On the Optimality of the Myopic Strategy in Last-Success Stopping Problems with Total Positivity

Core Concepts

This paper leverages the theory of total positivity to demonstrate that the myopic strategy is optimal for last-success stopping problems with unimodal payoffs or unimodal one-step lookahead payoffs.

Abstract

• Bibliographic Information: Derbazi, Z. (2024). On a connection between total positivity and last-success stopping problems. arXiv preprint arXiv:2411.07103v1.
• Research Objective: This paper investigates the optimality of the myopic strategy in last-success stopping problems, focusing on the relationship between total positivity and the unimodality of stopping and continuation payoffs.
• Methodology: The paper employs the Markovian method of optimal stopping and utilizes the properties of total positivity, particularly the variation-diminishing property of totally positive matrices. It analyzes the Markov chain embedded in the success epochs of Bernoulli trials and examines the conditions under which the stopping problem becomes monotone.
• Key Findings: The research establishes that the myopic strategy is optimal when either the stopping payoffs or the one-step lookahead (continuation) payoffs are unimodal. It demonstrates that the unimodality of one type of payoff implies the unimodality of the other, leveraging the variation-diminishing property of the transition matrix. The paper also proves that for oscillating payoffs, the problem is not monotone, and the myopic strategy does not correspond to a threshold rule.
• Main Conclusions: The study provides new proofs for the optimality of the myopic strategy in ℓth-to-mth last-success problems based on total positivity arguments. It highlights the usefulness of total positivity theory in analyzing and solving optimal stopping problems.
• Significance: This research contributes to the field of optimal stopping theory by providing a novel perspective on the optimality of the myopic strategy in last-success problems. It establishes a clear connection between total positivity and the unimodality of payoffs, offering a new tool for analyzing and solving similar problems.
• Limitations and Future Research: The paper primarily focuses on last-success problems with independent Bernoulli trials. Future research could explore the applicability of these findings to more general optimal stopping problems with dependent trials or other stochastic processes. Additionally, investigating the computational aspects of finding optimal thresholds for specific payoff structures could be a promising direction.

Stats

The success probability of the kth trial in the harmonic Bernoulli success probability model is given by pk = w1 / (w1 + w2 + k - 1), where w1 > 0 and w2 ≥ 0.

Quotes

"In last success problems, verifying the unimodality of the difference between the payoffs f and g is a common method for proving the optimality of the myopic strategy."
"The theory of total positivity underpins our work, particularly the reliance on unimodality-preserving transformations."
"It is common to see the terms myopic strategy and threshold strategy employed interchangeably. In many cases, the myopic strategy turns out to be a threshold rule. However, this is not always true."

Key Insights Distilled From

On a connection between total positivity and last-success stopping problems

by Zakaria Derb... at arxiv.org 11-12-2024

https://arxiv.org/pdf/2411.07103.pdf

Deeper Inquiries

How can the concepts of total positivity and unimodality be applied to optimal stopping problems in continuous time, such as those involving Brownian motion or other diffusion processes?

Extending the concepts of total positivity and unimodality to continuous-time optimal stopping problems, particularly those involving Brownian motion or diffusion processes, presents exciting challenges and opportunities. Here's a breakdown of potential approaches and considerations:
1. Transition Kernels and Total Positivity:

In discrete-time Markov chains, the transition matrix P plays a crucial role. For continuous-time processes like Brownian motion, we work with transition kernels or transition probabilities. These kernels, often denoted as p(s, t; x, y), represent the probability density of the process transitioning from state x at time s to state y at time t.
Challenge: Defining total positivity for continuous kernels is more intricate than for matrices. One approach is to discretize the time and state spaces, approximate the transition kernel with a matrix, and analyze the total positivity of this matrix. As the discretization becomes finer, we can study the limiting behavior.
Alternative: Explore generalizations of total positivity to integral operators, as transition kernels can be viewed as integral operators acting on functions.
2. Unimodality of Value Functions:

Unimodality of the value function in continuous time is a powerful property. If the value function, which represents the expected payoff under the optimal stopping rule, is unimodal, it suggests the existence of a "threshold-like" stopping strategy.
Challenge: Establishing unimodality for diffusion processes can be challenging. Techniques from stochastic calculus, such as Itô's formula and Dynkin's formula, might be helpful in analyzing the properties of the value function and proving unimodality under certain conditions.
3. Specific Examples and Applications:

Option Pricing: In finance, pricing American options (which have early exercise features) is a classic optimal stopping problem. Investigating the total positivity of the underlying asset price process's transition kernel could offer insights into the structure of optimal exercise strategies.
Sequential Hypothesis Testing: In statistical signal processing, deciding when to stop collecting data based on a noisy signal is an optimal stopping problem. Total positivity and unimodality could be relevant in analyzing the performance of different stopping rules.
Key Considerations:

Regularity Conditions: The success of applying these concepts often relies on regularity conditions of the underlying diffusion process, such as smoothness of the transition kernel or growth properties of the payoff function.
Numerical Methods: When analytical solutions are elusive, numerical methods become essential. Developing efficient algorithms for solving continuous-time optimal stopping problems while leveraging total positivity and unimodality is an active research area.

