A minimal logic for interactive epistemology

Abstract

We propose a minimal logic for interactive epistemology based on a qualitative representation of epistemic individual and group attitudes including knowledge, belief, strong belief, common knowledge and common belief. We show that our logic is sufficiently expressive to provide an epistemic foundation for various game-theoretic solution concepts including “1-round of deletion of weakly dominated strategies, followed by iterated deletion of strongly dominated strategies” ($\hbox {DWDS}^1\hbox {-IDSDS}$) and “2-rounds of deletion of weakly dominated strategies, followed by iterated deletion of strongly dominated strategies” ($\hbox {DWDS}^2\hbox {-IDSDS}$). Axiomatization and complexity results for the logic are given in the paper.

Appendices

Appendix 1: Proof of Theorem 1

Let $i \in Agt$. Then:

• 1. $s _{ i} \not \in \mathsf { SD}^{ {\varGamma }} _i$ if and only if there exists an epistemic game model $M = ( W, \mathcal {E}_{1 } , \ldots , \mathcal {E}_{n} , \mathcal {B}_{1 } , \ldots , \mathcal {B}_{n} , \mathsf {ch}_{1} , \ldots , \mathsf {ch}_{n} , \mathcal {V})$ and a world $w\in W$ such that $M,w \models \mathsf { WRat}_{ i} ( s _{ i} )$;

• 2. $s _{ i} \not \in \mathsf { WD}^{ {\varGamma }} _i$ if and only if there exists an epistemic game model $M = ( W, \mathcal {E}_{1 } , \ldots , \mathcal {E}_{n} , \mathcal {B}_{1 } , \ldots , \mathcal {B}_{n} , \mathsf {ch}_{1} , \ldots , \mathsf {ch}_{n} , \mathcal {V})$ and a world $w\in W$ such that $M,w \models \mathsf {PRat}_{ i} ( s _{ i} )$.

Proof

We only prove the second item, as the first item can be proved in a very similar way.

As to the left-to-right direction, let us assume that $s _{ i} \not \in \mathsf { WD}^{ {\varGamma }} _i$. We are going to build an epistemic game model $M = ( W, \mathcal {E}_{1 } , \ldots , \mathcal {E}_{n} , \mathcal {B}_{1 } , \ldots , \mathcal {B}_{n} , \mathsf {ch}_{1} , \ldots , \mathsf {ch}_{n} , \mathcal {V})$ and a world $w\in W$ such that $M,w \models \mathsf {PRat}_{ i} ( s _{ i} )$. The model M is defined as follows:

• $W = \{w_s {:}s \in S _{ } \}$,

• For all $i \in Agt$, $\mathcal {E}_{i }= \mathcal {B}_{i } =\{(w_s, w_{ s' }) {:}w_s, w_{ s' }\in W \text { and } s _{ } [i] = s _{ } '[i] \}$,

• For all $w_s \in W$ and for all $i \in Agt$, $\mathsf {ch}_{ i } (w_s) = s _{ } [i]$,

• For all $w_s \in W$, $\mathcal {V}(w_s) = Atm$.

It is straightforward to verify that $M, w_s \models \mathsf {PRat}_{ i} ( s _{ i} )$ for all $w_s \in W$ such that $\mathsf {ch}_{ i} (w_s ) = s _{ i}$.

As to the right-to-left direction, we prove it by reductio ad absurdum. Let us suppose that $s _{ i} \in \mathsf { WD}^{ {\varGamma }} _i$. Moreover, let $M = ( W, \mathcal {E}_{1 } , \ldots , \mathcal {E}_{n} , \mathcal {B}_{1 } , \ldots , \mathcal {B}_{n} , \mathsf {ch}_{1} , \ldots , \mathsf {ch}_{n} , \mathcal {V})$ be an epistemic game model and let $w\in W$ such that $M,w \models \mathsf {PRat}_{ i} ( s _{ i} )$. $s _{ i} \in \mathsf { WD}^{ {\varGamma }} _i$ means that there exists $s _{ i} ' \in S _{ i}$ such that $\forall s _{ -i } \in S _{- i} , U _{i} ( s _{ i} , s _{- i} ) \le U _{i} ( s _{ i}' , s _{- i} )$ and $\exists s _{ -i } ' \in S _{- i} \text { such that } U _{i} ( s _{ i} , s _{- i} ) < U _{i} ( s _{ i}' , s _{- i} )$. By the Condition (C4) on epistemic game models and the definition of perfect rationality, the latter implies that $M,w \models \lnot \mathsf {PRat}_{ i} ( s _{ i} )$. This is in contradiction with the initial assumption. $\square$

Appendix 2: Proof of Theorem 4

Let

$$\begin{aligned} \mathsf {ComplAss} \ \mathop {=}\limits ^{\mathtt {def}} \ \bigwedge _{i \in Agt } \bigwedge _{ s _{-i } \in S _{-i } {:} s _{-i } \in S _{-i, 1} ^{ DWDS ^2- IDSDS } } \widehat{ \mathsf { K}}_{ i} ( choose _{ - i} ( s _{ -i } ) \wedge \mathsf {AllPRat}_{ -i } ). \end{aligned}$$

Then:

$$\begin{aligned} \models \mathsf { CB}_{ } ( \mathsf {ComplAss} \wedge \mathsf {AllPRat}_{ Agt } \wedge \bigwedge _{ i \in Agt } \mathsf { SB}_{i} \ \mathsf {AllPRat}_{ -i } ) \rightarrow \bigvee _{ s _{ } \in S _{ } ^{ DWDS ^2- IDSDS } } choose _{ } ( s _{ } ) \end{aligned}$$

Proof

Instead of proving

$$\begin{aligned} \models \mathsf { CB}_{ } ( \mathsf {ComplAss} \wedge \mathsf {AllPRat}_{ Agt } \wedge \bigwedge _{ i \in Agt } \mathsf { SB}_{i} \ \mathsf {AllPRat}_{ -i } ) \rightarrow \bigvee _{ s _{ } \in S _{ } ^{ DWDS ^2- IDSDS } } choose _{ } ( s _{ } ) \end{aligned}$$

we simply prove that for all $s _{ } \not \in S _{ } ^{ DWDS ^2- IDSDS }$:

$$\begin{aligned} \models \mathsf { CB}_{ } ( \mathsf {ComplAss} \wedge \mathsf {AllPRat}_{ Agt } \wedge \bigwedge _{ i \in Agt } \mathsf { SB}_{i} \ \mathsf {AllPRat}_{ -i } ) \rightarrow \lnot choose _{ } ( s _{ } ) \end{aligned}$$

Indeed, we have the following valid equivalence:

$$\begin{aligned} \models \bigvee _{ s _{ } \in S _{ } ^{ DWDS ^2- IDSDS } } choose _{ } ( s _{ } ) \leftrightarrow \bigwedge _{ s _{ } \not \in S _{ } ^{ DWDS ^2- IDSDS } } \lnot choose _{ } ( s _{ } ) \end{aligned}$$

The proof is by induction. The proof of the inductive case goes exactly as the proof of the inductive case in the proof of Theorem 3.

