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Decision-Making Under Complete Uncertainty: You Will Regret Not Being Greedy

Kristijan Atanasov, Mehmet Ismail, Frederik Mallmann-Trenn

TL;DR

The paper tackles decision-making under complete uncertainty by modeling a one-shot game between a DM and Nature, examining the worst-case regret of the greedy strategy that selects the highest observed average rating. It establishes that, in the 2-product, 2-rating setting, the greedy rule achieves a worst-case regret of at most $\frac{1}{8}$ for $m=1$, is optimal in the sense of minimizing worst-case regret, and that this regret vanishes as the number of observations per product grows, with a computable bound for zero-regret probability. It further shows that the greedy strategy outperforms Thompson Sampling in both finite and asymptotic regimes and validates these findings empirically using Google restaurant reviews. The results provide theoretical guarantees for the efficacy of simple greedy decisions under severe ambiguity, with practical implications for sampling and decision guidance in uncertain environments. The work also outlines future directions for handling unequal sampling, scaling to more products and ratings, and refining comparisons with established exploration-exploitation algorithms.

Abstract

In this paper, we propose a probabilistic game-theoretic model to study the properties of the worst-case regret of the greedy strategy under complete (Knightian) uncertainty. In a game between a decision-maker (DM) and an adversarial agent (Nature), the DM observes a realization of product ratings for each product. Upon observation, the DM chooses a strategy, which is a function from the set of observations to the set of products. We study the theoretical properties, including the worst-case regret of the greedy strategy that chooses the product with the highest observed average rating. We prove that, with respect to the worst-case regret, the greedy strategy is optimal and that, in the limit, the regret of the greedy strategy converges to zero. We validate the model on data collected from Google reviews for restaurants, showing that the greedy strategy not only performs according to the theoretical findings but also outperforms the uniform strategy and the Thompson Sampling algorithm.

Decision-Making Under Complete Uncertainty: You Will Regret Not Being Greedy

TL;DR

The paper tackles decision-making under complete uncertainty by modeling a one-shot game between a DM and Nature, examining the worst-case regret of the greedy strategy that selects the highest observed average rating. It establishes that, in the 2-product, 2-rating setting, the greedy rule achieves a worst-case regret of at most for , is optimal in the sense of minimizing worst-case regret, and that this regret vanishes as the number of observations per product grows, with a computable bound for zero-regret probability. It further shows that the greedy strategy outperforms Thompson Sampling in both finite and asymptotic regimes and validates these findings empirically using Google restaurant reviews. The results provide theoretical guarantees for the efficacy of simple greedy decisions under severe ambiguity, with practical implications for sampling and decision guidance in uncertain environments. The work also outlines future directions for handling unequal sampling, scaling to more products and ratings, and refining comparisons with established exploration-exploitation algorithms.

Abstract

In this paper, we propose a probabilistic game-theoretic model to study the properties of the worst-case regret of the greedy strategy under complete (Knightian) uncertainty. In a game between a decision-maker (DM) and an adversarial agent (Nature), the DM observes a realization of product ratings for each product. Upon observation, the DM chooses a strategy, which is a function from the set of observations to the set of products. We study the theoretical properties, including the worst-case regret of the greedy strategy that chooses the product with the highest observed average rating. We prove that, with respect to the worst-case regret, the greedy strategy is optimal and that, in the limit, the regret of the greedy strategy converges to zero. We validate the model on data collected from Google reviews for restaurants, showing that the greedy strategy not only performs according to the theoretical findings but also outperforms the uniform strategy and the Thompson Sampling algorithm.

Paper Structure

This paper contains 22 sections, 5 theorems, 47 equations, 2 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Results Overview
  3. Industry Use-Cases
  4. Related Work
  5. The Model
  6. States
  7. Observations
  8. Strategies
  9. Payoff
  10. Regret
  11. Strategies for the Decision Maker
  12. Upper Confidence Bound Algorithm
  13. Thompson Sampling
  14. Maximum Regret for the Greedy Strategy
  15. With one observation per product
  16. ...and 7 more sections

Key Result

Proposition 1

For $n_d=2$, $n_r=2$ and $m=1$, the worst-case regret of the greedy strategy is $\frac{1}{8}$.

Figures (2)

  • Figure 1: Maximum regret of the greedy strategy choosing between two products as number of observations increase up to 20
  • Figure 2: Regret of the proposed strategies when tested on Google reviews data for restaurants. The regret of the greedy strategy is barely visible as it is below all other strategies.

Theorems & Definitions (10)

  • Definition 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Corollary 1: Optimality of greedy
  • Theorem 4
  • proof
  • Proposition 5
  • proof