Kasparov is an idiot on anything that isn't chess. You have to make a destinction between a current event like the crisis in Ukraine and a strategy like Russia's "zone of privileged interests".
To a crisis you react with tactics. Texas hold 'em describes that much better than chess.
A strategy on the other hand isn't a short term project. The Monroe doctrine is the key US strategy that tried to push any other powers out of the Americas. It was started in 1823. Obama claimed to have abandoned it in 2013. We will see.
China is also in a very long term project with its first and second island chains. The nine dashed line indea was made public in 1947 by the Republic of China. Both China's have adopted that idea today (yes both the PLC and Taiwan claim all or most of the SCS) They are currently in the sand castle building phase.
If you want to describe strategy as chess, then it is an extremely drawn out game - interupted by current events and played by different people over decades with variing degrees of competence and enthusiasm.
But, it's not a two player game; and it typically doesn't end in a complete victory. We have a much better game to describe strategy. It's called risk, and just like in real life you get a bonus for achieving hegemony of a region. The US as the only real hegemon is free to roam around and fuck shit up without having to fear any competitor at home. And just like in real life the other players try to keep a foothold in your region. To prevent you from getting the "bonus". Russia can't go attack Afghanistan for example if there are hostile tank divisions in Ukraine. The US on the other hand has nothing to fear from Mexico or Canada and if it doesn't work out they call it a day and go home.
Of course, reality is more complex than a game and Britain just went for hegemony over the sea, and could therefore ignore its own region while fucking shit up everywhere else. and the Soviet Union just thought that they could afford a war in Afghanistan.
It's actually not looking so bad for Putin is this slow motion game of risk. Belarus, Kazakhstan, Armenia are EEU members, and in the future Kyrgistan will join. Ukraine was pretty close, but that didn't work out, but he prevented a fast integration with the west, so the next Russian president can try again in part because the EU is and will be preoccupied with internal problems caused by a badly implemented common currency without automatic stabilizers.
The only scenario that chess describes somewhat OKish is a Napoleonic battle where the general stands on a hill and can see both his and the enemy's forces and make decisions accordingly. It's pretty useless in today's day and age where the OODA loop dominates everything and all the sides don't strife for a decisive victory on the battlefield, but information dominance which even with much smaller numbers leads to an almost automatic victory. It's chess where one side sees everything and the other can't even control its own troops. Checkmate right from the start.