Boolean Observation Games

We introduce Boolean Observation Games, a subclass of multi-player finite strategic games with incomplete information and qualitative objectives. In Boolean observation games, each player is associated with a finite set of propositional variables of which only it can observe the value, and it controls whether and to whom it can reveal that value. It does not control the given, fixed, value of variables. Boolean observation games are a generalization of Boolean games, a well-studied subclass of strategic games but with complete information, and wherein each player controls the value of its variables. In Boolean observation games, player goals describe multi-agent knowledge of variables. As in classical strategic games, players choose their strategies simultaneously and therefore observation games capture aspects of both imperfect and incomplete information. They require reasoning about sets of outcomes given sets of indistinguishable valuations of variables. An outcome relation between such sets determines what the Nash equilibria are. We present various outcome relations, including a qualitative variant of ex-post equilibrium. We identify conditions under which, given an outcome relation, Nash equilibria are guaranteed to exist. We also study the complexity of checking for the existence of Nash equilibria and of verifying if a strategy profile is a Nash equilibrium. We further study the subclass of Boolean observation games with `knowing whether' goal formulas, for which the satisfaction does not depend on the value of variables. We show that each such Boolean observation game corresponds to a Boolean game and vice versa, by a different correspondence, and that both correspondences are precise in terms of existence of Nash equilibria.


Introduction
Reasoning about strategic agents is an important problem in the theory of multi-agent systems and game-theoretic models and techniques are often used as a tool in such analysis.Strategic games [38] is a classic and well-studied framework that models one-shot multi-player games where agents make their choice simultaneously.While it discards the underlying dynamic aspects of the game, it forms a simple and intuitive formalism to analyse and reason about the strategic behaviour of agents.From the perspective of computer science and artificial intelligence, one of the main drawbacks of strategic games is that the explicit representation of the payoff (or utility) function is exponential in the number of players and the strategies available for each player.In many applications, compact representation of the underlying game model is highly desirable.
Various approaches have been suggested to achieve compact representation of games and these mainly involve imposing restrictions on the payoff functions.For instance, constraining the payoff functions to be pairwise separable [34,14] results in the well-studied class of games with a compact representation, called polymatrix games.Additively separable hedonic games [8] form another subclass of strategic games with pairwise separable payoff functions which can be used to analyse coalition formation in multi-agent systems.It is also possible to achieve compact representation by explicitly restricting the dependency of payoff functions to a "small" number of other agents (or neighbourhood) as done in graphical games [35].
An alternative approach to imposing quantitative constraints on payoffs is to restrict the payoffs to qualitative outcomes which are presented as logical formulas.An example are "extensive" games played on graphs where the goal formulas can specify the evolution of play with a combination of temporal and epistemic specifications.Although originally defined as two-player perfect information games motivated by questions in automata theory and logic, these models are now sophisticated to reason about multi-player games and imperfect information [15,6,25].Boolean games [29], a subclass of strategic games with complete information where objectives are expressed as Boolean formulas, is also a wellstudied framework with such qualitative outcomes.
In Boolean games, each player controls a disjoint subset of propositional variables where their strategies correspond to choosing values for these variables and each player's goal is specified by a Boolean formula over the set of all variables.While the model was originally defined to analyse two-player games, the framework has been extended in many directions.
Multi-player, non-zero-sum Boolean games are studied in [27,12].In [29,27] Boolean games are modelled as imperfect information games by taking the uncertainty over the other player's actions as an information set, as in [43].In [18,12,19] the computational properties of Boolean games are adressed, in [11] graphical dependency structures for Boolean games and their implications for various structural and computational properties, and in [33] mixed strategy Nash equilibria and related computational questions.The issue of equilibrium selection is considered in [1].Iterated Boolean games [24,23] model repeated interaction between players with temporal goals specified in linear time temporal logic (LTL).Partial ordering of the run-time events in terms of a dependency graph on propositions is studied in [13].
Epistemic Boolean games, wherein goal formulas may be epistemic, were proposed in [2,32].Both works combine the control of variables with the observation of variables (or formulas), where some of this is strategic and some is given with the game.This hybrid setting allows the authors to continue to analyse these epistemic Boolean games as complete information strategic form games. Realizing epistemic objectives depends on the valuation of variables resulting from strategic action.
In this paper, we introduce Boolean observation games as a qualitative model to analyse and reason about a subclass of strategic games with incomplete information.In Boolean observation games, players control whether and to whom they reveal (announce) the value of propositional variables that can only be observed by them.This constitutes a multiplayer game model with concise representation where players have (qualitative) epistemic objectives.It is incomplete information because realizing the objectives depends on a given fixed valuation that the players cannot control.Players do not know what that valuation is and therefore do not know what game they play.Realizing epistemic objectives depends on the unknown valuation of variables that is independent from strategic action.(We should note that such incomplete games of imperfect information can also be modelled as complete games of imperfect information by assuming an initial random move of a player 'nature' determining the valuation.)Since Boolean observation games define a subclass of strategic games, they form an ideal framework to analyse interactive situations that incorporate aspects of both imperfect as well as incomplete information games.Please consider the following examples.
Example 1 (A West Side Story).Tony and Maria (or was Romeo and Juliet? or Shanbo and Jingtai?) are in love with each other.But they have not declared their love to each other yet.This is risky business, as they are both uncertain about the feelings of the other one.Surely, given that they both love each other their objective is to get to know that.But they consider it possible that the other person does not love them in which case they might prefer not to declare their love.Their personalities are different in that respect.What Tony wants to know depends on how his feelings (being in love / not being in love) relate to the other person's: if they match, he wants the other person to know, otherwise, he doesn't.Whereas what Maria wants to know only depends on the other person's feelings: if the other one is in love, she wants the other one to know her true feelings and otherwise not.
Given their state of mind and their personalities, should they declare their love to each other?
Let Tony be player 1 and Maria be player 2, and let p 1 represent 'Tony is in love' and p 2 represent 'Maria is in love'.Propositions p 1 and p 2 are both true and remain so forever after.They cannot be controlled.The objectives are (where K i p j means 'player i knows p j '): They each have two strategies: declare their feelings (revealing the value of p i ), or not.It is a cooperative game as they can both win.It is not so clear whether there (hopefully) is an equilibrium strategy profile allowing them to declare their love to each other.As p 1 and p 2 are true, it is an equilibrium when they both announce that, as K 1 p 2 ∧ K 2 p 1 is then true and they both win (the other three strategy profiles result in both losing, including another equilibrium namely when both don't declare).But Tony considers it possible that ¬p 2 in which case announcing p 1 , and Maria's behaviour being equal, goal K 1 p 2 ∧ ¬K 2 p 1 will fail.In that case he should have kept his mouth shut to have them win.Given the uncertainty over the game he has to reason about not two but four strategies for Maria: depending on whether she is in love or not, whether she would show her feelings or not.What he will do given this information set of two indistinguishable outcomes, also depends on his risk aversity.If he's an optimist, he might still go for it.But if he's a pessimist, maybe better not.Maria's considerations are not dissimilar, but recall she has a different personality (the goals are asymmetric).Example 11 on page 15 will reveal it all.
Example 2 (A game of pennies that do not match).Consider two players Even and Odd both having a penny.They also both have a dice cup wherein they put their penny, shake the cup, and then put it on the table and watch privately whether their penny is heads or tails.Now they decide whether to inform the other player of the result, or not.If they both do or if they both don't, Even wins.So if either 2 know or 0 know.One might say that their state of knowledge is then 'even'.Otherwise, Odd wins.So for the outcome it only matters whether they know that the penny is heads or tails, it not matter whether it is heads or tails.What should they do?
We let Even be player 1 and Odd be player 2, and we let p 1 stand for 'Even's penny is heads' whereas p 2 represents 'Odd's penny is heads.'The goals are therefore (where Kw i p j abbreviates K i p j ∨ K i ¬p j and means 'player i knows whether p j ): What should they do?On first sight this seems quite straightforward as the outcome does not depend on the valuation of p 1 and p 2 .If Even and Odd both announce the result of their throw with the penny, Odd would then have done better not to make that announcement.But if that were to have happened, Even would have done better not to announce either.And so on.There is no equilibrium.Or is there?Yes, there is.And it is pure.Example 12 on page 15 will reveal it all.
Our framework of Boolean observation games clearly builds upon [2,32] but a main difference is that these are complete information games whereas ours are incomplete information games.Thus we have very different strategies.Players do not control the values of variables, but they control whether they reveal the fixed values of variables that only they can observe.In that respect our framework also builds on the public announcement games of [4,3].They only allow strategies that are public announcements wherein the same information is revealed to all players.However, they permit announcing any epistemic formula, not merely propositional variables.A more detailed comparison with all these approaches is only possible after having given our framework in detail and is therefore in a later Section 7.
Our games are strictly qualitative and thus abstract from truly Bayesian approaches [30] with probabilities.To determine equilibrium we compare information sets, called 'expected outcomes'.As the expected outcome may not be a value and the relation may not be a total order, our work is therefore in ordinal game theory [20,17,5].
Our contributions.We analyse both structural and computational properties of Boolean observation games.We study various equilibrium notions based on different interpretations of profitable deviations from information sets, and we provide existence results for such equilibria.We identify various fragments of Boolean observation games including one where the goals are 'knowing whether formulas' of which the realization does not depend on the valuation.We show that this fragment corresponds to Boolean games in terms of existence of equilibrium outcomes.We also provide complexity results for the natural questions of verification and checking of emptiness of equilibrium outcomes in Boolean observation games.
Overview of content.Section 2 provides technical preliminaries needed to define Boolean observation games, that are then defined in the subsequent Section 3. Section 4 presents the a correspondence between Boolean games and Boolean observation games.Section 5 provides various results for the existence of Nash equilibria and Section 6 contains the results on the computational complexity of determining whether a strategy profile is an equilibrium, and whether equilibria exists.Section 7 gives a more in-depth comparison with other epistemic Boolean games.

Introduction
In this section we introduce an auxiliary notion that is a complete information strategic game, which is played with strategies that are epistemic actions, that has epistemic formulas as goals and for which we propose a greatly simplified epistemic logic, and where outcomes are the truth values of those goals.Boolean observation games, that are incomplete information strategic games with more complex strategies and outcomes, will then be defined in the next section.The logic is simple in order to ensure a compact representation allowing to obtain complexity results comparable to those for Boolean games.Some logical details that are fairly elementary but that might distract from the game theoretical content that is our focus, are deferred to the Appendix.

