Nuel Belnap on Indeterminism and Free Action pp 291314  Cite as
Connecting Logics of Choice and Change
Abstract
This chapter is an attempt at clarifying the current scene of sometimes competing action logics, looking for compatibilities and convergences. Current paradigms for deliberate action fall into two broad families: dynamic logics of events, and STIT logics of achieving specified effects. We compare the two frameworks, and show how they can be related technically by embedding basic STIT into a modal logic of matrix games. Amongst various things, this analysis shows how the attractive principle of independence of agents’ actions in STIT might actually be a source of high complexity in the total action logic. Our main point, however, is the compatibility of dynamic logics with explicit events and STIT logics based on a notion that we call ‘control’—and we present a new system of dynamicepistemic logic with control that has both. Finally, we discuss how dynamic logic and STIT face similar issues when including further crucial aspects of agency such as knowledge, preference, strategic behavior, and explicit acts of choice and deliberation.
Keywords
Logic of action Stit theory Dynamic logic Dynamicepistemic logic Strategic games Control1 Introduction: Logical Frameworks for Agency
The STIT logic of Belnap et al. (2001) and its variants have proven fruitful tools to help philosophers and computer scientists explore their intuitions about agency and social interaction. These logics provide a framework to reason about choices, abilities and actions of agents, all placed in a temporal setting. And further issues lie just below the surface: what agents know or believe at the time of choice, how they act based on preferences, and engage in deliberate strategic interaction (cf. Horty 2001).
But STIT is not the only game in town. Many logical paradigms are active in the above territory, and they often show clear similarities. This calls for analysis and reflection. For instance, van Benthem and Pacuit (2006) relate the major varieties of epistemic temporal logics, coming from mathematical logic, computational logic, and studies of agency. Continuing in this line, van Benthem et al. (2009) prove representation theorems linking dynamicepistemic models with epistemictemporal ones, making it possible to enlist ideas from one logic in the service of the other. In the case of STIT, too, much has been done to clarify its connections with other frameworks. In fact, Belnap et al. (2001) already pointed out links with earlier work of Chellas (1992), to which one can add the neighborhood logics of ability in (Brown 1988, 1992). Moreover, connections have been found with coalition logic (Broersen et al. 2006b) and alternatingtime temporal logic (Broersen et al. 2006a), while Lorini and Schwarzentruber (2010) relates STIT to logics for strategic and extensive games—a line that we will continue in this chapter (cf. also Herzig and Lorini 2010). Finally, Ciuni and Zanardo (2010) shows how STIT extends wellknown logics of branching time.
Our aim in this chapter is to continue in the latter vein, and connect STIT models further with modal models for action from the realm of propositional dynamic logic (PDL), modal game logics (see van Benthem 2014), and dynamicepistemic logic DEL (van Benthem 2011). We start by addressing an initial barrier to making any comparison between these different logical frameworks.
STIT logics are primarily intended as logics of ontic freedom and indeterminacy while the logical systems we discuss in this chapter are focused on epistemic uncertainty (i.e., knowledge about what will happen next). The heart of our comparison is the simple observation that the basic STIT modality turns out to be precisely the “knowledge” modality found in many epistemicallyoriented logical systems. Importantly, however, we are not suggesting that all discussion about “agency” and agents making choices in an indeterministic world can or should be replaced with an analysis of what the agents know about their own choices and the consequences of their actions in an indeterministic world, or vice versa. Our point is simply that similar logical frameworks are open to different interpretations. The goal is not to argue for the primacy of any single interpretation, but rather to demonstrate how two different perspectives on modeling rational agency can lead to similar insights. This is in line with a broader goal. The arena of logics for agency appears to be moving from an initial stage of a “Battle of the Sects” to a more detached understanding of both similarities and relative advantages of different paradigms, leading to a more unified sense of purpose and methodology.
2 Preliminaries: The STIT Framework
In this section, we introduce the basic STIT framework. We will be very brief, only touching on the key notions we need later in this chapter. For more information, the reader is invited to consult (Horty 2001; Belnap et al. 2001; Horty and Belnap 1995; Balbiani et al. 2008).
STIT structures STIT models are based on branchingtime frames, structures \(\langle T, <\rangle \) where \(T\) is a nonempty set of “moments”, and \(<\) is a strict partial order on \(T\) without backwards branching: for all \(m,m',m''\), if \(m'<m\) and \(m''<m\), then either \(m'\le m''\) or \(m''\le m'\) (where \(x\le y\) iff \(x<y\) or \(x=y\)). A history is a maximal linearly ordered subset of \(T\). Let \(\mathsf {Hist}\) denote the set of all histories and for \(t\in T\), \(H_t=\{h\in \mathsf {Hist}\ \ t\in h\}\) is the set of histories containing moment \(t\).
At each moment, there is a choice available to the agent. Let \(\mathcal {A}\) be the set of agents. Formally, the choices available to agent \(i\) at moment \(t\) are represented by a partition \(Choice_i^t\) on the set \(H_t\) of histories containing \(t\). Let \(Choice_i^t(h)\) denote the cell containing \(h\). Since \(Choice_i^t\) is a partition, we have for each \(i\in \mathcal {A}\) and \(t\in T\), \(Choice_i^t\ne \emptyset \) and \(\emptyset \not \in Choice_i^t\). In addition, the choice partitions of the agents must satisfy one additional condition:
Independence For all \(t\in T\) and all \(s_t:\mathcal {A}\rightarrow \wp (H_t)\) with \(s_t(i)\in Choice_i^t\), \(\bigcap _{i\in \mathcal {A}} s_t(i)\ne \emptyset \).
Now we define a STIT model as a tuple \(\langle T, <, \mathcal {A}, Choice, V\rangle \), where \(\langle T, <\rangle \) is a branchingtime frame, \(\mathcal {A}\) is a finite set of agents, \(Choice\) is a function assigning to each \(i\in \mathcal {A}\) and \(t\in T\) a partition on \(H_t\) satisfying \(\mathsf {Independence}\), and \(V\) is a function assigning to each atomic proposition a set of history/moment pairs (\(V:\mathsf {At}\rightarrow \wp (T\times \mathsf {Hist})\)).

