Strategy Composition in Dynamic Games with Simultaneous Moves

Sujata Ghosh

, Neethi Konar

and R. Ramanujam

Indian Statistical Institute, Chennai, India

Institute of Mathematical Sciences, Chennai, India

Keywords:

Dynamic Games, Simultaneous Moves, Repeated Normal Form Games, Top-down Strategizing, Strategy

Logic.

Abstract:

Sometimes, in dynamic games, it is useful to reason not only about the existence of strategies for players, but

also about what these strategies are, and how players select and construct them. We study dynamic games

with simultaneous moves, repeated normal form games and show that this reasoning can be carried out by

considering a single game, and studying composition of “local” strategies. We study a propositional modal

logic in which such reasoning is carried out, and present a sound and complete axiomatization of the valid

formulas.

1 INTRODUCTION

Game theory provides a normative view of what ra-

tional players should do in decision making scenarios

taking into account their beliefs and expectations re-

garding other players’ (decision-makers’) behaviour.

In this sense, the reasoning involved is more about

games, rather than within the games themselves. The

entire game structure is known to us, and we can

predict how rational players would play, so that we

can ask whether rational play could / would result in

equilibrium strategy proﬁles, whereby players would

not deviate from such strategic choices. Strategies of

players in a game are assumed to be complete plans of

actions that prescribe a unique move at each playable

position in the game.

When games are ﬁnite, and strategies are complete

plans, each player has only ﬁnitely many strategies to

choose from and the study of normal form games sim-

ply abstracts strategies and sets of choices and studies

the effect of each player making a choice simultane-

ously. Probabilistic (or mixed strategies) become the

focus of such a study, and equilibrium theory then as-

signs probabilities to expected modes of player be-

haviour.

Even if we retain the structure of games, and

study them as trees of possible sequences of player

moves, as in extensive form games, a backward induc-

tion procedure (BI procedure (Osborne and Rubin-

stein, 1994)) can be employed to effectively compute

optimal strategies for players, leading to predictions

of stable play by rational players. Questions of how

players may arrive at selecting such strategies and

playing them, and their expectations of other players

symmetrically choosing such strategies are (rightly)

glossed over.

And, when the moves are simultaneous, for ex-

ample, in the case of inﬁnitely repeated normal form

games, even though there exists an equilibrium strat-

egy (the grim strategy) (Rasmusen, 2007), Folk theo-

rem tells us that under certain reasonable conditions,

the claim that a particular behaviour arises is mean-

ingless in such a game.

1.1 Strategies as Partial Plans

For dynamic games with simultaneous moves, one

can think of both individual and group strategies, and

the BI-procedure works bottom up on the game tree,

and assumes players who have the computational and

reasoning ability required to work it out (and hence

each player can assume that other rational players do

likewise). If the game is ﬁnite but consists of a tree

of large size, such an assumption is untenable. Strate-

gizing during play (rather than about the entire game

tree) is meaningful in such situations and we need to

consider players who are decisive and active agents

but limited in their computational and reasoning abil-

ity.

Unlike the BI-procedure, strategizing during

The notion of players whose rationality is also limited

in some way is interesting but more complex to formal-

ize; for our considerations perfectly rational but resource

bounded players sufﬁce.

624

Ghosh S., Konar N. and Ramanujam R.

Strategy Composition in Dynamic Games with Simultaneous Moves.

DOI: 10.5220/0006205106240631

In Proceedings of the 9th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2017), pages 624-631

ISBN: 978-989-758-220-2

play follows the ﬂow of time and hence works top

down. Hence, unless a player has access to the entire

subtree issuing at a node, she cannot compute opti-

mal strategies, however well she is assured of their

existence. It is in fact for this reason that although

the determinacy of chess was established by Zermelo

(Zermelo, 1913) the game remains fascinating to play

as well as study even today.

Indeed, resource limited players working top

down are forced to strategize locally, by selecting

what part of the past history they choose to carry in

their memory, and how much they can look ahead in

their analysis. In combinatorial games, complexity

considerations dictate such economizing in strategy

selection. Predicting rational play by resource limited

players is then quite interesting.

