= Friedman–Moschovakis coding game
{c}
The <Friedman–Moschovakis coding game> turns strategies into codes for sets of ordinals. An explicit $\omega_1$ instance makes its mechanism visible. Fix $A\subseteq\omega_1$. Let a real $y$ code a countable <well-order> together with a marked subset of its domain. Write $E(y,z)$ when the rank of the <well-order code> $z$ is the rank of a marked element in that coded order. A suitable $y$ can therefore code $A\cap\eta$ for any particular countable ordinal $\eta$.
Players I and II produce reals $x,y$. Player II wins if $x\notin\mathrm{WF}$, or if $y$ correctly codes membership in $A$ through rank $\lVert x\rVert$: for every $z\in\mathrm{WF}$ with $\lVert z\rVert\leq\lVert x\rVert$,
$$
E(y,z)\ \Longleftrightarrow\ \lVert z\rVert\in A.
$$
A winning strategy for Player I would produce only well-order codes. Its possible outputs form an <analytic set>, so their ranks are bounded by the <boundedness theorem for well-order codes>. Player II could then play one real coding $A$ through a larger countable rank, defeating the strategy. Under the <axiom of determinacy>, Player II consequently has a winning strategy.
Such a strategy determines at most one $A$: play any code of an ordinal $\xi$ against it and inspect its response at rank $\xi$. Strategies themselves are coded by reals, giving the coding consequence for subsets of $\omega_1$. The general <Friedman–Moschovakis coding lemma> replaces these countable-order codes and analytic boundedness by the corresponding uniform coding and covering machinery; independently choosing unrelated codes for every initial segment does not supply that uniformity.<Descriptive Set Theory by Yiannis N. Moschovakis (2009)>[§7D, Coding Lemmas7D.5–7D.6 and their game-theoretic applications]{ref}
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