Could there be scenarios where a non-myopic strategy outperforms the myopic strategy in last-success problems, even when the payoff structure doesn't strictly satisfy the unimodality conditions?

Yes, it's certainly possible for non-myopic strategies to outperform myopic strategies in last-success problems, even when the unimodality conditions on the payoff structure aren't strictly met. Here's why:
1. Information Horizon:

Myopic strategies, by definition, have a limited information horizon; they only consider the immediate next step.
Non-myopic strategies can incorporate information about the future beyond the next immediate step.
2. Scenarios Where Non-Myopic Strategies Excel:

Oscillating Payoffs with Predictable Patterns: Imagine a last-success problem where the payoffs exhibit an oscillating pattern, but the oscillations are not strictly unimodal. If there's a predictable structure to these oscillations (e.g., a repeating cycle), a non-myopic strategy could exploit this pattern to achieve a higher expected payoff. It might be beneficial to wait for a more favorable point in the cycle, even if the immediate next step seems suboptimal.
Changing Success Probabilities: If the success probabilities of the Bernoulli trials are not constant and follow a known pattern or trend, a non-myopic strategy could adapt to these changes more effectively. For instance, if the success probabilities are initially low but increase over time, a non-myopic strategy might wait longer before stopping.
3. Complexity vs. Performance:

The trade-off is often between the complexity of the strategy and its potential performance gain.

Myopic strategies are simple to implement but might miss opportunities when more global information is available.
Non-myopic strategies can be more complex to design and compute, but they hold the potential for higher payoffs in specific scenarios.
Example:
Consider a last-success problem with a payoff structure that alternates between high (H) and low (L) values, but with a slight upward trend: H, L, H+ε, L+ε, H+2ε, L+2ε, ... A myopic strategy might get stuck repeatedly selecting the low values, while a non-myopic strategy could recognize the trend and wait for the higher payoffs.

If we consider a game where the objective is to minimize the expected time to stopping while ensuring a certain minimum probability of selecting a success within the last m trials, how would the optimal strategy change, and what role would total positivity play in its analysis?

This modified objective introduces an interesting twist to the classic last-success problem. Let's break down the optimal strategy and the role of total positivity:
1. Modified Objective:

You want to minimize the expected time to stopping (which encourages earlier stopping) while still maintaining a minimum probability (let's call it α) of capturing a success within the last m trials.
2. Optimal Strategy - A Balancing Act:

The optimal strategy will need to balance these two competing goals.

Stopping too early might satisfy the minimum probability requirement but could lead to a longer waiting time on average.
Stopping too late might reduce the waiting time in cases of early successes but risks missing the opportunity to stop within the last m trials and failing to meet the probability threshold.
3. Threshold-Based Strategy:

It's reasonable to expect that the optimal strategy will still have a threshold-like structure. However, the threshold will likely be dynamic, depending on the remaining trials and the current estimate of achieving the minimum probability α.
4. Role of Total Positivity:

Analyzing Transition Dynamics: Total positivity can still be helpful in understanding the underlying dynamics of the system. The transition probabilities of the embedded Markov chain (as discussed in the context) would influence how quickly you approach the desired probability α as you observe more trials.
Unimodality and Threshold Structure: If certain unimodality properties hold (even if modified to account for the dual objective), they might provide insights into the structure of the optimal stopping regions. For example, unimodality could imply that once it becomes optimal to stop, it remains optimal to stop for all subsequent trials.
5. Mathematical Formulation:

You could formulate this problem more rigorously using dynamic programming. Let V(k, p) be the expected time to stopping under the optimal strategy, given that you are at trial k and the current probability of having selected a success within the last m trials is p.
The Bellman equation would capture the trade-off:V(k, p) = min { 1 + E[V(k+1, p')], E[V(k+1, p'')] }

The first term represents the expected time if you continue to the next trial.
The second term represents the expected time if you stop at the current trial.
p' and p'' are the updated probabilities of success within the last m trials based on whether you continue or stop, respectively.
6. Computational Challenges:

Solving this modified problem will likely be more computationally intensive than the classic last-success problem due to the additional state variable (p) and the dual objective.
In summary, total positivity and unimodality, while potentially needing adjustments to accommodate the dual objective, can still offer valuable tools for analyzing the structure of the optimal strategy and understanding the transition dynamics in this modified last-success problem.