When game situations involve uncertainty, as in-

evitably happens in the case of games with large struc-

ture or large number of players, such top down strate-

gizing is further necessitated by players having only

a partial view of not only the past and the future but

also the present as well. Once again, we are led to

the notion of a strategy as something different from

a complete plan, something analogous to a heuris-

tic, whose applicability is dictated by local obser-

vations of game situations, for achieving local out-

comes, based on expectations of other players’ locally

observed behaviour. The notion of locality in this de-

scription is imprecise, and pinning it down becomes

an interesting challenge for a formal theory.

As an example, consider a heuristic in chess such

as pawn promotion. This is generic advice to any

player in any chess game, but it is local in the sense

that it fulﬁls only a short term goal, it is not an advice

for winning the game. A more interesting example

is the heuristic employed by the computer Deep Blue

against Gary Kasparov (on February 10, 1996) threat-

ening Kasparov’s queen with a knight (in response to

Kasparov’s 11th move). The move famously slowed

down Kasparov for 27 minutes, and was later hailed

as an important strategy.

The point is that such

strategizing involves more than “look-ahead”.

1.2 Strategy Selection

The foregoing discussion motivates a formal study

of strategies in extensive form games, where we go

beyond looking for existence of strategies for play-

ers to ensure desired outcomes, but take into ac-

count strategy structure and strategy selection as well.

This work was initiated in (Ramanujam and Simon,

2008b), and we extend it to incorporate extensive

https://en.wikipedia.org/wiki/Deep Blue versus

Kasparov, 1996, Game 1

form games with simultaneous moves, so that they too

can be studied through ‘top-down strategizing’. Re-

lating strategy structure with game structure was in-

dependently taken up in (Ghosh, 2008) and (Ramanu-

jam and Simon, 2008a), see (Ghosh and Ramanujam,

2012) for a survey of such work.

When we reason about the existence of strategies

for players to ensure desired outcomes, we can sim-

ply formalize the collection of strategies as a set and

a rational player can be depended upon to pick the

right one from the set and play it. In this case, rea-

soning about strategies amounts to assigning names

to strategies and we can speak of player A playing a

strategy σ to ensure outcome α from game position

s. Such a player, in effect, chooses between outcomes

and we reason about players’ consideration of choices

by other players.

In the case of a resource limited player who re-

lies on partial and local plans, choice of strategies

can be seen as composition of local plans to make

‘more global’ plans, and in this sense we can ascribe

structure to strategies. As an example, consider a

chess player who strategizes locally to capture either

a knight or a bishop, and makes further conditional

plans based on the success of either attempt, while

at the same time formulating backup plans to counter

unforeseen disasters along the way. In such a situ-

ation, each player can be seen as composing partial

strategies, and exercising selections at each stage of

the composition. Such strategy structure would then

include hypothesizing about partial strategies of other

players (as witnessed by their moves) as well. This

mutual recursion in strategy structure and selection

has been explicated in the logical study of (Ramanu-

jam and Simon, 2008b) and in this paper we extend

the approach to a class of games with simultaneous

moves.

1.3 Related Work

When an extensive form game is presented as a ﬁnite

or inﬁnite tree, strategies constitute selective quantiﬁ-

cation over paths. When every edge of the tree corre-

sponds to a normal form game, we obtain a concur-

rent game structure. The temporal evolution of such

structures is studied in the pioneering work on Alter-

nating time temporal logic (ATL) (Alur et al., 2002).

In this logic, one reasons about groups of players hav-

ing a strategy to force certain outcomes. Since the

game played at one node of the tree is essentially dif-

ferent from that at any other node in the tree, strategiz-

ing is local at any node in the basic framework. These

can be construed as subgames and hence ATL can be

also seen as a logic of game composition. Further,

Strategy Composition in Dynamic Games with Simultaneous Moves

625

named strategies can be introduced as in extensions

of ATL with explicit treatment of strategies (such as

in (Chatterjee et al., 2007; van der Hoek et al., 2005;

Walther et al., 2007;

Agotnes, 2006)). However, they

principally reason about the existence of functional

strategies in both normal form and extensive form

games. For a detailed survey, see (Bulling et al.,

2015).

Strategy composition arises from a different per-

spective. The point of departure here is in working

with the heuristic notion of strategies as partial plans,

and studying compositional structure in strategies. In

this sense the contrast of ATL to this work is akin to

that of temporal logics to process logics (which incor-

porate dynamic logic into temporal reasoning).