Strategies consisting of players revealing observations
Let N = {1, . . ., n} be a finite set of players i and P a finite set of (propositional) variables such that (P i ) i∈N defines a partition of P .The set P i is the set of variables p i observed by player i (that is, of which player i, and only player i, observes the value).A valuation is a subset v ⊆ P , where p i ∈ v means that p i is true and p i / ∈ v means that p i is false.The set P(P ) of all valuations is denoted V .
A strategy for player i is a function s i : N → P(P i ) that assigns to each player j the set s i (j) ⊆ P i of variables that player i reveals (announces) to player j.We require that s i (i) = P i .Let S i denote the set of all strategies of player i.A strategy profile is a member s of S = S 1 × • • • × S n .The set P i (s) = {p j ∈ P | p j ∈ s j (i)} consists of the variables revealed to i in s.As s i (i) ⊆ P i , P i ⊆ P i (s).For i ∈ N, we denote the n-tuple s as (s i , s −i ) where s −i represents the (n − 1)-tuple of the strategies of other players.Strategy s ∅ i is such that for all j ∈ N with j = i, s i (j) = ∅.This means that nobody reveals anything to anyone.Strategy s ∀ i is such that for all i, j ∈ N, s i (j) = P i .This means that everybody reveals everything and to everyone.
Given i ∈ N and strategy profile s, the observation relation ∼ s i on V is defined as, for v, w ∈ V : Observation relation ∼ s i encodes the informative effect of s.For ∼ s ∅ i we write ∼ i .This is the initial observation relation.We further note that P i (s ∀ ) = V for any player i, so that ∼ s ∀ i is the the identity relation =.A ∼ s i equivalence class, defined as , is also called an information set (of player i given valuation v and observation relation ∼ s i ).Inasfar as strategies consist of each player i selecting a subset P ′ i of her variables P i , these are like the strategies in Boolean games.However we interpret this differently: player i does not make the variables in P ′ i true, but reveals the value of the variables in P ′ i according to a fixed valuation v. Another departure (or generalization) from Boolean games is that different variables are revealed to different agents.This is because we felt that more interesting game theoretical results could be obtained for such a generalization, and because more interesting communicative scenarios could then be treated with the game theoretical machinery.
Example 3. We assume a strategy profile to take place in some instantaneous, synchronous, fashion, such as, when s 1 (2) = {p 1 , q 1 }, s 1 (3) = {p 1 , q 1 }, and s 1 (4) = ∅, player 1 informing player 2 and player 3 that p 1 and q 1 are both true, and such that player 4 observes this without being party to the message content (for example, 1 whispering to 2 and 3).In other words, player 4 knows that player 1 informs player 2 and player 3 whether p 1 and q 1 , but player 4 remains uncertain of the value of p 1 and q 1 , so does not know that 1 informs 2 and 3 that p 1 and q 2 .1 Now consider s ′ 1 that is like s 1 except that s 1 (3) = {p 1 , q 1 } as well.This is the public announcement of p 1 and q 1 by player 1 to all players.
What if for example And what about s 2 , s 3 and s 4 ?This cannot be done instantaneously.But we can ensure independence: all players commit to their s i before they execute it, and not after they see what variables are revealed to other players before it is their turn to reveal.Instead of whispering we can all have prepared closed envelopes adressed to all others on which is written for example, 'from player 1 to player 2: contains the truth about p 1 and p 2 '.All envelopes are collected blindly and then put on the table for all to see and are then handed out.
Such forms of communication are known as semi-public announcement [44], see Appendix A.2 on dynamic epistemic logic for details.

Goals that are epistemic formulas
The language of epistemic logic is defined as follows, where i ∈ N and p i ∈ P i .
Here, ¬ is negation, ∨ is disjunction, and K i ϕ stands for 'player i knows ϕ.' Other propositional connectives are defined by abbreviation, and also Ki α := ¬K i ¬α (player i considers α possible), and Kw i α := K i α ∨ K i ¬α (player i knows whether α).The members of L K are goals and may as well be called, suiting our purposes formulas.
The following fragments of L K also play a role, where i, j ∈ N and p i ∈ P i .
The language L B of the the Booleans is the fragment of L K without K i modalities.In the language L Kw of knowing whether formulas (Kw formulas) the constructs Kw j p i play the role of propositional variables.The fragments L K nnf and L Kw nnf are those of the negation normal form (nnf) of respectively L K and L Kw , where the language L + of the positive formulas is the fragment of L K nnf without Ki modalities (corresponding to a universal fragment of first-order logic).Note that L Kw and L Kw nnf are really propositional languages, not modal languages.
Apart from the above fragments yet another fragment plays a role in our contribution, namely that of the self-positive goals.The self-positive goal formulas are defined as L self+ := j∈N L j+ , where each L j+ is given by the following BNF, wherein i, k ∈ N and k = j.
Note that L + is a fragment of L j+ , namely the fragment where all occurrences of K k are positive, and that L j+ is a fragment of L K nnf , namely the fragment wherein all occurrences of K j are positive.In a self-positive goal for agent j, j's objective is to (get to) know others' variables and others' knowledge and ignorance, although other players may either know or remain ignorant of j's knowledge.This implies that j's goal also cannot be for others to know j's ignorance.A larger number of communicative scenarios seem to have self-positive goals than merely positive goals: it seems fairly typical that you wish others to remain ignorant even when you are only interested in obtaining (factual) knowledge.
The inductively defined semantics of L K formulas are relative to a valuation v and a strategy profile s, where i ∈ N and For v, s ∅ |= α we write v |= α.This is a bit sneaky: by definition this represents what players know after the strategy profile is executed wherein nobody reveals anything, but we can therefore just as well let it stand for what players initially know, before anything has been revealed.We let s |= α denote "for all v ∈ V , v, s |= α," and |= α denote "for all s ∈ S, s |= α".In our semantics, K i p i , K i ¬p i , and Kw i p i are always true (equivalent to the trivial assertion ⊤).We therefore informally assume that they do not occur in goal formulas.
We note that our epistemic semantics is not the usual one for the epistemic language, interpreted on arbitrary Kripke models, but a greatly simplified epistemic semantics dedicated to reason about strategies that are joint revelations of observed variables.We do not even use the word 'model' !And we do not allow announcements (revelations) of other information than variables!In Appendix A.2 we show how (valuation, strategy) pairs induce multi-agent Kripke models.All these simplifications are in order to obtain a smooth comparison with Boolean games and with comparable complexities, unlike the higher complexities common in multi-agent epistemic reasoning.
We continue with some elementary properties of this simple logical semantics, in the form of propositions.
Similarly, each formula in L Kw is equivalent to a formula in L Kw nnf .Proof.This well-known result in modal logic for L K is shown by induction on formula structure, using the equivalences ¬¬α ↔ α, ¬(α ∨ β) ↔ (¬α ∧ ¬β) and ¬K i α ↔ Ki ¬α.For L Kw , as this is essentially a propositional and not a modal language, we only need to use the first equivalence.
The basic but lengthy proof of this proposition is in Appendix A.1.Prop. 5 says in other words, that if v, s |= ϕ for some v ∈ V , then v, s |= ϕ for all v ∈ V .Proposition 6.For any α ∈ L Kw , |= α ↔ K i α.
Proof.Let valuation v and strategy profile s be given.Assume v, s |= α.Then from Prop. 5 it follows that for all w ∈ V , w, s |= α.Therefore, in particular, w, s |= α for all w ∼ s i v, which is by definition v, s |= K i α.Now assume v, s |= K i α.From v ∼ s i v and the semantics of knowledge now directly follows v, s |= α.
As v and s were arbitrary, we have shown |= α ↔ K i α.
As a consequence each formula in the fragment Kw i p j | ¬α | α ∨ α | K i α is equivalent to a formula in L Kw , in other words, knowledge can then be eliminated.This explains why we defined the fragment L Kw without an inductive clause for knowledge.
Knowledge cannot generally be eliminated from a language with knowing whether variables.For example, Anne (1) may know whether Bill (2) passed the exam (p 2 ), but Bill may be uncertain whether she knows.So we have Kw 1 p 2 ∧¬K 2 Kw 1 p 2 .Props. 5 and 6 (and the subsequent Prop.7) do not hold for knowing whether fragments on arbitrary Kripke models.Proposition 7.For all i, j, k ∈ N: |= Kw i Kw j p k .
Proof.Formula Kw i Kw j p k is by definition equivalent to K i Kw j p k ∨ K i ¬Kw j p k .From Prop. 6 it follows that this is equivalent to Kw j p k ∨ ¬Kw j p k which is a tautology.Therefore, in our very simple epistemic logic it is common knowledge whether a player knows a variable.This reflects the dynamics of revealing variables.Suppose all players hold cards named p 1 , q 1 , p 2 , . . . on the back side and the value 0 or 1 on the front (face) side.You may not know that your neighbour has shown to your other neighbour that the value of the card p 1 is 1 (true).But but you know whether your neighbour has shown card p 1 to your other neighbour.You saw it happen.

Pointed Boolean observation games
A pointed Boolean observation game (pointed observation game) is a pair (G, v), denoted G(v), where v ∈ V and where G is a triple (N, (P i ) i∈N , (γ i ) i∈N ), where all γ i ∈ L K .The players' strategies in the pointed observation game are the strategies s i ∈ S i .The players' goals in the pointed observation game are the γ i ∈ L K .Given i ∈ N, the outcome function u i : V × S → {0, 1} of a pointed observation game is defined as: A strategy profile s is a Nash equilibrium of G(v) iff for all i ∈ N and s ′ i ∈ S i we have . That is, no player has a profitable deviation from s in G(v), which would therefore be a s The pointed observation game is an auxiliary notion, matching the intuition that after revealing variables a player wins when her goal has become true.The game is one of complete information because the valuation is known to you, the reader.But the valuation is typically not known to the players.It already uses the parameters of the Boolean observation game that we will now define in the next section.
Example 8. We recall Example 1.We summarily describe a pointed Boolean observation game and its equilibria, where a fuller development is only given in Example 11.Consider pointed game G(v) with G = ({1, 2}, ({p 1 }, {p 2 }), ({γ 1 , γ 2 }) where γ 1 , γ 2 are as in Example 1, and where valuation v = {p 1 , p 2 } (both are in love).The strategies are to reveal nothing or to reveal all, that is: ) is an equilibrium strategy profile of the pointed game G(v), with outcome 1 for both players.This is the only way to make K 1 p 2 ∧ K 2 p 1 true.However, both players not announcing their variable is also an equilibrium with outcomes 0.

Introduction
We will now define the Boolean observation game.A Boolean observation game is an incomplete information strategic form game with uniform strategies (uniform functions from valuations to strategies) and expected outcomes (information sets of outcomes), whereas the auxiliary notion of a pointed observation game is a complete information strategic form game with strategies and with (Boolean-valued) outcomes.

Boolean observation games
This section contains the crucial game theoretical notions of our contribution.

