Identifying the Player Responsible for Deviating from a Prescribed Strategy Profile

Core Concepts

If players are supposed to follow a prescribed strategy profile to reach a target set, and the target is not reached due to a deviation by one player, an outside observer can identify the deviating player.

Abstract

The paper studies a problem where a group of players are supposed to follow a prescribed strategy profile to reach a given target set. If the target is not reached due to a deviation by one of the players, the goal is to identify the deviating player.
The key insights are:

The authors formally define the concept of a "blame function" that can identify the deviating player if the target set is not reached.

They show that if the probability of reaching the target set under the prescribed strategy profile is at least 1-ε, then the goal is 2√(|I|-1)ε-testable, meaning there exists a blame function that correctly identifies the deviator with high probability.

If the probability of reaching the target set under the prescribed strategy profile is 1, then the goal is 0-testable, meaning the deviator can be identified with certainty.

The authors provide explicit constructions of blame functions for two specific examples - the "Adjacent Ones" problem and the "Avoiding the Origin in a Random Walk" problem.

The general proof uses a game-theoretic approach, considering a zero-sum game between an adversary (who chooses the deviating player and strategy) and a statistician (who tries to identify the deviator).

The results have applications in game theory, where identifying deviators is important for implementing punishment strategies in dynamic games with Nash equilibria.

Stats

The probability of reaching the target set under the prescribed strategy profile is at least 1-ε.
The number of players is |I|.

Quotes

None.

Key Insights Distilled From

Identifying the Deviator

by Noga Alon,Be... at arxiv.org 04-03-2024

https://arxiv.org/pdf/2203.03744.pdf

Deeper Inquiries

How can the blame function be efficiently computed in practice, especially for problems with finite but long horizons

In practice, computing the blame function efficiently for problems with finite but long horizons can be challenging due to the exponential growth in the number of possible sequences as the horizon increases. One approach to address this issue is to use dynamic programming techniques to optimize the computation process. By breaking down the problem into smaller subproblems and storing intermediate results, dynamic programming can help reduce redundant calculations and improve efficiency. Additionally, heuristic methods or approximation algorithms can be employed to speed up the computation process while still providing reasonably accurate results. Furthermore, leveraging parallel computing or distributed systems can help distribute the computational load and expedite the computation of the blame function for problems with long horizons.

What are the implications of allowing multiple players to deviate simultaneously, rather than just a single deviator

Allowing multiple players to deviate simultaneously, rather than just a single deviator, can significantly complicate the identification process. In such scenarios, the blame function needs to be adapted to handle the possibility of multiple deviators. This may involve assigning probabilities or weights to each player based on their likelihood of deviating and the impact of their deviations on the outcome. The identification of multiple deviators may require more sophisticated algorithms and statistical methods to accurately pinpoint the responsible players. Moreover, the implications of multiple deviators can lead to increased uncertainty and complexity in determining the true cause of the deviation, potentially affecting the overall reliability of the blame function.

Can the techniques developed in this paper be extended to settings where the players' actions are not fully observable by the outside observer

Extending the techniques developed in the paper to settings where the players' actions are not fully observable by the outside observer introduces additional challenges. In such cases, the observer may have limited or incomplete information about the players' strategies and deviations, making it harder to accurately identify the deviators. One approach to address this limitation is to incorporate probabilistic models or Bayesian inference methods to infer the players' actions based on the observed outcomes. By leveraging statistical techniques and machine learning algorithms, it may be possible to estimate the players' strategies and deviations even in scenarios with partial observability. However, the complexity of the inference process and the uncertainty introduced by incomplete information may require more advanced modeling and computational methods to achieve accurate identification of deviators.