# Representing Strategies

## Abstract

Quite some work in the ATL-tradition uses the differences between various types of strategies (positional, uniform, perfect recall) to give alternative semantics to the same logical language. This paper contributes to another perspective on strategy types, one where we characterise the differences between them on the syntactic (object language) level. This is important for a more traditional knowledge representation view on strategic content. Leaving differences between strategy types implicit in the semantics is a sensible idea if the goal is to use the strategic formalism for model checking. But, for traditional knowledge representation in terms of object language level formulas, we need to extent the language. This paper introduces a strategic STIT syntax with explicit operators for knowledge that allows us to charaterise strategy types. This more expressive strategic language is interpreted on standard ATL-type concurrent epistemic game structures. We introduce rule-based strategies in our language and fruitfully apply them to the representation and characterisation of positional and uniform strategies. Our representations highlight crucial conditions to be met for strategy types. We demonstrate the usefulness of our work by showing that it leads to a critical reexamination of coalitional uniform strategies.

## 1Introduction

To make way for strategic reasoning on the syntactic level we need to know how to represent the various types of strategies that have been proposed in the literature. In this paper, our aim in particular will be to provide syntactic counterparts for the various strategy types proposed in the -tradition, such as positional strategies [2] and uniform strategies [10]. Our proposed language, an extension of strategic including temporal and epistemic modalities and action types, is sufficiently expressive for representing positional and uniform strategies. ^{1}

Our representation of uniform positional strategies enhances the understanding of a *coalition’s* uniform strategy. Our result naturally invites different ways to distribute relevant strategic knowledge. This observation complements [1] where it is argued that a coalition’s uniform strategy is imprecise only in the “mode” of the coalition’s knowledge, referring either to common, distributive or mutual knowledge.

To further explain our view and approach, we will subsequently answer the following three questions:

what

*is*a strategy?how to represent the

*performance*of a strategy?how to

*characterize*a rule-based strategy?

**what is a strategy?**

It is remarkable how strong the notion of a strategy varies throughout the literature on strategic reasoning. In frameworks^{2}^{3}^{4}

**how to represent the performance of a strategy?**

To express that a strategy is actually performed, we use insights from frameworks. Other frameworks express strategy performance only implicitly, safely tucked away under path quantifiers () or in quantifiers in the central modalities (Dynamic logic). This follows from the observation that he main operators and in these systems are interpreted as ’coalition is able to ensure ’ and ’after executing action , holds’. This reveals that enables one to reason about strategic ability and Dynamic Logics support reasoning about the results of actions, but not reasoning about the

*performance*of actions or strategies here and now.

**how to characterize a rule-based strategy?**

Strategies are typically communicated in the form of condition-action rules. Therefore it makes sense to also logically represent them in that form using a suitable language. Recently, has been extended to enable reasoning about rule-based strategies in [17]. One of the main themes there is the representation of (semantic) strategies by formulas of their proposed language, which includes rule-based strategies. A different study in [16] evaluates formulas at game-strategy pairs thereby combining aspects of

*Game Logic*

^{5}

*structure*of the strategy. The logic includes two types of conditional strategies and , which are interpreted as ’player chooses move whenever holds’ and ’player sticks to the specification given by if on the history of play, all moves made by conform to ’, respectively. The in the first formula is restricted to boolean combinations of propositional letters, and the second formula represents that the other players

*have acted*in accordance with . We, however, do not want to commit ourselves to these restrictions in the language or regarding the conditions; Most importantly, to represent uniform strategies epistemic conditions have to be allowed.

The paper is organized as follows: In Section ? the well-known Concurrent Epistemic Game Models (see [2] and [8]) are introduced to provide the basis for our semantics. In contrast to the usual -divide between path and state formulae, a key idea from ^{6}*the current strategy profile* is implemented in Section ?. In parallel, a logical language is introduced including action types and temporal, epistemic and agency operators. As is common in frameworks, the structure of a strategy can be described by specifying the (temporal) properties it ensures. This allows us to view a rule-based strategy as a set of condition-effect rules in Section ?, which are then used in Section 5 to investigate how this logical language can be fruitfully applied to represent strategy types. Certain types of rule-based strategies are used to represent positional and uniform strategies, thereby uncovering crucial conditions for these semantic strategy types. Some novel implications are drawn on coalition’s uniform strategies by appealing, not to different “modes” of coalitional knowledge but, to ways of distributing the relevant strategic knowledge. We conclude in Section ?, proof sketches are to be found in the appendix.

## 2Concurrent Game Models

In this section we introduce Concurrent Game Models (see [2]) and Concurrent Epistemic Game Models (see [8]). Our treatment will be roughly in line with [1] although our notation on histories will differ to neatly support the syntactic approach in the next section.

