Breaking the Rules
==================
By Steve Gardner

1. Retractable and non-retractable errors

The question I wish to address in this essay is an easy one to state:
how should play proceed in a game when the rules have been broken?
This is a question which even relatively simple games must address in
some way, for whenever people engage in any rule-governed activity, the
possibility of the rules being broken must always be present. However,
we shall see that although it is a quite straightforward matter to
deal with breaches of the rules in simple games, for more complex
games, this problem becomes one of considerable subtlety.

It is natural that this aspect of game play should in its turn be
governed by rules. However, these rules are of a different character
to the rules which govern the ordinary processes of play, for they
speak not only about those processes, but also about the rules which
govern them, and how they are to be applied. Hence it is natural to
refer to the rules which govern play when the (other) rules have been
broken as *meta-rules*. I will follow this usage for the rest of
this essay.

For simple games, the meta-rules will usually be implicit. Consider
one of the simplest and most widely known games, tic-tac-toe. Like
many games, it is played on a grid, and a legal move, it is generally
understood, consists of placing a marker, an 'X' or an 'O', in one of
the squares of the grid.  It is not a legal move to place a marker
straddling two or more squares. But what if, to your surpise, your
opponent were to play in this fashion? 

Let us begin our discussion by assuming that the illegality of this
move is immediately detected. Although this will seem a trivial point
in relation to tic-tac-toe, it will later assume considerable
significance, so bear with me. Later we shall be able to relax this
assumption, and see what ensues.

A very natural response to this kind of attempted move, which I
shall call variously a breach of the rules, a violation of the rules,
or just an error, would be to point out that it is not according to
the rules of tic-tac-toe, and to demand that it be retracted. (It is
irrelevant for our purposes exactly how one might go about proving
that this move is in fact illegal according to the rules of tic-tac
-toe. Let us stipulate that you are able to demonstrate this to the
satisfaction of your opponent. Perhaps she is a small child to whom
you are teaching the game, who will take your word in these matters
as authoritative.) What happens then? Well, if your opponent agrees
that her move was illegal, then she retracts it.

But what is meant by 'retraction'? There are two possible senses to
distinguish: it might mean that the move is considered to have been
played, and then taken back; or it might mean that the move is
considered not to have been played at all. 'Retraction', as I am using 
the word, has the second of these senses. For those who find it difficult
to distinguish these cases, we can think of the difference in terms
of what a recorder of the moves of the game would normally record. 
Consider for example the illegal opening move in chess: 1. Nb3. Let
us imagine tham I am playing chess with you and that I make this
illegal move. You point out that this move is illegal, and I agree,
so I put the knight back on b1 and instead play 1. Nc3. A recorder of
the moves of this chess game will record this as 1. Nc3, not as 1.
Nb3 (illegal move) Nb1 (retraction) Nc3.  This is in accordance with
the convention that, for ordinary purposes, *only the legal moves of a
game count as part of the game*. For instance, if you want to
reconstruct the chess game which we played subsequent to my illegal
move, you will not include the illegal move as a part of your
reconstruction. You will not begin by saying, "Well, first you played
your knight to b3, then you took it back to b1, then you played it to
c3...". You will quite rightly take it that the game begins with my
move 1. Nc3. So, when I say that a move is retracted, I shall mean
that it is considered not to have taken place, that it is not considered
to have been a part of the game. We shall see that does not prevent the
rules of a game from recognizing, and if appropriate, penalizing, an
*attempt* to make such a move.

Returning to our tic-tac-toe example: your opponent makes an illegal
move by placing her marker straddling two squares. You point out
that this move is illegal. She retracts the move, and (abstracting
away from complications to be introduced presently), makes another,
legal, move instead. That the players follow this course of action
and not some other (which is perfectly possible), is dictated by a
meta-rule of tic-tac-toe, which must say, in effect:

      Markers which are not placed legally on the grid, as described
      in other rules, shall be removed from the grid, and the player
      whose turn it was shall continue by making a legal play, if one
      is possible.

