This chapter presented a RL model of statistical discrimination using the BRA algorithm developed in chapter2. It aimed to reproduce experimental results and asked whether these results could be generalised with an RL- based model. Thus, the question was, whether, starting from empirical observation, there is a general mechanism that could explain the emergence of statistical discrimination.
The RL model was compared with both theoretical and empirical re- sults from statistical discrimination games. Several model variants were simulated to find out which setup and parameter setting can reproduce the patterns of Fryer Jr. et al (2005)’s classroom experiment best. One model (variant II) reproduced with a relative large discrimination the aggregate as well as behavioural patterns. Using this setup, some more scenarios were simulated to analyse the properties of the model further.
Similar to game-theoretic and experimental results, the RL model shows that statistical discrimination can exist. Whether it occurs is, however, path-dependent. The scenarios simulated in this chapter were not capable of creating a setting in which statistical discrimination emerges on the average. Some differences to the theoretical as well as experimental literature can be highlighted:
– The relationship between θ and employment level could not be ob- served. Increasing the likelihood of a positive test result does not increase the number of investing workers and thus their hiring rates.
– Initial beliefs do not necessarily influence the outcome in the long run. That is, employer behaviour can adjust very quickly if worker behaviour changes, e.g. due to an intervention such as increasing access to human capital.
– In the RL model, discrimination emerged although no knowledge about market outcomes was available, whereas in the experiment knowledge about market outcomes was public. Thus, for a belief to emerge it may require even less publicly shared information. By look- ing closer at the rules that emerged during the simulations, it could be shown that the behavioural patterns of the RL model are nevertheless similar as in the classroom experiment: If in doubt, hire a worker if he or she comes from the group that is believed to be more productive; do not hire if he or she comes from the group that is assumed to be less productive. In some simulation runs, this results in a state in which workers of the preferred group get hired even if they signal low productivity.
– In the RL model, worker behaviour is the driving factor for generating discrimination. Discrimination can only emerge if the groups stick to different investment behaviour after employer change their policies.
In summary, the RL model was shown to be a good approximation of actual human behaviour in the experiments. While results of experiments and simulations are similar, the RL model cannot confirm all the relation- ships postulated by theory. Furthermore, a general rule capable of creating discrimination could not be found.
Network Formation
4.1
Introduction
Networks are an important paradigm for modelling social and economic relationships. How members of a society are connected to each other de- termines behaviour and welfare. Through connections to other persons, important resources can be accessed and used for one’s own purpose.
A useful distinction is between social and personal networks. Personal networks are comprised of the relations an individual has, e.g. relatives, friendships, acquaintances. Social networks are an aggregation of individual networks.
The structural properties of networks have long been the topic of network analysis in Sociology. The following stylized facts about empirical personal networks can be drawn from this literature:
– Personal networks are small; the closer the contact, the smaller the network. In personal relationships, these contacts are usually family and a few very close friends. Building on existing research, Hamill
and Gilbert (2009) also note that maintaining close relationships is costly, requiring resources as time and effort. This naturally limits the extent of close relationships a person can maintain.
– The distribution of the degree (the number of links a node in the net- work has) of personal networks is unequal. Some individuals are more sociable, and so have more relationships than most other people in the network. Empirically, distributions of personal networks are typically right-skewed, or ’fat-tailed’. This has been observed, for example, for co-authorship data: There are some economists who appear in many co-authored papers, while the majority has only few co-author rela- tionships (Jackson 2008; p.60).
– Members of personal networks tend to share the same characteristics (homophily). Contacts between similar people are more likely than among dissimilar people. This can be described by the cluster coeffi- cient. This coefficient determines, in principle, the likelihood that two nodes share the same links. Thus, personal networks have characteris- tically large cluster coefficients, as compared to, say, a social network. This phenomenon has already been observed by Granovetter (1973).
Social networks, on the other hand, are much less connected than indi- vidual networks, i.e. they have a low overall network density (the ratio of all links relative to all possible links). In larger groups, it is simply impossible to know most other people. Nevertheless, most individuals in a society can be reached within a few steps. This property of small average path lengths (the number of nodes between any pair of nodes in the network) and small diameters (the largest distance between any two nodes in the network) has been captured in the notion of small worlds. This phenomenon became widely known by Milgram’s experiments (e.g. Watts 2004; Milgram and
Travers 1969). In these experiments, persons had to route letters to per- sons in other states whom they did not directly know, passing the letter to the target themselves or to someone they thought is likely to know that person. About a quarter of the letter reached their targets. Drawing on such insights, Milgram suggested that there are in general six degrees of separation, i.e. anyone in a society is linked to anyone else with just six intermediaries.