We fix a concurrent game structure . An action profile is used to determine a successor of a state using the transition function . The set of the available action profiles is denoted by , consequently the set of *possible successors* of is the set of states where ranges over . An infinite sequence of states from is called a *play* if is a successor of for all positions . denotes the -th component in , and denotes the initial sequence, or *history*, of .

A *perfect recall strategy for an agent* is a function that maps every history to an action type . A *positional (aka memoryless) strategy for an agent* is a function that maps every state to an action type. A *perfect recall strategy for a coalition* , also called a coalitional strategy for , is a function mapping each agent to a perfect recall strategy . Positional strategies for coalitions are defined analogously. A *strategy profile* is a coalitional strategy for . A coalitional strategy *extends* , notation , if and only if and for every . Given a strategy profile, we often write for the coalitional strategy for satisfying .

The set of outcome plays of a strategy for at a history is the set of all plays such that and, for every , there is an action profile satisfying for all and .

Our formal results rely on the notion of *play-equivalence*:

It is standard to model the agent’s incomplete information by extending concurrent game models:

These indinstinguishability relations are straightforwardly extended to histories by: iff and for every we have . Then we introduce a third strategy type:

a

*uniform strategy for an agent*is a perfect recall strategy satisfying: for all histories , , if then .

A *uniform coalitional strategy for coalition* is a function mapping each agent to a uniform strategy .

In the remainder, we mean “perfect recall strategies” when writing “strategies”, unless otherwise specified.

## 3Strategic language

In the previous section we outlined the models that provide the basis for the semantics of our logical enterprise. In the current section we introduce our logical framework, which is inspired by [5]. ^{7}*the current strategic course of action* (inspiration from [5]). Finally, some crucial observations on the resulting logical formalism are presented by reviewing the underlying models.

Given a CEGM with and , these formulas will be evaluated at tuples consisting of a strategy profile , a play such that , and a position . This means that the truth of formulas is evaluated with respect to a current state , a current history , a current future , and a current strategy profile . Obviously, by incorporating the current strategy profile into the worlds of evaluation we get a semantic explication of the performance of a strategy.

The central agency operator is the modality which stands for ’the coalition strategically sees to it that holds’. Relative to a tuple the modality is interpreted as ’the coalition is in the process of executing thereby ensuring the (temporal) condition ’. In addition, the language includes temporal modalities and which are interpreted, relative to a tuple , as ’ holds in the next moment after on ’ and ’ holds on all future moments after on ’, respectively. In contrast to this longitudinal dimension of time, the language includes a temporal modality for historical necessity. The modality is interpreted, relative to a tuple , as ’ holds on any tuple at ’. This highlights that the truth of does not depend on the dynamic aspects represented by the current future and the current strategy profile, we call such formulas *moment-determinate*. Finally, we include epistemic modalities , one for each agent, which are interpreted as ’agent knows that ’. The presented syntax and semantics are formally connected by the truth conditions for the syntactic clauses:

With the semantics in place, we gather some crucial observations:

Because the truth of a propositional letter only depends on the current state, it is not surprising that the truth of any propositional formula only depends on the current state. Therefore, we will often write instead of for a propositional formula . This is connected to the familiar divide in syntax between state and path formulas.

As mentioned before, a formula is moment-determinate if . The truth of such formulas only depends on the history, so we will often write instead of for such formulas.

Formulas of the form are moment-determinate. In particular, an agent does not know what he is doing.

The truth of a coalitional action type only depends on the current strategy of that coalition, i.e. .

The formula expresses that a coalitional action type is executable at a state.

Observe that only the truth conditions for the -operator and the action types involve the current -strategy profile. It is clear that adding (at least one of) these is necessary to express that a certain strategy is performed. The action types are inherited from a bottom-up perspective on strategies with the action types as atomic building blocks. In contrast, the -operator incorporates a top-down view in that a strategy is described by the properties it ensures.

## 4Rule-based strategies

A rule-based strategy consists of rules. Such a rule is composed of a condition and an effect, thereby incorporating the intuition that a rule is triggered under certain conditions and has a certain effect:

Performing a rule-based strategy means that in case a rule is triggered one ensures that the corresponding effect is realized:

The formula is interpreted, relative to a tuple , as ’coalition is in the process of executing strategy thereby ensuring that the conditionals are met’. Informally, it means that coalition is currently performing a strategy that ensures that in case a condition holds he performs a strategy ensuring the corresponding effect.

Although the nested operator may be puzzling at first sight, it makes perfect sense. To argue in favour we break the formula down. A rule of the rule-based strategy is formalized as , but one should not forget that here we intend to formulate that a coalition is acting according to such a rule-based strategy. This is expressed by the second operator, which guarantees that one is acting accordingly not only at the current play, but also at all plays in . To formalize that a coalition is performing a rule-based strategy, we add the operator to express that it is henceforth acting according to strategy .

