FREE IMPROVISATION AND GAME THEORY : AN INTRODUCTION AND A FEW
EXPERIMENTS
First thing first : what’s Game Theory ? Here’s a short but explicit definition : Game theory is
a branch of applied mathematics that is used in the social sciences (most notably economics),
biology, engineering, political science, computer science (mainly for artificial intelligence),
and philosophy. Game theory attempts to mathematically capture behavior in strategic
situations, in which an individual's success in making choices depends on the choices of
others.
Before starting, we’ve to make a distinction between parametric situations and strategic
situations. Rational choice theory deals with the first ; Game Theory with the second. In a
parametric situations, I’m the only rational agent around : the outcome depends mainly on my
decision ; of course, there is a lot of stuff that are uncertains, that’s why rational choice theory
has a lot to do with probability theory. In strategic situations on the other hand, there are more
than one rational agents interacting, and I have to anticipate their possible choices before I
make mine. Game theory gives the rules of rational choice in a such situation.
Although music is not mentionned in the definition above, one can clearly see a link between
improvisation and the kind of situations Game Theory is supposed to modelize. Indeed,
collective improvisation, especially free, non-idiomatic, generative… (whatever the name) is
probably one of the highest possible interactive situation. Since there is no standard, no
raga/tala… before I start playing, there is nothing to tell me what to do, what is right to do,
what is of aesthetic value. Only one thing : the aesthetic value of what I will play depends on
what the other musicians will play ; and that’s the same thing for each musicians : no one
know what the other will play, yet what I do will be judged in regard of what the other has
played (or is playing). So we have a typical interactive situation ; there is no action from x on
y without an action from y on x. And this is also a strategic situation : from the many musical
possibilities, the wide range of expressions I can summon, I have to choose, I have to make a
choice : and the success of this choice depends on the choices of others (in other words, the
quality of the choice is not absolute but relative).
But the idea of mixing together music and game theory, while not new, is still one of a strange
kind. How can Game Theory, a theoretical tool whose main goal is to modelize interactive
situations between economically rational agents, help us producing music (that’s one thing) or
thinking about music (that’s another thing) ?
Iannis Xenakis, the father of this mixture, was more in this former perspective. Game Theory
was conceived as a compositionnal tool to generate unheard structures. Let’s take an
example : Duel is a piece for 2 orchestras with 2 conductors : each conductor has a set of
musical strategies (partly written, partly stochastic) for his orchestra and there is a matrix,
which is given with the score, for reading the result (the number of points you win/loose) of
playing Strategy X (as chosen by Orchestra 1) over Strategy Y (as chosen by Orchestra 2). At
the end, you can count the points and tell wich orchestra won. A matrix is just a double-entry
table where you can get the payoffs for each player in respect of the strategies chosen by each
player.
For the Duel piece, if there were only two strategies available (« silence » or « playing long
notes »), it would be something like that :
Orchestra 2
Strategy A (Silence) Strategy B (long notes)
Strategy A (Silence)
result POOR: -5, -5
result OK: 0, +5
Orchestra 1
Strategy B (long notes) result OK: +5, 0
result GOOD: +3, +3
Here’s how to read this matrix :
When Strategy Orchestra 1 plays Strategy A (Silence) and Orchestra 2 decides also to tacet,
there will be silence that is qualified by the matrix as POOR, and each player looses "5
points".
When long notes (Strategy B) are played by Orchestra 1 and Orchestra B chooses to tacet or
vice versa, the musical result is qualified as OK, the player that plays gains "5 points", while
the other gets "0".
The result qualified as GOOD is when both orchestras play long notes (Strategy B), each
player gains "3 points".
Without knoowledge of game-theory it seems obvious that the best choice for each Orchestra
is to play Strategy B (long notes). This is a very simple example, but it gets far more complex
when you have more possible strategies.
Note that the numeric values are somewhat arbitrary ; their goal is to make explicit the
aesthetic value of the possible musical results, in a ranking decided by the composer. So if the
piece works, if the matrix is well done, good players, who don’t care about music but about
making points, will make good music… !
The idea was new and exciting, but the musical result was less impressive than the theoretical
foundations used to build the piece, probably because it was too demanding both for the
conductors and the orchestras…
There are three main ideas to keep in mind when thinking of Xenakis’ approach :
1) Game Theory is used in a normative, prescriptive way. The matrix tells the players
what to do, or at least what they should do if they were rational enough to determine
the correct set of musical moves to make ; of course the game, and then the matrix, is
complex, so there is no room for strictly dominant pure strategies so you have to use
mixed strategies, that is to say strategies that assign one probability to chose this
strategy to each available strategy : it’s a sort of meta-strategy, and you need
probability theorie (and let a computer do the maths) to find the equilibrium points.
That’s why each performance of Duel is different from another one : because the
solution of the game is too complex for actual humans. But de recto, the matrix tells
what is good, what is bad (not in a moral way, obviously), in one word, what is
rational.
