# Social Networks and Choice Set Formation in Discrete Choice Models

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Unordered Multiple Choice and Choice Sets

## 3. Social Networks and Choice Set Formation

#### 3.1. Modeling Social Networks

**A**, where an element ${a}_{nm}=1$ if decision maker n is connected to decision maker m $(\forall n\ne m)$, and ${a}_{nm}=0$ otherwise. We assume the diagonal of

**A**is equal to zero reflecting the fact that decision makers are not socially connected to themselves. We also assume that every decision maker has at least one connection.

**A**. Decision maker n may be connected to m while m is not connected to n. This implies that

**A**is a directed network. The relevance of this property will be evident in the next sections when we add social network effects into the IAL model. This property indicates that, if ${a}_{nm}=1$ and ${a}_{mn}=0$, decision maker m influences decision maker n, however, n does not influence m. Therefore, the n-th row of

**A**represents n’s social connections, in other words, it indicates all decision makers with a social influence on n.

**W**be a row-normalization of

**A**such that its elements are ${w}_{nm}={a}_{nm}/{\sum}_{m}{a}_{nm}$. The n-th row of

**W**represents a distribution of weights that the row decision maker assigns to the connections with the column decision makers. This normalization implies that all connections of a decision maker’s network have identical weight.5 However, our CSF model allows for respondents to adjust the relative strengths of own and network characteristics. This is described in the next section.

#### 3.2. Social Network Effects on Alternative Availability

**q**be a $N\times J$ matrix representing the quality perception profile of the decision makers in the social network

**W**. Its $(n,j)$ element ${q}_{n,j}$ is individual n’s quality perception about alternative j. The matrix

**Wq**is therefore the quality profile of the decision makers’ social networks.6 We assume the deterministic component of availability to be a convex combination of own and network quality perceptions. Considering the entire network, this component can be written as

**Wq**). The availability of alternative j to decision maker n is determined by comparing element $(n,j)$ of the matrix (8) with the random quality threshold determined by ${\tau}_{0}+{\xi}_{nj}$. Specifically, availability is probabilistically defined in our network model as

#### 3.3. Social Network Effects on Availability and Utility

**W**. As a result, the DNE model replaces utility (5) with

## 4. Estimation of Welfare Impacts

## 5. Monte Carlo Experiments

**U**(0, 1) to construct ($2000\times 3$) matrix that collects ${p}_{nj}$ and ${q}_{nj}$, for $n=1,...,2000$ and $j=1,2,3$.10 This set of price and quality data are utilized in all replications of all experiments.

**W**, we split our population of 2000 decision makers into 10 groups of 200 individuals representing 10 social networks. This structure mimics a situation in which the econometrician has access to network data on distinct social groups and builds a block diagonal matrix

**W**in which each block corresponds to a social network, e.g., a school, groups of coworkers in companies offices, or a neighborhood.

**A**(see Section 3.1), we develop a latent variable model in which a link is observed (${Y}_{n,m}=1$) if latent link ${Y}_{n,m}^{*}\ge 0$:

**A**. Figure 1 shows the scatter plot of latent link (${Y}_{n,m}^{*}$) and inverse homophily (${H}_{n,m}$), making it evident that decision makers with similar quality perceptions are more likely to be connected in the social network. Figure 2 shows the histogram of (${Y}_{n,m}^{*}$). Dyads above the dashed line in Figure 1, or in the shaded are to the right of the dashed line in Figure 2, are socially connected and corresponds to a network density of 0.22.11 Once

**A**is generated, it is straightforward to compute

**W**and calculate network quality $\tilde{q}$.

**G**(0,1). For simplicity, we assume that V takes a linear functional form $V={x}^{\prime}\beta $, where x is a vector of alternative attributes. In the SNE model, attributes are alternative-specific constants, price, and own quality. Network quality is added as an attribute in the DNE model (see Table 1).