\(\mathcal {M},t/\,h\models p\) iff \(t/\,h\in V(p)\)

\(\mathcal {M},t/\,h\models \lnot \varphi \) iff \(\mathcal {M},t/\,h\not \models \varphi \)

\(\mathcal {M},t/\,h\models \varphi \wedge \psi \) iff \(\mathcal {M},t/\,h \models \varphi \) and \(\mathcal {M},t/\,h \models \psi \)

\(\mathcal {M},t/\,h\models \Box \varphi \) iff \(\mathcal {M},t/\,h' \models \varphi \) for all \(h'\in H_t\)

\(\mathcal {M},t/\,h\models [i\ \mathsf {stit}]\varphi \) iff \(\mathcal {M},t/\,h' \models \varphi \) for all \(h'\in Choice_i^t(h)\)

\(\mathcal {M},t/h\models [i\ {\mathsf {dstit}}] \varphi \) iff \(\mathcal {M},t/h' \models \varphi \) for all \(h'\in Choice_i^t(h)\) and there is a \(h''\in H_t\) such that \(\mathcal {M},t/h''\models \lnot \varphi \)

The \(\mathbf{{S5}}\) axioms for \(\Box \) and \([i\ \mathsf {stit}]\): \(\bigcirc (\varphi \rightarrow \psi )\rightarrow (\bigcirc \varphi \rightarrow \bigcirc \psi )\), \(\bigcirc \varphi \rightarrow \varphi \), \(\bigcirc \varphi \rightarrow \bigcirc \bigcirc \varphi \), \(\lnot \bigcirc \varphi \rightarrow \bigcirc \lnot \bigcirc \varphi \), for \(\bigcirc \in \{\Box ,[i\ \mathsf {stit}]\}\)

\(\Box \varphi \rightarrow [i\ \mathsf {stit}]\varphi \)

\(\left( \bigwedge _{i\in \mathcal {A}}\Diamond [i\ \mathsf {stit}]\varphi _i\right) \rightarrow \Diamond \left( \bigwedge _{i\in \mathcal {A}}[i\ \mathsf {stit}]\varphi _i\right) \)

