Covering of an infinite game

A covering consists of a new game tree, a length-preserving prefix map to the original tree, and maps taking strategies of each player to strategies of that same player. Every play following a projected strategy lifts to a play following the original strategy, with identical projection or a premature terminal loss for that strategy's player. Terminal outcomes inherited from the original tree are respected. The strategy maps are local in depth. These conditions transfer every winning strategy, for any payoff pulled back by the prefix map. A $k$-covering leaves all positions, outcomes and strategy choices through depth $k$ unchanged.

In a sequence of game coverings whose unchanged initial depths tend to infinity, each finite game-tree level eventually stabilizes. These stable levels form a limit game. Compositions of prefix projections and depth-local strategy maps give limit projections and strategy maps. Lift a compatible play coherently through successive coverings. If no premature loss occurs, stabilization of its finite prefixes supplies a compatible limit play. If a premature loss occurs, subsequent lift lengths cannot increase and eventually stabilize, supplying a losing terminal limit play. This proves the lifting condition for the limit covering. It makes countably many pulled-back payoffs clopen simultaneously in the covering proof of Borel determinacy theorem.

A covering of an infinite game unravels a payoff when its inverse image is a clopen subset of the new infinite-play space. A closed payoff can be unravelled by letting Player I announce a subset of the first exit positions, and letting Player II accept that announcement or challenge one announced position. Accepted plays stop at exit positions, while challenged plays are forced through the challenged exit. Strategy-transfer maps make this more than an ordinary continuous representation of the payoff. Depth-stabilizing inverse limits of coverings handle countable unions in a transfinite induction.

After a fixed even-depth prefix, Player I announces a move and a set $X$ of first exit positions from a closed set of infinite plays. Player II accepts, or challenges an exit $r\in X$. Acceptance follows the old game until the first exit, where I wins exactly when that exit belongs to $X$. A challenge forces play through $r$ and then resumes the old game. Infinite accepted plays remain in the closed payoff; infinite challenged plays are outside it. Thus the pulled-back payoff is clopen. For an I-strategy, transfer by initially accepting and switching to the challenged simulation at an exit in $X$. For an II-strategy, announce the set of exits that this strategy never challenges under any announcement; it must accept this set. An exit outside it permits a challenged simulation, while an exit inside it is a premature loss for II. These maps prove the game covering lifting condition, rather than merely continuity of the projection.