[Paper Review] A Time-Inconsistent Dynkin Game: from Intra-personal to Inter-personal Equilibria

Yu‐Jui Huang, Zhou Zhou|arXiv (Cornell University)|Jan 2, 2021

Capital Investment and Risk AnalysisEconomics, Econometrics and Finance49 references12 citations

TL;DR

This paper introduces a novel framework for nonzero-sum Dynkin games under non-exponential discounting by integrating intra-personal equilibria—where each player reconciles time-inconsistent preferences across her current and future selves—into inter-personal equilibria via a two-level game-theoretic structure. It establishes the existence of both soft and sharp inter-personal equilibria through alternating iterative procedures, demonstrating that a firm's coercive power in real options negotiation critically depends on relative impatience levels, with highly impatient firms potentially becoming coerced in equilibrium.

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.

Motivation & Objective

• To formalize and analyze inter-personal equilibria in nonzero-sum Dynkin games under time inconsistency due to non-exponential discounting.
• To bridge intra-personal equilibrium (self-consistent strategies across time) with inter-personal equilibrium (Nash equilibrium between agents).
• To develop constructive iterative procedures for computing optimal intra-personal and inter-personal equilibria.
• To investigate the economic implications of time inconsistency in real options negotiation between two firms.

Proposed method

• Defines intra-personal equilibrium as a fixed point of a value-optimizing operator, ensuring consistency across current and future selves.
• Introduces individual iterative procedures (3.4) to compute optimal intra-personal equilibria for each player.
• Proposes an alternating iterative procedure (4.2) where players update their strategies in turn based on the other’s current policy.
• Uses Zorn’s lemma and convergence arguments to prove existence of soft inter-personal equilibria.
• Applies the method to a two-firm real options model with state-dependent payoffs and non-exponential discounting.
• Employs analytical techniques including connectedness of stopping sets and inequalities involving discount parameters β1, β2, α1, α2, u, K, N.

Experimental results

Research questions

• RQ1 Under what conditions does a soft inter-personal equilibrium exist in a time-inconsistent nonzero-sum Dynkin game?
• RQ2 Can the alternating iterative procedure (4.2) converge to a sharp inter-personal equilibrium, where each player selects her optimal intra-personal equilibrium?
• RQ3 How do differences in impatience (β1 vs. β2) affect the outcome of a real options negotiation?
• RQ4 What determines whether a firm can successfully coerce the other into stopping first in a negotiation?
• RQ5 Why might a highly impatient firm fail to coerce a patient firm and instead be forced to stop first?

Key findings

• An inter-personal equilibrium exists under appropriate conditions, proven via Zorn’s lemma and convergence of the alternating iterative procedure.
• The alternating iterative procedure (4.2) converges to a soft inter-personal equilibrium, which is not necessarily sharp.
• In the real options model, if Firm 1 is more impatient (β1 > β2), its attempt to coerce Firm 2 by refusing to stop (S0 = ∅) may fail.
• When Firm 1 is sufficiently impatient and Firm 2 sufficiently patient, the equilibrium outcome is ((0, y∗₁] ∩X, ∅), meaning Firm 1 stops while Firm 2 never stops.
• The coercer (Firm 1) may become the coerced, as the iterative process leads to Firm 1 taking the stopping policy (0, y∗₁] ∩X despite starting with S0 = ∅.
• The existence of a sharp inter-personal equilibrium is not guaranteed in general, as shown by explicit counterexamples where the limit is only soft.