Modus Ponens and Necessitation for \(\Box \).
Now that we have all major components of STIT on the table, we will discuss its semantics and axiomatics in relation to other approaches for studying agency coming from the “dynamic logic family”. We will not define these other frameworks in any detail, but refer the reader to the literature on dynamic logic, game logics, and dynamicepistemic logics cited in this chapter.
3 Modeling Choice Situations
3.1 The Modal Heart of Choice
The key idea of STIT in these models may be called control: the equivalence relations represent the extent to which agents control outcomes by their choices. The product axiom says that no agent can prevent any other agent from making any of her choices. There is more to this condition than meets the eye. For instance, assume that agent \(1\) has a singleton choice somewhere. Since 2’s choices must always overlap with this singleton, and different choices are disjoint, it follows that 2 has only one choice set.
So, basic STIT logic is a nice simple multi\(\mathbf{{S5}}\)extension. This first natural connection with modal logic shows that we are at least generally in the same world as modal logics of action.^{4}
3.2 An Initial Comparison with Modal Logics of Action
Broadly speaking, there are two general views about how to model the actions available to an agent. The first is the view found in STIT as presented in Sects. 2 and 3.1 above. Let us now consider the second view, that of modal and dynamic logics of actions (see Harel et al. 2000 for a discussion), which is also the main model of action in Situation Calculus (Reiter 2001), Automata Theory, Decision Theory and Game Theory. Its general idea is to think of actions as transitions moving between different “states of the system”. This happens again in standard modal models \(\mathcal {M}=\langle W, \{R_a\}_{a\in \mathsf {Act}}, V, s\rangle \), with worlds in \(W\) viewed as states of some process (\(s\) is the initial state), and labeled transition relations \(R_a\subseteq W\times W\) for each action label \(a\in \mathsf {Act}\). Each relation \(R_a\) indicates the possible executions of the basic action \(a\). Modal languages over these models then describe possible effects of actions, while real dynamic logics also have an explicit language for speaking about complex actions defined by means of sequential composition, conditional choice, or iteration.^{5} We use the phrase PDL scenarios for this family of paradigms.
At a first glance, these are very different views. While both perspectives acknowledge variety in possible outcomes of actions, they also have structure that the other lacks. In actionlabeled approaches, the primary emphasis is on actions or events themselves and their properties, of which a description of outcome states seems only one. For instance, dancing a tango involves many features in addition to its end state: we would trivialize the process by just having an end state of ‘having danced a tango’. On the other hand, many daily actions expressed in natural language are largely defined by just postconditions on their outcomes, witness ‘opening the door’ or ‘posting a letter’. In that sense, STIT’s approach to describing actions is very natural.
We now proceed to a more technical comparison of the two styles. But to do so, we need some further touches. For a start, our simple modal picture of STIT choice situations takes out all of temporal structure. However, for a comparison with PDL, it seems more concrete to view the above ‘worlds’ as steps emanating from some root toward next states in a tree, a snapshot of an ongoing decision process. The actual world is then the actual transition from the root to some next state:
What this suggests is introducing a richer modal language for basic STIT, referring also to the two stages: ‘now’ and ‘next’. This motivates the NEXTSTIT of Broersen (2011), and we will also encounter this setting in the DELstyle logic of Sect. 6. But right now, we continue in a semantic mode with worlds viewed intuitively as transitions.
Likewise, in order to compare STIT with PDL, we must also clarify the intuitive interpretation of PDLstyle models. In particular, there are two broad views in the literature. One is that of transition models as abstract processes or machines, the other as unraveled temporal executions. On the process view, worlds are states in a process, and the relations indicate possible transitions. On this view, the model is a sort of automaton, perhaps in a very compact form, where many different transition relations can go from one state to the same next state. By contrast, the second view of PDLmodels is one of unraveled temporal execution. Intuitively, once a process starts working, it produces a temporal universe of executions, being histories of successive admissible actions (cf. Clarke et al. 2000; Clarke and Emerson 1981 for this view). For the usual modal languages of action, the difference between the two views does not matter, since the execution tree is just a bisimilar unraveling of the process. And vice versa, we can think of a process as a sort of bisimulationcontracted essence of what can happen in the execution tree. But in our present setting, comparing with STIT seems to favor the temporal execution view.^{6}
We therefore continue with the temporal view, where for simplicity, all event labels are taken to be unique.^{7} Like with the above basic STIT, we will not take the full temporal models here, but just the snapshots of a onestep action. A PDL action scenario is a set of labeled transitions from some initial state \(s\), each leading to a different successor state. This can be viewed as an obvious special “oneshot” case of the earliermentioned transition models.
3.3 Merging the Two Perspectives on Action
Our goal in this chapter is not to reduce STIT models to PDL models, or vice versa. We find it more rewarding to show connections between the two perspectives leading to merged systems.
For better focus, we start with the singleagent case. Consider a simple STIT choice situation with two states \(W= \{w_1,w_2\}\) and two equivalence classes: \([w_1]=\{w_1\}\) and \([w_2]=\{w_2\}\). Thus, there are two choices for the agent, which we label \(c_1\) and \(c_2\), respectively. A simple corresponding PDL action scenario has two transitions from the root state \(s\) labeled by \(c_1\) and \(c_2\), respectively:
