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The topology of interactions has been proved very influential in the results of models based on learning and evolutionary game theory. This paper is aimed at investigating the effect of structures ranging from regular ring lattices to random networks, including small-world networks, in a model focused on property distribution norms. The model considers a fixed and finite population of agents who play the Nash bargaining game repeatedly. Our results show that regular networks promote the emergence of the equity norm, while less-structured networks make possible the appearance of fractious regimes. Additionally, our analysis reveals that the speed of adoption can also be affected by the network structure.

The emergence, prevalence, and collapse of social norms in groups have attracted scientists from a wide range of disciplines [

In the field of Economics, there are plenty of transactions that are commonly regulated by means of habits, repeated interaction, community enforcement, social pressure, trust or reputation, rather than by formal contracts [

Learning game theory provides a useful framework to analyse this type of norms formally [

Relaxing the assumption of global interaction and using sophisticated learning rules usually reduces the analytic tractability of the models and accentuates the relative usefulness of computer simulation for exploration and analysis. Given the explicit correspondence between players in the model and computational entities in the simulation, those players are naturally implemented as agents in an agent-based model [

Concretely, in the case of property distribution norms, interactions are often modelled as Nash bargaining games (also known as Nash demand games) [

AEY’s model has been extended to understand the effect of spatial structure. In particular it has been analysed in regular square lattices with a fixed finite population of tagged agents [

The scientific origin of small-world research is attributed to the pioneering work of Pool and Kochen [

Models of dynamical systems embedded in small-world networks display different global behaviour due to enhanced signal-propagation speed, computational power, and synchronizability [

In this paper we have extended the analysis of dynamic norm diffusion in a population considering AEY’s model as a framework. We have analysed the influence of the small-world topology on the results of the game. To this aim, we have organized the paper as follows: first, we briefly explain the extensions and modifications that we have performed on AEY’s original model and the main properties of the network generator mechanism based on the Watts-Strogatz algorithm [

The model proposed in this paper is based on AEY’s model [

Payoff matrix of the Nash demand game.

(0,0) | (0,0) | ||

(0,0) | (50,30) | ||

(30,50) | (30,30) |

Agents are endowed with a memory (of size

The influence of some parameters of the model (such as the number of agents, the memory size, the payoff matrix, or the decision rule) has been thoroughly analysed in [

In a later extension of the model [

The model that we present in this paper is an extension of AEY’s model [

In the model implemented in this paper we use the Watts-Strogatz algorithm [

Network structure for several values of the probability of rewiring.

In our model, the network is created at the beginning of each run and remains fixed thereafter. At each time period, all the agents are selected in a random order to play the Nash Demand Game with one of their (randomly selected) neighbours. It is important to note that each time period consists of

In the subsequent experiments we will show how the probability of rewiring (and thus the properties of the resulting network) affects the regimes that can be reached in the AEY’s game.

Notice that, unlike in previous works [

Before doing a computational exploration of the agent-based model, it is particularly interesting to conduct a previous analysis using the framework of Markov Chains [

As previously explained, an interaction between two agents is modelled as a Nash demand game [

The system dynamics are determined by the presence or absence of errors in agents’ decisions. In the absence of errors, that is,

When we assume local interactions, that is, an agent can only play with her neighbours in the interaction network, besides the EQ state, there are two other absorbing states corresponding to the inequitable strategies (In networks with more than one component, there can be more types of absorbing states. In these cases, because each component is independent of the others, the absorbing state of the system is defined as the combination of the absorbing states reached by each component.). This happens when there are two separated groups of agents, in terms of the network, in which the individuals of one of the groups expect the others will demand

An inequitable state in a triplet (a), and in an even cycle of 6 agents (b) with its corresponding bipartite representation (c). Agents who demand high (H) are depicted in light blue, while those ones who demand low (L) are in light yellow. Note that for inequitable states to be absorbing, the interaction network has to be bipartite (second and third figures), that is, it should not have odd cycles.

When errors are possible in the agents’ decisions, (Errors refer to the noisy response explained in the model’s section.), that is,

The asymptotic behaviour is not very useful if we want to apply the model to real situations in which the “long run” is a very vague concept. For that reason, it is interesting to pay attention to the transient dynamics too, following the guidelines proposed by Axtell et al. [

All the states and persistent regimes defined in the previous section correspond to a set of Markovian states of the system. However, in order to complete the analysis and discussion of the model, we also need to characterize some of the individual states in which an agent can be from the point of view of the agent’s behaviour, that is, which of the three possible decisions

It is well known that many real social networks show a significant propensity to form groups or clusters of agents more densely interconnected among them than what could be expected by pure randomness [

The effect of a mutation within a triplet (a), and on a nonclustered triad (b), when agents are initially coordinated in the equity norm and the mutant changes her demand from M (light green) to H (light blue).

The first case represents a triplet of agents initially following the equity norm, when one of them changes (mutates) her demand from

Obviously, the analysis is not so trivial if the network is bigger and much more complex, but the intuition, inferred from these simple examples, is that the equity norm is much more robust against random mutations when agents are clustered than when they are not. Consequently, we should expect that the evolution of the bargaining (under the hypothesis of the model proposed in this work) tends to reach the EQ regime more frequently in networks with higher clustering. The design of experiments and the computer simulations described in the next section aim to confirm this intuition.

