**Appendix**

In this appendix, I precisely describe the definitions and the theorems mentioned in the main text.

(Ap.1) Definition and Characterization of BDO-Logic

We use not, &, or,), and ⇔ as meta-language expressions of logical connectives. Let Cn(X) be an abbreviation for the deductive closure of X, where X is a set of FOsentences. Thus, Cn(X) contains every FO-sentence that can be logically deduced from Cn(X). Furthermore, for set X of FO-sentences, we define the consistency of X as follows: consistent(X) ⇔ there is no FO-sentence φ such that φ∈Cn(X) and ¬φ∈Cn (X). Here, we assume that each of BB, DB, and OB is a set of FO-sentences.


(d1) If φ is a FO-sentence and BDOA is a BDO-system, then each of BA φ, MA φ, OA φ, FA φ, PA φ, and DA φ is a sentence in BDO-Logic.

(d2) If each of Φ and Ψ is a sentence in BDO-Logic, then each of not Φ, Φ & Ψ, Φ or Ψ, Φ) Ψ, and Φ ⇔ Ψ is a sentence in BDO-Logic.

(d3) Every sentence in BDO-Logic satisfies (Ap.1.d1) or (Ap.1.d2).

	- (e1) [Belief] BAφ ⇔ φ∈ Cn(BBA).
	- (e2) [Possibility] MAφ ⇔ consistent(BBA∪{φ}).
	- (e3) [Obligation] OAφ ⇔ [φ∈Cn(BBA∪OBA) & not (φ∈Cn(BBA))].
	- (i1) Desire-Space(A, BDOA)={φact(A): DA φact(A)}.
	- (i2) Obligation-Space(A, BDOA)={φact(A): OA φact(A)}.
	- (i3) Prohibition-Space(A, BDOA)={φact(A): FA φact(A)}.
	- (i4) Permission-Space(A, BDOA)={φact(A): PA φact(A) }.

(Ap.2) Dynamic Belief-Desire-Obligation Logic (Dynamic BDO-Logic)

We can update a BDO-system hBB, DB, OBi by updating BB or DB or OB. We call the framework that allows this kind of updates Dynamic BDO-Logic. A BDO-system in Dynamic BDO-Logic contains information about its stage. We write a BDO-system of Dynamic BDO-Logic as follows: BDO(k) = hBB(k), DB(k), OB(k)i.

(Ap.3) Formalization of Games (Example: Tic-Tac-Toe)

Here, by using example of Tic-Tac-Toe, I demonstrate how Dynamic BDO-Logic can be applied to descriptions of game processes. The normative system for Tic-Tac-Toe is a BO-system hBBttt, OBttti. There are two types of propositions, namely action tapes and state types:

Action type: placing(X, mark(X), s, k). State type: occupied(s, m, k), turn(X, s), vacant(s, k), opponent(X, Y), end(k), won(X, k), play(X, g).

Here, X and Y are used as variables of players, s, s1, … are used as variables of positions, m is used as a variable of marks, k and n are used as variables of game stages, and g is a variable for games. For the sake of readability, I use Many-Sorted Logic instead of FO-Logic. However, all formulas of Many-sorted Logic can be translated into formulas of FO-Logic. I use a function mark(X), where there are two values for this function, namely O and �.

a. Belief base BBttt consists of the following elementary theory for Tic-Tac-Toe: BBttt = {(ET1), (ET2), (ET3), (ET4), (ET5), (ET6)}.

(ET1) There are some trivial stipulations that can be expressed in FO-Logic. For the sake of understandability, we verbally summarize these conditions as follows: (1) There are exactly two players who are opponents each other. (2) We use O and � as two marks of players (∀X (mark(X) = O ∨ mark(X) = �). (3) If one player wins in Tic-Tac-Toe, then her opponent loses. (4) When it is turn of a player, it is not turn of her opponent.

(ET2) [Persistence] ∀k ∀n ∀s ∀m (occupied(s, m, k) ∧ k ≤ n ! occupied(s, m, n)). (ET3) ∀k ∀s (vacant(s, k)\$ ¬ ∃ m occupied(s, m, k)).

(ET4) The definition of victory in Tic-Tac-Toe can be given in FO-Logic. However, for the sake of understanderbility, we verbally express the condition for

victory: The player who succeeds in placing three of their marks in a horizontal, vertical, or diagonal row wins the game.

(ET5) ∀k (end(k) \$ (∃X won(X, k) ∨¬ ∃ s vacant(s, k)).

(ET6) [Effect] ∀k ∀X ∀Y ∀s ((placing(X, mark(X), s, k) ∧ opponent(X,

Y)) ! (occupied(s, mark(X), k + 1) ∧ turn(Y, k + 1))).

b. Obligation base OBttt is defined as follows: OBttt = {(OB1), (OB2), (OB3), (OB4)}.

(OB1) <sup>∀</sup><sup>X</sup> <sup>∀</sup>k (¬ end(k) <sup>∧</sup> turn(X, k) ! <sup>∃</sup> =1s (placing(X, mark(X), s, k) <sup>∧</sup> vacant (s, k))). (OB2) ∀X ∀k ¬∃ s (placing(X, mark(X), s, k) ∧¬ vacant(s, k)). (OB3) ∀X ∀k (¬ turn(X k) ! ¬ ∃ s placing(X, mark(X), s, k)). (OB4) ∀X ∀k (end(k) ! ¬ ∃ s placing(X, mark(X), s, k)).

c. The normative system for Tic-Tac-Toe is hBBttt, OBttti.

(Ap.4) By using Dynamic BDO-Logic, game processes of Tic-Tac-Toe can be described. Here, we describe game processes of Tic-Tac-Toe in general.

