minimimaximi

Minimaximi is a neologism used in decision theory and game theory to describe a hybrid optimization principle that blends minimax and maximin ideas. The term, formed by repeating and combining “minimax” and “maximin,” appears in theoretical discussions and introductory teaching as a thought experiment rather than an established method. It is typically discussed as a framework for balancing worst-case safeguards with favorable-case opportunities.

Definition and formalization: Let A be a set of actions and U a set of uncertain outcomes with loss function L(a,u). A minimaximi approach can be defined in two related ways. One is to minimize a weighted or convex combination of the worst-case loss and the best-case loss: F_w(a) = w · max_{u} L(a,u) + (1−w) · min_{u} L(a,u), with w in [0,1]. A second view treats minimaximi as an iterative process that alternates between minimax and maximin updates to the strategy until a fixed point or satisfactory equilibrium is reached.

Properties and limitations: Because the composite objective blends extreme outcomes, convergence and uniqueness are not guaranteed in general. The behavior depends on the structure of L, the action set A, and the outcome set U. In some settings the method yields robust but conservative decisions; in others it may overemphasize favorable outcomes or fail to reflect a decision maker’s true preferences.

Applications and reception: As a teaching device, minimaximi helps illustrate trade-offs between robustness and opportunism. It has appeared in discussions of robust optimization, risk assessment, and AI safety thought experiments. However, unlike classical minimax or maximin, minimaximi has limited formal consensus, and practitioners typically prefer well-defined criteria with clear axioms.

See also: minimax, maximin, saddle point, robust optimization, decision theory.