Let us prove the base case first.

Base case. For all $s _{ } \not \in S _{2} ^{ DWDS ^2- IDSDS }$ we prove that:

(A):

$\models \mathsf { CB}_{ } ( \mathsf {ComplAss} \wedge \mathsf {AllPRat}_{ Agt } \wedge \bigwedge _{ i \in Agt } \mathsf { SB}_{i} \ \mathsf {AllPRat}_{ -i } ) \rightarrow \lnot choose _{ } ( s _{ } )$

To prove (A), it is sufficient to prove the following validity (B), as we have the following validity:

$$\begin{aligned} \models \mathsf { CB}_{ } ( \mathsf {ComplAss} \wedge \mathsf {AllPRat}_{ Agt } \wedge \bigwedge _{ i \in Agt } \mathsf { SB}_{i} \ \mathsf {AllPRat}_{ -i } ) \rightarrow \\ ( \mathsf {ComplAss} \wedge \mathsf {AllPRat}_{ Agt } \wedge \bigwedge _{ i \in Agt } \mathsf { SB}_{i} \ \mathsf {AllPRat}_{ -i } ) \end{aligned}$$

For all $s _{ } \not \in S _{2} ^{ DWDS ^2- IDSDS }$ we have that:

(B):

$\models ( \mathsf {ComplAss} \wedge \mathsf {AllPRat}_{ Agt } \wedge \bigwedge _{ i \in Agt } \mathsf { SB}_{i} \ \mathsf {AllPRat}_{ -i } ) \rightarrow \lnot choose _{ } ( s _{ } )$

And to prove (B), it is sufficient to prove that if $s _{ } [i] \not \in S _{i,2} ^{ DWDS ^2- IDSDS }$ then:

(C):

$\models \Big ( \bigwedge _{ s _{-i } \in S _{-i } {:} s _{-i } \in S _{-i, 1} ^{ DWDS ^2- IDSDS } } \widehat{ \mathsf { K}}_{ i} ( choose _{ - i} ( s _{ -i } ) \wedge \mathsf {AllPRat}_{ -i } ) \wedge \mathsf {PRat}_{ i } \wedge$ $\mathsf { SB}_{i} \mathsf {AllPRat}_{ -i } \Big ) \rightarrow \lnot choose _{i } ( s _{ } [i] )$

Let us prove (C) by reductio ad absurdum. We assume that $s _{ } [i] \not \in S _{i,2} ^{ DWDS ^2- IDSDS }$ and $M,w \models \bigwedge _{ s _{-i } \in S _{-i } {:} s _{-i } \in S _{-i, 1} ^{ DWDS ^2- IDSDS } } \widehat{ \mathsf { K}}_{ i} ( choose _{ - i} ( s _{ -i } ) \wedge \mathsf {AllPRat}_{ -i } ) \wedge \mathsf {PRat}_{ i } \wedge \mathsf { SB}_{i} \ \mathsf {AllPRat}_{ -i } \wedge choose _{ i} ( s _{ } [i] )$ for some arbitrary epistemic game model $M = ( W, \mathcal {E}_{1 } , \ldots , \mathcal {E}_{n} , \mathcal {B}_{1 } , \ldots , \mathcal {B}_{n} , \mathsf {ch}_{1} , \ldots , \mathsf {ch}_{n} , \mathcal {V})$ and world w in M. We are going to show that these two facts are inconsistent.

The rest of the proof makes use of the following Lemma 1.

Lemma 1

Let $M = ( W, \mathcal {E}_{1 } , \ldots , \mathcal {E}_{n} , \mathcal {B}_{1 } , \ldots , \mathcal {B}_{n} , \mathsf {ch}_{1} , \ldots , \mathsf {ch}_{n} , \mathcal {V})$ be an epistemic game model, let $w \in W$ and let $s _{-i } , s _{-i } ' \in S _{-i }$. Then, if

• 1. $M,w \models \bigwedge _{ s _{-i } \in S _{-i } {:} s _{-i } \in S _{-i, 1} ^{ DWDS ^2- IDSDS } } \widehat{ \mathsf { K}}_{ i} ( choose _{ - i} ( s _{ -i } ) \wedge \mathsf {AllPRat}_{ -i } ) \wedge \mathsf { SB}_{i} \mathsf {AllPRat}_{ -i }$

• 2. $s _{-i } \in S _{-i, 1} ^{ DWDS ^2- IDSDS }$ and $s _{-i } ' \not \in S _{-i, 1} ^{ DWDS ^2- IDSDS }$

then $\varPi _{w,i } ( choose _{ -i } ( s _{-i } ) ) > \varPi _{w,i } ( choose _{-i } ( s _{ -i } ') )$.

Proof

Assume that $M,w \models \bigwedge _{ s _{-i } \in S _{-i } {:} s _{-i } \in S _{-i, 1} ^{ DWDS ^2- IDSDS } } \widehat{ \mathsf { K}}_{ i} ( choose _{ - i} ( s _{ -i } ) \wedge \mathsf {AllPRat}_{ -i } ) \wedge \mathsf { SB}_{i} \ \mathsf {AllPRat}_{ -i }$ and that $s _{-i } \in S _{-i, 1} ^{ DWDS ^2- IDSDS }$ and $s _{-i } ' \not \in S _{-i, 1} ^{ DWDS ^2- IDSDS }$. By Proposition 1, $M,w \models \mathsf { SB}_{i} \ \mathsf {AllPRat}_{ -i }$ implies that for all $v \in || \mathsf {AllPRat}_{ -i } ||_{w {, } i }$ and for all $u \in || \lnot \mathsf {AllPRat}_{ -i } ||_{w {, } i }$, $\pi _i(u) < \pi _i(v)$. Thus, by the second item of Theorem 1 and the fact that $M,w \models \widehat{ \mathsf { K}}_{ i} ( choose _{ - i} ( s _{ -i } ) \wedge \mathsf {AllPRat}_{ -i } )$ and that $s _{-i } \in S _{-i, 1} ^{ DWDS ^2- IDSDS }$ and $s _{-i } ' \not \in S _{-i, 1} ^{ DWDS ^2- IDSDS }$, it follows that there exists $v \in || choose _{ -i } ( s _{-i } ) ||_{w {, } i }$ such that, for all $u \in || choose _{ -i } ( s _{-i } ') ||_{w {, } i }$, $\pi _i(u) < \pi _i(v)$. The latter implies that $\varPi _{w,i } ( choose _{ -i } ( s _{-i } ) ) > \varPi _{w,i } ( choose _{-i } ( s _{ -i } ') )$. $\square$