There are two ways in which a rule-based strategy can be unsatisfactory: (a) the agent might not be able to perform a certain rule-based strategy, or (b) the action description given by a certain rule-based strategy can be underspecified. So a rule-based strategy can be viewed as a partial perfect recall strategy which is defined at a history if and only if it is possible to act accordingly and there is but one way to do so.^{8}

Can a given rule-based strategy be viewed as a partial strategy? There is a straightforward way to attempt this whenever the conditions are moment-determinate , i.e. for each . ^{9}*partial* coalition strategy by:

iff | . |

The first conjunct says that the coalition is able to act accordingly, whereas the second conjunct says that performing action profile is the only way to do so.

Clearly, this partial coalition strategy is defined at a history if the following conditions hold:

- (1)
there is a such that ,

- (2)
there is at least one such that , and

- (3)
there is at most one such that .

The failure of (1) and the failure of (3) signify that the rule-based strategy is underspecified either because no rule has been triggered or because there are multiple ways to act accordingly. The failure of (2), however, indicates a practical inconsistency or a conflict in the rule-based strategy , because it implies that there is no way to act accordingly.

Before proceeding, we extend to pertain also to partial strategies:

To investigate the perfect recall strategies represented by a rule-based strategy, we use the partial strategy it defines:

This establishes a crucial connection between the syntactic notion of performing a rule-based strategy and the semantic notion of a (partial) perfect recall strategy. In the following subsections we use this link to represent positional and uniform strategies up to play-equivalence by rule-based strategies.

The notion of play-equivalence stems from the views in our formalism. From a perspective a strategy is identified by the futures it allows, so two play-equivalent strategies not only *appear* to be same strategy, they *are* the same strategy.

## 5Representation Results

### 5.1Representing positional strategies

In this subsection, we prove that rule-based strategies can be used to represent positional strategies. For that purpose we introduce a specific type of rule-based strategies:

Because the effects are of the form , there can be at most one way to act according to a proposition-action strategy whenever one of the conditions is triggered. This motivates our definition of *completeness*; a notion that plays a key role in our findings in the correspondence between proposition-action strategies and positional strategies:

This shows that performing a complete proposition-action strategy implies that one is performing a strategy that is play-equivalent to a positional strategy. In a sense, this means that the strategies represented by complete proposition-action strategies are positional strategies.

The converse does not hold in general, which can be shown by providing a CGM containing a positional strategy differing at two propositionally equivalent states. So to prove the converse we have to restrict our investigation to CGMs in which enough states are propositionally definable:

This shows that, under certain semantic constraints, a given positional strategy is represented by a complete proposition-action strategy. Thereby we can move strategic reasoning in the semantics about positional strategies to reasoning on the syntactic level about proposition-action strategies. In conclusion, we show that in a common class of CGMs, complete proposition-action strategies correspond to positional strategies up to play-equivalence:

This corollary uncovers that completeness is an underlying condition for positional strategies. Although this discovery is unsurprising and intuitive, it shows that our language is able to express such underlying intuitions.

### 5.2Representing uniform strategies

Here we prove that rule-based strategies are useful for representing uniform strategies, focussing on individuals’ uniform strategies, using a type of rule-based strategies:

This result is similar to Proposition ? on positional strategies. Here we see that whenever an agent performs a knowledge-action strategy of which he knows both that it is complete and that he is henceforth able to act accordingly, the agent is performing a strategy that is play-equivalent to a uniform strategy. This means that, under certain syntactically representable epistemic conditions, a syntactically characterised knowledge-action strategy corresponds with a uniform strategy in the semantic structures.

The converse does not hold, as can be shown by providing a CEGM containing a uniform strategy that differs at two distinguishable propositionally equivalent states. But the mismatch runs deeper because of the restrictions on the conditions of knowledge-action strategies. To obtain a correspondence result in line with Corollary ? we believe that the language has to be extended with temporal modalities referring to the past and the conditions of knowledge-action strategies have to be modified accordingly. To keep the current exposition accessible we leave this for another occasion. In spite of these simplifications, a representation result for uniform positional strategies can be proven:

Note that this proposition starts with a uniform *positional* strategy. This result establishes that, under certain semantic restrictions, a uniform positional strategy is represented by a knowledge-action strategy of which one knows both that it is complete and that one can henceforth act accordingly. This shows that strategic reasoning in the semantics on uniform positional strategies can be diverted to reasoning on the syntactic level about knowledge-action strategies.

Our representation result reveals crucial underlying conditions for uniform positional strategies, which are expressible in our language. Indeed, the previous proposition shows that, under certain model restrictions, performing a uniform positional strategy implies that one is performing a knowledge-action strategy and one knows both that this knowledge-action strategy is complete and that one is henceforth able to act accordingly. Revealing such underlying conditions enhances our understanding and triggers further questions; two of such inquiries are discussed below.