Generalizing, the meta-rule says, essentially, this: 

(1)   Illegal moves are retracted, and play then continues according
      to other rules as if the illegal move had not been played. 

This is a meta-rule for many simple games, and can even serve
reasonably well in some quite complex games. Of course, for complex
games, the meta-rules dealing with violations are much more likely to
be explicitly included in the rules of the games themselves. This is
only natural, since having more complicated rules makes it more
likely that some players will misunderstand them, and hence try to
move in ways which are in fact illegal. It is perhaps in order to
provide an incentive to players to learn the rules quickly, and hence
to expedite the smooth flow of the game by reducing attempts at
illegal moves, that many games introduce complications of (1), along
the lines of

(2)   Illegal moves are retracted, and the player making the illegal
      move suffers a game penalty. With this exception, play then
      continues as if the illegal move had not been played.

Meta-rule (2) shows that it is not essential, when errors occur, that
the course of play be completely unaffected by them. Meta-rule (2)
can serve as the template for meta-rules even for highly evolved,
complex and sophisticated games. What is more, (2) can itself be
further complicated into

(3)   Illegal moves are retracted, and the player making the illegal
      move either suffers a game penalty, or is forced to play in some
      specified manner, or both. Play then continues according to other
      rules.

A meta-rule like (3) is already sophisticated enough to be used in
chess: if a player moves a piece illegally, we may require that she
lose 1 minute on her clock, and must make a legal move with the piece
she attempted to move illegally, if that is possible. Play then
continues normally.

Note that in moving from (2) to (3) we have moved still further away
from a treatment of error which tries to restore the former game
state precisely to whatever it was before the error was made (which
was our response to errors made playing tic-tac-toe). Nevertheless,
meta-rules (1) to (3) still have in common the concept of *retraction*, 
and I will refer to errors which are treated in this way as *retractable 
errors*.

The question now arises: are all errors retractable? It is easy to 
see that they are not. There are some errors the nature of which
guarantee that they are not retractable. For example, consider a game
whose rules require that an action be performed within a certain time.
Then clearly, failure to perform that action within the required time
constitutes a breach of the rules, and merits any penalties which are
incurred for such breaches. But the idea of retracting this error is
a nonsense; that would require that we somehow change the past.
Similarly, if the rules of a game require that certain information
remains secret from some or all of the players, then the revelation
of this information to those players likewise constitutes a breach of
the rules that can be penalized, but which cannot be undone. Errors
such as these, which cannot in principle be retracted, I shall call,
unimaginatively, *non-retractable errors*. Examples of non-retractable
errors are such things as fouls in various sports (soccer, basketball,
snooker, etc), or misplays in card games such bridge or poker.

The meta-rules for dealing with non-retractable errors will, of
course, say nothing about retraction. They will say simply 

(4)   A player who commits a (non-retractable) illegal move incurs
      a game penalty, or is forced to play in some specified manner,
      or both. Play then continues according to other rules.


2. Error detection and retractability

So far, so good. We appear to have a nice, clear-cut distinction
between retractable and non-retractable errors. We can use the
distinction to classify all the different errors which can arise in
game play, and the distinction will also tell us the appropriate
kinds of meta-rules which our games should include for dealing with
them. 

However, this pleasing result was achieved only under an important
assumption: that errors are detected as soon as they occur. When we
relax this assumption, a worrying new phenomenon emerges, one which
will require us to complicate our account of breaking the rules. For
if we allow that errors may go undetected for some time, then we must
face the possibility that when an error is detected (if it ever is),
that it may be practically impossible, or at the very least
undesirable, for it to be retracted. For non-retractable errors, this
poses no problem, since we do not expect them to be retracted in any
case. But for retractable errors, this possibility raises new
questions, and it is therefore on retractable errors that I now wish
to concentrate.