This seems straightforward, but we have not yet found the real structure that we need. To get at this, consider a STIT choice situation with the same two states, but now only one equivalence class \([w_1]=[w_2]=\{w_1,w_2\}\). Now the agent only has one choice \(c\). We cannot label the two transitions by \(c\) now, since that gives a PDL model with the same event, and it is unclear how this would fit the scenario. Here the difficulty is not that we cannot label the transitions: We can introduce different events for them, say \(e\) and \(f\). In fact, this makes sense even in STIT, since histories consist of events, and as we said earlier, if two histories are different, this is because different events take place on them. But this still does not address the matter of the choice structure, and crucially related to this: how we interpret the branching in our PDL model.^{8} What emerges here is an ambiguity in the usual talk about PDL models. In particular, what do branchings mean? Sometimes, people talk as if these are conscious choices a process or an agent can make, sometimes as if they are variations that cannot be predicted. What we need to distinguish the two senses is precisely the notion provided by STIT, that of control. In our first scenario, the two labels \(c_1\) and \(c_2\), when added to events, divide them into two control equivalence classes. In the second scenario, adding the label \(c\) to both \(e\) and \(f\) indicates how the events belong to the same control class. The agent cannot choose between the events.
We end with two more general comments about this encounter.
More on interpretations of PDL The confrontation with STIT leads to some useful clarification. We already mentioned the two main views of models as representing ‘process’ structure versus ‘execution space’. We also discussed a major ambiguity in how one interprets branching. As a final point, we mention the issue of events versus actions. There is a lot of loose talk in the PDL literature about action and choice. For instance, backandforth clauses in bisimulation are justified by looking at ‘internal choices’ that a process has, and one often talks about events in PDL models as actions performed by agents.^{9} But really, PDL talks about arbitrary events, all further meaning for these has to be supplied additionally in different settings. In particular, actions by agents are events with special further structure, and if it matters, these need to be made explicit. The above case of ‘control’ is one clear instance.^{10}
A caveat about framework comparison In this section, we have engaged in highlevel framework comparison. But rarefied air can exaggerate ideological differences, and it is important to also think of applied experience. In modeling practice, framework differences often prove much less dramatic than expected, as is wellknown from the fact that the same real process can often be specified very happily in quite different computational paradigms. For instance, in our setting, dealing with concrete scenarios of choices and actions requires an explicit modeler’s decision as to individuating states and actions: formal frameworks themselves do not tell us how to do that. But then, differences between STIT and PDL tools may just amount to different legitimate decisions on how one individuates actions.^{11} The problem of individuating actions has been discussed extensively in philosophy (see footnote 3 on pg. 588 of Horty and Belnap (1995) for a concise explanation), and it also shows in modeling practice in computer science.
This brings us to an important philosophical issue which we have thus far swept under the rug. In PDL models, actions are labels of transitions and this basic sorting of transitions by their labels seems to suggest a particular ontology of actions and events. In STIT models, there is no such sorting and, indeed, the only way to characterize an action is by reference to the outcomes. This raises an important question for the philosophical logician: Does adopting PDL as a logic of actions force one to take sides in philosophical debates about the ontology of events and actions? Our response is to bracket this question since we feel that both STIT and PDL models are open to a wide range of philosophical interpretations, regardless of the original intended interpretation of these logical frameworks. However, we certainly admit that this rather mathematical “formal modeling” view is itself controversial and we welcome (and enjoy) debates on this issue. Nonetheless, we hope that the comparative points we are making in this chapter still make sense.
4 A Merged System: Matrix Game Logic
Now, we want to make our comparisons and merges more concrete by looking at a concrete modal logic that already existed independently, and that turns out to shed some additional light on the semantic and axiomatic aspects of STIT meeting PDL.
Choices and pair events Let us return to the STIT choice situation for two agents. There is an actual world with the choices that were actually made. It makes sense to think of the worlds here as pairs of actions chosen. Note that each world \(w\) can be mapped to a unique pair of equivalence classes containing it, one for each agent, and by the product axiom, this map to pairs of equivalence classes is surjective. What we do not know is whether the map is injective, and indeed it may not be, unless we modify the product axiom to require that different choices for all the agents have singleton intersections. The latter constraint says that all slack in choices has been explained by introducing enough agents—perhaps including the ‘environment’ to take up all remaining slack. There is some simple arithmetic involved here. Assume that our model is finite. The product axiom with the singleton clause forces all equivalence classes for agent \(1\) to have the same size \(n\), as they need room for representatives of all choices of \(2\). The total size will be \(n\times k\), with \(k\) the fixed size for \(2\) that exists similarly. But this suggests a viewpoint in terms of “matrix models” for joint actions that is wellknown from logics of games in strategic form (cf. Osborne and Rubinstein 1994). We will develop this analogy here, using a logic proposed in (van Benthem 2007) that provides a particularly apt comparison for STIT, while also doing full justice to the PDL perspective.^{12}
4.1 Modal logic of matrix games
Games induce natural models for epistemic, doxastic and preference logics, as well as conditional logics and temporal logics of action. See van der Hoek and Pauly (2006) for an overview of many such systems. Our discussion just takes a small slice.
Recall the definition of a strategic game for a set of players \(N\): (1) a set \(A_i\) of actions for each \(i\in N\), and (2) a utility function or preference ordering on the set of outcomes. For simplicity, one often identifies the outcomes with the set \(S\) \(=\) \(\Pi _{i\in N} A_i\) of strategy profiles. Given a strategy profile \(\sigma \in S\) with \(\sigma =(a_1,\ldots ,a_n)\), \(\sigma _i\) is the \(i\)th projection (i.e., \(\sigma _i=a_i\)) and \(\sigma _{i}\) lists the choices of all agents except agent \(i\): \(\sigma _{i}=(a_1,\ldots ,a_{i1},a_{i+1},\ldots ,a_n)\).

