  All-Against-One (AAO) games are a special class of multi-player games where
all players except one are in direct conflict with the remaining player. In the
case of Linear Quadratic Differential (LQD) games, the AAO structure can be
used to describe a situation where all players except one are trying to
regulate the state of the system (drive it towards the origin) while the
opposing player is trying to de-regulate the state (drive it away from the
origin). Similar to the standard LQD games, the closed-loop Nash strategies in
the AAO LQD games are expressed in terms of the solutions of a set of coupled
matrix Riccati differential equations. However, conditions for the existence of
a solution to these equations are different and more challenging in the AAO
case. In this paper, we derive conditions for existence, definiteness, and
uniqueness of a solution to these equations as well as conditions for
boundedness of the resulting state trajectory to ensure that the opposing
player fails in accomplishing its objective. Finally, we consider two options
for designing Nash strategies for the players and we illustrate the results
with an example of 3-against-1 pursuit-evasion differential game.
