Following from the previous post on why it’s hard to make friends where I listed communication as a major factor in making and keeping friends, I shall now explore what I consider to be the second major factor — perceived effort.

I shall start by describing a brief model. I had originally started out with more math in mind, but ended up feeling too sleepy to write much else (seriously, LaTeX on WordPress is a bitch, should have written in LyX and compiled it, but oh well), so forgive me for the slipshod work on the modeling.

A brief description is this: Being able to make friend someone depends on your efforts in communication as perceived by the other party.

## A Brief Model

Let us start by imagining a room full of people who do not yet know each other. Everyone has a set of actions they can perform in terms of making friends.

$Let \ C \ = \ \{Person_1, \ Person_2 \ ... \ Person_n\} \\ \\ Let \ A_c \ = \ \{Action_1, \ Action_2 \ ... \ Action_k\} \ \forall \ c \in \ C$

Everyone expends effort (we denote effort as f) in performing communication actions. For some people, initiating a chat with a stranger is easy. Not so for others (I am personally terrified of talking to strangers).

$Let \ AE_c \ = \ \{ Effort_1, \ Effort_2 \ ... \ Effort_k \} \ \forall \ c \in \ C; \ \ effort \in \mathbb{R}_+ \\ \\ f_c: {\bf A_c} \ \rightarrow {\bf AE_c} \ \forall \ c \in C;$

In this simplified model, everyone has in their minds, a perceived-effort function. This perceived-effort function is a device that summarily describes how effort in communication is perceived by the person, and can be directly mapped onto a utility function that is monotonic and quasi-concave (that is to say we assume the person is not batshit crazy).

$f_p : \mathbb{R}^n_+ \ \rightarrow \ \mathbb{R} \ \ \forall \ \ {\bf f} \ \in \ \mathbb{R}^n_+ \\ \\ g: f_p({\bf f}) \ \rightarrow \ u({\cdot})$

For example, person B initiates chat with person A and person C. Person A may think that it’s extremely sweet of person B to initiate chat, and it is considered a good level of effort. Person C however, doesn’t think that person B has put in enough effort by merely initiating a chat.

Now, let’s define a friendship as a relationship between two people. These two people take turns to play the roles of effort-maker and effort receiver in a repeated game (with high signalling and cumulative utility) until such that both parties consider each other friends (this axiom can be loosened in later works).

This leads us to a concept of a criterion to consider the opposite party as a friend. Let’s call this criterion the Minimum-Effort Criterion and define it as the minimum perceived cumulative effort of a effort-maker as perceived by the effort-receiver in order for the effort-receiver to consider the effort-maker a friend. In simple terms, person A only considers person B to be a friend if person A perceives person B to have made the minimum level of effort over a period of time.

### An example of the model in action

So how would an interaction look like between two strangers? Something like this:

Now, to clarify the picture above. Consider Player A and Player B. To make things simple, both Player A and Player B have a minimum-effort criterion of 60. If both players perceive the other party to have made a minumum cumulative effort of 60, they’re considered friends.

Keen readers may have already noticed that different actions are perceived differently by different players. This is how each player perceives effort from the other player based solely on their actions:

 Actions\Players Player A Player B Phone 0.5(x) 0.8(x) IM 0.2(x) 0.2(x) Email 1(x) 0.8(x) Meetup/RSVP 2(x) 1(x) Visitation 2(x) -

The number on the edges of the tree indicates how much effort is expended upon to take that action. For example, it takes a lot of effort for Player A to arrange a meetup (50), while for Player B, responding to a meetup takes the same amount of effort as visiting (30). Both players perceive IMs as low-effort communications, and as such lots of IM conversation needs to happen before they can become friends.

In fact, if I were to give one single equation that defines making friends, it’d be this:

$\displaystyle \sum\limits_{a \in A} N \times f_a | c \ > \ Minimum-Effort \ Criterion$

Simply put: the product of the number of interactions and the perceived-effort from those interactions have to be higher than the minimum-effort criterion for a friendship to be formed. If you were to perform only low perceived-effort actions like IM, you’d have to do a lot to keep up with it to form a friendship.

