Part IV. Games on networks
IV-1. The nxn game on a random network
1. Goal
The goal of this chapter is to learn how to implement models where players are connected in a network (see figure 1). A network is a set of nodes and a set of links.[1] Links connect pairs of nodes. In our models, the nodes in the network will be the players, so each link connects two players.
In this book, we will only use undirected links (which denote symmetric relations such as “being a sibling of”). Nonetheless, in NetLogo it is equally easy to implement models with directed links (for asymmetric relations, such as “being a parent of”), and also models with both types of links.
Here, we will use networks to limit the information that players can access. We assume that players can only interact with their link-neighbors (i.e. those with whom the player shares a link), so link-neighbors are the only players that a player can observe or play with. In this sense, networks define local neighborhoods of interaction, potentially different for each player.
By using networks, we will be able to generalize all the models previously developed in this book. Note that in Part II we implemented models where every player could observe and play with every other player. Such models can be interpreted as network models where players are connected through a complete network, i.e., a network where everyone is linked to everybody else. In Part III, we implemented models with spatial structure, i.e., models where players were embedded on a 2-dimensional grid and they could only interact with their spatial neighbors. Those models can be perfectly interpreted as network models. For instance, the spatial model where we used Von Neumann neighborhoods of radius 1 corresponds to a square lattice network. In this Part IV, we will learn to implement models where players are connected through any arbitrary network.
2. Motivation. A single-optimum coordination game
Consider the following 2-player 2-strategy single-optimum coordination game (which we discussed in chapter I-2):
| Player 2 | |||
| Player 2 chooses A | Player 2 chooses B | ||
| Player 1 | Player 1 chooses A | 1 , 1 | 0 , 0 |
| Player 1 chooses B | 0 , 0 | 2 , 2 |
If you explore the dynamics of this game with the last model we developed in Part II, i.e. nxn-imitate-if-better-noise-efficient, you will see that a population of 100 agents using the imitate if better rule with low noise (e.g. noise = 0.03), starting with 70 A-strategists and 30 B-strategists, will almost certainly approach the inefficient state where everyone is choosing strategy A, and spend most of the time around it. In the video below, strategy A corresponds to strategy 0 (orange) and strategy B corresponds to strategy 1 (green).
Note that in every model developed in Part II, every player can observe and play with every other player (i.e., the network of potential interactions is complete). Now imagine that instead of assuming that everyone can interact with everyone, we