From $M,w \models \bigwedge _{ s _{-i } \in S _{-i } {:} s _{-i } \in S _{-i, 1} ^{ DWDS ^2- IDSDS } } \widehat{ \mathsf { K}}_{ i} ( choose _{ - i} ( s _{ -i } ) \wedge \mathsf {AllPRat}_{ -i } ) \wedge \mathsf { SB}_{i} \mathsf {AllPRat}_{ -i }$, by Lemma 1, it follows that:

(D):

if $s _{-i } ' \in S _{-i, 1} ^{ DWDS ^2- IDSDS }$ and $s _{-i } '' \not \in S _{-i , 1} ^{ DWDS ^2- IDSDS }$ then $\varPi _{w,i } ( choose _{ -i } ( s _{ -i } ' ) ) > \varPi _{w,i } ( choose _{ -i } ( s _{-i } '' ) )$.

$s _{ } [i] \not \in S _{i,2} ^{ DWDS ^2- IDSDS }$ implies that:

(E1):

$s _{ } [i] \not \in S _{i,1} ^{ DWDS ^2- IDSDS }$ or

(E2):

$s _{ } [i] \in S _{i,1} ^{ DWDS ^2- IDSDS }$ and $s _{ } [i] \not \in S _{i,2} ^{ DWDS ^2- IDSDS }$

We split the proof in the two subcases: (E1) and (E2).

Proof for the case (E1).

$s _{ } [i] \not \in S _{i,1} ^{ DWDS ^2- IDSDS }$ implies that:

(F1):

There is $s _{ i}' \in S _{i} ^{ DWDS ^2- IDSDS }$ such that: (1) $s _{ i}' \ne s _{ } [i]$ and (2) $\langle s _{ } [i] , s _{- i} ' \rangle <_i \langle s _{ i}' , s _{- i} ' \rangle$ for some $s _{- i} ' \in S _{- i} ^{ DWDS ^2- IDSDS }$ and (3) $\langle s _{ } [i] , s _{- i} '' \rangle \le _i \langle s _{ i}' , s _{- i} '' \rangle$ for all $s _{- i} '' \in S _{-i} ^{ DWDS ^2- IDSDS }$.

From (F1), by the Condition (C4) on epistemic game models, it follows that:

(G1):

There are $s _{ i}' \in S _{i} ^{ DWDS ^2- IDSDS }$ and $s _{- i} ' \in S _{- i} ^{ DWDS ^2- IDSDS }$ and $v \in W$ such that: (1) $s _{ i}' \ne s _{ } [i]$ and (2) $w \mathcal {E}_{i } v$ and (3) $M,v \models choose _{-i } ( s _{- i} ' )$ and (4) $\langle s _{ } [i] , s _{- i} ' \rangle <_i \langle s _{ i}' , s _{- i} ' \rangle$ and (5) for all $u \in W$ such that $w \mathcal {E}_{i } u$ and for all $s _{- i} '' \in S _{- i} ^{ DWDS ^2- IDSDS }$: if $M, u \models choose _{-i } ( s _{- i} '' )$ then $\langle s _{ } [i] , s _{- i} '' \rangle \le _i \langle s _{ i}' , s _{- i} '' \rangle$.

But (G1) is in contradiction with $M, w \models \mathsf {PRat}_{ i }$ and $M,w \models choose _{ i} ( s _{ } [i] )$.

Proof for the case (E2).

(E2) implies that:

(F2):

There is $s _{ i}' \in S _{i,1} ^{ DWDS ^2- IDSDS }$ such that: (1) $s _{ i}' \ne s _{ } [i]$ and (2) $\langle s _{ } [i] , s _{- i} ' \rangle <_i \langle s _{ i}' , s _{- i} ' \rangle$ for some $s _{- i} ' \in S _{- i,1} ^{ DWDS ^2- IDSDS }$ and (3) $\langle s _{ } [i] , s _{- i} '' \rangle \le _i \langle s _{ i}' , s _{- i} '' \rangle$ for all $s _{- i} '' \in S _{-i,1} ^{ DWDS ^2- IDSDS }$.

By the Condition (C4) on epistemic game models, (F2) together with $M, w \models \mathsf {PRat}_{ i }$ and $M,w \models choose _{ i} ( s _{ } [i] )$ imply that:

(G2):

There are $s _{- i} ' \in S _{- i,1} ^{ DWDS ^2- IDSDS }$ and $s _{- i} '' \not \in S _{- i,1} ^{ DWDS ^2- IDSDS }$ such that

$\varPi _{w,i } ( choose _{-i } ( s _{-i } ' ) ) \le \varPi _{w,i } ( choose _{ -i} ( s _{ -i } '' ) )$.

But (G2) is in contradiction with (D). This proves the base case.

Let us now prove the inductive case.

Inductive case. For $m>1$, we assume that if $s _{ } \not \in S _{m} ^{ DWDS ^2- IDSDS }$ then:

(Inductive Hypothesis):

$\models \mathsf { CB}_{ } ( \mathsf {ComplAss} \wedge \mathsf {AllPRat}_{ Agt } \wedge \bigwedge _{ i \in Agt } \mathsf { SB}_{i} \ \mathsf {AllPRat}_{ -i } ) \rightarrow \lnot choose _{ } ( s _{ } )$

We are going to prove that if $s _{ } \not \in S _{m+1} ^{ DWDS ^2- IDSDS }$ then:

(A):

$\models \mathsf { CB}_{ } ( \mathsf {ComplAss} \wedge \mathsf {AllPRat}_{ Agt } \wedge \bigwedge _{ i \in Agt } \mathsf { SB}_{i} \ \mathsf {AllPRat}_{ -i } ) \rightarrow \lnot choose _{ } ( s _{ } )$

Let us take an arbitrary epistemic game model M and world w and assume that $M,w \models \mathsf { CB}_{ } ( \mathsf {ComplAss} \wedge \mathsf {AllPRat}_{ Agt } \wedge \bigwedge _{ i \in Agt } \mathsf { SB}_{i} \ \mathsf {AllPRat}_{ -i } )$ and $M,w \models choose _{ } ( s _{ } )$. We are going to show that $s _{ } \in S _{m+1} ^{ DWDS ^2- IDSDS }$.

