Intrinsic Robustness of Prophet Inequality to Strategic Reward Signaling

Abstract: Prophet inequality concerns a basic optimal stopping problem and states that simple threshold stopping policies -- i.e., accepting the first reward larger than a certain threshold -- can achieve tight $\frac{1}{2}$-approximation to the optimal prophet value. Motivated by its economic applications, this paper studies the robustness of this approximation to natural strategic manipulations in which each random reward is associated with a self-interested player who may selectively reveal his realized reward to the searcher in order to maximize his probability of being selected. We say a threshold policy is $\alpha$(-strategically)-robust if it (a) achieves the $\alpha$-approximation to the prophet value for strategic players; and (b) meanwhile remains a $\frac{1}{2}$-approximation in the standard non-strategic setting. Starting with a characterization of each player's optimal information revealing strategy, we demonstrate the intrinsic robustness of prophet inequalities to strategic reward signaling through the following results: (1) for arbitrary reward distributions, there is a threshold policy that is $\frac{1-\frac{1}{e}}{2}$-robust, and this ratio is tight; (2) for i.i.d. reward distributions, there is a threshold policy that is $\frac{1}{2}$-robust, which is tight for the setting; and (3) for log-concave (but non-identical) reward distributions, the $\frac{1}{2}$-robustness can also be achieved under certain regularity assumptions.

• The paper characterizes optimal information revealing strategies under threshold stopping policies when rewards are strategically signaled by self-interested players.
• It demonstrates that even with strategic behavior, threshold policies can achieve tight approximations, including (1 - 1/e)/2 for general distributions and 1/2 for IID and log-concave cases.
• The analysis employs rigorous techniques such as reduction to binary-support distributions and convexity arguments, offering insights applicable to economic decision-making.

Intrinsic Robustness of Prophet Inequality to Strategic Reward Signaling: A Synopsis

The paper "Intrinsic Robustness of Prophet Inequality to Strategic Reward Signaling" by Wei Tang, Haifeng Xu, Ruimin Zhang, and Derek Zhu, explores the strategic aspect of the prophet inequality problem in the context of optimal stopping theory. The authors explore the robustness of threshold stopping policies when each random variable is associated with a self-interested player who may strategically signal their reward to maximize their chance of being selected by the searcher.

Key Contributions

The paper introduces a new angle to the classic prophet inequality by considering scenarios where players may behave strategically. Several substantial findings are presented:

1. Characterization of Optimal Information Revealing Strategies:
   • The optimal information revealing strategy for each player under a given threshold policy is characterized. If the threshold is not greater than the player's expected reward, the player uses a no-information strategy. Otherwise, the player reveals information based on a specific threshold structure.
2. Intrinsic Robustness for General Distributions:
   • For arbitrary reward distributions, the authors demonstrate that a threshold policy with a threshold equal to $\sfrac{OPT}{2}$ achieves a $(1 - \sfrac{1}{e})/2$-approximation even when players are strategic. This approximation is shown to be tight, marking the intrinsic robustness of such policies.
3. Enhanced Robustness for Specific Distribution Families:
   • IID Distributions: For identical reward distributions, the $T^*$ threshold policy, where $T^*$ satisfies certain equilibrium conditions, achieves a $\sfrac{1}{2}$-approximation in the strategic setting. This result is optimal for these distributions.
   • Log-concave Distributions: Under certain regularity conditions, such as having a sufficiently large number of players, threshold policies can achieve $\sfrac{1}{2}$-robustness for log-concave distributions.

Methodological Approach

The authors apply rigorous mathematical analysis to characterize the strategies and provide proofs for the robustness bounds. The methodology includes:

• Reduction to Binary-Support Distributions: By reducing the problem to an analysis of binary-support distributions, the authors simplify the strategic interaction, making the analysis tractable.
• Tightness Analysis: To establish the tightness of their results, the authors construct specific hard instances that demonstrate the lower bounds of the approximation ratios.
• Convexity Analysis: For log-concave distributions, they employ convexity properties to derive conditions under which robustness can be guaranteed.

Theoretical and Practical Implications

The paper's findings bridge online decision-making theory and strategic behavior in game theory. The implications are twofold:

• Theoretical: The results generalize the prophet inequality framework to a more realistic setting where self-interested agents can exhibit strategic behavior. This extension advances the theory of optimal stopping problems by incorporating a strategic dimension.
• Practical: The insights are valuable in economic scenarios where entities sequentially disclose information for optimal selection, such as hiring processes, venture capital investment, and real estate transactions. The robust policies proposed can be practically applied to ensure that strategic misreporting by agents does not significantly deteriorate the searcher's performance.

Future Directions

Several open questions emerge from this work:

1. Beyond Threshold Policies: While the paper focuses on threshold policies, exploring dynamic or adaptive policies that may offer better robustness could be an intriguing direction.
2. Necessity of Regularity Conditions: Investigating whether the regularity conditions (e.g., convexity requirements for log-concave distributions) can be relaxed without compromising robustness remains an important open question.
3. Alternative Measurements: Examining other performance metrics beyond the competitive ratio, such as regret minimization, in the presence of strategic behavior would provide a richer understanding of the interplay between strategy and optimal stopping.

In summary, this paper makes significant strides in understanding the robustness of prophet inequalities under strategic behavior, laying the groundwork for future exploration and application in strategic decision-making contexts.