Our goal in this subsection is to determine the vertices of resilience ω and ω+1, i.e., those from which Player 0 can win even under an unbounded, but finite number of disturbances, and under an infinite number of disturbances, respectively. We first show how to determine vertices of resilience ω+1.
Recall that, by definition, the resilience of a vertex is in ω+2. Moreover recall that we have determined the vertices with resilience less than ω in the previous section. Hence, by determining those vertices with resilience ω+1, we obtain that those ver- tices not determined to have finite resilience in the previous section, nor determined to have resilience ω+1 in this section, have resilience ω. Thus, we have then determined the resilience of all vertices.
Intuitively, recall that a vertex has resilience ω+1 if and only if Player 0 can win the game from that vertex even if infinitely many disturbances occur. Thus, we give Player 1 control over the disturbance edges, as there cannot be more than infinitely many disturbances during a play.
Example 5.20. Consider again the parity game with disturbances from : Figure 5.2.
: Sec. 5.1, Page 152
Intuitively, in order to give Player 1 control over the occurrence of disturbances, for each vertex v of Player 0, we add a vertex v, which belongs to Player 0, and we give the original vertex v to Player 1. Upon entering vertex v, Player 1 may then choose to traverse an outgoing disturbance edge of v, or he may decide to give control to Player 0 by moving to vertex v. From v, Player 0 can then pick a non-disturbance edge of v to continue along.
We show the game without disturbances resulting from giving control over the dis- turbances to Player 1 in Figure 5.3. We observe that the winning region of Player 0 corresponds to the vertices of resilience ω+1 in the game of Figure 5.2. Dually, the winning region of Player 1 corresponds to the vertices of resilience at most ω, i.e.,
the vertices vI through vVI with finite resilience identified in: Example 5.12 and the : Sec. 5.2, Page 156 vertices v0Iand v0II with resilience ω.
From v00I, Player 0 wins even if Player 1 controls whether the disturbance edge is traversed from v00I, as both v00I and v00II have color zero. On the other hand, giving Player 1 control over the disturbance edges lets him win from v0I, as he can use the disturbance edge incident to v0Iinfinitely often to move to v0II, which has color one. 4
In the following, we prove this intuition of handing control over the disturbance to Player 1 to be correct for determining the vertices of resilience ω+1. To this end, we transform the arena of the game so that at a vertex of Player 0, first Player 1 gets to chose whether he wants to take one of the disturbance edges and, if not, gives control to Player 0, who is then able to use a standard edge. The resulting game is a game without disturbances: Since all disturbances are now under the control of Player 1, the corresponding disturbance edges are normal edges in the resulting game.
Again, since we want to obtain a uniform approach to characterizing vertices of
resilience ω+1 that works for all winning conditions discussed in this work, we
consider an arbitrary game with disturbances (A, Win).
Given a game with disturbancesG = (A, Win) with A = (V, V0, V1, E, D), we de-
fine the rigged game Grig = (A0, Win0) with A0 = (V0, V00, V10, E0) such that V00 = Def.rigged game
{v |v∈ V0}, V10 = V, and V0 = V00 ∪V10. The set E0 of edges is the union of the following sets:
• D: Player 1 uses a disturbance edge.
• {(v, v) | v∈V0}: Player 1 does not use a disturbance edge and yields control to Player 0.
• {(v, v0) | (v, v0) ∈E and v∈ V0}: Player 0 has control and picks a standard edge. • {(v, v0) | (v, v0) ∈E and v∈ V1}: Player 1 takes a standard edge.
Further, Win0 = {ρ∈ (V0)ω |h(ρ) ∈Win} where h : (V0)ω → Vω is the homomor- phism induced by h(v) = v and h(v) = e for every v ∈ V. In particular, we con- struct Grig to be a game without disturbances. Hence, plays in Grig do not contain additional vertices denoting whether or not a disturbance edge has been taken.
We illustrate the construction of the rigged game from the parity game with dis-
turbances shown in : Figure 5.2 in Figure 5.3. The following lemma generalizes and : Sec. 5.1, Page 152
formalizes the observation of Example 5.20 that W0(Grig)characterizes the vertices of resilience ω+1 inG.
Lemma 5.21. Let v be a vertex of a game G. We have v ∈ W0(Grig)if and only if rG(v) =
ω+1.
Proof. We first show the implication from left to right. Let Player 0 winGrigfrom v, say with winning strategy σ0. In order to construct a strategy σ witnessing rG(v) =ω+1, we inductively translate play prefixes w inG into play prefixes t0(w)inGrigsatisfying the following invariant:
t0((v0, b0) · · · (vj, bj))starts in v0 and ends in vj
We begin by defining t0(v0, b0) =v0. To define t0((v0, b0) · · · (vj, bj) · (vj+1, bj+1)), we consider several cases:
• If bj+1 = 1, then the play traverses the disturbance edge (vj, vj+1), i.e., we have
(vj, vj+1) ∈D. We mimick this move by defining
t0((v0, b0) · · · (vj, bj) · (vj+1, bj+1)) =t0((v0, b0) · · · (vj, bj)) ·vj+1 .
• If bj+1 = 0 and vj ∈ V0, then we have (vj, vj+1) ∈ E, i.e., the play did not traverse a disturbance edge and instead allowed Player 0 to pick the standard edge(vj, vj+1)to traverse. We mimick this move by defining
t0((v0, b0) · · · (vj, bj) · (vj+1, bj+1)) =t0((v0, b0) · · · (vj, bj)) ·vj·vj+1 .
