## ⓘ Gross substitutes (indivisible items)

In economics, gross substitutes is a class of utility functions on indivisible goods. An agent is said to have a GS valuation if, whenever the prices of some items increase and the prices of other items remain constant, the agents demand for the items whose price remain constant weakly increases.

An example is shown on the right. The table shows the valuations in dollars of Alice and Bob to the four possible subsets of the set of two items: {apple, bread}. Alices valuation is GS, but Bobs valuation is not GS. To see this, suppose that initially both apple and bread are priced at $6. Bobs optimal bundle is apple+bread, since it gives him a net value of $3. Now, the price of bread increases to $10. Now, Bobs optimal bundle is the empty bundle, since all other bundles give him negative net value. So Bobs demand to apple has decreased, although only the price of bread has increased.

The GS condition was introduced by Kelso and Crawford in 1982 and was greatly publicized by Gul and Stacchetti. Since then it has found many applications, mainly in auction theory and competitive equilibrium theory.

### * 1.1. Definitions * Gross Substitutes GS

The original GS definition is based on a price vector and a demand set.

- The demand set D u, p {\displaystyle Du,p} is the set of all demands.
- Given a utility function u {\displaystyle u} and a price vector p {\displaystyle p}, a set X {\displaystyle X} is called a demand if it maximizes the net utility of the agent: u X − p ⋅ X {\displaystyle uX-p\cdot X}.
- A price vector p {\displaystyle p} is a vector containing a price for each item.

The GS property means that when the price of some items increases, the demand for other items does not decrease. Formally, for any two price vectors q {\displaystyle q} and p {\displaystyle p} such that q ≥ p {\displaystyle q\geq p}, and any X ∈ D u, p {\displaystyle X\in Du,p}, there is a Y ∈ D u, q {\displaystyle Y\in Du,q} such that Y ⊇ { x ∈ X | p x = q x } {\displaystyle Y\supseteq \{x\in X|p_{x}=q_{x}\}} Y contains all items in X whose price remained constant.

### * 1.2. Definitions * Single Improvement SI

The SI condition says that a non-optimal set can be improved by adding, removing or substituting a single item. Formally, for any price vector p {\displaystyle p} and bundle X ∉ D u, p {\displaystyle X\notin Du,p}, there exists a bundle Y {\displaystyle Y} such that u Y − p ⋅ Y > u X − p ⋅ X {\displaystyle uY-p\cdot Y> uX-p\cdot X}, | X ∖ Y | ≤ 1 {\displaystyle |X\setminus Y|\leq 1} and | Y ∖ X | ≤ 1 {\displaystyle |Y\setminus X|\leq 1}.

### * 1.3. Definitions * No Complementaries NC

The NC condition says that every subset of a demanded bundle has a substitute. Formally: for any price vector p {\displaystyle p} and demanded bundles X, Y ∈ D u, p {\displaystyle X,Y\in Du,p}, and for every subset X ′ ⊆ X {\displaystyle X\subseteq X}, there is a subset Y ′ ⊆ Y {\displaystyle Y\subseteq Y} such that: X ∖ X ′ ∪ Y ′ ∈ D u, p {\displaystyle X\setminus X\cup Y\in Du,p}

If the valuation function is monotone, then GS implies SI and SI implies NC and NC implies GS, so these three conditions are equivalent.

### * 1.4. Definitions * M# Concave MX

The MX condition comes from convex analysis. It says that for all sets X, Y {\displaystyle X,Y} and for every item x ∈ X {\displaystyle x\in X}, at least one of the following must be true:

- there exists an item y ∈ Y {\displaystyle y\in Y} such that u X + u Y ≤ u X ∖ { x } ∪ { y } + u Y ∖ { y } ∪ { x } {\displaystyle uX+uY\leq uX\setminus \{x\}\cup \{y\}+uY\setminus \{y\}\cup \{x\}}.
- u X + u Y ≤ u X ∖ { x } + u Y ∪ { x } {\displaystyle uX+uY\leq uX\setminus \{x\}+uY\cup \{x\}}, or -

The M#-concavity property is also called M#-exchange property. It has the following interpretation. Suppose Alice and Bob both have utility function u {\displaystyle u}, and are endowed with bundles X {\displaystyle X} and Y {\displaystyle Y} respectively. For every item that Alice hands Bob, Bob can hand at most one item to Alice, such that their total utility after the exchange is preserved or increased.

SI implies MX and MX implies SI, so they are equivalent.

