Simulation of Social Interaction
(The Cocktail Party)
- And You Are?
- Bond, James Bond…
By Josh Anderson and Eldar Nigmatullin.
Introduction
We provide an analysis of the spatial dynamic behavioral outcome of a group of people gathered for a cocktail party when individual utility exhibits social interaction effects. Such interaction effects might cause extremely complicated dynamics of a system as a result of non-linear spatio-temporal interactions among individuals even when the decision making process of each individual is governed by simple rules. The spatial dynamics of an individual moves are determined by several factors: her a-priori preferences about other party participants, presence of groups of individuals, differing affinity for various individuals, and timing of conversations.
Environment Description and the Model
There are N partygoers gathered in a square room. Our goal is to model the spatial dynamics of individual moves within the room during the party. For this purpose we represent the room space as a two dimensional array with individuals each occupying one cell. Each individual is assigned a unique identifying number to allow move tracking. All individuals are aware of the others’ locations.
Our key assumptions are:
- Pre-existing friendship structure
- No cooperative decisionmaking
- Non-adaptive agents
- Additive, factor based utility function
- Popularity effects on utility
- Non-linear spatial discounting
- Diminishing returns to continuous
exposure to individuals
We assume the environment of individual (agent-level) decisionmaking. Each agent faces a discrete choice problem, choosing to move into one of the surrounding 8 floor locations (if unoccupied) or remaining at the current location. We assume that individuals possess well-defined utility function (see below) and that their choice is aimed at utility maximization. At the same time we assume that agents are myopic in the sense that they look only one period ahead.
In our model each person “i” has a priori taste for other people. We denote it by “Likes”: each other individual “j” is associated with a number L(i,j) ranging from –50 to 50 expressing the degree of “being liked” by the person “i”. Big negative number corresponds to the situation when a person does not like this individual, and positive number represents the case when a person likes this individual. Matrix L summarizes pre-existing friendship structure among individuals.
The utility derived by individual A from being located in a particular cell (x,y) is given by
where summation is over all “i” different from A and
U[A,”i”](x,y) is given by
where f(A,j) is the reciprocal of physical distance between A and “i” squared, second summation is over all “k” in the immediate neighborhood of “j” (that is adjacent cells), time(A,”i”) is the time in rounds passed from the starting the conversation of A with “i”, g(.) is a linearly decreasing function. The second term in the bracketed expression represents the spillover effect: people with positive “Likes” next to the person “j” add extra utility associated with this person and vice versa. In our model individuals exhibit time dependent preferences. The last term represents the temporal flow of utility derived from conversation. Initially conversation is rewarding, but soon decreasing returns appear and eventually no value is derived from talking to the individual. The timer starts as soon as person A enters the immediate neighborhood of individual “i.”
Methods
The specified model was implemented in Java. A class for party-goers and the room were developed. A UML diagram is below:
The source code is here:
Results and Conclusions
Simulations were run with varing parameter sets and room sizes. As expected, increasing the rate of diminishing returns to conversation created more movement between agents. Biasing the affinities agents hold for each other downward created repulsion effects, biasing it upwards resulted in one large cluster of individuals. Increasing the parameter that controlled the return to participating in a group made groups more likely to form. Some of the results are here:
|
|
Grid size |
Neighborhood Affinity |
Time Discount Factor |
Maximum agent affinity |
Time factor |
Affinity bias |
|
20 x 20 |
0.7 |
25 |
15 |
2 |
0.2 |
|
|
20 x 20 |
0.8 |
25 |
20 |
3 |
0.4 |
|
|
15 x 15 |
3 |
25 |
20 |
3 |
0.7 |
|
|
15 x 15 |
2 |
25 |
20 |
3 |
0.2 |
|
|
15 x 15 |
0.7 |
15 |
10 |
5 |
0.7 |
The dynamics of a system are quite sensitive to the change in the preexisting friendship structure. If all spillovers are negative then agents are trying to spread evenly within a room, while in case of positive spillover agents are “stacking over” in one spot. In case of the presence of mixed (both negative and positive) spillovers the system behaves sporadically.
Extensions and Future Work
There are several possible extensions that deserve careful consideration.
Adaptive Learning - One possible extension will be to introduce adaptive learning and time varying preferences Lij’s into the model.
Aggregate Utility – The natural question arises about the aggregate utility of a party (sum of individual utilities of agents, say). Is it possible to find the dynamic path of a system that maximizes aggregate utility?
Moving Costs / Status Quo Premium – One plausible assumption will be to introduce the costs associated with moving. Or introduce some premium derived from preserving status quo, i.e. not staying at the same time.
Exiting and Entrance – The natural extension will be to introduce several rooms with independent parties into the original setup and to allow agents to choose between being in one room versus the other. The idea is that all interactions (and consequently utility flows) are shut down when agents are in different rooms. At the same time presence of rooms structure reduces the cost of move.
Increasing Foresight Horizon – Currently we model agents as myopic persons choosing their next step only by comparing their utility values at the next possible steps. Since the system is perfectly deterministic it is possible to make agents more rational by extending their foresight horizon. Who knows what happens in this case?
Limited Scanning Area– One can argue that the location of all individuals is not a common knowledge and it is fair enough to introduce limited scanning area when individuals know of location of other agents and respectively derive utility only in the case if these agents are in some sufficiently small neighborhood. This extension is very close to the one of introduction of several rooms.
Effect of Initial Conditions – A very challenging task will be to classify all possible patterns of dynamic behavior of the system depending on the initial conditions. Initial conditions here are the pre-existing friendship structure and initial allocation of agents ini the room.