The stit frameworks of (Horty, 2001) work with

notions such as “agent sees to it that” a particular

condition holds, and automatically refers to agents

having strategies to achieve certain goals. Exten-

sive form game versions of stit have been discussed

in (Broersen, 2009; Broersen, 2010), where strate-

gies are considered as sets of histories of the game.

Each such history gives a full play of the game, and

hence, only total strategies are taken into account.

See (Broersen and Herzig, 2015) for a detailed sur-

vey. We note here that such reasoning may be relevant

for strategy logics with more detailed agency. Strate-

gies as move recommendations of players based on

game description language are considered in (Zhang

and Thielscher, 2015).

Our work, which is an extension of the work

done in (Ramanujam and Simon, 2008b), is closest in

spirit to logical studies on games such as (Benthem,

2002; Benthem, 2007), (Harrenstein et al., 2003),

(Bonanno, 2001). However, rather than formalizing

the notion of backward induction and its epistemic en-

richments, we study top down reasoning in the same

basic framework as in these logics.

An aspect that comes out of our studies of compo-

sitional strategy structures is the fact that we are able

to model players’ responses to other players while

playing a game, even in the case of games with simul-

taneous moves, which occurs naturally in repeated

normal form games. Thus one can study the differ-

ent strategic responses of the players leading to dif-

ferent outcomes. Simple games or strategies are com-

bined to form complicated structures which provide

a way to describe actual plays of a game - how a

player with limited resources, without knowing how

a game might proceed in the future, can actually go

about playing the game.

1.4 Contributions

Thus the main contributions of this paper are the fol-

lowing:

• We propose a logical language in which strate-

gies are partial plans, have compositional struc-

ture and we reason about agents employing such

strategies to achieve desired outcomes. The focus

is on games with simultaneous moves (also known

as concurrent games structures).

• We present a complete Hilbert-style axiomatiza-

tion of this logic. A decision procedure can be

extracted with some more work, as demonstrated

in (Ramanujam and Simon, 2008b).

The syntax for composing strategies presented

here is not proposed to be deﬁnitive, but merely

illustrative. When we build libraries of strategies

employed in games that programs modelling player

agents make use of, we will have a more realistic un-

derstanding of compositional structure in strategies.

We note here that plays in many popular dynamic

board games with simultaneous moves (e.g. Robo-

Rally) are actually based on heuristic strategizing and

local plans, and a library of strategies will aid the

game developers and the general game playing crowd

towards providing a better game-playing and strategy-

developing experience. In what follows we will use

the repeated normal form game, Iterated Prisoner’s

Dilemma (Rasmusen, 2007) to explicate the concepts

introduced.

2 PRELIMINARIES

We start with deﬁning dynamic games with simulta-

neous moves, the basic underlying structure for this

work, and what is meant by player-strategies in such

games. Let N = {1,2,. ..,n} be a non-empty set of

players. For each player i ∈ N, we associate a ﬁnite

set Γ

, the set of symbols which constitute player i’s

actions. Let

Γ = Π

i∈N

, the cartesian product of the

’s over all i ∈ N. Throughout the text we will denote

the elements of

Γ by γ. For each i ∈ N, let γ

−i

denote

the tuple (γ

,. .. ,γ

i−1

,γ

i+1

,. .. ,γ

), and

−i

denote the

set of all such γ

−i

’s.

In case of Iterated Prisoner’s Dilemma (IPD), we

have that N = {1, 2}. For each player i ∈ N, we as-

sociate a ﬁnite set Γ

= {c

(confess),d

(defect)}, and

Γ = Π

i∈N

Game Arena with Simultaneous Moves. A

game arena with simultaneous moves is a tu-

ple G = (W,→, w

,χ) such that W is the set of

game positions, w

is the initial game position,

ICAART 2017 - 9th International Conference on Agents and Artiﬁcial Intelligence

626

χ(w) = χ

(w) × ... × χ

(w), for any w ∈ W , with

: W → 2

\ {

0} giving the set of possible actions

that an agent can take when she is at a certain game

position, and the move function →: W ×

Γ → W

satisﬁes that for all w, v ∈ W , if w

−→ v, then

γ[i] ∈ χ

(w).

