Towards a Logic for Conditional Local Strategic Reasoning

Abstract

We consider systems of rational agents who act in pursuit of their individual and collective objectives and we study the reasoning of an agent or an external observer about the consequences from the expected choices of action of the other agents based on their objectives, in order to assess the reasoner’s ability to achieve his own objective.

To formalize such reasoning we introduce new modal operators of conditional strategic reasoning and use them to extend Coalition Logic in order to capture variations of conditional strategic reasoning. We provide formal semantics for the new conditional strategic operators, introduce the matching notion of bisimulation for each of them and discuss and compare briefly their expressiveness.

The work of Valentin Goranko was supported by a research grant 2015-04388 of the Swedish Research Council. The work of Fengkui Ju was supported by the Major Program of the National Social Science Foundation of China (NO. 17ZDA026). We thank the reviewers for some helpful remarks.

Appendix: Some Examples

Example 1

The game model $\mathcal {M}$ below has two players, $\mathsf {a}$ and $\mathsf {b}$. Each has two actions at state $s_0$: $a_1, a_2$, resp. $b_1, b_2$.

It can be verified that $\mathcal {M}, s_0 \Vdash \langle \!\langle \mathrm {\mathsf {a}}\rangle \!\rangle _\mathsf {c}(p; \langle \mathrm {\mathsf {b}} \rangle q)$, while $\mathcal {M}, s_0 \not \Vdash [\mathrm {\mathsf {b}}] q$. Thus, an agent may have only conditional ability to achieve its goal.

Example 2

The game model $\mathcal {M}$ below has two players, $\mathsf {a}$ and $\mathsf {b}$.

It can be verified that $\mathcal {M}, s_0 \Vdash [\mathrm {\mathsf {a}}]_\mathsf {dr}(p; \langle \mathrm {\mathsf {b}} \rangle q)$. However, $\mathcal {M}, s_0$ does not satisfy the $\mathsf {ATL^*}$ formula $[\![\mathrm {\mathsf {a}}]\!] (\mathrm {X} p \rightarrow \langle \!\langle {\mathrm {\mathsf {b}}}\rangle \!\rangle \mathrm {X} q)$, hence these are not equivalent.

Also, $\mathcal {M}, s_0 \not \Vdash [\mathrm {\mathsf {a}}]_\mathsf {dd}(p; \langle \mathrm {\mathsf {b}} \rangle q)$. However, if the outcomes of $({a_2}, {b_1})$ and $({a_2}, {b_2})$ are swapped, then $[\mathrm {\mathsf {a}}]_\mathsf {dd}(p; \langle \mathrm {\mathsf {b}} \rangle q)$ becomes true at $s_0$ in the resulting model.

Example 3

The game model $\mathcal {M}$ below involves two players: $\mathsf {a}$ and $\mathsf {b}$. It can be verified that $\mathcal {M}, s_0 \Vdash [\mathrm {\mathsf {a}}]_\mathsf {dr}(p; \langle \mathrm {\mathsf {b}} \rangle q)$ but $\mathcal {M}, s_0 \not \Vdash [\mathrm {\mathsf {a}}]_\mathsf {dd}(p; \langle \mathrm {\mathsf {b}} \rangle q)$.

Example 4

The game models $\mathcal {M}_1$ and $\mathcal {M}_2$ below involve three players: $\mathsf {a}$, $\mathsf {b}$, $\mathsf {c}$. It can be verified that:

• (1) The relation $\beta = \{ (s_i,t_i) \mid i = 0,1,2,3 \}$ is an alternating bisimulation between $\mathcal {M}_1$ and $\mathcal {M}_2$ (cf. [1]).

• (2) $\mathcal {M}_1, s_0 \Vdash \langle \!\langle \mathrm {\mathsf {a}}\rangle \!\rangle _\mathsf {c}(p; \langle \mathrm {\mathsf {b}} \rangle q)$ but $\mathcal {M}_2, t_0 \not \Vdash \langle \!\langle \mathrm {\mathsf {a}}\rangle \!\rangle _\mathsf {c}(p; \langle \mathrm {\mathsf {b}} \rangle q)$.

Example 5

The game models $\mathcal {M}_1$ and $\mathcal {M}_2$ below involve two players: $\mathsf {a}$ and $\mathsf {b}$. It can be verified that:

• (1) The relation $\beta = \{ (s_i,t_i) \mid i = 0,1,2,3 \}$ is an alternating bisimulation between $\mathcal {M}_1$ and $\mathcal {M}_2$ (cf. [1]).

• (2) $\mathcal {M}_1, s_0 \Vdash [\mathrm {\mathsf {a}}]_\mathsf {c}(p | q)$ but $\mathcal {M}_2, t_0 \not \Vdash [\mathrm {\mathsf {a}}]_\mathsf {c}(p | q)$.