Boolean observation game.
A Boolean observation game (or observation game) is a triple G = (N, (P i ) i∈N , (γ i ) i∈N ), where all γ i ∈ L K .Formula γ i is the goal (objective) of player i.They are played with uniform strategies and the payoffs are expected outcomes.Both will now be defined.
Uniform strategy.A uniform strategy for player i ∈ N is a function So, uniform means the same for all indistinguishable valuations, which is different from globally uniform, which means the same for all valuations.Let S i denote the set of uniform strategies of player i, and S = S 1 × • • • × S n the set of uniform strategy profiles.Let S g i and S g denote the set of globally uniform strategies of player i and the set of globally uniform strategy profiles respectively.Given a valuation v, a uniform strategy profile s determines a strategy profile s(v) = (s 1 (v), . . ., s n (v)).Note that (s(v) i , s(v) −i ) = (s i , s −i )(v).For i ∈ N and s i ∈ S i , we define ṡi ∈ S g i as: for all v ∈ V , ṡi (v) = s i .Similarly for s ∈ S we define ṡ ∈ S g as the globally uniform strategy profile such that for all v ∈ V , ṡ(v) = s.It follows from the definition that every globally uniform strategy profile s ∈ S g is of the form ṡ for some strategy profile s ∈ S.
Expected outcome.Given i ∈ N, the expected outcome function is a function u i : V × S → {0, 1} * that is uniform in V , and defined as u i (v, s) = (u i (w, s(w))) w∼ i v .So, expected outcome u i (v, s) is a vector of outcomes u i (w, s(w)) for each valuation w in the information set of player i.In our setting where outcomes are 0 (lose) or 1 (win) this vector is a bitstring.
As far as nomenclature is concerned, we are putting the reader on the wrong foot, as a uniform strategy is not a kind of strategy (as defined in the previous section), nor is expected outcome a kind of (binary valued) outcome.However, we are in good company: game theory is not a theory, and an artificial brain is not a brain.So we hope the reader will allow us this slight abuse of language.
Outcome relation and Nash equilibrium.To define the notion of an equilibrium in observation games, we need to first define a comparison relation between uniform strategy profiles.Note that unlike in classical strategic games, the expected outcome function in observation games generates a vector of outcomes.Therefore, there is no canonical definition for the comparison relation.We define an outcome relation > over vectors of outcomes and write u i (v, s) > u i (v, s ′ ) for "player i prefers s i over s ′ i in the information set containing v"; we also say that s i is a profitable deviation from s ′ i .This outcome relation may not be a total order.We therefore prefer not to use notation ≤ to compare the bitstrings that are outcome sets, as it is ambiguous whether x ≤ y means (x < y or x = y) or x ≯ y (and even when defined as either one or the other, it seems unkind to the reader).
Given an outcome relation >, a uniform strategy profile is a Nash equilibrium if no player has a profitable deviation.
A uniform strategy profile s is a Nash equilibrium of G iff for all i ∈ N, s ′ i ∈ S i and v ∈ V , we have that u i (v, (s ′ i , s −i )) ≯ u i (v, s).Given an observation game G, NE (G) denotes its Nash equilibria, and among those NE g (G) denotes the globally uniform Nash equilibria.
Also, a uniform strategy The optimist, pessimist and realist outcome relations are total orders, as it suffices to assign a number to the information set constituting an expected outcome.The maximal outcome relation is not a total order.We let NE pess (G), NE opt (G), NE real (G), and NE max (G) denote the Nash equilibria under the pessimist, optimist, realist and maximal outcome relation, respectively.
Example 9. Let us consider an abstract example where a player has to choose between expected outcomes (bitstrings) 00, 10, 01, 11.We then get (where clustered bitstrings means equally preferred): The > max relation is not a total order.It is neither antisymmetric nor transitive.For example, we have that 10 > max 01 but also 01 > max 10.It is not transitive because 01 > max 10 > max 01 however 01 ≯ max 01.
The outcome relations that we have proposed are qualitative versions of well-known criteria in decision theory and Bayesian reasoning.None assume a probability distribution, however, all assume a strictly positive probability for each valuation.
• The optimist outcome relation is the max instantiation (as there is only one maximal value) of the minimax regret decision criterion [41].With respect to the highest possible outcome in the information set, a lower possible outcome in the information set (which can only be 0 instead of 1) would cause regret if this were to happen.
• The pessimist outcome relation is the min instantiation of the maximin or Wald decision criterion [47].We then choose the information set with the best worst outcome.This outcome relation has been used to model uncertainty in voting (with similar considerations involving Nash equilibria and dominance) in [16,45,9].
• The realist outcome relation is a qualititative version (lack of justification to rule out any outcome) of a random decision in Bayesian terms, also known as the insufficient reason or Laplace decision criterion, or as the principle of indifference [36,Chapter IV].
Instead of taking the sum of the outcomes in the information set we could of course have normalized this so it adds up to 1, suggesting an even distribution of probability mass.Such scaling is irrelevant for our purposes of determining Nash equilibria and dominance, wherein we only need to compare outcomes.That comparison relation remains the same.
This outcome relation was used in [4,3] to determine equilibria of similar incomplete information games, but where more complex formulas than mere variables could be 'revealed' (however, they could only be publicly announced).An issue for the relatist outcome relation is whether bisimilar game states (that therefore satisfy the same goals for all players) should be counted once or twice. 3On the one hand, if two game states are bisimilar this is justification / sufficient reason to rule out one of them, according to Laplace.On the other hand these bisimilar game states might have originated from playing strategy profiles (executing epistemic actions) in initial game states that were non-bisimilar.It is relevant to observe this as we note that this phenomenon cannot occur in our simpler setting involving observation relations.
• The notion of Nash equilibrium for the maximal outcome relation has an interesting interpretation.A maximal Nash equilibrium is a uniform strategy profile where no player has a profitable deviation even if the player has complete information about the game.There is an equivalent formulation of maximal Nash equilibrium as a qualitative version of ex-post equilibrium [7], which we show in Proposition 10.
As already observed, the maximal outcome relation is not a total order.However, the maximal outcome relation satisfies the property that all outcomes can be compared: The disjunction in the consequent is inclusive, both may hold (we recall that 10 > max 01 as well as 01 > max 10).To require this property is common in ordinal game theory [20].
Various of the above outcome relations have also been considered in [40].
In the remaining sections we focus on optimist, pessimist and maximal Nash equilibrium and not on realist Nash equilibrium.We use the operational definition of maximal Nash equilibrium given by the correspondence in Proposition 10.It is easy to see that a maximal Nash equilibrium is also an optimist, pessimist, and realist Nash equilibrium.In that sense the maximal outcome relation is the strongest notion, resulting in the smallest number of equilibria for a game (if any).

Various classes of observation games, and examples
With all the technical tools now at our disposal, very different observation games are of specific interest.We can distinguish them by which outcome relation they employ, and independently by the shape of the epistemic goals.Concerning goals it is useful to distinguish the following.
• In two-player zero-sum games γ i = ¬γ j , and in two-player symmetric games γ i = γ j .
In cooperative games i∈N γ i is consistent.Example 11 is symmetric.Example 12 is zero-sum (and therefore not consistent).Communicative scenarios obeying the Gricean cooperative principle are clearly consistent observation games (and might still be considered games inasfar as people want to outdo each other in being informative).
Whereas security protocol settings with eavesdroppers (consider observing an SMS code that you were sent to confirm a bank transfer) tend to be zero-sum; that is, a generalization of zero-sum: the objectives of the principals are the opposite of those of the eavesdroppers.
We do not have theoretical results for zero-sum or symmetric games.
• In knowing-whether observation games (knowing-whether games, Kw games) all goals γ i are in L Kw .In knowing-whether games the outcome does not depend on the valuation.Whether some Kw i p j is true only depends on player j revealing p j to player i, and does not depend on the valuation, because the truth of p j does not depend on the value of p j .
Section 4 is entirely devoted to knowing-whether games, and Section 5 contains results on existence of equilibria.They related well to the usual Boolean game.Not surprisingly, as the outcome does not depend on the valuation, they also also score better on the computational complexity of determining whether a uniform strategy profile is a Nash equilibrium, or whether Nash equilibria exist than other classes of observation game.That will be investigated in Section 6.4.
• A guarded goal for player i has shape γ i = K i α.In observation games with guarded goals the players know whether they have achieved their objective after playing the game.Whereas in games where the goals are not guarded they may not and need an oracle to inform them of the outcome (such as, when standing in front of an ATM teller, the bank's interface informing them).If goals are guarded, Nash equilibria always exists for the optimist and the pessimist outcome relation, as formulated and shown in Theorem 28 in Section 5.
• In games where all γ i are so-called positive formulas (in the fragment L + where negations do not bind K i modalities), a player's goal is never to remain ignorant of a fact, or even for other players to remain ignorant.Under such circumstances revealing all you know is a dominant strategy.This is therefore rather restricted.
• More interesting than positive goals are the observation games with self-positive goals, wherein you merely want to increase your factual knowledge (and of the knowledge of others) but may wish to preserve other players' ignorance.(We recall prior Subsection 2.3.)We provide a result for self-positive goals in Corollary 29 in Section 5.
For all these, results on existence of equilibria and complexity also depend on which outcome relation is used, as already occasionally listed above.
Last but not least one can consider iterated observation games with temporal eventuality goals, where players successively reveal more and more of their observed variables.An example are (successive) question-answer games wherein the strategic aspect is what variable(s) to ask another player(s) to reveal, which seems of particular interest for strategic negotiation (if you give me this, I'll give you that.In small steps).All these come with specific questions on compact representation and existence of equilibria.
We defer the investigation of iterated games and question-answer games to future research.In this work we focus on knowing-whether games and their relation to Boolean games, on the existence of equilibria for various outcome relations (where the realist outcome relation plays no role), and on complexity results for some of our variations.
We now continue with some detailed examples.
Example 11.Recall Example 1 (page 3) and Example 8. We now give full details.Consider the observation game G where N = {1, 2}, P 1 = {p 1 }, P 2 = {p 2 } and the (symmetric) goals: As there are only two players and each player observes a single variable the strategies are to reveal nothing or to reveal all, that is: , and s ∀ 2 .For each valuation v the pointed observation game G(v) has an equilibrium where both players get outcome 1.For example, if p 1 and p 2 are both true, then both players revealing (announcing) that is an equilibrium with outcome 1 for both players.However, both players not announcing their variable is also an equilibrium with outcome 0.
Let us now determine equilibria for G, with uniform strategies instead of strategies, and let us consider the different outcome relations.
Thus, for all s ∈ S and for all v ∈ V , min u 1 (v, s) = 0.The situation is symmetric for player 2. Therefore, for all s ∈ S, s ∈ NE pess (G).
• optimist.Similarly, for all s ∈ S, for all v ∈ V and for all i ∈ {1, 2}, max u i (v, s) = 1.Therefore, for all s ∈ S, s ∈ NE opt (G).
• realist.In this example, max s i = Σs i , so that also, for all s ∈ S, s ∈ NE real (G).
• maximal.NE max (G) = ∅.There are no maximal Nash equilibria, because every information set for both players always contains a win and a lose, so if they were to know the real valuation, one of those is not an equilibrium for the pointed game.
Possibly, the equilibria depend on what we called the 'personalities of Tony and Maria', that is on the shape of the goals?We considered two different personalities that therefore allow four different goals, but (the reader can check that) none makes a difference for any of the four outcome relations, as the property that each information set contains a win and a lose persists throughout such transformations.The best is always win, and the worst is always lose.However for other 'personalities' (for lack of a better term) this need not be, for example, change ¬K 1 p 2 ∧ ¬K 2 ¬p 1 in the third conjunct into ¬K 1 p 2 ∧ K 2 ¬p 1 (we removed one negation symbol).It is now dominant for player 1 to announce the value of p 1 in the information set wherein p 1 is false.
Example 12. Recall Example 2 on page 4 about the pennies that do not match.We can now model this as a knowing-whether Boolean observation game G where N = {1, 2}, P = P 1 ∪ P 2 with P 1 = {p 1 }, P 2 = {p 2 } and For i = 1, 2, player i has strategy s ∅ i wherein she reveals nothing ('hide p i ') and strategy s ∀ i wherein she reveals the value of p i .Irrespective of the valuation, in the strategy profiles , player 2 has a profitable deviation in the corresponding pointed observation game.Similarly, in , player 1 has a profitable deviation.Thus it can be verified that NE max (G) = ∅.Also, within the set of all globally uniform strategy profiles, G does not have a Nash equilibrium for the pessimist and optimist outcome relation.
However, this game has a Nash equilibrium with uniform strategies that are not globally uniform, for the pessimist and for the optimist outcome relation.Consider the uniform strategy profile s = (s 1 , s 2 ) where in s 1 , player 1 reveals p 1 to 2 when p 1 is true and hides p 1 from 2 when p 1 is false, and in s 2 player 2 reveals p 2 to 1 when p 2 is true and hides p 2 from 1 when p 2 is false.It is easy to see that s ∈ NE pess (G) and s ∈ NE opt (G).