2) This is heteronomic music. That means we are using other laws than the law of music
(laws of tonality in western occidental music, laws of rythmic constructions in carnatic
music…) in order to build the music.
3) This is agonistic music wich is of course not the main kind of music ! But not an
inexistant one either (some examples of traditional music in Madagascar, or closer
from us, the trade-four in early jazz, or hip-hop battles)… Players are opponents and
must try to outwit themselves. If one wins, the other looses.
In our edition of Zam Lab, we’ve tried to built our own matrix with 4 strategies available for
each of the 2 players : play static, develop, play the same than the other, play the opposite
than the other. While the result was musically very interesting, it was also difficult to make
sense from the payoffs written in the matrix : you didn’t always clearly saw the link between
the musical outcome and the payoff outcome… In fact, to build this kind of matrix is clearly a
composer job, because your matrix must be done in such a way that the good improviser get
the high payoff, and that’s quite uneasy. Moreover, as the game is agonistic, you have to
make difference between the players’payoffs (it’s not just : the result is beautiful, let’s give
this two players the same number of points), so you have to take account of the risky-aspect
of their strategies : a player who chooses a risky strategy should be rewarded if it’s a good
musical choice in the context, punished otherwise ; conversely, a musician who chooses an
« easy » strategy, which works in a lot of musical context (like a pedal point or something like
that), will earn less points with that.
But somehow, we’re now off the territory of free improvisation : there are rules, and so
something like a score (the matrix). So it seems obvious that we can’t use Game Theory to
make free improvisation !
So how can we use Game Theory to think about free improvisation ? First, it seems clear that
we need to have a descriptive use of Game Theory, not a normative one. So the question is :
can we really modelize with Game Theory what is happening in actual improvisation ?
One possible answer is to see a free impro as a game of coordination, that is a game where
players manage to make their actions consistents. In other words, they have to agree on
something common.
Formally, here’s a possible definition : In game theory, coordination games are a class of
games with multiple pure strategy Nash equilibria in which players choose the same or
corresponding strategies. Coordination games are a formalization of the idea of a
coordination problem, which is widespread in the social sciences, including economics,
meaning situations in which all parties can realize mutual gains, but only by making mutually
consistent decisions. A common application is the choice of technological standards.
A typical case for a coordination game is choosing the side of the road upon which to drive, a
social standard which can save lives if it is widely adhered to. In a simplified example,
assume that two drivers meet on a narrow dirt road. Both have to swerve in order to avoid a
head-on collision. If both swerve to the same side they will manage to pass each other, but if
they choose different sides they will collide. In this case there are two pure Nash equilibria:
either both swerve to the left, or both swerve to the right. In this example, it doesn't matter
which side both players pick, as long as they both pick the same
This is a problem of pure coordination, because both players have the same preferences about
the Nash equilibrium outcome. There is different kinds of pure coordination game : matching
game (pick the same), Hi-Lo Game (one of the equilibrium is better than the other, for
example we’re all old bluesmen, and of all the possible way to play music together, we
strictly prefer to play a Blues in B-Flat).
The main problem is how we think about improvisers’preferences. Player’s preferences in
Game Theory are his ordinal ranking of all possible outcomes than can be actual as the result
of the strategic interaction. For example, at this time I would like more than anything else to
play a ballad, or something like that, but what I would really not like is ending playing In C
from Terry Riley or something like that. And so on for each player.
If each players has a different way to ranking outcomes, different preferences (wich seems to
be the most common case), but still want to make his action consistent with other’s actions,
then we have a game of impure coordination. That’s the game of the Battle of the Sexes :
In this game both players prefer engaging in the same activity over going alone, but their
preferences differ over which activity they should engage in. Player 1, the woman, prefers
that both go to the movies while player 2, the man, prefers that they both stay at home
watching football on TV.
In improvisation, that would mean : OK we want to play together, and not each of us in his
own corner, but I will prefer to play that kind of music, and you will prefer this kind and so
on… Of course, here it’s very radical (the idiom we choose, or the genre) ; but impure
coordination can be in much more subtle things.
We’ve shown that aspect of impure coordination in Zam Lab by playing what we’ve called
« The Confession Game » : a quartet was improvising, and it was recorded. At the end, the
music was rewinded, and each musician listened to it on his headphones while speaking about
it on a mic, especially stating what he was liking, what he was not liking, which direction he
would have wanted the music to take… And so, it was quite clear that
improvisers’preferences were differents, but that they were nevertheless trying to play
together in achieving a common goal.
In fact, pure coordination seems purely speculative as for free improvisation, cause it seems
crazy to imagine that musicians have the same preferences on everything. But it’s only crazy
if we maintain the individualistic view from Game Theory to think about free improvisation.
We can also see this bunch of musicians as a group, or more specifically as a team. A team is
simply a group of people with a common goal, and so common preferences. That doesn’t
mean that all the musicians in this team are clones or robots. But when they start playing
together, they constitute themselves as a team, and they engage themselves in a specific mode
of reasonning, known as team reasonning. The team becomes the main agent, with its proper
preferences (wich, of course, are not the same from the musicians per se preferences).
