# Convex hull of strategies vs set of all probability distributions over strategies in definition of rationalizable strategies

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 $$\Pi(S_j^{k-1})$$.

• I suspect what they mean is that they consider $\prod_{j\neq i}CH(S_j^{k-1})$ and not $CH\left(\prod_{j\neq i}S_j^{k-1}\right)$. These are different sets where the latter allows for correlation between players.
– user42421
Commented Nov 14, 2022 at 16:52