Hybrid Network Models¶
- Most networks involve both choice and chance in formation
- What are the relative roles?
- Random/Strategic models can be too extreme
- Can we see relative roles in homophily?
Application - Homophily¶
- Group A and Group B form fewer cross race friendships than would be expected given population mix
- Is it due to structure: few meetings?
- Is it due to preferences of group A?
- Is it due to preferences of group B?
Currarini, Jackson, Pin(09,10)¶
- Utilities specified as a function of friendships
- Meeting process that incorporates randomness
- Allow both utilities and meeting process to depend on types
Revealed Preference Theory¶
- Common to Consumer Theory
- Use it in mapping social/friendship choices to!
- Different information than surveys on racial attitudes
Model¶
Types: $i \in \{1,...,k\}$
$s_i$ = # same-type friends $d_i$ = # different-type friends
$U_i = (s_i + \gamma_i d_i)^{\alpha}$ utility to type i
- $\gamma_i$ is the preference bias
- $\alpha < 1$ captures diminishing returns
Individual Choice¶
$t_i$ number of friends -- proportional to time spent socializing -- i is "type"
$q_i$ fraction of friends that will be of own type
$t_i$ maximize $(q_it_i + \gamma_i (1-q_i)t_i)^\alpha - ct_i$
- Solution: $t_i = (\alpha /c)^{1/(1-\alpha)}(q_i + \gamma_i(1 - q_i))^{\alpha/(1-\alpha)}$
- Add noise for particular agent a of type i: $t_{ai} = (\alpha /c)^{1/(1-\alpha)}(q_i + \gamma_i(1 - q_i))^{\alpha/(1-\alpha)} + \epsilon_\alpha$
How to identify preference parameters from data?¶
$t_{ai} = (\alpha /c)^{1/(1-\alpha)}(q_i + \gamma_i(1 - q_i))^{\alpha/(1-\alpha)} + \epsilon_\alpha$
This is observed directly in the data and will vary with $q_i$
If $\gamma_i < 1$ then this is increasing in $q_i$, so more fraction of people I'm meeting of my own type, I should form more friendships, and so that's going to allow us to begin to fit what $\gamma_i$ is. So the idea is $t_{ai}$ should be a function of $q_i$, and how quickly it varies with $q_i$ is going to be dependent on what $\gamma_i$ is
Larger Group = More Friends¶
Meeting Process¶
Where do $q_i$ s come from?
Randomness in meetings, but also have $q_i$ s determined by the decisions of the agents
Bias in Meeting Process¶
$q_i$ rate at which type i meets type i $1-q_i$ rate at which type i meets other types
$q_i = (stock_i)^{1/\beta_i}$
$\beta_i = 1$ "unbiased": $q_i = stock_i$
$\beta >1$ meet own types faster than stocks
Meeting Process¶
$q_i = (stock_i)^{1/\beta_i}$
$beta_i = 1$ if $stock_i = 1/2$ then $q_i = (1/2)^{1/1} = 1/2$
$beta_i = 2$ if $stock_i = 1/2$ then $q_i = (1/2)^{1/2} = 0.707$
$beta_i = 7$ if $stock_i = 1/2$ then $q_i = (1/2)^{1/7} = 0.906$
Equilibrium Conditions¶
$t_i$ maximize $(q_it_i + \gamma_i (1-q_i)t_i)^\alpha - ct_i$
$stock_i = w_i t_i / \sum w_j t_j $ fraction type i in the meeting
$q_i = (stock_i)^{1/\beta_i}$ meetings determinded by stocks
$q_i^{\beta_i} = stock_i$ and $\sum stock_i = 1$ imply that $\sum q_i^{\beta_i} = 1$ (balanced meetings)
atomless population (ignore individual errors)
Fitting the model¶
$t_i - \epsilon_i= (\alpha /c)^{1/(1-\alpha)}(q_i + \gamma_i(1 - q_i))^{\alpha/(1-\alpha)}$
$\sum q_i^{\beta_i} - \epsilon = 1$
Eliminate (unobserved) costs¶
$t_i - \epsilon_i= (\alpha /c)^{1/(1-\alpha)}(q_i + \gamma_i(1 - q_i))^{\alpha/(1-\alpha)}$
$(t_i - \epsilon_i) / (t_j - \epsilon_j) = (q_i + \gamma_i(1 - q_i))^{\alpha/(1-\alpha)} / (q_j + \gamma_j(1 - q_j))^{\alpha/(1-\alpha)}$
$t_i(q_j + \gamma_j(1 - q_j))^{\alpha / (1-\alpha)} - t_j(q_i + \gamma_i(1 - q_i))^{\alpha/(1-\alpha)} = error$
Fitting Technique¶
Search on grid of biases in preferences and meetings ($\alpha_i, \beta_i, \gamma_i$):
For each network (school) and specification of biases, calculate an error in terms of total deviation from fitting equations ($t_i, q_i$)
Sum squared errors across networks (schools)
Choose biases to minimize (weighted) sum of squared error
