I'm reading the book "Multiagent systems: Algorithmic, game theoretic and logical foundations" by Leyton-Brown and Shoham. Since this is a game theory question I thought it was best to ask it here.
My question concerns the use of the convex hull of a set of strategies in the definition of rationalizable strategies. The authors make a point that we could not have used the set of all probability distributions over a set of strategies instead, but I am unable to understand the point they are making. Here's an excerpt:
We now define rationalizability more formally. First we will define an infinite sequence of (possibly mixed) strategies $S_i^0, S_i^1, S_i^2,...$ for each player $i$. Let $S_i^0 = S_i$; thus, for each agent $i$, the first element in the sequence is the set of all $i$'s mixed strategies. Let $CH(S)$ Denote the convex hull of a set $S$: the smallest convex set containing all the elements of $S$. Now we define $S_i^k$ as the set of all strategies $s_i \in S_i^{k-1}$ for which there exists some $s_{-i} \in \Pi_{j\neq i} CH(S_j^{k-1})$ such that for all $s_i' \in S_i^{k-1}, u_i(s_i, s_{-i}) \geq u_i(s'_i, s_{-i})$. That is, a strategy belongs to $S_i^k$ if there is some strategy $s_{-i}$ for the other players in response to which $s_i$ is at least as good as any other strategy from $S_i^{k-1}$. The convex hull operation allows i to best respond to uncertain beliefs about which strategies from $S_j^{k-1}$ player $j$ will adopt. $CH(S_j^{k-1})$ is used instead of $\Pi(S_j^{k-1})$, the set of all probability distributions over $S_j^{k-1}$, because the latter would allow consideration of mixed strategies that are dominated by some pure strategies for $j$. Player $i$ could not believe that $j$ would play such a strategy becase such a belief would be inconsistent with $i$'s knowledge of $j$'s rationality.
Now we define the set of rationalizable strategies for player $i$ as the intersection of the sets $S_i^0, S_i^1,S_i^2...$.
I have highlighted the part I don't understand in bold.
My reasoning is as follows: Say that a set of strategies $S = \{s, s' \}$. Each strategy is just a probability distribution over actions, so they can be seen as vectors containing real numbers in $[0,1]$, with $L_1$ norm $1$. Then as far as I understand $CH(S) = \{ \lambda s + (1-\lambda)s' : \lambda \in [0,1] \}$ while the set of probability distributions $\Pi(S) = \{\mathcal{P}: \mathcal{P}(s) = \lambda, \mathcal{P}(s') = 1-\lambda, \lambda \in[0,1] \}$. Ignoring the fact that the sets do not contain objects of the same type, I don't see the practical difference: there is a bijection between $\Pi(S)$ and $CH(S)$. Why then would $\Pi(S_j^{k-1})$ allow consideration of mixed strategies that are dominated by some pure strategies for $j$, while this apparently is not the case when using $CH(S_j^{k-1})$?
Note: I have read the answer to this related question about why the convex hull is needed in the definition, but it does not touch the difference between $CH(S_j^{k-1})$ and $\Pi(S_j^{k-1})$.
