# Logics for Strategic Reasoning of Socially Interacting Rational Agents: An Overview and Perspectives

## Abstract

**:**

## 1. Introduction

- The STIT-based approach, building on the theory of “Seeing To It That” (STIT), originating from the seminal work of Belnap and Perloff [3]. For exploring the closer links of the present paper with STIT, the reader is referred to [13] and the overview chapter [14] on using STIT for strategic reasoning, as well as to [15] for temporal extensions of STIT and applications to normative reasoning, to [16] for using STIT for reasoning about social influence, and to [17] for providing semantics of temporal STIT in concurrent game models used in this paper.

#### Structure of the paper

## 2. Technical Preliminaries

#### 2.1. Concurrent Game Models

**agents**$\mathsf{Agt}=\{{\mathsf{a}}_{1},\dots ,{\mathsf{a}}_{n}\}$, also called here

**players**, and a fixed countable set of

**atomic propositions**$\mathsf{Prop}$. Subsets of $\mathsf{Agt}$ will also be called

**coalitions**. The

**Cartesian product**of a family of sets ${\left\{{X}_{\mathsf{a}}\right\}}_{\mathsf{a}\in \mathsf{Agt}}$ is the set of tuples $({x}_{1},\dots ,{x}_{n})$ where ${x}_{i}\in {X}_{{\mathsf{a}}_{i}}$ or each $i=1,\dots ,n$, denoted, as usual, by ${\mathsf{\Pi}}_{\mathsf{a}\in \mathsf{Agt}}{X}_{\mathsf{a}}$.

**Definition 1.**

**(strategic) game form for the set of agents**$\mathsf{Agt}$

**over the set of outcomes**$\mathsf{O}$ is a tuple

- $\mathsf{Act}$ is a nonempty set of
**actions**; - $\mathsf{act}:\mathsf{Agt}\to {\mathcal{P}}^{+}\left(\mathsf{Act}\right)$ is a mapping assigning to each $\mathsf{a}\in \mathsf{Agt}$ a nonempty set $\mathsf{act}\left(\mathsf{a}\right)$ of
**actions available to the player**$\mathsf{a}$; - $\mathsf{out}:{\mathsf{\Pi}}_{\mathsf{a}\in \mathsf{Agt}}\mathsf{act}\left(\mathsf{a}\right)\to \mathsf{O}$ is a map assigning to every
**action profile**$\zeta \in {\mathsf{\Pi}}_{\mathsf{a}\in \mathsf{Agt}}\mathsf{act}\left(\mathsf{a}\right)$ a unique**outcome**in $\mathsf{O}$.

**Definition 2.**

**concurrent game model (CGM)**is a tuple

- $\mathsf{S}$ is a nonempty set of
**(game) states**; - $\mathsf{Act}$ is a nonempty set of
**actions**; - $\mathfrak{g}$ is a
**game map**, assigning to each state $w\in \mathsf{S}$ a strategic game form$\mathfrak{g}\left(w\right)=(\mathsf{Act},{\mathsf{act}}_{w},\mathsf{S},{\mathsf{out}}_{w})$ over the set of outcomes $\mathsf{S}$; - $\mathsf{Prop}$ is a countable set of
**atomic propositions**; - $\mathsf{L}:\mathsf{S}\to \mathcal{P}\left(\mathsf{Prop}\right)$ is a
**labeling function**, assigning to every state in $\mathsf{S}$ the set of atomic propositions true at that state.

**locally available actions**for $\mathsf{a}$ in w. We can now define the

**global action function**$\mathsf{act}:\mathsf{Agt}\times \mathsf{S}\to {\mathcal{P}}^{+}\left(\mathsf{Act}\right)$ by setting $\mathsf{act}(\mathsf{a},w)\u2a74{\mathsf{act}}_{w}\left(\mathsf{a}\right)$. We also define the set ${\mathsf{Act}}_{\mathsf{a}}\u2a74{\bigcup}_{w\in \mathsf{S}}{\mathsf{act}}_{w}\left(\mathsf{a}\right)$ of

**possible actions**for $\mathsf{a}$.

- An action profile $\zeta \in {\mathsf{\Pi}}_{\mathsf{a}\in \mathsf{Agt}}{\mathsf{Act}}_{\mathsf{a}}$ is
**available at the state**w if it consists of locally available actions, i.e., if $\zeta \in {\mathsf{\Pi}}_{\mathsf{a}\in \mathsf{Agt}}{\mathsf{act}}_{w}\left(\mathsf{a}\right)$.The set of all action profiles that are available at w will be denoted by ${\mathsf{ActProf}}_{w}$. - ${\mathsf{out}}_{\mathcal{M}}$ is the
**global outcome function**assigning to every state w and an action profile $\zeta \in {\mathsf{ActProf}}_{w}$ the unique**outcome**${\mathsf{out}}_{\mathcal{M}}(w,\zeta )\u2a74{\mathsf{out}}_{w}\left(\zeta \right)$.When $\mathcal{M}$ is fixed by the context, it will be omitted from the subscript. - More generally, given a coalition $C\subseteq \mathsf{Agt}$, a
**joint action**for C in $\mathcal{M}$ is a tuple of individual actions ${\zeta}_{C}\in {\prod}_{\mathsf{a}\in C}\mathsf{act}\left(\mathsf{a}\right)$. In particular, for any action profile $\zeta \in {\mathsf{\Pi}}_{\mathsf{a}\in \mathsf{Agt}}\mathsf{act}\left(\mathsf{a}\right)$, ${\zeta |}_{C}$ is the joint action obtained by restricting $\zeta $ to C. The joint action ${\zeta |}_{C}$ is**locally available at the state**w iff every individual action in it is locally available for the respective agent in w. - For any $w\in \mathsf{S}$, $C\subseteq \mathsf{Agt}$, and joint action ${\zeta}_{C}$ that is available at w, we define
**the set of possible outcomes from the application of the joint action**${\zeta}_{C}$**at the state**w:$$\mathsf{Out}[w,{\zeta}_{C}]=\left\{u\in \mathsf{S}\mid \mathrm{there}\mathrm{exists}\phantom{\rule{3.33333pt}{0ex}}\zeta \in {\mathsf{ActProf}}_{w}{\mathrm{such}\mathrm{that}\zeta |}_{C}={\zeta}_{C}\mathrm{and}\mathsf{out}(w,\zeta )=u\right\}.$$In the special case when $C=\mathsf{Agt}$, ${\zeta}_{\mathsf{Agt}}$ is a full available action profile, so $\mathsf{Out}[w,{\zeta}_{\mathsf{Agt}}]$ consists of the single outcome $\mathsf{out}(w,{\zeta}_{\mathsf{Agt}})$.

**Remark 3.**

**Example 4**

**Adam**and

**Eve**, live in hotel “Life”. It has (at least) three rooms: $R1,R2,andR3$. Initially, both Adam and Eve are in room $R1$. Every day, each of them is able to choose either to stay in the same room (action $\mathsf{stay}$, denoted by $\mathsf{s}$), or to move to another room (action $\mathsf{move}$, denoted by $\mathsf{m}$), with some restrictions described further. Whichever room each of them chooses, they stay there for the night, and then each chooses again to either stay in the same room, or move to another room, as specified in the model. In some cases, they also have the option to “retreat” (action $\mathsf{retreat}$, denoted by $\mathsf{r}$).

- The set of agents is $\mathsf{Agt}=\{\mathit{Adam},\mathit{Eve}\}$.
- The set of game states is $\mathsf{St}=\{{A}_{1}{E}_{1},{A}_{1}{E}_{2},{A}_{2}{E}_{1},{A}_{2}{E}_{2},{A}_{3}{E}_{1},{A}_{1}{E}_{3},{A}_{3}{E}_{2},{A}_{2}{E}_{3}\}$ and the names of the states represent the current locations of the agents, e.g., ${A}_{1}{E}_{2}$ means that
**Adam**is in $R1$ and**Eve**is in $R2$, etc. Thus, the names of the states can also be interpreted as pairs of atomic propositions saying “**Adam**is in room X” and “**Eve**is in room Y”. These atomic propositions, however, will not feature in the example. - There are three atomic propositions of interest for us: $\mathsf{Prop}=\{T,{H}_{A},{H}_{E}\}$, where:
- -
- T, which is true in a state iff
**Adam**and**Eve**are in the same room (“together”), i.e., in ${A}_{1}{E}_{1}$ and ${A}_{2}{E}_{2}$, as indicated in their labels; - -
- ${H}_{A}$, meaning “
**Adam**is happy”, true in the states where ${H}_{A}$ is listed in the label; - -
- ${H}_{E}$, meaning “
**Eve**is happy”, true in the states where ${H}_{E}$ is listed in the label.

- The global set of actions is $\mathsf{Act}=\{\mathsf{stay},\mathsf{move},\mathsf{retreat}\}$, denoted in the figure, respectively, by $\mathsf{s},\mathsf{m},\mathsf{r}$. The transitions caused by the respective ordered pairs of actions are indicated on the figure (the first for
**Adam**, and the second for**Eve**). For example, if at state ${A}_{1}{E}_{1}$ both**Adam**and**Eve**choose to stay, then the transition, labeled with (**s**,**s**), is looping back to the same state, whereas if**Adam**chooses to stay and**Eve**chooses to move, the transition, labeled with (**s**,**m**), leads to the state ${A}_{1}{E}_{2}$, etc. These transitions also define the global outcome function $\mathsf{Out}$. - The global action function $\mathsf{act}$ (defining the available actions to every agent at every state) can be readily extracted from the figure. For example, $\mathsf{act}(\mathit{Adam},{A}_{1}{E}_{2})=\{\mathsf{s},\mathsf{m}\}$, $\mathsf{act}(\mathit{Eve},{A}_{1}{E}_{2})=\{\mathsf{s},\mathsf{m},\mathsf{r}\}$, etc.
- Lastly, the labeling function $\mathsf{L}:\mathsf{St}\to \mathcal{P}\left(\mathsf{Prop}\right)$ is explicitly given in the figure, where the label of each state is given in $\{\dots \}$.

#### 2.2. Plays and Strategies in Concurrent Game Models

**partial play**, or a

**history**in $\mathcal{M}$ is either an element of $\mathsf{S}$ or a finite word of the form:

**(initial) path induced by**h and denoted $\mathsf{path}\left(h\right)$.

**(memory-based) strategy for player**$\mathsf{a}$ is a map ${\sigma}_{\mathsf{a}}$ assigning to each history $h={w}_{0}{\zeta}_{0}\dots {\zeta}_{n-1}{w}_{n}$ in $\mathsf{Hist}\left(\mathcal{M}\right)$ an action ${\sigma}_{\mathsf{a}}\left(h\right)$ from $\mathsf{act}(\mathsf{a},{w}_{n})$. A strategy ${\sigma}_{\mathsf{a}}$ is

**path-based**, if it assigns actions only based on the path generated by the history (not taking into account the intermediate action profiles), i.e., ${\sigma}_{\mathsf{a}}\left(h\right)={\sigma}_{\mathsf{a}}\left({h}^{\prime}\right)$ whenever $\mathsf{path}\left(h\right)=\mathsf{path}\left({h}^{\prime}\right)$. A strategy ${\sigma}_{\mathsf{a}}$ is

**memoryless**, or

**positional**, if it assigns actions only based on the current (last) state, i.e., ${\sigma}_{\mathsf{a}}\left(h\right)={\sigma}_{\mathsf{a}}\left({h}^{\prime}\right)$ whenever $\mathit{l}\left(h\right)=\mathit{l}\left({h}^{\prime}\right)$.

**joint strategy**for C in the model $\mathcal{M}$ is a tuple ${\mathsf{\Sigma}}_{C}$ of individual strategies, one for each player in C. For every joint strategy ${\mathsf{\Sigma}}_{C}$ and a history h, we denote by ${\mathsf{\Sigma}}_{C}\left(h\right)$ the joint action prescribed by ${\mathsf{\Sigma}}_{C}$ on h.

