FUNCIONES DE LAS ESTRATEGIAS

Given ˜g, the negotiation network, players bargain on prices under a standard Nash protocol8. Linking announcements being uncoordinated can lead to ˜g’s for which some of the linked pairs exhibit no individual gains from trade, in which case no trade takes place and the corresponding link dissolves. In addition, the network formation protocol described above is compatible with negotiation networks in which sellers are linked to more than one buyer. However, the capacity constraints of the sellers restrict the possible networks to those that exhibit one-to-one links only. CallO(˜g) =g0an operator mapping from the states space to itself, with g0 being the network arising from ˜g after deleting all the links that exhibit no gains from trade for at least one of the players, within the constrains imposed on the network.

The gains from trade in this dynamic game include both the period profits, π, and the future discounted value of being in a given network, V. Per-period contracts tij;g0,

between any linked pair underg0 then satisfy the following generic condition:

tij;g0 ∈argmax_˜ t [πb_i(g0,˜tg0) +βV_ib(g0)]−[πb_i(g00,tσ g00) +βV_ib(g00)] bij × [π_js(g0,˜tg0) +βV_js(g0)]−[π_js(g00,tσ g00) +βV_js(g00)] bji (3.3)

The surplus for the buyer is defined as the difference in current and future payoffs between trading with the seller she is linked to or not doing so and bij and bji are the

corresponding bargaining parameters, which naturally add up to one. In this context, Nash bargaining parameters equal to 0.5, for instance, give an equal division of the surplus. Setting bij = 0 gives the Nash-Bertrand pricing solution in the competition

upstream.

The first term of the downstream player’s surplus, [πb_i(g0,˜tg0) +βV_ib(g0)], contains the

relationship payoffs, with g0 being the stable network arising after ˜g is formed and

˜_tg0 ={˜t,tσ₋

ij;g0}, with optimisation over ˜ttaking all other prices, optimally determined,

as given. The second term in the buyer’s surplus in3.3, [π_ib(g00,tσ_g00) +βV_ib(g00)] , contains

the counterfactual payoffs for the buyer, if the relationship was broken. Let g00 be the counterfactual network and tσ_g00 its associated prices. Different assumptions on g00 and

prices adjustments after disagreement imply alternative ways of endogenising the players outside options.

As inLee and Fong(2013), the pricing problem isnetworked in the sense that the surplus over which bilateral bargain takes place depends on other bilateral negotiations taking place in the graph. For games with one player only in one side,Hart and Tirole(1990);

Horn and Wolinsky(1988);Segal and Whinston(2003) propose approaches that describe those interactions across simultaneous bilateral negotiations. Similarly,de Fontenay and Gans (2014) generalise the framework to two-sided large games. Like in Lee and Fong

(2013), the pricing problem is as welldynamic: the framework captures the period effects of failed agreements together with the impact of disagreement in the continuation values

V for the bargaining pairs. Finally, outside options areendogenously determined, as they are not given by a fixed counterfactual outcome but by the outcomes potentially reached under network-specific renegotiations and re-linking.

The framework inLee and Fong (2013) definesg00 to beg₋0 _ij, equivalent to deleting link

ij in networkg0 and tσ_g00 =tσ_−ij,g0 ={tσ_g0\tσ_ij;g0}, such that in the current period, a price

for pair ij is not defined (as g00 does not include the pair) and the contracts between all other pairs remain unchanged. In other words, disagreement points imply that all the contracts that involve the rest of the players in the negotiation network are binding, so after i and j disagree, no contemporaneous changes in tg0_−ij are allowed for. This

setup is consistent with the idea that bargaining takes place simultaneously for all linked pairs. As a consequence, other buyers who might have chosen supplierj under the new circumstance, cannot do so immediately and adjustments of this type will take time. In the conceptualisation ofHorn and Wolinsky(1988) this equilibrium can be interpreted as the Nash equilibrium across many Nash bargains. As explained in Crawford and Yurukoglu (2012), this is equivalent to considering a simultaneous moves game, where a player is conformed by a pairij, whose strategy istij,g0 and whose payoff is the Nash

product of i and j’s surpluses. Then, the bargaining problem is solved as the Nash equilibrium of that game. Such a device rules out the possibility of a player exploiting an informational asymmetry due to the order in which negotiations take place. So, if

j is bargaining at the same time with buyers i and k, j has no information advantage about the outcome of the process with kwhen bargaining with iand viceversa.

Given the simultaneity in all the bargains and the fact that disagreements are off- equilibrium events, Lee and Fong(2013) leave prices for all other pairs fixed upon dis- agreement in the current period. An alternative specification would be to defineg00to be the network that arises after deleting ij fromg0, allowing for all the pairs to renegotiate prices in the new setting, closer to the non-binding contracts setting described inStole and Zweibel (1996). This is the alternative I propose here. To observe the difference this approach makes in the generation of disagreement points a small static example is presented in Appendix H. In short, consider a seller that is bargaining simultaneously

with two buyers of which he will choose one. One possibility would be that when bar- gaining with the first buyer, the disagreement payoffs of the seller were those of trading with second buyer at the price this could agree to pay under the current situation. But actually, if the seller was to break the link with the first buyer, the conditions in which she would bargain with the second buyer are those in which the seller’s outside option was not trading at all.

Given the per-period profits defined in 3.1, the gains for buyer i from trading with j

depend on the continuation values for V(g0) and V(g00), on the negotiated tij and an

outside price for the garment,xg00 =x 9.

S_ijb(g0) = [(−tij(g0) +xg00+ρ_ij)q_i+β(V_ib(g0)−V_ib(g00))] (3.4)

While each buyer negotiates with at most one seller in any network, as a result of the uncoordinated linking decisions in the first stage, sellers might participate of multiple simultaneous bargains at a given point in time. Given the sellers’ (capacity) constrains to trade with one buyer only, using3.2, the gains from trade for sellerj when bargaining with buyer iunder network g0 is given by:

S_ijs(g0) = (tij(g0)−mi)qi+βVjs(g0)− max k∈B,k6=i

{[g_kj0(kj)×[(tkj(g0(kj))−mk)qk+βVjs(g

0(kj)_{)]] ,}

βV_js(g0(∅j))} (3.5)

In some abuse of notation, g0(kj)=g₋0 _ij\ {nj:g_nj0 = 1,n6=k}, so the outside option for seller j negotiating with buyer iis the best over the alternative partners she has under

g0 or not trading in the current period at all.