To begin with, the analogy is massively flawed: like there is a "ball" and one side can only "get ready" while the other prepares to "serve". And then it goes one side OR the other until one side "wins" and the other "loses" - everything.
Making sure we don't have a nuclear war isnt a game. And if you want to game it, it should be more like like some constructive endeavor where the result isn't zero sum. If this goes zero sum, even by accident, it will be zero sum for billions who arent' privy to the benefits and private jet planes..only to the dying.
The worst thing with a bad analogy is it cripples the imagination and paints people into corners where stupid things take precedence over longer term, win-win solutions.
As you ought to know, there is a MOUNTAIN of evidence that it was the United States that interrupted the negotiation process which Russia and Ukraine had begun on their own early in the spring.
There are witnesses that Boris Johnson, acting apparently at the nudging of the United States and its NATO puppet "partnership" suddenly dropped everything and went to talk directly with Zelinsky. Within days, Zelinsky /Ukraine clearly changed its position. Shortly thereafter, the regular diplomatic meetings stopped and the surge of BILLIONS from the US to ENCOURAGE prolonging the war (and probably padding Zelinsky's pockets pretty well) increased.
Anyway, your comment is totally inappropriate, it seems to me. A horrible model hobbles the imagination. This isn't a "game". Even if you want to argue it is, it isn't a game where "tit for tat" and lobbing a ball back and forth until one party loses and the other takes all is an appropriate metaphor.
That said, I'd still like to know why you'd say "ball is in Putin's court". Is that because the US will fry the planet and make humanity go extinct before it will deprive our "Military Industrial Complex" /Chicken Hawks of expanded profits? Is that the game? Is that the "ball"? Is that appropriate?
Why don't we use cooperating to grow a garden as the metaphor, instead? Would that open other possibilities for you?