• If bj+1 =0 and vj ∈V1, then the play traversed the standard edge(vj, vj+1) ∈E. We mimick this move by defining
t0((v0, b0) · · · (vj, bj) · (vj+1, bj+1)) =t0((v0, b0) · · · (vj, bj)) ·vj+1 .
It is easy to see that the invariant is satisfied in any case. Also, we lift t0 to infinite plays by taking limits as usual.
Using this translation, we define a strategy σ for Player 0 inG via σ(v0· · ·vj) =σ0(t0((v0, b0) · · · (vj, bj)) ·vj) ,
where b0 = 0 and where for every j0 > 0, bj0 = 1 if and only if vj0 6= σ(v0· · ·vj0−1),
i.e., we reconstruct the consequential disturbances as described in: Section 5.1.1. A
: Page 151
straightforward induction shows that for every play ρ= (v0, b0)(v1, b1)(v2, b2) · · · inG that is consistent with σ, the play t0(ρ) is consistent with σ0. Hence, t0(ρ) ∈ Win0 for every ρ starting in v. Furthermore, we have h(t0(ρ)) = v0v1v2· · · ∈ Win, as t0(ρ) ∈ Win0. Thus, ρ = (v0, b0)(v1, b1)(v2, b2) · · · is winning for Player 0. As we have no restriction on the number of disturbances in ρ, the strategy σ is(ω+1)-resilient from v. Hence, rG(v) =ω+1, which concludes the proof of the implication from left to right. It remains to show the implication from right to left. To this end, let rG(v) = ω+1, i.e., Player 0 has an (ω+1)-resilient strategy σ from v in G. We define a winning strategy σ0 for Player 0 from v in Grig. This time, we inductively define a translation t of play prefixes in Grig into play prefixes in G. Since all vertices in G correspond to vertices of Player 1 inGrig, it suffices to consider those prefixes that start and end in V10. For these, we construct t to satisfy the following invariant:
If π starts in v0 and ends in vj, then t(π) starts in v0 and ends in vj as well.
Recall that Grig is a game without disturbances. Thus, plays in Grig do not contain bits indicating whether a disturbance edge has been traversed and we have to recon- struct them from the traversal of the vertices v and v. We define t(v0) = (v0, 0) and consider several cases for the inductive step:
• First, assume we have a prefix of the form v0· · ·vj·vj+1 for some vj ∈ V0 ⊆ V10, i.e., the move of Player 1 simulates traversing the disturbance edge(vj, vj+1) ∈D. Then, we define
t(v0· · ·vj·vj+1) =t(v0· · ·vj) · (vj+1, 1) .
• Next, assume we have a prefix of the form v0· · ·vj·vj+1 for some vj ∈V1 ⊆V10, i.e., the move of Player 1 simulates traversing the standard edge (vj, vj+1) ∈ E. Then, we define
t(v0· · ·vj·vj+1) =t(v0· · ·vj) · (vj+1, 0) .
• Finally, the last case is a prefix of the form v0· · ·vjvj·vj+1for some vj ∈V00, i.e., the move of Player 0 simulates traversing the standard edge(vj, vj+1) ∈E. Then, we define
t(v0· · ·vjvj·vj+1) =t(v0· · ·vj) · (vj+1, 0) .
The invariant is satisfied in any case. Also, we again lift t to infinite plays via limits. Now, we define a strategy σ0 for Player 0 inGrigvia
σ0(v0· · ·vjvj) =σ(t(v0· · ·vj)) ,
where, for the sake of notational convenience, we assume that σ ignores the bits indi- cating whether or not a disturbance edge was taken present in t(v0· · ·vj).
The restriction to play prefixes ending in a vertex of the form v suffices, as these are the only vertices of Player 0 in Grig. A straightforward induction shows that for every play ρ that is consistent with σ0, the play t(ρ)is consistent with σ. Hence, t(ρ) satisfies the winning condition if ρ starts in v, as σ is (ω+1)-resilient from v. Let t(ρ) = (v0, b0)(v1, b1)(v2, b2) · · ·. Then, v0v1v2· · · ∈ Win. Now, h(ρ) = v0v1v2· · · implies ρ∈ Win0. Thus, σ0 is a winning strategy for Player 0 from v, which concludes the implication from right to left.
In the proof of Lemma 5.21 we construct an ω+1-resilient strategy for Player 0 inG
from a winning strategy for her inGrig. It is easy to see that this construction preserves positionality, which gives rise to the following corollary.
Corollary 5.22. If Player0 has a positional winning strategy that is winning for her from all
vertices in W0(Grig), then she has a positional strategy that is ω+1-resilient from all vertices v inG with rG(v) =ω+1.
At this point, we are able to compute the resilience of those vertices v with rG(v) ∈ω due to Lemma 5.17 and Lemma 5.18. Furthermore, we can identify the vertices v with rG(v) = ω+1 due to Lemma 5.21. Since, for each vertex v, we have rG(v) ∈
ω∪ {ω, ω+1}, all vertices not characterized by the statements above have resilience ω.
Thus, we have shown how to determine the resilience of all vertices in G and how
to construct strategies witnessing the respective resilience out of winning strategies for variants of the underlying game without disturbances. In the following section, we show how to combine these results to effectively compute an optimally resilient strategy for Player 0 inG.