Consider the pure strategy Markov Perfect Equilibrium of the game to be a set of strategies σ∗ such that for any i, network g and shocksi:
σi∗(g, i) =argmaxai∈Ai[ai,i+v
σ∗
i (ai, g)] (3.10)
, where the restrictions in the price bargaining problem and stability of the network are satisfied: so given Vσ∗, period contracts tσg∗ satisfy the Nash problem in 3.3, with 3.4
and 3.5defining players’ surplus for all stableg11 12.
It can be seen that a strategy profileσis Markov Perfect if there is no playeriand alter- native strategyσ0isuch that playeriprefersσi0 overσi, when all other players are playing
σ−i. This is, ∀i,∀g and ∀σi0 alternative strategies: Viσ(g, i|σi, σ−i) ≥ Viσ(g, i|σi0, σ−i).
The reader is referred to Maskin and Tirole’s original paper for definitions and proofs (Maskin and Tirole,1988). Intuitively, this equilibrium concept implies (i) that for each state of the world, optimal policies are chosen by all players given their beliefs on the future structure of the network and (ii) that those beliefs are consistent with rivals’ behaviour.
Lee and Fong(2013) argue that the above can be re-written in the space of probabilities, following the procedures in the work byAguirregabiria and Mira(2002). WithPσ∗, the conditional probability corresponding to the MPE σ∗, an analogous fixed point of the best response probability function describes the solution of the system. To re-express the problem in the space of probabilities, note that πi, Vi and %i(g0|g, ai) depend on
players’ strategies only through Pi, the associated probabilities. Also, by the definition
of Pi and σ∗, Pi∗(ai|g) =
R
I{ai =σ∗i(g, i)}fi(i)di, so equilibrium probabilities are a
fixed point Λ(P∗) =P∗ with
Λi(ai|g;P−i) =
Z
I{ai=argmaxa∈Ai(a,i+v
P∗
i (a, g))}fi(i)di (3.11)
with vP being choice specific value functions derived from vσ and defined in terms of
conditional choice probabilitiesP∗. In the terms of Aguirregabiria and Mira(2007), Λi
constitutes thebest response probability function for agentiand it is continuous (given the assumptions on f) in the choice set, so that existence is guaranteed by Brower’s
Theorem. Under standard regularity conditions, existence in this type of game with dynamic strategic interaction and incomplete information is guaranteed (Aguirregabiria
11Buyers and sellers interactions over the infinite horizon could induce several complex behavioural
patterns compatible with other equilibrium concepts, like more unrestricted concepts of subgame per- fection. I follow the literature in restricting attention to Markov Perfect Equilibria in pure strategies (which in turn, in our context is also subgame perfect).
12
The focus on pure-strategy equilibria only follows Aguirregabiria and Mira’s argument, according to which a mixed strategy equilibrium in a complete information game can be interpreted as a pure strategy in the game with incomplete information, such that the probability distribution of players’ actions is the same under the two equilibria, as shown in Harsanyi’s ”Purification Theorem” (1973) (Aguirregabiria and Mira,2007).
and Mira,2007;Doraszelski and Satterthwaite,2010). More specifically, followingAguir- regabiria and Mira (2007) to prove existence, at least one fixed point of the mapping is to be found. This, in turn implies showing that the mapping is continuous in the compact space of probabilities. It is sufficient to show that the choice-specific value functions are continuous inP for allg. This will follow from the shape of the per-period function, uniqueness in the pricing problem, continuity of prices onP and continuity of
π in prices 13.
For the purpose of this paper, as in many other applications of models of industry dynamics for empirical estimations, the discussions around existence is two-fold (Do- raszelski and Pakes, 2007): one aspect of it, is connected to whether the computation algorithms that find an equilibrium, sometimes required for estimation purposes, actu- ally converge to policies that satisfy, with error, the equilibrium conditions of the model; the second aspect is whether such conditions are guaranteed to hold exactly under the assumptions of the model. With regards to the former, most papers exploiting itera- tive algorithms to find MPE do converge, even when the proved sufficient conditions for existence are not satisfied (Ackerberg et al., 2007). Regarding the latter, starting from the general framework in Ericson and Pakes (1995), various assumptions have been presented to guarantee existence in the context of specific applications, ranging from allowing for mixed strategies to the more widespread alternative of introducing incomplete information, for example, as firm-specific privately known scrap values or entry costs (Doraszelski and Pakes,2007; Doraszelski and Satterthwaite, 2010; Ericson and Pakes,1995). Lee and Fong’s existence assumption (A.3.1. in their paper) involves allowing for a buyer - seller shock after the negotiation network is formed, drawn inde- pendently over time and all players from a known distribution with full support, such that small changes in conditional choice probabilities don’t trigger discontinuous jumps in the choice specific value functions. In their application, however, their algorithm con- verges to an equilibrium when those shocks are assumed away. The computer exercises in section 3.4show that this is also the case for the game presented here.
For arbitrary parameters, multiple equilibria is likely to arise in our context, as in most games of the same class with best responses being non-linear functions of rivals’ actions. A discussion of additional assumptions that have been imposed to guarantee uniqueness in similar games can be found inDoraszelski and Pakes (2007). However, these are not applicable to our setting over the whole of the parameters space and I will need to return to multiplicity issues when estimating the game.
13And in our context, given the value functions, this simplifies to a system of linear equations, in