**(global) strategy profile**$\mathsf{\Sigma}$ is a joint strategy for the grand coalition $\mathsf{Agt}$, i.e., an assignment of a strategy to each player. We denote the set of all strategy profiles in the model $\mathcal{M}$ by ${\mathsf{StratProf}}_{\mathcal{M}}$ and the set of all joint strategies for a coalition C in $\mathcal{M}$ by ${\mathsf{StratProf}}_{\mathcal{M}}\left(C\right)$. Thus, ${\mathsf{StratProf}}_{\mathcal{M}}={\mathsf{StratProf}}_{\mathcal{M}}\left(\mathsf{Agt}\right)$.

**play induced by**$\mathsf{\Sigma}$

**at**$w\in \mathsf{S}$ is the unique infinite word

**path induced by**$\mathsf{\Sigma}$

**at**w, denoted $\mathsf{path}(\mathsf{\Sigma},w)$.

- Consider the strategy ${\sigma}_{1}$ which prescribes the action $\mathsf{s}$ if Adam and Eve are currently together, or else $\mathsf{m}$, if possible, otherwise, again, $\mathsf{s}$. If both Adam and Eve follow that strategy starting from ${A}_{1}{E}_{1}$, the induced play is$\mathsf{play}({A}_{1}{E}_{1},({\sigma}_{1},{\sigma}_{1}))={A}_{1}{E}_{1}(\mathsf{s},\mathsf{s}){A}_{1}{E}_{1}(\mathsf{s},\mathsf{s})\dots $Respectively, the induced play starting from ${A}_{1}{E}_{2}$ is$\mathsf{play}({A}_{1}{E}_{2},({\sigma}_{1},{\sigma}_{1}))={A}_{1}{E}_{2}(\mathsf{m},\mathsf{m}){A}_{2}{E}_{1}(\mathsf{m},\mathsf{m}){A}_{1}{E}_{2}(\mathsf{m},\mathsf{m})\dots $
- Consider the strategy ${\sigma}_{2}$ which prescribes the action $\mathsf{s}$ if the player is currently happy, or else $\mathsf{r}$ if possible, otherwise $\mathsf{m}$, if possible, otherwise, $\mathsf{s}$.If both Adam and Eve follow ${\sigma}_{2}$ starting from ${A}_{1}{E}_{1}$, the induced play is$\mathsf{play}({A}_{1}{E}_{1},({\sigma}_{2},{\sigma}_{2}))={A}_{1}{E}_{1}(\mathsf{m},\mathsf{m}){A}_{2}{E}_{2}(\mathsf{s},\mathsf{s}){A}_{2}{E}_{2}(\mathsf{s},\mathsf{s})\dots $
- If Adam follows ${\sigma}_{1}$ and Eve follows ${\sigma}_{2}$, the induced play starting from ${A}_{1}{E}_{1}$ is$\mathsf{play}({A}_{1}{E}_{1},({\sigma}_{1},{\sigma}_{2}))={A}_{1}{E}_{1}(\mathsf{s},\mathsf{m}){A}_{1}{E}_{2}(\mathsf{m},\mathsf{r}){A}_{1}{E}_{3}(\mathsf{m},\mathsf{s}){A}_{2}{E}_{3}(\mathsf{m},\mathsf{s}){A}_{1}{E}_{3}(\mathsf{m},\mathsf{s})\dots $
- Lastly, if Adam follows ${\sigma}_{1}$ and Eve follows the strategy ${\sigma}_{3}$ which prescribes the action $\mathsf{s}$ only if both players are currently happy or if no other action is available, or else $\mathsf{r}$ if possible, otherwise $\mathsf{m}$, then the induced play starting from ${A}_{1}{E}_{1}$ is this happy ending:$\mathsf{play}({A}_{1}{E}_{1},({\sigma}_{1},{\sigma}_{3}))={A}_{1}{E}_{1}(\mathsf{s},\mathsf{m}){A}_{1}{E}_{2}(\mathsf{m},\mathsf{r}){A}_{1}{E}_{3}(\mathsf{m},\mathsf{m}){A}_{2}{E}_{2}(\mathsf{s},\mathsf{s}){A}_{2}{E}_{2}(\mathsf{s},\mathsf{s})\dots $

**set of outcome plays induced by the joint strategy**${\mathsf{\Sigma}}_{C}$

**at**w is the set of plays

## 3. A Variety of Strategic Abilities

- Strictly competitive and unconditional, where all agents, respectively, coalitions, act only in pursuit of their own goals and can be assumed to regard all others either as adversaries, or as behaving randomly. An alternative way of thinking here is that these are abilities of a given agent, respectively, coalition, to achieve goals independently of the actions of all other agents. Both interpretations make good sense in the context of this paper. The typical claim for such kind of local (immediate, one step) abilities is:“The coalition A has a joint action to ensure satisfaction of the coalitional goal of A in every outcome state that may result from that joint action”.This is the informal semantic reading of the strategic operator ${\left[A\right]}_{{}_{\phantom{\rule{-0.166667em}{0ex}}}}$ in the coalition logic $\mathsf{CL}$ [4,5], which will be presented in Section 4.Respectively, here is a typical long-term strategic ability claim:“The coalition A has a joint strategy to ensure satisfaction of the coalitional goal of A in every outcome play resulting from A following that strategy”.
- Competitive, but conditional on the other agents’ expected actions, where coalitions (respectively, agents) still act only in pursuit of their own goals, but, when deciding on their course of action, they take into account the goals and the respective expected actions of the other players, so they are not treated as adversaries (or, behaving randomly), but as rational agents pursuing their own goals. Here is a typical such claim expressing conditional strategic ability:“For some (or, every) joint action of the coalition A that ensures satisfaction of its goal ${\gamma}_{A}$, the coalition B has a joint action of its own to ensure satisfaction of its goal ${\gamma}_{B}$.There are (at least) two naturally arising readings of such conditional claims, as “proactive ability” and as “reactive ability”, and two respective versions of local strategic operators formalizing them. These were first introduced in [27] (respectively, as “de dicto” and “de re” abilities) and further studied in [28]. Conditional abilities will be discussed in more detail in Section 6.
- Socially cooperating abilities, where agents and coalitions still act in pursuit of their own goals, but when deciding on their course of action take into account the goals of other agents in the system and make allowance, if possible, for their satisfaction, too. Thus, agents and coalitions are assumed, i.e., not only rational but also cooperative, whenever possible to reconcile their interests with those of the others. Two natural examples of strategic operators formalizing such abilities are:
- (a)
- the “cooperative ability” operator ${\mathsf{O}}_{c}$, again introduced and studied in [27,28], which, when applied to (disjoint) coalitions A and B with respective goals $\varphi $ and $\psi $, formalizes the statement saying that“$\mathrm{A}$ has a joint action ${\sigma}_{\mathrm{A}}$ which guarantees the satisfaction of $\varphi $ and enables $\mathrm{B}$ to also apply a joint action ${\sigma}_{\mathrm{B}}$ that guarantees $\psi $”.This will be presented in more detail in Section 6.
- (b)
- The “socially friendly” coalitional operator SF, introduced and studied in [25], which is a somewhat more general version of ${\mathsf{O}}_{c}$, informally says that“A has a joint action ${\sigma}_{A}$ that guarantees satisfaction of its goal and also enables the complementary coalition $\overline{A}$ to realize any one of a list of secondary goals by applying a respectively suitable joint action”.The operator SF and the logic $\mathsf{SFCL}$ built on it will be presented in more detail in Section 7.

- Abilities for cooperation, protecting the interests of agents and coalitions. These capture the idea, complementary to that of $\mathsf{SFCL}$, that, while socially responsible rational agents and coalitions contribute with their individual and collective actions to the society of all agents, they wish to perform that in a way that protects their individual and collective interests and goals. Such abilities are expressed by means of the coalitional goal assignment operator$\langle \phantom{\rule{-0.166667em}{0ex}}[\xb7]\phantom{\rule{-0.166667em}{0ex}}\rangle $, introduced in [25] as a local operator and extended with temporalized long-term goals in [26]. A coalitional goal assignment $\gamma $ is a mapping which assigns a goal formula to each possible coalition, and the operator $\langle \phantom{\rule{-0.166667em}{0ex}}\left[\gamma \right]\phantom{\rule{-0.166667em}{0ex}}\rangle $ formalizes the claim that there is a strategy profile $\sigma $ of $\mathsf{Agt}$, the restriction of which to every coalition $\mathrm{C}$ is a joint action ${\sigma}_{\mathrm{C}}$ that guarantees the satisfaction of its goal $\gamma \left(\mathrm{C}\right)$ regardless of any possible behavior of the remaining agents. The coalitional goal assignment operator and the logics $\mathsf{LCGA}$ and $\mathsf{TLCGA}$ built on it will be presented in more detail in Section 8.

## 4. Coalition Logic and Unconditional Local Strategic Reasoning

#### 4.1. The Basic Logic for Coalitional Strategic Reasoning $\mathsf{CL}$

#### 4.2. Expressing Claims about Strategic Abilities in $\mathsf{CL}$

- 1.
- $$\left[\mathbf{Eve}\right]{H}_{E}\to \neg \left[\mathbf{Adam}\right]\neg {H}_{E}$$If
**Eve**has an action ensuring that she becomes happy, then**Adam**cannot prevent**Eve**from reaching a state where she is happy. - 2.
- The statement above naturally generalizes (with $\mathrm{Win}$ being an atomic proposition) to$$\left[\mathsf{i}\right]\mathrm{Win}\to \neg \left[\mathsf{Agt}\phantom{\rule{-0.166667em}{0ex}}\backslash \phantom{\rule{-0.166667em}{0ex}}\left\{\mathsf{i}\right\}\right]\neg \mathrm{Win}$$If the agent $\mathsf{i}$ has an action to guarantee reaching a “winning” outcome, then the coalition of all other agents cannot prevent $\mathsf{i}$ from reaching a “winning” outcome.It should be intuitively clear that this statement expresses a valid principle of $\mathsf{CL}$. Indeed, we will see shortly that this is the case.
- 3.
- $$\neg \left[\mathbf{Adam}\right]T\phantom{\rule{0.166667em}{0ex}}\wedge \neg \left[\mathbf{Eve}\right]T\phantom{\rule{0.166667em}{0ex}}\wedge \left[\{\mathbf{Adam},\mathbf{Eve}\}\right]T$$Neither
**Adam**nor**Eve**has an action ensuring that they stay together, but they have a joint action ensuring that. - 4.
- $$\neg \left[\mathbf{Adam}\right]{H}_{A}\phantom{\rule{0.166667em}{0ex}}\wedge \neg \left[\mathbf{Eve}\right]{H}_{E}\phantom{\rule{0.166667em}{0ex}}\wedge \left[\{\mathbf{Adam},\mathbf{Eve}\}\right]({H}_{A}\wedge {H}_{E})$$Neither
**Adam**nor**Eve**has an action ensuring happiness for himself/herself but they have a joint action ensuring happiness for both. - 5.
- $$\neg \left[\mathbf{Adam}\right]({H}_{A}\wedge \neg \left[\mathbf{Eve}\right]\phantom{\rule{0.166667em}{0ex}}{H}_{E})$$
**Adam**cannot act so as to ensure at the outcome state that both**Adam**is happy and**Eve**does not have an action to ensure reaching her happiness. - 6.
- $$\left[\{\mathbf{Adam},\mathbf{Eve}\}\right](\left[\mathbf{Adam}\right]{H}_{A}\to \left[\mathbf{Eve}\right]\phantom{\rule{0.166667em}{0ex}}{H}_{E})$$
**Adam**and**Eve**can act jointly so that at the outcome state**Adam**has an action to ensure reaching his happiness only if**Eve**has an action to ensure reaching her happiness. - 7.
- $$\left[\mathbf{Adam}\right]\left[\mathbf{Eve}\right]({H}_{A}\wedge {H}_{E}\wedge \left[\mathbf{Adam}\right]{H}_{E}\wedge \left[\mathbf{Eve}\right]\phantom{\rule{0.166667em}{0ex}}{H}_{A})$$
**Adam**can act to ensure (at the outcome state), that**Eve**can then act to ensure that they are both happy and that each of them can act so as to keep the other happy.

**Adam**and

**Eve**have a joint strategy to stay together forever, or to eventually reach happiness. For that, a more expressive logic is needed, also involving temporal operators. It is introduced in Section 5.

