In sociology, the notion of power is a central issue. In the presence of a social network, one may ask whether an individual’s power is primarily derived from the individual’s exceptional attributes, or whether it is based on the individual’s pivotal position in the network.

Here we develop two mathematical notions, stability and balance, that allow us to quantify power in network structures.

## Network Exchange under Controlled Conditions

The vague idea of power in a network can be made precise in the form of the following experiment.

People are place on the nodes of a small graph, representing a social network. Each edge of the graph carries a “social value” of, say, €1 and the nodes joined by that edge can negotiate over how that money is to be divided between them. After a certain time limit agreements must have been reached. Under the one-exchange rule, each node can only be involved in a successful exchange with at most one neighbor. Hence, a node stops negotiating once an agreement on how to split the euro has been reached with one of its neighbours.

Such an experiment can be run with real people, on different graphs, and for multiple rounds. We seek a theoretical framework that predicts the outcome of such negotiations on a given graph.

Examples. The networks we wish to understand in this way include straight line graphs consisting of $2$, $3$, or more nodes:

Or graphs containing triangles, like:

## The Nash Bargaining Solution

In the simplest example of a $2$-node graph, two people, $A$ and $B$, argue about how to split €1 between them. We further assume that $A$ has an outside option of $x$ and $B$ has an outside option of $y$. This means that $A$ can decide to not split the euro with $B$ and take $x$ instead. In the same way, $B$ can decide against a deal with $A$ and take $y$ instead. Presumably, $A$ and $B$ won’t come to an agreement over the splitting of the euro, if either of them has an outside option that exceeds their proposed share of the euro. As then no agreement can be reached at all if $x + y > 1$, we assume that $x + y \leq 1$.

Then $A$ expects a share of at least $x$ and $B$ expects a share of at least $y$. The negotiations then are over how to split the surplus $s = 1 - (x + y)$. Naturally one would expect them to split this in halves.

Nash Bargaining Solution. When $A$ and $B$ negotiate over splitting a euro, with an outside option of $x$ for $A$ and an outside option of $y$ for $B$ (such that $x + y \leq 1$), the Nash bargaining outcome assigns

• $x + \frac12s = \frac12(x-y+1)$ to $A$, and
• $y + \frac12s = \frac12(y-x+1)$ to $B$,

where $s = 1 - (x + y)$ is the surplus.

It can be shown that this solution arises naturally as an equilibrium, when the bargaining process is studied as a game.

## Stability

An outcome of a network exchange consists of two things:

• A matching on the nodes, specifying who exchanges with whom. Recall that a matching is a subset of the edges of the network that involves each node at most once. This matching is not necessarily perfect.

• A value attached to each node, indicating the node’s payoff from the exchange. The sum of values on two matched nodes this equals $1$, whereas the value of any unmatched node is $0$.

Examples. The following examples of outcomes are either stable or unstable:

Here, $B$ would be willing to switch if $C$ offered more than the $\frac12$ which $B$ is currently getting.

Here, $C$ can no longer make an offer that would be attractive for $B$.

Here, $B$ and $C$ could improve their lot by agreeing to split a euro between them.

This is a stable outcome.

This is also a stable outcome.

Most networks allow a wide range of possible outcomes. An outcome is considered stable if no node $X$ can offer a node $Y$ a deal that makes both $X$ and $Y$ better off.

We define an instability in an outcome as a situation where two nodes have the opportunity and the incentive to disrupt the current outcome.

Definition (Instability). An instability in an outcome, consisting of a matching and values for the nodes, is a non-matching edge, joining nodes with values $x$ and $y$ such that $x + y < 1$.

Note how the two nodes forming an instability have the opportunity to disrupt, as they are connected by a non-matching edge. And they have the incentive to disrupt, as the 1 euro on their joining edge provides them with more combined value than they currently have.

Definition (Stability). An outcome of a network exchange is stable if it does not contain any instabilities.

Stable outcomes are those outcomes that one would expect to see in practice.

There are, however, networks that have no stable outcome at all, for example the triangle network.

## Balance

The notion of balance refines the notion of stability: when a network allows several stable outcomes, only some will be balanced. Balance makes use of the earlier idea of the Nash bargaining solution. Here, the outside options are provided by the other nodes in the network.

Examples.

The all-$\frac12$ outcome is not balanced, as the values on matching edges do not represent the Nash bargaining solution, if alternative agreements are considered as outside options.

This outcome is balanced, as both matching edges do represent Nash bargaining solutions with respect to outside options provided by alternative arrangements.

Shifting the payoffs further to the more central nodes $B$ and $C$, results in an unbalanced outcome.

Balanced Outcome. An outcome, consisting of a matching and node values, is balanced if, for each edge in the matching, the values on its endpoints represent the Nash bargaining solution for those two nodes, relative to outside options provided by the values on the other nodes of the network.

Note that every balanced outcome is necessarily stable, as no two nodes have an incentive to disrupt the current outcome.

Example.

A balanced outcome on the stem graph.

Chapter 12 of the book contains a more detailed discussion of network bargaining in practice and in theory.