Matriz de análisis de los diarios de campo

Various algorithms have been suggested to compute Markov Perfect Equilibria 15. The one proposed here follows the broad structure in the seminal work byPakes and McGuire

(1994): each iteration has a starting value, that is updated according to the equilibrium conditions of the game, the “distance” between the updated value and the starting value is evaluated and if these are not close enough, according to a pre-specified cutoff, a new iteration starts with the updated values as new starting points. Updating occurs at every possible state of the world and for every point in the policy and value functions. The starting functions in each iteration (either a guess for the fist iteration or last round’s updated result) governing players’ behaviour and transitions over states are taken as the

true policies for rival firms and transitions, so updating value functions for each player constitutes a single-agent dynamic programming problem. The most suitable cutoff, naturally, depends on the implemented algorithm.

Unlike most of the applications of algorithms computing MPE of similar games, the

networked structure of the game proposed here implies that the profit functions cannot be computedoff-line and fed into the algorithm. Instead, because the equilibrium prices depend on future values, which in turn depend on equilibrium prices, prices and stage profits would need updating within each iteration as well. Within this structure, Lee and Fong propose an iterative procedure that, applied to our setting, can be informally described by the steps below.

Consider the profit equations for buyers and sellers, as described by equations 3.1 and

3.2. Start by fixing the exogenous components of the model: the size of the game, B

and S; prices of inputs, {mi}, final prices in the buyers’ domestic markets, {ri}, and

each buyer’s demand, {qi}; and the (exogenous) outside price for the buyers xi = x.

Choose a suitable set of parameters for the game: costs of linking, cand c; bargaining parameters,bij for all players iand j; and matching qualitiesρij for all playersiand j.

1. Start with an initial guess of conditional choice probabilities,P0 of sizeB× |Ai| ×

|G|, period contracts, tP0 of size B ×S× |G|, and value functions VP0 of size (B+S)× |G|16_.

2. Start the following iteration procedure until convergence:

15

In the brief comments here I exclude algorithms that re-express the problem as a system of non- linear equations and solve for it, as for large games like the one at hand, the advantages of Gaussian methods have been discussed extensively (Doraszelski and Pakes,2007;Pakes and McGuire,1994)

16_{The procedure presented in the next subsection has been run initialising the algorithm with random}

CCPs, prices and values, as well as arbitrarily fixed arrays. Convergence is achieved in the relevant areas of the parameter space irrespective of the initialisation

(a) Use CCPs to construct the transition matrix for states%P(g0|g) and the con- ditional transitions, %σ_i(g0|ai, g), for every action, player and state.

(b) GiventP and VP find the sellers’ best response to each possible negotiation network, this is, out of all the buyers linked to the seller under a given network, identify the relationship that gives largest payoffs - treat no-linking as an alternative.

(c) Compute players period payoffs under each possible network in all standing bilateral relations.

(d) Use the state transitions and the period payoffs of the standing relations for each player to generate the value functions. For simplicity, I perform this via value function iteration.

(e) For each negotiation network (both standing and non-standing relations) com- pute the outside options for each player in each bilateral negotiation she faces, taking all other prices as given, using the value functions computed above. (f) Given the bargaining parameters, for each linked pair solve the Nash Bargain-

ing problem to obtain prices. All pairs that exhibit no gains from trade for at least one of the parties in trade are unstable. Note that in our context, the simultaneous Nash Bargaining problems reduce to a system of (simultaneous) linear equations.

(g) Update the CCPs for each buyer. In my application, I assume follows a Type I Extreme Value distribution, which makes the updating stage straight- forward, obtaining CCPs as ratios of transformed inter-temporal profits17. (h) Feed in the CCPs updated in (2.g) and the prices and stability rule obtained

in (2.f) to start a new iteration in (2.a).

3. Perform step (2) until convergence is achieved. In Lee and Fong convergence is evaluated element-wise as |VPτ+1 −VPτ| < ω with ω pre-specified, with τ

denoting iterations of step (2) (Lee and Fong,2013). I set ω= 10(−6)_1+|1_Vτ_| which

is equivalent to using the sup-norm criterion discussed in Doraszelski and Pakes

(2007) 18.

The algorithm proposed here (again, following Lee and Fong (2013)) is not a contrac- tion mapping so convergence is not guaranteed. Moreover, like in Pakes and McGuire

17_{The convenience of this parametric choice for computational purposes is immediate. The criticism}

around the Independence of Irrelevant Alternatives observed early in Debreu (1960), does not hold in the dynamic context, as the choice specific value functions depend on all other alternatives via future payoffs, even when the stage profits are a function of the (one) current action only.

18_{For robustness, the smaller exercises were run twice, once evaluating convergence on the values}

and once evaluating convergence on the CCPs. No differences in the equilibrium reached were found, although the convergence in CCPs was found less smooth (reasonably enough in the context of our application) and slower, as the sup norm is evaluated over larger arrays: CCPs are player, action and state specific.

(1994) systematic non-convergence and cycling can arise due to the characteristics of the game. Like in the classic entry-exit example, choices in the linking stage are such that discontinuities in future values can arise, particularly when more than one buyer clump

choosing the same seller, bringing the future values discontinuously down for (some of the) buyer(s) causing the corresponding links to break inducing another discontinuous jump in values. One way of solving this is introducing additional randomness in the linking stage. A second issue inducing cycling patterns in the convergence path is a non-uniqueness feature immediately associated to the specification of the game: con- sider two buyers linking with one seller (gij = gkj = 1); in the game presented here,

there are occasions in which either of the links could be individually sustainable provided the other link breaks, and an iteration starting with both firms linking, can induce a best response of none of them linking, so each buyer would hold the belief that the rival is not linking, and insist herself on re-forming the link, going back to the starting point. One possible way of solving this type of cycling pattern is to introduce an ordering in which the breakage occurs (in line withPakes and McGuire’s type of solution). This is, indeed, naturally embedded in our game. In the presence of enough heterogeneity affecting the surplus each seller makes with each individual link, only the link with higher surplus is kept. A random device picks out one link only in the unlikely case of a tie occurring.