Extensive Form Game Tree. Given a game arena

G = (W,→, w

,χ), we can associate its tree unfold-

ing, also referred to as the extensive form game tree,

= (S,⇒,s

,X , λ), where (S, ⇒) is a tree rooted at

with edges labelled by members of

Γ, X : S → 2

and λ : S → W is such that:

- λ(s

) = w

- ∀s,s

∈ S,if s

=⇒ s

, then λ(s)

−→ λ(s

- if λ(s) = w and w

−→ w

, then ∃s

∈ S s.t. s

=⇒ s

and λ(s

) = w

- X(s) = χ(λ(s))

We have X

(s) = χ

(λ(s)). We deﬁne moves(s) = {γ ∈

Γ : ∃s

∈ S, such that s

=⇒ s

}, and moves(s)

is the

set of ith projections of the members of moves(s).

When we are given the tree unfolding T

of a

game arena G and a node s in it, we deﬁne the re-

striction of T

to s, denoted by T

to be the subtree

obtained by retaining the unique path from root s

s and the tree rooted at s.

Let us come back to our example. The game arena

in case IPD is a tuple G = (W,→,w

,χ) such that

W = {w}, w

= w is the initial and only game posi-

tion, χ(w) = χ

(w) × χ

(w), with χ

: w 7→ Γ

giving

the set of possible actions that an agent can take when

she is at a certain game position. The arena consists of

a single game state and four loops labelled by (c

), (d

) and (d

). The tree unfolding of

this arena gives us the IPD. Let us now deﬁne strate-

gies in the following.

Strategies. Let a game be represented by G = (W,→

,χ). Let T

be the tree unfolding of the game

arena G and s be a node in it. A strategy for player i at

node s, µ

is given by T

 µ

= (S

,⇒

,s, X

) which

is the subtree of T

containing the unique path from

root s

to s and is the least subtree satisfying the fol-

lowing property: ∀s ∈ S

,∃ a unique γ

∈ Γ

such that,

∀γ

−i

∈

−i

, ∀s

with s

(γ

,γ

−i

)

====⇒ s

, s

(γ

,γ

−i

)

====⇒

. X

the restriction of X to S

. The idea is that we pick

a single action for player i and all possible actions

for other players and consider those tuples of moves

corresponding to each game position in the subtree

rooted at s.

For example, the tuples of actions (c

) and (c

)

at each node of the IPD tree constitute a strategy tree

for player 1. Let Ω

denote the set of all strategies of

player i in G. Given a game tree T

and a node s in it,

let ρ

: s

=⇒ s

.. .

=⇒ s

= s denote the unique path

from s

to s. In what follows we restrict the number

n of elements in N to 2 for convenience. The ensuing

discussion and results would follow similarly (with

minor modiﬁcations) for any arbitrary n ≥ 2.

3 STRATEGY LOGIC

We now propose a strategy logic (SL) to reason about

such strategies and their compositions and to describe

what these strategies can ensure in the positions where

they are enabled. We give the syntax in two levels, as

in (Ramanujam and Simon, 2008b), the ﬁrst level con-

sists of the strategy speciﬁcation language, and the

second level provides a syntax for reasoning in games

with simultaneous moves.

3.1 Strategy Speciﬁcations

We ﬁrst provide a speciﬁcation language to describe

such strategies. These strategies can be given as ad-

vices from the point of view of an outsider advising

the players how to play in such games, and also can

be considered from the players’ perspectives at game

positions, given the facts that hold in such positions.

A propositional syntax with certain past formulas is

used to describe observations at game positions.

Observation Syntax. Let P be a countable set of

propositions. Then we deﬁne,

φ ∈ Φ : p | ¬φ | φ

∨ φ

| 3

−

where p ∈ P. Here, 3

−

φ denotes that φ has happened

sometime in the past and can be evaluated in terms of

ﬁnite sequences.

Speciﬁcation Syntax. For i = 1,2, the set of strategy

speciﬁcations, Strat

(P) is given by,

σ ∈ Strat

(P) : [φ 7→ γ

]

| σ

+ σ

| σ

· σ

| π ⇒

σ,

where, φ ∈ Φ, and π ∈ Strat

(P). Here

i = 2(1) if

i = 1(2).