$M,w \models \mathsf { CB}_{ } ( \mathsf {ComplAss} \wedge \mathsf {AllPRat}_{ Agt } \wedge \bigwedge _{ i \in Agt } \mathsf { SB}_{i} \ \mathsf {AllPRat}_{ -i } )$ implies $\models \mathsf {AllPRat}_{ Agt }$ which in turn implies:

(A):

$\models \bigwedge _{i \in Agt } \mathsf { SRat}_{ i}$

Moreover, we have the following validity by the property $\models \mathsf { CB}_{ Agt } \varphi \rightarrow \mathsf { B}_{i } \mathsf { CB}_{ Agt } \varphi$ for every $i \in Agt$:

(B):

$\models \mathsf { CB}_{ } ( \mathsf {ComplAss} \wedge \mathsf {AllPRat}_{ Agt } \wedge \bigwedge _{ i \in Agt } \mathsf { SB}_{i} \ \mathsf {AllPRat}_{ -i } ) \rightarrow \bigwedge _{i \in Agt } \mathsf { B}_{i } \mathsf { CB}_{ } ( \mathsf {ComplAss} \wedge \mathsf {AllPRat}_{ Agt } \wedge \bigwedge _{ i \in Agt } \mathsf { SB}_{i} \ \mathsf {AllPRat}_{ -i } )$

Therefore, from $M,w \models \mathsf { CB}_{ } ( \mathsf {ComplAss} \wedge \mathsf {AllPRat}_{ Agt } \wedge \bigwedge _{ i \in Agt } \mathsf { SB}_{i} \ \mathsf {AllPRat}_{ -i } )$ we infer that:

(C):

$M,w \models \bigwedge _{i \in Agt } \mathsf { B}_{i } \mathsf { CB}_{ } ( \mathsf {ComplAss} \wedge \mathsf {AllPRat}_{ Agt } \wedge \bigwedge _{ i \in Agt } \mathsf { SB}_{i} \ \mathsf {AllPRat}_{ -i } )$

By the inductive hypothesis, Axiom K and the rule of necessitation for the belief operator $\mathsf { B}_{i }$, from (C) it follows that if $s _{ } ' \not \in S _{m} ^{ DWDS ^1- IDSDS }$ then:

(D):

$M,w \models \bigwedge _{i \in Agt } \mathsf { B}_{i } \lnot choose _{ } ( s _{ } ' )$

From (B), (D) and $M,w \models choose _{ } ( s _{ } )$ it follows that for every $i \in Agt$ and for all $s _{ i}' \in S _{ i}$ either there is $s _{} ' \in S _{ m} ^{ DWDS ^1- IDSDS }$ such that $\langle s _{ i}' , s _{- i} ' \rangle <_i \langle s _{ } [i], s _{- i} ' \rangle$ or for all $s _{} ' \in S _{ m} ^{ DWDS ^1- IDSDS }$ we have $\langle s _{ i}' , s _{- i} ' \rangle \le _i \langle s _{ } [i], s _{- i} ' \rangle$. The latter implies that for every $i \in Agt$ we have $s _{ } [i] \in S _{i, m+1} ^{ DWDS ^1- IDSDS }$ which is equivalent to $s _{ } \in S _{ m+1} ^{ DWDS ^1- IDSDS }$. $\square$

Appendix 3: Proof of Theorem 5

Let $s _{ } \in S _{ }$. Then:

• 1. If $s _{ } \in S _{ } ^{ IDSDS }$ then there exists an epistemic game model $M = ( W, \mathcal {E}_{1 } , \ldots , \mathcal {E}_{n} , \mathcal {B}_{1 }$, $\ldots , \mathcal {B}_{n} , \mathsf {ch}_{1} , \ldots , \mathsf {ch}_{n} , \mathcal {V})$ and a world $w\in W$ such that $M,w \models choose _{ } ( s _{ } ) \wedge \mathsf { CB}_{ } \ \mathsf {AllWRat}_{ Agt }$;

• 2. If $s _{ } \in S _{ } ^{ DWDS ^1- IDSDS }$ then there exists an epistemic game model $M = ( W, \mathcal {E}_{1 } , \ldots , \mathcal {E}_{n} , \mathcal {B}_{1 } , \ldots , \mathcal {B}_{n} , \mathsf {ch}_{1} , \ldots , \mathsf {ch}_{n} , \mathcal {V})$ and a world $w\in W$ such that $M,w \models choose _{ } ( s _{ } ) \wedge \mathsf { CB}_{ } \ \mathsf {AllPRat}_{ Agt }$;

• 3. If $s _{ } \in S _{ } ^{ DWDS ^2- IDSDS }$ then there exists an epistemic game model $M = ( W, \mathcal {E}_{1 } , \ldots , \mathcal {E}_{n} , \mathcal {B}_{1 } , \ldots , \mathcal {B}_{n} , \mathsf {ch}_{1} , \ldots , \mathsf {ch}_{n} , \mathcal {V})$ and a world $w\in W$ such that $M,w \models choose _{ } ( s _{ } ) \wedge \mathsf { CB}_{ } ( \mathsf {ComplAss} \wedge \mathsf {AllPRat}_{ Agt } \wedge \bigwedge _{ i \in Agt } \mathsf { SB}_{i} \ \mathsf {AllPRat}_{ -i } )$.

Proof

We only prove the third item, as the first item and the second item can be proved in a very similar way.

We build the following epistemic game model $M = ( W, \mathcal {E}_{1 } , \ldots , \mathcal {E}_{n} , \mathcal {B}_{1 } , \ldots , \mathcal {B}_{n} , \mathsf {ch}_{1} , \ldots , \mathsf {ch}_{n} , \mathcal {V})$:

• $W = \{w_s {:}s \in S _{ } \}$,

• For all $i \in Agt$, $\mathcal {E}_{i }= \{(w_s, w_{ s' }) {:}w_s, w_{ s' }\in W \text { and } s _{ } [i] = s _{ } '[i] \}$,

• For all $i \in Agt$, $\mathcal {B}_{i } = \{(w_s, w_{ s' }) {:}w_s \mathcal {E}_{i } w_{ s' } \text { and } s _{-i } ' \in S _{-i, 1} ^{ DWDS ^2- IDSDS } \}$,

• For all $w_s \in W$ and for all $i \in Agt$, $\mathsf {ch}_{ i } (w_s) = s _{ } [i]$,

• For all $w_s \in W$, $\mathcal {V}(w_s) = Atm$.

