Abstract
We present a new notion of game equivalence that captures basic powers of interacting players. We provide a representation theorem, a complete logic, and a new game algebra for basic powers. In doing so, we establish connections with imperfect information games and epistemic logic. We also identify some new open problems concerning logic and games.
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Acknowledgments
We thank audiences in Amsterdam and at TARK 2017 Liverpool for their feedback, in particular, Valentin Goranko and Paolo Turrini. We also thank the two referees of this paper for their careful and helpful comments.
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Appendices
Appendix A: Selected Game-Theoretical Definitions
Definition A.1
Let a tree \(\mathcal {T}\) over a given set Σ be a non-empty and prefix closed subset of the set Σ∗ of finite words over the alphabet Σ. In particular the empty word ε belongs to every tree and is called the root of the tree.
Standard concepts like branches and leaves of trees are defined as usual.
Definition A.2
An extensive game \(\mathcal {G}\) for a finite set of players A with outcomes in the set O is a tuple \((\mathcal {T},t,o,{\Pi })\) where \(\mathcal {T}\) is a tree over some set Σ, t a map from non-leaf nodes of \(\mathcal {T}\) to A, o a map from branches of \(\mathcal {T}\) to O, and π a partition of \(\mathcal {T}\) subject to the following condition: for any pair w, v within the same partition cell of π, we have t(n) = t(n′) and for all i ∈Σ:
If all partition cells of π are singletons we call \(\mathcal {G}\) a game of perfect information, and we omit π.
Maximal branches of \(\mathcal {T}\) will also be called full matches, and prefixes of maximal branches are called partial matches.
A strategy for player a ∈ A is a map σ : t− 1[a] →Σ where w ⋅ σ(w) is a child of w for each w with t(w) = a, and σ(w) = σ(w′) whenever w, w′ are in the same partition cell in π. A strategy profile is a tuple (σa)a∈A of one strategy for each player in A. A strategy profile p determines a unique full match guided by the strategies of each player, so we can speak of the outcome of the profile p and denote it by o(p). Generally, we say that a full match m of \(\mathcal {G}\) is guided by the strategy σ for a if for every prefix w of m such that t(w) = a, σ(w) is also a prefix of m. Match(σ) is the set of σ-guided matches.
\(\mathbb {G}(A,O)\) is the set of games for players A with outcomes in O. For two-player games we call the players E and A. We set \(\overline {\mathsf {E}} = \mathsf {A}\) and \(\overline {\mathsf {A}} = \mathsf {E}\).
We have not attributed payoffs to matches in a game or preferences over the outcomes, but rather (and more generally) outcomes from some fixed set. In this sense we are dealing with game forms rather than proper games.
Definition A.3
Let \(\mathcal {G} = (\mathcal {T},t,o,{\Pi })\) be a game with outcomes in O. A set P ⊆ O is a power of player a ∈ A in the game \(\mathcal {G}\) if there is a strategy σ for a in G such that o(m) ∈ P for every σ-guided match m. Given a player a ∈ A we let \(P_{a}(\mathcal {G})\) denote the set of powers of a in \(\mathcal {G}\).
Two games \(\mathcal {G}_{1},\mathcal {G}_{2} \in \mathbb {G}(A,O)\) are power equivalent, \(\mathcal {G}_{1} \sim \mathcal {G}_{2}\), if for all a ∈ A: \(P_{a}(\mathcal {G}_{1}) = P_{a}(\mathcal {G}_{2})\). If \(P_{a}(\mathcal {G}_{1}) = P_{a}(\mathcal {G}_{2})\) for some specific a ∈ A, we write \(\mathcal {G}_{1} \sim _{a} \mathcal {G}_{2}\).
Appendix B: Soundness of Principles for Basic Powers
Proof of Fact 4.1
For Non-Emptiness, pick a player P. By definition of an extensive game, the set of strategies available to each player is non-empty: in particular, if t− 1[P] = ∅ then the empty function is the unique available strategy for P. So pick an arbitrary strategy σ for P. Then the set {o(m)∣m ∈Match(σ)} is a basic power for P.
For Consistency, let P be a basic power for E and Q a basic power for A. Then there are strategies σ, τ for E, A respectively such that P = {o(m)∣m ∈Match(σ)} and Q = {o(m)∣m ∈Match(τ)}. Hence o(p) ∈ P ∩ Q for the strategy profile p = (σ, τ).
Finally, for Exhaustiveness, suppose that P is a basic power of Player P and that x ∈ P. Pick a strategy σ for P such that P = {o(m)∣m ∈Match(σ)}. Since x ∈ P, there is a σ-guided match m such that o(m) = x. We define a strategy τ for Player \(\overline {\mathsf {P}}\) as follows: if a position \(u \in t^{-1}[\overline {\mathsf {P}}]\) belongs to m then u cannot be an endpoint of he game, since t(u) is defined, and hence, because m is a full match, there must be some unique child v of u such that v also belongs to m. Set τ(u) = v. If \(u \in t^{-1}[\overline {\mathsf {P}}]\) does not belong to m then define τ(u) arbitrarily. Then m is a τ-guided match, so x = o(m) ∈ Q where Q = {o(m)∣m ∈Match(τ)}, which is a basic power of \(\overline {\mathsf {P}}\). □
Appendix C: Completeness of Dynamic Game Logic
We denote the axiom system in Section 8.1 by Ax2 and write Ax2 ⊩ φ to say that formula φ is provable in this axiom system. We also write φ ⊩Ax2ψ for Ax2 ⊩ φ → ψ. We sometimes drop the reference to Ax2 to keep notation cleaner.