Does ensuring a property by performing a uniform strategy *entail* that knowing that performing this uniform strategy ensures that property? According to our representation result, this translates to questioning whether logically entails (where is a knowledge-action strategy). It turns out that this indeed fails since “in order to identify a successful strategy, the agents must consider not only the courses of action, starting from the current state of the system, but also from states that are indistinguishable from the current one.” [1] Uniform strategies are therefore not faithful to the expectation that “the agent has enough control and knowledge to identify and execute a strategy that enforces [a certain property] .” [1] An agent has this control and knowledge if and only if there is a knowledge-action strategy satisfying . This discussion highlights the flexibility of our syntactical approach to correct the flaw of uniform strategies.

What *is* a coalition’s uniform strategy? Formally, it is a tuple of individuals’ uniform strategies; intuitively, it is intended to capture a coalition’s control and knowledge to identify and execute a strategy that enforces a certain property . Does a coalition’s uniform strategy meet this intuition? No, it does not. In [1] it is argued that “there are several different “modes” in which [a coalition] can know the right strategy”, pointing to a choice between common, mutual, or distributed knowledge of the right coalitional strategy . ^{10}*identify* the coalitional strategy . Indeed, a coalition’s uniform strategy merely requires every member to know their part of the coalitional strategy . So the *object* of the members’ knowledge differs. Second, since every member knows their part of a coalition’s uniform strategy , it follows that a coalition’s uniform strategy entails that the coalition has distributed knowledge of the right coalitional strategy , positioning a coalition’s uniform strategy between the modes of mutual and distributed knowledge of the right coalitional strategy. Third, this knowledge is, however, distributed in a very particular way, namely by every member knowing *their own part* . For instance, whenever knows ’s part and knows ’s part , then they have distributed knowledge of even though it is not a uniform strategy. A coalition’s uniform strategy hence does not correspond to any of the “modes” in [1]. It seems that this distinctive way of distributing knowledge solicits a comparison, not with the proposed modes in [1] but, with different ways of distributing knowledge of a coalitional strategy .

## 6Conclusion

We have shown that, under certain model restrictions, a strategy that is play-equivalent to a positional strategy *corresponds* to a complete proposition-action strategy. Thereby we have established a firm correspondence between a semantic strategy type and a syntactic one.

In our research on individuals’ uniform strategies, we have proven that any knowledge-action strategy of which one knows both that one can henceforth act accordingly and that it is complete *represents* a strategy that is play-equivalent to a uniform strategy. Conversely, under certain semantic restrictions, a uniform positional strategy *is represented by* a knowledge-action strategy of which one knows both that one can henceforth act accordingly and that it is complete. This latter result exposes the implicit conditions of uniform positional strategies.

The current enterprise is a crucial first step in facilitating strategic reasoning at the syntactic level. By representing several semantic strategy types and drawing novel conceptual implications we have shown the fruitfulness of our syntactic approach to enhance our understanding of semantic strategy types.

## AAppendix: Proof sketches of propositions

1. Follows from the fact that is defined at a history iff there is exactly one way to act according to at that history. 2. Follows straightforwardly from 1. and property 5(b) in Definition ?.

It is easy to show that the partial strategy defined by is defined on and positional for histories in . This partial positional strategy can be trivially extended to a positional strategy, thereby proving the proposition.

Let , and let us denote the propositional formula defining by for each . The proposition-action strategy can be used to prove the proposition.

The partial strategy defined by is defined and uniform on histories in .

Analogous to Proposition ?.

### Footnotes

- Recently, a syntactic characterization of uniform strategies in
*Epistemic Strategy Logic*was presented [11]. Our representation differs in that we use rule-based strategies. - See the seminal work [2] and extensions such as [7], and
*Strategy Logic*[6] and [12]. - See the seminal work [3], [9] and the recent extension to strategic action [5].
- See [4].
- See the original work [14] and the overview [15].
- is first introduced in [5]. It is an extension of basic frameworks to a strategic and multi-agent setting.
- Whereas [5] uses models based on the -tradition, here Concurrent Game Models are used for interpreting the language.
- This resembles the informal notion of deterministic strategies in [17]: “move recommendations are always unique” for deterministic strategies.
- If the conditions are not moment-determinate “a choice of an agent, at a given point of a play, may depend on choices other agents can make in the future or in counterfactual plays“ (cf. the study on ”behavioral strategies” in [13]).
- They also mention the option that “the strategy can be identified by” (altered notation) a leader, headquarters committee, or consulting company. Our representation result suggests that the syntactical counterpart of these “modes” is straightforward by replacing ’s with the respective group knowledge in Proposition ?. We will not pursue this suggestion in further detail here.

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