taken at time t, and final state s' at time t + 1, Pr(S t + 1 = s'|S t = s, A t = a) = P(s, a, s').

The given definition of P assumes that the world dynamics obey the Markov assumption.

## 2.1.3 A Gent O Bservations: N A

{{i}iEa is a set of observations that each agent, i, can experience of its world, implicitly defining a combined observation, n a = n i eg a j . 

to indirect evidence of the state (e.g., sensor readings) and actions of other agents (e.g.,
movement of other helicopters). In the original team-theoretic framework, the information structure that represented the observation process of the agents was a set of deterministic functions, Oi : S → Ωj.

Extension of Allowable Information Structures:  O a  We extend the information structure representation to allow for uncertain observations. We use a general stochastic model, borrowed from the partially observable Markov decision process model (Smallwood &
Sondik, 1973), with a joint observation function: O a (s, a, w) = Pr( Ω i , = ω ( S i = s, A i - 1 = a). This function models the sensors, representing any errors, noise, etc. In some cases, we can separate this joint distribution into individual observation functions: O a = Nico Oil where Oi (s, a, w) = Pr( S, = w | S t = s, Ai-1 = a ). Thus, the probability distribution specified by O a forms the richer information structure used in our model. We can make useful distinctions between different classes of information structures: Collective Partial Observability This is the general case, where we make no assumptions on the observations.

Collective Observability There is a unique world state for the combined observations of the team: ∀ω ∈ Ωα ; ∃ s ∈ S such that ∀ s ′ ̸ = s ′ Pr(Ω ′ = ω | S ′ = s ′ ) = 0. The set of domains that are collectively observable is a strict subset of the domains that are collectively partially observable.

Individual Observability There is a unique world state for each individual agent's observations: Vw E Ri, Is E S such that Vs' F s, Pr(Ri = w|Sf = s') = 0. The set of domains that are individually observable is a strict subset of the domains that are collectively observable.

Non-Observability The agents receive no feedback from the world: ∃ω ∈ Ω i , such that Vs E S and Va E Ag, Pr(R; = w|S t = s, A t -1 = a) = 1. This assumption holds in open-loop systems, which come under frequent consideration in classical planning (Boutilier, Dean, & Hanks, 1999).

## 2.1.4 Policy (Strategy) Space

π i A is a domain-level policy (or strategy, in the original team theory specification) to map an agent's belief state to an action. In the original formalism, the agent's beliefs correspond directly to its observations (i.e., TiA : Ri -> Ai). Extension to Richer Belief State Space: Ba We generalize the set of possible strategies to capture the more complex mental states of the agents. Each agent, i E α , forms a belief state, U, E Bi, based on its observations seen through time t, where Bi circumscribes