For any two structures $\mathcal{M}$ and $\mathcal{N}$ in the same first-order language $\mathcal{L}$ and any ordinal $\theta$, let $G_\theta(\mathcal{M},\mathcal{N})$ be the two-player game of perfect information of length $\theta$ such that in round $\alpha$ player I plays an element $x_\alpha\in \mathcal{M}$ and then player II plays an element $y_\alpha \in \mathcal{N}$, and player II wins if and only if the augmented structures $(\mathcal{M}, x_\alpha)_{\alpha < \theta}$ and $(\mathcal{N}, y_\alpha)_{\alpha < \theta}$ have the same theory in the language obtained from $\mathcal{L}$ by adding $\theta$ constant symbols.

Note that if there is an elementary embedding $j$ of $\mathcal{M}$ into $\mathcal{N}$ then player II has winning strategies in $G_\theta(\mathcal{M},\mathcal{N})$ for all $\theta$ obtained by letting $y_\alpha = j(x_\alpha)$. Conversely, if player II has a winning strategy in $G_\theta(\mathcal{M},\mathcal{N})$ where $\theta$ is the cardinality of $\mathcal{M}$ then we may obtain an elementary embedding of $\mathcal{M}$ into $\mathcal{N}$ by letting $j(x_\alpha) = y_\alpha$ where $(x_\alpha:\alpha < \theta)$ is an enumeration of $\mathcal{M}$. The existence of a winning strategy for player II in the case $\theta = \omega$ is weaker: it is equivalent to the existence of an elementary embedding of $\mathcal{M}$ into $\mathcal{N}$ in every generic extension of $V$ by the poset $\text{Col}(\omega,\mathcal{M})$.

Let us define a cardinal $\kappa$ to be *$\theta$-strategically measurable* if
there is an ordinal $\kappa'$ and a transitive set $N$ such that $\kappa < \kappa' \in N$ and letting $\mathcal{M} = (H_{\kappa^+}; \mathord{\in},\kappa, \xi)_{\xi < \kappa}$ and $\mathcal{N} = (N; \mathord{\in}, \kappa', \xi)_{\xi < \kappa}$, player II has a winning strategy in the game $G_\theta(\mathcal{M},\mathcal{N})$.

Remarks:

$2^\kappa$-strategic measurability is equivalent to measurability.

If $0^\sharp$ exists then every Silver indiscernible is $\omega$-strategically measurable in $L$. (Use $j:L \to L$ and the absoluteness of existence of winning strategies for closed games of length $\omega$).

Every $(\omega+1)$-strategically measurable cardinal is a Ramsey cardinal and a limit of Ramsey cardinals. (Proof given below.)

Questions:

- Is ($\omega+1$)-strategic measurability equivalent to measurability?
- If not, what is its consistency strength?
- Have these games been studied before for $\theta > \omega$? Do they have a name?

Proof of remark 3:

We use a winning strategy for player II in the game $G_{\omega+1}(\mathcal{M},\mathcal{N})$ (or just the nonexistence of a winning strategy for player I) to build increasing sequences $(M_n, n<\omega)$ and $(\mu_n, n<\omega)$ such that $M_n \prec H_{\kappa^+}$, $\left| M_n\right| = \kappa$, $\mu_n$ is an $M_n$-normal ultrafilter on $\mathcal{P}(\kappa)\cap M_n$, and $M_n, \mu_n \in M_{n+1}$. In round $n$ player I plays an enumeration of $\mathcal{P}(\kappa)\cap M_n$ in order type $\kappa$ and uses player II's response to define $\mu_n$. Then letting $M_\omega = \bigcup_{n<\omega}M_n$ and $\mu_\omega = \bigcup_{n<\omega} \mu_n$, we see that $\mu_\omega$ is a weakly amenable $M_\omega$-normal ultrafilter on $\mathcal{P}(\kappa)\cap M_\omega$. Moreover it is countably complete (in the sense of nonempty intersection) because otherwise player I can win by playing a counterexample to countable completeness in round $\omega$. Since we can take $M_0$ to contain any given subset of $\kappa$, it follows that $\kappa$ is a Ramsey cardinal. Because $M_\omega \prec H_{\kappa^+}$ we can reflect Ramseyness below $\kappa$.

(See Gitman, Ramsey-like cardinals for the relationship between Ramsey cardinals and weakly amenable countably complete ultrafilters.)

Further remarks:

- The proof of Ramseyness outlined above resembles the "filter games" introduced by Holy and Schlicht and further studied by Nielsen and Welch.
- Defining ($\omega+1$)-strategic strongness in an analogous way, I think I can prove that it is equiconsistent with strongness. (
~~And again the argument only requires the nonexistence of a winning strategy for player I.~~*The previous sentence is wrong; I will add more details later.*) However, the proof goes through ${\bf\Sigma}^1_4$ generic absoluteness and does not seem to generalize to other strategic large cardinals.