Knowing-whether Boolean observation games
In this section we show a correspondence between knowing-whether Boolean observation games (Kw games) and Boolean games.We provide polynomial time reductions that convert a Boolean game to a Kw game and vice-versa.
We first recall the definition of Boolean game.We then show that every Boolean game defines a Kw Boolean observation game, and that every Kw Boolean observation game defines a Boolean game.These embeddings are different, the first is not the converse of the second.
We further show a utility preserving equivalence between strategies in Boolean games and equivalence classes of globally uniform strategies in Kw games ( Lemmas 18,22).As a consequence, we prove a correspondence between the existence of Nash equilibria in Boolean games and the existence of maximal Nash equilibria in Kw games, and for both reductions (Theorems 19,24).
We finally show that there always exists a pessimist equilibrium for 2-player Kw games, but not for Kw games in general: we give an 8-player Kw game without a Nash equilibrium.
Recall that for any v ∈ V and α ∈ L Kw , v, s |= α iff s |= α (Prop.).This justifies writing u i (s) for the outcome u i (v, s) of a pointed Kw game.Now consider a globally uniform strategy profile ṡ ∈ S g .As s), this justifies writing u i ( ṡ) for the expected outcome of a such a Kw game.

Boolean games
Boolean games have the same parameters as Boolean observation games but simpler strategies.A Boolean game is denoted B to distinguish it from a Boolean observation game G.
A Boolean game is a tuple B = (N, (P i ) i∈N , (γ i ) i∈N ) where all γ i ∈ L B (all goals are Boolean).For i ∈ N, a strategy v i for player i is a (local) valuation v i ⊆ P i , where, slightly abusing notation, we identify a strategy profile For Boolean games, the outcome function is denoted u B to distinguish it from the outcome function u of pointed Boolean observation games.We define Equilibrium is as for pointed observation games: a strategy profile v ∈ V is a Nash equilibrium in B if for all i ∈ N and Given B, its Nash equilibria are denoted NE (B).Let us emphasize the difference between Boolean games and Boolean observation games.In Boolean observation games, as in Boolean games, a player i selects a subset v i of her local variables P i .However, in Boolean observation games this subset may be a different subset s i (j) ⊆ P i for each other player j.Also, in Boolean games, excuting strategy v i means that the p i ∈ v i become true whereas the p i ∈ P i \ v i become false.Whereas in Boolean observation games, executing strategy with component s i (j) means that the p i ∈ s i (j), that alaready have an observed truth value, are revealed (to j).

Boolean games to knowing-whether games
We construct a Kw game denoted G B from a Boolean game B as follows.Let B = (N, (P i ) i∈N , (γ i ) i∈N ).Then G B := (N, (P i ) i∈N , (β i ) i∈N ) where each β i := λ(γ i ) is defined as follows.Let i + := i + 1 for i = 1, . . ., n − 1 and n + := 1.Then λ : L B → L Kw is inductively defined as: for all i, p i ∈ P i , λ(p i ) := Kw i + p i , and (trivially) λ(¬α) := ¬λ(α) and λ(α Then G B has the same variables p 1 , q 1 , p 2 , p 3 but different goals, namely Kw 2 p 1 ↔ Kw 1 p 3 for player 1, Kw 1 p 3 → Kw 2 p 1 for player 2, and ¬Kw 2 p 1 → Kw 3 p 2 for player 3. In the Boolean game, for player 1 to obtain her goal p 1 ↔ p 3 , she has to make p 1 true and player 3 has to make p 3 true.In the Kw game, in order to achieve the same (in order to execute the same strategy), she has to reveal p 1 (well, the value of p 1 , although what value it is does not matter) to player 2, and player 3 has to reveal her p 3 .More precisely, the strategies for 1,2,3 should be: Given a strategy profile v ∈ V for B, we define strategy profile ; and for all j = i, i + , s v i (j) = ∅.Notation s v is not to be confused with notation s(v) for uniform profiles s.Lemma 14.Let G B be the Kw game associated with the Boolean game B. For all i ∈ N, for all w ∈ V , Proof.This is shown by induction where only the base case is not-trivial.For that, we have that s w |= Kw i + p i iff w |= p i by definition of the embedding.
Therefore, for any i ∈ N and v ∈ V : . This correspondence allows us to relate Nash equilibria in B to Nash equilibria in G B .The result uses an interesting property of NE max in Kw games (this property does not hold for, e.g., the pessimist outcome relation).
Proof.Consider an arbitrary v ∈ V and let s = s(v).Suppose that s ∈ NE max (G), we claim that the globally uniform strategy profile ṡ ∈ NE max (G).Suppose not.Then there exists i ∈ N, w ∈ V and s ′ i ∈ S i such that u i (w, (s ′ i , ṡ−i (w))) > u i (w, ṡ(w)).This implies that w, (s ′ i , ṡ−i (w)) |= γ i and w, ( ṡ(w)) |= γ i .Since for all j ∈ N, ṡj (w) = s j (v), we have w, Also, since w, ṡ(w) |= γ i , we have v, ṡ(w) |= γ i and by definition of ṡ, we have v, s(v) |= γ i .Therefore, s ∈ NE max (G) which gives the required contradiction.
An equivalence relation over global strategy profiles.Recall that every s ∈ S g is of the form ṡ where s ∈ S(G B ).We define an equivalence relation over S g in G B as follows.
For i ∈ N, ṡi ≡ i ṫi iff s i (i + ) = t i (i + ).For ṡ, ṫ ∈ S g , we define ṡ ≡ ṫ iff for all i ∈ N, ṡi ≡ i ṫi .Let S g /≡ denote the set of equivalence classes and [s] denote the equivalence class containing s ∈ S g .
Proof.Let s = ṡ and t = ṫ.For all i ∈ N, since t ∈ [s], we have s i (i + ) = t i (i + ).By induction of the structure of γ i , we can prove the following: for all v ∈ V , for all i ∈ N and for all An outcome preserving bijection.We now show that there is an outcome preserving bijection χ between strategy profiles in B and equivalence classes in S g /≡.For a Boolean game B, and Lemma 18.Given a Boolean game B, the function χ : V → S g /≡ is a bijection.
Proof.Given ṡ ∈ S g , consider v ∈ V defined as follows: for all i ∈ N and p i ∈ P i , p i ∈ v iff p i ∈ s i (i + ).We then have χ(v) = [ ṡ] and therefore χ is onto.For v, w ∈ V such that v = w, there exists i ∈ N , there exists p i ∈ P i such that p i ∈ v and p i ∈ w.Thus, for χ(v) = [ ṡ] and χ(w) = [ ṫ], we have ṡ ≡ i ṫ, which implies that ṡ ≡ ṫ.Therefore, χ is a bijection.
Consequently, we can prove a correspondence between Nash equilibria existence: Proof.(⇐) We argue that if w ∈ NE (B) then ṡ w ∈ NE max (G B ). Suppose not, then there exists i ∈ N, v ∈ V and ).From Lemmas 14, 17 and 18 it follows that ).We claim that w ∈ NE (B).Suppose not, then there exists i ∈ N and Note that by definition w = w ′ .From Lemma 14 it follows that for all v we have u . From Lemmas 14, 17 and 18 it follows that Other ways to get a Kw game from a Boolean game.Let us imagine our n players sitting round a table numbered in clockwise fashion.In the embedding λ : every player i reveals the value of her observed variable p i to her left neighbour (while other players observe her doing that).There are many other embeddings that would serve equally well to obtain our results.For example, every player i could reveal her variable to her right neighbour.This would be a λ ′ with basic clause where i − is i − 1 except for 1 − := n.A more interesting embedding would be every player publicly announcing p i to all other players.We then have a λ ′′ for which

Knowing-whether games to Boolean games
Kw game to Boolean game.We now construct a Boolean game denoted B G from a knowing-whether Boolean observation game G. Let G = (N, (P i ) i∈N , (γ i ) i∈N ).Assume that the goals γ i do not contain trivial constituents Kw i p i . 4Then where for all i ∈ N, Q i = {Kw j p i | p i ∈ P i , i = j}.We view Kw j p i for each i and j, with i = j as atomic propositions in Observation.Both G and B G are defined over the same set of players and goal formulas.
The number of variables in Thus given G, the associated Boolean game B G can be constructed in polynomial time.Also, note that B G B = B and G B G = G, the constructions are unrelated.
Example 20.As an illustration to construct a Boolean game from a Kw game, let us take the Kw game just constructed in Example 13.We recall that The Boolean game B G B constructed from that has the same goals but has more variables, namely Kw i p j for all i, j ∈ N with i = j and for all p j ∈ P i , that is: Kw Let W = P(Q) be the set of valuations over Q.We define a function η : S g → W and argue that it is a bijection which is outcome equivalent.Given ṡ ∈ S g , define w = η( ṡ) as follows: for i ∈ N, Kw j p i ∈ η( ṡ) i iff p i ∈ s i (j).
Lemma 21.Let B G be the Boolean game associated with the Kw game G.For all i ∈ N, for all s ∈ S and for all γ i , s Proof.This is shown by induction using as the base case that s |= Kw j p i , iff η( ṡ) |= Kw j p i The other cases are trivial.
It therefore also follows, similarly to the above, that u i ( ṡ) = u i (s) = u B i (w s ).
Lemma 22.Given a Kw game G, let B G be the associated Boolean game.The function η : S g → W is a bijection.
Proof.For an arbitrary w ∈ W , consider ṡ ∈ S g defined as follows.For all i ∈ N, and for all p i ∈ P i , p i ∈ s i (j) iff i = j or Kw j p i ∈ w i .By definition, η( ṡ) = w and thus η is onto.Consider ṡ, ṫ ∈ S g where ṡ = ṫ.Then there exists i, j ∈ N with i = j and there exists p i ∈ P i such that p i ∈ s i (j) and p i ∈ t i (j).This implies that Kw j p i ∈ η( ṡ) i and Kw j p i ∈ η( ṫ) i .Therefore η is a bijection.
Non-global uniform strategies as mixed strategies for Boolean games.We allow ourselves a little detour.We can straightforwardly adjust the function η mapping globally uniform strategy profiles of the Kw game to valuations that are strategy profiles of the Boolean game, to a function mapping arbitrary uniform strategy profiles of the Kw game G to mixed strategy profiles of the Boolean game B G .We simply define the 'revised η' on the level of strategy profiles s ∈ S. Given a uniform strategy profile s ∈ S, for each s ∈ S such that s(v) = s for some v ∈ V , we let π(s) is the probability that a valuation is mapped in s, given s.We can now define a mixed strategy profile w s of the Boolean game B G is as the one executing each s ∈ S with probability π(s).We defer the investigation of embeddings into a mixed equilibrium to future research and for now restrict ourselves to a relevant example, finally closing the loop with matching pennies.
Example 23.Once more we recall Example 2 on page 4 about the pennies that do not match, already further developed in Example 12, wherein it was shown that this game does not have a Nash equilibrium with globally uniform strategies for the pessimist and optimist outcome relation, but has a Nash equilibrium with uniform strategies that are not globally uniform: the uniform strategy profile s = (s 1 , s 2 ) where in s 1 , player 1 reveals p 1 to 2 when p 1 is true and hides p 1 from 2 when p 1 is false, and in s 2 player 2 reveals p 2 to 1 when p 2 is true and hides p 2 from 1 when p 2 is false.
When translating this game into a Boolean game, we can now observe that this uniform strategy s becomes a mixed strategy η(s) where player 1 randomly chooses between revealing or hiding her propositional variable Kw 2 p 1 and where player 2 randomly chooses between revealing or hiding his propositional variable Kw 1 p 2 .To realize that this is indeed random it is important to observe that in Example 2 the probability of observing p 1 or ¬p 1 was determined by Even flipping its penny before privately watching the outcome under the dice cup, and similarly for p 2 and Odd.So after all, for those who may have wondered, there was a reason for setting up the experiment just like that.
We continue with a relevant result for the maximal outcome relation.Proof.(⇐) Suppose NE max (G B ) = ∅.By Lemma 15 there exists s ∈ S g such that s ∈ NE max (G B ).Let w = η(s), we argue that w is a Nash equilibrium.Suppose not, there exists i ∈ N, there exists Consider the globally uniform strategy profile t = η −1 (w ′ i , w −i ) (this is well defined by Lemma 22).From Lemmas 21 and 22 it follows that u . This implies that w ∈ NE (B G ) which is a contradiction.

