Risk-Seeking Strategic Robustness in Games: A Substitutes-Complements Rule, and Where It Breaks

Strategic risk aversion, formalized through a fictitious-adversary construction, has recently been shown to induce a “free lunch” in a shared-effort game: more robustness can strictly improve equilibrium welfare, in contrast to classical robust optimization. We study this phenomenon in a tractable class of two-player quadratic games, and introduce the natural risk-seeking counterpart of the construction, in which the fictitious adversary is replaced by a fictitious ally. Within this game class, we show that whether risk aversion, risk seeking, or neither attitude helps is governed by a single, classical structural property: whether the game exhibits strategic substitutes or complements, and that the losing attitude always strictly hurts welfare instead. We illustrate the theory on three examples: trading, a shared-effort aggregative game, and Bertrand price competition.