Blutner: "Some Aspects of Optimality in Natural Language" (2000)

This paper by Reinhard Blutner is the one that introduced the idea of "bidirectional" optimality theory. Bidirectional optimality theory is bidirectional because it doesn't just optimize an input parse or an output expression, but does both at the same time. The whole speech situation thus defines a little game, and Blutner defines an equilibrium concept for such speaking/hearing games.

The Phenomenon in Focus

The observation that drives Blutner's idea is that non-standard forms tend to designate non-standard referents. So the straightforward formulation I killed him designates a stereotypical killing-event, while I caused him to die designates an atypical one (p. 9).

This can be explained economically if we assume that being ambiguous is more "costly" than being brief. Then the pairing (s, t')—i.e., unmarked sentence with marked meaning—is suboptimal because it makes the unmarked sentence ambiguous. It also makes sense information-theoretically, because you want to reserve the short sentences for the frequently occurring referents.

Blutner's Idea

It seems that Blutner wants to arrive at this conclusion, too, but from a different angle. Imagine that we still only have two signals and two interpretations, and we put them into a table like this:

	t	t'
s	–m, –m	–m, +m
s'	+m, –m	+m, +m

Here, being marked is taken to be bad, so we can imagine that people try to choose cells with as many occurrences of "–m" as possible. This means that the top left cell is the best one, corresponding to a pairing of unmarked form with unmarked meaning.

We accordingly take the whole first row and first column out of the game, since they have now been coupled with something. We are then left with a reduced subgame:

	t	t'
s	–m, –m	–m, +m
s'	+m, –m	+m, +m

In this subgame, both players have but a single option, so this obviously becomes optimal. We thus couple the marked form with the marked meaning, and we're done.

The same little game could of course be played with more signals and more meanings. We would then couple forms to meanings in increasing order of  markedness, until one of the sets had been exhausted. We could also play it with a single meaning and two forms, as his fury/furiosity example points out.

A More Formal Version of The Idea

In an attempt to capture the dynamics of this process, Blutner suggests the following definition (p. 11), here in a slightly reformulated version:

(s,t) is super-optimal if s is a candidate reading of t, and if
Q: There is no other pair (s',t) that satisfies I as well as u(s',t) ≥ u(s,t).
I: There is no other pair (s,t') that satisfies Q as well as u(s,t') ≥ u(s,t).

A different way of putting this is to translate it into an update procedure:

Let a table of numbers be given.

Randomly write crosses in some cells and nothing in others. This is your start configuration.

Then repeat the following loop:

    For each cell c:

        Find the set of competitors; these are the cells that
        are in the same row or the same column as c and are
        marked with a cross in the current configuration.

        If the number in c is larger than the number in all
        of these competing cells, give it a cross in the
        following configuration.

    If the last configuration is equal to the next, halt.

I realize this is not as romantic as a circular definition, but it is easier to apply. For instance, let's imagine we're starting with the following table:
 

8	9	7
3	2	1
5	6	4

If we start with a completely empty start configuration (no crosses in any cells), then we can apply the procedure 6 times before we arrive at the following configuration of crosses:

	×
		×
×

At this point, we have indeed reached a fixed point: No crossed cell can out-compete another crossed cell. Note that this is achieved by having the characteristic one-to-one mapping between forms and meanings.