\(\sigma \sim _i \sigma '\) iff \(\sigma _i=\sigma _i'\): this epistemic relation represents player \(i\)’s “view of the game” at the ex interim stage where \(i\)’s choice is fixed but the choices of the other players’ are unknown,

\(\sigma \approx _i\sigma '\) iff \(\sigma _{i}=\sigma '_{i}\): this relation of “action freedom” (a term taken from Seligman (2010)) gives the alternative choices for player \(i\) when the other players’ choices are fixed.

\(\sigma \models [\sim _i]\varphi \) iff for all \(\sigma '\), if \(\sigma \sim _i\sigma '\) then \(\sigma '\models \varphi \).

\(\sigma \models [\approx _i]\varphi \) iff for all \(\sigma '\), if \(\sigma \approx _i\sigma '\) then \(\sigma '\models \varphi \).
What is the deductive power of the basic modal logic of strategic games? As before, we restrict attention to twoplayer games. First, given the nature of our relations, the separate logics are standard modal \(\mathbf{{S5}}\) for epistemic outlook and action freedom. In addition, the interaction of these modalities validates further laws. In particular, the above fact about the universal modality is reflected in the following law:
the equivalence \([\sim _i][\approx _i]\varphi \leftrightarrow [\approx _i][\sim _i]\varphi \) is valid in all matrix game models.
This validity depends on, and in fact it expresses, the geometrical “grid property” of game matrices that, if one can go on a path \(x \sim _i y \approx _i z\), then there also exists a point \(u\) with \(x \approx _i u \sim _i z\). We will discuss what this feature means in some more detail in Sect. 4.3.
This concludes our brief introduction to the modal logic of matrix games. For details and further issues, the reader is referred to (van Benthem 2014).
4.2 STIT in Modal Matrix Logic
Fact 4.1.
Our translation embeds STIT logic faithfully into the modal logic of full matrix games.
Proof.
First consider the direction from STIT theoremhood to modal game logic. Our translation validates the earlier STIT axioms, where the action modality refers to all consequences of the choice actually made, while the freedom modality looks at all alternative histories passing through the current profile. In particular, the quantifier combination employed in the Freedom axiom now becomes derivable through the theorems that are derivable for the STIT modality plus the existential modality \(E\) defined as \(\langle \approx _1\rangle \langle \approx _2\rangle \):
Fact 4.2.
The formula \((E [\sim _1] \varphi \ \wedge \ E [\sim _2]\psi )\rightarrow E(\varphi \wedge \psi )\) is derivable in multi\(\mathbf{{S5}}\) plus the commutation law for the two modalities.
Conversely, to prove that the embedding is faithful, we need to refute each nonvalid STIT law in our matrix models. To do so, take any STIT temporal countermodel in the sense of Sect. 2, and note that it suffices to look at the current moment and the next moments only (recall, that our STIT language does not contain temporal modalities). Furthermore, without loss of generality, we assume that this model is finite. More precisely, as in Sect. 3.1, we can abstract a finite twoagent basic STIT \(\mathbf{{S5}}\)model out of the temporal structure by letting histories be worlds, and defining agent’s equivalence relations respecting their choice partitions. Now, the historic necessity operator is the universal modality while the two STIT modalities are the modalities for the equivalence relations. The last step is to show that we can transform this model into a matrix model.
If the intersections of the equivalence classes, one from each agent, are singletons, then we are done. Otherwise, we proceed as follows. A cell is an intersection of the agents’ equivalence classes (i.e., \(C=[w]_1\,\cap \,[v]_2\) for some states \(w\) and \(v\)). Since the model is finite, there are finitely many cells and each cell has only finitely many states in them. Furthermore, by the independence assumption, each cell is nonempty. Let \(m\) be the number of elements in the largest cell. Without loss of generality, we can assume that all cells contain exactly \(m\) states (this may require adding copies of states to the model).
Organize the cells so that they form a matrix where each row contains all the cells making up a 1equivalence class and each column contains all cells making up a 2equivalence class. Label each cell by its position in the matrix (so, the pair \((x,y)\) corresponds to the cell in row \(x\) and column \(y\)). There may be more than one way to organize the cells so that the rows correspond to a 1equivalence class and the columns correspond to a 2equivalence class. Our construction does not depend on the choice of labeling. For the remainder of the proof, fix such a \(r\times c\) matrix.
Now, construct an \(m\times m\) matrix for each cell. Fix a cell \(C\) labelled with \((x,y)\) containing states \(w_1,\ldots , w_m\). Worlds in the new model with be 4tuples \((i,j,x,y)\) where \((i,j)\) denotes the position in the matrix and \((x,y)\) denotes the cell containing the world. Formally, let \((i,j,x,y)\) be a copy of \(w_{i+{j1} \mod m}\). So, for example, if \(m=3\), then the world \((2,3,x,y)\) is a copy of \(w_1\). Note that each row and each column contains a copy of all the worlds in \(C\).

