I have a difficulty in understanding a given definition for convex independence:
A set of beliefs $\beta$ of an agent $i$ satisfy convex independence if beliefs of no type $t_i$ can be represented as a convex combination of other types $t'_i \neq t_i$
I read the Wikipedia article about convex combinations and I think I get what it means. Thus, also the definition of convex independence should be clear but I can't make sense of the given example:
example of convex independence
- why does convex independence fail for the bottom example? I can't see how I can create a convex combination (of the red row?) to match another row (belief)
- what is the fastest way to check for convex independence in such a case? Checking each row vs. the other ones would take quite some time
Notation:
$\beta_i(.) = \pi_j(.)$: belief of player i (expressed as a probability) about player $j$ being of a certain type given his own type
$\theta_1^i$: type „$\theta_1$“ of player $i$
The example you show has a failure of convex independence because you can write $\beta_i(\theta^i_2) = (\beta_i(\theta^i_3) + \beta_i(\theta^i_2))/2$. Ignoring the obscure economic decision theory notation, what they say is simply that $(1/5, 3/10, 5/10) = ((1/5, 1/5, 3/5) + (1/5, 2/5, 2/5))/2$. The fastest way to check for independence by hand is by Gaussian elimination.