While a large literature about the structural properties of networks exists, much less has been written about the dynamic aspects. Many concepts of how and why people relate to each other are based on chance (homophily, social or regional closeness, etc.), but they do not conceptualise the cre- ation and maintenance of relationships as choice. For example, Barabasi’s preferential attachment model (Barabasi and Albert 1999) simply assumes a higher probability of linking to already well-connected persons in the so- ciety. However, if links, as stated above, are assumed to be costly, persons have to make implicit or explicit decisions about who they want to be friends with.
The concept of strategic network formation models the decisions on the micro-perspective explicitly. Strategy should here be understood not liter- ally, but in the sense that individuals tend to form mutually beneficially relationships and drop relationships that are not (Jackson 2008; p.153).
Viewing the formation of connections in such a way allows to model networks as the outcome of a game. Goyal (2007) summarises the main features of strategic network formation as follows:
– Strategic network formation can be modelled as a game in which play- ers decide to link or not to link to each other, depending on some
value function of the network and an allocation rule that distributes the value among the players.
– It is based on assumptions of complete information (players know each other and the payoff structure).
– Networks have some form of externality; that is, for individual players the structure of the network itself influences their utility.
Communication networks represent a commonly used network model in Economics. Communication networks model relationships among individ- uals that exhibit some benefit to the members of the network, typically in the form of information flows. The benefit depends on the number of other persons a member is linked to; the more persons in the network, the higher its value to the individual. In its simple form, utility is a linear function of the number of other players in the personal network. A more realistic form assumes decay in value the more distant the source of information is. Es- tablishing and maintaining direct links is costly because it typically involves some effort. Individual utility depends, then, on the relationship between costs and benefits. On the aggregate, then, this relationship determines the shape of the networks that can form. Such a general model can cover many interesting social and economic settings where the structure of the networks influences the well-being of the members, for example, friendships, work re- lationships, but also research partnerships between firms.
Although strategic network formation focuses on the micro perspective, it is also possible to generate large-scale networks. Jackson and Rogers (2005), for instance, present a spatial variation of a communication net- work game. In this model, players are distributed on islands. Costs for connecting to near-by players are low and high for connecting to distant
players. Jackson and Rogers (2005) show that with certain cost settings, the resulting network exhibits small world properties. The intuition is that players form links with most of their close neighbours, but economise on dis- tant links. It is, nevertheless, still beneficial to maintain the distant links which provide the only chance to access the benefits of more players. Simi- larly, the residential segregation model ofSchelling(1971) could be seen as a prototype of a network model combining chance and choice: Green and red members of a society move randomly and meet other members. Depending on their preference for living in a same-colour neighbourhood, they decide to relocate or stay. The result is a society that is clustered into same colour neighbourhoods.
In recent years, several experiments with strategic network formation have been conducted in order to compare the theoretic predictions with empirical data. Few of them are based on the partially cooperative network model ofJackson and Wolinsky(1996; JW). In this model, links are formed only if both involved players agree. More research is relatedBala and Goyal (2000)’s non-cooperative version, where links can also be established uni- laterally.
As this overview illustrates, strategic network formation can be seen as a different and complementary way for generating personal and social net- works dynamically. There have also been experiments to evaluate the pre- dictive power of network formation models. So far, however, no experience- based model of network formation exists. In general, RL models have been found to predict experimental data better (see chapter 2). The purpose of this chapter is to provide such a model for network games in order to bridge the gap between theory and experimental evidence. It focuses on the level of personal networks alone.
Section4.2 first introduces notation and definitions. Section4.3 shortly discusses the relevant theoretical, section 4.4 the experimental literature. The RL model is then described in section4.5. The simulations are analysed in section 4.6. In section4.8, a modified version of the RL model is used to compare the results to the laboratory experiments conducted byConte et al (2009). The main question is whether the RL model can predict the outcome of network formation processes better than the equilibrium prediction.
Relating the RL network model to the general learning approach as discussed in chapter 2, it represents the case-based variant of BRA. Using definition 4 developed in section 2.4.4, it can be described with: k > 1,
|Ak| > 1, ∩n
k=1
Lk
l = ∅ and succ(Lk0) = ∅ for a number k of cases. It is
assumed that all players know each other, so that k players represent the
k cases. There is no dynamic extraction of rules. A variant of BRA where the case distinctions are allowed to evolve dynamically is shortly presented in section 4.7.