\((i,j,x,y)\sim _1^0 (i,j',x,y)\) for all \(j,j'\le m\)

\((i,j,x,y)\sim _2^0 (i',j,x,y)\) for all \(i,i'\le m\)

\((i,j,x,y)\sim _1^0 (i,0,x,y+1)\), where the addition is taken modulo m

\((i,j,x,y)\sim _2^0 (0,j,x+1,y)\), where the addition is taken modulo m
 1.
If \((i,j,x,y)\sim _1' (i',j',x',y'),\) then \(i'=i\) and \(x'=x\). If \(y'=y,\) then \(w_{i+(j1) \mod m}\) and \(w_{i+(j'1) \mod m}\) are both in the cell labeled by \((x,y)\), and so \(w_{i+(j1) \mod m} \sim _1 w_{i+(j'1) \mod m}\). If \(y'\ne y\), then \(w_{i+(j1) \mod m}\) and \(v_{i+(j'1) \mod m}\) are in different cells. However, we still have \(w_{i+(j1) \mod m}\sim _1 v_{i+(j'1) \mod m}\) since we assume that cells in the same row are in the same 1equivalence class.
 2.
If \((i,j,x,y)\sim _2' (i',j',x',y')\) then \(j'=j\) and \(y'=y\). If \(x'=x\), then \(w_{i+(j1) \mod m}\) and \(w_{i+(j'1) \mod m}\) are both in the cell labeled by \((x,y),\) and so \(w_{i+(j1) \mod m} \sim _2 w_{i+(j'1) \mod m}\). If \(x'\ne x\), then \(w_{i+(j1) \mod m}\) and \(v_{i+(j'1) \mod m}\) are in different cells. However, we still have \(w_{i+(j1) \mod m}\sim _2 v_{i+(j'1) \mod m}\) since we assume that cells in the same column are in the same 2equivalence class.
The last thing we need to check is that the intersection of agents’ equivalence classes are singletons. Suppose that \((i_0,j_0,x_0,y_0)\sim _1' (i',j',x',y')\), \((i_0,j_0,x_0,y_0) \sim _1' (i'',j'',x'',y'')\), \((i_1,j_1,x_1,y_1)\sim _2' (i',j',x',y')\) and \((i_1,j_1,x_1,y_1)\sim _2' (i'',j'', x'',y'')\). Then, by construction, \(i'=i''=i_0\), \(x'=x''=x_0\) and \(j'=j''=j_1\) and \(y'=y''=y_1\). Hence, \((i',j',x',y')=(i'',j'',x'',y'')\), as desired.
We have shown that our translation is both correct and faithful.^{16} QED
This proof exploits the fact that matrix game models are close to the multi\(\mathbf{{S5}}\) models for basic STIT defined in Sect. 3.1. Still, the geometrical matrix perspective is useful, since it links up with a body of existing results. We will see a number of examples as we proceed.
4.3 Complexity and Correlation
While the preceding embedding makes sense, it does embed STIT in a system whose behavior is potentially complex. Richer modal logics of matrix games may well be unaxiomatizable and worse. The reason is the above commutation law for the two equivalence relations. While this may look like a pleasant structural feature of matrices, its logical effects are delicate. It is wellknown that the general logic of bimodal languages plus a universal modality on ‘grid models’ with two immediate successor relations is not decidable, and not even axiomatizable: indeed, it is “\(\Pi _1^1\)complete” (cf. Halpern and Vardi 1989; Marx 2007; Gabbay et al. 2003; van Benthem and Pacuit 2006). The reason is that grid structure can be exploited to encode computations of Turing machines on successive rows, or geometrical “tiling problems” of known high complexity.
Now, it is not clear whether our most basic modal game logic falls into this trap, since our models only have two equivalence relations, one horizontal and one vertical. Indeed, its closeness to STIT may suggest that it remains decidable–even though this does not follow from our earlier embedding result, that went in the opposite direction. Still, Halpern and Vardi (1989) and Spaan (1990) show high complexity of modal logics on grid models with reflexive transitive relations, using an encoding trick with alternating proposition letters.^{17}
This potential high complexity, while not directly threatening to STIT, does raise an interesting issue in modeling action. A standard way of defusing high complexity results is by allowing more models. In the present setting, the resulting structures are general game models where certain strategy profiles may be absent. Then general modal game logic becomes much simpler, being just multiagent modal \(\mathbf{{S5}}\) without any connecting axioms (van Benthem 1997).^{18}
Now this is not just a technical move: “profile gaps” encode something interesting, namely correlations between behavior of agents. In a general game model, if player \(i\) changes her move, then the only available profiles for this may now be ones where some other player \(j\) has changed his move as well. Game theorists have studied correlations extensively: cf. (Aumann 1987; Brandenburger and Friedenberg 2008). But the same notion has come up in logic, since correlations provide “information channels” where the behavior of one agent can carry information about that of another (Barwise and Seligman 1997). And more recently, generalized forms of such dependencies have become the focus of attention in “dependence logics” (Väänänen 2007). In other words, independence may be costly, and the Product Axiom that seemed the pride of STIT may eventually stand in the way, being just an extreme case of a more sophisticated theory of agent behavior.^{19}
In the rest of this chapter, we look at extensions of the current framework with features that seem essential to rational agency, and that have been the subject of study in dynamic logics.
5 The Roles of Knowledge
Our connection between STIT and matrix games introduced a notion of knowledge, of agents that have decided, but do not know yet what the others have chosen. Knowledge is not mentioned explicitly in STIT framework, but it seems to be lurking behind the scenes here. In fact, it is present in more than one way: choice and action naturally come with varieties of knowledge. Here is how this can happen, even in the simple setting that we have considered.
Consider a onestep action. Before I have made my choice, I only know that one of the available future histories will occur: and in that sense, the STIT tree modality already acts as a form of knowledge about how the whole future can unfold. This knowledge can be significant, since the tree encodes the “protocol” of all possible runs of the current process.
Next, right after I have chosen my action, I know what I am going to do, but I still do not know what the others will do, and this was the sort of knowledge based on personal decisions that was made explicit in the matrix models for games of Sect. 4.
Finally, once both our actions have actually taken place, agents do know what was chosen, if we assume that they observe these actions publicly. Knowledge from observation of events is a major source of information in a temporal world. It is is often encoded in epistemic uncertainty relations between moments of time that are used to model informationdriven processes, such as games with imperfect information (cf. Binmore 2009; Parikh and Ramanujam 2003; Fagin et al. 1995). As for its driving forces, updating knowledge from public observation or more private sources is the key topic in dynamicepistemic logics (van Benthem 2011).
It is natural to add epistemic operators of all these sorts to logics of decision and action, and in fact, this is happening in logics of games (cf. van Benthem 2014). Many kinds of knowledge relevant to action scenarios are local, having to do with what agents know temporarily as they make a choice. But more global “procedural knowledge” about the future of the process is essential, too, and then the trees of STIT may lose their grip. If I know something about your space of possible strategies, the informational situation will need “STIT forests” rather than trees to distinguish the alternatives (cf. van Benthem et al. 2009). The same complication arises in genuine multiagent scenarios. One cannot assume that agents know everything about others, and to cope with this variation, again, models have to be complicated beyond the basic STIT format.
Pursuing these matters is beyond the scope of this chaper, but explicit modeling of knowledge seems inescapable in a serious theory of choice and action. For a discussion along these lines, see Pacuit and Simon (2011) for a logical system that merges ideas from STIT and PDL while explicitly representing the agents’ knowledge. We see it as one virtue of our linking up STIT and PDL that experiences in the latter area can then be enlisted for the former. Our next section will present a case study, of one particular dynamic epistemic logic with added STIT features.
6 Dynamic Epistemic Logic Meets STIT
Temporal trees with epistemic features may be viewed as a record of actions unfolding over time, while marking local uncertainties (or information) that agents had. If we want to understand the dynamics that gives rise to such a record, we need an account of information update in a temporal universe. A typical system where PDLstyle events and knowledge come together is dynamic epistemic logic (DEL). We assume the reader is familiar with its basics, and so, we only give the key definitions here (see van Benthem 2011 for more details and motivation).
The basics of DEL update The basic structures are epistemic models, tuples \(\langle W, \{R_i\}_{i\in I}, V\rangle \) with \(W\) a (finite) set of worlds, \(R_i\subseteq W\times W\) an equivalence relation, and \(V: \mathsf {At}\rightarrow \wp (W)\) a valuation function marking at which worlds the atomic propositions in \(\mathsf {At}\) are true. Over these models the basic language of epistemic logic \(\mathcal {L}_{EL}\) can be interpreted, including universal modalities \(K_i\varphi \) for “agent \(i\) knows that \(\varphi \). This much is completely standard.
The central idea of dynamic epistemic logic is now to describe social interaction, including agents’ uncertainty about the events they witness, in socalled event models. These are tuples \(\mathbf{E} =\langle E, \{S_i\}_{i\in I},\mathsf{pre}\rangle \) with \(E\) a (finite) set of basic events, \(S_i\subseteq E\times E\) is an uncertainty relation, and \(\mathsf{pre}:E\rightarrow \mathcal {L}_{EL}\) assigns to each event \(e\in \ E\) a formula that serves as a precondition for that event.

