On the Necessity of Rules in Ensemble Coordination

1. 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

2. Contents • A Coordination Problem in a Music Ensemble • Modeling Coordination in Terms of Traditional Epistemic Logic • Modeling in Terms of Variable Frame Theory • Modeling in Terms of Logic for Intentions • Conclusion

3. Modeling Coordination in Terms of Traditional Epistemic Logic An idealized passage from a fictitious score

4. 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.

5. 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 Loboe={phrase1,phrase2} Lviolin={phrase3, phrase4} Lcello={phrase 5, phrase6} (We could also add a state Λ to each set Li, denoting that nothing is played, but we choose to omit this here for clarity.)

6. 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.

7. Modeling Coordination in Terms of Traditional Epistemic Logic • In our system, we can think of the runs as different performances • We define a player i’s 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)= r´i(m)

8. Examples of different possible runs (performances) rlateoboe 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)

9. Examples of different possible runs (performances) rlateviolin 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)

10. Examples of different possible runs (performances) rviolinwaits 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)

11. Examples of different possible runs (performances) rscorevar1 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)

12. Examples of different possible runs (performances) rscorevar2 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)

13. The only way in which the players can distinguish the runs from each other at m=3, 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.

14. 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.

15. Modeling in Terms of Variable Frame Theory • To simplify our initial example, let us consider two possible strategies: • ”Wait” – corresponding to rviolinwaits • ”Don’t 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

16. 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 ”Don’t Wait” as ”more rhythmical” • Let ”Wait” be symbolized by x1 and “Don’t Wait” by x2

17. Modeling in Terms of Variable Frame Theory • Frhythm={keeps the piece going rhythmically,…}, where E(keeps the piece going rhythmically) = {x2} • Fmelody={melodic,…}, where E(melodic) = {x1} • Fthing={thing} where E(thing) = {x1, x2} • We have the universal frame for the coordination game: F={Fthing, Frhythm, Fmelody…}

18. 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

19. 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 0.3*0.5*1 = 0.15. • His chances of coordinating with the others on ”choose the melodic” would be 0.7*0.6*1 = 0.42

20. 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.

21. 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.

22. 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 Roy’s dissertation is that if someone forms an intention to achieve one or more outcomes, she sticks to her intention for as long as possible.

23. 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

24. 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):

25. 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.

26. 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.

27. 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).

28. 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.

29. 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.

30. 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.

31. 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.

32. 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, everyone’s 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.

33. 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 other’s different interpretations of the norms, and how this can be compared to more general, non-musical scenarios.