It is routine to check that, for all $w_s \in W$ and for all $i \in Agt$:

(A):

$s _{ } [i] \in S _{i,1} ^{ DWDS ^2- IDSDS }$ iff $M, w_s \models \mathsf {PRat}_{ i}$.

The right-to-left direction follows from Theorem 1 above. The left-to-right direction follows from the construction of the previous epistemic game model M and the definition of $\mathsf {PRat}_{ i}$.

By the previous item (A), from the definition of $\mathsf { SB}_{i}$ and the construction of the previous epistemic game model M, for all $w_s \in W$ and for all $i \in Agt$, we have:

(B):

$M, w_s \models \mathsf { SB}_{i} \ \mathsf {AllPRat}_{ -i }$.

By the previous item (A), from the definition of $\mathsf {ComplAss}$ and the construction of the previous epistemic game model M, for all $w_s \in W$, we have:

(C):

$M, w_s \models \mathsf {ComplAss}$.

By the previous item (A), from the construction of the previous epistemic game model M, we have that:

(D):

If $s _{ } \in S _{1} ^{ DWDS ^2- IDSDS }$ then $M, w_{ s' } \models \mathsf {AllPRat}_{ Agt }$ for all $w_{ s' } \in \mathcal {B}_{ } ^+(w_{ s})$.

Now, let $s _{ } \in S _{ } ^{ DWDS ^2- IDSDS }$. Hence, $s _{ } \in S _{1} ^{ DWDS ^2- IDSDS }$. Thus, from the previous items (B), (C) and (D) it follows that $M, w_s \models choose _{ } ( s _{ } ) \wedge \mathsf { CB}_{ } ( \mathsf {ComplAss} \wedge \mathsf {AllPRat}_{ Agt } \wedge \bigwedge _{ i \in Agt } \mathsf { SB}_{i} \ \mathsf {AllPRat}_{ -i } )$. $\square$

Appendix 4: Proof of Theorem 6

The set of validities of the logic ${\textsf {LEG}}$ relative to the class of epistemic game models is completely axiomatized by the principles given in Fig. 4.

Proof

Proving that the axioms given in Fig. 4 are sound with respect to the class of epistemic game models (EGMs) and that the inference rules preserve validity is just a routine task and we do not give it here.

As to completeness, the proof consists of three steps.

Step 1 Let us first observe that the ${\textsf {LEG}}$ semantics in terms of epistemic game models is equivalent to a ${\textsf {LEG}}$ semantics in terms of multi-relational Kripke models in which formulas $choose _{i} ( s _{i} )$ and $poss _{i}$ are seen as atomic propositions.

Definition 10

(Multi-relational Kripke model) A multi-relational Kripke model (MKM) for the strategic game ${\varGamma }= ( Agt , \{ S _{i} \}_{i \in Agt }, \{ U _{i} \}_{i \in Agt } )$ and for the set of atomic propositions $Atm$ is a tuple $M = ( W, \{ \mathcal {E}_{i } \}_{i \in Agt } , \{ \mathcal {B}_{i} \}_{i \in Agt }, \pi )$ where $W, \{ \mathcal {E}_{i } \}_{i \in Agt }$ and $\{ \mathcal {B}_{i} \}_{i \in Agt }$ are as in Definition 1 and $\pi$ is the valuation function:

$$\begin{aligned} \pi : W \longrightarrow 2^{ Atm \cup {\varPhi }\cup {\varPsi }} \end{aligned}$$

with ${\varPhi }_i = \left\{ choose _{i} ( s _{i} ) {:} s _{i} \in S _{i} \right\}$, ${\varPhi }= \bigcup _{ i \in Agt } {\varPhi }_i$ and ${\varPsi }= \left\{ poss _{i} {:}i \in Agt \right\}$, which satisfies the following conditions:

(C3 $^*$):

For all $w,v \in W$, for all $i \in Agt$ and for all $s _{i} \in S _{ i }$: if $choose _{i} ( s _{i} ) \in \pi ( w )$ and $w \mathcal {E}_{i} v$ then $choose _{i} ( s _{i} ) \in \pi ( v )$;

(C4 $^*$):

For all $w \in W$, for all $i \in Agt$ and for all $s _{ -i } \in S _{ -i }$: there is $u \in W$ such that $w \mathcal {E}_{i } u$ and $choose _{ j } ( s _{ j} ) \in \pi ( u )$ for all $j \in Agt \setminus \left\{ i \right\}$;

(C5):

For all $w \in W$ and for all $i \in Agt$: $\pi _i (w)$ is a singleton with $\pi _i (w) = \pi (w) \cap {\varPhi }_i$;

(C6):

For all $w \in W$ and for all $i \in Agt$: if $poss _{i} \in \pi (w)$ then $w \mathcal {B}_{i} w$.

The truth conditions of formulas in $\mathcal {L}_{{\textsf {LEG}}}({\varGamma }, Atm )$ relative to MKMs are like the truth conditions relative to EGMs except for formulas $choose _{i} ( s _{i} )$ and $poss _{i}$, which are interpreted by means of the valuation function $\pi$ as follows:

$$\begin{aligned} \begin{array}{lll} M,w \models poss _{i} &{} \text { iff } &{} poss _{i} \in \pi (w) \\ M,w \models choose _{i} ( s _{i} ) &{} \text { iff } &{} choose _{i} ( s _{i} ) \in \pi (w) \end{array} \end{aligned}$$

We write $\models _{ MKM } \varphi$ to mean that the ${\textsf {LEG}}$-formula $\varphi$ is valid relative to the class of MKMs.

We have the following equivalence result:

Lemma 2

Let $\varphi \in \mathcal {L}_{{\textsf {LEG}}}({\varGamma }, Atm )$. Then, $\models \varphi$ iff $\models _{ MKM } \varphi$.

Step 2 The second step consists in introducing the class of weak multi-relational Kripke models (WMKMs) that are like MKMs except that they do not necessarily satisfy Conditions (C3 $^*$), (C4 $^*$) and (C5).

We write $\models _{ WMKM } \varphi$ to mean that the ${\textsf {LEG}}$-formula $\varphi$ is valid relative to the class of WMKMs.

Moreover, for any finite set ${\varDelta }$ of ${\textsf {LEG}}$-formulas, we write ${\varDelta }\models _{ WMKM } \varphi$ to mean that $\varphi$ is a logical consequence of the set of formulas ${\varDelta }$ relative to the class of WMKMs. That is, ${\varDelta }\models _{ WMKM } \varphi$ iff, for every weak multi-relational Kripke model M, if $M, w \models \bigwedge _{ \psi \in {\varDelta }} \psi$ for all $w \in W$, then $M, w \models \varphi$ for all $w \in W$.