The main idea is to use the above constructs to

specify properties of strategies as well as to combine

them to describe a play of the game, say. For in-

stance the interpretation of a player i’s speciﬁcation

[φ 7→ γ

]

where φ ∈ Φ, is to choose move γ

at every

game position where φ holds. At positions where φ

does not hold, the strategy is allowed to choose any

enabled move. The constructs ‘+’ and ‘·’ correspond

Strategy Composition in Dynamic Games with Simultaneous Moves

627

to ‘or’ and ‘and’ respectively. The strategy speciﬁca-

tion σ

+σ

says that the strategy of player i conforms

to the speciﬁcation σ

or σ

. The construct σ

· σ

says that the strategy conforms to speciﬁcations σ

and σ

. The strategy speciﬁcation π ⇒

σ says that if

player

i’s strategy conforms to the speciﬁcation π in

the history of the game, then play σ.

Semantics. Let V : S → 2

be a valuation. Then a

model is given by M = (T

,V ), where T

is an ex-

tensive form game tree (deﬁned in Section 2) and V

is a valuation. Let M be a model and s be a node in

it. Let ρ

: s

=⇒ s

.. .

=⇒ s

= s denote the unique

path from s

to s. We deﬁne for all k ∈ {0,. .., m},

- ρ

,k  p iff p ∈ V (s

- ρ

,k  ¬φ iff ρ

,k 2 φ,

- ρ

,k  φ

∨ φ

iff ρ

,k  φ

or ρ

,k  φ

- ρ

,k  3

−

φ iff there exits a j : 0 ≤ j ≤ k such that

, j  φ.

For any σ

∈ Strat

(P), we can deﬁne σ

(s) (the set of

player i actions at node s conforming to the strategy

speciﬁcation σ

) as follows:

([φ 7→ γ

]

)(s) =



{γ

} i f ρ

,m  φ and γ

∈ X

(s),

(s) otherwise;

(σ

+ σ

)(s) = σ

(s) ∪ σ

(s),

(σ

· σ

)(s) = σ

(s) ∩ σ

(s).

(π ⇒

σ)(s) =



σ(s) i f ∀ j : 0 ≤ j < m,γ

∈ π(s

)

(s) otherwise.

Here, X

(s) denotes the set of moves enabled for a

player i at s, γ

denotes an action of player

i, and π(s

)

denotes the set of actions enabled at s

for player

i by

the strategy speciﬁcation π.

Given a game tree T

and a node s in it, and a

strategy speciﬁcation σ ∈ Strat

(P), we deﬁne T

σ =

,⇒

,s, X

) to be the subtree of T

containing the

unique path from root s

to s and is the least subtree

satisfying the following property: ∀s ∈ S

,∃γ

∈ Γ

such that, ∀γ

∈ Γ

, ∀s

, s

=⇒ s

, where γ = (γ

,γ

) iff

∈ σ(s). X

is the restriction of X to S

In IPD one can consider strategies like always

cooperate or always defect, which can be easily de-

scribed in the syntax as follows:

- always cooperate - [> 7→ c]

- always defect - [> 7→ d]

Consider the following non-trivial strategy:

- Start by choosing d (defect)

- As long as the opponent cooperates, cooperate, or

as long as the opponent defects, defect

This strategy can be represented as follows:

[root 7→ d]

.(([> 7→ c]

⇒

[> 7→ c]

) + ([> 7→ d]

⇒

[> 7→ d]

)). Here, root can be considered as an atomic

formula, true only at the root node. A commonly

known strategy is given in the following. If both

players choose this strategy in the inﬁnite IPD, then

one can ﬁnd a simple perfect equilibrium (Rasmusen,

2007).

The Grim Strategy

- Start by choosing d

- Continue to choose d unless some player has cho-

sen c, in which case, choose c forever.

We will show how this Strategy Logic, SL can de-

scribe such strategies. Let us now provide a language

to reason with such strategies.

3.2 Logic

We are now ready to propose a logic, SL for describ-

ing strategic reasoning in games with simultaneous

moves.

SL syntax The syntax of SL is deﬁned as follows:

α ∈ Θ : p ∈ P | ¬α | α

∨α

| hγiα | hγiα | 3

−

α | σ

→

where, γ

∈ Γ

, σ

∈ Strat

(P), and φ ∈ Φ.

The formula hγiα says that after a γ move α holds,

and hγiα says that α holds before the γ move was

taken. The formula σ

asserts, at any game po-

sition, that the strategy speciﬁcation σ

for player i

suggests that the move γ

can be played at that posi-

tion. The formula σ

→

φ says that from this position,

there is a way of following the strategy σ

for player i

so as to ensure the outcome φ. These two modalities

constitute the main constructs of our logic.

