Lecture 2: Büchi condition
Teachers: Wieslaw Zielonka
Games and Graphs
- $G ≝ (V, E)$
- $V ≝ V_A \sqcup V_B$
- $A$: Alice, $B$: Bill
- $Paths$:
-
the set of finite paths in $G$
- $Paths^∞$:
-
the set of infinite paths
- $W_A ⊆ Paths^∞$:
-
the set of infinite paths winning for Alice
- $W_B = Paths^∞ \backslash W_A ⊆ Paths^∞$:
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the set of infinite paths winning for Bill
- Strategy $σ$ for Alice:
-
is a mapping $σ: V^\ast V_A ⟶ V$ such that \(∀ x \, ≝ \, v_1 ⋯ v_n ∈ V^\ast V_A \qquad (v_n, σ(x)) ∈ E\)
- $(σ, τ)$-strategy profile:
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if $σ$ (resp. $τ$) is strategy for Alice (resp. Bill).
- A path $p ≝ v_1 ⋯ v_n$ is consistent with the strategy $σ$ of Alice:
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if for each $i$ such that $v_i ∈ V_A$, $v_{i+1} = σ(v_1 ⋯ v_i)$
If we fix an initial vertex $v ∈ V$, then there exists a unique infinite path $p(v, σ, τ)$ which starts at $v$ and is consistent with $σ$ and $τ$.
- $σ$ is a winning strategy for Alice for the initial vertex $v$:
-
if each infinite path $p$ strating at $v$ and consistent with $σ$ belongs to $W_A$.
For a given initial vertex $v$, $σ, τ$, or none of them, may be a winning strategy.
- The game is determined:
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if for each initial vertex, one of the players has a winning strategy.
Memoryless/positional strategies
- The strategy $σ$ is memoryless (or positional):
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iff for all $x ≝ yv ∈ V^\ast V_A$: \(σ(x) = σ(v)\)
Ex: If $σ$ and $τ$ are memoryless and the graph is finite: all the plays are “lasso” loops in the graph.
Normally, when considering games on graphs, one looks for a partition:
\[V = \underbrace{Win_A}_{\text{vertices winning for Alice}} \sqcup Win_B \sqcup (V \backslash (Win_A ∪ Win_B))\]Moreover, usually: people look for memoryless strategy that are independent on the initial vertex.
- $OneStep_A(X)$:
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set of vertices such that, strating from one of these vertices, Alice can reach $X$ in exactly one move.
\[OneStep_A(X) \, ≝ \, \lbrace v ∈ V_A \; \mid \; ∃ w ∈ X; (v, w)∈E \rbrace\\ ∪ \lbrace v ∈ V_B \; \mid \; ∀w ∈ V, (v,w)∈E ⟹ w ∈ X \rbrace\]
and:
\[\begin{cases} X_0 ≝ X \\ X_{i+1} ≝ \overline{OneStep}_A(X_i) \end{cases}\]Then
\[X ⊆ \overline{OneStep}_A(X)\]and if $G$ is finite, there exists $k$ such that $X_{k+1} = X_k$.
\[Reach_A(X) \; ≝ \; \bigcup_i X_i\]For all $v ∈ V$,
\[rank(v) \; ≝ \; \min \lbrace i \; \mid \;v ∈ X_i \rbrace \qquad \min ∅ ≝ ∞\]Let $v ∈ Reach_A(X)$ such that $rank(v) > 0$.
If
- $v ∈ V_A$, then there exists $w$ s.t. $(v, w) ∈ E$ and $rank(w) < rank(v)$
- $v ∈ V_B$, then for all $w$ s.t. $(v, w) ∈ E$, $rank(w) < rank(v)$
Büchi condition
Büchi: Alice wins iff the play visits infinitely often a fixed set $R$ of vertices.
If $X ⊆ Y$:
- $OneStep_A(X) ⊆ OneStep_A(Y)$
- $Reach_A(X) ⊆ Reach_A(Y)$
So the following mapping is clearly monotone: $Y ⟼ OneStep_A(Reach_A(Y))$
And on top of that:
\[Reach_A(Y) \backslash Y ⊆ OneStep_A(Reach_A(Y)) ⊆ Reach_A(Y)\]NB: $Reach_A(Y) \backslash Y$ because there may exist some $v ∈ Y$ s.t. $v ∉ OneStep_A(Reach_A(Y))$, that is: for which Bill has a strategy to leave $Reach_A(Y)$ in one step.
\[ϕ(Y) \; ≝ \; OneStep_A\Big(Reach_A( \overbrace{Y ∩ R}^{\rlap{\text{fixed set of vertices}}})\Big)\]Let $W$ s.t. $ϕ(W) = W$. Then Alice has a strategy to visit $R$ infinitely often.
