Downside Risk-Aware Equilibria for Strategic Decision-Making

TL;DR

This work introduces Downside Risk-Aware Equilibria (DRAE), a game-theoretic solution concept that prioritizes downside risk through lower partial moments rather than variance. By extending normal-form games with environmental states and defining utility as $u = ER - \gamma \cdot Risk$, it demonstrates existence and computability of DRAE via Stochastic Fictitious Play, and shows that DRAE can achieve significantly lower downside risk while preserving expected rewards compared to Nash equilibria and prior risk-aware approaches. Across synthetic, asset, and product-portfolio experiments, DRAE proves robust to exogenous risk and supports higher-order risk preferences, illustrating practical gains for risk-averse decision-making in finance and economics. The paper also connects DRAE to existing equilibrium concepts and outlines future extensions to temporal settings using reinforcement learning and policy-space response oracles.

Abstract

Game theory has traditionally had a relatively limited view of risk based on how a player's expected reward is impacted by the uncertainty of the actions of other players. Recently, a new game-theoretic approach provides a more holistic view of risk also considering the reward-variance. However, these variance-based approaches measure variance of the reward on both the upside and downside. In many domains, such as finance, downside risk only is of key importance, as this represents the potential losses associated with a decision. In contrast, large upside "risk" (e.g. profits) are not an issue. To address this restrictive view of risk, we propose a novel solution concept, downside risk aware equilibria (DRAE) based on lower partial moments. DRAE restricts downside risk, while placing no restrictions on upside risk, and additionally, models higher-order risk preferences. We demonstrate the applicability of DRAE on several games, successfully finding equilibria which balance downside risk with expected reward, and prove the existence and optimality of this equilibria.

Figures (5)

• Figure 1: The normalised Euclidean distance between the DRAE and RAE equilibrium mixed strategies (over 50 seeds) as the skewness of the payoff distribution increases.
• Figure 2: Variance (top row) versus Downside risk (bottom row) across the various environments (columns) for a range of gamma values $\circ$ ($\gamma$ increasing as we move right-to-left). In each case, the $x$-axis shows the expected return, while the $y$-axis is the risk measure.
• Figure 3: Comparison of the expected return value for DRAE over a large range of threshold values $\tau$ on synthetic games.
• Figure 4: Equilibrium LPM under varying levels of exogenous risk (through adding more risky states to $\mathcal{S}$).
• Figure 5: The impact of higher-order preferences $d \in \{1.5,2,2.5,3,3.5,4\}$ on the mean risk (across all $d$).

Theorems & Definitions (4)

• Definition 1: DRAE
• proof
• proof
• proof