Utility functions on indivisible goods

Key takeaways

Utility functions on indivisible goods belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Utility functions on indivisible goods to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Utility functions on indivisible goods from memory before moving on to harder problems.

Reference excerpt

Some branches of economics and game theory deal with indivisible goods, discrete items that can be traded only as a whole. For example, in combinatorial auctions there is a finite set of items, and every agent can buy a subset of the items, but an item cannot be divided among two or more agents. It is usually assumed that every agent assigns subjective utility to every subset of the items. This can be represented in one of two ways:

An ordinal utility preference relation, usually marked by ≻ {\displaystyle \succ } . The fact that an agent prefers a set A {\displaystyle A} to a set B {\displaystyle B} is written A ≻ B {\displaystyle A\succ B} . If the agent only weakly prefers A {\displaystyle A} (i.e. either prefers A {\displaystyle A} or is indifferent between A {\displaystyle A} and B {\displaystyle B} ) then this is written A ⪰ B {\displaystyle A\succeq B} . A cardinal utility function, usually denoted by u {\displaystyle u} . The utility an agent gets from a set A {\displaystyle A} is written u ( A ) {\displaystyle u(A)} . Cardinal utility functions are often normalized such that u ( ∅ ) = 0 {\displaystyle u(\emptyset )=0} , where ∅ {\displaystyle \emptyset } is the empty set. A cardinal utility function implies a preference relation: u ( A ) > u ( B ) {\displaystyle u(A)>u(B)} implies A ≻ B {\displaystyle A\succ B} and u ( A ) ≥ u ( B ) {\displaystyle u(A)\geq u(B)} implies A ⪰ B {\displaystyle A\succeq B} . Utility functions can have several properties.

Monotonicity Monotonicity means that an agent always (weakly) prefers to have extra items. Formally:

For a preference relation: A ⊇ B {\displaystyle A\supseteq B} implies A ⪰ B {\displaystyle A\succeq B} . For a utility function: A ⊇ B {\displaystyle A\supseteq B} implies u ( A ) ≥ u ( B ) {\displaystyle u(A)\geq u(B)} (i.e. u is a monotone function). Monotonicity is equivalent to the free disposal assumption: if an agent may always discard unwanted items, then extra items can never decrease the utility.

Additivity

Additivity (also called linearity or modularity) means that "the whole is equal to the sum of its parts." That is, the utility of a set of items is the sum of the utilities of each item separately. This property is relevant only for cardinal utility functions. It says that for every set A {\displaystyle A} of items,

u ( A ) = ∑ x ∈ A u ( x ) {\displaystyle u(A)=\sum _{x\in A}u({x})}

assuming that u ( ∅ ) = 0 {\displaystyle u(\emptyset )=0} . In other words, u {\displaystyle u} is an additive function. An equivalent definition is: for any sets of items A {\displaystyle A} and B {\displaystyle B} ,

u ( A ) + u ( B ) = u ( A ∪ B ) + u ( A ∩ B ) . {\displaystyle u(A)+u(B)=u(A\cup B)+u(A\cap B).}

An additive utility function is characteristic of independent goods. For example, an apple and a hat are considered independent: the utility a person receives from having an apple is the same whether or not he has a hat, and vice versa. A typical utility function for this case is given at the right.

Submodularity and supermodularity

Submodularity means that "the whole is not more than the sum of its parts (and may be less)." Formally, for all sets A {\displaystyle A} and B {\displaystyle B} ,

u ( A ) + u ( B ) ≥ u ( A ∪ B ) + u ( A ∩ B ) {\displaystyle u(A)+u(B)\geq u(A\cup B)+u(A\cap B)}

In other words, u {\displaystyle u} is a submodular set function. An equivalent property is diminishing marginal utility, which means that for any sets A {\displaystyle A} and B {\displaystyle B} with A ⊆ B {\displaystyle A\subseteq B} , and every x ∉ B {\displaystyle x\notin B} :

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Utility functions on indivisible goods

Start with the simplest possible case. Write down what Utility functions on indivisible goods claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Utility functions on indivisible goods before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Utility functions on indivisible goods ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Utility functions on indivisible goods

In research: Utility functions on indivisible goods appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Utility functions on indivisible goods in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Utility functions on indivisible goods is common in secondary-school and first-year university syllabi. It links to neighbouring topics Combinatorial optimization, Utility function types, so understanding it makes those chapters shorter.
In everyday life: Look for Utility functions on indivisible goods outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Utility functions on indivisible goods in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Utility functions on indivisible goods means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Utility functions on indivisible goods out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Utility functions on indivisible goods in simple terms?

Some branches of economics and game theory deal with indivisible goods, discrete items that can be traded only as a whole. For example, in combinatorial auctions there is a finite set of items, and every agent can buy a subset of the items, but an item cannot be divided among two or more agents.

Why does Utility functions on indivisible goods matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Utility functions on indivisible goods?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Utility functions on indivisible goods.