#### 4.3. Formal Semantics of CL

**Truth of a CL-formula**$\psi $

**at a state**s

**in**$\mathcal{M}$, denoted $\mathcal{M},s\vDash \psi $, as in modal logic, is defined by structural induction on formulae via the clauses:

- $\mathcal{M},s\vDash p$ iff $p\in \mathsf{L}\left(s\right)$, for $p\in \mathsf{Prop}$.
- $\mathcal{M},s\vDash \neg \varphi $ iff $\mathcal{M},s\u22ad\varphi $.
- $\mathcal{M},s\vDash ({\varphi}_{1}\vee {\varphi}_{2})$ iff $\mathcal{M},s\vDash {\varphi}_{1}$ and $\mathcal{M},s\vDash {\varphi}_{2}$.
- $\mathcal{M},s\vDash \left[C\right]\varphi $ iff there exists a joint action ${\sigma}_{C}$ available at s,such that $\mathcal{M},u\vDash \varphi $ for each $u\in \mathsf{Out}[s,{\sigma}_{C}]$.

- $\mathcal{M},{A}_{1}{E}_{1}\vDash \phantom{\rule{4pt}{0ex}}\neg \left[\mathbf{Adam}\right]T$; $\mathcal{M},{A}_{1}{E}_{1}\vDash \phantom{\rule{4pt}{0ex}}\neg \left[\mathbf{Eve}\right]T$.
- $\mathcal{M},{A}_{1}{E}_{1}\vDash \phantom{\rule{4pt}{0ex}}\left[\{\mathbf{Adam},\mathbf{Eve}\}\right]T$.
- $\mathcal{M},{A}_{1}{E}_{1}\vDash \phantom{\rule{4pt}{0ex}}\neg \left[\mathbf{Adam}\right]{H}_{A}\wedge \neg \left[\mathbf{Eve}\right]{H}_{E}\wedge \left[\{\mathbf{Adam},\mathbf{Eve}\}\right]({H}_{A}\wedge {H}_{E})$.
- $\mathcal{M},{A}_{1}{E}_{1}\vDash \left[\{\mathbf{Adam},\mathbf{Eve}\}\right](\left[\mathbf{Eve}\right]{H}_{E}\phantom{\rule{-0.166667em}{0ex}}\wedge \phantom{\rule{-0.166667em}{0ex}}\neg \left[\mathbf{Adam}\right]{H}_{A}\phantom{\rule{-0.166667em}{0ex}}\wedge \phantom{\rule{-0.166667em}{0ex}}\left[\{\mathbf{Adam},\mathbf{Eve}\}\right]\phantom{\rule{-0.166667em}{0ex}}({H}_{A}\phantom{\rule{-0.166667em}{0ex}}\wedge \phantom{\rule{-0.166667em}{0ex}}{H}_{E}))$.

#### 4.4. Axiomatizing the Validities of $\mathsf{CL}$

**(Logically) valid**if $\mathcal{M},s\vDash \varphi $ for every CGM $\mathcal{M}$ and a state $s\in \mathcal{M}$.**Satisfiable**if $\mathcal{M},s\vDash \varphi $ for some CGM $\mathcal{M}$ and a state $s\in \mathcal{M}$.

- $\mathsf{Agt}$-maximality: $\neg \left[\xd8\right]\phantom{\rule{0.166667em}{0ex}}\neg \phi \to \left[\mathsf{Agt}\right]\phantom{\rule{0.166667em}{0ex}}\phi $This axiom says that the grand coalition $\mathsf{Agt}$ can act collectively so as to guarantee any goal that is satisfied in at least one outcome state. Note that the validity of this axiom presupposes that the models are deterministic, implying that the coalition of all agents $\mathsf{Agt}$ can enforce any particular possible outcome.
- Safety: $\neg \left[C\right]\phantom{\rule{0.166667em}{0ex}}\perp $No coalition has the ability to ensure the falsum will become true.
- Liveness: $\left[C\right]\phantom{\rule{0.166667em}{0ex}}\top $Every coalition has the ability to ensure the truth will become true.
- Superadditivity: for any ${C}_{1},{C}_{2}\subseteq \mathsf{Agt}$ such that ${C}_{1}\cap {C}_{2}=\xd8$:$$(\left[{C}_{1}\right]\phantom{\rule{0.166667em}{0ex}}{\phi}_{1}\wedge \left[{C}_{2}\right]\phantom{\rule{0.166667em}{0ex}}{\phi}_{2})\to \left[{C}_{1}\cup {C}_{2}\right]\phantom{\rule{0.166667em}{0ex}}({\phi}_{1}\wedge {\phi}_{2})$$This axiom scheme says that two disjoint coalitions, each of which has a joint action to guarantee satisfaction of their own goal, can join forces (simply by each of them following their respective joint action) to guarantee satisfaction of both goals.
- $\left[C\right]\phantom{\rule{0.166667em}{0ex}}$-monotonicity Rule:$$\frac{{\phi}_{1}\to {\phi}_{2}}{\left[C\right]\phantom{\rule{0.166667em}{0ex}}{\phi}_{1}\to \left[C\right]\phantom{\rule{0.166667em}{0ex}}{\phi}_{2}}$$

- Outcome Monotonicity: $\left[C\right]\phantom{\rule{0.166667em}{0ex}}({\phi}_{1}\wedge {\phi}_{2})\to \left[C\right]\phantom{\rule{0.166667em}{0ex}}{\phi}_{1}$.This is derived directly by applying the $\left[C\right]\phantom{\rule{0.166667em}{0ex}}$-monotonicity rule.
- Coalition Monotonicity: $\left[{C}_{1}\right]\phantom{\rule{0.166667em}{0ex}}\phi \to \left[{C}_{1}\cup {C}_{2}\right]\phantom{\rule{0.166667em}{0ex}}\phi $.This is derived by applying the superadditivity axiom scheme to the coalitions ${C}_{1}$ and ${C}_{2}\backslash {C}_{1}$ with respective goals $\phi $ and ⊤. Note that this validity together with $\mathsf{Agt}$-maximality also implies the validity of $\left[\mathsf{Agt}\right]\phantom{\rule{0.166667em}{0ex}}\phi \vee \left[\mathsf{Agt}\right]\phantom{\rule{0.166667em}{0ex}}\neg \phi $, for any formula $\phi $.
- Coalition complementarity: $\left[C\right]\phantom{\rule{0.166667em}{0ex}}\phi \to \neg \left[\overline{C}\right]\phantom{\rule{0.166667em}{0ex}}\neg \phi $, where $\overline{C}=\mathsf{Agt}\backslash C$.This is derived by applying the superadditivity axiom scheme to the coalitions C and $\overline{C}$ with respective goals $\phi $ and $\neg \phi $, and then the safety axiom scheme.

## 5. Logics for Unconditional Long-Term Strategic Reasoning

“The coalition C has a joint strategy to guarantee achievement of the coalitional goal of C in every outcome play resulting from C following that strategy”.

#### 5.1. The Alternating-Time Temporal Logic ${\mathsf{ATL}}^{*}$

where $\varphi $ is any formula of ${\mathsf{ATL}}^{*}$. The language of ${\mathsf{ATL}}^{*}$ is formally defined as follows. It involves two sorts of formulae:“The coalition $\mathrm{C}$ has a joint strategy that guarantees the satisfaction of the goal $\varphi $ on every outcome play induced by that joint strategy“,

**state formulae**, that are evaluated at game states, and

**path formulae**, that are evaluated on game plays. These are, respectively, defined by the following grammars, where $p\in \mathsf{Prop}$ and $\mathrm{C}\subseteq \mathsf{Agt}$:

State formulae: | $\phi $ | $\u2a74$ | $p\mid \top \mid \neg \phi \mid (\phi \vee \phi )\mid \langle \phantom{\rule{-0.166667em}{0ex}}\langle \mathrm{C}\rangle \phantom{\rule{-0.166667em}{0ex}}\rangle \gamma $. |

Path formulae: | $\gamma $ | $\u2a74$ | $\phi \mid \neg \gamma \mid (\gamma \vee \gamma )\mid \mathsf{X}\phantom{\rule{0.166667em}{0ex}}\gamma \mid \mathsf{G}\phantom{\rule{0.166667em}{0ex}}\gamma \mid (\gamma \cup \gamma )$. |

#### 5.2. Formal Semantics of $\mathsf{ATL}$

**Truth of an**$\mathsf{ATL}$

**-formula**$\psi $

**at a state**s

**in**$\mathcal{M}$, denoted $\mathcal{M},s\vDash \psi $, is defined by structural induction on formulae. The propositional cases are similar to in $\mathsf{CL}$, and the only new clauses are those for the strategic operators. They all follow the same pattern, which is essentially the semantic clause for $\langle \phantom{\rule{-0.166667em}{0ex}}\langle \cdots \rangle \phantom{\rule{-0.166667em}{0ex}}\rangle $ in ${\mathsf{ATL}}^{*}$:

$\mathcal{M},s\vDash \langle \phantom{\rule{-0.166667em}{0ex}}\langle \mathrm{C}\rangle \phantom{\rule{-0.166667em}{0ex}}\rangle \gamma $ iff there exists a joint strategy ${\mathsf{\Sigma}}_{C}$ for C such that $\gamma $ is true on every play in $\mathsf{Plays}(s,{\mathsf{\Sigma}}_{C})$, i.e., every play starting at s and induced by ${\mathsf{\Sigma}}_{C}$.

- $\mathcal{M},s\vDash \langle \phantom{\rule{-0.166667em}{0ex}}\langle \mathrm{C}\rangle \phantom{\rule{-0.166667em}{0ex}}\rangle \mathsf{X}\phantom{\rule{0.166667em}{0ex}}\phi $ iff there exists a joint strategy ${\mathsf{\Sigma}}_{C}$ such that $\mathcal{M},{s}_{1}\vDash \phi $ for every play ${s}_{0},{s}_{1},\dots \in \mathsf{Plays}(s,{\mathsf{\Sigma}}_{C})$.
- $\mathcal{M},s\vDash \langle \phantom{\rule{-0.166667em}{0ex}}\langle \mathrm{C}\rangle \phantom{\rule{-0.166667em}{0ex}}\rangle \mathsf{G}\phantom{\rule{0.166667em}{0ex}}\phi $ iff there exists a joint strategy ${\mathsf{\Sigma}}_{C}$ such that $\mathcal{M},{s}_{i}\vDash \phi $ for every play ${s}_{0},{s}_{1},\dots \in \mathsf{Plays}(s,{\mathsf{\Sigma}}_{C})$ and $i\ge 0$.
- $\mathcal{M},s\vDash \langle \phantom{\rule{-0.166667em}{0ex}}\langle \mathrm{C}\rangle \phantom{\rule{-0.166667em}{0ex}}\rangle \left(\phi \phantom{\rule{0.166667em}{0ex}}\mathsf{U}\phantom{\rule{0.166667em}{0ex}}\psi \right)$ iff there exists a joint strategy ${\mathsf{\Sigma}}_{C}$ such that for every play ${s}_{0},{s}_{1},\dots \in \mathsf{Plays}(s,{\mathsf{\Sigma}}_{C})$ there is $i\ge 0$ for which $\mathcal{M},{s}_{i}\vDash \psi $ and $\mathcal{M},{s}_{j}\vDash \phi $ for all j such that $0\le j<i$.

