Decision Theory

Decision Theory studies how to choose the optimal action under uncertainty by comparing expected payoffs across possible actions.

1. Core Objective

A probability model is ultimately used to support decision-making. The standard task is:

  • list the available actions
  • specify the possible outcomes
  • assign probabilities to outcomes
  • define the payoff for each action-outcome pair
  • compute expected value for each action
  • choose the action with the highest expected payoff, or solve for the parameter range under which each action is optimal

2. Standard Framework

The expected value of an action is

$$\mathbb{E}[\text{action}] = \sum_{\text{outcomes}} P(\text{outcome}) \times \text{payoff}(\text{action}, \text{outcome})$$

The optimal decision is obtained by comparing expected values across actions.

3. Basic Elements

A decision problem is defined by:

  • Actions: the choices available to the decision-maker
  • Outcomes or States of Nature: the uncertain future scenarios
  • Probabilities: the probability distribution over outcomes
  • Payoff Function: the utility, profit, cost, or loss associated with each action-outcome pair

4. Standard Procedure

4.1 Define the actions, outcomes, probabilities, and payoffs

Write down the full decision table.

4.2 Compute expected value for each action

For each action, sum probability times payoff across all outcomes.

4.3 Compare actions

Solve inequalities such as

$$\mathbb{E}[A_i] > \mathbb{E}[A_j]$$

to determine the parameter region where one action is better than another.

5. Example: Server Allocation

Suppose a new app is about to launch.

Actions

  • $A_1$: rent a standard server
  • $A_2$: rent a high-performance server

Outcomes

  • $S_1$: low traffic
  • $S_2$: traffic surge

Probabilities

Let

$$P(S_2)=p, \qquad P(S_1)=1-p$$

Payoff Matrix

Payoff is net profit in thousands:

  • $$U(A_1,S_1)=100$$
  • $$U(A_1,S_2)=-50$$
  • $$U(A_2,S_1)=20$$
  • $$U(A_2,S_2)=200$$

6. Expected Value Calculation

For $A_1$,

$$\mathbb{E}[A_1] = P(S_1)U(A_1,S_1) + P(S_2)U(A_1,S_2)$$$$\mathbb{E}[A_1] = (1-p)\cdot 100 + p\cdot (-50) = 100 - 150p$$

For $A_2$,

$$\mathbb{E}[A_2] = P(S_1)U(A_2,S_1) + P(S_2)U(A_2,S_2)$$$$\mathbb{E}[A_2] = (1-p)\cdot 20 + p\cdot 200 = 20 + 180p$$

7. Optimality Condition

To find when $A_2$ is better than $A_1$, solve

$$\mathbb{E}[A_2] > \mathbb{E}[A_1]$$$$20 + 180p > 100 - 150p$$$$330p > 80$$$$p > \frac{80}{330} \approx 0.2424$$

8. Decision Rule

  • If $$p > 0.2424$$, choose $A_2$
  • If $$p < 0.2424$$, choose $A_1$
  • If $$p = 0.2424$$, both actions are equivalent in expected value

9. Exam Pattern

Typical questions in Decision Theory are structurally fixed: compute the expected value of each action, compare them, and determine the parameter range for which each action is optimal.