1. General Idea

  • Utility = satisfaction, usefulness, or value you get from consuming a good or outcome.
  • Diminishing utility = the more you consume, the less additional satisfaction you gain from each extra unit.
  • In math terms: the utility function is concave. $U”(x) < 0$

2. Example in Economics

  • First slice of pizza → very satisfying.
  • Second slice → still good, but less extra satisfaction.
  • Fifth slice → hardly adds any joy, maybe even negative.

So:

$MU = \frac{\Delta U}{\Delta x}$

where $MU$ = marginal utility decreases as $x$ increases.

3. Application in Decision Theory

  • Risk-averse agents are modeled with diminishing marginal utility of wealth.
  • Example: Going from $0 → $1,000 feels huge, but $1,000 → $2,000 feels smaller in impact.
  • That’s why people buy insurance: they prefer guaranteed small losses over risky large losses.

4. Connection to Information Retrieval (DCG)

  • In DCG, the discount factor (logarithm) is a form of diminishing utility:
    • A relevant document at rank 1 is extremely valuable.
    • At rank 10, it still helps, but much less.
    • At rank 100, its contribution is almost negligible.

Formally: $\frac{1}{\log_2(i+1)}$

decreases as $i$ increases → each additional relevant item further down the list has diminishing impact.

5. Real-world Intuition

  • Everyday consumption: water, food, entertainment.
  • Finance: diminishing utility of wealth.
  • Search results: diminishing utility of relevance at lower ranks.
  • Learning: first hours of study give big improvement, but after long hours, each extra hour adds less.

Summary:
Diminishing utility = each extra unit of something gives less additional value than the previous one. It’s a core concept in economics (marginal utility), risk theory, and evaluation metrics like DCG, where the value of relevant items decreases with position.