A Time-Inconsistent Dynkin Game: from Intra-personal to Inter-personal Equilibria

TL;DR

This paper introduces a framework for analyzing nonzero-sum Dynkin games with non-exponential discounting, combining intra-personal and inter-personal equilibrium concepts, and applies it to a real options negotiation scenario.

Contribution

It develops a novel approach to inter-personal equilibrium in nonzero-sum Dynkin games considering time inconsistency and intra-personal reasoning, with explicit existence proofs.

Findings

Inter-personal equilibrium exists under certain conditions.

Negotiation power depends on firms' impatience levels.

Explicit equilibria are derived for a real options negotiation model.

Abstract

This paper studies a nonzero-sum Dynkin game in discrete time under non-exponential discounting. For both players, there are two levels of game-theoretic reasoning intertwined. First, each player looks for an intra-personal equilibrium among her current and future selves, so as to resolve time inconsistency triggered by non-exponential discounting. Next, given the other player's chosen stopping policy, each player selects a best response among her intra-personal equilibria. A resulting inter-personal equilibrium is then a Nash equilibrium between the two players, each of whom employs her best intra-personal equilibrium with respect to the other player's stopping policy. Under appropriate conditions, we show that an inter-personal equilibrium exists, based on concrete iterative procedures along with Zorn's lemma. To illustrate our theoretic results, we investigate a two-player real options valuation problem: two firms negotiate a deal of cooperation to initiate a project jointly. By deriving inter-personal equilibria explicitly, we find that coercive power in negotiation depends crucially on the impatience levels of the two firms.