The definition of stratified Pareto improvement doesn’t seem right to me. You are trying to solve the problem that there are too many Pareto optimal outcomes. So, you need to make the notion of Pareto improvement weaker. That is, you want more changes to count as Pareto improvements so that less outcomes count as Pareto optimal. However, the definition you gave is strictly stronger than the usual definition of Pareto improvement, not strictly weaker (because condition 3 has equality instead of inequality). What it seems like you need is dropping condition 3 entirely.
The definition of almost stratified Pareto optimum also doesn’t make sense to me. What problem are you trying to solve? The closure of a set can only be larger than the set. Also, the closure of an empty set is empty. So, on the one hand, any stratified Pareto optimum is in particular an almost stratified Pareto optimum. On the other hand, if there exists an almost stratified Pareto optimum, then there exists a stratified Pareto optimum. So, you neither refine the definition of an optimum nor make existence easier.
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.
The definition of stratified Pareto improvement doesn’t seem right to me. You are trying to solve the problem that there are too many Pareto optimal outcomes. So, you need to make the notion of Pareto improvement weaker. That is, you want more changes to count as Pareto improvements so that less outcomes count as Pareto optimal. However, the definition you gave is strictly stronger than the usual definition of Pareto improvement, not strictly weaker (because condition 3 has equality instead of inequality). What it seems like you need is dropping condition 3 entirely.
The definition of almost stratified Pareto optimum also doesn’t make sense to me. What problem are you trying to solve? The closure of a set can only be larger than the set. Also, the closure of an empty set is empty. So, on the one hand, any stratified Pareto optimum is in particular an almost stratified Pareto optimum. On the other hand, if there exists an almost stratified Pareto optimum, then there exists a stratified Pareto optimum. So, you neither refine the definition of an optimum nor make existence easier.
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.