Network Formation and Identification with Missing Links
If people choose their neighbors, an observed link reflects incentives as well as a possible channel for spillovers. The formation equation must then be studied together with the outcome equation. These notes compare estimated equilibrium beliefs in (Leung 2015), joint link and activity choices in (Battaglini, Patacchini, and Rainone 2022), and repeated risk sharing in (Bramoullé and Kranton 2007). The calculations distinguish responses on a fixed network from responses that also change the network.
1. Estimate equilibrium beliefs before preferences
In an incomplete-information link-formation game, an agent chooses links using private payoff shocks and beliefs about others’ choices. Equilibrium requires those beliefs to agree with the choice probabilities generated by the model.
An equilibrium supplies choice probabilities that can be estimated from network and covariate patterns. The two-step method uses those estimates as beliefs in a payoff index, then estimates preferences. Its computational advantage is that each candidate preference parameter does not require solving the entire equilibrium again.
The main consistency result relies on the paper’s symmetry, distributional, identification, and regularity restrictions. It does not follow simply because there are order \(n^2\) possible dyads. Dyads sharing an agent can be dependent, and the first-step estimation error enters the second-step distribution.
2. An equilibrium calculation
A related model links a vector of outcomes or activities \(E\) to the network through
For a fixed \(G\), if the spectral radius of \(\delta G\) is below one,
Each successive term transmits the initial disturbances along longer walks through the network. Here \(G\) is fixed. If links are chosen, the same outcome equation remains necessary, but the link choices must also satisfy their own optimality conditions.
For a link payoff \(\alpha\delta E_jg_{ij}\) and convex cost
the interior first-order condition gives
An upper capacity \(\overline g\) caps the optimal link at \(\overline g\), subject to the sign and concavity conditions. These best responses must be solved together with the outcome equation. Replacing that equation by an unweighted degree sum would discard both the neighbors’ activity levels and the exogenous term.
3. Repeated sharing is another mechanism
To see the repeated-sharing intuition, consider \(x^{(t+1)}=Px^{(t)}\) on a connected component. If \(P\) is a primitive doubly stochastic matrix, then
The limit sends each initial vector to its component average. Two conditions explain the equal sharing: double stochasticity preserves the total, and primitivity ensures convergence. Connectivity alone gives neither conclusion; periodic exchange can oscillate, and a row-stochastic matrix can converge to unequal weights. Stochastic sharing models require their corresponding assumptions.
4. What these models imply for empirical work
The formation model also matters for identification. People with similar unobserved characteristics may choose each other, so conditioning on adjacency does not remove selection. If some links are unobserved, an observation model is needed as well. The missing-link note derives the resulting mixture over latent exposures.
5. Differentiate the link choice and check the corners
Holding the relevant activity \(E_j\) fixed in the link payoff, let \(b=\alpha\delta E_j\) and \(\theta=\theta_{ij}>0\). The derivative of the cost is
For \(b>0\), equating marginal benefit and marginal cost gives \(g=\theta^{1+\lambda}b^\lambda\). The cost is strictly convex on the positive interior, so this is the unique interior optimum. With \(0\leq g\leq\overline g\), the best response is zero when \(b\leq0\), the interior value when it lies below capacity, and \(\overline g\) otherwise.
Strict convexity settles the link choice for a given activity. It leaves open whether the network and activities have a unique joint solution, since higher activity may increase links, which in turn raise activity. That feedback is a separate fixed-point problem.
6. The resolvent separates paths from feedback
For comparison, hold the network fixed. The spectral-radius condition makes repeated substitution converge to the Neumann series
The \((i,j)\) entry of \(G^r\) sums products of link weights along all length-\(r\) walks from \(i\) to \(j\). Walks can revisit vertices. The effect of a primitive disturbance on \(E_i\) includes each such route, discounted by \(\delta^r\).
A differential calculation makes the amplification explicit. From \((I-\delta G)E=\epsilon\), a fixed-\(\delta\) perturbation gives
so
For a change in one link \(g_{ab}\) with exogenous \(\epsilon\), the derivative is \(\delta E_b(I-\delta G)^{-1}e_a\). The link first changes activity at its origin through the destination’s activity, then that change propagates through the resolvent. If \(G\) itself responds to \(E\), its differential must also satisfy the link first-order conditions; treating \(dG\) as externally fixed would omit that endogenous feedback.
A symmetric two-person special case shows how much this distinction can matter. Suppose both activities equal \(E\), both chosen links equal \(g(E)=cE^\lambda\), and both primitive shocks equal \(\epsilon\). The simultaneous equilibrium equation reduces to
At a positive interior equilibrium where \(1-(\lambda+1)\delta cE^\lambda\ne0\), implicit differentiation gives
Holding the network fixed would instead give \(1/(1-\delta g)\). The additional \(\lambda\delta g\) in the equilibrium derivative comes from the response of link intensity to activity. Indeed, \(d(gE)=g\,dE+E\,dg=(1+\lambda)g\,dE\).
