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INFORMATION SETS

  • Information Set Notation: 𝓘_{t,s,p} represents the information set at the beginning of perch p in stage s during period t, where p ∈ {≺, ∼, ≻} denotes the perch using relational symbols that convey temporal ordering:
  • ≺ (precedes) = arrival perch — the agent has arrived but not yet observed shocks or made decisions
  • ∼ (concurrent) = decision perch — the agent is at the point of making a decision
  • ≻ (succeeds) = continuation perch — the decision has been made and the stage concludes
  • Definition: 𝓘_{t,s,p} typically includes the history of observed states and actions up to the beginning of perch p in stage s during period t, along with any common knowledge (model parameters, functional forms).

Information Flow Through a Stage

Each stage has two shock spaces (see "Shock Spaces and Timing" in Recursive Problem Definition): - 𝒵ₐᵥ: Shocks realizing in the arvl→dcsn transition - 𝒵ᵥₑ: Shocks realizing in the dcsn→cntn transition

The information available at each perch depends on which shocks have realized:

Perch State Shocks Realized Information Set
≺ (arvl) xₐ neither 𝓘_{t,s,≺} =
∼ (dcsn) xᵥ ζₐᵥ only 𝓘_{t,s,∼} = 𝓘_{t,s,≺} ∪
≻ (cntn) xₑ both ζₐᵥ and ζᵥₑ 𝓘_{t,s,≻} = 𝓘_{t,s,∼} ∪

When a shock space is trivial (𝒵 = {∅}), the corresponding information update is empty.

Shock Timing and Expectations

The location of expectation operators depends on which shock spaces are non-trivial:

Pre-decision shocks only (𝒵ₐᵥ non-trivial, 𝒵ᵥₑ trivial): - Agent observes ζₐᵥ before choosing 𝜋 - Expectation E_{ζₐᵥ}[·] is taken at the arrival perch - Decision is made with full knowledge of ζₐᵥ

Post-decision shocks only (𝒵ₐᵥ trivial, 𝒵ᵥₑ non-trivial): - Agent chooses 𝜋 without observing ζᵥₑ - Expectation E_{ζᵥₑ}[·] is taken at the decision perch, inside the max - Continuation state xₑ depends on both 𝜋 and ζᵥₑ

Both shock spaces non-trivial: - Agent observes ζₐᵥ, then chooses 𝜋, then ζᵥₑ realizes - Nested expectations: E_{ζₐᵥ}[max_𝜋 E_{ζᵥₑ}[·]]

  • Optimal Choice Function: The optimal choice function depends on the information set: π(t, i, κ, xᵥ, 𝓘_{t,s,∼}).

  • Value Function: The value function also depends on the information set: 𝒱ₜ(s, xᵥ, 𝓘_{t,s,∼}).

  • Modified Bellman Equation (Example - Simplified):

  • If ζ is observed:

𝒱ₜ(s, xᵥ, 𝓘_{t,s,∼}) = max_{c ∈ Π(κ, xᵥ)} [r(κ, 𝜋, xᵥ) + β E[𝒱ₜ(s₊, gᵥₑ(κ, xᵥ), 𝓘₊) | 𝓘_{t,s,∼}]]

  • 𝓘₊ represents the updated information set after observing the next state xᵥ and taking action 𝜋.

  • E[⋅ | 𝓘_{t,s,∼}] is the expectation conditional on the current information set.

  • If ζ is not observed:

𝒱ₜ(s, xₐ, 𝓘_{t,s,≺}) = max_{c ∈ Π(κ, xᵥ)} E[r(κ, 𝜋, xᵥ) + β 𝒱ₜ(s₊, gᵥₑ(κ, xᵥ), 𝓘₊) | 𝓘_{t,s,≺}] - 𝓘₊ represents the updated information set after observing the next state and taking action 𝜋. - E[⋅ | 𝓘_{t,s,≺}] is the expectation conditional on the arrival perch's information set. - E[⋅ | 𝓘_{t,s,∼}] is the expectation conditional on the decision perch's information set. - They are the same for an agent who has not observed ζ

MODIFIED EXPRESSIONS INCORPORATING INFORMATION SETS

If information sets were incorporated into the core notation, the following expressions would be modified:

  1. Notation Reference:
  2. 𝒱ₜ(s, xᵥ) would become 𝒱ₜ(s, xᵥ, 𝓘_{t,s,∼})

  3. π(t, i, κ, xᵥ) would become π(t, i, κ, xᵥ, 𝓘_{t,s,∼})

  4. Population Dynamics:

  5. Sequential Stages: 
    Γᵥₑ(t, i, κ): μᵥ → μₑ, where μₑ(C) = μᵥ({xᵥ ∈ 𝓧ᵥ(κ) | gᵥₑ(κ)(xᵥ, π(t, i, κ, xᵥ, 𝓘_{t,s,∼})) ∈ C})

  6. Branching Stages: 

    Γᵥₐ₊(t, i, κ): μᵥ → (μₐ₊)ₐ₊ ∈ J̄(κ), where μₐ₊(B) = ∫𝓧ᵥ(κ) p(j | κ, xᵥ, π(t, i, κ, xᵥ, 𝓘_{t,s,∼})) 𝟙{gᵥₐ₊(κ)(xᵥ, π(t, i, κ, xᵥ, 𝓘_{t,s,∼})) ∈ B} dμᵥ(xᵥ)

  7. Bellman Equation: The standard Bellman equation V(x) = max_{𝜋 ∈ Π(x)} [r(x, 𝜋) + β E[V(x₊) | x, 𝜋]] would become: 

    V(x, 𝓘) = max_{𝜋 ∈ Π(x)} [r(x, 𝜋) + β E[V(x₊, 𝓘₊) | x, 𝜋, 𝓘]]

Note: These modifications make explicit the dependence on information sets but do not change the fundamental structure of the model.