User:SlamDiego/Independence (decision theory)

Ramsey
/Something surely goes here./

Strong Independence Axiom
A slightly stronger propostion than Savage's Sure-Thing Principle is the Strong Indepence Axiom:
 * Alternative a is weakly preferred (i.e., preferred or indifferent) to alternative b if and only if the lottery which consists of alternative a with probability p and alternative  c with probability 1 - p is weakly preferred to the lottery where b is obtained with probability p and c is obtained with probability 1 - p, for all positive probabilites p and all alternatives c.

Formal statement
Let a lottery with possible outcomes $$X_1$$, $$X_2$$, et cetera, each $$X_n$$ having associated probability $$p_n$$, be denoted
 * $$\left\langle(X_1 ,p_1 ),(X_2 ,p_2 ),\ldots\right\rangle$$

strict preference be denoted with “$$\succ$$” and indifference with “$$\sim$$”. Then the strong independence axiom is
 * $$(X\succcurlyeq Y)~\Leftrightarrow~\left\{[\left\langle(X,p),(Z,1-p)\right\rangle\succcurlyeq\left\langle(Y,p),(Z,1-p)\right\rangle]\Leftarrow(p>0)\right\}\forall(X,Y,Z,p)$$

In the context of quantified utility, strong independence implies that the utility of a lottery is linear in the probabilities:
 * $$U\left\langle(X_1 ,p_1 ),(X_2 ,p_2 ),\ldots\right\rangle~=~\sum_{i\,=\,1}^n [p\cdot U(X_i )]$$

Separability
Let weak preference ($$\succ\cup\sim$$) be denoted with “$$\succcurlyeq$$”.

Mixture separability
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Replacement separability
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Monotonicity
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Substitutibility
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Weak Independence Axiom

 * $$(X\sim Y)~\Leftrightarrow~[\left\langle(X,p),(Z,1-p)\right\rangle\sim\left\langle(Y,r),(Z,1-r)\right\rangle]\exists r\forall(X,Y,Z,p)$$