Pessimist Nash equilibria for knowing-whether games
We close the section with results for Kw games for the pessimist outcome relation.Example 12 shows that in the Kw fragment a maximal Nash equilibrium is not guaranteed to exist.
It is natural to ask if a similar observation holds for pessimist Nash equilibrium.We show that for two-player Kw games, a pessimist Nash equilibrium always exists (Proposition 25).
However, for general Kw games existence is not guaranteed.Example 26 gives an 8-agent Kw game without a Nash equilibrium.
Proposition 25.All two-player Kw games have a pessimist Nash equilibrium.
Proof.We construct a uniform strategy profile (s * 1 , s * 2 ) as follows.For i ∈ {1, 2}, let ı denote the player such that ı = i.If i ∈ {1, 2} has a uniform strategy s i that is dominant, then set s * i (v) = s i for all v ∈ V and let s * ı be the best response to s * i .It can be verified that s * as defined above is a Nash equilibrium.
If neither player has a uniform strategy that is dominant, then we have the following • For all t 2 ∈ S 2 , there exists t 1 ∈ S 1 such that v, (t 1 , t 2 ) |= γ 2 for all v ∈ V .For two player games there is a bijection between the set strategies S i and the set of (local) valuations V i (one can think of a strategy as deciding for each proposition whether to reveal to the other player).Therefore for each t 1 and s 2 as described above, we can set s * 1 (v 1 ) = t 1 and s * 2 (v 2 ) = s 2 appropriately for some v 1 and v 2 .To see that (s * 1 , s * 2 ) is a Nash equilibrium, note that for all i ∈ {1, 2} and for all v ∈ V , min u i (v, s * ) = 0. Also, for all v ∈ V , min u i (v, (s ′ i , s * −i )) = 0 due to the above condition.
However, for more than two players a Kw game need not have a pessimist Nash equilibrium.We present a counterexample for eight players.
Example 26.Consider the observation game G where N = {1, 2, . . ., 8} and P i = {p i } for i ∈ N. Player 8 acts as an "observer" whose goal γ 8 = ⊤.To specify the goals of the other players we use the following formulas.
The main idea is to exploit that player 1 controls a single variable.Therefore in any uniform strategy player 1 can choose to satisfy at most two of A, B, C, D. E.g., "when p 1 is true reveal p 1 to 3 and 4 (A), when p 1 is false reveal p 1 to 4 but not to 3 (C).For players {2, . . ., 7} the goal formulas are: The goals of the players (except 8) involve assertions about whether players 2, . . ., 7 reveal the proposition that they control to player 8 along with whether 1 reveals p 1 to players 3 and 4. For the purpose of this example, note that for all j ∈ {2, . . ., 7}, for all s j ∈ S j and k = 8, the value of s j (k) is irrelevant.To show that NE pess (G) = ∅ we first argue that for all uniform strategy profiles s ∈ S, if s 1 is globally uniform then s ∈ NE pess (G).In other words, no uniform strategy profile in G with a globally uniform strategy for player 1 can be a pessimist Nash equilibrium.Consider an arbitrary s ∈ S where s 1 = ṡ1 ∈ S g 1 .Uniform strategy ṡ1 satisfies exactly one of the formulas A, B, C, D. From the goal formulas we can see that there exists a nonempty subset of players X ⊆ {2, . . ., 7} and for all j ∈ X, min u j (v, ( ṡ1 , ṡ∀ j , s N −{1,j} )) = 1 for all v. Thus if s ∈ NE pess (G) then u j (v, s) = 1 for all j ∈ X and for all v. From the goal formulas it also follows that there exists ṡ′ 1 = ṡ1 such that min u 1 (v, ( ṡ′ 1 , ṡ∀ j , s N −{1,j} )) = 1 for all j ∈ X and for all v.
In Table 1 we list all such possibilities.The first column in Table 1 lists the formula in A, B, C, D that is satisfied by a globally uniform strategy ṡ1 .The second column lists the players j ∈ {2, . . ., 7} who can ensure an outcome 1 with ṡ∀ j given ṡ1 .The third column gives the corresponding formulas in A, B, C, D that player 1 should satisfy to achieve an outcome 1.For example suppose ṡ1 satisfies A (first row), players 2, 5 and 6 can reveal their proposition to player 8 to ensure an outcome of 1 given ṡ1 .Player 1 can then choose to satisfy D (corresponding to α 1 ), C (corresponding to α 4 ), B (corresponding to α 5 ), respectively to achieve an outcome of 1.Using Table 1 it can be verified that any s ∈ S where s 1 is a globally uniform strategy is not a pessimist Nash equilibrium.
Next, note that since player 1 controls a single proposition p 1 , any uniform strategy (of player 1) can satisfy at most two of A, B, C, D. E.g., "when p 1 is true reveal p 1 to 3 and 4 (A), when p 1 is false reveal p 1 to 4 but not to 3 (C).Using an argument similar to the one above we can show that if s ∈ NE pess (G) then min u 1 (v, s) = 1 for all v ∈ V .
In Table 2 we list all the possible combinations of the formulas in A, B, C, D that player 1 can possibly satisfy in any uniform strategy.In the second column in Table 2 we list the minimal set of players X which satisfies the condition: if for all j ∈ X, s j = ṡ∀ j then u 1 (v, s) = 1 for all v ∈ V .In other words, if all the players in X reveal their proposition to player 8 then the outcome for player 1 under the strategy s 1 is 1 for all v. From the goal formulas γ 1 , . . ., γ 7 we can verify that for every such set X, there is a player k ∈ X who can ensure an outcome of 1 by not revealing p k to player 8.

Existence of Nash equilibrium
Given Example 26, a natural question is to ask whether there are fragments of observation games where existence of a Nash equilibrium is guaranteed.An initial step would be to consider observation games where the goal formulas for all players are restricted to the positive fragment of L K .For this fragment, we have the following result.
Proposition 27.Let G = (N, (P i ) i∈N , (γ i ) i∈N ) be an observation game where γ i ∈ L + for all i ∈ N. Then NE (G) = ∅ for all outcome relations.
Proof.Observe that when γ i ∈ L + for all i ∈ N, the globally uniform strategy ṡ∀ i (public announcement by player i of P i ) is dominant for all i ∈ N. Thus ṡ∀ ∈ NE (G) for any outcome relation.
In this section, we present a more general structural result that identifies a class of observation games in which a Nash equilibrium is guaranteed to exist.Our results show that the existence of equilibrium crucially depends on the combination of positive/negative epistemic assertions made by players in their goal formulas (and goals with positive formulas can be viewed as a simple case).
We assume that the goals are in negation normal form.For i, j ∈ N (where j may be i) and γ i in L K nnf , we define x j i (γ i ) for x ∈ {+, −}.Intuitively, + j i (γ i ) and − j i (γ i ) encodes the fact that player i makes a positive and negative (respectively) epistemic assertion about a variable assigned to player j in the goal formula γ i .Formally, for atoms we have: + j i (p j ) and + j i (¬p j ).For the inductive clauses, for + j i and − j i (and where k ∈ N): . For a node i, we now define type(i) ⊆ {+, −, c+, c−}.For x ∈ {+, −}, x ∈ type(i) if there is j ∈ N with j = i such that x i j (γ i ) and cx ∈ type(i) if x i i (γ i ).That is, + and − is in type(i) if player i makes a positive and negative (respectively) epistemic assertion about a variable assigned to player j in γ i .Similarly, c+ and c− is in type(i) if player i makes a positive and negative (respectively) epistemic assertion about its own variable in γ i .For example, consider the goal formula γ i for i ∈ N given in Example 12.We have type(i) = {+, −, c+, c−}.In fact, Theorem 32 given below shows that it is crucial that |type(i)| = 4 in this example.
We define the following subsets of N.
For the proofs in this section, we also find it useful to define an ordering over the set of strategy profiles.Let X ⊆ N and s X , t X ∈ S X .We say that s X t X if for all i ∈ X and j ∈ N, t i (j) ⊆ s i (j).
Output: A uniform strategy profile s ∈ NE pess (G).
Using the "type" classification of goal formulas we have the following existence results.

NE opt (G) = ∅.
Proof.Part 1. NE pess (G) = ∅.We can argue that Algorithm 1 always terminates and constructs a uniform strategy profile s ∈ NE pess (G).Note that the sets X + , X − \ X + , W l , W + and W − form a partition of N.
In each iteration of the outer loop in Algorithm 1 (steps 7 -13), the size of the set Y (v) or Z(v) strictly increases for some v ∈ V .We also have that for all v ∈ V , 0 ≤ |Y (v)| ≤ |N| and 0 ≤ |Z(v)| ≤ |N|.It follows that Algorithm 1 always terminate.Let s be the strategy profile constructed by Algorithm 1. From the description of the procedure, it can be verified that s is a uniform strategy profile.We now argue that s ∈ NE pess (G).
Note that for all i ∈ X + , s ∅ i is a dominant uniform strategy and for all i ∈ X − , s ∀ i is a dominant uniform strategy.Therefore, for all v ∈ V , for all i ∈ X + ∪ X − and for all Therefore, by the choice of s i made in line 4 of Algorithm 1, we have for all i ∈ W l , for all v ∈ V , for all ).Now consider a player i ∈ W + .For v ∈ V , suppose s i (v) is assigned a value in the while loop (steps 9-10).Let s k denote the resulting strategy profile after this assignment (step 10) and Z k denote the value of Z in the corresponding iteration.By definition of the while loop, for all w such that v ∼ i w, for all s W − \Z k (w) , (s i , Consider a player i ∈ W − .For v ∈ V , suppose s i (v) is assigned a value in the while loop (steps [11][12].Let s k denote the resulting strategy profile after this assignment (step 12) and Y k denote the value of Y in the corresponding iteration.By definition of the while loop, for all w such that v ∼ i w, for all s W + \Y k (w) , (s i , Now suppose there exists v ∈ V and i ∈ W + such that i / ∈ Y (v) (on termination of the repeat loop, steps 7-13).By definition, for all s i , there exists w with v ∼ i w and there exists t W − \Z(w) such that (s i , s . Suppose there exists v ∈ V and i ∈ W + such that i / ∈ Z(v).Using a similar proof as above and using the fact that s j (v)(i) = ∅ for all j ∈ W − \ Z(v) we can argue that all s i , there exists a w with v ∼ i w such that (s i , s To show that NE opt (G) = ∅, we modify Algorithm 1 to reflect the optimist decision rule.This is achieved by changing the conditional in both the While loops (line 9 and line 11) as follows: The result in Theorem 28 is tight in the sense that there exist observation games where |type(i)| = 4 for i ∈ N and NE pess (G) = ∅.This is illustrated in Example 26.
An interesting corollary of Theorem 28 is for self-positive goals: my objective is never to remain ignorant of other's variables even when it may be for others to remain ignorant.(We recall their definition in Section 2.3.)Corollary 29.Let G = (N, (P i ) i∈N , (γ i ) i∈N ) be an observation game where all (γ i ) i∈N are guarded and self-positive, then NE pess (G) = ∅ and NE opt (G) = ∅.
For NE max (G) we show a weaker result (Theorem 30) which can be strengthened for Kw games (Theorem 32).
Theorem 30.Let G = (N, (P i ) i∈N , (γ i ) i∈N ) be an observation game where the goal formulas Proof.Let G = (N, (P i ) i∈N , (γ i ) i∈N ) be an observation game.Let X + = {i ∈ N | c+ ∈ type(i)} and X − = {i ∈ N | c− ∈ type(i)}.Consider the uniform strategy profile s defined as follows.
• For i ∈ X + ∩ X − , we define s i using the iterative procedure: for v ∈ V where s i (v) is not defined, if there exists s ∈ S such that u i (v, s) = 1 then set s i (w) = s i for all w : v ∼ i w.Otherwise set s i (w) = s ∅ i for all w : v ∼ i w.
• For all i ∈ X + , for all v ∈ V , let Note that for all i ∈ X + , s ∅ i is a dominant uniform strategy.For all i ∈ X − , s ∀ i is a dominant uniform strategy and for all i ∈ (X + ∪X − ), both s ∅ i and s ∀ i are dominant uniform strategies.
Since the goal formulas are guarded, we have for all i ∈ N and for all v ∈ V , v, s(v) |= γ i iff w, s(v) |= γ i for all w : v ∼ i w.Also, for all i ∈ X + ∩ X − , with type(i) ≤ 2, we have that Lemma 31.Algorithm 2 always terminates and it satisfies the following properties.
• After each iteration of the while loops, steps 9-10 and steps 11-12, the strategy profile s constructed is a globally uniform strategy profile.
Example 34.Consider the two-player game where P 1 = {p 1 , q 1 } and P 2 = {p 2 }.Let the goal formulas be: Here s j i (p i )(v) = ⊤ is viewed as player i revealing the variable p i to player j under the valuation v.We assume that the Boolean function s j i (p i ) is represented as a propositional formula β j i (p i ) over the propositions P .It is well known that every such Boolean function can be represented as a propositional formula, in the worst case the size of s j i (p i ) can be exponential in k.In this section we address the computational complexity of the following two basic algorithmic questions.
• Verification.Given an observation game G and a uniform strategy profile s ∈ S, is s ∈ NE x (G) for an outcome relation x ∈ {pess, opt, max}?
• Emptiness.Given an observation game G is NE x (G) = ∅ for an outcome relation x ∈ {pess, opt, max}?
We show that the verification and emptiness questions are PSPACE-complete and NEXPTIME-complete respectively for the maximal outcome relation.We also show that for the pessimist and optimist outcome relations, the verification and emptiness questions are in PSPACE and NEXPTIME respectively.To obtain these results it is crucial to establish the complexity of the model checking problem of the logic L K .The following result shows that the model checking problem is PSPACE-complete.It follows directly from Proposition 2 in [2]5 .Theorem 35.Given α ∈ L K along with a strategy profile s ∈ S and a valuation It is well known that the model checking problem for epistemic logic formulas over Kripke structures (for example, formulas of multi-agent S5) can be solved in polynomial time [21,26].Note that in our setting, a Kripke structure is not explicitly part of the input, rather the underlying relational structure is compactly presented in terms of the valuation v and strategy s.This is the reason for PSPACE-hardness of the model checking problem.