\(W'=\{(w,e)\in W\times E\ \ \mathcal {M},w\models \mathsf{pre}(e)\}\);

\((w,e)R'_i(w',e')\) iff \(wR_iw'\) and \(eS_ie'\); and

\((w,e)\in V'(p)\) iff \(w\in V(p)\)
Our aim in this section is just to show how, in line with our analysis of Sect. 3, STIT ideas of control fit quite well in this PDL stronghold.
The complete dynamic logic of this expanded system lies embedded in the base logic of DEL in an obvious manner. Its laws for the new control operator will be essentially those of STIT. But what we obtain in this way is a much richer logic of onestep information flow plus an explicit account of agents’ choices of actions where relevant. However, it should be noted that this logic still runs on the usual analysis of DEL’s standard dynamic modality.
One crucial feature is that, unlike standard DEL logics, this new system does not have a modality reflecting its dynamic control relations in the static epistemic base models \(\mathcal {M}\).^{22} One might think of this negatively as limiting the logical status of control, reflecting its ephemeral nature. Our own more positive view is that this feature makes event models really come into their own, as carrying crucial information that is sui generis.
While our proposal for merging enriches DEL with STIT ideas, what good does it do in the opposite direction? One effect is that we now have a logic that describes both steps in the fork models of Sect. 3, before and after the choice. Thus, it is a logic of choosing and moving ahead, like the NEXTSTIT system of Broersen (2011). But the main virtue is that, given the long experience in DEL, our merged system plugs STIT into the world of private versus public information, imperfect information games, and much more.
Dynamifying STIT But the DEL perspective also suggests a more radical move, affecting our view of the scenarios that motivated STIT in the first place. STIT is a logic of deliberate choice and action, but remarkably, it does not analyze any of these activities explicitly, recording only their outcomes.^{23} By contrast, the DEL methodology follows a main principle of Logical Dynamics:
Where there is a change, there is an event.
Taking this line, can we ‘dynamify’ STIT in DEL style? What are the main events that take place in a choice scenario? Here are the main stages as we see them:
deliberation, decision, action, and observation
In a first deliberation stage, we analyze our options, and find optimal choices. Next, at the decision stage, we make up our mind and choose an action of our own. Then at the action stage, everyone acts publicly, and this gets observed, something that we can also model as a separate observation stage, though things happen simultaneously.
All these stages can be analyzed using DELstyle models. Perhaps the easiest is the final stage, where an event model will do with all possible events, marking the actual one, and giving agents the right amount of observational powers: totally public in STIT, perhaps more mixed in other settings. But the intermediate stage, too, invites event models. We can have pair events with control relations, as we just introduced in Sect. 6, and then get the matrix models of Sect. 4 as an output. Finally, modeling the initial deliberation stage is more complex, since many factors can weigh in here that are not represented in basic STIT models, such as agents’ preferences over outcomes. Still, there is a growing body of work on deliberation analyzed in terms of DELstyle updates (van Benthem 2007; Pacuit and Roy 2011), and this might inform an account of deliberation that seems a natural companion to any logic of “deliberate action”.
7 Further Directions
There are many followup topics to our analysis, of which we mention three.
The first is the addition of agents’ preferences. Clearly, this further structure is crucial to the game logics of Sect. 4, and existing modal systems do incorporate preferences in order to define and reason about notions like ‘best response’, Nash equilibrium, and rational behavior generally (cf. van Benthem 2011, 2014 for such notions analyzed in DEL). In particular, the interplay of actions and preferences has already been studied in the matrix logics of Sect. 4, using techniques from (Liu 2012; van Benthem et al. 2009).^{24}
Adding preferences seems a necessity for STIT as well, since rational agency is about best actions rather than just any actions, and agents may also prefer ensuring \(\varphi \) rather than \(\psi \) for other reasons, including deontic norms. This need not be a simple matter, since best action is not just a matter of, say, finding Paretooptimal simultaneous choices for all agents. As we know from game theory, more complex deliberation methods are needed, such as iterated removal of dominated actions.^{25} All this seems a happy marriage with STIT, and indeed, many of the relevant issues are addressed in Horty’s book (Horty 2001).
The next extension would be the study of longterm temporal evolution. Our logics so far described single steps in a larger process, but it has long been acknowledged that the proper stage for studying agency is that of a linear or branchingtime temporal logic (Fagin et al. 1995; Parikh and Ramanujam 2003). The same is true for STIT, and one question that seems of interest is whether our onestep event models with control relations can be related systematically to epistemic temporal universes via representation theorems extending those of van Benthem et al. (2009).
Our final topic is strategic interactive behavior. We started our presentation of STIT with its basic properties for agents’ choices: for each agent, these formed a partition of all possible outcomes (call this the Partition property), and also, any two choices for different agents have to overlap. This level of stating constraints is similar to that of representation theorems for games characterizing players’ strategic powers, forcing the game to end in certain sets of outcomes by playing one of their strategies against any counterplay of the opponent. The latter type of result, however, usually refers to powers in a longer extensive game that can take many individual steps. For instance, van Benthem (2001) characterizes players’ powers in finite determined twoplayer games in terms of three constraints: Monotonicity (powers are upward closed), Consistency (any two powers of different players overlap), and Determinacy (if a set of outcomes is not a power for one of the players, then its complement is a power for the other player). Of these three, Determinacy is typically lost in the STIT setting of simultaneous action. Nevertheless, it seems significant that there are extended representation results for players’ powers in extensive games with imperfect information that require only Monotonicity and the typical STIT constraint of Consistency (cf. again van Benthem 2001).^{26}
We end with just one simple observation. What happens to the key STIT constraints when we consider iterated simultaneous action? Most importantly, the crucial property of Partition disappears, and the reason is very instructive. When we make consecutive choices, our available strategies get enriched. In a onestep scenario, agents could only choose one of their actions ab initio. But now, they can have strategies letting their next action depend on the observed behavior of the other agents. A standard example of this is the famous strategy Tit for Tat in evolutionary game theory: one copies the opponent’s preceding move. Hence, the strategies available at the second level do not just consist of choosing an action uniformly, they can depend on the behavior of others. It is easy to see that the disjointness property for sets of outcomes (i.e., the powers matching these strategies) are no longer disjoint.^{27} On the other hand, this richer set of strategies does depend crucially on a special feature of the STIT scenario, namely the public observation of everyone’s moves. If there were no such observation, then players’ could not make their choices dependent on what others have done, and we would get a simple product model of two consecutive actions that does satisfy the Partition condition. Put differently, onestep simultaneous action does not allow for sequential dependence of actions, though it may allow for correlation as we saw in Sect. 4. But it is precisely the observation feature built into STIT that does make more sophisticated dependent behavior possible as actions get repeated.
