  Subgame perfect equilibria are specific Nash equilibria in perfect
information games in extensive form. They are important because they relate to
the rationality of the players. They always exist in infinite games with
continuous real-valued payoffs, but may fail to exist even in simple games with
slightly discontinuous payoffs. This article considers only games whose outcome
functions are measurable in the Hausdorff difference hierarchy of the open sets
(\textit{i.e.} $\Delta^0_2$ when in the Baire space), and it characterizes the
families of linear preferences such that every game using these preferences has
a subgame perfect equilibrium: the preferences without infinite ascending
chains (of course), and such that for all players $a$ and $b$ and outcomes
$x,y,z$ we have $\neg(z <_a y <_a x \,\wedge\, x <_b z <_b y)$. Moreover at
each node of the game, the equilibrium constructed for the proof is
Pareto-optimal among all the outcomes occurring in the subgame. Additional
results for non-linear preferences are presented.