For retractable errors, a new distinction is forced upon us: between
those retractable errors which may be retracted in practice, and
those which may only be retracted in theory, but which in practice
are not retractable. This last class of errors, the errors
retractable in theory but not in practice, seems to occupy a kind of
middle ground between the retractable and non-retractable errors, and
so threatens to undermine our neat distinction. Furthermore, the
existence of this class of errors leaves us wondering whether or not
we ought to say of them that they are considered to be a part of the
game. For recall that it was the distinguishing mark of retractable
errors that they are retracted, that is, considered not to have been
a part of the game. But if we are to have theoretically retractable
errors which are not in fact retracted, that would appear to leave us
in the uncomfortable position of admitting that some illegal moves
are to be considered to have been a part of the game. I shall later
argue that this position is not as uncomfortable as I have made it
sound, but first a little more needs to be said to clarify the
foregoing.

In particular there is the matter of why there should be a class of 
not practically retractable errors at all. Specifically, why is the
existence of these errors related to the time of detection? The
answer is intuitively obvious, but worth spelling out: it is that
during the delay between the time the error is committed and the
time it is detected, the game goes on. By the time the error is
detected, the game may have evolved to a state which either makes it
very difficult to return to the state of the game at the time the
error occurred, or undesirable to do so, or both. It will be
difficult in proportion to the difficulty or inconvenience of
reconstructing the state of the game at the time the error occurred,
and this difficulty will depend on the number of different kinds of
entities described by the rules, and the complexity of the
relationships between them. And it may be undesirable to return to
the game state at the time the error was committed, not just because
of the practical difficulties involved in reconstructing that state
of the game, but also because the evolved state, including the error,
might now hold some intrinsic interest for the players of the game.
It may be this game state, error or no error, which now holds
interest for them, and from which they now wish to continue the game.

They may, however, feel some disquiet about doing so. For it seems
that to ignore the existence of a retractable breach of the rules,
and continue playing the game regardless, runs counter to every
instinct about why people play games in the first place. That
instinct seems to tell us that respect for the rules of play must be
absolutely paramount; that the rules may not be ignored for the sake
of convenience; that if players do not try to adhere to the rules,
then their activity is without meaning.

This disquiet, although powerful, is misplaced. To forestall
immediate objections, I should add that I don't think there is
anything wrong with the intuitions which drive it. Rather, I think
the disquiet is produced by a failure to interpret broadly enough the
phrase 'the rules of the game', as it appears above in my attempt to
capture our intuitions. In particular, the worry can emerge only if
'the rules of the game' are being thought of as explicitly not
including (some of) the meta-rules for dealing with errors.

An example will help us see thing more clearly. Consider again you
and I, playing chess. Imagine that about ten moves into the game, I
accidentally knock a piece off the edge of the board, and that,
distracted by my thoughts of the game, I absent-mindedly misplace the
piece back on a different square than the one from which it came.
However, so absorbed are we in our contest that neither of us notice
the error -- until, some 30 moves later, you suddenly exclaim "Good
heavens! However did that piece come to be there? You must have
misplaced it back on the board when you knocked it off!" Somewhat
embarrassed, I agree that this must have been what happened. How
ought we to proceed?

If this were a tournament game, with much hanging on the outcome,
then the meta-rules are clear about this situation. The game must be
replayed from the point at which the error occurred. This is the way
in which the bodies which administer tournament chess play respect
the important intuitions I outlined above: by including a meta-rule
like (1), (2) or (3) above in the rules for tournament chess. In
the language I have introduced, we might say that in tournament chess,
nearly all errors are retractable, in practice as well as in theory.

However, in a social game, things may very well go quite differently.
Suppose that you and I are especially interested in the current
position. Then we may agree to play on from that position. The
question is: can a purist legitimately argue that in this case we
are simply ignoring the rules of chess as a matter of convenience
to ourselves? Can she argue that we have, notwithstanding our 
intuitions about the importance of respect for the rules, included
an illegal move as part of our game?

I don't think so. I think instead that we simply play according to
a different meta-rule than the one found in the rules for tournament
play. Instead, we employ a meta-rule for not practically retractable
errors, one which is more appropriate for the context of social play:

(5)  Illegal moves are retracted, unless they go undetected for a
     sufficiently long period. In that case, they are considered
     to have been legal moves.

