[Paper Review] Time Consistent Stopping For The Mean-Standard Deviation Problem --- The Discrete Time Case

Erhan Bayraktar, Jingjie Zhang+1 more|arXiv (Cornell University)|Feb 23, 2018

Auction Theory and Applications16 references3 citations

TL;DR

This paper formulates the infinite-horizon mean-standard deviation stopping problem in discrete time using subgame perfect Nash equilibrium to achieve time consistency. It introduces liquidation strategies—where stopping rights are divisible—and proves that equilibrium liquidation strategies always exist, though optimal equilibria may not be unique or even exist.

ABSTRACT

Inspired by Strotz's consistent planning strategy, we formulate the infinite horizon mean-variance stopping problem as a subgame perfect Nash equilibrium in order to determine time consistent strategies with no regret. Equilibria among stopping times or randomized stopping times may not exist. This motivates us to consider the notion of liquidation strategies, which lets the stopping right to be divisible. We then argue that the mean-standard deviation variant of this problem makes more sense for this type of strategies in terms of time consistency. It turns out that an equilibrium liquidation strategy always exists. We then analyze whether optimal equilibrium liquidation strategies exist and whether they are unique and observe that neither may hold.

Motivation & Objective

• To resolve time-inconsistency in the mean-variance optimal stopping problem by formulating it as a subgame perfect Nash equilibrium.
• To address the failure of existence of equilibria among pure or randomized stopping times in infinite-horizon discrete-time settings.
• To introduce liquidation strategies as a novel class of mixed strategies that allow divisible stopping rights and restore time consistency.
• To analyze the existence, uniqueness, and optimality of equilibrium liquidation strategies under the mean-standard deviation criterion.
• To demonstrate that while optimal equilibria may not exist or be unique, Pareto optimal equilibria do exist.

Proposed method

• Formulate the mean-standard deviation problem as a subgame perfect Nash equilibrium to ensure time consistency.
• Introduce liquidation strategies, where the stopping time is represented as a continuous allocation of stopping rights across states.
• Define the equilibrium condition as a fixed-point problem: a strategy η is an equilibrium if it maximizes the liquidation payoff at each state x.
• Use the operator T that maps a strategy η to the optimal response ξ(·) = h_x(η) under fixed η, and solve for fixed points.
• Characterize equilibrium strategies via the condition h_x(η) = η(x), where h_x(η) is derived from a quadratic optimization of the mean-standard deviation criterion.
• Apply graphical and analytical methods to verify existence and uniqueness of equilibria in specific examples, including solving for intersections of curves g_x(a,b) = x.

Experimental results

Research questions

• RQ1 Do equilibria exist among pure or randomized stopping times in the infinite-horizon discrete-time mean-standard deviation problem?
• RQ2 Can liquidation strategies resolve time-inconsistency in the mean-standard deviation stopping problem?
• RQ3 Under what conditions does an optimal equilibrium liquidation strategy exist, and is it unique?
• RQ4 Can Pareto optimal equilibrium liquidation strategies be guaranteed to exist?
• RQ5 How do the mean and variance of the liquidated payoff depend on the strategy parameters in specific Markov processes?

Key findings

• Equilibria among pure or randomized stopping times may not exist in the infinite-horizon discrete-time mean-standard deviation problem.
• Liquidation strategies, which allow divisible stopping rights, always admit at least one equilibrium.
• An optimal equilibrium in the sense of pointwise dominance may fail to exist.
• Even when optimal equilibria exist, they may not be unique, as demonstrated by multiple equilibrium points on the boundary of the parameter space.
• In Example 4.1, there exists exactly one intersection point of the curves h_1(a,b) = a and h_7(a,b) = 7, corresponding to a unique equilibrium liquidation strategy.
• In Example 4.2, the existence of equilibrium liquidation strategies is confirmed through graphical analysis of the curves g_11(a,b) = 11 and g_17(a,b) = 17, which intersect at five points, all of which are verified as equilibria.