#### 5.1. Evaluating Parameter Estimates

#### 5.2. Evaluating Welfare Estimates

## 6. Results

#### 6.1. SNE Experimets

#### 6.1.1. SNE Parameter Estimates

#### 6.1.2. SNE Market Shares

#### 6.1.3. SNE Welfare Estimates

#### 6.2. DNE Experimets

#### 6.2.1. DNE Parameter Estimates

#### 6.2.2. DNE Market Shares

#### 6.2.3. DNE Welfare Estimates

## 7. Discussion

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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^{1}Ben-Akiva and Lerman [28] offer a discussion about the properties of the Gumbel distribution.^{2}The primitive set can be thought of as the MNL’s global choice set.^{3}The literature offers several approaches to incorporate choice sets in discrete choice models. For example, the Constrained Multinomial Logit introduces a penalty to the utility of alternatives outside the consumer’s consideration set (Martinez et al. [29]). Other papers examine the role of choice set formation with a focus on advertising. Sovinsky Goeree [30] presents a discrete-choice model of limited consumer information; where advertising influences the set of products from which consumers choose to purchase. Draganska and Klapper [31] examine the dual role of advertising, consumer preferences and choice set formation, and propose an approach to disentangle the two effects using individual-level data on brand awareness (i.e., consideration sets).^{4}The graph-theoretic approach proves to be useful when modeling networks of multiple relationships. Jackson and Watts [36] and Jackson and Rogers [37] are examples of papers that use the graph-theoretic approach. Neilson and Wichmann [17] use sociometric approach to develop a valuation model in which social networks influence the utility that individuals obtain from public goods. The sociometric approach is also used by DeGroot [38] and DeMarzo et al. [39] to model social influence in unidimensional opinion models.^{5}Theoretically, this standardized normalization can be easily relaxed by allowing connections to have different weights. The standardized normalization captures the extensive margin of social effects. In practice, it requires pairwise data about the existence of social links. A more sophisticated normalization may capture the intensive margin of social effects; however, it would require data not only on the existence of social links, but also on the strengths of social contacts.^{6}To see why, recall that the diagonal of the matrix**W**is equal to zero.^{7}We implicitly assume that there is one social network influencing both choice set formation and utility. This is just a simplification assumption and it is straightforward to extend this model to one in which the network that influences choice set formation in the first stage is different from the network that influences choice in the second stage.^{8}The subscripts n (indexing the decision maker) and j (indexing the alternative) are suppressed to simplify notation.^{9}In fact, Equation (14) corresponds to the $E[CV]$ expression for the MNL model.^{10}Both prices and quality differ not only between alternatives but also between individuals capturing decision-maker heterogeneity in quality perceptions and prices (due to, for example, accessibility).^{11}Network density is the share of the potential links in a network that are actual links.^{12}The first stage alternative rule-out process determines the distribution of choice sets. Indexing choice sets by $j=1,2,3$, the universe of choice sets is $\{1\}$, $\{2\}$, $\{3\}$, $\{1,2\}$, $\{1,3\}$, $\{2,3\}$, $\{1,2,3\}$, and the empty set. Therefore, there is a possibility that none of the three alternatives satisfies the availability condition and choices are therefore not observed. In order to keep the sample size constant throughout the simulation, we redraw the error ${\xi}_{n}$ for individual n if all alternatives are ruled out. The parametrization of the experiment was such that the re-drawing was done only in approximately 3%–5% of the replications. This procedure enables us to compare estimates from different models with the assurance that (possibly observed) biases or inefficiencies are not driven by sample size differences.^{13}Note that the chosen alternative after the project may be the same alternative chosen before the policy, therefore ${\mathbf{x}}_{chosen,n,r}^{1}$ may (or may not) be different from ${\mathbf{x}}_{chosen,n,r}^{0}$, and the same is true for the error term.^{14}The one exception is the estimate of ${\tau}_{0}$ in the experiment with ${\tau}_{1}=0.2$.^{15}Refer to the Supplementary Materials for a discussion about the distribution of choice sets.^{16}“Available estimates” refers to estimates obtained using pre-project (real, not hypothetical) data.^{17}Recall that the IAL ignores network effects and does not estimate the ${\beta}_{\mathrm{NET}}$ and ${\tau}_{1}$ parameters.^{18}Experiments with ${\alpha}_{1}=0.2$ are experiments where (${\beta}_{\mathrm{OWN}}=4$, ${\beta}_{\mathrm{NET}}=1$), i.e., 20% of the combined quality effect ($\beta =5$) is assigned to network quality (see Equation (11)). Similarly, ${\alpha}_{1}=0.5$ corresponds to (${\beta}_{\mathrm{OWN}}=2.5$, ${\beta}_{\mathrm{NET}}=2.5$).^{19}Note that the top two panels of Table 7 present estimates of experiments in which the DGP has (${\alpha}_{1}=0.2,{\tau}_{1}=0.2$) and (${\alpha}_{1}=0.2,{\tau}_{1}=0.8$), and the bottom two panels presents results for the DGPs with (${\alpha}_{1}=0.8,{\tau}_{1}=0.2$) and (${\alpha}_{1}=0.8,{\tau}_{1}=0.8$).

