On the Necessity of Rules in Ensemble Coordination

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On the Necessity of Rules in Ensemble Coordination

• Søren R. Frimodt-Møller, PhD Fellow, Institute
  for Philosophy, Education and the Study of
  Religions, University of Southern Denmark. Ghent,
  February 19, 2009

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Contents

1. A Coordination Problem in a Music Ensemble
2. Modeling Coordination in Terms of Traditional
   Epistemic Logic
3. Modeling in Terms of Variable Frame Theory
4. Modeling in Terms of Logic for Intentions
5. Conclusion

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Modeling Coordination in Terms of Traditional
Epistemic Logic
An idealized passage from a fictitious score
Bar 1 Bar 2 Bar 3 Bar 4 Bar 5
oboe Phrase1 Phrase1 Phrase2 Phrase1 Phrase1
violin Phrase3 Phrase3 Phrase3 Phrase4 Phrase3
cello Phrase5 Phrase5 Phrase5 Phrase5 Phrase6
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Modeling Coordination in Terms of Traditional
Epistemic Logic

• In the spirit of Fagin et al Reasoning About
  Knowledge, MIT Press 1995, let us model the
  performance situation as a multi-agent system,
  more specifically a system of information states
  developing over time.
• Intuitively, we let the label of the musical
  phrase denote the information state of a player
  playing that phrase.
• We model time as a stepwise development where
  each step is the length of an arbitrary bar in
  the score.

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Modeling Coordination in Terms of Traditional
Epistemic Logic

• We call the information state of a given player
  i, the local state of i. For each i, we have a
  set of local states Li, such that
• Loboephrase1,phrase2
• Lviolinphrase3, phrase4
• Lcellophrase 5, phrase6
• (We could also add a state ? to each set Li,
  denoting that nothing is played, but we choose to
  omit this here for clarity.)

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Modeling Coordination in Terms of Traditional
Epistemic Logic

• We define a global state as a set of the local
  states of each player at a given time point m,
  m?0,1.
• r(m) (soboe, sviolin, scello), where si is the
  local state of player i.
• The function r is called a run and describes a
  development of the global state over time.
• The multi-agent system can be described as a set
  of runs over the set of possible global states.

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Modeling Coordination in Terms of Traditional
Epistemic Logic

• In our system, we can think of the runs as
  different performances
• We define a player is local state at a given
  time m in a given run r as ri(m).
• We say that i cannot distinguish between two
  global states r(m) and r(m), if i has the same
  local (information!) state at both of these
  global states, r(m) i r(m), if ri(m) ri(m)

