Decision Analysis

Dr. James Doss-Gollin

Thu., Oct. 10

Decision Theory

Today

  1. Decision Theory

  2. Cost-Benefit Analysis

  3. Multi-Criteria Decision Analysis

Goals

We want to compare alternatives \(a\) in some set of possible alternatives \(\mathcal{A}\)

  • Pick the best one
  • Inform others about trade-offs
  • Learn about the true nature of the problem

Rationality

  1. Utility is the value of a choice, in abstract units. More utility is better.
  2. Axioms of rationality:
    • Completeness: For any pair of alternatives \(a_i\) and \(a_j \in \mathcal{A}\), either \(a_i \geq a_j\) or \(a_j \geq a_i\) or both.
    • Transitivity: If \(a_i \geq a_j\) and \(a_j \geq a_k\), then \(a_i \geq a_k\).
    • Independence: If \(a_i \geq a_j\), then for any \(a_k\), \(a_i + a_k \geq a_j + a_k\).

The World is Uncertain

Breakout

If we are evaluating whether a proposed bridge is a good investment, what are some factors that might impact the utility of each alternative?

  1. We consider many states of the world \(s \in \mathcal{S}\)
  2. We use probabilities to quantify our uncertainty: \(p(s)\)
  3. Utility depends on the state of the world: \(u(a, s)\)

Expected Utility

  1. We can calculate the expected utility of an alternative \(a\) as: \[ u(a) = \mathbb{E}[u(a, s)] = \int_{s \in \mathcal{S}} u(a, s) p(s) \]
  2. Idealized rational decision-makers behave as though they are maximizing utility (@ Savage, 1954), though this is not a good description of actual human behavior (Ellsberg, 1961)

Cost-Benefit Analysis

Today

  1. Decision Theory

  2. Cost-Benefit Analysis

  3. Multi-Criteria Decision Analysis

Objectives

  1. Compare multiple alternatives on an “apples-to-apples” basis
    • Often, using money as the common denominator
  2. Make underlying assumptions explicit

Many Engineering Decisions Involve Time

  1. When building infrastructure, costs and benefits often occur at different times
    • Costs now, benefits later
  2. Money now is more valuable than money later
    • Opportunity cost of capital
    • People in the future will be richer
    • Uncertainty, which many people dislike, grows with time

Net Present Value

Put all costs and benefits into a common unit (typically present dollars)

\[ \text{NPV}(a) = \sum_{t=0}^T (1 - \gamma)^t u(a) \] where \(\gamma\) is the discount rate

\[ \text{NPV}(a) = \mathbb{E} \left[ \sum_{t=0}^T (1 - \gamma)^t u(a, s_t) \right] \]

Optimization

We can define the best alternative as \[ a^* = \arg \max_a \text{NPV}(a) \]

There are many algorithms and solution techniques, depending on the problem.

Multi-Criteria Decision Analysis

Today

  1. Decision Theory

  2. Cost-Benefit Analysis

  3. Multi-Criteria Decision Analysis

Many Costs and Benefits are Difficult to Monetize

Breakout

What are some costs and benefits that are difficult to monetize?

  1. Value of ecosystems
  2. Health and safety
  3. Aesthetics
  4. Social and cultural values

Rubrics

Category 0 points 1 point 2 points 3 points
Environmental Impact Severe negative impact Moderate negative impact Minimal impact Positive impact
Cost Efficiency Greatly exceeds budget Slightly over budget Within budget Under budget
Community Benefit No benefit Minor benefit Moderate benefit Significant benefit
Technical Feasibility Not feasible Challenging but possible Feasible with effort Easily implementable
Long-term Sustainability Not sustainable Sustainable for <5 years Sustainable for 5-10 years Sustainable for >10 years

Pitfalls

  1. Rubrics can be highly subjective in unclear ways
    • Objectivity is a good aim, but not attainable – be transparent
  2. Arbitrary weights and criteria

Multiobjective Optimization

Rather than find a single best alternative, map trade-offs between multiple objective functions

Project 2

Use a multi-criteria rubric to evaluate your proposed transportation improvements

  • Justify your criteria and weights
  • Use the rubric to evaluate alternatives
  • Justify your scoring

References

Ellsberg, D. (1961). Risk, ambiguity, and the Savage axioms. The Quarterly Journal of Economics, 75(4), 643–669. https://doi.org/10.2307/1884324
Savage, L. J. (1954). Foundations of statistics. New York: Wiley.