The connectives ∧, ⊃ and the formulas [γ]α,

[γ]α are deﬁned as usual, α = ¬3

−

¬α, hNiα =

γ∈

hγiα, [N]α = ¬hNi¬α, hPiα =

γ∈

hγiα,

[P]α = ¬hPi¬α, root = ¬hPi>, enabled

∈Γ

h(γ

,γ

)i>, enabled

∈Γ

h(γ

,γ

)i>.

SL semantics Let M be a model as described in Sec-

tion 3.1 and s be a node in it. Let ρ

be s

=⇒

.. .

=⇒ s

= s, as earlier. The truth deﬁnition of

the SL formulas is given as follows:

- M, s  p iff p ∈ V (s),

ICAART 2017 - 9th International Conference on Agents and Artiﬁcial Intelligence

628

- M, s  ¬α iff M, s 2 α,

- M, s  α

∨ α

iff M, s  α

or, M,s  α

- M, s  hγiα iff ∃s

∈ S s.t. s

=⇒ s

and M,s

 α,

- M, s  hγiα iff m > 0, γ = γ

and M,s

m−1

 α,

- M, s  3

−

α iff ∃ j : 0 ≤ j ≤ m s.t. M,s

 α,

- M, s  σ

iff γ

∈ σ

(s),

- M, s ` σ

→

φ iff for all s

∈ T

σ

with s ⇒

∗

have M,s

 φ ∧ enabled

, where enabled

γ∈

(hγi> ∧ σ

The Grim Strategy (followed by player 1, say) de-

scribed earlier can be represented as follows: (root ⊃

(hd

i> ∨ hd

i>)) ∧ (hPi3

−

(([> 7→ c

]

+ [> 7→

]

) ⊃ (hc

i> ∨ hc

i>)). We now pro-

vide the main technical result of the paper, that is a

sound and complete axiomatization of SL.

4 A COMPLETE

AXIOMATIZATION

The following axioms and rules provide a sound and

complete axiomatization for SL. A proof sketch is

provided below.

- (A

) tautologies in classical propositional logic.

- (A

)(a)[γ](α

⊃ α

) ⊃ ([γ]α

⊃ [γ]α

)

(b)[γ](α

⊃ α

) ⊃ ([γ]α

⊃ [γ]α

)

- (A

)(a)hγiα ⊃ [γ]α

(b)hγiα ⊃ [γ]α

(c)hγi> ⊃ ¬hγ

i>, for all γ

6= γ

- (A

)(a)α ⊃ [γ]hγiα

(b)α ⊃ [γ]hγiα

- (A

)(a)3

−

root

(b)  α ≡ (α ∧ [P]  α)

- (A

)(a)enabled

≡ [φ 7→ γ

]

, γ

∈ Γ

(b)[φ 7→ γ

]

≡ ¬φ ∧ enabled

, γ

6= γ

- (A

)(a)(σ

+ σ

) :

≡ σ

∨ σ

(b)(σ

.σ

) :

≡ σ

∧ σ

(c)(π 7→

σ) :

≡ (((hγi> ⊃ hγi(π :

)) ⊃

(σ :

))∧(¬(hγi> ⊃ hγi(π :

)) ⊃ enabled

))

- (A

) σ

→

φ ⊃ (φ∧  ∧ enabled

), where 

is σ

⊃ [γ](σ

→

φ).

- (A

)(a)

∈Γ

enabled

(b)

∈Γ

enabled

Inference Rules

M.P.

α, α ⊃ β

[γ]α

α ⊃ [P]α

α ⊃ α

α ∧ σ

⊃

∈Γ

[γ

,γ

]α, α ⊃ enabled

. α ⊃ φ

α ⊃ σ →

α ∧ σ :

⊃

∈Γ

[γ

,γ

]α, α ⊃ enabled

. α ⊃ φ

α ⊃ σ →

The axioms (A

) and rules G

and G

show that

the modalities [γ] and [γ] are normal modalities. Ax-

ioms (A

) account for uniquely labelled moves. Ax-

ioms (A

) show that the modalities [γ] and [γ] are con-

verses of each other. Axioms (A

) and rule P take

care of the past modality. Axioms (A

) deal with the

atomic strategy advices, whereas axioms (A

) deal

with the complex ones. For example, axiom (A

(c))

