Solving value functions where the state variables are information sets typically arises in dynamic programming problems, particularly in economics and operations research. These problems often involve decision-making under uncertainty, where the decision-maker's current state is defined by the information available to them, rather than a simple numerical state.
Here’s a general approach to tackle such problems:
Steps to Solve Value Functions with Information Set State Variables
1. Define the Problem:
- Clearly specify the decision problem, including the objective function, decision variables, constraints, and the nature of uncertainty.
2. Model the Information Set:
- Identify what constitutes the information set. This could include past states, actions, observations, or signals that inform the current decision.
3. Formulate the Value Function:
- The value function ( V(I_t) ) represents the maximum expected utility (or other objective) achievable from the current information set ( I_t ) onward.
- Typically, this involves a recursive relationship, often expressed as: [ V(I_t) = \max_{a_t} \left{ u(I_t, a_t) + \beta \mathbb{E}[V(I_{t+1}) \mid I_t, a_t] \right} ] where ( u ) is the utility function, ( a_t ) is the action, ( \beta ) is the discount factor, and the expectation is over future states given current information and actions.
4. Dynamic Programming or Bellman Equation:
- Use dynamic programming techniques to solve the Bellman equation. This typically involves:
- Backward Induction: Start from a terminal period and solve backwards.
- Value Iteration: Iteratively update the value function until convergence.
- Policy Iteration: Iteratively improve the policy and evaluate its value.
