Unanimity is principle from Voting Theory that is the same as Dominance:
If a candidate $a$ is always preferred by the majority to $b$, then we can say that $b$ is dominated by $a$ and never consider $b$ again
Pareto-Optimal Solutions
In Multi-Objective Optimization and Multi-Criteria Decision Aid there could be many "best" solutions - these solutions are called the Pareto-Optimal solutions
Consider this example:
-
- the two alternatives along the blue line form the pareto-optimal set
-
- the solutions in blue circles also form the pareto-optimal set
Dominance
- for the second examples we can say that $b$ dominates $c$:
- $b$ has the same level of quality, but it is cheaper
- we can remove all dominated solutions from the solution space and this will give us the Pareto-optimal set of solutions
- in MOO this is also called the set of efficient solutions
- dominance
- $a$ dominates $b$ $\iff \forall i: f_i(a) \geqslant f_i(b)$ and $\exists i: f_j(a) > f_j(b)$
- i.e. for all criteria $a$ is at least as good as $b$, but there's at least one criteria at which $a$ is strictly better than $b$
This is not always good. Consider this example
- you're looking for an apartment to rent
- you consider price and distance to work (want to minimize both)
-
- in this case $c$ is dominated by $a$ and $b$:
- $b$ is very cheap, $a$ is very close
- $c$ is a good compromise, but it's dominated
Another example
- suppose we're choosing a car
- there are 4 criteria: price, power, consumption, comfort
- there are 6 alternatives
|
Price |
Power |
Consumption |
Comfort
|
Avg A. |
18 |
75 |
8 |
3
|
Sport |
18.5 |
110 |
9 |
2
|
Avg B. |
17.5 |
85 |
7 |
3
|
Lux 1 |
24 |
90 |
8.5 |
5
|
Exonomic |
12.5 |
50 |
7.5 |
1
|
Lux 2 |
22.5 |
85 |
9 |
4
|
We see that Avg B is always better than Avg. A
- then nobody will ever choose Avg A: A is dominated by B
No other alternative can be eliminated this way
This principle is as well applied in the Game Theory
Notation:
- $A_i$ - set of strategies for player $i$
- $u_i$ - the utility function of $i$
- $a, b \in A_i$ - two strategies in $A_i$
- denote $A_{-i}$ as the set of all strategies for other players
- Strict dominance
- $a$ strictly dominates $b$ if $\forall c \in A_{-1}: u_i(a, c) > u_i(b, c)$
- in other words: $a$ strictly dominates $b$ is for every action that other players can take, the action $a$ gives $i$ better payoff than $b$
- Weak dominance
- $a$ (weakly) dominates $b$ if $\forall c \in A_{-1}: u_i(a, c) \geqslant u_i(b, c)$
If $a$ dominates all other strategies $b$ of the player $i$ then it's dominant
Pareto Optimality
In Game Theory there's also a notion of Pareto Optimality
Suppose you see a game as an outside observer, not a player
- Can we say that one outcome $O$ is better than some other outcome $O'$?
Pareto-Dominance
- suppose there's one outcome $O$ that is as good as some other outcome $O'$ for all players
- but there's one agent $i$ who strictly prefers $O$ to $O'$
- then $O$ is considered better than $O'$
- and $O$ pareto-dominates $O'$
Pareto-Optimality
- outcome $O^*$ is pareto-optimal if there is no other outcome that pareto-dominates it
- a game can have more than one pareto-optimal outcome
- for Zero-Sum Games every outcome is pareto-optimal
Problems
Close Contenders
$b$ is a close contender if
- it's dominated by some alternative $a$
- but $b$ is still a good choice
Consider this example:
- $A = \{a, b, c, d\}$
- $A^* = \{a, c, d\}$ - efficient (pareto-optimal) set of solutions
$c$ |
$e_1$ |
$e_2$ |
$e_3$ |
$...$ |
$e_{100}$
|
$a$
|
100 |
100 |
100 |
... |
100
|
$b$
|
99 |
99 |
99 |
... |
99
|
$c$
|
101 |
0 |
0 |
... |
0
|
$d$
|
0 |
101 |
0 |
... |
0
|
But we see that $c$ and $d$ aren't that good, but both are in $A^*$
- i.e. we should have taken $b$ instead them
Sources