says that the response speciﬁcation suggests the move

iff whenever the opponent strategy speciﬁcation π

suggests γ

in the history of the game, player i strat-

egy σ suggests γ

, and if not, γ

has to be enabled at

the game position. Finally, the (A

) axiom (if φ is

ensured under the strategy σ then φ holds, σ is en-

abled and if σ suggests γ

at the game position, then

after every γ move containing γ

, φ is ensured under

the strategy σ) and the I

and I

rules give the nec-

essary and sufﬁcient conditions for ensuring φ under

the strategy σ

, and (A

) ensures inﬁniteness of each

branch of the game tree. Note that we are assuming

game trees without leaf nodes. This is not a matter of

principle but rather of convenience. We can incorpo-

rate ﬁnite branches to our game trees, thereby having

leaf nodes and change the axiomatization accordingly.

The completeness proof will work.

The validities of these axioms and rules can be

checked routinely. We now provide a proof sketch

for the completeness.

4.1 Completeness

To prove completeness, we have to prove that every

consistent formula is satisﬁable. We mention all the

intermediate propositions and lemmas to prove the re-

sult without providing any detailed proof due to lack

of space.

Let α

be a consistent formula and let W denote

the set of maximal consistent sets (mcs). We use w, w

to range over mcs’s. Since α

is consistent, there ex-

ists an mcs w

such that α

∈ w

Strategy Composition in Dynamic Games with Simultaneous Moves

629

Deﬁne a transition relation on mcs’s as follows:

(γ

,γ

)

−−−→ w

iff {hγ

,γ

iα| α ∈ w

} ⊆ w. For a for-

mula α, let cl(α) denote the subformula closure of α.

In addition to the usual download closure, we also re-

quire that 3

−

root ∈ cl(α) and σ →

φ ∈ cl(α) implies

that φ,enabled

∈ cl(α).

Let AT denote the set of all maximal consistent

subsets of cl(α), referred to as atoms. Each t ∈ AT is

a ﬁnite set of formulas, we denote the conjunction of

all formulas in t by

t. For a nonempty subset X of AT ,

we denote by

X the disjunction of all

t, t ∈ X. Deﬁne

a transition relation on AT as follows: t

(γ

,γ

)

−−−→ t

iff

t ∧ hγ

,γ

is consistent. Call an atom t a root atom if

there does not exist any atom t

such that t

(γ

,γ

)

−−−→ t for

some (γ

,γ

) in

Γ. Note that t

= w

∩ cl(α

) ∈ AT .

Proposition 4.1 There exist t

,. .. ,t

∈ AT and

(γ

,γ

),. .. ,(γ

,γ

) ∈

Γ with (k ≥ 1) such that

(γ

,γ

)

−−−→ t

k−1

.. .

(γ

,γ

)

−−−→ t

, where t

is a root atom.

Lemma 4.2 For every t

∈ AT , the following are

equivalent:

∧ hγ

,γ

is consistent.

2) hγ

,γ

∧

is consistent.

Lemma 4.3 Consider the path t

(γ

,γ

)

−−−→

k−1

.. .

(γ

,γ

)

−−−→ t

where t

is a root atom. Then:

(i) For all j ∈ {0,··· ,k − 1}, if [γ

,γ

]α ∈ t

and t

j+1

(γ

,γ

)

−−−→ t

, then α ∈ t

j+1

(ii) For every j ∈ {0,·· · ,k − 1}, if

hγ

,γ

iα ∈ t

and t

j+1

(γ

,γ

)

−−−→ t

, then

(γ

,γ

) = (γ

,γ

) and α ∈ t

j+1

, and

(iii) For all j ∈ {0,· ·· ,k − 1}, if 3

−

α ∈ t

, then

there exists i such that j ≤ i ≤ k and α ∈ t

Thus there exist mcs’s w

,· ·· ,w

, and

(γ

,γ

),· ·· , (γ

,γ

) ∈

Γ such that w

(γ

,γ

)

−−−→

k−1

.. .

(γ

,γ

)

−−−→ w

, where w

∩ cl(α

) = t

. This

path deﬁnes a tree T

= (S

,⇒

) rooted at s

where S

= {s

,· ·· ,s

} and for all j ∈ {0,··· ,k}, s

is labelled by the mcs w

k− j

. The relation is ⇒

deﬁned as usual From now on we will denote α ∈ s

whenever α ∈ w, w being the mcs associated with s,

where s is a tree node.