#### 5.3. Expressing Claims about Strategic Abilities in $\mathsf{ATL}$

- 1.
- $$\langle \phantom{\rule{-0.166667em}{0ex}}\langle \mathbf{Eve}\rangle \phantom{\rule{-0.166667em}{0ex}}\rangle \mathsf{F}\phantom{\rule{0.166667em}{0ex}}{H}_{E}\to \neg \langle \phantom{\rule{-0.166667em}{0ex}}\langle \mathbf{Adam}\rangle \phantom{\rule{-0.166667em}{0ex}}\rangle \mathsf{G}\phantom{\rule{0.166667em}{0ex}}\neg {H}_{E}$$If
**Eve**has an action ensuring that she eventually becomes happy, then**Adam**cannot prevent**Eve**forever from reaching a state where she is happy.It should be intuitively clear that this statement expresses a valid principle of $\mathsf{ATL}$. Indeed, this is the case. - 2.
- $$(\phantom{\rule{-0.166667em}{0ex}}\langle \phantom{\rule{-0.166667em}{0ex}}\langle \mathbf{Yin}\rangle \phantom{\rule{-0.166667em}{0ex}}\rangle \mathsf{G}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{Safe}\wedge \langle \phantom{\rule{-0.166667em}{0ex}}\langle \mathbf{Yin}\rangle \phantom{\rule{-0.166667em}{0ex}}\rangle \mathsf{F}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{Goal})\phantom{\rule{-0.166667em}{0ex}}\to \phantom{\rule{-0.166667em}{0ex}}\langle \phantom{\rule{-0.166667em}{0ex}}\langle \mathbf{Yin}\rangle \phantom{\rule{-0.166667em}{0ex}}\rangle (\mathrm{Safe}\phantom{\rule{0.166667em}{0ex}}\cup \phantom{\rule{0.166667em}{0ex}}\mathrm{Goal})$$If $\mathbf{Yin}$ has a strategy to keep the system in safe states forever and has a strategy to eventually achieve its goal, then $\mathbf{Yin}$ has a strategy to keep the system in safe states until it achieves its goal.The formula above is not logically valid. Indeed, the strategies of $\mathbf{Yin}$ to keep the system in safe states forever and to eventually achieve its goal may be incompatible.
- 3.
- $$(\phantom{\rule{-0.166667em}{0ex}}\langle \phantom{\rule{-0.166667em}{0ex}}\langle \mathbf{Yin}\rangle \phantom{\rule{-0.166667em}{0ex}}\rangle \mathsf{G}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{Safe}\phantom{\rule{0.277778em}{0ex}}\wedge \langle \phantom{\rule{-0.166667em}{0ex}}\langle \mathbf{Yang}\rangle \phantom{\rule{-0.166667em}{0ex}}\rangle \mathsf{F}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{Goal})\phantom{\rule{-0.166667em}{0ex}}\to \phantom{\rule{-0.166667em}{0ex}}\langle \phantom{\rule{-0.166667em}{0ex}}\langle \{\mathbf{Yin},\mathbf{Yang}\}\rangle \phantom{\rule{-0.166667em}{0ex}}\rangle (\mathrm{Safe}\phantom{\rule{0.166667em}{0ex}}\cup \phantom{\rule{0.166667em}{0ex}}\mathrm{Goal})$$If $\mathbf{Yin}$ has a strategy to keep the system in safe states forever and $\mathbf{Yang}$ has a strategy to eventually reach a goal state, then $\mathbf{Yin}$ and $\mathbf{Yang}$ together have a strategy to stay in safe states until a goal state is reached.Assuming that $\mathbf{Yin}$ and $\mathbf{Yang}$ are distinct agents, the formula above is logically valid, unlike the previous one. Indeed, this is due to the independence of the actions of the two agents, and hence of their abilities to execute their respective strategies.
- 4.
- $$\langle \phantom{\rule{-0.166667em}{0ex}}\langle \mathrm{A}\rangle \phantom{\rule{-0.166667em}{0ex}}\rangle \mathsf{F}\phantom{\rule{0.166667em}{0ex}}\langle \phantom{\rule{-0.166667em}{0ex}}\langle \mathrm{B}\rangle \phantom{\rule{-0.166667em}{0ex}}\rangle \mathsf{G}\phantom{\rule{0.166667em}{0ex}}\neg \phi $$The coalition A has a joint strategy to eventually ensure that the coalition B has a strategy to prevent φ from ever happening.This example raises the question of how the semantics works in the case of nested strategic operators. Suppose, the coalitions A and B intersect and $\mathsf{a}$ is an agent in both of them. Then, the claim of the external strategic operator $\langle \phantom{\rule{-0.166667em}{0ex}}\langle \mathrm{A}\rangle \phantom{\rule{-0.166667em}{0ex}}\rangle $ requires, inter alia, existence of a strategy for the agent $\mathsf{a}$ within a joint strategy ${\sigma}_{A}$ for A that guarantees the eventual satisfaction of the subformula $\langle \phantom{\rule{-0.166667em}{0ex}}\langle \mathrm{B}\rangle \phantom{\rule{-0.166667em}{0ex}}\rangle \mathsf{G}\phantom{\rule{0.166667em}{0ex}}\neg \phi $. However, when evaluating that subformula, to justify its truth, one has to identify a respective joint strategy ${\sigma}_{B}$ for $\langle \phantom{\rule{-0.166667em}{0ex}}\langle \mathrm{B}\rangle \phantom{\rule{-0.166667em}{0ex}}\rangle $. Now, the question arises whether the strategy of $\mathsf{a}$ within ${\sigma}_{B}$ should not be assumed to be already fixed by ${\sigma}_{A}$ or, conversely, whether the strategy of $\mathsf{a}$ within ${\sigma}_{A}$ should not be assumed to be already fixed by ${\sigma}_{B}$. Note that the standard formal semantics for ${\mathsf{ATL}}^{*}$ (in particular, of $\mathsf{ATL}$) presented here does not impose any such constraints, but, rather, treats these strategies independently. That is, the standard semantics of ${\mathsf{ATL}}^{*}$does not commit the agents in A to the strategies they adopt in order to bring about the truth of the formula $\langle \phantom{\rule{-0.166667em}{0ex}}\langle \mathrm{A}\rangle \phantom{\rule{-0.166667em}{0ex}}\rangle \gamma $. That creates some conceptual issues with the very concept of “strategy”, independently addressed in different ways in [38,39,42,43], where several proposals were made in order to incorporate strategic commitment or uncommitment and persistent strategies in the syntax and semantics of ${\mathsf{ATL}}^{*}$. These raise a number of still open technical problems regarding constructing provably complete axiomatizations, proving decidability, and designing decision procedures for the variations of $\mathsf{ATL}$ and ${\mathsf{ATL}}^{*}$ mentioned above.

## 6. The Logic of Conditional Strategic Reasoning ConStR

#### 6.1. Conditional Strategic Reasoning: An Informal Discussion

For some/every action of Alice that guarantees achievement of her goal ${\gamma}_{A}$,

Bob has/does not have an action of his own to guarantee achievement of his goal ${\gamma}_{B}$.

#### 6.1.1. Case 1: Bob Does Not Know Alice’s Goal or Actions

#### 6.1.2. Case 2: Bob Knows Alice’s Goal and Possible Actions

“Whichever way Alice acts towards achieving her goal ${\gamma}_{A}$,

Bob can act so as to ensure achievement of his goal ${\gamma}_{B}$”.

#### 6.1.3. Reactive and Proactive Ability

**Remark 5.**

#### 6.1.4. Case 3: Assuming Alice’s Cooperation

#### 6.2. Modal Operators for Conditional Strategic Reasoning

**(**${\mathsf{O}}_{\alpha}$**)**- ${\left[\mathrm{A}\right]}_{\alpha}({\gamma}_{\mathrm{A}};\langle \mathrm{B}\rangle {\gamma}_{\mathrm{B}})$ means that the coalition $\mathrm{B}\backslash \mathrm{A}$ of agents who are in $\mathrm{B}$ but not in $\mathrm{A}$ has a joint action ${\sigma}_{\mathrm{B}\backslash \mathrm{A}}$ such that if $\mathrm{A}$ applies any joint action that guarantees the truth of ${\gamma}_{\mathrm{A}}$, then $\mathrm{B}\backslash \mathrm{A}$ can ensure the truth of ${\gamma}_{\mathrm{B}}$ by applying ${\sigma}_{\mathrm{B}}$.This operator formalizes the notion of a coalition’s proactive ability, discussed in the special case of single-agent coalitions in Section 6.1.3, respectively, to the game-theoretic notion of conditional α-effectivity, hence the notation. Note that the agents in $\mathrm{B}\cap \mathrm{A}$ (if any) are assumed to act on behalf of $\mathrm{A}$ in its pursuit of ${\gamma}_{\mathrm{A}}$.
**(**${\mathsf{O}}_{\beta}$**)**- ${\left[\mathrm{A}\right]}_{\beta}({\gamma}_{\mathrm{A}};\langle \mathrm{B}\rangle {\gamma}_{\mathrm{B}})$ means that for any joint action ${\sigma}_{\mathrm{A}}$ of $\mathrm{A}$ that guarantees the truth of ${\gamma}_{\mathrm{A}}$, when applied by $\mathrm{A}$ there is a joint action ${\sigma}_{\mathrm{B}\backslash \mathrm{A}}$ that guarantees ${\gamma}_{\mathrm{B}}$ when additionally applied by $\mathrm{B}\backslash \mathrm{A}$. Note that ${\left[\mathrm{A}\right]}_{\beta}(\perp ;\langle \mathrm{B}\rangle {\gamma}_{\mathrm{B}})$ is vacuously true for any $\mathrm{A}$, $\mathrm{B}$, and ${\gamma}_{\mathrm{B}}$, as then there cannot be such joint actions ${\sigma}_{\mathrm{A}}$ that enable satisfying ⊥. This may sounds odd, but it is no special phenomenon in $\mathsf{ConStR}$, as the same effect occurs in FOL with universal quantification over an empty set of objects.This operator formalizes a claim of the ability of the coalition $\mathrm{B}$ to choose a suitable joint action so as to achieve the goal ${\gamma}_{\mathrm{B}}$ assuming that $\mathrm{A}$ acts so as to achieve the goal ${\gamma}_{\mathrm{A}}$, if $\mathrm{B}$ is to choose their joint action after $\mathrm{B}$ learns the joint action of $\mathrm{A}$. In this case, the actions of the agents in $\mathrm{B}\cap \mathrm{A}$ (if any) are assumed to be already fixed by ${\gamma}_{\mathrm{A}}$.This corresponds to the notion of agents’ reactive ability discussed in Section 6.1.3, respectively, to the game-theoretic notion of conditional β-effectivity, hence the notation.
**(**${\mathsf{O}}_{c}$**)**- ${\langle \mathrm{A}\rangle}_{\mathsf{c}}({\gamma}_{\mathrm{A}};\langle \mathrm{B}\rangle {\gamma}_{\mathrm{B}})$ means that $\mathrm{A}$ has a joint action ${\sigma}_{\mathrm{A}}$ which, when applied, guarantees the truth of ${\gamma}_{\mathrm{A}}$ and enables $\mathrm{B}\backslash \mathrm{A}$ to apply a joint action ${\sigma}_{\mathrm{B}\backslash \mathrm{A}}$ that guarantees ${\gamma}_{\mathrm{B}}$ when additionally applied by $\mathrm{B}\backslash \mathrm{A}$.This operator formalizes Case 3 discussed in Section 6.1, where $\mathrm{A}$ knows the goal of $\mathrm{B}$ and can choose to cooperate with $\mathrm{B}$ by selecting an action among those that ensure satisfaction of ${\gamma}_{\mathrm{A}}$ which is also suitable for $\mathrm{B}$.

#### 6.3. The Logic $\mathsf{ConStR}$: Language and Some Definable Operators

- The coalitional operator from $\mathsf{CL}$ is definable by means of each of ${\mathsf{O}}_{c}$, ${\mathsf{O}}_{\alpha}$, ${\mathsf{O}}_{\beta}$ as follows:
- (${\mathsf{O}}_{\alpha}$)
- $\left[\mathrm{C}\right]\varphi \u2a74{[\xd8]}_{\alpha}(\top ;\langle \mathrm{C}\rangle \varphi )$.This claims an unconditional ability of $\mathrm{C}$ to choose an action that guarantees $\varphi $.
- (${\mathsf{O}}_{\beta}$)
- $\left[\mathrm{C}\right]\varphi \u2a74{[\xd8]}_{\beta}(\top ;\langle \mathrm{C}\rangle \varphi )$, or $\left[\mathrm{C}\right]\varphi \u2a74{\left[\overline{\mathrm{C}}\right]}_{\beta}(\top ;\langle \mathrm{C}\rangle \varphi )$.The only strategy of the empty coalition guarantees the satisfaction of ⊤.
- (${\mathsf{O}}_{c}$)
- $\left[\mathrm{C}\right]\varphi \u2a74{\langle \mathrm{C}\rangle}_{\mathsf{c}}(\varphi ;\langle \mathrm{C}\rangle \varphi )$ or $\left[\mathrm{C}\right]\varphi \u2a74{\langle \mathrm{C}\rangle}_{\mathsf{c}}(\varphi ;\langle \mathrm{C}\rangle \top )$.