The following Proposition 4 highlights that the validity problem relative to the class of MKMs is reducible to the logical consequence problem relative to the class of WMKMs.

Proposition 4

Let

$$\begin{aligned} {\varDelta }_0 =&\{\bigvee _{ s _{i} \in S _{i} } choose _{i} ( s _{i} ) : i \in Agt \} \cup \\&\{ choose _{i} ( s _{i} ) \rightarrow \lnot choose _{i} ( s _{i}' ) : i \in Agt \text { and } s _{i} , s _{i}' \in S _{i} \text { with } s _{i} \ne s _{i}' \} \cup \\&\{ choose _{i} ( s _{i} ) \rightarrow \mathsf { K}_{i} choose _{i} ( s _{i} ) : i \in Agt \text { and } s _{i} \in S _{i} \} \cup \\&\{ \widehat{\mathsf { K}}_{i} choose _{-i } ( s _{ -i } ) : i \in Agt \text { and } s _{ -i } \in S _{ Agt \setminus \{ i\} } \} \end{aligned}$$

Then, for every ${\textsf {LEG}}$-formula $\varphi$, $\models _{ MKM } \varphi$ iff ${\varDelta }_0 \models _{ WMKM } \varphi$.

Proof

We just need to observe that the (global) axioms in ${\varDelta }_0$ force a weak multi-relational Kripke models to satisfy Conditions (C3 $^*$), (C4 $^*$) and (C5). That is, M is a WMKM in which the formula $\bigwedge _{ \psi \in {\varDelta }_0 } \psi$ is true (i.e., $M,w \models \bigwedge _{ \psi \in {\varDelta }_0 } \psi$ for all w in M) iff M is a MKM. Therefore, the class of WMKMs in which the formula $\bigwedge _{ \psi \in {\varDelta }_0 } \psi$ is true coincides with the class of MKMs. $\square$

The following Proposition 5 highlights that, thanks to the common knowledge modality $\mathsf { CK}_{ }$, the logical consequence problem relative to the class of WMKMs can be reduced to the validity problem relative to the class of WMKMs.

Proposition 5

For every ${\textsf {LEG}}$-formula $\varphi$ and for every finite set ${\varDelta }$ of ${\textsf {LEG}}$-formulas, ${\varDelta }\models _{ WMKM } \varphi$ iff $\models _{ WMKM } \mathsf { CK}_{ } \bigwedge _{ \psi \in {\varDelta }} \psi \rightarrow \varphi$.

Step 3 The third step consists in providing an axiomatization result for ${\textsf {LEG}}$ relative to the class of WMKMs.

Lemma 3

The set of validities of the logic ${\textsf {LEG}}$ relative to the class of WMKMs is completely axiomatized by the groups of axioms (1), (2), (3) and (4) and by the rules of inference (6) and (7) in Fig. 4.

Proof

The proof just consists in adapting the completeness proof by Kraus and Lehmann (1988). Kraus and Lehmann provide an axiomatization result for the logic of belief, knowledge, common belief and common knowledge whose language is defined by the following grammar:

$$\begin{aligned} \varphi {:}{:}{=}p \mid \lnot \varphi \mid \varphi \wedge \psi \mid \mathsf { K}_{i} \varphi \mid \mathsf { B}_{i} \varphi \mid \mathsf { CK}_{ } \varphi \mid \mathsf { CB}_{ } \varphi \end{aligned}$$

and which is interpreted over structures that are like WMKMs except that they do not necessarily satisfy Condition (C6). Their axiomatization consists of the groups of axioms (1), (2) and (4), the rules of inference (6) and (7) and Axioms (3a), (3b), (3d), (3e), (3f) and (3g) in Fig. 4. It is a routine task to slightly modify Kraus and Lehmann’s proof by adding Condition (C6) in the semantics and proving that the resulting structures are completely axiomatized by their axioms and inference rules plus Axiom (3c).⁴ $\square$

The last element we need for proving Theorem 6 is the following Proposition 6. Let $\vdash _{ {\textsf {LEG}}} \varphi$ and $\Vvdash _{ {\textsf {LEG}}} \varphi$ mean, respectively, that the ${\textsf {LEG}}$-formula $\varphi$ is provable via the groups of axioms (1), (2), (3), (4) and (5) and the rules of inference (6) and (7) in Fig. 4 and that the ${\textsf {LEG}}$-formula $\varphi$ is provable via the groups of axioms (1), (2), (3) and (4) and the rules of inference (6) and (7) in Fig. 4.

Proposition 6

For every ${\textsf {LEG}}$-formula $\varphi$, if $\Vvdash _{ {\textsf {LEG}}} \mathsf { CK}_{} \bigwedge _{ \psi \in {\varDelta }_0 } \psi \rightarrow \varphi$ then $\vdash _{ {\textsf {LEG}}} \varphi$, where ${\varDelta }_0$ is defined as in Proposition 4.

Proof

Suppose $\Vvdash _{ {\textsf {LEG}}} \mathsf { CK}_{} \bigwedge _{ \psi \in {\varDelta }_0 } \psi \rightarrow \varphi$. Hence, $\vdash _{ {\textsf {LEG}}} \mathsf { CK}_{} \bigwedge _{ \psi \in {\varDelta }_0 } \psi \rightarrow \varphi$.

By the inference rule (7) (viz. necessitation for $\mathsf { CK}_{}$) and the group of axioms (5), we have $\vdash _{ {\textsf {LEG}}} \bigwedge _{ \psi \in {\varDelta }_0 } \mathsf { CK}_{} \psi$. By Axiom 2(e), we can derive $\vdash _{ {\textsf {LEG}}} \mathsf { CK}_{} \bigwedge _{ \psi \in {\varDelta }_0 } \psi$.

Consequently, by the inference rule (6) (viz. modus ponens), we have that $\vdash _{ {\textsf {LEG}}} \varphi$.

$\square$

Propositions 4, 5 and 6 together with Lemmas 2 and 3 are sufficient to prove Theorem 6.

Suppose that $\models _{ {\textsf {LEG}}} \varphi$. Hence, by Lemma 2, $\models _{ MKM } \varphi$. Hence, by Propositions 4 and 5, $\models _{ WMKM } \mathsf { CK}_{} \bigwedge _{ \psi \in {\varDelta }_0 } \psi \rightarrow \varphi$. By Lemma 3, it follows that $\Vvdash _{ {\textsf {LEG}}} \mathsf { CK}_{} \bigwedge _{ \psi \in {\varDelta }_0 } \psi \rightarrow \varphi$. Hence, by Proposition 6, $\vdash _{ {\textsf {LEG}}} \varphi$. $\square$

Appendix 5: Proof of Theorem 7

The satisfiability problem of ${\textsf {LEG}}$ relative to the class of epistemic game models is decidable in exponential time.