8 Conclusion
In this chapter, we have lightly compared the STIT approach to choice and action with that offered by dynamic logic, broadly conceived (including dynamicepistemic logics). We found that, despite differences in style and presentation, these frameworks are much more congenial than is often thought. Indeed, key ideas from STIT about actions and control merged well with modal logics of games, and in particular, they led to natural dynamicepistemic logics of information and events that incorporate the crucial STIT notion of control. We have only proposed a few such bridges here, without any sustained development, suggesting how ideas might flow across, and further directions pursued. Even so, we hope to have put to rest some views about vast chasms separating STIT and PDL that are sometimes found in the literature.
We are by no means the first to have observed the compatibility of STIT and ideas from the world of PDL and DEL. Notably, Horty articulates many of the idea sketched in this chapter in his important book (Horty 2001). Also, Xu (2010, 2012) are interesting examples of STIT systems that have borrowed notions of action and strategy from the PDL tradition to form richer frameworks for strategic agency. We see our analysis as making a small push in the same direction.
Finally, we recall an earlier point made at the start of our analysis. A paradigm is not just a set of definitions of structures and axioms for reasoning. It is also a belt of applications, in the terminology of Kuhn (1962), a growing family of successful “exemplars”. This makes frameworks harder to compare and merge, since their success does not just depend on their formal backbone, but also on the “art of modeling” that has been invested by skilled practitioners. In a practical setting, choices between paradigms may just be choices of taste and lifestyle, and these of course will not be affected much by theoretical analysis. Still, tastes can at least be diversified—and we hope to have contributed at least to what is on the menu in the logical study of deliberate action.
Footnotes
 1.
 2.
Truth for these operators is defined as usual: \(\mathcal {M},w\models [\,{i}\,]\varphi \) iff for all \(v\in W\), if \(w\sim _i v\) then \(\mathcal {M}, v\models \varphi \), and \(\mathcal {M},w\models E\varphi \) iff for some \(v\in W\), \(\mathcal {M},v\models \varphi \).
 3.
As observed in Balbiani et al. (2008), this principle can also function as a product axiom by itself. Also interderivable with our version of the product axiom is the strongerlooking \((E[\,{1}\,]\varphi \wedge E[\,{2}\,]\psi )\rightarrow E([\,{1}\,]\varphi \wedge [\,{2}\,]\psi ).\), for which Roberto Ciuni has proposed an interesting epistemic interpretation.
 4.
These simple modal equivalence models show up when studying many aspects of rational agency: they work for specifying ranges of knowledge, issues in the logic of questions, etc.
 5.
This is just a first intuitive pass. We will have occasion to spell out things further later on.
 6.
However, the process view of PDL may be closer to the dynamics of agents making choices and performing actions. We do not claim that our take in this chapter is the only way to go.
 7.
This uniqueness is standard modeling practice in many temporal formalisms: if histories differ at a point, then there should be a difference in the next event.
 8.
We could view the branching as “nondeterminism” in PDL, but this does not clarify the issues very much. Nondeterminism usually means that a process has several options, ‘ways of doing \(c\)’, but that is not the situation in the STIT model: it is not up to the agent to nondeterministically chose one or the other transition.
 9.
The same is true in the dynamic epistemic logic literature: notice the terminology ‘action models’ versus ‘event models’ for its core update rules.
 10.
 11.
As a concrete example, suppose there are two histories \(h, h'\) where an agent refrains from choosing either. Presumably, refraining means she could have made a choice for \(h\) or for \(h'\). One way of viewing this involves three actions: choosing \(h\), choosing \(h'\), or ‘leaving things be’: \({h, h'}\). This would violate the disjointness constraint of STIT. But we can also individuate events differently, with four histories: one where \(h\) is chosen, one where \(h'\) is chosen, and two copies of these except for the fact that no choice was made.
 12.
 13.
One can also have more abstract worlds in socalled ‘models of games’, as is usual in epistemic game theory, see (Aumann 1999)—but this generality is not needed in what follows.
 14.
For example, a proposition \(p_i^a\) might say “agent \(i\) plays action \(a\) in the current profile”—but atomic propositions could also encode utility values for players.
 15.
As noted in (van Benthem 2007), another interesting feature of our models is that ‘distributed knowledge’ \(D_G \varphi \) for a group of players accesses those profiles where only players outside the group still have options.
 16.
Note that the construction given in this proof is only needed because the singleton intersection property (the intersection of all the agents equivalence classes are singletons) is not definable in our language. However, if the language contains group STIT operators \([G\ \mathsf {stit}]\varphi \) meaning that the group \(G\) can see to it that \(\varphi \) is true, then the singleton intersection property is definable via the formula \([\mathcal {A}\ \mathsf {stit}](\varphi \vee \psi )\rightarrow [\mathcal {A}\ \mathsf {stit}]\varphi \vee [\mathcal {A}\ \mathsf {stit}]\psi \), where \(\mathcal {A}\) is the set of all agents. Furthermore, the argument would be very different once we consider STIT formulas with temporal operators shifting moments along histories, as is suggested in Sect. 7.
 17.
Such encodings also work with two equivalence relations and common knowledge in one dimension of the grid model, while time provides the other dimension. See van Benthem and Pacuit (2006) for an extensive survey.
 18.
For a concrete counterexample, note that the formula in Fact 4.2 is not valid on such models. Suppose that \(\approx _i\) and \(\sim _i\) are arbitrary equivalence relations for each \(i\). Consider a model where \(w\approx _1 v\) and \(w\approx _2 v'\) with \(v\ne v'\), and both \(v\) and \(v'\) are deadend states (i.e., we only have \(v\sim _1 v\) and \(v'\sim _2 v'\)). Suppose that \(\varphi \) is true at \(v\) only and \(\psi \) is true at \(v'\) only. Then the antecedent is true, but the consequence is not.
 19.
Also relevant to the issue of generalized “profile models” is recent work by Roberto Ciuni on connections between generalized STIT models and notions of effectivity in games, and on actions whose effects are only given probabilistically: Ciuni (2013).
 20.
A pointed epistemic model is a model with a distinguished state intended to represent the “actual” state of the world.
 21.
This may have to be modified when we want some events to just happen without agency. Also, there are problems of intuitive interpretation for control in privateinformation scenarios, but we ignore these here.
 22.
This makes our DEL system with control different, e.g., from DEL logics of questions with issue relations: see van Benthem and Minicǎ (2012).
 23.
This output orientation on choice is of course precisely the official STIT view of actions.
 24.
Many interesting new problems arise in this area. One is finding a formalization of basic gametheoretic reasoning that makes sense for rational action generally: as initiated in (van Benthem 2007). Another unresolved issue is whether introducing preference structure increases the computational complexity of the modal logic of action, an issue known as the “price of rationality”.
 25.
Thus one might first iteratively prune a given choice situation in this way, and only follow the standard STITstyle format once an equilibrium has been reached.
 26.
 27.
It is an interesting problem whether some special properties remain for STIT powers. In particular, the temporal logic of Ciuni and Zanardo (2010) seems relevant to analyzing these matters, including the special constraints imposed on STIT models if we do insists on the above properties of powers for single agents, or groups of these: cf. Zanardo (2013).
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