Let’s take an example in a family who discusses how they should spend the summer ; when it
comes to walk, they prefer walks of 6 miles or so to ones wich are much shorter or much
longer, and prefer wel-marked but uncrowded paths to ones wich are either more rugged or
more popular. When they say that they prefer that kind of walks, does it mean that each of
them prefer this ? Not quite. What I’ve called their preferences are not so very different from
those than the father have as an individual, or the mother. But the father’s ideal walk would
be somewhat longer than 6 miles, along rougher and less well-marked paths than they prefer
as a family. So « We prefere x to y » is not equivalent to « Each of us prefers x to y » ; but it
does not mean it makes no sense at all.
So when we’re doing collective free improvisation, there’s reasons to think that we’re
individually engaged in what we can call a team-directed reasonning, wich give birth to a
team agency, team preferences and a proper team reasonning. Of course, one of the main
difficulty with that approach is to gain team-confidence, that is to be sure that each one of the
group of people we see as a team is engaged in the same kind of team-directed reasonning.
So team-reasonning is like a framing effect. Instead of seeing the improvisation as « What
sould I do ? », I see it as « What should we do ? », I imagine which is the best combination of
actions for the team, and I play the strategy wich is my part in this combination.
If this idea is correct, then we can see a free improvisation as a game of pure coordination.
The problem now becomes an equilibrium-selection problem. Wich equilibrium to choose ?
We have to introduce the concept of Focal Points, first introduced by Thomas Schelling in his
book The Strategy of Conflict. A focal point is something which is salient to the cognition of
the players ; this salience can be of three main types :
1) Primary salience : if someone asks me « choose any number », I’ll say « 7 », because
it’s my magic number. So something with primary salience has a special meaning and
a special cognitive priority for an agent.
2) Secondary salience : something has secondary salience if I think it has primary
salience for another player. For exemple if I try to guess which number you have
chosen, and I answer « 13 », « 13 » has secondary salience for me (that is, I think it’s
probably the most common answer, the number with the greatest primary salience)
3) Schelling salience : something has Schelling salience if it can be generated by
following a rule of deduction wich is non-ambiguous. For example, if we’re both
asked to chose one number, and that we should try to choose the same, there’s good
probability that we both choose « 1 ». Why ? Because « 1 » as a unique feature : it’s
the first of all the numbers.
A focal point is like a mix between this three types of salience, depending of the context of
the coordination. The idea is that we can see a free improvisation as a focal-point game. As
we’ve seen, there are many many different equilibria in a coordination game. But some are
more salient than others : they are focal points. If the improvisers are really engaged in a sort
of team-reasoning, then they should coordinate on this focal points, because they are all trying
to maximize their probability of sucessfull coordination.
Much of our experiments were designed upon this conceptual articulation Primary
Salience/Secondary Salience/ Schelling Salience. We build three different situations :
1) A memory game : two players improvise freely for 1 min. Then, at a given signal, they
must try to play exactly what they’ve just played. Of course, it’s almost impossible !
The idea is to see wich moments or points of the improvisation are salient to the
memory of the players, wich are the more vivid. It’s like a concrete measurement of
primary salience in real time.
2) A coordination game : two players improvise simultaneously with headphones (so
they don’t hear each other, but only themselves) for 3 min. Then, at a given signal,
they start to hear each other, and they must now quit their solos and make a proper
duo. We can observe how the coordination is made in real time. Each musician has his
own energy , and his own memory of what just happened. But they have suddenly to
act as a team in order to find right away (in a few seconds or so!) the proper focal
point which will make the coordination possible.
3) A Solos/Tutti settings : we ask 3 musicians to make 3 short solos (1 min), one at a
time. Then, at a given signal, they must freely improvise a trio which takes as a point
of departure one of the previous solo. Here, we were trying to show wich cognitive
procedure was used for coordination between secondary salience type of reasonning
(that is the guessing procedure : knowing these musicians, they will prefer this solo, so
I should go with it) and Schelling salience reasonning (if they collectively decide than
one of the solo had a very unique and special feature). Not surpinsingly, the results
were very mixed, so it’s difficult to say which procedure is more likely to be used…
So we’ve tried to use Game Theory as a descriptive tool to think about improvisation, and
we’ve made some experiments to highlight the coordination devices that were implied by
every free improvisations : are they team preferences ? Are we playing a game of pure
coordination or a game of impure coordination ? How is salience generated ? Is it only a
feature of a common cultural environnement, or can it be build in real time during the
improvisation, even with people from very different musical backgrounds (as it was the
case in this edition of Zam Lab) ? Game Theory is a very powerful conceptual tool, and it
can probably help us thinking about free improvisation, and analyzing the cognitive
procedures underlying free improvisation, mainly under the category of coordination.
This edition of Zam Lab was a great example of what can be done in this new and original
field !
Clément CANONNE (December 08)
Université Jean Monnet de Saint-Etienne/ Ecole Normale Supérieure Lettres et Sciences Humaines