These game pieces for networked music performance are  based on probabilistic graphical models (PGMs), Bayesian inference and  game-theoretical approaches to systemic free improvisation.

I have worked on two main models: Adaptive Markov Networks (or Adaptive Markov Random Fields…same egg different wrapping…) and Pairwise Markov Networks.  The first are borrowing ideas from adaptive dynamical systems and economics (relying heavily on Thomas Nunn’s relational functions), while the second are a repeated synchronous Bayesian game for pairwise musical interactions, using a far-fetched and uncomfortable analogy involving terrorism.

Markov Networks (MNs) are undirected and possibly cyclic graphs. In the context of the natural interactions between free improvisers, MNs are more appropriate than directed graphs (Bayesian Networks), as they allow influences and inferences to flow in both directions. A formal definition can be stated as follows: a Markov Network is a random field S, which is a collection of indexed random variables (either discrete or continuous) where any variable  F_{i} is independent of all other variables in S. Such a network also satisfies the Markov property, which states that no matter what path the system took to get to the current state, the transition probability from that state to the next will be independent from such path.

The simplest class of MNs is the pairwise MN, an example of which is depicted in the following example:


Figure 1. A pairwise Markov Network

In my musical implementation, the nodes of above graph represent four players, which, by virtue of playing together, influence each other. Since there is no strictly conditioning and/or conditioned variable, as one would have in a Bayesian Network, the notion of factor, hereinafter indicated as φ, will come in handy for defining the interactions between the nodes (players). Factors also go under the names of affinity functions, compatibility, soft constraints, and they generalise the idea of the local predisposition and willingness of any pair of nodes to take a joint assignment. Skipping a few steps for the sake of this article’s readability, the normalised probability distribution will be expressed by:

P_{\Phi}(X_{1},..,X_{n}) = (1/Z_{\Phi})P_{\Phi}(X_{1},..,X_{n})

where the partition function  Z_{\Phi} is equal to:

Z_{\Phi}= \sum_{X_{1},..,X_{n}}P_{\Phi}(X_{1},..,X_{n})

Simply put, two or more variables (players, in this case) are connected whenever they appear in the same scope of a given factor. However, it would be impossible to infer the factorisation from the graph. In this sense, influence can flow along any active trail/edge. I find this model an exquisite abstraction of a typical interactive and dynamically assigned scenario amongst music improvisers, where alliances and joint assignments are formed, undertaken, updated, abandoned, in continuous real-time.

The decision of employing MN as a model for improvised musical interaction follows on previous experiments of mine, carried out in regards to focal points, Schelling’s salience  and Markov Chains. The aforementioned experiments, pointed at the need to move towards an increased complexity of inter-relations and a decreased complexity of the instructions/constraints, as a step in the direction of allowing for more prompt and reactive environments for the player to operate in. Unlike the literature that has dealt with models based on either probabilistic graphical models or automata theory, the two models here presented are systemic frameworks for and between human players and no musical output is generated by the machine.

The two models follow directly from the example above regarding an induced MN, in that they formally map players to a type and each type to a set of weighted strategies, or affinity preferences. The potential of this model lies in the fact that local distributions are not reflected or retrievable by the global graph and in that each of the players’ screen might depict a different locality of connections.

The interaction model is described by a graph with vertices representing players. Each player is assigned a musical personality, what in Bayesian terms would be referred to as a type. Such type is private information and it is not shared amongst the players. This very fact implies that particular care needs to be taken when deciding on the spatial physical distribution of the players, in that they ought not to be able to see each other’s screen.

Each of these different types has an optimal local pairwise counter type or a best response strategy associated with it, and ideally each player will try to infer the others’ type in order to achieve such optimal joint assignment. Players can only musically interact with players they are connected to.

The two models are different in that the players abide by different ‘rules of the game’ and have, in each, different goals and action/strategy space.

The theory behind these models for systemic improvisation can get quite involved, so enjoy (or not!) the videos instead.