This meta-rule for dealing with not practically retractable errors
differs from meta-rules (1) to (4) above in a crucial respect: it
mentions move legality. The meta-rules we have considered to this
point have no effect on what constitutes a legal move; they only tell
us how to proceed when the rules have been broken. Meta-rule (5), on 
the other hand, does say something substantial about which moves are
legal: it tells us that if a breach of the rules goes undetected long
enough, then by virtue of its having remained undetected, it in
effect inherits the status of a legal move. A recorder of the moves
of the game would be required to record it as having been a part of
the game.

The effect of this meta-rule, then, is to subtly alter the meaning
of 'legal move' in the game to which it applied. So to the extent
that this meta-rule (or one like it) is accepted for social chess,
the meaning of 'legal move' in social chess is correspondingly
altered. The effect of the alteration is to add a rider to the other
definitions of which moves are legal, one which says that moves which
would otherwise be illegal are legal if the illegality goes
undetected long enough.

The answer to the question of how long is long enough will of course
vary from game to game. It is perfectly possible (although rarely
done in practice), to explicitly include meta-rules in the rules for
a game which answer this question. More often, such rules will be
implicit, perhaps even negotiated among the players when a previously
undetected error is detected. There may well be disagreements among the
players about whether long enough has passed since the time the error
was committed to grant the move, otherwise illegal, the status of a
legal move which meta-rule (5) confers on it. But this is no concern
of mine. Such disagreements may be settled according to whatever
conventions are in place for settling disagreements among players
(the majority decides, or the player most familiar with the rules
adjudicates, or the player providing the materials for the game 
adjudicates, etc), or even according to other rules for settling
disagreements, if such there be. If the disagreement persists and
cannot be resolved, then certainly further play will be difficult,
if not impossible, for the players to undertake. But none of this
poses a threat to the formulation of meta-rule (5) as the right
kind of meta-rule to deal with not practically retractable errors.

There is an obvious objection to my treatment of the not practically
retractable errors, and to my formulation of meta-rule (5) above. 
The objection is to the way in which meta-rule (5) confers legality
on otherwise illegal moves, if they go undetected long enough. To
some, it will seem that this is undesirable, just because it makes
legal otherwise illegal moves, and furthermore that it is unnecessary, 
because meta-rule (5) could be reformulated without mentioning move
legality. Hence the objection is to the effect that the meta-rule for
not practically retractable errors would be better formulated as 

(5') Illegal moves are retracted, unless they go undetected for a
     sufficiently long period. In that case, they are allowed to
     stand, although they are still considered to have been illegal.

Of course, if one agrees that it is preferable not to grant moves which
are otherwise illegal the status of legal moves under any circumstances,
then the above argument will appear compelling. On the other hand,
taking this path forces one to confront squarely the fact that the
non-retraction of a retractable error will entail the incorporation
into the game state of an confessedly illegal move. It is clear that
we are faced with a trade off between on the one hand making some
illegal moves legal (as in meta-rule (5)), with the price that we
weaken to some extent the definition of 'legal move' to include
otherwise illegal moves which go undetected for a sufficently long
period, and on the other hand insisting on maintaining the strength of
whichever definition of 'legal move' is in play (as in meta-rule
(5')), at the cost of occasionally admitting that moves contrary to
the rules were allowed in the course of play. I find this latter
alternative distasteful, although by so saying I concede that the
matter is to some extent a matter of taste. For my part, the central
intuition which I want to preserve intact is the one about the 
conventions for recording moves: that only the legal moves of the
game count as part of the game, and hence that a recorder of the
game should write down all and only these moves. If that means that
we retrospectively grant legality to illegal moves which it would 
be undesirable or impractical to retract, then so be it. 


Steve Gardner                     |  Some people lose their sense of
Dept. of Philosophy, Monash Uni.  |  perspective. I've lost my sense
gardner@aurora.cc.monash.edu.au   |  of scale.          -- Will Self