- ${\langle \mathrm{A}\rangle}_{\mathsf{c}}({\gamma}_{\mathrm{A}};\langle \mathrm{B}\rangle {\gamma}_{\mathrm{B}})$ is equivalent to $\langle \phantom{\rule{-0.166667em}{0ex}}\langle \mathrm{A}\rangle \phantom{\rule{-0.166667em}{0ex}}\rangle ({\gamma}_{\mathrm{A}}\wedge \langle \phantom{\rule{-0.166667em}{0ex}}\langle \mathrm{B}\backslash \mathrm{A}\rangle \phantom{\rule{-0.166667em}{0ex}}\rangle {\gamma}_{\mathrm{B}})$ in ${\mathsf{ATL}}^{*}$.
- The negated ${\mathsf{O}}_{c}$ operator $\neg {\langle \mathrm{A}\rangle}_{\mathsf{c}}(\varphi ;\langle \mathrm{B}\rangle \psi )$ says that every joint action of $\mathrm{A}$ that, when applied, guarantees the truth of $\varphi $, would prevent $\mathrm{B}$ from acting additionally so as to guarantee $\psi $. This formalizes the conditional reasoning scenario where the goals of $\mathrm{A}$ and $\mathrm{B}$ are conflicting, so whichever way $\mathrm{A}$ acts towards its goal would block $\mathrm{B}$ from acting to guarantee achievement of $\mathrm{B}$’s goal.
- ${\left[\mathrm{A}\right]}_{\beta}(\top ;\langle \mathrm{B}\rangle \psi )$ essentially formalizes the case when the agents in $\mathrm{B}$ are not informed about the goal of $\mathrm{A}$, but have to choose their action after learning the action of $\mathrm{A}$.
- On the other hand, $\left[\mathrm{A}\right]\left(\varphi \right|\psi )\u2a74{\left[\mathrm{A}\right]}_{\beta}(\varphi ;\langle \xd8\rangle \psi )$, also equivalent to ${\left[\mathrm{A}\right]}_{\alpha}(\varphi ;\langle \xd8\rangle \psi )$, says that any joint strategy of $\mathrm{A}$ that guarantees $\varphi $ to be true also guarantees $\psi $ to be true. That formalizes the claim of an observer who knows both the goal $\varphi $ and the possible joint actions of $\mathrm{A}$, that the outcome from the joint action of $\mathrm{A}$ will also satisfy $\psi $.
- $\langle \mathrm{A}\rangle \left(\varphi \right|\psi )\u2a74\neg [\mathrm{A}\left]\right(\varphi |\neg \psi )$ says that there is a joint strategy of $\mathrm{A}$ that ensures the truth of $\varphi $ and also enables the satisfaction of $\psi $.$\langle \mathrm{A}\rangle \left(\varphi \right|\psi )$ is also definable as ${\langle \mathrm{A}\rangle}_{\mathsf{c}}(\varphi ;\langle \overline{\mathrm{A}}\rangle \psi )$, where $\overline{\mathrm{A}}=\mathsf{Agt}\backslash \mathrm{A}$.
- The coalitional operator $\left[\mathrm{A}\right]$ from $\mathsf{CL}$ is a special case of the above: $\left[\mathrm{A}\right]\varphi \u2a74\langle \mathrm{A}\rangle \left(\varphi \right|\top )$.

#### 6.4. Formal Semantics of $\mathsf{ConStR}$

**ordered join of**${\sigma}_{\mathrm{A}}$

**and**${\sigma}_{\mathrm{B}}$ to be the joint action ${\sigma}_{\mathrm{A}}\uplus {\sigma}_{\mathrm{B}}$ for $\mathrm{A}\cup \mathrm{B}$ which equals ${\sigma}_{\mathrm{A}}$ when restricted to $\mathrm{A}$ and equals to ${\sigma}_{\mathrm{B}}$ when restricted to $\mathrm{B}\backslash \mathrm{A}$. Thus, in particular, ${\sigma}_{\mathrm{A}}\uplus {\sigma}_{\mathrm{B}}={\sigma}_{\mathrm{A}}$ for any $\mathrm{B}\subseteq \mathrm{A}\subseteq \mathsf{Agt}$ and ${\sigma}_{\mathrm{A}}\uplus {\sigma}_{\mathrm{B}}={\sigma}_{\mathrm{A}}\cup {\sigma}_{\mathrm{B}}$ when $\mathrm{A}$ and $\mathrm{B}$ are disjoint.

- $\mathcal{M},s\vDash {\left[\mathrm{A}\right]}_{\alpha}(\varphi ;\langle \mathrm{B}\rangle \psi )$ iff
- $\mathrm{B}$ has a joint action ${\sigma}_{\mathrm{B}}$ such that for every joint action ${\sigma}_{\mathrm{A}}$ of $\mathrm{A}$,if $\mathcal{M},u\vDash \varphi $ for every $u\in \mathsf{Out}[s,{\sigma}_{\mathrm{A}}]$, then $\mathcal{M},u\vDash \psi $ for every $u\in \mathsf{Out}[s,{\sigma}_{\mathrm{A}}\uplus {\sigma}_{\mathrm{B}}]$.

- $\mathcal{M},s\vDash {\left[\mathrm{A}\right]}_{\beta}(\varphi ;\langle \mathrm{B}\rangle \psi )$ iff
- for every joint action ${\sigma}_{\mathrm{A}}$ of $\mathrm{A}$ such that $\mathcal{M},u\vDash \varphi $ for every $u\in \mathsf{Out}[s,{\sigma}_{\mathrm{A}}]$,$\mathrm{B}$ has a joint action ${\sigma}_{\mathrm{B}}$ (generally, dependent on ${\sigma}_{\mathrm{A}}$) such that$\mathcal{M},u\vDash \psi $ for every $u\in \mathsf{Out}[s,{\sigma}_{\mathrm{A}}\uplus {\sigma}_{\mathrm{B}}]$.

- $\mathcal{M},s\vDash {\langle \mathrm{A}\rangle}_{\mathsf{c}}(\varphi ;\langle \mathrm{B}\rangle \psi )$ iff
- $\mathrm{A}$ has a joint action ${\sigma}_{\mathrm{A}}$, such that $\mathcal{M},u\vDash \varphi $ for every $u\in \mathsf{Out}[s,{\sigma}_{\mathrm{A}}]$ and$\mathrm{B}$ has a joint action ${\sigma}_{\mathrm{B}}$ such that $\mathcal{M},u\vDash \psi $ for every $u\in \mathsf{Out}[s,{\sigma}_{\mathrm{A}}\uplus {\sigma}_{\mathrm{B}}]$.

**Remark 6.**

**valid in**$\mathsf{ConStR}$, denoted ${\vDash}_{\mathsf{ConStR}}\varphi $, iff $\mathcal{M},u\vDash \varphi $ for every concurrent game model $\mathcal{M}$ and state $u\in \mathcal{M}$; $\varphi $ is

**satisfiable in**$\mathsf{ConStR}$, iff $\mathcal{M},u\vDash \varphi $ for some concurrent game model $\mathcal{M}$ and state $u\in \mathcal{M}$.

**Proposition 7.**

- 1.
- ${\vDash}_{\mathsf{ConStR}}{\left[\mathsf{a}\right]}_{\alpha}(p;\langle \mathsf{b}\rangle q)\to {\left[\mathsf{a}\right]}_{\beta}(p;\langle \mathsf{b}\rangle q)$.
- 2.
- ${\u22ad}_{\mathsf{ConStR}}{\left[\mathsf{a}\right]}_{\beta}(p;\langle \mathsf{b}\rangle q)\to {\left[\mathsf{a}\right]}_{\alpha}(p;\langle \mathsf{b}\rangle q)$.
- 3.
- ${\u22ad}_{\mathsf{ConStR}}{\langle \mathsf{a}\rangle}_{\mathsf{c}}(p;\langle \mathsf{b}\rangle q)\to {\left[\mathsf{a}\right]}_{\beta}(p;\langle \mathsf{b}\rangle q)$.
- 4.
- ${\u22ad}_{\mathsf{ConStR}}{\langle \mathsf{a}\rangle}_{\mathsf{c}}(p;\langle \mathsf{b}\rangle q)\to {\left[\mathsf{a}\right]}_{\alpha}(p;\langle \mathsf{b}\rangle q)$.
- 5.
- ${\u22ad}_{\mathsf{ConStR}}{\left[\mathsf{a}\right]}_{\alpha}(p;\langle \mathsf{b}\rangle q)\to {\langle \mathsf{a}\rangle}_{\mathsf{c}}(p;\langle \mathsf{b}\rangle q)$.
- 6.
- ${\u22ad}_{\mathsf{ConStR}}{\left[\mathsf{a}\right]}_{\beta}(p;\langle \mathsf{b}\rangle q)\to {\langle \mathsf{a}\rangle}_{\mathsf{c}}(p;\langle \mathsf{b}\rangle q)$.
- 7.
- ${\vDash}_{\mathsf{ConStR}}(\left[\mathrm{A}\right]p\wedge {\left[\mathsf{a}\right]}_{\beta}(p;\langle \mathsf{b}\rangle q))\to {\langle \mathsf{a}\rangle}_{\mathsf{c}}(p;\langle \mathsf{b}\rangle q)$.

**Proof.**

- Immediately from the semantic definitions, as $\exists \forall $ implies $\forall \exists $.
- A counter-model is shown further, in Example 8.
- A counter-model is shown in Example 8.
- Follows from 1 and 3 above.
- This holds for the trivial reason that if in a given model the agent $\mathsf{a}$ has no action that ensures the truth of p in the given state, then the antecedent is vacuously true, whereas the consequent is false there.
- Likewise.
- An easy semantic exercise.

**Example 8.**

- 1.
- $\mathcal{M},{s}_{0}\u22ad\left[\mathsf{b}\right]q$, whereas $\mathcal{M},{s}_{0}\vDash {\langle \mathsf{a}\rangle}_{\mathsf{c}}(p;\langle \mathsf{b}\rangle q)$.Thus, an agent may have only conditional ability to achieve its goal.
- 2.
- $\mathcal{M},{s}_{0}\u22ad{\left[\mathsf{a}\right]}_{\alpha}(p;\langle \mathsf{b}\rangle q)$. Indeed, neither ${b}_{1}$ nor ${b}_{2}$ ensures q against both choices ${a}_{1}$ and ${a}_{2}$ of $\mathsf{a}$. Thus, $\mathsf{b}$ does not have a uniform action to ensure q against any action of $\mathsf{a}$that ensures p.
- 3.
- $\mathcal{M},{s}_{0}\vDash {\left[\mathsf{a}\right]}_{\beta}(p;\langle \mathsf{b}\rangle q)$. Indeed, $\mathsf{a}$ has two actions at state ${s}_{0}$ to ensure p: ${a}_{1}$ and ${a}_{2}$. For each of them, $\mathsf{b}$ has an action to ensure q: choose ${b}_{2}$ if $\mathsf{a}$ chooses ${a}_{1}$, and choose ${b}_{1}$ if $\mathsf{a}$ chooses ${a}_{2}$.Therefore, ${\u22ad}_{\mathsf{ConStR}}{\left[\mathsf{a}\right]}_{\beta}(p;\langle \mathsf{b}\rangle q)\to {\left[\mathsf{a}\right]}_{\alpha}(p;\langle \mathsf{b}\rangle q)$. Thus, the claim made by the proactive ability operator ${\mathsf{O}}_{\alpha}$ is stronger than the claim made by the reactive ability operator ${\mathsf{O}}_{\beta}$.
- 4.
- On the other hand, if the outcomes of $({a}_{2},{b}_{1})$ and $({a}_{2},{b}_{2})$ are swapped, then ${\left[\mathsf{a}\right]}_{\alpha}(p;\langle \mathsf{b}\rangle q)$ becomes true at ${s}_{0}$in the resulting model.
- 5.
- Furthermore, clearly, ${\left[\mathsf{a}\right]}_{\alpha}(p;\langle \mathsf{b}\rangle q)\to {\left[\mathsf{a}\right]}_{\beta}(p;\langle \mathsf{b}\rangle q)$ is a valid formula in $\mathsf{ConStR}$.
- 6.
- If the model is modified by making p true also at ${s}_{6}$, then in the resulting model ${\mathcal{M}}^{\prime}$ we have ${\mathcal{M}}^{\prime},{s}_{0}\vDash {\langle \mathsf{a}\rangle}_{\mathsf{c}}(p;\langle \mathsf{b}\rangle q)$ and ${\mathcal{M}}^{\prime},{s}_{0}\u22ad{\left[\mathsf{a}\right]}_{\beta}(p;\langle \mathsf{b}\rangle q)$.Therefore, ${\u22ad}_{\mathsf{ConStR}}{\langle \mathsf{a}\rangle}_{\mathsf{c}}(p;\langle \mathsf{b}\rangle q)\to {\left[\mathsf{a}\right]}_{\beta}(p;\langle \mathsf{b}\rangle q)$.
- 7.
- Note also that$(\mathcal{M},{s}_{0})$ does not satisfy the $\mathsf{ATL}$ formula $\left[\phantom{\rule{-0.166667em}{0ex}}\right[\mathsf{a}\left]\phantom{\rule{-0.166667em}{0ex}}\right](\mathrm{X}p\to \langle \phantom{\rule{-0.166667em}{0ex}}\langle \mathsf{b}\rangle \phantom{\rule{-0.166667em}{0ex}}\rangle \mathrm{X}q)$ (where $\left[\phantom{\rule{-0.166667em}{0ex}}\right[\mathrm{C}\left]\phantom{\rule{-0.166667em}{0ex}}\right]\varphi \u2a74\neg \langle \phantom{\rule{-0.166667em}{0ex}}\langle \mathrm{C}\rangle \phantom{\rule{-0.166667em}{0ex}}\rangle \neg \varphi $); hence, it is not equivalent to ${\left[\mathsf{a}\right]}_{\beta}(p;\langle \mathsf{b}\rangle q)$.