I personally believe that as one grows older, the minimum effort criterion increases. As mentioned by Laura Carstensen in the original New York Times article that inspired this post,

…[S]he suggests, this is because people have an internal alarm clock that goes off at big life events, like turning 30. It reminds them that time horizons are shrinking, so it is a point to pull back on exploration and concentrate on the here and now. “You tend to focus on what is most emotionally important to you,” she said, “so you’re not interested in going to that cocktail party, you’re interested in spending time with your kids.”

## Discussions

The tree above is merely a very simple perceived-effort based model of making friends. More advanced stuff can most certainly be done, for example incorporating Bayesian information into the tree, allowing for signaling; or defining decay functions into the perceived efforts; or introducing situational modifiers to the perceived efforts, although that will be left as an exercise to the reader, as this is merely a back-of-the-envelope type of exploratory work. I personally feel that as it is though, the model can explain a few other theses on making friends.

### On the shared experience thesis

This model explains the shared experience thesis of making friends to some degree. The shared experience thesis of making friends is essentially this: adults make friends only after sharing an experience together. Many books were written on this topic. The Way of Men by Jack Donovan for example, focuses on a small subset of experiences — namely shared struggle and purpose. Over here in Australia, there is the concept of mateship that expounds the very same thesis.

This is not unexpected. By way of having a shared experience, people communicate more, and more often than not, the experience forces one to expend more effort, or makes one perceive the other’s efforts more clearly.

Take a traumatic experience like… Speed for example (I used Speed as an example because I keep remembering a conversation between Sandra Bullock’s character and Keanu Reeves’ character about relationships borne out of trauma never works). The amount of effort expended by the main characters of the movie towards one another was extraordinary throughout the traumatic episode. As such, friendships are able to be formed over a short period of time.

That said, there is a better model of friendship that explains the shared experience thesis much better. See DeScioli and Kurzban (2009) The Alliance Hypothesis for Human Friendship for more information.

### On the ‘click’ thesis and the quality of interactions

Timothee Boucher suggested this thesis. It essentially says that people will be friends if they ‘click’. That is to say they share interests. The perceived-effort model of making friends does not preclude the ‘click’ thesis.

In fact, a fairly common objection to this model is that it doesn’t take into account the quality of an interaction. For example, a very intellectual session may happen over IM, and the participants of the IM conversation feel connected to one another after. By being directly mappable to a utility function, the perceived-effort function in its own encapsulates the quality of the interaction. That means that if an interaction brings about high utility to a person, the perceived-effort too will be high.

The perceived-effort function too encapsulates a person’s nature. For example, an introvert may perceive email to be a better form of communication than meeting up in a pub. If person A tells person B who is an introvert to meet and discuss project plans in a pub, person B may think that person A is not putting enough effort to accomodate his needs.

Thought this way, the model explains the click thesis of making friends quite well.

### On the community thesis

An oft mentioned thesis of making friends is having a community to ‘belong’ in. However it is true that one can belong in a community and not feel any kinship. Church is often used as an example of how the community thesis works. Let’s analyze the claim using the perceived-effort model as a framework.

Most people dress up nicely to go to church. That is effort expended. By going to church, a shared community location, you are signaling to everyone within the same community that you are willing to put effort. As such, people will perceive you to have put more effort in communicating with one another. So this model also explains the community thesis fairly well.

## Future Work

As mentioned, this is actually very rough, back-of-an-envelope work. I am merely positing this model as a model to make friends. It is not THE model of making and keeping friends. It is very simplified (that’s what models are anyway), and if anyone wants to have fun adding stuff like decay functions to perceived-effort or making a Bayesian tree out of the repeated game, go ahead.

In the next post, I will show you how Strangers for Dinner is applying the Perceived-Model of making friends to its design. One of those “how Strangers for Dinner was made” articles is long overdue. The long story short is that if you host a dinner party, you’re signaling to your guests that you’re willing to put in effort to be friends.

## Errata

RESERVED FOR MAKING ERRATAS

Strangers for Dinner helps you make friends over dinner.
Host or join a dinner party with 5 other perfect strangers every month and make new, better friends.