A Multinomial Structure under PSMt - A Multinomial Choice Model under PSMt

3.4 A Multinomial Choice Model under PSMt

3.4.1 A Multinomial Structure under PSMt

From this section, I set p = 2 to simplify the model for ease of explanation. When p is bigger than two but still small, the results in this section can be easily

extended with only several more steps.9 _{I also assume that the utility of forming} a multigraph is transferable.

PSM provides no-deviation conditions for each pair. When there are a total of p simultaneous networks, the cardinality of Yi j (the set of all possible links between i and j) is 2p_{. Hence, PSM provides 2}p_{− 1}_{no-deviation conditions for} each pair. When p = 2, there are only four possible combinations of link types for each pair of agents. Once one of four combinations, say ‘type-1 link only’, is chosen by i and j, PSM provides the following 22_{− 1}_{conditions; (1, 0) % (0, 0),} (1, 0)_{% (0, 1) and (1, 0) % (1, 1).}10

Recall that mu(s)i ( j|Y (−s) i j = a

(t)

i j)is i’s marginal utility from a type-s link with j, when they have relations a(t)_{i j}. In addition, let mu(1,2)_i ( j)be i’s marginal utility by adding both types of links at the same time. I describe how to construct moment inequalities under PSM with transferable utility.

First, consider the case that i and j choose Yi j = (a(1)_{i j} , a(2)_{i j} ) = (1, 1). It implies that (1, 1) is weakly preferred to any other combinations, (0, 1), (1, 0), and (0, 0). Hence, I have [(1, 1)_{% (0, 1)] ⇒ mu}(1)_i ( j|a(2)_{i j} = 1) + mu(1)_j (i|a(2)_{i j} = 1) ≥ 0, (3.5) [(1, 1)_{% (1, 0)] ⇒ mu}(2)_i ( j|a(1)_{i j} = 1) + mu(2)_j (i|a(1)_{i j} = 1) ≥ 0, (3.6) [(1, 1)% (0, 0)] ⇒ mu(1,2)i ( j)+ mu (1,2) j (i) ≥ 0 (3.7)

9_{Even when p is large, the results in this section can be extended. However, practical imple-}

mentation can be difficult. For example, when p= 20, there are more than a million choices for

each pair.

10_{I use the weak preference relation % in the sense that utility of one choice is as great as that}

When i and j choose (a(1)i j , a (2)

i j ) = (1, 0), similarly there are following three conditions: [(1, 0)_{% (1, 1)] ⇒ mu}(2)_i ( j|a(1)_{i j} = 1) + mu(2)_j (i|a(1)_{i j} = 1) ≤ 0, (3.8) [(1, 0)_{% (0, 1)] ⇒ mu}(1)_i ( j|a(2)_{i j} = 0) + mu(1)_j (i|a(2)_{i j} = 0) ≥ mu(2)_i ( j|a(1)_{i j} = 0) + mu(2)_j (i|a(1)_{i j} = 0), (3.9) [(1, 0)_{% (0, 0)] ⇒ mu}(1)_i ( j|a(2)_{i j} = 0) + mu(1)_j (i|a(2)_{i j} = 0) ≥ 0. (3.10) The case of (a(1)i j , a (2)

i j )= (0, 1) is symmetric to the above. That is,

[(0, 1)_{% (1, 1)] ⇒ mu}(1)_i ( j|a(2)_{i j} = 1) + mu(1)_j (i|a(2)_{i j} = 1) ≤ 0, (3.11) [(0, 1)% (1, 0)] ⇒ mu(1)i ( j|a (2) i j = 0) + mu (1) j (i|a (2) i j = 0) ≤ mu(2)_i ( j|a(1)_{i j} = 0) + mu(2)_j (i|a(1)_{i j} = 0), (3.12) [(0, 1)_{% (0, 0)] ⇒ mu}(2)_i ( j|a(1)_{i j} = 0) + mu(2)_j (i|a(1)_{i j} = 0) ≤ 0. (3.13)

Finally, when a pair i j chooses (a(1)_{i j} , a(2)_{i j} )= (0, 0), I have

[(0, 0)% (1, 1)] ⇒ mu(1,2)i ( j)+ mu (1,2)

j (i) ≤ 0, (3.14)

[(0, 0)_{% (1, 0)] ⇒ mu}(1)_i ( j|a(2)_{i j} = 0) + mu(1)_j (i|a(2)_{i j} = 0) ≤ 0, (3.15) [(0, 0)_{% (0, 1)] ⇒ mu}(2)_i ( j|a(1)_{i j} = 0) + mu(2)_j (i|a(1)_{i j} = 0) ≤ 0. (3.16)

Since each pair makes a decision among multiple (2p

) alternatives, and the set of relations chosen gives the maximum sum of utilities among all alternatives, the model has a multinomial structure. However, the model is not ex- actly a typical multinomial choice model, e.g. multinomial probit, because the marginal utilities depend on on Y−i jand may have non-additively separable εi j.

3.4.2 A Multinomial Choice Model under PSMt

In the end of Section 3.3.2., I briefly mentioned the benefits of using pairwise stability of a multigraph. In this section, I will show that the structural model is equivalent to a typical multinomial choice model under PSMt and additional assumptions. It will be done through three steps. First, I impose additive sep- arability of εi j and demonstrate that for any values of εi j ∈ R2

p₋₁

, a set of links of a pair is uniquely determined given the rest of the multigraph and observed characteristics. Second, I show that all pairs’ links decisions are separate from each other due to the additional assumption of myopic agents. Finally, I explain the irrelevance of other pairwise stable multigraphs.

Assumption 3.2. (i) εi j is additively separable.

(ii) ε is a continuously distributed r.v. with everywhere positive density w.r.t. Lebesgue measure.