#### 6.5. On the Axiomatization and Decidability for $\mathsf{ConStR}$

**(**$\mathsf{ConStR}$**1)**- ${\left[\mathrm{A}\right]}_{\alpha}(\varphi ;\langle \mathrm{B}\rangle \psi )\to {\left[\mathrm{A}\right]}_{\beta}(\varphi ;\langle \mathrm{B}\rangle \psi )$.
**(**$\mathsf{ConStR}$**2)**- $(\left[\mathrm{A}\right]\varphi \wedge {\left[\mathrm{A}\right]}_{\beta}(\varphi ;\langle \mathrm{B}\rangle \psi )\to {\langle \mathrm{A}\rangle}_{\mathsf{c}}(\varphi ;\langle \mathrm{B}\rangle \psi )$.

**-monotonicity rule**:

## 7. The Socially Friendly Coalition Logic SFCL

#### 7.1. Socially Friendly Coalitional Operators

**SF**- The
**socially friendly coalitional operator SF**takes any positive number of formulae $\varphi ,{\psi}_{1},\dots ,{\psi}_{k}$ as arguments and places them together into a single formula as follows:$$\left[C\right](\varphi ;{\psi}_{1},\dots ,{\psi}_{k}).$$I will call the formula $\varphi $ above the**primary goal**of the formula (and of the coalition C) and the formulae ${\psi}_{1},\dots ,{\psi}_{k}$—the**secondary goals**of the formula.The intuitive meaning of the formula $\left[C\right](\varphi ;{\psi}_{1},\dots ,{\psi}_{k})$ is that“C has a joint action ${\sigma}_{C}$ that guarantees satisfaction of ϕ and also enables the complementary coalition $\overline{C}$ to satisfy any one of the goals ${\psi}_{1},\dots ,{\psi}_{k}$ by applying a respectively suitable joint action”.The operator**SF**is a multiagent extension of the modal operator $\square ({\psi}_{1},\dots ,{\psi}_{k};\varphi )$ (note the different order of the arguments) in the instantial neighborhood logic (INL) introduced and studied in [45].The special case of the “socially friendly coalitional operator” with a single secondary goal ${\left[\mathrm{A}\right]}_{{}_{\phantom{\rule{-0.166667em}{0ex}}}}(\varphi ;\psi )$ is equivalent to the operator $\langle \mathrm{A}\rangle \left(\varphi \right|\psi )$ defined in Section 6.In particular, the strategic operator of $\mathsf{CL}$ is also definable here, as $\left[C\right]\varphi \u2a74\left[C\right](\varphi ;\top )$. **SF1**- A refinement of
**SF**: $\left[C;{C}_{1},\dots ,{C}_{n}\right](\varphi ;{\varphi}_{1},\dots ,{\varphi}_{n})$, meaning:“C has a collective action ${\sigma}_{C}$ that guarantees ϕ, and is such that, when fixed,each ${C}_{i}$ has a collective action that guarantees ${\varphi}_{i}$”.This definition presumes that if C intersects with ${C}_{i}$, then the agents in $C\cap {C}_{i}$ are already committed to ${\sigma}_{C}$. On the other hand, the collective actions claimed to exist for each ${C}_{i}$ need not be compatible, similar to in the intuitive semantics of $\left[C\right](\varphi ;{\psi}_{1},\dots ,{\psi}_{k})$, which is a special case of SF1, where ${C}_{1}=\dots ={C}_{n}=\mathsf{Agt}\backslash C$. **SF2**- $\left[\langle {C}_{1}\rangle {\varphi}_{1};\dots ;\langle {C}_{k}\rangle {\varphi}_{k}\right]$ meaning: “${C}_{1}$ has a collective action to guarantee ${\varphi}_{1}$, and given that action… ${C}_{k}$ has a collective action to guarantee ${\varphi}_{k}$”.This is a sequential version of SF1 where the coalitions ${C}_{1},...,{C}_{k}$ are arranged in decreasing priority order.

**SF**only.

#### 7.2. Syntax and Semantics of the Logic $\mathsf{SFCL}$

**truth of an**$\mathsf{SFCL}$

**-formula at a state**s

**of a concurrent game model**$\mathcal{M}$ inductively, as in $\mathsf{CL}$, with the following main clause:

- $\mathcal{M},s\vDash \left[\mathrm{C}\right](\varphi ;\mathsf{\Psi})$ iff
- there exists a joint action ${\sigma}_{C}$ of $\mathrm{C}$ available at s,
- such that $\mathcal{M},u\vDash \varphi $ for each state u in its outcome set $O[s,{\sigma}_{C}]$,
- and for each $\psi \in \mathsf{\Psi}$ there is a state $v\in O[s,{\sigma}_{C}]$ such that $\mathcal{M},v\vDash \psi $.

#### 7.3. Example 1: Negotiating the Family Budget

**Example 9.**

- $CK$ (cheap kitchen), $EK$ (expensive kitchen), $K=CK\vee EK$ (any kitchen);
- $CC$ (cheap car), $AC$ (average car), $EC$ (expensive car), $C=CC\vee AC\vee EC$ (any car);
- $CT$ (cheap trip), $ET$ (expensive trip), $T=CT\vee ET$ (any trip).

- Ann can afford to pay for an expensive kitchen and then let the others choose some car or some trip, formally: $\left[\mathsf{Ann}\right](EK;C,T)$.
- Alternatively, Ann can settle with a cheap kitchen, and then let the others choose between some car plus some trip, or an expensive car, or an expensive trip; formally: $\left[\mathsf{Ann}\right](CK;CC\wedge CT,EC,ET)$.
- However, if Ann opts for an expensive kitchen, then the family cannot afford an average car plus a trip; formally: $\neg \left[\mathsf{Ann}\right](EK;AC\wedge T)$.

- $\left[\mathsf{Bill}\right](C;K,T)$;
- $\left[\mathsf{Bill}\right](CC;EK,ET)$.
- However, $\neg \left[\mathsf{Bill}\right](EC;AK\wedge T)$.

- $\left[\right\{\mathsf{Ann},\mathsf{Bill}\left\}\right](C\wedge EK;CT)$;
- $\left[\right\{\mathsf{Ann},\mathsf{Charlie}\left\}\right](CK\wedge ET;CC)$;
- $\left[\right\{\mathsf{Bill},\mathsf{Charlie}\left\}\right](AC\wedge CT;K)$.

#### 7.4. Example 2: Job Applicants

Banana: | Megasoft: | Fakebook: | ||

1. Alice, 2. Bob, 3. Carl. | 1. Diana, 2. Bob. | 1. Alice, 2. Diana. |

- $\left[A\right]({e}_{A};{e}_{B}\wedge {e}_{D},{e}_{C}\wedge {e}_{D},{e}_{B}\wedge {e}_{C})$Indeed, Alice can take the offer from Fakebook and leave it to the others to decide on the other two positions. Note that Bob can then choose either offer and enable either Carl or Diana to obtain a job, but can also decline both offers and thus enable both of them to obtain a job.
- $\left[A\right](\top ;{e}_{B}\wedge {e}_{C}\wedge {e}_{D})$Alternatively, Alice can act selflessly by declining both offers, and thus enable all others to obtain a job (by Bob choosing Fakebook).
- $\neg \left[C\right]({e}_{C};\top )\wedge \left[C\right](\top ;{e}_{C})$Carl cannot be sure to obtain a job, but all others can be sure of this.
- $\neg \left[D\right]({e}_{D};{e}_{B}\wedge {e}_{C})\wedge \left[D\right]({e}_{D};{e}_{B},{e}_{C})$Diana cannot be sure to obtain a job and enable the others to see to it that both Bob and Carl obtain a job, but she can ensure that she obtains a job (by accepting the offer from Megasoft) and then the others ensure that either Bob or Carl obtains a job, too.
- $\left[\{A,D\}\right]({e}_{A}\wedge {e}_{D};{e}_{B},{e}_{C})$Alice and Diana together can ensure that they both obtain a job and then either of the other two can obtain a job, too, up to their choice.
- $\left[\{A,D\}\right]({e}_{A}\wedge {e}_{D}\wedge \neg {e}_{B}\wedge \neg {e}_{C})$Alice and Diana can also be mean and act together so that only they obtain jobs, by accepting the respective offers from Banana and Megasoft and leaving the Fakebook position unfilled.

#### 7.5. Socially Friendly Coalition Logic SFCL: A System of Axioms

**(INL5)**- $\left[C\right](\varphi ;\mathsf{\Psi})\to \left[C\right](\varphi \wedge \neg \theta ;\mathsf{\Psi})\vee \left[C\right](\varphi ;\mathsf{\Psi},\theta )$This is a valid scheme. Indeed, for any formula $\mathsf{\theta}$, the strategy for C that ensures $\varphi $ and enables each formula in the list $\mathsf{\Psi}$ either also ensures $\neg \theta $, in which case it can be added conjunctively to $\varphi $, or else enables $\theta $, in which case it can be added to the list $\mathsf{\Psi}$.

**$\left[C\right]$-monotonicity**:

**SF**, including

**SF1**and

**SF2**. To my knowledge, they have not been explored and no technical results are even conjectured for them yet, except for some special cases explored in the context of conditional strategic reasoning in Section 6.

## 8. The Logic of Local Coalitional Goal Assignments (LCGA)

#### 8.1. The Coalitional Goal Assignments Operator

**coalitional goal assignments operator**takes a list of coalitions ${C}_{1},\dots ,{C}_{k}$ and a list of formulae representing their goals ${\varphi}_{1},\dots ,{\varphi}_{k}$ and produces a formula of the type:

**coalitional goal assignment**is a mapping $\gamma :\mathcal{P}\left(\mathsf{Agt}\right)\to \mathsf{\Gamma}$, where $\mathsf{\Gamma}$ is a set of goal formulae (which may, but need not be, the full logical language under consideration). Thus, for every coalition C, $\gamma \left(C\right)$ expresses the goal of C.

#### 8.2. Syntax and Semantics of the Logic $\mathsf{LCGA}$

**truth of an $\mathsf{LCGA}$-formula at a state**s

**of a concurrent game model**$\mathcal{M}$. The only new semantic clause is the one for $\langle \phantom{\rule{-0.166667em}{0ex}}[\xb7]\phantom{\rule{-0.166667em}{0ex}}\rangle $:

“There is a strategy profile of the grand coalition in which all agents participate with their individual strategies in such a way that, each agent or coalition guarantees the satisfaction of its own goal against any possible deviations of all other agents, thus protecting its individual (respectively, coalitional) interests.”

