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Active Inference in Multiagent Systems 79

the space of decision vectors locally (i.e., each agent samples its own subset of
decision vector variables). Max-marginal estimates include variable mar-
ginals used by agents to sample the decision space:

Þ∝ Y ðÞ,
b i d i ¼ max p dj bð m j!i d i
ðÞ
dn d i j2NiðÞ
fg
as well as factor marginals, used to adapt a team’s structure:
Y

Þ∝φ d j
b j d j ¼ max p dj bð j m i!j d i ðÞ
i2NjðÞ
dn d j fg
With these quantities, we compute a Bethe approximation to the free-
energy function (Yedidia et al., 2005):
F Bethe b, mÞ ¼ E Bethe b, mÞ H Bethe b, mÞ,
ð
ð
ð
where the first component is a negative expected utility computed as

P
b j d j c j d j , and the second component is the entropy
E Bethe b, mð Þ ¼
d j P P
P P
b
b
H Bethe (b,m)¼(n i 1) i d i i (d i )lnb i (d i ) j d j j (d j )lnb j (d j ), where
n i ¼jN(i)j is the number of factors d j that the variable i is involved in.
Minimizing the free energy is achieved when the team finds all possible
(maximally varying) marginals with the highest utility.
4.3.4 Adapting Team Structure
The team structure is represented by a model variable m, which affects the
perceptions and decisions the team jointly produces. In addition, this struc-
ture also constrains how the information flows and is incorporated in the
organization, including where the belief message can be sent, what commu-
nication delays and transmission errors are incurred, which of the messages
are used to update decisions, and the concomitant computation workload
incurred by team agents. This problem of team structure design can be for-
mulated (and solved) as a network design problem (Feremans, Labb e, &
Laporte, 2003).
Decision decomposition and the corresponding BP message calculations
represent the internal computational workload incurred by agents, representing
the collaboration process required to solve a decision problem. Specification
of the messages passed between decision and factor nodes in a factor graph
define the external communication workload of the agents. The problem of
structuring a team can be formulated as the alignment of decision decom-
position (“the task network”) and the agent network (“team structure”)
to properly balance internal and external workloads. We model the impact