Number of Decision Makers (N) | 2000 |

Number of Global Alternatives (Alternatives in B) | 3 |

Random Terms | |

Parameters of the Logistic Distribution (mean, scale) | 0, 10 |

Parameters of the Gumbel Distribution (location, scale) | 0, 1 |

Attributes of Each Alternative | |

SNE model | price, own quality |

DNE model | price, own quality, and network quality |

Value of True Parameters of the SNE Experiments | |

Fixed CSF threshold parameter (${\tau}_{0}$) | 0.5 |

Varying degree of social interaction in CSF | ${\tau}_{1}\in \{0.2,0.4,0.5,0.6,0.8\}$ |

Fixed utility parameters (ASC1, ASC2, ${\beta}_{p}$, ${\beta}_{\mathrm{OWN}}$, ${\beta}_{\mathrm{NET}}$) ${}^{\mathrm{a}}$ | 2, 1, −3, 5, 0 |

Value of True Parameters of the DNE Experiments | |

Fixed CSF threshold parameter (${\tau}_{0}$) | 0.5 |

Varying degree of social interaction in CSF | ${\tau}_{1}\in \{0.2,0.4,0.5,0.6,0.8\}$ |

Fixed utility parameters (ASC1, ASC2, ${\beta}_{p}$, β) ${}^{\mathrm{b}}$ | 2, 1, −3, 5 |

Varying degree of social interaction in choice | ${\alpha}_{1}\in \{0.2,0.5,0.8\}$ ^{c} |

Number of Replications (datasets within an experiment) | 400 |

SNE: | ${\mathit{\tau}}_{1}$ | ||||
---|---|---|---|---|---|

0.2 | 0.4 | 0.5 | 0.6 | 0.8 | |

ASC1 {2} | 2.0068 (0.0512) | 2.0046 (0.0453) | 2.0057 (0.0438) | 1.9988 (0.0470) | 1.9892 (0.0397) |

ASC2 {1} | 0.9986 (0.0580) | 0.9956 (0.0547) | 1.0029 (0.0495) | 1.0061 (0.0432) | 1.0012 (0.0387) |

Price {−3} | −3.0217 (0.0573) | −3.0112 (0.0553) | −2.9991 (0.0523) | −3.0182 (0.0483) | −3.0135 (0.0427) |

Quality {5} | 5.0148 (0.0583) | 5.0115 (0.0630) | 4.9913 (0.0579) | 5.0081 (0.0478) | 5.0521 (0.0431) |

${\tau}_{0}$ {0.5} | 0.5003 (0.0380) | 0.4999 (0.0410) | 0.5000 (0.0390) | 0.5001 (0.0365) | 0.5011 (0.0370) |

${\tau}_{1}$ {varied} | 0.1988 (0.0206) | 0.4023 (0.0363) | 0.4992 (0.0413) | 0.5994 (0.0401) | 0.8052 (0.0404) |

μ {10} | 10.0404 (0.0425) | 10.0680 (0.0426) | 10.0199 (0.0426) | 10.0279 (0.0390) | 10.0270 (0.0416) |

IAL: | ${\mathbf{\tau}}_{\mathbf{1}}$ | ||||

0.2 | 0.4 | 0.5 | 0.6 | 0.8 | |

ASC1 {2} | 2.0001 (0.0599) | 1.9671 (0.0653) | 1.9051 (0.0817) | 1.8170 (0.1146) | 1.6460 (0.2157) |

ASC2 {1} | 0.9917 (0.0751) | 0.9590 (0.0908) | 0.9310 (0.1118) | 0.8798 (0.1436) | 0.7718 (0.2671) |

Price {−3} | −3.0289 (0.0704) | −2.9707 (0.0867) | −2.8956 (0.0974) | −2.7878 (0.1153) | −2.6186 (0.1992) |