#### 8.3. Some Observations and Examples of the Expressiveness of $\mathsf{LCGA}$

- The operator ${\langle \mathrm{A}\rangle}_{\mathsf{c}}(\varphi ;\langle \mathrm{B}\rangle \psi )$, introduced in Section 6, is definable in terms of $\langle \phantom{\rule{-0.166667em}{0ex}}[\xb7]\phantom{\rule{-0.166667em}{0ex}}\rangle $: ${\langle \mathrm{A}\rangle}_{\mathsf{c}}(\varphi ;\langle \mathrm{B}\rangle \psi )=\langle \phantom{\rule{-0.166667em}{0ex}}[A\u25b9\varphi ,\mathrm{A}\cup \mathrm{B}\u25b9\psi ]\phantom{\rule{-0.166667em}{0ex}}\rangle $. Nevertheless, it has a different motivation and intuitive interpretation there.
- The fragment ${\mathsf{SFCL}}_{\mathsf{1}}$ of $\mathsf{SFCL}$ (subsuming $\mathsf{CL}$) embeds into $\mathsf{LCGA}$ by defining $\left[C\right](\varphi ;\psi )$ equivalently as $\langle \phantom{\rule{-0.166667em}{0ex}}[C\u25b9\varphi ,\mathsf{Agt}\u25b9\psi ]\phantom{\rule{-0.166667em}{0ex}}\rangle $.However, this does not generalize to ${\left[C\right]}_{{}_{\phantom{\rule{-0.166667em}{0ex}}}}(\varphi ;{\psi}_{1},\dots ,{\psi}_{k})$ for $k\ge 2$.Conversely, the operator $\langle \phantom{\rule{-0.166667em}{0ex}}[{C}_{1}\u25b9{\varphi}_{1},\dots ,{C}_{n}\u25b9{\varphi}_{n}]\phantom{\rule{-0.166667em}{0ex}}\rangle $ cannot be expressed in terms of the $\mathsf{SFCL}$ operators ${\left[C\right]}_{{}_{\phantom{\rule{-0.166667em}{0ex}}}}(\varphi ;{\psi}_{1},\dots ,{\psi}_{k})$, either.These nonexpressiveness claims can be proved by using the respective notions of bisimulations introduced for each of these operators in [25].
- $\langle \phantom{\rule{-0.166667em}{0ex}}[{C}_{1}\u25b9{\varphi}_{1},{C}_{1}\cup {C}_{2}\u25b9{\varphi}_{2},\dots ,{C}_{1}\cup \dots \cup {C}_{k}\u25b9{\varphi}_{k}]\phantom{\rule{-0.166667em}{0ex}}\rangle $ is equivalent to the sequential version SF2 of the operator SF, mentioned in Section 7.1 as ${\left[\langle {C}_{1}\rangle {\varphi}_{1};\dots ;\langle {C}_{k}\rangle {\varphi}_{k}\right]}_{{}_{\phantom{\rule{-0.166667em}{0ex}}}}$, where the coalitions ${C}_{1},...,{C}_{k}$ and their goals are arranged in decreasing priority order.
- On the other hand, $\langle \phantom{\rule{-0.166667em}{0ex}}[{C}_{1}\u25b9{\varphi}_{1},{C}_{1}\u25b9{\varphi}_{2},\dots ,{C}_{k}\u25b9{\varphi}_{k}]\phantom{\rule{-0.166667em}{0ex}}\rangle $ is essentially expressible, up to a natural transformation of the models and semantics, in the group strategic STIT (cf. [13,14]), as $\diamond (\left[{C}_{1}\phantom{\rule{4pt}{0ex}}\mathsf{stit}\right]\mathsf{X}\phantom{\rule{0.166667em}{0ex}}{\varphi}_{1}\wedge ...\wedge \left[{C}_{k}\phantom{\rule{4pt}{0ex}}\mathsf{stit}\right]\mathsf{X}\phantom{\rule{0.166667em}{0ex}}{\varphi}_{k})$. This observation (suggested by a reviewer) leads to a new computationally well-behaved fragment of the group strategic STIT, which is known to be not only undecidable, but even nonaxiomatizable; cf. [14].

**Example 10.**

#### 8.4. Axiomatic System for $\mathsf{LCGA}$

**(Triv)**$\langle \phantom{\rule{-0.166667em}{0ex}}\left[{\gamma}^{\top}\right]\phantom{\rule{-0.166667em}{0ex}}\rangle $, where ${\gamma}^{\top}$ is the trivial goal assignment, mapping each coalition to ⊤.**(Safe)**$\neg \langle \phantom{\rule{-0.166667em}{0ex}}[\mathsf{Agt}\u25b9\perp ]\phantom{\rule{-0.166667em}{0ex}}\rangle $Even the grand coalition of all agents cannot ensure the falsum to become true.**(Merge)**$\langle \phantom{\rule{-0.166667em}{0ex}}[{C}_{1}\u25b9{\theta}_{1}]\phantom{\rule{-0.166667em}{0ex}}\rangle \wedge \dots \wedge \langle \phantom{\rule{-0.166667em}{0ex}}[{C}_{n}\u25b9{\theta}_{n}]\phantom{\rule{-0.166667em}{0ex}}\rangle \to \langle \phantom{\rule{-0.166667em}{0ex}}[{C}_{1}\u25b9{\theta}_{1},...,{C}_{n}\u25b9{\theta}_{n}]\phantom{\rule{-0.166667em}{0ex}}\rangle $,where ${C}_{1},\dots ,{C}_{n}$ are pairwise disjoint.This axiom scheme generalizes the superadditivity axiom of coalition logic (cf. Section 4.4.) It is valid, because, if the coalitions ${C}_{1},\dots ,{C}_{n}$ are pairwise disjoint, then they can join together their collective strategies for their respective coalitional goals into one strategy profile that ensures satisfaction of all these collective goals.**(GrandCoalition)**$\langle \phantom{\rule{-0.166667em}{0ex}}\left[\gamma \right]\phantom{\rule{-0.166667em}{0ex}}\rangle \to (\langle \phantom{\rule{-0.166667em}{0ex}}[\gamma [\mathsf{Agt}\u25b9(\phi \wedge \psi \left)\right]]\phantom{\rule{-0.166667em}{0ex}}\rangle \vee \langle \phantom{\rule{-0.166667em}{0ex}}[\gamma [\mathsf{Agt}\u25b9(\phi \wedge \neg \psi \left)\right]]\phantom{\rule{-0.166667em}{0ex}}\rangle )$, where $\gamma \left(\mathsf{Agt}\right)=\phi $.This axiom scheme is valid because any strategy profile for $\mathsf{Agt}$ generates a unique successor state. If a state formula $\psi $ is true in it, then it can be added to the coalitional goal of $\mathsf{Agt}$, otherwise its negation can be added to it.**(Case)**$\langle \phantom{\rule{-0.166667em}{0ex}}\left[\gamma \right]\phantom{\rule{-0.166667em}{0ex}}\rangle \to (\langle \phantom{\rule{-0.166667em}{0ex}}\left[\gamma [C\u25b9(\phi \wedge \psi )]\right]\phantom{\rule{-0.166667em}{0ex}}\rangle \vee \langle \phantom{\rule{-0.166667em}{0ex}}[\gamma {|}_{C}\left[(\mathsf{Agt}\u25b9\neg \psi ]\right]\phantom{\rule{-0.166667em}{0ex}}\rangle )$, where $\gamma \left(C\right)=\phi $.This scheme says that for any coalition C, state formula $\psi $, and a strategy profile $\mathsf{\Sigma}$, either its projection ${\mathsf{\Sigma}}_{C}$ to C ensures the truth of $\psi $ in all successor states enabled by ${\mathsf{\Sigma}}_{C}$—in which case $\psi $ can be added to the goal of C enforced by $\mathsf{\Sigma}$—or, otherwise, $\neg \psi $ is true in some of these successor states, in which case it can be added to ${\gamma |}_{C}$ as the goal of the grand coalition $\mathsf{Agt}$ enforced by $\mathsf{\Sigma}$.**(Con)**$\langle \phantom{\rule{-0.166667em}{0ex}}\left[\gamma \right]\phantom{\rule{-0.166667em}{0ex}}\rangle \to \langle \phantom{\rule{-0.166667em}{0ex}}\left[\gamma \right[C\u25b9(\phi \wedge \psi )\left]\right]\phantom{\rule{-0.166667em}{0ex}}\rangle $ where $\gamma \left(C\right)=\phi $ and $\gamma \left({C}^{\prime}\right)=\psi $ for some ${C}^{\prime}\subseteq C$.Given any coalition C and subcoalition ${C}^{\prime}$, this scheme says that the goal of ${C}^{\prime}$ can be added for free to the goal of C. Indeed, if there is any strategy profile $\mathsf{\Sigma}$ that ensures that C and ${C}^{\prime}$ can force their respective goals, then $\mathsf{\Sigma}$ also ensures that C can force the conjunction of these goals.

#### 8.5. The Temporal Logic of Coalitional Goal Assignments $\mathsf{TLCGA}$

**state formulae**and

**path formulae**, and to define their sets by mutual induction, as follows:

**$\langle \phantom{\rule{-0.166667em}{0ex}}\langle \mathrm{C}\rangle \phantom{\rule{-0.166667em}{0ex}}\rangle \theta \equiv \langle \phantom{\rule{-0.166667em}{0ex}}[C\u25b9\theta ]\phantom{\rule{-0.166667em}{0ex}}\rangle $**, for any $\theta \in \mathsf{PathFor}$.

#### 8.6. An Aside: Equilibria and Co-Equilibria

**co-equilibrium**was proposed in [26], meaning a strategy profile ensuring that every player (and coalition) achieves their private goal, even if all other agents deviate. Existence of a co-equilibrium is expressed precisely by the $\mathsf{TLCGA}$ formula $\langle \phantom{\rule{-0.166667em}{0ex}}[{\mathsf{a}}_{1}\u25b9{\varphi}_{1},\dots ,{\mathsf{a}}_{k}\u25b9{\varphi}_{k}]\phantom{\rule{-0.166667em}{0ex}}\rangle $.

#### 8.7. An Example

**Example 11.**

- c is the atomic proposition “all animals have crossed the river”.
- e is the atomic proposition “some sheep are eaten”.

## 9. Basic Strategy Logic: A Unifying Formalism for Strategic Reasoning

**strategy logics**, the first (two-agents) version of which was introduced in [21], and later extended and further studied in [22,23,24]. The common feature of these strategy logics is that they formally treat strategies as primitive objects, similar to elements of standard structures for first-order logic. I will refer to these generically as

**standard strategy logic**, denoted $\mathsf{SSL}$.

**basic strategy logic**, further denoted $\mathsf{BSL}$. It builds on some temporalized language for expressing goals, such as $\mathsf{LTL}$ (cf., e.g., [36] (Chapter 6)) that defines a

**set of goal formulae**$\mathsf{\Gamma}$ which are evaluated on plays in concurrent game models. $\mathsf{BSL}$ extends $\mathsf{\Gamma}$ with standard Boolean connectives, with variables associated with agents and ranging over strategies for them, by means of strategy assignments, and with quantification over such variables within the formulae. More precisely, with each agent $\mathsf{a}$, the language of $\mathsf{BSL}$ associates a

**strategy variable for $\mathsf{a}$**, denoted by ${\mathsf{x}}_{\mathsf{a}}$. Let $\mathsf{StrVar}$ be the set of strategy variables for all agents. Then, for every nonempty coalition of agents $C=\{{\mathsf{c}}_{1},...,{\mathsf{c}}_{m}\}$, I will use ${\mathsf{x}}_{C}$ to denote the tuple of variables $({\mathsf{x}}_{{\mathsf{c}}_{1}},...,{\mathsf{x}}_{{\mathsf{c}}_{m}})$. Furthermore, I will write $\forall {\mathsf{x}}_{\mathrm{C}}$ as a shorthand for the string $\forall {\mathsf{x}}_{{\mathsf{c}}_{1}}...\forall {\mathsf{x}}_{{\mathsf{c}}_{m}}$, and likewise for $\exists {\mathsf{x}}_{\mathrm{C}}$.