Proposition C.1 (Soundness)
IfAx2 ⊩ φ,thenφis valid on all neighborhood models for games with forcing relations as defined earlier.
Proof
We consider only soundness of the reduction axioms. Soundness of the axiom for dual − is immediate from the forcing definition for game dual reversing the roles of the players: in fact, the axiom encodes that definition. The same is true for the initial choice operation 〚∪〛 whose axiom is immediate from the game construction. For the dual construction 〚∩〛, we reason as follows.
Suppose that \(\mathfrak {M},w \Vdash \langle \pi _{1}\cap \pi _{2}\rangle ({\Psi };\varphi )\). Then there is some set Z such that \((w,Z) \in R_{\pi _{1} \cap \pi _{2}}\), Z ⊆ 〚φ〛 and Z ∩ 〚ψ〛 ≠ ∅ for all ψ ∈Ψ. Hence Z is of the form Z1 ∪ Z2 where \((w,Z_{1}) \in R_{\pi _{1}}\) and \((w,Z_{2}) \in R_{\pi _{2}}\). Let Θ1 = {ψ ∈Ψ∣Z1 ∩ 〚ψ〛 ≠ ∅}, and let Θ2 = {ψ ∈Ψ∣Z2 ∩ 〚ψ〛 ≠ ∅}. Then, since Z = Z1 ∪ Z1, we have Ψ = Θ1 ∪Θ2. Furthermore, we get
as required. The converse direction is proved in a similar manner.
Next, we consider sequential composition. For one direction of the equivalence, suppose that \(\mathfrak {M},w \Vdash \langle \pi _{1} ; \pi _{2}\rangle (\psi _{1},...,\psi _{n}; \varphi )\). Then there is some set Z with \((w,Z) \in R_{\langle \pi _{1} ; \pi _{2}\rangle }\), Z ⊆ 〚φ〛 and Z ∩ 〚ψi〛 ≠ ∅ for each ψi. By definition of the composition operator, we find a set Y with \((w,Y) \in R_{\pi _{1}}\) and a family of sets F such that \((Y,F) \in \overline {R}_{\pi _{2}}\) and \(Z = \bigcup F\). So for each v ∈ Y there is some Sv ∈ F with \((v,S_{v})\in R_{\pi _{2}}\), and we get Sv ⊆ 〚φ〛 so \(\mathfrak {M},v \Vdash \langle \pi _{2}\rangle \varphi \). Also, for each ψi there is some Si ∈ F with Si ∩ 〚ψi〛 ≠ ∅, and there must be some v ∈ Y with \((v,S_{i}) \in R_{\pi _{2}}\), hence \(\mathfrak {M},v \Vdash \langle \pi _{2}\rangle (\psi _{i};\varphi )\). It follows that \(\mathfrak {M},w \Vdash \langle \pi _{1}\rangle (\langle \pi _{2}\rangle (\psi _{1};\varphi ),...,\langle \pi _{2}\rangle (\psi _{n};\varphi );\langle \pi _{2}\rangle \varphi )\) as required.
Conversely, suppose that \(\mathfrak {M},w \Vdash \langle \pi _{1}\rangle (\langle \pi _{2}\rangle (\psi _{1};\varphi ),...,\langle \pi _{2}\rangle (\psi _{n};\varphi );\langle \pi _{2}\rangle \varphi )\). Then there exists some set Y such that \((w,Y) \in R_{\pi _{1}}\), Y ⊆ 〚〈π2〉φ〛 and Y ∩ 〚〈π2〉(ψi; φ)〛 ≠ ∅ for each i ∈{1,..., n}. Let:
Since Y ⊆ 〚〈π2〉φ〛, we get \((Y,F) \in \widetilde {R}_{\pi _{2}}\), so \((w,\bigcup F) \in R_{\pi _{1} ; \pi _{2}}\). Furthermore, since Y ∩ 〚〈π2〉(ψi; φ)〛 ≠ ∅ for each i ∈{1,..., n}, it follows that \(\bigcup F \cap [\![{\psi _{i}}]\!] \neq \emptyset \) for each i ∈{1,..., n}. Hence \(\mathfrak {M},w \Vdash \langle \pi _{1} ; \pi _{2}\rangle (\psi _{1},...,\psi _{n}; \varphi )\) as required. □
Next, by applying the sound reduction axioms, a standard argument shows that, for every consistent formula φ of there is a provably equivalent equivalent formula φt in INL, which is then satisfiable by Theorem 6.1.
The preceding argument establishes the following result.
Theorem C.2 (Completeness)
A formulaφofis valid on all neighborhood models iffAx2 ⊩ φ.
Also, the finite model property and decidability carry over from INL:
Theorem C.3
The logic has the finite model property and is decidable.
This simple completeness argument no longer works with iteration, cf. [3].
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van Benthem, J., Bezhanishvili, N. & Enqvist, S. A New Game Equivalence, its Logic and Algebra. J Philos Logic 48, 649–684 (2019). https://doi.org/10.1007/s10992-018-9489-7
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DOI: https://doi.org/10.1007/s10992-018-9489-7