Verification
In the rest of this section, we refer to valuations over various sets of variables and therefore find it convenient to use the following notations.Let A be a finite set of variables.We use V (A) to denote the set of all valuations over A.
Theorem 36.Given an observation game G = (N, (P i ) i∈N , (γ i ) i∈N ) and a uniform strategy profile s ∈ S, Proof.We can argue that the complement of the problem is in PSPACE.That is, given G and s ∈ S, the problem is to verify if s ∈ NE max (G).This can be solved with the following two steps: For step 1 note that the size of a strategy |s i | = O(nk).So the triple (i, v, s) which forms a possible witness to the fact that s ∈ NE max (G) has a polynomial representation.By Lemma 35 it follows that step 2 can be solved in PSPACE.Since PSPACE is closed under complementation and NPSPACE = PSPACE due to Savich's Theorem, the membership in PSPACE follows.
To show PSPACE-hardness, we give a reduction from the model checking problem for L K .That is, given α ∈ L K , a strategy profile s ∈ S and a valuation v ∈ V we construct an observation game G and a uniform strategy s as follows.Let P (α) denote the set of variables occurring in α and p 1 ∈ P (α) be an arbitrary fixed variable.Let q be a variable such that r ∈ Voc(α).
The set of players N = {1, 2}.P 1 = P (α) and P 2 = {q}.To define the goal formulas we make use of the following notations.Let δ v denote the Boolean formula over P 1 which uniquely characterises the valuation v.That is, δ v := p∈v p ∧ p / ∈v ¬p.For the (fixed) variable p 1 ∈ P 1 , we define the formula flip(q) as follows.
In the case of pessimist and optimist outcome relations, the following computational upper bounds for the verification question are relatively straightforward.
Theorem 37. Given an observation game G = (N, (P i ) i∈N , (γ i ) i∈N ) and a uniform strategy profile s ∈ S, checking if s ∈ NE x (G) is in PSPACE where x ∈ {pess, opt}.
Proof.Observe that by Theorem 35, , it suffices to perform the following two steps.
• Guess a player i, a valuation v and a strategy s ′ i ∈ S i .
For step 1 note that the size of the strategy |s i | = O(nk).Thus the triple (i, v, s i ) that forms a possible witness to the fact that s ∈ NE pess (G) (respectively, s ∈ NE opt (G)), has a polynomial representation.By the observation above, step 2 can be solved in PSPACE.Since PSPACE is closed under complementation and NPSPACE = PSPACE, the membership in PSPACE follows.

Emptyness
Next we address the complexity of checking for emptiness of maximal Nash equilibria in observation games.We find it useful to introduce the following definitions.Let A = {a 1 , . . ., a l } and B = {b 1 , . . ., b l } be two finite sets of variables where |A| = |B| and let ζ : A → B be a bijection.For valuations v 1 ∈ V (A) and v 2 ∈ V (B), we say that We also define the formula ).Given a uniform strategy s i and a set Z ⊆ P i , we say that i can be viewed as a natural generalisation of S g i by parameterising the uniform strategies on the set Z.
An NEXPTIME-complete problem.We now show that given an observation game G, checking if NE max (G) is empty is NEXPTIME-complete.To prove the hardness, we give a reduction from the Dependency quantifier Boolean formula game (Dqbfg) [31, p.87].Dqbfg involves a three player game with players 1, 2 and 3.There are four finite sets of variables which are mutually disjoint, X 2 , X 3 , A 2 and A 3 along with a Boolean formula ϕ over the variables For the rest of this section we use L B to denote the set of Boolean formulas over the variables X ∪ A. Players' strategies are defined as follows.
• Player 3: a strategy t 3 : In other words, a strategy for player 1 is to select a valuation for variables in X. Player 2 chooses a valuation for variables in A 2 and his strategy can depend on the valuation for variables in X 2 .Similarly, a strategy for player 3 is to choose a valuation for variables in A 3 which can depend on the valuation of variables in X 3 .
For player i ∈ {1, 2, 3} let T i denote the set of strategies and T the set of strategy profiles.It is easy to observe that a strategy profile t = (t 1 , t 2 , t 3 ) defines a valuation over the set of variables X ∪ A. For a formula α ∈ L B we then have the natural interpretation for t |= α.Given strategies t 2 ∈ T 2 and t 3 ∈ T 3 , we say that the pair (t 2 , t 3 ) is a winning strategy for the coalition of players 2 and 3 if for all t 1 ∈ T 1 , (t 1 , t 2 , t 3 ) |= ¬ϕ.
An instance of Dqbfg is then given by the tuple H = ((X i ) i∈{2,3} , (A i ) i∈{2,3} , ϕ) and the associated decision problem is to check if the coalition of players 2 and 3 have a winning strategy in H.
The reduction.Given an instance of Dqbfg H = ((X i ) i∈{2,3} , (A i ) i∈{2,3} , ϕ), we construct an observation game G H = (N, (P i ) i∈N , (γ i ) i∈N ) as follows.The set of players N = {1, 2, 3}.For i ∈ {2, 3}, let Y i be a copy of the variables in X i , so For the rest of this section, we use V and L K to denote the set of all valuations and the set of all formulas over the variables in the observation game G H respectively (so We also define the bijection ζ : X → Y as the function that maps each variable in X i to its corresponding copy in Y i .Formally, let 1 for all j ∈ {1, . . ., l} and and ζ(x j 2 ) = y j 2 for all j ∈ {1, . . ., h}.To simplify notation, we denote cons ζ by cons and C ζ by C for this fixed bijection ζ.
In order to define the goal formulas, we first inductively define a function λ : L B → L K that transforms ϕ to a formula in L K as follows.
Properties of G H .It is easy to see that the resulting observation game G H is polynomial in the size of H.We first make the following observations about G H . Lemma 39.Let G H be the observation game corresponding to H and let s ∈ S. If there Proof.Suppose there exists v ∈ V such that cons(v . By Example 12, we have that there exists i ∈ {2, 3} such that v, s(v) |= ψ i and there exists s i ∈ S i such that v, (s i , Proof.For i ∈ {2, 3}, the claim clearly holds for formulas ψ i and C(X i , Y i ).Thus for γ i , the claim follows by a simple induction on ϕ.
Next, we show that the structure of G H ensures the existence of certain restricted type of maximal Nash equilibria.Let R denote the set of uniform strategy profiles s ∈ S that satisfy the following conditions: • for i ∈ {2, 3}, s i ∈ S Y i i .
In other words, R consists of the set of all uniform strategy profiles s such that s 1 is globally uniform and for i ∈ {2, 3}, s i is globally Y i -uniform.Lemma 41.If NE max (G H ) = ∅ then there exists s * ∈ NE max (G H ) such that s * ∈ R.
Proof.For players i ∈ {2, 3} we define an equivalence relation [v] i denote the equivalence class containing the valuation v and c i v ∈ [v] i denote a fixed valuation which is interpreted as the canonical representative element in the equivalence class [v] i .
Suppose s ∈ NE max (G H ). Consider the uniform strategy profile s * ∈ R defined as follow.
• For player 1, fix a valuation w ∈ V and let s * 1 (v) = s 1 (w) for all v ∈ V .
• For players i ∈ {2, 3}, for all v ∈ V , We claim that s * ∈ NE max (G H ). Suppose not, then there exists i ∈ {2, 3}, there exists w ∈ V , there exists s i ∈ S i such that u i (w, (s i , s * −i (w))) > u i (w, s * (w)).Then w, (s i , s * −i (w)) |= γ i and w, s * (w) |= γ i .Now consider the valuation u defined as follows: u ∩ P 1 = w ∩ P 1 and for i ∈ {2, 3}, u ∩ P i = c i w ∩ P i .By definition of u we have that u ∩ (X ∪ Y ) = w ∩ (X ∪ Y ) and therefore, w ∼ = i u.From the definition of s * it follows that s * (w) = s(u).
Since w, (a i , s Strategy translation.Note that by the construction of G H , the strategies of player 1 are irrelevant in terms of existence of maximal Nash equilibria.Player 1 can ensure a utility of 1 by choosing any strategy.We now define two functions which translate strategies of players 2 and 3 between H and G H .For the rest of the section we make use of the following concise notation.For i = 2, let i + = 3 and for i = 3, let i + = 2. For i ∈ {2, 3}, let χ i : T i → S Y i i be the function that translates every strategy t i of player i in H to a globally Y i -uniform strategy s i = χ i (t i ) in G H as defined below.
For i ∈ {2, 3}, let µ i : S Y i i → T i be the function that translates every globally Y i -uniform strategy s i of player i in H to a strategy t i = µ i (s i ) in G H as defined below.
Note that since s i ∈ S Y i i , µ i is well defined.
Lemma 42.For all i ∈ {2, 3} and for all s i ∈ S Y i , let s ′ i = χ i (µ i (s i )).For all s 1 , s ′ 1 ∈ S 1 , for all i ∈ {2, 3} and for all v ∈ V such that cons(v Proof.For i ∈ {2, 3}, the claim clearly holds for the formulas ψ i and C(X i , Y i ).Thus for γ i , the claim follows by induction on ϕ.
Lemma 43.For all α ∈ L B , for all t ∈ T , for all i ∈ {2, 3} and for all v ∈ V such that t ∩ X = v ∩ X and cons(v Proof.For i ∈ {2, 3}, let s i = χ i (t i ).The proof is by induction on the structure of α where the interesting cases involve the three base cases.
• α = p ∈ X.Then we have λ(p) = p and the following sequence of equivalences.
• α = p ∈ A 2 .Then we have λ(p) = Kw 3 p and the following sequence of equivalences.
• α = p ∈ A 3 .Then we have λ(p) = Kw 2 p and the following sequence of equivalences.
Lemma 44.Let H = ((X i ) i∈{2,3} , (A i ) i∈{2,3} , ϕ) be an instance of Dqbfg and G H the associated observation game.The coalition of players 2 and 3 have a winning strategy in Proof.Let (t 2 , t 3 ) be a winning strategy for the coalition of players 2 and 3 in H.By definition of a winning strategy, for all t 1 ∈ T 1 , we have (t 1 , t 2 , t 3 ) |= ¬ϕ.Let t = (t 1 , t 2 , t 3 ) and consider the observation game G H .Note that in G H , by the definition of player 1's goal γ 1 , we have for all v ∈ V and for all s ∈ S, u 1 (v, s(v)) = 1.Now consider an arbitrary valuation v ∈ V .There are two cases to consider.Case 1. Suppose there exists i ∈ {2, 3} such that cons(v ∩ X i , v ∩ Y i ) does not hold.By semantics, for all s ∈ S and for all i ∈ {2, 3} we have v, Case 2. Suppose for all i ∈ {2, 3}, cons(v ∩ X i , v ∩ Y i ) hold.By semantics we have for all s ∈ S, for all i ∈ {2, 3}, v, . By definition of t ′ , we have t ′ ∩ X = v ∩ X.Since (t 2 , t 3 ) is a winning strategy for players 2 and 3 in H, we have (t ′ 1 , t 2 , t 3 ) |= ¬ϕ.By Lemma 43 v, (s 1 , χ 2 (t 2 ), χ 3 (t 3 ))(v) |= λ(¬ϕ).
In the context of Kw games, as an immediate consequence of Proposition 5 we get that the model checking question for the fragment L Kw is in polynomial time.
Corollary 49.Given α ∈ L Kw along with a strategy profile s ∈ S and a valuation v ∈ V , checking if v, s |= α is in PTIME.
We then have the following results for the complexity of verification and emptiness in Kw games.
Theorem 50.Given a Kw game G = (N, (P i ) i∈N , (γ i ) i∈N ) and a uniform strategy profile s ∈ S, checking if s ∈ NE max (G) is co-NP-complete.
Proof.Membership in co-NP follows immediately from Corollary 49.For hardness, we show a reduction from the corresponding verification problem in Boolean games which is: given a Boolean game B and a strategy profile v in B, to check if v ∈ NE (B).By Theorem 47 this problem is known to be co-NP-complete.
Given a Boolean game B and a strategy profile w in B, let G B and ṡ w be the corresponding observation game and the globally uniform strategy profile in G B as defined in Section 4.2.We argue that w ∈ NE (B) iff ṡ w ∈ NE max (G B ).
(⇒) This direction is exactly the same as the first part of the proof of Theorem 19.Suppose w ∈ NE (B) and ṡ w ∈ NE max (G B ). Then there exists i ∈ N, v ∈ V and t i ∈ S i such that u i (v, (t i , ṡ w −i (v))) > u i (v, ṡ w (v)).Let w ′ = χ −1 ([ ṫi , s −i ]) From Lemmas 14, 17 and 18 it follows that u B i (w ′ ) = u i (v, ( ṫi , From Lemma 14 we have that