(Indeed:
\[Reach_A(W ∩ R) \backslash (W ∩ R) ⊆ W = OneStep_A(Reach_A((W ∩ R))) ⊆ Reach_A((W ∩ R))\]so $Reach_A(W ∩ R) \backslash (W ∩ R) = ∅$, and $Reach_A(W ∩ R) ⊆ W ∩ R$)
Let
\[Y_0 ≝ V, \; Y_1 ≝ ϕ(Y_0), \; … , \, Y_{i+1} = ϕ(Y_i)\]Then by induction: this sequence is decreasing: $Y_{i+1} ⊆ Y_i \; ∀i$.
So $\bigcap\limits_{i} Y_i$ is a fixed point of $ϕ$. And it is the greatest fixed point. Indeed, if $W$ is another such fixed point. Then $W ⊆ V$, and by applying $ϕ$ over and over, we get the result.
\[Y_{i+1} ∪ (Y_i ∩ R) = Reach_A(Y_i ∩ R)\]Path consistent with Bill’s strategy at $Y_i \backslash Y_{i+1}$
\[p = v_1 ⋯\]- either $∀ i ≥ 2$, $v_i ∉ R$
- or there exists $j≥2$ s.t. $v_j ∈ Y_i^c$ and all vertices $v_2, …, v_{j-1}$ do not belong to $R$.
Forgetting the first moment, the index corresponds to the maximal number of times that Alice can force the visit to $R$.
Example
Imagine a pie that two players share to eat.
- Player 1 proposes a proportion $x_1$ of the pie to player 2
- Player can either accept, in which the proportions gotten are $(1-x_1, x_1)$, or rejects.
- etc… (as described below)
digraph {
rankdir=TB;
1 -> 2[label="P1 proposes x_1"];
2 -> "(1-x_1, x_1)"[label="P2 accepts"];
2 -> 0[label="P2 rejects"];
0 -> "(0, 0)"[label="1 - δ"];
0 -> "2'"[label="δ"];
"2'" -> "1'"[label="P2 proposes x_2"];
"1'" -> "(x_2, 1-x_2)"[label="P1 accepts"];
"1'" -> "0'"[label="P1 rejects"];
"0'" -> 1[label="δ"];
"0'" -> "(0, 0)'"[label="1-δ"];
}
With proba 1, the game stops.
Nash equilibria: for each $0 ≤ x ≤ 1$, the proportion $(x, 1-x)$ is a Nash equilibrium. Indeed: P1 never accepts less than $x$, and never proposes more than $1-x$. Symmetrically for P2.
Can we find a subgame-perfect equilibrium? (where each time P1 (resp. P2) plays, he/she makes the same proportion).
Backward induction: begin at node $1$ and go backward (along the $δ$ arrow):
- Expectation for each player: $(z, 1-z)$
- At $0’$, expectation: $(1-δ)×(0, 0) + δ(z, 1-z) = (δz, δ(1-z))$
- At $2’$: P2 proposes $(zδ, 1-zδ)$, so that P1 accepts and P2’s payoff is $1-zδ > (1-z)δ$
- etc…
digraph {
rankdir=TB;
"1 | (1-(δ-zδ²), δ-zδ²)" -> 2[label="P1 proposes x_1"];
2 -> "(1-x_1, x_1) | (1-(δ-zδ²), δ-zδ²)"[label="P2 accepts"];
2 -> "0 | (zδ², δ-zδ²)"[label="P2 rejects"];
"0 | (zδ², δ-zδ²)" -> "(0, 0)"[label="1 - δ"];
"0 | (zδ², δ-zδ²)" -> "2' | (zδ, 1-zδ)"[label="δ"];
"2' | (zδ, 1-zδ)" -> "1'"[label="P2 proposes x_2"];
"1'" -> "(x_2, 1-x_2) | (zδ, 1-zδ)"[label="P1 accepts"];
"1'" -> "0' | (zδ, (1-z)δ)"[label="P1 rejects"];
"0' | (zδ, (1-z)δ)" -> 1[label="δ"];
"0' | (zδ, (1-z)δ)" -> "(0, 0)'"[label="1-δ"];
}
So now:
\[(z, 1-z) = (1-δ+zδ^2, δ-zδ^2)\]Therefore:
\[z = \frac{1-δ}{1-δ^2} = \frac{1}{1+δ}\]
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