Quality {5} | 4.9830 (0.0710) | 4.9823 (0.0856) | 4.9689 (0.0857) | 4.9383 (0.0980) | 4.8769 (0.1361) |

${\tau}_{0}$ {0.5} | 0.5025 (0.0430) | 0.5000 (0.0522) | 0.4953 (0.0569) | 0.4847 (0.0643) | 0.4525 (0.1517) |

μ {10} | 8.9504 (0.1057) | 7.7838 (0.2216) | 7.2074 (0.2793) | 6.6611 (0.3339) | 5.3400 (0.4660) |

**Table 3.**SNE DGP—Market shares of True, SNE, and IAL models, before and after the policy, and market share change.

(${\mathit{\tau}}_{1}$ = 0.2) | |||||||||
---|---|---|---|---|---|---|---|---|---|

Consumer Choice | Before | After | Change | ||||||

True | SNE | IAL | True | SNE | IAL | True | SNE | IAL | |

{1} | 0.407 | 0.407 (0.015) | 0.407 (0.014) | 0.245 | 0.245 (0.032) | 0.251 (0.036) | −0.163 | −0.162 (0.022) | −0.157 (0.044) |

{2} | 0.327 | 0.326 (0.013) | 0.327 (0.015) | 0.644 | 0.643 (0.014) | 0.630 (0.023) | 0.317 | 0.317 (0.017) | 0.304 (0.044) |

{3} | 0.266 | 0.266 (0.020) | 0.266 (0.018) | 0.111 | 0.112 (0.031) | 0.119 (0.075) | −0.155 | −0.155 (0.019) | −0.147 (0.050) |

(${\mathbf{\tau}}_{\mathbf{1}}$ = 0.8) | |||||||||

Consumer Choice | Before | After | Change | ||||||

True | SNE | IAL | True | SNE | IAL | True | SNE | IAL | |

{1} | 0.402 | 0.401 (0.020) | 0.403 (0.020) | 0.224 | 0.221 (0.045) | 0.248 (0.114) | −0.179 | −0.179 (0.018) | −0.154 (0.137) |

{2} | 0.325 | 0.325 (0.012) | 0.324 (0.022) | 0.678 | 0.681 (0.016) | 0.615 (0.093) | 0.353 | 0.355 (0.024) | 0.291 (0.177) |

{3} | 0.273 | 0.274 (0.024) | 0.273 (0.026) | 0.098 | 0.098 (0.021) | 0.136 (0.393) | −0.175 | −0.176 (0.038) | −0.137 (0.218) |

Models | ${\mathit{\tau}}_{1}$ | ||||
---|---|---|---|---|---|

0.2 | 0.4 | 0.5 | 0.6 | 0.8 | |

True | 0.3784 | 0.3895 | 0.3956 | 0.4025 | 0.4181 |

SNE | 0.3728 (0.0814) | 0.3814 (0.0843) | 0.3865 (0.0842) | 0.3906 (0.0747) | 0.4076 (0.0737) |

IAL | 0.3649 (0.0984) | 0.3721 (0.1113) | 0.3822 (0.1166) | 0.3950 (0.1311) | 0.4254 (0.2066) |

**Table 5.**DNE DGP—Mean DNE estimates of the degree of social interaction in choice set formation (CSF) (${\tau}_{1}$), the degree of social interaction in choice (${\alpha}_{1}$), and the combined quality effect (β).

Estimates of ${\mathit{\tau}}_{1}$ | ${\mathit{\tau}}_{1}$ | ||||
---|---|---|---|---|---|

${\mathit{\alpha}}_{\mathbf{1}}$ | 0.2 | 0.4 | 0.5 | 0.6 | 0.8 |

0.2 | 0.2001 (0.0130) | 0.4011 (0.0308) | 0.5005 (0.0408) | 0.5974 (0.0411) | 0.8018 (0.0393) |

0.5 | 0.2015 (0.0178) | 0.4015 (0.0327) | 0.5002 (0.0376) | 0.5980 (0.0401) | 0.8008 (0.0363) |

0.8 | 0.1996 (0.0137) | 0.4029 (0.0309) | 0.4999 (0.0321) | 0.5990 (0.0359) | 0.8021 (0.0355) |

