e. Obligation space and permission space.

(e1) Obligation-Space(X, BDOX+Y(k)) = {∃=1s (placing(X, mark(X), s, k) ∧ vacant(s, k)): BX(k) (¬end(k) ∧ turn(X, k))}. This can be proved based on (Ap.1.i2) and (OB1). (e2) Obligation-Space(X, BDOX+Y(k)) = {¬∃s (placing(X, mark(X), s, k): BX(k) turn(X, k)}. This can be proved based on (Ap.1.i2) and (OB3).

(e3) Permission-Space(X, BDOX+Y(k)) = {placing(X, mark(X), s, k): PX(k) placing(X, mark(X), s, k)}.

f. Schema for dynamic development of Tic-Tac-Toe

(f1) [Action in a continuing stage] In case: BX + Y(k)¬ end(k) & BX + Y(k) (turn(X, k) ∧ opponent(X, Y)). From (Ap.4.e) follows: Obligation-Space(X, k) = {∃=1s (placing(X, mark(X), s, k) ∧ vacant(s, k))} & Obligation-Space(Y, k) = {¬∃s (placing(Y, mark(Y), s, k)} & Permission-Space(X, k) = PLACING(X, mark(X), List(k), k) & Permission-Space(Y, k) = Ø. We assume here that X desires to place mark(X) in position s1. Thus, DBX(k) = DBX(k - 2) ∪ {placing(X, mark(X), s1, k)}. As special cases, we define: DBA(�2) = DBA & DBB(�1) = DBB. Then, X decides to perform this action and both X and Y confirm that this action is performed. Thus, we set: BBx+Y(k + 1) = BBX+Y(k) ∪ {placing(X, mark(X), s1, k)} & List(k + 1) = List(k) – {s1}. Then, based on (ET6), we can infer: BX + Y(k + 1) (occupied(s1, mark(X), k) ∧ turn(Y, k + 1)). (f2) [Action in the terminal stage] In case: BX + Y(k) end(k). The game ends

here. Because of (OB4), it holds: Permission-Space(X, k) = Ø & Permission-Space(Y, k) = Ø.

g. [Translation into Propositional Logic] Each FO-sentence in BBttt and in OBttt is reducible to a propositional sentence, because the ranges of all variables in them are finite. For example, ∀x φ(x) can be translated into φ(a1) ∧ … ∧ φ(ak) and ∃x φ(x) can be translared into φ(a1) ∨ … ∨ φ(ak), where a1, … , ak are names for objects whose set is a range of variable x.

(Ap.5) [Description of game developments] The following sequence of formulas describes the game development discussed in Subsection 3.1: hplacing(A, O, [1,1], 0), placing(B, �, [2,2], 1), placing(A, O, [3,3], 2), placing(B, �, [1,3], 3), placing(A, O, [3,1], 4), placing(B, �, [3,2], 5), placing(A, O, [2,1], 6)i. I call this sequence of actions "the record of a game". In general, a game of Tic-Tac-Toe can be described by a structure in form: hhBBG(0), OBGi, hBBG(1), OBGi, … , hBBG(k), OBGii. I call this structure "the game history". In general, this game history can be constructed for any game that is describable by Dynamic BDO-Logic. For example, a game history can be constructed for a baseball game and a soccer game.

(Ap.6) [Role-specific normative spaces] For any agent A in group G, role-specific obligation, prohibition, and permission space are characterized as follows (here, we assume that φact(A) is a sentence, which expresses that A performs certain action).


*Game Theory - From Idea to Practice*
