(followup to: A correlated analogue to reflective oracles)
Motivation
Suppose players are playing in a correlated equilibrium using a reflective oracle distribution. How does the equilibrium they play in vary as a function of the parameters of the game, or of the players’ policies? It turns out that the set of equilibria is a Kakutani map of the parameters to the game. This is a lot like it being a continuous map.
This might make it possible for players to reason about the effects of their policy on the equilibrium that they play (since the equilibrium is now a Kakutani map of the players’ policies).
Definitions
Let \(k\) be some natural number. We will consider reflective oracle distributions whose queries are parameterized on some vector in \(\theta \in \mathbb{R}^k\).
To do this, let the Turing machines \(\mathcal{M}\), instead of outputting a raw query, output a continuous function from \(\theta \in \mathbb{R}^k\) to the query. (The details of representing continuous functions don’t seem that important). The reflectivity condition on oracle distributions is now relative to \(\theta\) (since the queries depend on \(\theta\)).
Define the map
\[\mathrm{ParamsToDistrs}(\theta) := \{D \in \mathcal{D}  D \text{ is reflective relative to $\theta$ }\}\]
which maps the parameters \(\theta\) to the set of reflective oracle distributions for \(\theta\).
Theorem: \(\mathrm{ParamsToDistrs}\) is a Kakutani map.
Proof:
From the previous post, we have:
 For each \(\theta\), \(\mathrm{ParamsToDistrs}(\theta)\) is nonempty (Theorem 1)
 For each \(\theta\), \(\mathrm{ParamsToDistrs}(\theta)\) is convex (Theorem 2)
So it is sufficient to show that \(\mathrm{ParamsToDistrs}\) has a closed graph.
Let \(\theta_1, \theta_2, ...\) be an infinite sequence of \(\mathbb{R}^k\) values converging to \(\theta_\infty\). Let \(D_1, D_2, ...\) be an infinite sequence of oracle distributions in \(\mathcal{D}\) converging to \(D_\infty\). Assume that for each natural \(n\), \(D_n\) is reflective relative to \(\theta_n\). We will show that \(D_\infty\) is reflective relative to \(\theta_\infty\); this is sufficient to show that \(\mathrm{ParamsToDistrs}\) has a closed graph.
Let \(M \in \mathcal{M}\). Let \(a\) be such that \(D_\infty(O(M) = a) > 0\). Then \(D_\infty(O(M) = a) > \epsilon\) for some \(\epsilon > 0\). By convergence, there is some \(N\) such that for all \(n \geq N\), \(D_n(O(M) = a) > \epsilon\). By reflectivity of each \(D_n\) relative to \(\theta_i\), we have, for each \(n \geq N\),
\[a \in \mathrm{Eval}(M)(\theta_n)(\mathrm{Condition}(D_n, M, a))\]
Let \(q^\theta := \mathrm{Eval}(M)(\theta)\). Rewriting the above statement:
\[a \in \arg\max_{i \in \{1, ..., l(q)\}} \mathbb{E}_{D_n}[q^{\theta_n}_i(O)]\]
Consider the set \(\{(\theta, D)  a \in \arg\max_{i \in \{1, ..., l(q)\}} \mathbb{E}_D[q^\theta_i(O)]\}\). This is the intersection of a finite number of sets of the form \(\{(\theta, D)  \mathbb{E}_D[q^\theta_a(O)  q^\theta_i(O)] \geq 0\}\). Each of these sets is closed because \((\theta, D) \mapsto \mathbb{E}_D[q^\theta_a(O)  q^\theta_i(O)]\) is continuous (so the preimage of the closed set \(\{x  x \geq 0\}\) is closed). Therefore the set \(\{(\theta, D)  a \in \arg\max_{i \in \{1, ..., l(q)\}} \mathbb{E}_D[q^\theta_i(O)]\}\) is closed.
In total, this is sufficient to show \(a \in \arg\max_{i \in \{1, ..., l(q)\}} \mathbb{E}_{D_\infty}[q^{\theta_\infty}_i(O)]\). Therefore \(D_\infty\) is reflective relative to \(\theta_\infty\), as desired.
\(\square\)
An immediate consequence of this theorem is that the set of correlated equilibria of a normalform game is a Kakutani map of the parameters of the game.
