to get a foot in the door for anti-goals as rules which can still do the work of a goal, which is to give meaning to choices and actions in an interactive system. An open question is whether such anti-goals are useful for single player games, whether they are useful but only in conjunction with game-terminating goals, or whether, though useful, we can always find a related normal goal which is superior from a design point of view. Hopefully, this essay provides a good jumping off point for those discussions.


Quick, Probabily Naive Thoughts about Turing Machines and Random Numbers

Here is a fact which is still blowing my mind, albeit quietly, from the horizon.

Turing Machines, the formalism which we use to describe computation, do not, strictly speaking, cover computational processes which have access to random values. When we wish to reason about such machines people typically imagine a Turing Machine with two tapes, one which takes on the typical role and another which contains an infinite string of random numbers which the machine can peel off one at a time.

Screen Shot 2016-05-30 at 9.42.43 AM

I know what you are all thinking: can’t I just write a random number generator and put it someplace on my turing machine’s tape, and use that? Sure, but those numbers aren’t really random, particularly in the sense that a dedicated attacker, having access to the output of your turing machine can in principle detect the difference between your machine and one with bona fide random numbers if it has access to your outputs. And, in fact, the question of whether there exists a random number generator which uses only polynomial time and space such that a polynomial time and space algorithm is unable to detect whether the numbers derive from a real random process or an algorithm is still open.

All that is really an aside. What is truly, profoundly surprising to me is this: a machine which has access to random numbers seems to be more powerful than one without random numbers. In what sense? There are algorithms which are not practical on a normal turing machine which become immanently practical on a turing machine with a random tape as long as we are able to accept a vanishingly small probability that the result is wrong. Algorithms about which we can even do delta/epsilon style reasoning: that is, we can make the probability of error as small as we like by the expedient of repeating the computation with new random numbers and (for instance) using the results as votes to determine the “correct answer.” This expedient does not really modify the big O complexity of algorithms.

Buridan’s Ass is a paradox in which a hungry donkey sits between two identical bales of hay and dies of hunger, unable to choose which to eat on account of their equal size. There is a strange sort of analogy here: if the Ass has a source of random numbers he can pick one randomly and survive. It is almost as if deterministic, finitist mathematics, in its crystalline precision, encounters and wastes energy on lots of tiny Ass’ Dilemmas which put certain sorts of results practically out of reach, but if we fuzz it up with random numbers, suddenly it is liberated to find much more truth than it was before. At least that is my paltry intuitive understanding.

Notes on `Quantum Computing Since Democritus, Chapter 1`

For a long time, I’ve been interested in the sorts of questions exemplified by the following example:

Suppose we are Isaac Newton or  Gottfried Leibniz. We have at our disposal two sources of inspiration: data, collected by intrepid philatelists like Tycho Brahe and something like theory, in the form of artifacts like Kepler’s Laws, Galileo’s pre-Newtonian laws of motion (for it was he who first suggested that objects in motion retain that motion unless acted upon), and a smattering of Aristotelian and post-Aristotelian intuitions about motion (for instance, John Philoponus’ notion that, in addition to the rules of motion described by Aristotle, one object could impart on another a transient impetus). You also have tables and towers and balls you can roll on them or drop from them. You can perform your own experiments.

The question, then, is how do you synthesize something like Newton’s Laws. Jokes about Newton’s extra-scientific interests aside, this is alchemy indeed, and an alchemy to which most training physicists receive (or at least I received) does not address itself.

Newton’s Laws are generally dropped on the first year physics student (perhaps after working with statics for awhile) fully formed:

First law: When viewed in an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by an external force.[2][3]
Second law: The vector sum of the external forces F on an object is equal to the mass m of that object multiplied by the acceleration vector aof the object: F = ma.
Third law: When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.

(this formulation borrowed from Wikipedia)

The laws are stated here in terms of a lot of subsidiary ideas: inertial reference frames, forces, mass. Neglecting the reference to mathematical structures (vector sums), this is a lot to digest: and it is hard to imagine Newton just pulling these laws from thin air.  It took the species about 2000 years to figure it out (if you measure from Zeno to Newton, since Newton’s work is in some sense a practical rejoinder to the paradoxes of that pre-Socratic philosopher), so it cannot be, as some of my colleagues have suggested, so easy to figure out.

A doctorate in physics takes (including the typical four year undergraduate degree in math, physics or engineering) about ten years. Most of what is learned in such a program is pragmatic theory: how to take a problem statement or something even more vague, identify the correct theoretical approach from a dictionary of possibilities, and then to “turn the crank.” It is unusual (or it was unusual for me) for a teacher to spend time posing more philosophical questions. Why, for instance, does a specific expression called the “Action,” when minimized over all possible paths of a particle, find a physical path? I’ve had a lot of physicist friends dismiss my curiosity about this subject, but I’m not the only one interested (eg, the introductory chapter of Lanczos’ “The Variation Principles of Mechanics”).

What I am getting to here, believe it or not, is that I think physicists are over-prepared to work problems and under-prepared to do the synthetic work of building new theoretical approaches to existing unsolved problems. I enjoy the freedom of having fallen from the Ivory Tower, and I aim to enjoy that freedom in 2016 by revisiting my education from a perspective which allows me to stop and ask “why” more frequently and with more intensity.

Enter Scott Aaronson’s “Quantum Computing Since Democritus,” a book whose title immediately piqued my interest, combining, as it does, the name of a pre-Socratic philosopher (the questions of which form the basis, in my opinion, for so much modern physics) with the most modern and pragmatic of contemporary subjects in physics. Aaronson’s project seems to accomplish exactly what I want as an armchair physicist: stopping to think about what our theories really mean.

To keep myself honest, I’ll be periodically writing about the chapters of this book – I’m a bit rusty mathematically and so writing about the work will encourage me to get concrete where needed.

Atoms and the Void

Atoms and the Void is a short chapter which basically asks us to think a bit about what quantum mechanics means. Aaronson describes Quantum Mechanics in the following way:

Here’s the thing: for any isolated region of the universe that you want to consider, quantum mechanics describes the evolution in time of the state of that region, which we represent as a linear combination – a superposition – of all the possible configurations of elementary particles in that region. So, this is a bizarre picture of reality, where a given particle is not here, not there, but in a sort of weighted sum over all the places it could be. But it works. As we all know, it does pretty well at describing the “atoms and the void” that Democritus talked about.

The needs of an introductory chapter, I guess, prevent him from describing how peculiar this description is: for one thing, there is never an isolated region of the universe (or at least, not one we are interested in, I hope obviously). But he goes on to meditate on this anyway by asking us to think about how we interpret measurement where quantum mechanics is concerned. He dichotimizes interpretations of quantum mechanics by where they fall on the question of putting oneself in coherent superposition.

Happily, he doesn’t try to claim that any particular set of experiments can definitely disambiguate different interpretations of quantum mechanics. Instead he suggests that by thinking specifically of Quantum Computing, which he implies gets most directly at some of the issues raised by debates over interpretation, we might learn something interesting.

This tantalizes us to move to chapter 2.