Comparison to other epistemic Boolean games
Boolean games with epistemic goals [2].In 'Boolean games with epistemic goals' [2] the set of variables forms a partition into n mutually disjoint subsets of the variables that can only be controlled by the n respective players.This is as usual in Boolean games, and therefore also the strategies played.A strategy profile is therefore a valuation of all variables.However, the goals are different: these are not merely Boolean goals whose satisfaction depends on this valuation but are epistemic goals whose satisfaction depends on what the players know about this valuation.This is where other variables come into play: each player has a 'visibility set' of Booleans: those are the propositions whose value that player can observe of the outcome valuation.This seems to beg some questions on logical closure, for example if p ∧ q could be in the visibility set but neither variable p nor variable q (where we note that the epistemic goal formulas have the usual compositional semantics, so K i (p ∧ q) is true if and only if K i p and K i q are true).However, a special case is when the visibility set consists of variables only, which [2] call atomic games, and this suffices for a comparison with our results.The visibility set determines what is known by the players and thus which epistemic goals are satisfied in a valuation.Because the players altogether control the value of all variables the game is not one of incomplete information (strategies do not depend on an unknown initial valuation) although it is one of imperfect information (over the outcome valuation).The authors then determine that model checking goal formulas is PSPACE-complete and that the existence of Nash equilibria is in PSPACE, although they do not show a lower bound.They also provide an interesting embedding of their epistemic Boolean games into the standard Boolean games by observing that an epistemic goal corresponds to an exponentially larger Boolean goal that is the disjunction of all valuations over which the epistemic goal is uncertain.For example, in some given game K i p may abbreviate (p ∧ ¬q) ∨ (p ∧ q).This is therefore a rather different embedding from our embedding of knowing-whether Boolean observation games into Boolean games wherein the goals remain the same but the set of variables (and thus valuations) is larger: we recall that a Kw game G for variables p i is transformed into a Boolean game B G for variables Kw j p i : the knowing-whether formulas are now considered atomic propositions.
The goals remain the same in our approach, because knowing-whether goals are Booleans in the language wherein Kw j p i are atomic propositions.
Epistemic Boolean games based on a logic of visibility and control [32].'Epistemic Boolean games based on a logic of visibility and control' [32] presents an expressive logical language and semantics for players controlling the value of propositional variables or observing the value of propositional variables.They also axiomatize this logic.They then use the logic to formalize game theoretical primitives, in particular the existence of equilibria, in an epistemic extension of Boolean games.This formalization allows them to determine the complexity of these games.The problems of determining whether a profile is Nash equilibrium as well as the existence of Nash equilibrium are both in PSPACE.Their language extension includes knowledge, common knowledge, and for control or observation of propositional variables they propose additional propositional variables.We not only have, for example, a variable p, but also S i p, for 'player i observes the value of p' and C i p for 'player i controls the value of p'.But also variables like C j S i p, for 'player j controls whether player i observes p', and so on for any stack of C j or S i predicates.The interest of these complex propositional variables is that they induce relational Kripke models or can be used to formalize strategies in Boolean games.
In the epistemic Boolean games of [32] the strategies assign values to variables that are stacks of S i binding some atom p (so without any C i or K i ), as in S i S j p, saying that i can see whether j can see the value of p, whereas the goals are epistemic formulas in the language for such atoms p (so without S i or C i ), as in K i p∧¬K j p. 6 One might say that their epistemic Boolean games essentially remain Boolean games, because the players still only control the value of variables, but this is only by a (quite smart!) stretch of the modeling imagination, because their Boolean variables hard-code arbitrarily complex higher-order multi-agent observations.However, these are not games of incomplete information.
The focus of [32] is the axiomatization of their logic of visibility and control (it also contains program modalities with primitive operations assigning values to variables).The game-theorical contribution is mainly 'proof of concept'.
Public announcement games [4].'Public announcement games' [4] and the related 'question answer games' [3] also present incomplete games of imperfect information.Expected outcomes are compared with the realist outcome relation.The value of variables is not controlled in any way in [4,3], the valuations are fixed.Public announcement games are not Boolean games, because the players' strategies are revelations of any formula, not merely of Booleans.One could consider a class of public announcement games wherein the strategies are restricted to announcing propositional variables only.We recall that public announcements are revelations of the same information to all players simultaneously.

Conclusions
We introduce Boolean observation games as a qualitative model which combines aspects of imperfect and incomplete information games.We study (mainly) three notions of Nash equilibrium based on outcome relations that compare sets of outcomes.Our main technical contributions include results for the existence of Nash equilibria, the computational analysis of Nash equilibria, as well as identifying knowing whether games, the fragment of observation games that precisely corresponds to Boolean games in terms of existence of Nash equilibria.A summary of our results are listed in Table 3.
There are many directions for further research.One could imagine a whole and ever widening range of qualitative incomplete information games of imperfect information.For the strategies, instead of merely revealing the value of propositional variables, we could consider revealing the value of any epistemic proposition (as already considered in [4,3] for the more complex Kripke models).Instead of having merely partitions (exhaustive and exclusive) of all variables, one could consider overlapping sets of variables (exhaustive but not exclusive, so more than one player may observe the same variables). 7Doing the same 6 As S i stacks are arbitrarily long, there is an infinite set of such atoms to consider.However, the partition among players controlling variables is of a finite subset only of that infinite set.This permits S i p but not p to be in that finite subset, which would rule out to determine the value of a goal K i p (as no player gives a value to p, that is, no player controls p).In their accompanying examples, the finite subset jointly controlled by all players is always subformula closed.This therefore seems an omitted requirement.for Boolean games would create the possibility of conflict, as not more than one player can control the value of a variable.But as many agents as you wish can make the same observation.
Another interesting extension to explore would be to consider iterated observation games, wherein players can gradually reveal more and more of their variables.This would be a generalization similar to that already studied for Boolean games in [24,23].It would involve epistemic temporal goals (or dynamic epistemic goals).Different from iterated Boolean games, in iterated Boolean observation games one can only reveal more and more variables in every round, until all have been revealed.
Yet another relevant direction is (epistemic) incentive engineering in Boolean observation games, similar to what is studied in Boolean games [48,42,28].final statement, v is arbitrary.Therefore, the initial statement v, s |= Kw i p j is equivalent to "for all w ∈ V , w, s |= Kw i p j ," in other words, to s |= Kw i p j .
Case negated atom: v, s |= ¬Kw i p j , iff (there are w, x ∈ V with w ∼ s v and x ∼ s v and such that w, s |= p j and x, s |= ¬p j ), iff (there are w, x ∈ V with w ∩ P i (s) = x ∩ P i (s) = v ∩ P i (s) and such that w, s |= p j and x, s |= ¬p j ), iff p j / ∈ P i (s).As in the previous case, the final statement is independent from v and therefore the initial statement is equivalent to s |= ¬Kw i p j .
Case conjunction:

A.2 Strategies as epistemic actions
In this section we compare our modelling and our results with related work in epistemic logic.We model strategy profiles as epistemic actions in a dynamic epistemic logic, where we also discuss an alternative semantics of strategies resulting in far larger models.The alternatives can be compared on their game theoretical implications, which may help to motivate our preference.
The situation wherein each player only observes the value of its own variables, corresponds to a Kripke model where the accessibility relation is the initial observation relation, and a strategy profile corresponds to an action model that, when executed in this Kripke model, results in an updated model wherein the accessibility relation is the observation relation (for that strategy profile).In this section we make precise how.It may serve to illustrate that our setting is very simple.This was why we were able to obtain modelling and computational results for Boolean observation games that are close or analogous to those for Boolean games.
An epistemic model (Kripke model) M is a triple (W, ∼, π) where W is an (abstract) domain of worlds or states, where ∼ is a collection of equivalence relations on W , one for each agent, denoted ∼ a (also known as indistinguishability relations), and where π is a valuation (function) mapping each state w ∈ W to the subset of the propositional variables P that are true in that state.A pointed epistemic model (M, w) is a pair consisting of an epistemic model and a state w ∈ W . Now consider the situation in our observation games where each of n players 1, . . ., n only observes the value of its own variables P i , but before they enact/play a strategy s i .We have implicitly modelled this as the strategy profile s ∅ wherein no player reveals any variable.We can identify this situation with the following epistemic model.
The initial observation model (IM, v), where IM = (V, ∼, π), is such that: • domain V is the set of valuations of P (V = P(P )); • for each player i ∈ N and valuations v, w ∈ V , v ∼ i w iff v ∩ P i = w ∩ P i ; • for each v ∈ V , π(v) = v.
Note that the relations are exactly as in interpreted systems [22].
Similarly, the result of playing strategy profile s ∈ S given valuation v ∈ V of observed variables, corresponds to an updated epistemic model.
The observation model (IM s , v), where IM s = (V, ∼ s , π), is such that V and π are as for IM, whereas in this case v ∼ s i w iff v ∩ P i (s) = w ∩ P i (s).We recall that P i (s) = {p ∈ P | there is a j ∈ N with p ∈ s j (i)}, the variables revealed to i in s, where by definition P i (i) = P i so that always P i ⊆ P i (s).
Surely more interestingly, we can model a strategy profile as an independent semantic primitive namely as an action model U such that v, s |= ϕ iff IM ⊗ U, (v, s) |= ϕ where the former is the satisfaction relation in our logical semantics for L K and the latter is the satisfaction relation in action model semantics.In order to establish that we first need to define action models and their execution (following details as in [10,46,37]).
An action model U is a triple (E, ≈, pre) where E is a domain of actions, for each player i = 1, . . ., n, ≈ i is an equivalence relation on E, and pre is a precondition function mapping each action e ∈ E to an executability precondition pre(e) that is a formula in some logical language L. The execution of an action model in an epistemic model M = (W, ∼, π) is then defined as the restricted modal product M ⊗ U = (W ′ , ∼ ′ , π ′ ) where W ′ = {(w, e) | w ∈ W, e ∈ E, M, w |= pre(e)}, where (w, e) ∼ ′ i (w ′ , e ′ ) iff w ∼ i w ′ and e ≈ i e ′ , and where π ′ (w, e) = π(w).
In the case of strategy profiles for observation games, the logical language of action model preconditions can be restricted to L B , the Booleans (the language required to describe preconditions is therefore simpler than the language L K to describe epistemic goals), and a rather simple action model corresponds to a strategy profile s.A strategy profile can be identified with the following action model.In the definition, δ v ∈ L B is the description of the valuation v, defined as δ v := p∈v p ∧ p / ∈v ¬p.A strategy profile action model U s is a triple (V, ∼ s , pre) where the set of actions is the set of valuations V , where for each i = 1, . . ., n, v ∼ s i w iff v ∩ P i (s) = w ∩ P i (s), and where for each action v ∈ V , pre(v) = δ v .
The domain of the strategy profile action model is therefore the same as the domain of an observation model, namely the set of all valuations.
In can be verified that IM ⊗ U s is isomorphic to IM s .This is fairly elementary.We note that each action can only be executed in a single world -IM, v |= δ v , so that the size of IM s is the same as the size of IM.Then, (v, v) ∼ i (w, w) iff, by definition of action model execution, v ∼ i v (in IM) and v ∼ s i w (in U s ), iff, by definition of these relations, v ∩ P i = w ∩ P i and v ∩ P i (s) = w ∩ P i (s).As the latter is a refinement of the former, the desired result that v ∩ P i (s) = w ∩ P i (s) follows.Finally, π ′ (v, v) = π(v) = v.And the valuations π do not change.
In fact, already U s is isomorphic to IM s (slightly abusing the notion, but when we identify valuations with their description).It should be noted that it is common that action models are isomorphic to updated models when executed in initial models consisting of all valuations (and representing some sort of initial maximal ignorance over those valuations).
As a word of warning: the 'actions' that are the points in our action model U s do not correspond to the strategies, that are sometimes also called actions.The action model 'action' combines the strategies of all players simultaneously, so they rather correspond to strategy profiles.
More succinct action models.A slightly more succinct modelling of strategy profiles as action models is conceivable, that is a quotient of the action model U s defined above with respect to variables that are not revealed by any player.Let us call this set P s , that is therefore defined as the complement of the set P s := {p ∈ P | ∃i, j ∈ [1..n], i = j, p ∈ s i (j)}.We can now redefine U s small as (P(P s ), ∼ s , pre) where in this case for any v, w ⊆ P s (so for partial valuations of atoms revealed by some agent only), v ∼ i w iff v ∩ P i (s) = w ∩ P i (s).This looks the same as before, but note that P i (s) may involve far more variables, namely in P s , than v and w, that are both restricted to P s .Also, still pre(v) = v for all v ∈ P s (and where pre(∅) = ⊤ in case P s = ∅).
Again, it is elementary to show that IM ⊗ U s small is isomorphic to IM s .We now have that IM, w |= pre(v) iff v ⊆ w.But in this case U s small is typically smaller than the resulting updated model IM s .The resulting IM s , as before, has the same domain as the initial model IM.
We now have, for example, that the action model corresponding to the 'reveal nothing' strategy profile s ∅ is the trivial singleton action model U s ∅ small with precondition ⊤ (as P s ∅ = ∅), and in this case IM ⊗ U s ∅ small is isomorphic to the initial observation model IM again: the relations ∼ i have not changed.

A.3 Strategies for weaker observations give bigger models
In our modelling, it is common knowledge to all players what variables have been revealed by who and to whom: the strategy profile s is common knowledge 'after the fact'.But, although I therefore know what variables are revealed by other players to yet other players, I still have not learnt the values of these variables.
For example: After player 1 reveals atom p 1 to player 2 and atom q 1 to player 3, player 2 knows whether p 1 and player 3 knows whether q 1 .Also, player 2 knows that player 3 knows whether q 1 , and player 3 knows that player 2 knows whether p 1 .
In a different modelling, each player only learns what variables have been revealed by other players to herself, and what variables she reveals to others.
For example: After player 1 reveals atom p 1 to player 2 and atom q 1 to player 3, player 2 knows whether p 1 and player 3 knows whether q 1 .However, player 2 does not know that player 3 knows whether q 1 , and player 3 does not know that player 2 knows whether p 1 .Player 2 also considers it possible that no variable has been revealed to 3, in which case 3 does not know whether q 1 .And similarly for player 3.
So, clearly, depending on which modelling one prefers, different goal formulas γ of observation games may be satisfied, and it will therefore affect the existence of Nash equilibria and what the optimal strategies are.
Let us first formalize this as an action model, and let us be explicit about the (rather different) updated model as well.The strategies s i and profiles s = (s 1 , . . ., s n ) remain the same, and thus also the P i (s), the set of atoms revealed to agent i.However, we can no longer define an updated observation model as one wherein only the indistinguishability relations have been changed, namely as v ∼ s i w iff v ∩ P i (s) = w ∩ P i (s), while keeping the domain (and the valuation).
Instead of models consisting of valuations (domain V ) we now need much larger models consisting of pairs (v, t) for valuations v and profiles t (domain V × S) and define: For all v, v ′ ∈ V and for all s, t, t ′ ∈ S and for all players i ∈ N: (v, t) ∼ s i (v ′ , t ′ ) if v ∩ P i (s) = v ′ ∩ P i (s) [same valuation inasfar observed], t i = t ′ i = s i [same variables revealed to others], and P i (s) = P i (t) = P i (t ′ ) [same variables revealed by others to you].
As a consequence, we cannot describe the initial observation model as the one wherein s ∅ is executed, because that would still blow up the model and introduce maximal uncertainty about what is revealed by who.So the initial observation model IM needs to be given separately (namely as the model already defined in Appendix A.2).However, once this is done, that is all.An action model can also be given for this modelling.
In this alternative modelling the players would remain far more ignorant about other players: optimist expected outcome would be more optimist, pessimist expected outcome would be more pessimist, realist expected outcome would quantify over a far larger set of possible outcomes.Basically, any epistemic feature is diluted.It therefore appeared to us that our preferred modelling provides more interesting results and variations.
Beyond that, the envisaged iterated Boolean observation games would become less meaningful for such strategies encoding weaker observations, as a player remains unaware of other players' increasing knowledge over such iterations, unless as a consequence of that player informing those other players.

B Representation and complexity
Theorem 35.Given α ∈ L K along with a strategy profile s ∈ S and a valuation v ∈ V , checking if v, s |= α is PSPACE-complete.

2
Four outcome relations.It remains to define the outcome relation.We propose four.optimist : Observation.Both B and G B are defined over the same set of players and variables.Note that for all i ∈ N, we have |β i | = O(|γ i |) where |β i | and |γ i | denote the size of (number of symbols in) β i and γ i respectively.Thus given B, the associated Kw game G B can be constructed in polynomial time.Example 13.We illustrate how to construct a Kw game G B from a Boolean game B. (We will not analyze the equilibria of the game, if any.) Consider

9 while
∃v ∈ V , ∃i ∈ W + \ Y (v), ∃s i , such that ∀w : w ∼ i v, ∀s W − \Z(w) , we have (s i , s W + \{i} (w), s Z(w) (w), s W − \Z(w) , s Wo (w)), w |= γ i do 10 ∀w : w ∼ i v, set s i (w) := s i ; Y (w) := Y (w) ∪ {i}; 11 while ∃v ∈ V , ∃i ∈ W − \ Z(v), ∃s i such that ∀w : w ∼ i v, ∀s W + \Y (w) , we have w) for all v ∈ V .Therefore w ∈ NE (B) which is a contradiction.(⇐)Suppose ṡ w ∈ NE max (G B ) and w ∈ NE (B).Then there exists i ∈ N and w Kw 3 p 2 .Therefore B G B has more variables than B. The constructions are not each other's converse.However, in order to realize the goals of B G B the players only need to assign a value to variables Kw i p j occurring in the goal formulas, so with respect to playing this game the extra variables do not play a role.After replacing Kw 2 p 1 by p 1 , etcetera for other variables ocurring in goal formulas, we recover the original Boolean game.

Table 2 :
Uniform strategies for player 1. Explanations are given in the text.
).In this game, both goals are Kw formulas.We have |type(1)| = 4 and |type(2)| = 3.It can be verified that NE max (G) = ∅.For an observation gameG = (N, (P i ) i∈N , (γ i ) i∈N ), let |N| = n, |P | = k and max i∈N |γ i | = m (where |γ i | denotes the number of symbols in γ i ).For i ∈ N, every strategy s i : N → P(P i ), can be represented in size O(nk).That is, both observation games and strategies have compact representations -linear in n, k and m.On the other hand, each uniform strategy s i : V → S i can be encoded as a tuple of Boolean functions (s j i (p i )) j∈N,p i ∈P i where each s j i (p i ) : P(P ) → {⊤, ⊥}.
* −i (w)) This implies that ṡ w ∈ NE max (G B ) which is a contradiction.Theorem 51.Given a Kw game G, checking if NE max (G) = ∅ is Σ p 2 -complete.Proof.Membership in Σ p 2 follows immediately from Corollary 16 and Theorem 50.For Σ p 2 -hardness, notice that the translation from observation games to Boolean games that we provide in Section 4.3 is polynomial time computable.Thus given an instance of an observation game G, we can construct a Boolean game B G in polynomial time.By Theorem 24, NE max

Table 3 :
Summary of results 7Kindly suggested by Paul Harrenstein.