Estimates of ${\mathit{\alpha}}_{\mathbf{1}}$ | ${\mathit{\tau}}_{\mathbf{1}}$ | ||||

${\mathit{\alpha}}_{\mathbf{1}}$ | 0.2 | 0.4 | 0.5 | 0.6 | 0.8 |

0.2 | 0.1998 (0.0395) | 0.2014 (0.0376) | 0.2007 (0.0401) | 0.2020 (0.0383) | 0.2012 (0.0318) |

0.5 | 0.5002 (0.0163) | 0.4997 (0.0144) | 0.4995 (0.0190) | 0.5001 (0.0153) | 0.5008 (0.0108) |

0.8 | 0.7978 (0.0063) | 0.7989 (0.0067) | 0.7992 (0.0055) | 0.7992 (0.0036) | 0.7997 (0.0025) |

Estimates of β | ${\mathit{\tau}}_{\mathbf{1}}$ | ||||

${\mathit{\alpha}}_{\mathbf{1}}$ | 0.2 | 0.4 | 0.5 | 0.6 | 0.8 |

0.2 | 5.0327 (0.0450) | 4.9930 (0.0388) | 5.0208 (0.0447) | 4.9847 (0.0449) | 4.9955 (0.0349) |

0.5 | 5.0119 (0.0309) | 5.0151 (0.0246) | 5.0017 (0.0257) | 4.9734 (0.0227) | 5.0177 (0.0216) |

0.8 | 4.9980 (0.0293) | 5.0397 (0.0290) | 4.9807 (0.0271) | 4.9858 (0.0245) | 5.0016 (0.0168) |

Price Parameter Estimates | ||||||
---|---|---|---|---|---|---|

{${\mathit{\beta}}_{\mathrm{OWN}}$, ${\mathit{\beta}}_{\mathrm{NET}}$} | Models | ${\mathit{\tau}}_{\mathbf{1}}$ | ||||

0.2 | 0.4 | 0.5 | 0.6 | 0.8 | ||

{4, 1} | DNE | −2.9827 (0.0495) | −3.0090 (0.0507) | −2.9933 (0.0475) | −3.0005 (0.0449) | −2.9981 (0.0471) |

IAL | −3.0054 (0.0659) | −2.9831 (0.0842) | −2.8890 (0.1009) | −2.7934 (0.1258) | −2.7541 (0.1909) | |

{2.5, 2.5} | DNE | −2.9979 (0.0480) | −3.0108 (0.0476) | −2.9912 (0.0432) | −3.0163 (0.0421) | −3.0011 (0.0357) |

IAL | −3.0791 (0.0771) | −3.0421 (0.0826) | −2.9807 (0.1032) | −2.9439 (0.1287) | −2.8302 (0.1613) | |

{1, 4} | DNE | −3.0064 (0.0488) | −2.9968 (0.0473) | −2.9926 (0.0416) | −3.0041 (0.0413) | −3.0035 (0.0334) |

IAL | −3.2701 (0.1088) | −3.1576 (0.0815) | −3.1677 (0.1062) | −3.1762 (0.1330) | −3.5588 (0.2295) | |

Own Quality Parameter Estimates | ||||||

{${\mathit{\beta}}_{\mathbf{OWN}}$, ${\mathit{\beta}}_{\mathbf{NET}}$} | Models | ${\mathit{\tau}}_{\mathbf{1}}$ | ||||

