Playing with pebbles on the multiverse

Please find a recording below of a mini-talk I gave during Set Theory in The United Kingdom @ Home on December 4, 2020. We generalize Ehrenfeucht–Fraïssé games from classic model theory to modal model theory with actuality.

Please note that it has come to my attention during the talk that Sam Adam-Day had considered modal Ehrenfeucht–Fraïssé games (without actuality) independently. I am grateful to him for our discussion after the talk, which helped me prove the following:

Player II has the winning strategy in the modal pebble game with actuality on structures $M$ and $N$ just in case these structures are equivalent with respect to the infinitary modal language with actuality $\mathcal{L}^{\Diamond,@}_{\infty,\omega}$.