# If a rational preference relation over simple lotteries $\succsim$ are convex then they satisfy independence?

Let´s say there is an uncertain situation with $$N$$ possible consequences $$C = \{C_1, . . . C_N\}$$. Assume that there is a rational preference relation $$\succsim$$ over simple lotteries.

I know that if $$\succsim$$ satisfies independence, then it is also convex, but is it true if $$\succsim$$ are convex then they satisfy independence? How can I show this (if the implication is true)

1. It's well known that if $$\succsim$$ satisfies independence, then it is also convex.

Since $$\succsim$$ satisfies independence, $$L\succsim L^{'} \iff \alpha L+(1-\alpha)L^{''}\succsim \alpha L^{'}+(1-\alpha)L^{''}$$ for all $$\alpha \in [0,1]$$ and $$L, L^{'}, L^{''}\in \mathfrak{L}$$

Convexity requires:

$$L\succsim L^{''}$$ and $$L^{'}\succsim L^{''} \Longrightarrow \alpha L$$ + $$(1-\alpha)L^{'} \succsim L^{''}$$ for all $$\alpha \in \left[0,1\right]$$ and $$L, L^{'}, L^{''}\in \mathfrak{L}$$

Pick any $$L, L^{'}, L^{''}\in \mathfrak{L}$$ with $$L \succsim L^{''}$$ and $$L^{'} \succsim L^{''}$$. By completeness, $$L \succsim L^{'}$$ or $$L^{'} \succsim L$$ or both. Without loss of generality, assume $$L \succsim L^{'}$$. Then, by independence, for all $$\alpha \in [0,1]:$$

$$\alpha L$$ + $$( 1- \alpha) L^{'} \succsim \alpha L^{'} + (1-\alpha)L^{'} = L^{'} \succsim L^{''}$$, which we wanted to show.

1. Is it true if $$\succsim$$ are convex then they satisfy independence?

On the other side, convexity does not imply independence. To see this, take a look at Figure (see below). Triangles are indifference curves and arrows show the direction in which utility increases. It is convex, but not independent.