Proof

The following Lemma 4 follows from the fact that ${\textsf {LEG}}$ is an extension of the logic of common knowledge whose satisfiability problem is ExpTime-complete Fagin et al. (1995).

Lemma 4

The satisfiability problem of ${\textsf {LEG}}$ relative to the class of epistemic game models is ExpTime-hard.

In order to prove that the satisfiability problem of ${\textsf {LEG}}$ is in ExpTime, we are going to embed ${\textsf {LEG}}$ into the decidable logic S5-${\text {PDL}}$, i.e., the variant of propositional dynamic logic ${\text {PDL}}$Harel et al. (2000) in which atomic programs are interpreted by means of equivalence relations.

Let $Atm ^+ = Atm \cup \{ choose _{i} ( s _{i} ) {:}i \in Agt \text { and } s _{i} \in S _{i} \} \cup \{ poss _{i} {:}i \in Agt \}$. The language $\mathcal {L}_{ \text {S5}-{\text {PDL}}}( Atm ^+, Agt )$ of S5-${\text {PDL}}$ is defined as follows:

$$\begin{aligned} \pi {:}{:}{=}&\; \sim _i \mid \pi _1 {;} \pi _2 \mid \pi _1 \cup \pi _2 \mid \pi ^* \mid ?\varphi \\ \varphi {:}{:}{=}&\; p \mid \lnot \varphi \mid \varphi \wedge \varphi \mid [\pi ] \varphi \end{aligned}$$

where p ranges over $Atm ^+$ and i ranges over $Agt$. The dual of the operator $[\pi ]$ is defined in the standard way as follows: $\langle \pi \rangle \varphi ~\mathop {=}\limits ^{\text{ def }}~\lnot [\pi ] \lnot \varphi$.

S5-${\text {PDL}}$ models are tuples $M = \langle W, \mathcal {R}_{ \sim _1} , \ldots , \mathcal {R}_{ \sim _n}, \mathcal {I} \rangle$ where:

• W is a set of worlds;

• Every $\mathcal {R}_{ \sim _i }$ is an equivalence relation on W;

• $\mathcal {I} : Atm ^+ \longrightarrow 2^W$ is a valuation function.

Binary relations for complex programs are defined in the standard way as follows:

$$\begin{aligned} \mathcal {R}_{ \pi _1 {;} \pi _2 }&= \mathcal {R}_{ \pi _1 } {;} \mathcal {R}_{ \pi _2 }\\ \mathcal {R}_{ \pi _1 \cup \pi _2 }&= \mathcal {R}_{ \pi _1 } \cup \mathcal {R}_{ \pi _2 }\\ \mathcal {R}_{ \pi ^* }&= ( \mathcal {R}_{ \pi } )^*\\ \mathcal {R}_{ ?\varphi }&= \{ (w,w) {:}w \in W \text { and } M,w \models \varphi \} \end{aligned}$$

Truth conditions of S5-${\text {PDL}}$ formulae are the standard ones for Boolean operators plus the following one for the operator $[\pi ]$:

$$\begin{aligned} M,w \models [\pi ] \varphi \text { iff } M, v \models \varphi \text { for all } v \in \mathcal {R}_\pi (w) \end{aligned}$$

For any formula $\varphi$ of the language $\mathcal {L}_{ \text {S5}-{\text {PDL}}}( Atm ^+, Agt )$, we write $\models _{ \text {S5}-{\text {PDL}}} \varphi$ if $\varphi$ is S5-${\text {PDL}}$ valid, that is, if $\varphi$ is true in all S5-${\text {PDL}}$ models (i.e., for all S5-${\text {PDL}}$ models M and for all worlds w in M, we have $M,w \models \varphi$). We say that $\varphi$ is S5-${\text {PDL}}$ satisfiable if $\lnot \varphi$ is not S5-${\text {PDL}}$ valid. Moreover, we shall say that $\varphi$ is a global logical consequence in S5-${\text {PDL}}$ of a finite set of global axioms ${\varDelta }= \{ \chi _1, \ldots , \chi _n \}$, denoted by ${\varDelta }\models _{ \text {S5}-{\text {PDL}}} \varphi$, if and only if for every S5-${\text {PDL}}$ model M, if ${\varDelta }$ is true in M (i.e., for every world w in M, we have $M,w \models \chi _1 \wedge \ldots \wedge \chi _n$) then $\varphi$ is true in M too (i.e., for every world w in M, we have $M,w \models \varphi$).

Proposition 7

The satisfiability problem of S5-${\text {PDL}}$ is in ExpTime.

Proof

The logic S5-${\text {PDL}}$ is polynomially embeddable into ${\text {PDL}}$ extended with converse, by simulating S5 program $\sim _i$ with the composite program $( x_{ i} \cup -x_{ i} )^*$ where $x_{ i}$ is an arbitrary atomic program interpreted by means of the binary relation $\mathcal {R}_{ x_{ i} }$, and $-x_{ i}$ is the converse of $x_{ i}$ . (Note indeed that relation $\mathcal {R}_{( x_{ i} \cup -x_{ i} )^*}$ is an equivalence relation.) The satisfiability problem of ${\text {PDL}}$ with converse has been proved to be ExpTime-complete Vardi (1985). It follows that the satisfiability problem of S5-${\text {PDL}}$ is in ExpTime. $\square$

We define the following translation from the ${\textsf {LEG}}$ language $\mathcal {L}_{{\textsf {LEG}}}({\varGamma }, Atm )$ to $\mathcal {L}_{ S5-{\text {PDL}}}( Atm ^+, Agt )$ for $p \in Atm$ and $i \in Agt$:

$$\begin{aligned} tr(p)&= p \\ tr( choose _{i} ( s _{i} ) )&= choose _{i} ( s _{i} ) \\ tr( poss _{i} )&= poss _{i} \\ tr( \lnot \varphi )&= \lnot tr( \varphi ) \\ tr( \varphi \wedge \psi )&= tr( \varphi ) \wedge tr( \psi ) \\ tr( \mathsf { K}_{i} \varphi )&= [ \sim _i ] tr(\varphi ) \\ tr( \mathsf { B}_{i} \varphi )&= [ \sim _i ] ( poss _{i} \rightarrow tr(\varphi ) )\\ tr( \mathsf { CK}_{ } \varphi )&= [ (\sim _1 \cup \ldots \cup \sim _n)^* ] tr( \varphi ) \\ tr( \mathsf { CB}_{ } \varphi )&= [ (\sim _1 {;} ? poss _{1} \cup \ldots \cup \sim _n {;} ? poss _{n} )^* ] tr( \varphi ) \end{aligned}$$

As the following proposition highlights, the validity problem in ${\textsf {LEG}}$ can be reduced to the problem of (global) logical consequence in S5-${\text {PDL}}$.

