D Private information and maximum delay - Bargaining and delay in trading networks

Consider again the path (i0, i₁, i₂, ..., i_k, i_k+1) that connects seller s (player i0) to buyer b (player ik+1). Players negotiate how to split the surplus via successive bilateral bargaining sessions in the following order: (ik, i_k+1), (i_k1, i_k), (i_k2, i_k1), ..., (i1, i₂), (i0, i₁). Suppose now that the players have private information. They are uncertain about each others’ discount factors. Player i’s discount factor lies in the range [Î_i, ^P_i], where 0 < Î_i  ^Pi < 1. The types are independently drawn from the interval [^P_i , Î_i] according to the probability distribution pi, i  N.

Lemma 6. Consider the sequence (ik, i_k+1), (i_k1, i_k), (i_k2, i_k1), ..., (i1, i₂), (i0, i₁) of k +1 bilateral bargaining sessions with private information in which the probability distributions are common knowledge and in which the period length shrinks to zero.

For any perfect Bayesian equilibria, the payo of player i_k+1l in each bilateral bargaining session (i_kl, i_k+1l) belongs to



This lemma follows from Watson (1998) Theorem 1. Whether or not all payos within the intervals given in Lemma 6 are possible depends on the distributions over types. As Watson (1998) stated ”each player will be no worse than he would be in equilibrium if it were common knowledge that he were his least patient type and the opponent were his most patient type. Furthermore, each player will be no better than he would be in equilibrium with the roles reversed”. Since we allow for general probability distributions over discount factors, multiplicity of perfect Bayesian equilibria (PBE) is not an exception (even when the game is almost with complete information).

The maximum number of bargaining periods player i_k+1l would spend nego-tiating in the bilateral bargaining session (i_kl, i_k+1l), I (m(i_kl, i_k+1l)), is given

Notice that I (m(i_kl, i_k+1l)) is simply the integer part of m(i_kl, i_k+1l). It is cus-tomary to express the players’ discount factors in terms of discount rates, ri > 0, and the length of the bargaining period, , according to the formula i = exp (rⁱ).

With this interpretation, player i’s type is identified with the discount rate ri, where ri  [r^Pi , r_iÎ]. We thus have that Î_i = exp(riÎ) and ^P_i = exp(r^Pi). Note that r_iÎ  ri^P since greater patience implies a lower discount rate. As  approaches zero, using l’Hopital’s rule we obtain that

D(i_kl, i_k+1l) = lim

0(m(i_kl, i_k+1l)· ) =  1 r_k+1l^P ·log

 r_kl^P

r_k+1lÎ · r_klÎ + r_k+1l^P r_kl^P + r_k+1lÎ

 , which is a positive, finite number. Notice that D(i_kl, i_k+1l) converges to zero as r_i^P and r_i^I become close. Given the equilibrium trading path (s, i1, i₂, ..., i_k, b), the maximum real delay time in reaching a global agreement is D(s, i1, i₂, ..., i_k, b) = D(s, i1) + D(i1, i₂) + ... + D(ik, b).

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