**goal formulae**$\mathsf{\Gamma}$ and a set of

**state formulae**$\mathsf{StateFor}$, defined as follows:

**strategy assignments**, which are functions defined on $\mathsf{StrVar}$ and assigning to every strategy variable ${\mathsf{x}}_{\mathsf{a}}$ a strategy of the respectively specified in the semantics type (e.g., memory-based or positional) for the agent $\mathsf{a}$. Given such strategy assignment $\mathbf{v}$ and a strategy ${\sigma}_{\mathsf{a}}$ for agent $\mathsf{a}$, I will denote by $\mathbf{v}[{\mathsf{x}}_{\mathsf{a}}\u2a74{\sigma}_{\mathsf{a}}]$ the modified strategy assignment which reassigns the value of ${\mathsf{x}}_{\mathsf{a}}$ to be ${\sigma}_{\mathsf{a}}$. Likewise, $\mathbf{v}[{\mathsf{x}}_{\mathrm{C}}\u2a74{\sigma}_{\mathrm{C}}]$ is defined for any coalition $\mathrm{C}$.

- $\mathcal{M},s,\mathbf{v}{\vDash}_{}\psi $, for $\psi \in \mathsf{\Gamma}$ iff $\mathcal{M},\mathsf{play}(s,\mathbf{v}){\vDash}_{}\psi $(assuming that truth of the goal formulae in $\mathsf{\Gamma}$ on plays has already been defined).
- $\mathcal{M},s,\mathbf{v}{\vDash}_{}\exists {\mathsf{x}}_{\mathsf{a}}\phi $ iff $\mathcal{M},s,\mathbf{v}[{\mathsf{x}}_{\mathsf{a}}\u2a74{\sigma}_{\mathsf{a}}]{\vDash}_{}\phi $ for some strategy ${\sigma}_{\mathsf{a}}$ for $\mathsf{a}$.

- $\mathsf{CL}$/${\mathsf{ATL}}^{*}$$$\tau \left(\left[\mathrm{C}\right]\phi \right)\u2a74\exists {\mathsf{x}}_{\mathrm{C}}\forall {\mathsf{x}}_{\overline{\mathrm{C}}}\tau \left(\phi \right),$$
- $\mathsf{ConStR}$
**(**${\mathsf{O}}_{\alpha}$**)**$$\tau \left({\left[\mathrm{A}\right]}_{\alpha}(\varphi ;\langle \mathrm{B}\rangle \psi )\right)\u2a74\exists {\mathsf{x}}_{\mathrm{B}\backslash \mathrm{A}}\forall {\mathsf{x}}_{\mathrm{A}}(\forall {\mathsf{x}}_{\overline{\mathrm{A}}}\tau \left(\varphi \right)\to \forall {\mathsf{x}}_{\overline{\mathrm{A}\cup \mathrm{B}}}\tau \left(\psi \right)).$$- Note that the joint strategy for $\mathrm{B}\backslash \mathrm{A}$ assigned to ${\mathsf{x}}_{\mathrm{B}\backslash \mathrm{A}}$ is supposed to ensure, together with the joint strategy for $\mathrm{A}$ assigned to ${\mathsf{x}}_{\mathrm{A}}$, the truth of the (translation of the) goal formula $\psi $ whenever the joint strategy for $\mathrm{A}$ assigned to ${\mathsf{x}}_{\mathrm{A}}$ ensures the truth of the (translation of the) goal formula $\varphi $ against any joint strategy of the agents in $\overline{\mathrm{A}}$, including those in $\mathrm{B}\backslash \mathrm{A}$.

- $\mathsf{ConStR}$
**(**${\mathsf{O}}_{\beta}$**)**$$\tau ({\left[\mathrm{A}\right]}_{\beta}(\varphi ;\langle \mathrm{B}\rangle \psi ))\u2a74\forall {\mathsf{x}}_{\mathrm{A}}(\forall {\mathsf{x}}_{\overline{\mathrm{A}}}\tau \left(\varphi \right)\to \exists {\mathsf{x}}_{\mathrm{B}\backslash \mathrm{A}}\forall {\mathsf{x}}_{\overline{\mathrm{A}\cup \mathrm{B}}}\tau \left(\psi \right)).$$- Note the difference with the previous translation, reflecting the difference between the semantics of (${\mathsf{O}}_{\alpha}$) and (${\mathsf{O}}_{\beta}$): the joint strategy for $\mathrm{B}$ now depends on the joint strategy for $\mathrm{A}$, and the strategies of the agents in $\mathrm{B}\cap \mathrm{A}$ may not differ from those already fixed in the latter.

- $\mathsf{ConStR}$
**(**${\mathsf{O}}_{c}$**)**$$\tau \left({\langle \mathrm{A}\rangle}_{\mathsf{c}}(\varphi ;\langle \mathrm{B}\rangle \varphi )\right)\u2a74\exists {\mathsf{x}}_{\mathrm{A}}(\forall {\mathsf{x}}_{\overline{\mathrm{A}}}\tau \left(\varphi \right)\wedge \exists {\mathsf{x}}_{\mathrm{B}\backslash \mathrm{A}}\forall {\mathsf{x}}_{\overline{\mathrm{A}\cup \mathrm{B}}}\tau \left(\psi \right)).$$- This formula captures the semantics of ${\mathsf{O}}_{c}$ in a straightforward way.

- $\mathsf{SFCL}$
**(SF)**$$\tau \left(\left[C\right](\varphi ;{\psi}_{1},\dots ,{\psi}_{k})\right)\u2a74\exists {\mathsf{x}}_{\mathrm{C}}\left(\forall {\mathsf{x}}_{\overline{\mathrm{C}}}\tau \left(\varphi \right)\wedge \underset{m=1}{\overset{k}{\bigwedge}}\exists {\mathsf{x}}_{\overline{\mathrm{C}}}\tau \left({\psi}_{m}\right)\right).$$- Likewise, the formula above formalizes precisely the semantic definition of SF.

- $\mathsf{SFCL}$
**(SF1)**$$\tau \left(\left[C;{C}_{1},\dots ,{C}_{k}\right](\varphi ;{\varphi}_{1},\dots ,{\varphi}_{k})\right)\u2a74\exists {\mathsf{x}}_{\mathrm{C}}\left(\forall {\mathsf{x}}_{\overline{\mathrm{C}}}\tau \left(\varphi \right)\wedge \underset{m=1}{\overset{k}{\bigwedge}}\exists {\mathsf{x}}_{{\mathrm{C}}_{m}\backslash \mathrm{C}}\forall {\mathsf{x}}_{\overline{\mathrm{C}\cup {\mathrm{C}}_{m}}}\tau \left({\varphi}_{m}\right)\right).$$- This formula says that C has a collective strategy (in particular, a collective action) assigned to ${\mathsf{x}}_{\mathrm{C}}$ that guarantees (the translation of) $\varphi $, and is such that, when fixed,each ${C}_{m}$ has a collective action (already fixed for the agents in ${\mathrm{C}}_{m}\cap \mathrm{C}$) that guarantees (the translation of) ${\varphi}_{m}$ against any behavior of the noncommitted agents, i.e., those in $\overline{\mathrm{C}\cup {\mathrm{C}}_{m}}$. This is precisely the semantics of SF1, defined in Section 7.1.

- $\mathsf{SFCL}$
**(SF2)**$\tau \left(\left[\langle {C}_{1}\rangle {\varphi}_{1};\dots ;\langle {C}_{k}\rangle {\varphi}_{k}\right]\right)\u2a74$- (for simplicity, assuming that ${C}_{1},...,{C}_{k}$ are pairwise disjoint)$$\phantom{\rule{0.0pt}{0ex}}\exists {\mathsf{x}}_{{C}_{1}}\left(\forall {\mathsf{x}}_{\overline{{C}_{1}}}\tau \left({\varphi}_{1}\right)\wedge \exists {\mathsf{x}}_{{C}_{2}\backslash {C}_{1}}\left(\forall {\mathsf{x}}_{\overline{{C}_{1}\cup {C}_{2}}}\tau \left({\varphi}_{2}\right)\wedge \dots \wedge \exists {\mathsf{x}}_{{C}_{k}\backslash ({C}_{1}\cup ...\cup {C}_{k-1})}\forall {\mathsf{x}}_{\overline{{C}_{1}\cup ...\cup {C}_{k}}}\tau \left({\varphi}_{k}\right)\right)\dots \right).$$This formula likewise expresses the semantics of the sequential version SF2 of SF1, defined in Section 7.1. To achieve this, every time it claims existence of a joint strategy of ${C}_{i}$ it assumes that the strategies of all agents in the already previously mentioned coalitions are already fixed, and that joint strategy must succeed against any behavior of the not-yet-committed agents only.

- $\mathsf{LCGA}$$$\tau \left(\langle \phantom{\rule{-0.166667em}{0ex}}\left[\gamma \right]\phantom{\rule{-0.166667em}{0ex}}\rangle \right)\u2a74\exists {\mathsf{x}}_{\mathsf{Agt}}\underset{\mathrm{C}\subseteq \mathsf{Agt}}{\bigwedge}\forall {\mathsf{x}}_{\overline{\mathrm{C}}}\tau \left(\gamma \left(\mathrm{C}\right)\right).$$
- Again, this formula captures the formal semantics of $\langle \phantom{\rule{-0.166667em}{0ex}}\left[\gamma \right]\phantom{\rule{-0.166667em}{0ex}}\rangle $ in a straightforward way.

## 10. Concluding Remarks: Outlook and Perspectives

- Adding agents’ knowledge in the semantics, and explicitly in the language, by assuming that the agents reason and act under imperfect information.
- Taking into account the normative aspects and constraints of the socially interactive context, including obligations, permission, and prohibitions, which socially responsible rational agents must respect in their strategic behavior.
- Analysis of the expressiveness and computational complexity of the basic strategy logic $\mathsf{BSL}$ which should eventually determine whether and to what extent $\mathsf{BSL}$ may be regarded as a viable alternative to the propositional modal approach behind the logical systems for strategic reasoning presented here.
- In particular, an interesting currently open question is whether there is a finite set of modal strategic operators with semantics that can be translated to $\mathsf{BSL}$ which provide expressive completeness, if not for the full language of $\mathsf{BSL}$, then at least for natural and reasonably expressive fragments of it.
- Completeness results for some of the systems of axioms mentioned here, including the three main fragments of $\mathsf{ConStR}$, the entire $\mathsf{ConStR}$, and $\mathsf{SFCL}$. Ultimately, a complete axiomatic system for $\mathsf{BSL}$, if possible, or for substantially rich fragments of it.
- Finite tree-model property and decidability results for the logical systems presented here, as well as other fragments of $\mathsf{BSL}$. In particular, it would be tableaux-based deductive systems and decision methods for the logics studied here, by adapting such systems developed in, e.g., [47,48].

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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1 | These remarks are prompted by an exchange and apparent disagreement with an anonymous reviewer of this paper, who argues that the logical frameworks discussed here do not really deal with goals, rationality, or sociality. In my understanding, the core of that disagreement is in the essentially different perspectives which we apparently have on the topic and on some basic concepts in this paper, as explained further in the text. |

**Figure 2.**The formula $\langle \phantom{\rule{-0.166667em}{0ex}}\langle \mathsf{a}\rangle \phantom{\rule{-0.166667em}{0ex}}\rangle (\mathsf{F}\phantom{\rule{0.166667em}{0ex}}p\wedge \mathsf{F}\phantom{\rule{0.166667em}{0ex}}q)$ is true at state ${s}_{0}$ only if the agent $\mathsf{a}$ can use some memory.

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**MDPI and ACS Style**

Goranko, V.
Logics for Strategic Reasoning of Socially Interacting Rational Agents: An Overview and Perspectives. *Logics* **2023**, *1*, 4-35.
https://doi.org/10.3390/logics1010003

**AMA Style**

Goranko V.
Logics for Strategic Reasoning of Socially Interacting Rational Agents: An Overview and Perspectives. *Logics*. 2023; 1(1):4-35.
https://doi.org/10.3390/logics1010003

**Chicago/Turabian Style**

Goranko, Valentin.
2023. "Logics for Strategic Reasoning of Socially Interacting Rational Agents: An Overview and Perspectives" *Logics* 1, no. 1: 4-35.
https://doi.org/10.3390/logics1010003