Group formation

Marc Reimann and Nobi Hanaki

Introduction

We study the group formation among a number of reporters who want to be among the first ones to report an unfolding scandal. Each reporter decides how much of his resource to allocate between investiging the scandal and trying to form a group, given the current size of his group.

Setup

The difficulty of uncovering a scandal is represented by parameter $p\in (0,1)$. Lower the $p$ it is more difficult to uncover the scandal. In each period, each agent allocates his resource between two activities: investigation, $r,$ and group formation, $1-r$. The decision is based on his strategy MATH and the size $k$ of the group he belongs to according to:


MATH

The idea is the following: when he belongs to a large group, he spends more effort trying to investigate the story. However, given the size of the group, how much resources an agent allocates to each activity differs. Agents adjust their strategy over time based on their experience. Given the resource an agent allocates for an investigation, he successfully uncovers the story with probability $pr^{\gamma }$ where $\gamma $ is the characteristic of an agent that represents his inherent ability.

The agent also meets with another agent, randomly drawn from the rest of the population, with probability $1-r$. Notice that it is possible to run into a person that already belongs to the same group. In this case, nothing happens. When two agents belong to different groups, they decide whether to form a bigger group. This decision is based on the opinion of the members in each group and both groups have to agree. Each group decides to form a bigger group with a probability $\rho $ where $\rho $ is the average$1-r$ of the members of the group. (Let $\rho _{1}$ and $\rho _{2}$ be the average for each group, then the probability of the two group forming a bigger group is simply MATH)

This process goes on until someone successfully uncovers the scandal. Whoever uncovers the scandal, earns all the reporters belonging to the same group the credit of being the first ones to report the story and the next round of the game begins. Note that the magnitude of the "fame" to be earned is constant, such that belonging to a bigger group, while increasing the probability to be first to uncover the scandal leads to a decrease in the fame for each member of the group.

Before starting to investigate a new scandal, each agent updates his strategy $\alpha $ by comparing the size of the group that uncovered the most recent story and his own group. If a larger group uncovered the story, then he will become more likely to allocate more resources to look for another agent and enlarge the group. Specifically, let $\widehat{k}$ be the size of the group reporting the scandal and $k$ be the size of the group the agent belongs to, then the agent next period strategy is:


MATH

$\delta \in (0,1)$ is the parameter that determines the adjustment speed. It can be viewed as inertia with respect to changing ones preferred group size.




Preliminary Result

We are asking the following questions: What are the effects of different distributions of individual ability, $\gamma $, on the system behavior? Does adjustment speed matter? What if adjustment speed diffesr from one agent to another?

We would like to answer these questions through computational experiments. While we still lack enough computational experimental results to make statistically sound statements, our preliminary observations lead to some insights.

To understand these consider the following: From a point of view of uncovering the scandal as fast as possible, each agent should allocate all his resources to investigating the story. In this situaton, no groups will form, and all agents work on their own.

Given, that the agents act purely self-interested, our simulations generally do not reflect this outcome (except, if alpha is sufficiently low initialized). Rather, the average group formation effort of agents becomes higher, i.e., agents are spending more resources to find partners, and as a result the average group size becomes larger. (Indeed, it reaches the maximum size of 20. This can be seen in figures 1 and 2. For these two figures, we run the simulation with $\gamma =1$, $\delta =0.05\,$, $p=0.001$, $\alpha _{Max}=10$ and an initial $\alpha =0.002.$ Figure 1 plots the dynamics of the average group size. As one can see, after 500 periods, a group size of 20 is the norm. Also, around that period, the average effort of the agents reaches seemingly a steady state of 0.14.


index__33.png Figure 1: Dynamics of Group Size ($\gamma =1$, $\delta =0.05\,$, $p=0.001$, $\alpha _{Max}=10$ and initial $\alpha =0.002.$)





index__40.pngFigure 2: Dynamics of average strategy $\alpha $ ($\gamma =1$, $\delta =0.05\,$, $p=0.001$, $\alpha _{Max}=10$ and initial $\alpha =0.002.$)

Future Direction

At this point, we have not completed enough sets of experiments to fully understand the effects of the model parameters on the behavior of the system. Currently we are producing more computational results to answer our questions.

In addition, we believe that there are several shortcomings in our current setup. In the real life, the credit of uncovering the story one receives given his group size should influence strategies and decision making. In the above setup, we assumed this aspect away. This will be the first thing to consider in extending the framework. Also, when the characteristics and abilities of individuals differs, strategy update can depend on an agents knowledge of other agents' abilities. This might generate interesting dynamics both in the individual strategies and group size.