Proposition 8

A ${\textsf {LEG}}$ formula $\varphi$ is ${\textsf {LEG}}$ valid, i.e., $\models _{ {\textsf {LEG}}} \varphi$, if and only if $tr(\varphi )$ is a logical consequence of the set of global axioms ${\varDelta }_0$ in S5-${\text {PDL}}$, i.e., ${\varDelta }_0 \models _{ \text {S5}-{\text {PDL}}} tr (\varphi )$, where:

$$\begin{aligned} {\varDelta }_0 =&\{\bigvee _{ s _{i} \in S _{i} } choose _{i} ( s _{i} ) : i \in Agt \} \cup \\&\{ choose _{i} ( s _{i} ) \rightarrow \lnot choose _{i} ( s _{i}' ) : i \in Agt \text { and } s _{i} , s _{i}' \in S _{i} \text { with } s _{i} \ne s _{i}' \} \cup \\&\{ choose _{i} ( s _{i} ) \rightarrow \mathsf { K}_{i} choose _{i} ( s _{i} ) : i \in Agt \text { and } s _{i} \in S _{i} \} \cup \\&\{ \widehat{\mathsf { K}}_{i} choose _{-i } ( s _{ -i } ) : i \in Agt \text { and } s _{ -i } \in S _{ Agt \setminus \{ i\} } \} \end{aligned}$$

Proof

(Sketch) ($\Rightarrow$) Take an arbitrary EGM $M = ( W, \mathcal {E}_{1 } , \ldots , \mathcal {E}_{n} , \mathcal {B}_{1 } , \ldots , \mathcal {B}_{n} , \mathsf {ch}_{1} , \ldots , \mathsf {ch}_{n} , \mathcal {V})$ which satisfies formula $\varphi$. We build a corresponding S5-${\text {PDL}}$ model $M' = \langle W, \mathcal {R}_{ \sim _1 } , \ldots , \mathcal {R}_{ \sim _n } , \mathcal {I} \rangle$ such that:

• For all $i \in AGT$, $\mathcal {R}_{ \sim _i } = \mathcal {E}_{i}$;

• For all $p \in Atm$, $\mathcal {I} (p) =\mathcal {V}(p)$;

• For all $i \in Agt$, for all $s _{i} \in S _{i}$ and for all $w \in W$, $w \in \mathcal {I}( poss _{i} )$ iff $w \in \mathcal {B}_{i } (w)$;

• For all $i \in Agt$ and for all $w \in W$, $w \in \mathcal {I}( choose _{i} ( s _{i} ) )$ iff $\mathsf {ch}_{i} (w) = s _{i}$.

By induction on the structure of $\varphi$, it is easy to check that $tr(\varphi )$ is satisfied by $M'$ and that for every $\chi \in {\varDelta }_0$, $M',w \models \chi$ for all $w \in W$.

($\Leftarrow$) Take an arbitrary S5-${\text {PDL}}$ model $M = \langle W, \mathcal {R}_{ \sim _1 } , \ldots , \mathcal {R}_{ \sim _n } , \mathcal {I} \rangle$ which satisfies $tr (\varphi )$ such that that for all $\chi \in {\varDelta }_0$ and for all $w \in W$, $M,w \models \chi$. We build a corresponding EGM $M' = ( W, \mathcal {E}_{1 } , \ldots , \mathcal {E}_{n} , \mathcal {B}_{1 } , \ldots , \mathcal {B}_{n} , \mathsf {ch}_{1} , \ldots , \mathsf {ch}_{n} , \mathcal {V})$ such that:

• For all $i \in AGT$, $\mathcal {E}_{i} = \mathcal {R}_{ \sim _i }$;

• For all $i \in AGT$, $\mathcal {B}_{i} = \{ (w,v) {:}w \mathcal {E}_{i} v \text { and } M,v \models poss _{i} \}$;

• For all $i \in Agt$, for all $s _{i} \in S _{i}$ and for all $w \in W$, $\mathsf {ch}_{i} (w) = s _{i}$ iff $w \in \mathcal {I}( choose _{i} ( s _{i} ) )$;

• For all $p \in Atm$, $\mathcal {V}(p) = \mathcal {I} (p)$.

Again, by induction on the structure of $\varphi$, it is easy to check that $\varphi$ is satisfied by $M'$. $\square$

From Propositions 7 and we obtain the following upper boud for the complexity of the satisfiability problem of ${\textsf {LEG}}$.

Lemma 5

The satisfiability problem of ${\textsf {LEG}}$ relative to the class of epistemic game models is in ExpTime.

Proof

It is a routine task to verify that the problem of global logical consequence in S5-${\text {PDL}}$ with a finite number of global axioms is reducible to the problem of validity in S5-${\text {PDL}}$. In particular, if ${\varDelta }= \{ \chi _1, \ldots , \chi _m \}$ we have ${\varDelta }\models _{ \text {S5}-{\text {PDL}}} \varphi$ if and only if $\models _{ \text {S5}-{\text {PDL}}} [ \mathbf {any} ^* ] (\chi _1 \wedge \ldots \chi _m) \rightarrow \varphi$ where $\mathbf {any}$ is the special program defined as $\mathbf {any} ~\mathop {=}\limits ^{\text{ def }}~(\bigcup _{i \in Agt } \sim _i \cup \equiv )$. Hence, by Proposition  and by the fact that ${\varDelta }_0$ is finite, it follows that a ${\textsf {LEG}}$ formula $\varphi$ is ${\textsf {LEG}}$ valid if and only if $\models _{ \text {S5}-{\text {PDL}}} [ \mathbf {any} ^* ] (\bigwedge _{ \chi \in {\varDelta }_0 } \chi ) \rightarrow tr(\varphi )$. Consequently, given the fact that tr is a polynomial reduction of ${\textsf {LEG}}$ to S5-${\text {PDL}}$ and given also the fact that the satisfiability problem of S5-${\text {PDL}}$ is in ExpTime (Proposition 7), it follows that the satisfiability problem of ${\textsf {LEG}}$ is also in ExpTime. $\square$

Theorem 7 follows from Lemmas 4 and 5. $\square$