Pareto Optimal基础

• Pareto Optimal

  • http://www.gametheory.net/dictionary/ParetoOptimal.html
  • Named after Vilfredo Pareto, Pareto optimality is a measure of efficiency. An outcome of a game is Pareto optimal if there is no other outcome that makes every player at least as well off and at least one player strictly better off. That is, a Pareto Optimal outcome cannot be improved upon without hurting at least one player. Often, a Nash Equilibrium is not Pareto Optimal implying that the players’ payoffs can all be increased (比如囚徒博弈中，Nash Equilibrium是二者都背叛，而二者都沉默的话可以使得二者的payoff都提升。实际上，囚徒博弈的Pareto Optimal正是二者都沉默).

• Pareto efficiency

  • https://en.wikipedia.org/wiki/Pareto_efficiency
  • Pareto efficiency, or Pareto optimality, is a state of allocation of resources in which it is impossible to make any one individual better off without making at least one individual worse off.
  • Pareto improvement is defined to be a change to a different allocation that makes at least one individual better off without making any other individual worse off, given a certain initial allocation of goods among a set of individuals. An allocation is defined as “Pareto efficient” or “Pareto optimal” when no further Pareto improvements can be made.
  • Pareto efficiency is a minimal notion of efficiency and does not necessarily result in a socially desirable distribution of resources: it makes no statement about equality, or the overall well-being of a society. Pareto Optimal不能保证equality或overall well-being of a society，比如社会财富都给一个人，这是Pareto Optimal，但不能保证公平，不是inefficient allocation.
  • It is important to note, however, that a change from a generally inefficient economic allocation to an efficient one is not necessarily a Pareto improvement. Thus, in practice, to ensure that nobody is disadvantaged by a change aimed at achieving Pareto efficiency, compensation of one or more parties may be required. For instance, if a change in economic policy eliminates a monopoly and that market subsequently becomes competitive, the monopolist will be made worse off and still the gain in efficiency may be superior than the loss to the monopolist, so that the monopolist could hypothetically be compensated for its loss while still leaving a net gain for others in the economy, allowing for a Pareto improvement.

  • Weak Pareto efficiency

    • A “weak Pareto optimum” (WPO) is an allocation for which there are no possible alternative allocations whose realization would cause every individual to gain. Thus an alternative allocation is considered to be a Pareto improvement only if the alternative allocation is strictly preferred by all individuals. When contrasted with weak Pareto efficiency, a standard Pareto optimum as described above may be referred to as a “strong Pareto optimum” (SPO).
    • Weak Pareto-optimality is “weaker” than strong Pareto-optimality in the sense that any SPO also qualifies as a WPO, but a WPO allocation is not necessarily an SPO. （WPO 包含SPO）

  • The notion of Pareto efficiency is also useful in engineering. Given a set of choices and a way of valuing them, the Pareto frontier or Pareto set or Pareto front is the set of choices that are Pareto efficient. By restricting attention to the set of choices that are Pareto-efficient, a designer can make tradeoffs within this set, rather than considering the full range of every parameter.

• Multi-objective optimization

  • https://en.wikipedia.org/wiki/Multi-objective_optimization
  • Multi-objective optimization (also known as multi-objective programming, vector optimization, multicriteria optimization, multiattribute optimization or Pareto optimization) is an area of multiple criteria decision making, that is concerned with mathematical optimization problems involving more than one objective function to be optimized simultaneously.
  • For a nontrivial multi-objective optimization problem, there does not exist a single solution that simultaneously optimizes each objective. In that case, the objective functions are said to be conflicting, and there exists a (possibly infinite) number of Pareto optimal solutions. A solution is called nondominated, Pareto optimal, Pareto efficient or noninferior, if none of the objective functions can be improved in value without degrading some of the other objective values.

  • Without additional subjective preference information, all Pareto optimal solutions are considered equally good (as vectors cannot be ordered completely).

  • (Pareto) dominate ：
    

  • The Pareto front of a multi-objective optimization problem is bounded by a so-called nadir objective vector and an ideal objective vector
    

  • Scalarizing multi-objective optimization problems