The purpose of this section is to describe how the equity norm emerges and spreads across the population in finite time (transient dynamics). In simple and abstract terms, the dynamic process evolves as follows: the population starts from a randomly initialized state; these random initial conditions make it likely that, initially, one or more agents adopt the equity norm and coordinate with each other in small groups that reinforce the norm; if this coordination process occurs quickly, and some of these equity nuclei are able to reach a critical size (which depends on the particular properties of the network they are embedded in), then they will be able to expand their limits and grow, making the equity norm spread across the whole population. Unlike other diffusion phenomena already studied in the literature [

In order to do so, we initially define a new unit of analysis called

Second, we need to determine a metric to measure the change in an equity nucleus after a complete interaction at each time period

An equity nucleus (dark and light green agents). The inner border is made up of the (light green) agents belonging to the nucleus who have one or more neighbours out of it. The outer border consists of the (light blue) agents not belonging to the nucleus who have one or more neighbours within it.

Note that any change in a nucleus must involve one of these two borders. A nucleus can grow by adding new members of the outer border who adopt the equity norm. Similarly, a nucleus can decrease as a consequence of losing members of the inner border who leave the norm. Obviously, the real nuclei dynamics might be a little different, since in each time period

Finally, we set a procedure to compute all these properties over a simulation run. Before a complete interaction at time period

In the ABM model proposed, agents are embedded in a small-world interaction network (SWN from now). We have chosen the small-world algorithm by Watts and Strogatz [

The parameterization of all scenarios studied in this paper corresponds to a model of

During a simulation run, we say that an agent follows the equity norm strongly whenever she has at least

As explained before, in the transient dynamics of the system, simulations often reach one of two expected regimes: the EQ regime or the FR regime. The first one corresponds to the emergence of the equity norm, while the second represents a confusing and disordered state in agents’ decisions that prevents any coordination in the bargaining. Now, the first question that arouses our interest is to understand how small-world networks condition the emergence of these regimes. To determine this influence we have computed the frequencies of both regimes when the rewiring probability

Above, the frequency of the EQ regime reached at the end of the simulations when the rewiring probability

The first inference that can be made from the results is that the emergence of the EQ regime depends significantly on the rewiring probability, and more concretely on the structure of the interaction network. In the case of regular ring lattices

In this section we characterize the diffusion process of the equity norm. We apply our particular approach based on observing the emergence and evolution of clusters of agents playing the norm-equity nuclei. We also try to correlate the dynamics of these nuclei with their network properties and estimate their expected change. We will see how the rewiring probability of the small-world networks conditions significantly not only the probabilities of the emergence of successful equity nuclei, but also their growing speed over the population.

The diffusion of the equity norm is quite similar to the movement of a wave of adopters in a population embedded in a social network. By randomness, one or more small groups of linked agents start to follow the equity norm (equity nuclei), and depending on their internal structure and the structure of the network that surrounds them, they have greater or lower probabilities of growing successfully by incorporating new members which modify the properties of the nuclei. Overall, if an equity nucleus reaches a critical size with particular properties, it will invade the population, but these properties will depend highly on the parameters of the interaction network.

We have analysed the observed dynamics of the equity nuclei by means of a gradient map obtained through computer simulation data (this procedure has also been used in [

Gradient maps of the observed dynamics of equity nuclei for different values of the rewiring probability. Each arrow represents the direction of the change in the size-clustering space, while the colour of the cells is the probability of growing in size, which can be interpreted as a measure of the speed in nuclei growth. When there is no simulation data for a particular combination of size and clustering, the corresponding square cell of the map is coloured in white.

In most cases of Figure

For regular ring lattice

On the other hand, for greater rewiring probabilities

We can summarize these inferences into the next statements: locally structured networks—in the sense of having more clustering—promote the emergence of the equity norm, while less locally structured networks facilitate the appearance of disordered or fractious states (according to the data of Figure

Figure

Above, the average of the time of convergence to the equity norm, and below the corresponding boxplot, when the rewiring probability

Finally, we have extended the computing analysis of the bargaining model by running other simulations in order to check the sensitivity of the results to changes in other parameters, mainly in the size of the population. Figure

The frequency of the EQ regime for different sizes of the interaction network. The qualitative results do not differ from those ones analysed in the previous sections for a population of 100 agents. Regular ring lattices and networks with low rewiring probability support the dominance of the equity norm, while more random networks contribute to the emergence of fractious states.

In this work we have addressed the effect of topologies of interaction ranging from regular ring lattices to random networks, including small-world networks on the Nash demand game in a finite population of agents. Our analysis shows that locally structured networks—in the sense of having more clustering—promote the emergence of the equity norm, while less locally structured networks facilitate the appearance of disordered or fractious states. At the same time, results indicate that the clustering of the network can slow down the diffusion of the equity norm making more difficult the process of adoption. For example, in the case of quasiregular ring lattices, an equity nucleus that invades the whole population always emerges, sooner or later, although the clustering of the network slows down the convergence to the norm. On the contrary, in more random networks, the probability of this event decreases with lower clustering values; although if an equity nucleus succeeds, the speed of the convergence to the norm is much faster. Our findings seem robust to the size of population and corroborate the influence of some properties of the interaction structure in learning and evolutionary games.

The authors would like to thank Dr. Luis R. Izquierdo for some advice and comments on this paper. The authors acknowledge support from the Spanish MICINN Projects CSD2010-00034 (SimulPast CONSOLIDER-INGENIO 2010), TIN2008-06464-C03-02 and DPI2010-16920, and by the Junta de Castilla y León, References BU034A08 and GREX251-2009.