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Examples of different possible runs (performances)
rlateoboe
m1 m2 m3 m4 m5
oboe Phrase1 Phrase1 Phrase1 Phrase2 Phrase1
violin Phrase3 Phrase3 Phrase3 Phrase4 Phrase3
cello Phrase5 Phrase5 Phrase5 Phrase5 Phrase6
rlateoboe(3) i rlateviolin(3) rlateoboe(3) i
rviolinwaits(3), rlateoboe(3) i rscorevar1(3),
rlateoboe(3) i rscorevar2(3) and (rlateoboe(3)
i rviolinwaits(3)
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Examples of different possible runs (performances)
rlateviolin
m1 m2 m3 m4 m5
oboe Phrase1 Phrase1 Phrase1 Phrase2 Phrase1
violin Phrase3 Phrase3 Phrase3 Phrase3 Phrase4
cello Phrase5 Phrase5 Phrase5 Phrase5 Phrase6
rlateoboe(3) i rlateviolin(3) rlateoboe(3) i
rviolinwaits(3), rlateoboe(3) i rscorevar1(3),
rlateoboe(3) i rscorevar2(3) and (rlateoboe(3)
i rviolinwaits(3)
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Examples of different possible runs (performances)
rviolinwaits
m1 m2 m3 mt mt1
oboe Phrase1 Phrase1 Phrase1 Phrase2 Phrase1
violin Phrase3 Phrase3 Phrase3 Phrase3 Phrase4
cello Phrase5 Phrase5 Phrase5 Phrase5 Phrase5
rlateoboe(3) i rlateviolin(3) rlateoboe(3) i
rviolinwaits(3), rlateoboe(3) i rscorevar1(3),
rlateoboe(3) i rscorevar2(3) and (rlateoboe(3)
i rviolinwaits(3)
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Examples of different possible runs (performances)
rscorevar1
m1 m2 m3 m4 m5
oboe Phrase1 Phrase1 Phrase1 Phrase1 Phrase1
violin Phrase3 Phrase3 Phrase3 Phrase4 Phrase3
cello Phrase5 Phrase5 Phrase5 Phrase5 Phrase6
rlateoboe(3) i rlateviolin(3) rlateoboe(3) i
rviolinwaits(3), rlateoboe(3) i rscorevar1(3),
rlateoboe(3) i rscorevar2(3) and (rlateoboe(3)
i rviolinwaits(3)
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Examples of different possible runs (performances)
rscorevar2
m1 m2 m3 m4 m5
oboe Phrase1 Phrase1 Phrase1 Phrase1 Phrase1
violin Phrase3 Phrase3 Phrase3 Phrase3 Phrase3
cello Phrase5 Phrase5 Phrase5 Phrase5 Phrase6
rlateoboe(3) i rlateviolin(3) rlateoboe(3) i
rviolinwaits(3), rlateoboe(3) i rscorevar1(3),
rlateoboe(3) i rscorevar2(3) and (rlateoboe(3)
i rviolinwaits(3)
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• The only way in which the players can
  distinguish the runs from each other at m3, is
  if they have common knowledge of a rule p
  (broadly, a consciousness that p is known by
  everyone and known to be known by everyone),
  where p determines which run is being executed if
  deviations from rscore occur.
• We take for granted that by knowing p, and that
  p is common knowledge, a player i will follow p.
• Problem This does not allow for any
  disagreement on the content of p. Intuitively,
  everyone must have the same idea of the central
  rules of the composition.

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Modeling in Terms of Variable Frame Theory

• According to Michael Bacharach Beyond Individual
  Choice Teams and Frames in Game Theory,
  Princeton 2006, we tend to reason in a way where
  we find it rational to choose the action that we
  think will most likely lead to coordination (if
  coordination is the object of the game), even if
  we are strictly speaking not sure that
  coordination will take place.

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Modeling in Terms of Variable Frame Theory

• To simplify our initial example, let us consider
  two possible strategies
• Wait corresponding to rviolinwaits
• Dont wait where everyone, including the
  player with the erroneous phrase continues
  according to the score
• The object of the game is coordination on
  either of the two strategies

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Modeling in Terms of Variable Frame Theory

• In the objective game the players have a 0.25
  chance of coordinating on the same strategy. But
  this is assuming that the players choose at
  random.
• Wait could for instance be described as more
  melodic and Dont Wait as more rhythmical
• Let Wait be symbolized by x1 and Dont Wait
  by x2

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Modeling in Terms of Variable Frame Theory

• Frhythmkeeps the piece going rhythmically,,
  where E(keeps the piece going rhythmically)
  x2
• Fmelodymelodic,, where E(melodic) x1
• Fthingthing where E(thing) x1, x2
• We have the universal frame for the coordination
  game FFthing, Frhythm, Fmelody

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Modeling in Terms of Variable Frame Theory

• Three different act-descriptions pick a thing
  (something), choose the option that keeps the
  piece going rhythmically or choose the melodic
• A complication compared to Three Cubes and a
  Pyramid three players instead of two
• Each player may assign different availabilities
  to the same frame for each of the two co-players

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Modeling in Terms of Variable Frame Theory

• Example The violin assigns voboe(Fmelody) 0.7,
  voboe(Frhythm) 0.3, vcello(Fmelody) 0.6 and
  vcello(Frhythm) 0.5.
• If the violin is right, the chance of
  coordinating with the others on choose the
  option that keeps the piece going rhythmically
  is 0.30.51 0.15.
• His chances of coordinating with the others on
  choose the melodic would be 0.70.61 0.42

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Modeling in Terms of Variable Frame Theory

• It seems that given his expectations of how the
  other players frame the situation, it would be
  rational for the violin to choose the melodic.
• Of course, the idea of possibility assessments is
  an idealized model of considerations musicians
  make while playing, but the idea captures
  important insights.

