16.1 Decision Trees and Expected Value

Decision trees are powerful tools for visualizing complex decision-making processes. They break down problems into manageable parts, using nodes and branches to represent choices, uncertain outcomes, and payoffs. This graphical approach helps clarify the decision-making process and identify optimal choices.

Expected value calculations are crucial in decision tree analysis. By working backwards from the tree's endpoints, we can determine the best course of action at each decision node. This method allows us to make informed choices based on probabilities and potential outcomes, maximizing our chances of success.

Decision Trees

Construction of decision trees

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• Decision trees graphically represent decision-making processes breaking down complex problems into manageable parts
• Components include:
  • Decision nodes (squares) indicate points where a decision must be made
  • Chance nodes (circles) represent uncertain outcomes or events
  • Branches connect nodes representing different choices or outcomes
  • Payoffs are numerical values assigned to final outcomes representing the consequences of each path
• Constructing a decision tree involves:
  • Identifying the main decision at the root of the tree
  • Determining possible choices or actions at each decision node and drawing branches for each option
  • Identifying uncertain events or outcomes after each decision and representing them with chance nodes
  • Assigning probabilities to each branch from a chance node reflecting the likelihood of each outcome (coin flip, weather forecast)
  • Determining payoffs for each final outcome and placing them at the end of corresponding branches (profit, loss)

Calculation of expected values

• Expected value (EV) measures the average outcome of a decision considering probabilities and payoffs of each possible result
• Calculating EV for a single chance node:
  • Multiply the probability of each outcome by its corresponding payoff
  • Sum the products of probabilities and payoffs
  • Formula: $EV = \sum_{i=1}^{n} (Probability_i \times Payoff_i)$
• Calculating EV for a decision tree:
  1. Start from the right side of the tree and work backwards
  2. At each chance node, calculate the EV using the formula above
  3. At each decision node, choose the alternative with the highest EV
  4. The EV at the root represents the overall EV of the best decision (investment, project)

Expected Value and Optimal Decisions

Determination of optimal decisions

• The optimal decision maximizes the expected value
• At each decision node, compare EVs of all alternatives
  • Choose the alternative with the highest EV
  • If EVs are equal, other factors (risk preference) may influence the decision
• Consider trade-offs between potential gains and losses
  • Higher payoffs may be associated with higher risks (lottery, stock market)
  • Decision-makers should balance potential rewards with potential downsides
• Sensitivity analysis assesses the robustness of the optimal decision
  • Test how changes in probabilities or payoffs affect EVs and the optimal choice

Interpretation of decision trees

• Analyzing the decision tree structure provides insights into the decision-making process
• Identifying critical uncertainties:
  • Look for chance nodes with a significant impact on EVs
  • Chance nodes with a wide range of outcomes or high variability in payoffs are more critical (weather, market conditions)
• Identifying key factors influencing the final decision:
  • Trace the optimal path from the root to the final outcome
  • Decision nodes along this path represent key factors leading to the optimal choice (product features, pricing)
  • Changes in probabilities or payoffs at these nodes may alter the optimal decision
• Conducting sensitivity analysis determines the robustness of the optimal decision to changes in input values
  • Vary probabilities and payoffs to see how they affect EVs and the optimal choice
  • Identify the range of values for which the optimal decision remains unchanged (break-even analysis)