theorem
staged
Sure Thing Principle
The sure-thing principle says that a common consequence in a lottery mixture does not affect strict preference between the non-common parts.
If independence holds, then for any lotteries $L_1,L_2,L_3,L_4$ and $\alpha\in[0,1]$, $$ [\alpha L_1,(1-\alpha)L_3] \succ [\alpha L_2,(1-\alpha)L_3] $$ if and only if $$ [\alpha L_1,(1-\alpha)L_4] \succ [\alpha L_2,(1-\alpha)L_4]. $$
Proof
Sketch
When $\alpha=0$, both sides compare a lottery with itself and strict preference is impossible. When $\alpha>0$, apply independence once to remove $L_3$ and once to insert $L_4$.
References
- [MSZ, Chapter 2, Exercise 2.12] Maschler, Solan, and Zamir, Game Theory. The sure-thing principle follows from independence.