It actually is a weakening. Because all changes can be interpreted as making some player worse off if we just use standard Pareto optimality, the second condition mean that more changes count as improvements, as you correctly state. The third condition cuts down on which changes count as improvements, but the combination of conditions 2 and 3 still has some changes being labeled as improvements that wouldn’t be improvements under the old concept of Pareto Optimality.
The definition of an almost stratified Pareto optimum was adapted from this , and was developed specifically to address the infinite game in that post involving a non-well-founded chain of players, where nothing is a stratified Pareto optimum for all players. Something isn’t stratified Pareto optimal in a vacuum, it’s stratified Pareto optimal for a particular player. There’s no oracle that’s stratified Pareto optimal for all players, but if you take the closure of everyone’s SPO sets first to produce a set of ASPO oracles for every player, and take the intersection of all those sets, there are points which are ASPO for everyone.
“the combination of conditions 2 and 3 still has some changes being labeled as improvements that wouldn’t be improvements under the old concept of Pareto Optimality.”
Why? Condition 3 implies that U_{RO,j} \leq U_{RO’,j}. So, together with condition 2, we get that U_{RO,j} \leq U_{RO’,j} for any j. That precisely means that this is a Pareto improvement in the usual sense.
It actually is a weakening. Because all changes can be interpreted as making some player worse off if we just use standard Pareto optimality, the second condition mean that more changes count as improvements, as you correctly state. The third condition cuts down on which changes count as improvements, but the combination of conditions 2 and 3 still has some changes being labeled as improvements that wouldn’t be improvements under the old concept of Pareto Optimality.
The definition of an almost stratified Pareto optimum was adapted from this , and was developed specifically to address the infinite game in that post involving a non-well-founded chain of players, where nothing is a stratified Pareto optimum for all players. Something isn’t stratified Pareto optimal in a vacuum, it’s stratified Pareto optimal for a particular player. There’s no oracle that’s stratified Pareto optimal for all players, but if you take the closure of everyone’s SPO sets first to produce a set of ASPO oracles for every player, and take the intersection of all those sets, there are points which are ASPO for everyone.
“the combination of conditions 2 and 3 still has some changes being labeled as improvements that wouldn’t be improvements under the old concept of Pareto Optimality.”
Why? Condition 3 implies that U_{RO,j} \leq U_{RO’,j}. So, together with condition 2, we get that U_{RO,j} \leq U_{RO’,j} for any j. That precisely means that this is a Pareto improvement in the usual sense.