MAKING FRIENDS I

TEN strangers find themselves in a new location. Each morning, everyone wakes up and chooses someone to visit. Initially the process is random, but the visits result in interactions, and a pleasant interaction with someone makes it more likely the visitor will visit that person next time. Here is a simple model of the reinforcement dynamics. Each person has a numerical weight for each other person, and visits a person with probability proportional to his or her weight for that person. If, initially, all weights are equal, the process is random and visiting probabilities are equal. During a visit an interaction takes place, giving the visitor and host a payoff. At the end of the day, visitors and hosts update their weights for each other by adding their respective payoffs from their interaction to their old weights. The next day the process continues with the updated visiting probabilities.

All kinds of interesting interactions are possible, with all kinds of interesting outcomes. But we will start by looking at a very uninteresting interaction. Each visitor has an equally pleasant interaction, which we represent as a payoff of 1. The accommodating hosts have a payoff of 0 so that only the visitors are reinforced. We will suppose that everyone has an initial weight of 1 for everyone else, so that visiting starts out quite at random.