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Modeling in Terms of Variable Frame Theory

• We have analyzed this case as if there were no
  rules determining what the musicians should do,
  only mutual expectations. This is most likely not
  the case.
• An idea of what is acceptable in the performance
  context (whether composition based or not) shapes
  (and limits) the set of possible actions the
  musician can expect from the other musicians.

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Modeling in Terms of Logic for Intentions

• The following analysis is inspired by the work of
  Olivier Roy in Thinking before Acting
  Intentions, Logic, Rational Choice. ILLC 2008.
• A basic notion in Roys dissertation is that if
  someone forms an intention to achieve one or more
  outcomes, she sticks to her intention for as long
  as possible.

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Modeling in Terms of Logic for Intentions

• A musician intends one or more sonic outcomes of
  the performance

finds a strategy profile (set of strategies) for
the whole group that has this outcome set as part
of its set of possible outcomes
plays his strategy according to that profile
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Modeling in Terms of Logic for Intentions

• What happens when he realizes that the other
  players are not following the same profile (when
  he perceives an incongruence with the profile he
  is following)?
• Assuming (for simplicity) that no one makes
  mistakes, and, just as importantly, that no one
  believes that anyone can make mistakes, one of
  the following lines of action are adopted by the
  musician, listed in order of preference (by the
  musician)

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Modeling in Terms of Logic for Intentions

• 1. He tries to find another profile
  accommodating his intentions that fits the
  possible strategies of the other players at time
  t (given their history of actions up to t) and
  makes sense of his own history of actions up to
  t.

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Modeling in Terms of Logic for Intentions

• 2. He tries to find another profile
  accommodating his intentions that fits the
  possible strategies of the other players at time
  t, but does not necessarily make sense of his own
  history of actions up to t.

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Modeling in Terms of Logic for Intentions

• 3. He tries to find another profile that does
  not necessarily accommodate his intentions, but
  makes sense of his own history of actions up to t
  as well as the histories of other players and is
  compatible with his beliefs of the intentions of
  the other musicians (that is, accommodates these
  intentions).

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Modeling in Terms of Logic for Intentions

• 4. He tries to find another profile that fits
  his expectations regarding the intentions of
  other players, but not necessarily their actions
  up to t.

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Modeling in Terms of Logic for Intentions

• He tries to find another profile that fits the
  possible strategies of the other players at time
  t, but not necessarily his own actions up to t.
  If possible, he chooses a profile that fits his
  expectations regarding the intentions of other
  players.

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Modeling in Terms of Logic for Intentions

• The order of 4 and 5 is debatable, since the
  intention set of the musician might only include
  outcomes of profiles that makes sense of all
  actions.

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Modeling in Terms of Logic for Intentions

• I am working on an analysis to show that if all
  musicians think in the same way and adjust their
  lines of action according to this prioritized
  order of actions, they will, by each adjustment
  made, reduce the number of possible profiles to
  choose from, thus gradually bettering their
  chances of agreeing on a strategy profile.

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Modeling in Terms of Logic for Intentions

• This analysis presupposes that for each
  performance situation, the number of profiles to
  choose from is finite and smaller than the set of
  all possible combinations of actions by the
  musicians. Otherwise, everyones actions will
  always fit into some profile at time t. What
  defines the set of possible strategy profiles to
  choose from is a standard or set of norms for the
  performance.

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Conclusion

• So far, all of my analysises entail the presence
  of norms guiding the performance. I have further
  work to do describing to which extent it matters
  whether the norms in question are common
  knowledge, if the musicians are aware of each
  others different interpretations of the norms,
  and how this can be compared to more general,
  non-musical scenarios.