For k ≥ 0, we can inductively deﬁne T

= (S

,⇒

) such that the past formulas at every node will

have witnesses as ensured by lemma 4.3(iii). Given

any T

, one can extend it to T

k+1

= (s

k+1

,⇒

k+1

)

by adding new nodes corresponding to formulas like

hγ

,γ

i> ∈ s, for some s ∈ S

, for which there exists

no s

∈ S

such that s

(γ

,γ

)

−−−→ s

. The new node would

correspond to the mcs w associated with s and some

other w

, where w

(γ

,γ

)

−−−→ w

Let T = (S,⇒,s

,X ) where S =

k≥0

, ⇒

k≥0

⇒

, and X : S → 2

is given as follows:

(γ

,γ

) ∈ X(s) iff enabled

,enabled

∈ w, where w

is the mcs associated with s. It follows that X

(s) =

{γ

: enabled

∈ w}. We deﬁne the model M = (T,V )

where V (s) = w ∩ P, where w is the mcs associated

with s. The following lemma proves some important

properties which are used in the later proofs.

Lemma 4.4 For every s ∈ S, we have:

(i) if [γ

,γ

]α ∈ s and s

(γ

,γ

)

===⇒ s

, then α ∈ s

;

(ii) if hγ

,γ

iα ∈ s, then there exists s

s.t. s

(γ

,γ

)

===⇒

and α ∈ s

;

(iii) if

[γ

,γ

]α ∈ s and s

(γ

,γ

)

===⇒ s, then α ∈ s

;

(iv) if hγ

,γ

iα ∈ s, then there exists s

such that

(γ

,γ

)

===⇒ s and α ∈ s

;

(v) if α ∈ s and s

⇒

∗

s, then α ∈ s

;

(vi) if 3

−

α ∈ s, then there exists s

such that s

⇒

∗

and α ∈ s

The following result takes care of boolean formulas.

Proposition 4.5 For all φ ∈ Φ, for all s ∈ S, φ ∈ s iff

,s  φ.

The following lemmas and propositions take care of

the strategy constructs.

Lemma 4.6 For all i, for all σ

∈ Strat

(P), for all

∈ Γ

, for all s ∈ S, σ

∈ s iff γ

∈ σ

(s).

Lemma 4.7 For all t ∈ AT, σ →

φ /∈ t implies that

there exists a path ρ

: t = t

(γ

,γ

)

−−−→ t

.. .

(γ

k−1

,γ

k−1

)

−−−−−−→ t

which conforms to σ such that one of the following

holds:

(i) φ /∈ t

(ii) moves(t

)

∩ σ(t

) =

Proposition 4.8 For all s ∈ S, σ →

φ ∈ s iff M,s 

σ →

φ.

Finally, with all the required lemmas and propositions

in hand, we can prove the following:

Theorem 4.9 For all α ∈ Θ, for all s ∈ S, α ∈ s iff

M,s  α.

Consequently, the logic SL is weakly complete. The

completeness proof suggests the following automata

theoretic decision procedure. For every strategy spec-

iﬁcation we can construct an advice automaton with

output, and given a formula φ, we can construct a tree

ICAART 2017 - 9th International Conference on Agents and Artiﬁcial Intelligence

630

automaton whose states are the consistent subsets of

subformulas of φ. These two automata are run in par-

allel. Since the number of strategy speciﬁcations is

linear in the size of φ, the size of the tree automa-

ton is doubly exponential in the size of φ. Emptiness

checking for the automaton is polynomial, thus yield-

ing us a decision procedure that runs in double expo-

nential time. This is a modiﬁcation of the procedure

presented in (Ramanujam and Simon, 2008b).

5 FUTURE WORK

In this work we present a sound and complete ax-

iomatization of a Strategy Logic, SL which is used

to model strategic reasoning in dynamic games with

simultaneous moves, in particular, in inﬁnite repeated

normal form games. It would be interesting to see

the precise connection with the framework developed

in (Ramanujam and Simon, 2008b), in particular,

whether the strategy logic for turn-based games can

be embedded in the logic proposed here. Dually, we

could also consider the game tree as obtained by com-

position from a collection of “small” game trees con-

stituting subgames and strategies as complete plans

on them to ensure local outcomes. We may investi-

gate such ideas in dynamic games with simultaneous

moves, based on the work in (Ramanujam and Simon,

2008a; Ghosh, 2008).

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