0.2 | 0.4 | 0.5 | 0.6 | 0.8 | ||

{4, 1} | DNE | 4.0323 (0.0550) | 3.9928 (0.0493) | 4.0200 (0.0544) | 3.9845 (0.0474) | 3.9949 (0.0425) |

IAL | 4.2493 (0.0903) | 4.3142 (0.1117) | 4.3561 (0.1244) | 4.3446 (0.1307) | 4.2888 (0.1635) | |

{2.5, 2.5} | DNE | 2.5074 (0.0408) | 2.5119 (0.0362) | 2.4995 (0.0346) | 2.4900 (0.0356) | 2.5068 (0.0300) |

IAL | 3.0340 (0.2148) | 3.1055 (0.2445) | 3.3559 (0.3465) | 3.4283 (0.3818) | 3.4482 (0.3896) | |

{1, 4} | DNE | 1.0045 (0.0192) | 1.0074 (0.0202) | 0.9949 (0.0189) | 0.9982 (0.0154) | 1.0000 (0.0103) |

IAL | 1.3030 (0.3030) | 1.2917 (0.2917) | 1.5384 (0.5385) | 1.9042 (0.9050) | 2.1963 (1.2043) | |

Network Quality Parameter Estimates | ||||||

{${\mathit{\beta}}_{\mathbf{OWN}}$, ${\mathit{\beta}}_{\mathbf{NET}}$} | Models | ${\mathit{\tau}}_{\mathbf{1}}$ | ||||

0.2 | 0.4 | 0.5 | 0.6 | 0.8 | ||

{4, 1} | 1.0004 (0.0075) | 1.0002 (0.0075) | 1.0008 (0.0076) | 1.0002 (0.0058) | 1.0007 (0.0055) | |

{2.5, 2.5} | DNE | 2.5044 (0.0261) | 2.5032 (0.0184) | 2.5023 (0.0220) | 2.4834 (0.0177) | 2.5109 (0.0164) |

{1, 4} | 3.9935 (0.0342) | 4.0323 (0.0348) | 3.9858 (0.0303) | 3.9876 (0.0272) | 4.0017 (0.0187) |

**Table 7.**DNE DGP—Market shares of True, DNE, and IAL models, before and after the policy, and market share change.

(${\mathit{\beta}}_{\mathbf{OWN}}$ = 4, ${\mathit{\beta}}_{\mathbf{NET}}$ = 1, ${\mathit{\tau}}_{1}$ = 0.2) | |||||||||
---|---|---|---|---|---|---|---|---|---|

Consumer Choice | Before | After | Change | ||||||

True | DNE | IAL | True | DNE | IAL | True | DNE | IAL | |

{1} | 0.410 | 0.410 (0.017) | 0.409 (0.014) | 0.238 | 0.238 (0.035) | 0.259 (0.089) | −0.171 | −0.172 (0.021) | −0.150 (0.125) |

{2} | 0.326 | 0.327 (0.014) | 0.327 (0.016) | 0.657 | 0.659 (0.013) | 0.620 (0.057) | 0.331 | 0.332 (0.016) | 0.293 (0.115) |

{3} | 0.264 | 0.264 (0.022) | 0.263 (0.020) | 0.104 | 0.104 (0.034) | 0.120 (0.153) | −0.160 | −0.160 (0.019) | −0.143 (0.104) |

(${\mathit{\beta}}_{\mathbf{OWN}}$ = 4, ${\mathit{\beta}}_{\mathbf{NET}}$ = 1, ${\mathit{\tau}}_{\mathbf{1}}$ = 0.8) | |||||||||

Consumer Choice | Before | After | Change | ||||||

True | DNE | IAL | True | DNE | IAL | True | DNE | IAL | |

{1} | 0.405 | 0.404 (0.020) | 0.406 (0.020) | 0.217 | 0.216 (0.046) | 0.258 (0.194) | −0.188 | −0.188 (0.017) | −0.147 (0.216) |

{2} | 0.325 | 0.326 (0.011) | 0.324 (0.023) | 0.693 | 0.694 (0.016) | 0.604 (0.129) | 0.368 | 0.368 (0.022) | 0.280 (0.239) |

{3} | 0.270 | 0.271 (0.025) | 0.270 (0.025) | 0.090 | 0.090 (0.026) | 0.138 (0.524) | −0.180 | −0.180 (0.035) | −0.133 (0.263) |

(${\mathit{\beta}}_{\mathbf{OWN}}$ = 1, ${\mathit{\beta}}_{\mathbf{NET}}$ = 4, ${\mathit{\tau}}_{\mathbf{1}}$ = 0.2) | |||||||||

Consumer Choice | Before | After | Change | ||||||

True | DNE | IAL | True | DNE | IAL | True | DNE | IAL | |

{1} | 0.415 | 0.415 (0.014) | 0.411 (0.017) | 0.215 | 0.216 (0.041) | 0.300 (0.395) | −0.200 | −0.200 (0.026) | −0.111 (0.447) |

