  Nash equilibrium is a key concept in game theory fundamental for elucidating
the equilibrium state of strategic interactions, finding applications in
diverse fields such as economics, political science, and biology. However, the
Nash equilibrium may not always align with the optimal or desired outcomes
within a system. This article introduces a novel game engineering framework
that tweaks strategic payoffs within a game to achieve a desired Nash
equilibrium while averting undesired ones. Leveraging mixed-integer linear
programming, this framework identifies intricate combinations of players and
strategies and optimal perturbations to their payoffs that enable the shift
from undesirable Nash equilibria to more favorable ones. We demonstrate the
effectiveness and scalability of our approach on games of varying complexity,
ranging from simple prototype games such as the Prisoner's Dilemma and
Snowdrift games with two or more players to complex game configurations with as
high as $10^6$ entries in the payoff matrix. These studies showcase the
capability of this framework in efficiently identifying the alternative ways of
reshaping strategic payoffs to secure desired Nash equilibria and preclude the
undesired equilibrium states. Our game engineering framework offers a versatile
toolkit for precision strategic decision-making with far-reaching implications
across diverse domains.
