A time consistent dynamic bargaining procedure in differential games with hterogeneous discounting

Abstract

We study cooperative solutions for differential games where players consume a common property resource. Players are asymmetric, in the sense that they have different preferences and, in particular, different time preferences. We propose a new time-consistent dynamic bargaining procedure for this class of games. This solution concept, which is defined as the time-consistent dynamic bargaining (TCB) solution, extends the recursive Nash bargaining solution introduced in Sorger (2006) to a continuous time setting. The underlying idea is that, in case of disagreement, the threat is that players will play a noncooperative Markov Perfect Nash equilibrium just during a very small period of time, since new negotiations can take place at every future moment and, in particular, immediately later. Conditions for interior TCB solutions are derived. To illustrate the results, two common property resource games are analyzed in detail.

Dynamic Bargaining and Time-Consistency in Linear-State and Homogeneous Linear-Quadratic Cooperative Differential Games

“…As a result, the actual threat is to act in a noncooperative way just during a very small period of time. In this section we extend the results in Castañer et al (2021) to the case of general discount functions.…”

“…If, on the contrary, new negotiations can take place in the future (something that is not unusual in real life situations), the threat losses credibility. The time-consistent dynamic bargaining solution proposed in Castañer et al (2021) addresses this issue, by taking into account that, in case of disagreement, new negotiations can take place at every future moment and, in particular, immediately later. As a result, the actual threat is to act in a noncooperative way just during a very small period of time.…”

“…The corresponding solution, that makes use of memory (trigger) strategies, satisfies the property of DIR (i.e., it is time-consistent and agreeable in the language of NTU cooperative differential games). Alternatively, for memoryless strategies, a time-consistent dynamic bargaining procedure has been recently introduced in Castañer et al [2021]. However, this solution concept, that is time-consistent and subgame perfect in the language of noncooperative differential games, linked to the issue of credibility of the announced strategy, may not satisfy the property of DIR.…”