Game Theory.

Xenakis criticizes aleatory music in Formalized Music pointing out that 1. the performer is a “highly conditioned being” and as such it will bring his/her background in making choices and 2. that the composer “commits an act of resignation” and the result is a substitution of authors. His way of creating a heteronomus music involving an external conflict is by using Game Theory (http://en.wikipedia.org/wiki/Game_theory). Emile Borel (in 1921) and John von Neumann (in 1944) were two mathematicians who established this field. In a two person game the players communicate only through their actions and according to a game matrix that is known to both of them. B

3 -2 1 [-2] A 3 3 2 [2] -4 2 5 [-4] 4 6 0 [0] [4] [6] [5] If player A chooses the second row and player B chooses column 3, player B pays A 2, the number at the intersection of that rows and that column. The game continues alternating between players: next A chooses row 3 resulting in a payment of 5, and so on. A prudent strategy will be for B to find the largest number in each column [4], [5], [6] and choose the columns with the smallest number. That way B will minimize his loss to at most [4]. At the same time, A will find the smallest numbers in each row and choose the row with the largest such number [2]. That way A will insure a minimum gain. The fundamental inequality for all rectangular matrices is: min [max aij] ≥ max [min aij] j i i j now, if: min [max aij] = max [min aij] = v j i i j the game has a saddle point and v is the game value as in the following example: B

6 -4 -5 [-5] A 4 3 8 [3] [6] [3] [8] the game has v = 3 which is the saddle point. Xenakis has used Game Theory in three works: Duel, for two orchestras (56 players), Stratégie, for two orchestras (82 players), and Linaia-Agon, for horn, trombone, and tuba. The first two works are discussed in Formalized Music. We looked in class at Linaia-Agon. Game Theory has many applications in social sciences and especially in economics. Nobel Prize mathematician John Nash was influential in developing such theories (see the movie A Beautiful Mind, 2001).