For nonnegative parameters, the fixed-network condition \(\delta g<1\) therefore does not by itself establish local stability of the joint activity-link iteration. The stronger derivative bound \((\lambda+1)\delta g<1\) does so in this scalar iteration. This calculation is an illustration of feedback within the displayed model, not a claim about global equilibrium uniqueness. It explains the empirical role of the formation equation: an exogenous link change and a policy that induces new links generally have different multipliers.
7. Why two-step estimation has a first-stage term
Let \(p\) collect equilibrium choice probabilities and let the second-step population moment be \(M(\theta,p)\), with \(M(\theta_0,p_0)=0\). If \(A=\partial_\theta M(\theta_0,p_0)\) is nonsingular, a local expansion gives
The derivative in brackets is a directional linear map if \(p\) is a function rather than a finite vector. The first term is sampling noise with beliefs fixed; the second carries estimated-belief error into preference estimates. They can be correlated because the same network informs both stages.
The first-stage term disappears only if the moment has the required orthogonality property. Substituting estimates into a likelihood does not create that property. Identification and regularity assumptions must determine both which beliefs are estimable and how their estimation errors enter the second step.
For a finite-dimensional first stage with an asymptotic linear representation, this expansion also identifies the variance correction. For a common normalization \(r_n\), suppose \(r_n\widehat M(\theta_0,p_0)\) and \(r_n(\widehat p-p_0)\) have joint limits \((Z_M,Z_p)\), and write \(B=\partial_pM\). The second-step limit is \(-A^{-1}(Z_M+BZ_p)\), with covariance
The variance must therefore include first-stage uncertainty and its covariance with the second-step score. A second-step Hessian generally misses both. Avoiding repeated equilibrium solutions simplifies computation, but does not justify treating estimated beliefs as known.
8. Counting dyads does not count independent observations
For an undirected network, consider a simple dyadic variable \(H_{ij}=U_i+U_j+e_{ij}\) for \(i<j\), with independent centered node shocks of variance \(\sigma_U^2\) and independent edge shocks of variance \(\sigma_e^2\). Its average over \(m=n(n-1)/2\) dyads is
Thus
There are order \(n^2\) dyads, but shared node shocks produce order \(n^{-1}\) variance. This example does not describe every formation model; it explains why a dyad count alone cannot justify a root-number-of-dyads rate.
9. Equal sharing needs both conservation and convergence
For \(x^{(t+1)}=Px^{(t)}\), row stochasticity preserves constant vectors: \(P\mathbf1=\mathbf1\). Column stochasticity preserves total holdings: \(\mathbf1'P=\mathbf1'\). If \(P\) is primitive, its powers converge to \(\mathbf1\pi'\), where \(\pi\) is the stationary left eigenvector. Double stochasticity makes \(\pi=\mathbf1/m\), giving equal sharing.
Without aperiodicity, the swap matrix on two vertices alternates holdings forever. Without column stochasticity, the matrix with rows \((0.9,0.1)\) and \((0.2,0.8)\) converges to the weighted average with weights \((2/3,1/3)\), not the arithmetic mean. These checks identify what connectivity alone leaves unresolved.
If initial holdings contain independent shocks of common variance \(\sigma^2\), equal sharing reduces each final shock variance to \(\sigma^2/m\). Weighted sharing instead gives \(\sigma^2\sum_i\pi_i^2\), minimized by equal weights. This elementary calculation connects the averaging mechanism to risk reduction while keeping its stochastic and graph assumptions explicit.
An elementary convergence proof for repeated sharing
For a primitive row-stochastic \(P\), some power \(P^r\) has every entry at least \(\epsilon>0\). Put \(\eta=m\epsilon\le1\) and \(J=\mathbf1\mathbf1'/m\). If \(\eta<1\), write \(P^r=\eta J+(1-\eta)R\) for another row-stochastic matrix \(R\). Any stochastic average lies between the smallest and largest coordinates, so, for \(\operatorname{osc}(x)=\max_i x_i-\min_i x_i\),
Iterating makes the oscillation tend to zero geometrically. The case \(\eta=1\) gives exact averaging after \(r\) steps. When \(P\) is also column-stochastic, every iterate has the same arithmetic mean as \(x\), so all its coordinates converge to that mean. The finitely many intermediate powers preserve the shrinking coordinate interval, which proves \(P^t x\to Jx\) for the full sequence, and hence \(P^t\to J\). This supplies the convergence step in the sharing example without mistaking connectivity for primitivity.