• Arena

  • arena: a directed graph $G=(V,E,V_1,V_2)$
    • s.t. $\forall v\in V\text{outdegree}(v)\ge 1$
    • 👨‍🏫: ownership of graph: must also be defined
    • $V_1\cup V_2=V$
    • $V_1\cap V_2=\varnothing$
    • 👨‍🏫 assume it to be finite (in this course), although it's not explicitly mentioned
  • game: arena $G=(V,E,V_1,V_2)$ and starting vector $v_0\in V$
    • owner of new vertex: gets to make next decision
    • goal: making move within one's own
    • others' goal: preventing the player $i$ from doing so
  • example

graph LR
    0(( ))
    1(( ))
    2( )
    3(v_0)
    0-->2; 0-->1; 1-->0; 1-->2; 2-->2; 3-->0; 3-->1

• strategy: strategy for player $i$:
  • function ${\sigma }_i:V^{\ast }\times V_i\to V$
  • if ${\sigma }_i(h,v)=v^{\prime },⟹(v,v^{\prime })\in E$
  • walk: can be done infinitely as all nodes have at lease one outdegree
• outcome: infinite walk on $G$ starting at $v_0$
  • use $O$ for the set of all outcomes
• if ${\sigma }_1,{\sigma }_2$: strategies for players
  • then $o({\sigma }_1,{\sigma }_2)\in O$: corresponding outcome

• Objective

  • objective for player 1: a strategies
    • ${\text{obj}}_1\subseteq O$
  • zero-sum game: obj2​=O</span>obj1​
    • i.e. victory of one: defeat of one
    • 👨‍🏫 non-zero sum game: also interesting
      • where loss of a player does not necessarily mean loss of another
  • strategy ${\sigma }_1$ for player 1: winning strategy if:
    • for every ${\sigma }_2$, we have $o({\sigma }_1,{\sigma }_2)\in {\text{obj}}_1$
    • i.e. regardless of other players' action, outcome is in your objective
    • 👨‍🏫: same for non-zero sum game
  • losing condition can also be defined
    • 👨‍🏫 but "lose" is not well defined for non-zero sum game...
  • game $G$: determined if, for every starting vertex $v_0$
    • either $P_1$ or $P_2$ (or: $\dots ,P_n$) has a winning strategy

• Hint for problem set 11

  • let there be a game w/ both players having no winning strategy
    • e.g. rock scissor paper intuitively
  • how can we make this in a graph game?
  • think about infinite objective

• Reachability game

  • player 1's objective: to reach $T$
    • ${\text{obj}}_1=◊T=\text{Reach}(T)=\{\bar{\omega }\in O|\exists i\bar{\omega }[i]\in T\}$
    • ${\text{obj}}_2=\square T^c=\text{Safe}(T^c)=\{\bar{\omega }\in O|\exists i\bar{\omega }[i]\in T^c\}$
      • 👨‍🏫 game: also can be called as safety game
    • a zero-sum game
  • 👨‍🏫 diamond notation: from linear-time temporal logic (LTL)
  • input: arena $G(V,E,V_1,V_2)$
    • target set $T\subseteq V$
  • output: ${\text{win}}_1,{\text{win}}_2$
  • rules
    1. $T\subseteq {\text{win}}_1$
    2. if $v\in V_1$ and $(v,u)\in E$ and $u\in {\text{win}}_1$
       • then $v\in {\text{win}}_1$
    3. if $v\in V_2$ and $\forall e=(v,u)\in E$, we have $u\in {\text{win}}_1$
       • then $u\in {\text{win}}_1$
  • algorithm:
    • $T_0=T$ $$\begin{array}{rl}{T}_1=T_0& \cup \{v\in V_1|\exists (v,u)\in Es.t.u\in T_0\}\\ & \cup \{v\in V_2|\forall (v,u)\in Es.t.u\in T_0\}\end{array}$$ $$\begin{array}{rl}{T}_{i+1}=T_i& \cup \{v\in V_1|\exists (v,u)\in Es.t.u\in T_i\}\\ & \cup \{v\in V_2|\forall (v,u)\in Es.t.u\in T_i\}\end{array}$$
  • $T_i$: set of starting vertices player 1 wins in $i$ steps
  • compute $T_0,T_1,T_2,\dots$
    • algorithm: terminates as there are finite no. of vertices
    • $T_i$ cannot increase infinitely
    • terminate when $T_k=T_{k+1}$
  • yet: we must also show that player 2 has strategy if: v0​∈V</span>Tk​
  • if $v_0\in V_1$:
    • it has no edge to $T_k$
      • if so: player 1 will go to $T_k$
  • if $v_0\in V_2$:
    • it has at least an edge to stay in V</span>Tk​
    • as: player 2 has no reason to run into defeat
    • player 2's strategy: simply
  • theorem: ${\text{win}}_1=\bigcup T_i$, ${\text{win}}_2=$
    • winning strategy for player 1: move into an inner circle ($T_i\to T_j\wedge i>j$)
      • always succeeds: as $T_i$ is expanded that way
    • win2​=v</span>(∪Ti​)
    • must also ensure: the game is memory-less
      • computation of $T_i$: dependent on arena, not history
  • $\text{Attr}(T):=\cup T_i$
  • 👨‍🏫 let's make it more complicated (even in last session)

