Lecture 12 Games on Graphs

The same “position or configuration” might recur in the game, but the (infinite) game tree does not reflect this.

Game graphs and their trees

A 2-player game graph, $G = (V, E, pl)$ consists of:

A (finite) set $V$ of vertices.
A set $E \subseteq V\times V$ of edges.
A partition $(V_1, V_2)$ of the vertices $V = V_1 \cup V_2$ into two disjoint sets belonging to players 1 and 2, respectively.

A game graph $G$ together with a start vertex $v_0 \in V$, defines a game tree $T_{v_0}$ given by:

Action alphabet $\Sigma = V$. Thus $T_{v_0} \subseteq V^*$.
$\epsilon \in T_{v_0}$, and $wv^{\prime\prime} \in T_{v_0}$, for $v^{\prime\prime} \in V$, if and only if
- $w = \epsilon$ and $(v_0, v^{\prime\prime}) \in E$, or
- $w = w^{\prime\prime}$, for some $v^\prime\in V$, and $(v^\prime, v^{\prime\prime})\in E$.
(If last vertex of any node $w$ is $v^\prime$, it have action $v^{\prime\prime}$ if and only if $(v^\prime, v^{\prime\prime})\in E$.)

Games on graphs

A game on graph, $\mathcal{G}_{v_0}$, is given by:

A finite game graph $G$, vertex $v_0\in V$, and payoff function $u: \Psi_{T_{v_0}} \mapsto \mathbb{R}$.

These together define a 2-player zero-sum PI-game with game tree $T_{v_0}$.

Note: Finite graphs are not in general determined.

History oblivious payoff

Suppose $\exist$ vertex $v^\prime$ of graph $G$ that is a “dead end”. E.g., in chess this could be “checkmate for Player I”.

There may be many ways to get to $v^\prime$, but the winner is the same for any finite play $wv^\prime \in V^\ast$. I.e., $u(wv^\prime) = u(w^\prime v^\prime)$, for all $wv^\prime, w^{\prime\prime} \in V^\ast$. So, the pay off is “history oblivious”.

What about for infinite plays $\pi$? We can think of $\pi$ as an infinite sequence $v_0v_1v_2\cdots$, where each $v_i \in V$. We use the notation $\pi \in V^\omega$.

For $\pi = v_0v_1v_2\cdots$, let

\[\inf(\pi) = \lbrace v\in V \mid \text{ for }\infty\text{-many } i \in \mathbb{N}, v_i = v\rbrace\]

\[\inf(\pi) = \lbrace v\in V \mid \forall n\in \mathbb{N}, \exists j\ge n, v_j = v \rbrace\]

I.e., the entry of the cycle in the infinite play $\pi$.

Let’s call payoff function $u()$ history oblivious (h.o.), if for all infinite plays $\pi\ \&\ \pi^\prime$, if $\inf(\pi) = \inf(\pi^\prime)$, then $u(\pi) = u(\pi^\prime)$, and for all finite complete plays $wv$ and $w^\prime v$,

\[u(wv) = u(w^\prime v)\]

Call a graph game h.o. if its payoffs are h.o.. $w$ and $w^\prime$ represent histories here.

Finitistic payoffs

Note that in chess, if the play $\pi$ is infinite, then the play is always a draw, i.e., $u(\pi) = 0$.

Let’s call an h.o. payoff function finitistic if for ALL infinite plays $\pi$ and $\pi^\prime$, $u(\pi) = u(\pi^\prime)$. Let’s call game on a graph $\mathcal{G}_{v_0}$ finitistic if its payoff function is.

So, in win-lose-draw finitistic games, infinite plays are either all wins, all losses, or all draws, for player 1.

Therefore, all finitistic games on graphs determined. In fact, more it true, for finitistic games there is always a memorylessly strategy for each player that achieves the value of the game, and we can efficiently compute these strategies.

Memoryless strategies and determinacy

Definition: For a game $\mathcal{G}_{v_0}$, a strategy $s_i$ for player $i$ is memoryless strategy if for all $wv, w^\prime v \in Pl^\prime_i$, $s_i(wv) = s_i(w^\prime v)$, and if $wv_0\in Pl_i^\prime$ then $s_i(wv_0) = s_i(\epsilon)$.

I.e., the strategy always makes the same move from vertex, regardless of the history of how it got there.

Let ${\rm MLS}_i$ denote the set of memoryless strategies for player $i$. ${\rm MLS}_i$ is a finite set, even if $S_i$ is not. In particular, if $m = \lvert Pl_i\rvert$ is the number of vertices belonging to player $i$, then $\lvert {\rm MLS}_i \rvert \leq \lvert \Sigma \rvert ^m$.

Definition: $\mathcal{G}_{v_0}$ is memorylessly determined if both players have memoryless strategies that achieve “the value”. I.e.,

\[\max_{s_1\in {\rm MLS}_1}\inf_{s_2\in S_2} u(s_1, s_2) = \min_{s_2\in {\rm MLS}_2}\sup_{s_1\in S_1} u(s_1, s_2)\]

Theorem A Finitistic games on finite graphs are memorylessly determined. Moreover, there is an efficient (P-time) algorithm to compute memoryless value-achieving strategies in such game.

The win-lose case: easy “fixed point” algorithm

We first prove the theorem for finitistic win-lose games via an easy bottom up fixed point algorithm.

$\text{Input}$: Game graph $G = (V, E, pl, v_0)$.

Assume w.l.o.g. all infinite plays are win for player 1 (other case is symmetric). “Dead end”: vertex with no outgoing edge.

$\text{Good} := \lbrace v\in V \mid v \text{ a dead end that wins for player 1}\rbrace$.
$\text{Bad}:= \lbrace v\in V \mid v \text{ a dead end that wins for player 2} \rbrace$

$\mathbf{Initialize}$: $\text{Win}_1 := \text{Good}; St_1 := \emptyset$;
$\mathbf{Repeat}$
1. $\mathbf{Foreach}\ v\notin \text{Win}_1$:
  1. $\text{If } (pl(v) = 1 \ \&\ \exist (v, v^\prime)\in E: v^\prime\in \text{Win}_1)$:
    
    $\text{Win}_1 := \text{Win}_1 \cup \lbrace v\rbrace; St_1 := St_1 \cup \lbrace v\mapsto v^\prime\rbrace$;
  2. $\text{If } (pl(v) = 2 \ \&\ \forall (v, v^\prime)\in E: v^\prime\in \text{Win}_1)$:
    
    $\text{Win}_1 := \text{Win}_1 \cup \lbrace v\rbrace$;
$\mathbf{Until} \text{ The set Win}_1 \text{ does not change}$;

Player 1 has a winning strategy iff $v_0 \in \text{Win}_1$. If so, $St_1$ is a memoryless winning strategy for player 1.