### * 1.5. Definitions * Strong No Complementaries SNC

The SNC condition says that, for all sets X {\displaystyle X} and Y {\displaystyle Y} and for every subset X ′ ⊆ X {\displaystyle X\subseteq X}, there is a subset Y ′ ⊆ Y {\displaystyle Y\subseteq Y} such that:

u X + u Y ≤ u X ∖ X ′ ∪ Y ′ + u Y ∖ Y ′ ∪ X ′ {\displaystyle uX+uY\leq uX\setminus X\cup Y+uY\setminus Y\cup X}The SNC property is also called M#-multiple-exchange property. It has the following interpretation. Suppose Alice and Bob both have utility function u {\displaystyle u}, and are endowed with bundles X {\displaystyle X} and Y {\displaystyle Y} respectively. For every subset X ′ {\displaystyle X} that Alice hands Bob, there is an equivalent subset Y ′ {\displaystyle Y} that Bob can handle Alice, such that their total utility after the exchange is preserved or increased. Note that it is very similar to the MC condition - the only difference is that in MC, Alice hands Bob exactly one item and Bob returns Alice at most one item.

Note: to check whether u has SNC, it is sufficient to consider the cases in which X ′ ⊆ X ∖ Y {\displaystyle X\subseteq X\setminus Y}. And it is sufficient to check the non-trivial subsets, i.e., the cases in which X ′ ≠ ∅ {\displaystyle X\neq \emptyset } and X ′ ≠ X ∖ Y {\displaystyle X\neq X\setminus Y}. And for these cases, we only need to search among bundles Y ′ ⊆ Y ∖ X {\displaystyle Y\subseteq Y\setminus X}.

Kazuo Murota proved that MX implies SNC.

It is obvious that SNC implies NC. Proof: Fix an SNC utility function u {\displaystyle u} and a price-vector p {\displaystyle p}. Let A, B {\displaystyle A,B} be two bundles in the demand-set D u, p {\displaystyle Du,p}. This means that they have the same net-utility, E.g., U p:= u p A = u p B {\displaystyle U_{p}:=u_{p}A=u_{p}B}, and all other bundles have a net-utility of at most U p {\displaystyle U_{p}}. By the SNC condition, for every A ′ ⊂ A {\displaystyle A\subset A}, there exists B ′ ⊆ B {\displaystyle B\subseteq B} such that u p A ∖ A ′ ∪ B + u p B ∖ B ′ ∪ A ≥ u p A + u p B = 2 ⋅ U p {\displaystyle u_{p}A\setminus A\cup B+u_{p}B\setminus B\cup A\geq u_{p}A+u_{p}B=2\cdot U_{p}}. But u p A ∖ A ′ ∪ B {\displaystyle u_{p}A\setminus A\cup B} and u p B ∖ B ′ ∪ A {\displaystyle u_{p}B\setminus B\cup A} are both at most U p {\displaystyle U_{p}}. Hence, both must be exactly U p {\displaystyle U_{p}}. Hence, both are also in D u, p {\displaystyle Du,p}.

We already said that NC implies GS which implies SI, and that SI implies MX. This closes the loop and shows that all these properties are equivalent there is also a direct proof that SNC implies MX.

### * 1.6. Definitions * Downward Demand Flow DDF

The DDF condition is related to changes in the price-vector. If we order the items by an ascending order of their price-increase, then the demand of a GS agents flows only downwards – from items whose price increase more to items whose price increased less, or from items whose price increased to items whose price decreased, or from items whose price decreased less to items whose price decreased more.

Formally, let p, q {\displaystyle p,q} be two price-vectors and let Δ:= q − p {\displaystyle \Delta:=q-p} be the price-increase vector. If an item x {\displaystyle x} is demanded under p {\displaystyle p} and not demanded under q {\displaystyle q}, then there is another item y {\displaystyle y} with Δ y < Δ x {\displaystyle \Delta _{y} k x ′ > k y {\displaystyle k_{x}> k_{y}} we can take k y ′ = k y {\displaystyle k_{y}=k_{y}} which makes the inequality:

u k x + u k y ≤ u k y + k x − k x ′ + u k x ′ {\displaystyle uk_{x}+uk_{y}\leq uk_{y}+k_{x}-k_{x}+uk_{x}}which is equivalent to:

u k x ′ +-uk_{y}}This follows from submodularity because k x ′ > k y {\displaystyle k_{x}> k_{y}}.

- includes both substitute goods and independent goods, and only rules out complementary goods. See Gross substitutes indivisible items Polterovich
- 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
- auction. B. Substitutes A car and a horse are sold in an auction. Alice only cares about transportation, so for her these are perfect substitutes she gets
- several items are sold, one after the other, to the same group of potential buyers. In a sequential first - price auction SAFP each individual item is sold
- It is conventionally measured as the percent rate of increase in real gross domestic product, or real GDP. Growth is usually calculated in real terms
- gods Allah is perceived by Muslims to be a unique, independent and indivisible being, who is utterly independent of and who precedes all of creation
- because of scarcity, and because a scarce resource such as a liver is indivisible Emanuel said that McCaughey took words out of context, omitting qualifiers