{2} | 0.325 | 0.326 (0.013) | 0.326 (0.017) | 0.699 | 0.698 (0.015) | 0.562 (0.195) | 0.373 | 0.373 (0.017) | 0.236 (0.368) |

{3} | 0.260 | 0.259 (0.018) | 0.263 (0.024) | 0.086 | 0.086 (0.042) | 0.137 (0.597) | −0.174 | −0.173 (0.016) | −0.125 (0.279) |

(${\mathit{\beta}}_{\mathbf{OWN}}$ = 1, ${\mathit{\beta}}_{\mathbf{NET}}$ = 4, ${\mathit{\tau}}_{\mathbf{1}}$ = 0.8) | |||||||||

Consumer Choice | Before | After | Change | ||||||

True | DNE | IAL | True | DNE | IAL | True | DNE | IAL | |

{1} | 0.412 | 0.411 (0.017) | 0.411 (0.019) | 0.194 | 0.193 (0.044) | 0.296 (0.527) | −0.218 | −0.218 (0.014) | −0.115 (0.472) |

{2} | 0.324 | 0.325 (0.011) | 0.325 (0.020) | 0.736 | 0.737 (0.013) | 0.558 (0.242) | 0.412 | 0.412 (0.016) | 0.233 (0.433) |

{3} | 0.264 | 0.264 (0.020) | 0.264 (0.028) | 0.070 | 0.070 (0.030) | 0.146 (1.080) | −0.193 | −0.194 (0.026) | −0.118 (0.390) |

{${\mathit{\beta}}_{\mathbf{OWN}}$, ${\mathit{\beta}}_{\mathbf{NET}}$} | Models | ${\mathit{\tau}}_{1}$ | ||||
---|---|---|---|---|---|---|

0.2 | 0.4 | 0.5 | 0.6 | 0.8 | ||

{4, 1} | True | 0.3882 | 0.3991 | 0.4015 | 0.4112 | 0.4269 |

DNE | 0.3904 (0.0772) | 0.3928 (0.0730) | 0.4003 (0.0792) | 0.4018 (0.0709) | 0.4156 (0.0764) | |

IAL | 0.3197 (0.1943) | 0.3264 (0.1948) | 0.3404 (0.1873) | 0.3517 (0.1854) | 0.3600 (0.2394) | |

${\mathit{\tau}}_{\mathbf{1}}$ | ||||||

0.2 | 0.4 | 0.5 | 0.6 | 0.8 | ||

{2.5, 2.5} | True | 0.4040 | 0.4151 | 0.4209 | 0.4268 | 0.4411 |

DNE | 0.4056 (0.0663) | 0.4124 (0.0609) | 0.4177 (0.0634) | 0.4167 (0.0614) | 0.4336 (0.0599) | |

IAL | 0.2372 (0.4162) | 0.2424 (0.4165) | 0.2633 (0.3753) | 0.2740 (0.3583) | 0.2923 (0.3835) | |

${\mathit{\tau}}_{\mathbf{1}}$ | ||||||

0.2 | 0.4 | 0.5 | 0.6 | 0.8 | ||

{1, 4} | True | 0.4224 | 0.4339 | 0.4397 | 0.4452 | 0.4582 |

DNE | 0.4245 (0.0637) | 0.4382 (0.0609) | 0.4373 (0.0576) | 0.4411 (0.0606) | 0.4515 (0.0496) | |

IAL | 0.1294 (0.6938) | 0.1297 (0.7012) | 0.1413 (0.6787) | 0.1640 (0.6349) | 0.1569 (0.6576) |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license ( http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wichmann, B.; Chen, M.; Adamowicz, W.
Social Networks and Choice Set Formation in Discrete Choice Models. *Econometrics* **2016**, *4*, 42.
https://doi.org/10.3390/econometrics4040042

**AMA Style**

Wichmann B, Chen M, Adamowicz W.
Social Networks and Choice Set Formation in Discrete Choice Models. *Econometrics*. 2016; 4(4):42.
https://doi.org/10.3390/econometrics4040042

**Chicago/Turabian Style**

Wichmann, Bruno, Minjie Chen, and Wiktor Adamowicz.
2016. "Social Networks and Choice Set Formation in Discrete Choice Models" *Econometrics* 4, no. 4: 42.
https://doi.org/10.3390/econometrics4040042