The Dynamic Specification

The system evolves withgτfollowing a Markov process with known transitionP(gτ+1|gτ,aτ). Assume conditional independence of the transitions between states, such that

P(gt+1, t+1|gt, t, at) = f(t+1)CDFg(gt+1|gt, at), with f(·) with finite first moment,

9_{An alternative specification that was explored defined the outside price for the buyer as depending}

on the number of available (unlinked) sellers in the counterfactual network, following xg00 = κ(S−

PB k PS jg 00 kj) −1 2 _{, withκa constant.}

continuous and differentiable twice. Note then thatτ affects the transition across states

g only via the choices of actionsa, but not directly10.

A Markov strategy for buyer i constitutes a mapping σi(g, i) :G×R|Ai| →Ai so the

buyer only observes the current network, or state, and its individual draw of action- specific shocks to chooseai. Note that only the current state is relevant in the mapping.

Consider stationary strategies only, such thati’s decision is the same intands, whenever

{gt_, t

i}={gs, si}.

In this setting, the conditional choice probability of action ai being chosen by i, when

the state of the world is gis given by:

P_iσ(ai|g) =P rob(σi(g, i) =ai)

Z

I{σi(g, i) =ai}f(i)di (3.6)

whereIis an indicator taking value one when the argument holds true and we integrate over all the possiblei’s. By3.6, the probability of agentiannouncingaidepends on the

current network andi, which is not observed by third parties. Therefore,Piσconstitutes

the probability that an agent different fromiattaches toichoosingai and it is then the

belief that a third party has oni’s choices (Aguirregabiria and Mira,2007).

With independent draws of and conditional on her own action, the probability that agentiassigns to the final negotiation network being g0, given that the observable state isg and other players’s strategies areσ is just the product of the correspondingP_kσ’s:

%σ_i(g0|ai, g) = X a−i∈Qk6=iAk Y k6=i P_kσ(a−i[k]|g) I{g˜(ai, a−i) =g0} (3.7)

wherea−i is a vector containing actionsak of all the playersk=6 iand a−i[k] is thekth

action in that vector. So the probability thatiattaches to networkg0 arising is just: (i) the sum over all the possible vectors collecting actionsakfor all the playersk6=i(which

amounts to all the possible combinations of the elements in theAk sets, so the product

of these); (ii) of the product over all the k’s of the probabilities that iassigns to each agent k 6= i playing action ak in the vector a−i, whenever the actions in a−i together

with actionai result in networkg0.

Denote with ci(g0|g) the linking cost for i when the starting state is g and i’s choice

corresponds to network g0. LetO(.) be the mapping defined above, such that whenever a network is proposed a new network eventually arises after all unstable links in the proposed network have been broken. Definingv_iσ(ai, g) as the current and future profits

10

net of the stochastic utility component, , if ichooses actionai when the state isg and

he behaves optimally in the future, we have in3.8 thechoice-specific value function:

v_iσ(ai, g) = X g0 %σ_i(g0|ai, g)(ci(g0|g) + [πib(g 00_,tσ g00) +βV_i(g00) :g00 =O(g0, Vσ)]) (3.8)

So the value for player i of choosing actionai is the sum of current and future payoffs

he would make under the different negotiation networks g0 compatible with action ai,

weighted by the probability i attaches to each of these g0 arising, conditional on the current state g and the chosen action. The current and future payoffs are then given by the cost from negotiating network g0 plus the expected payoffs, current and future, attained under the stable network that arises from negotiation network g0.

At each state g, the corresponding value function (the integrated Bellman equation) is defined as: V_iσ(g) = Z [maxai∈Ai(ai,i+v σ i(ai, g))]f(i)di (3.9)

which represents i’s current and future profits at the beginning of each period, before the’s are drawn, given that the state network isg and everybody is playing strategies according to σ−i.

Note that under the assumptions of additive separability of the private information component in the players payoff functions and the conditional independence of the tran- sitions between states, the dynamic programming problem is fully characterised by the Bellman equation in3.9, which in turn is analogous to a static discrete choice problem, with choice specific (intertemporal) values instead of period profits (Rust,1994). For a fixed set of payoffs{π(·)}the equation above is a contraction mapping and has a uniqueV_iσ that solves it for any givenσ, under the assumptions of finiteness of the state space and the restrictions imposed on the error term and its relation to stage profits (Aguirregabiria and Mira,2002).