(iii) εi j = (εi j,1, εi j,2, · · · , εi j,2p₋₁)is i.i.d. across pairs, but not i.i.d. across the

alternatives.

mation. It is a strong assumption, but prevalent in the literature of empirical network formation, e.g. Christakis et al. (2010). From Assumption 3.2.(i), the base utility and the additional utility are written as

u(s)_{i j} := u(s)(Wi j, Vi j, Xi j)+ p X t=1 γ(t,s)X k a(t)_ika(t)_jk + ε(s)_{i j} , and δ(s,t) i j := δ (s,t) (Wi j, Vi j, Xi j)+ ε(s,t)_{i j} .

Then, the sum of marginal utilities in (3.5)-(3.16) are written as, for example,

mu(1)_i ( j|a(2)_{i j} = 1) + mu(1)_j (i|a(2)_{i j} = 1) = u(1)(Wi j, Vi j, Xi j)+ u(1)(Wi j, Vi j, Xji) +2 2 X t=1 γ(t,s)X k a(t)_ika(t)_jk + δ(s,t)(Wi j, Vi j, Xi j) +δ(s,t)_(W i j, Vi j, Xji)+ 2ε (1) i j + 2ε (1,2) i j .

For notational simplicity, I use εi j(Yi j)to denote the sum of unobservables which corresponds to Yi j, for example, εi j(1, 1) = ε(1)_{i j} + ε(2)_{i j} + ε(1,2)_{i j} . Then, I have

The right hand side of (3.17) is the same form as that of a typical multinomial choice model.

Now consider a dynamic process of multigraph formation as in Sections 3.2 and 3.3.2. In the process, a pair of individuals is chosen at each period, and the pair revises its current links given the current multigraph. The link formation (or revision) in each period occurs as a two people cooperative game. When they revise, they do not consider the future changes in the multigraph, i.e. agents are myopic. Hence, a pair chooses a set of links Yi j, which maximizes the sum of their utilities given Wi j, Vi j, Xi j, Y−i jand εi j. Lemma 1 shows that there is a unique prediction for each Yi j given Wi j, Vi j, Xi j and Y−i j for all realizations of εi j ∈ R2

p₋₁

Lemma 3.4.1. The utility function is defined as (3.1). Let Assumption 2 hold. Then,

given Wi j, Vi j, Xi j and Y−i j, each pair’s links decision Yi j is uniquely determined under PSMt for all values of εi j ∈ R2

p₋₁

Proof. Consider only the case with p = 2 for ease of exposition. It can be easily seen that the following four regions of ε corresponding to (3.5)-(3.7), (3.8)-(3.10), (3.11)-(3.13), and (3.14)-(3.16) are disjoint and compose R3_.

Once the rest of the multigraph is fixed, each pair’s links decision Yi jis uniquely predicted given Wi j, Vi j and Xi j for any realizations of εi j. However, Lemma 1 does not mean that a pairwise stable multigraph Y is uniquely determined given W, V and X for all ε = (ε12, · · · , εn−1,n). This result is noticeably different from that of a two by two simultaneous-move entry game in the literature (see for example, Tamer (2003)), where multiple equilibria, i.e. multiple predicted outcomes, occur for some regions of unobservables. In the link formation game, all n(n − 1) pairwise outcomes are uniquely determined. It is mainly because of the nature of the game. In the link formation game of a pair, two individuals

the maximum utility. In the entry game, on the other hand, two firms separately make an entry decision in a market without cooperation.

The remaining problem is the simultaneity among al pairs’ links decisions, (Y12, Y13, · · · , Yn−1,n). At this point the benefit of pairwise stability of a multigraph arises. When checking pairwise stability of a multigraph, I do not need to consider no-deviation conditions for a set of individuals larger than a pair. Checking pairwise stability of each pair’s current links decision Yi jonly requires comparison of utilities between Y and Y − Yi j+Y_{i j}0. Hence, if a multigraph is pairwise stable, it is possible to consider the multigraph as if it is formed by myopic agents. This assumption of myopic agents makes all pairwise decisions separate across pairs. To see this, consider a pair i j with current links Yi j on a multigraph Y. In order to form pairwise stable relations Yi j, the pair i j has to check the dif- ference between Ui(Y − Yi j+Yi j0)+Uj(Y − Yi j+Yi j0)and Ui(Y)+Uj(Y)for all Yi j0 , Yi j. This comparison does not take into account future changes in the rest of the multigraph Y−i j. In other words, the pair i j compares the sum of their utilities from Y − Yi j+Yi j with the sum of utilities from Y. Thus, the rest of network Y−i jis fixed when they make a decision, and there is no simultaneity among Yi j’s. The utility parameters in the strategic formation of a multigraph can be obtained by checking pairwise stability conditions for each pairwise decision Yi j separately, given Wi j, Vi j, Xi j and Y−i j.

Finally I discuss the irrelevance of other pairwise stable multigraphs in the set of all pairwise stable multigraphs, say PSM. Since pairwise stability of a multigraph is not an equilibrium solution concept, the set PSM is not a set of equilibrium multigraphs. Rather, it is a set of multigraphs that are outcomes of many different games. For the dynamic process of multigraph formation ex-

plained in Section 3.3.2, a multigraph has a corresponding history of meetings. Given a specific history, the same multigraph is always realized. By the implicit assumptions of the dynamic formation process and the independence between meeting and preference, other pairwise stable networks corresponding to different histories are irrelevant to the identification of the utility parameters.11

Now, if there are no endogenous variables, the identification problem is re- duced to that of multinomial choice models. If one imposes a distributional assumption on ε such as the normal or the logistic distribution, the parameter vector θ is point-identified under suitable location and scale normalizations.12 _I consider the case with an endogenous variable formally in the next section.

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