Representing type spaces as signal allocations

Consider a set of agents uncertain about the state in some finite state space $$\Omega $$ Ω . A type space $$\left( \varvec{T},Q\right) $$ T , Q that describes the agents’ information consists of a finite product set $$\varvec{T}=T_{1}\times \cdots \times T_{n}$$ T = T 1 × ⋯ × T n , and a probability distribution $$Q\in \Delta \left( \Omega \times \varvec{T}\right) $$ Q ∈ Δ Ω × T . Alternatively, a signal allocation assigns to each agent i a signal $$\pi _{i}$$ π i , a finite partition of $$\Omega \times X$$ Ω × X where X is a measurable space endowed with a non-atomic probability measure. Every signal allocation induces a type space in which the types in $$T_{i}$$ T i are the elements of $$\pi _{i}$$ π i . We establish two results. First, every type space is equivalent to one that is induced by a signal allocation. Second, encoding of type spaces into signal allocations can be done myopically, one agent at a time.