• Büchi game

  • goal: to visit $v\in T$ infinitely many times
  • ${\text{obj}}_1=\text{Bu¨chi}(T)=\square ◊T=\{\bar{\omega }\in O|\exists i_1<i_2,\dots \forall j\bar{\omega }[i_j]\in T\}$
  • ${\text{obj}}_2=\text{coBu¨chi}(T^c)=◊\square T^c=\{\bar{\omega }\in O|\bar{\omega }\text{ visiting }T\text{ finite times}\}$
  • input: arena $G=(V,E,V_1,V_2)$
    • target set $T\subseteq V$
  • output: ${\text{win}}_1,{\text{win}}_2$
  • 👨‍🏫 whether it's determinant or not: not shown
    • from there: show that ${\text{win}}_1={\text{win}}_2^c$
  • $A_1={\text{Attr}}_1(T)$, $B_1=A_1^c$
    • known: $B_1\subseteq {\text{win}}_2$
  • $C_1={\text{Attr}}_2(B_1)$
    • known: $C_1\subseteq {\text{win}}_2$
  • if: the game starts outside $C_1$:
    • player 1 doesn't want to land $C_1$
    • i.e. for all v∈V</span>C1​, v has at least one edge not going C1​
      • 👨‍🏫 reduce the problem by removing all edge going into C1​ / vertices in C1​
        • remainder: still a valid game
  • total algorithm
    • $G_0=G$
      • $A_1={\text{Attr}}_1(T,G_0)$
      • $C_1={\text{Attr}}_2(A_1^c,G_0)$
    • $G_1=G-C_1$
      • $A_2={\text{Attr}}_1(T,G_1)$
      • $C_2={\text{Attr}}_2(A_2^c,G_1)$
    • $G_i=G_{i-1}-C_i$
      • $A_{i+1}={\text{Attr}}_1(T,G_i)$
      • $C_{i+1}={\text{Attr}}_2(A_{i+1}^c,G_i)$
    • algorithm: terminates as there are finite vertices
  • what if: stuck? i.e. $C_k=\varnothing$
    • then, let G′:=(V</span>∪Ci​,Ek​)
    • $A_{k+1}={\text{Attr}}_1(T,G^{\prime })$
      • $=G^{\prime }$
    • $C_{k+1}$: empty
  • within $A_{k+1}$: one can reach $T$ arbitrarily (infinitely) many times
  • theorem: ${\text{win}}_2=\cup C_i$, win1​=V</span>∪Ci​
    • player 2: simply staying within the smallest inner circle possible
      • = memory-less
    • player 1: don't step outside of smaller game
      • don't join ∪Ci​
        • = memory-less
      • if it's already in ∪Ci​: just pick any
        • as player 1 will lose anyway
  • what if: we have combinations of the objectives?
    • e.g. player 1 want to reach both $T_1,T_2$?
      • once, or infinitely many times?
    • 👨‍🏫 def. on your final, and it's non-trivial
      • 👨‍🏫 solution available in Wikipedia, but involves extra concepts outside syllabus