Strategies made explicit in Dynamic Game Logic

Sujata GhoshCenter for Soft Computing Research, Indian Statistical Institute, Kolkata

Department of Mathematics, Visva-Bharati, Santiniketansujata t@isical.ac.in

Abstract

We propose an explicit logic of strategies (SDGL) inthe Dynamic Game Logic (DGL) framework and providea complete axiomatization for this logic. Some discussionsare put forward regarding SDGL and DGL, raising aninteresting issue about their combination.

1. Introduction: cudos for strategies

Many events that happen in our daily life can be thoughtof as games. In fact, besides the ‘games’ in the literal sense,our day-to-day dialogues, interactions, legal procedures,social and political actions, biological phenomena - allthese can be viewed as games together with their goals andstrategies. The theory of games has its various applicationsin the areas of economics, logic, computer science aswell as linguistics. Games play a very important role inmodelling intelligent interaction. In Rubinstein’s words, ‘Iview game theory as an analysis of concepts used in so-cial reasoning when dealing with situations of conflict’ [16].

As evident from the existing literature, much of gametheory deals with strategic equilibriums. Various equi-librium theories have been developed till date both forzero-sum as well as non zero-sum games, starting from theinitial concept of Nash [12], which have their implicationsin the studies of the society. They help in providing a ‘planof action’ to the agents participating in the state of affairs,which could be articulated as ‘games’, when faced withstrategic decision making in situations of conflict.

Over the past few decades a lot of work has been donein the epistemic foundations of game theory, studyingthe formal logics of knowledge and belief. The formalsystems expressing players’ knowledge and beliefs aboutthemselves as well as their competitors were looked atin much details - a tremendous amount of work is stillgoing on. But a very related and relevant issue - players’

strategies/plan of actions to play the game, which they baseon their epistemic states almost have rarely been lookedupon, until very recently. To mention a few, [15] proposesa logic of strategies in games over finite graphs, whereas[17] makes strategies explicit in Alternating-time TemporalLogic. The incorporation of ‘strategies’ within the logicallanguage would very well aid in the currently popularventures into social choice mechanism designs.

Strategies of the players playing the game form a basicingredient of game theory, whether looked upon from thewinning point of view or from the best-response one. Alot of other issues like the rationality of the players, theirgoals and preferences are also very important issues, butthey are outside the scope of this work, though we plan toincorporate them in the future.

Our main goal in this work is to incorporate explicitnotions of strategies in the framework of Dynamic GameLogic (DGL) [13]. Not unlike other logics talking aboutgame and coalition structures [1, 14], DGL suffers from‘!-sickness’ : the detailed level of game structures gettingsuppressed by existential quantifiers of “having a strategy”[7]. We intend to provide a logic (SDGL) that makes thegame structures explicit to a great extent.

In general, strategies are partial transition relations andhence dynamic modal logic provides a good framework totalk about them, as mentioned in [4, 5]. But the main chal-lenge here is to combine the strategy calculus together withthe game calculus. As one can easily guess, the constructsof Propositional Dynamic Logic [11] play an importantrole in achieving this amalgamation. In this regard, weshould mention that, a lot of discussions and proposals havealready been put forward by van Benthem [9, 8]. This effortcan be looked upon as a follow-up of one of these proposals.

After providing a brief overview of DGL in the next sec-tion, we propose a logic for strategizing DGL (SDGL)in section 3 with a complete axiomatization. Section 4

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provides some discussions over the two logics DGL andSDGL, with several pointers towards future work men-tioned in the last one.

2. Dynamic Game Logics : an overview

We now give a brief review of DGL, the dynamic gamelogic of two-person sequential games in this section, whichwas first proposed in [13], and further developed by [14],[6], [10] and others. DGL talks about ‘generic’ gameswhich can be played starting from any state s on the ‘gameboards’ and the semantics is based on the ‘forcing relations’describing the powers each player has to end in a set of finalstates, starting from a single initial state.

s!iGX : player i has a strategy for playing game G

from state s onwards, whose resulting final statesare always in the set X , whatever the other play-ers choose to do.

To exemplify, let us move onto real extensive games foronce. Consider the game tree:

E

!!!!!!

!""""

"""

A

#####

$$$$$

A

#####

$$$$$

1 2 3 4

In this game, player E has two strategies, forcing the sets ofend states {1, 2}, {3, 4}, while player A has four strategies,forcing one of the sets {1, 3}, {1, 4}, {2, 3}, {2, 4}.

These forcing relations satisfy the following two simpleset-theoretic conditions [3]:

(C1) Monotonicity: If s!iGX and X ! X !, then s!i

G X !.

(C2) Consistency: If s!EGY and s!A

GZ, then Y , Z overlap.

In the semantics of DGL as proposed in [13, 14], anotherextra condition is assumed :

(C3) Determinacy: If it is not the case that s!EGX , then,

s!AGS-X , and the same for A vis-a-vis E.

Both [13] and [14] talks about determined games. Thissimplifies things a lot, but fails to express the roles ofthe players in the games. The dynamic logic for non-determined games was studied extensively in [10] whichalso introduced the notion of parallel games in the syntax.For the present work the concurrent game construct has notbeen dealt with. The iteration operation for repeated play aspresent in [13] has also not been considered here. We onlyconsider the following constructs which form new games:

choice (G " G!), dual (Gd), and sequential composition(G; G!). The readers could easily guess the intuitive mean-ings of these constructs. For the sake of continuation to thenext section, in what follows, the DGL for non-determinedgames has been briefly discussed. To start with, it should benoted here that the players’ powers have a recursive struc-ture in the complex games:

Fact 2.1 Forcing relations for players in complex sequen-tial two-person games satisfy the following equivalences:

s!EG"G! X iff s!E

G X or s!EG!X

s!AG"G! X iff s!A

GX and s!AG!X

s!EGdX iff s!A

GXs!A

GdX iff s!EGX

s!iG;G!X iff #Z : s!i

GZ and for all z $ Z, z!iG!X.

The basic models that play the role of game boards aredefined as follows:

Definition 2.2 A game model is a structure M = (S, {!ig |

g $ !}, V ), where S is a set of states, V is a valuationassigning truth values to atomic propositions in states, andfor each g $ !, !i

g ! S % P(S). We assume that for each g,the relations are upward closed under supersets (the earlierMonotonicity), while also, the earlier Consistency conditionholds for the forcing relations of the players A, E. "

The language of DGL (without game iteration) is definedas follows:

Definition 2.3 Given a set of atomic games ! and a set ofatomic propositions ", game terms # and formulas $ aredefined inductively:

# := g | $? | #; # | # " # | #d

$ := & | p | ¬$ | $ ' $ | (#, i)$,

where p $ ", g $ ! and i $ {A,E}."

The truth definition for formulas $ in a model M at astate s is standard, except for the modality (#, i)$, which isinterpreted as follows:

M, s |= (#, i)$ iff there exists X : s!i!X and

*x $ X : M, x |= $.

The complete axiomatization of this logic has been pro-posed and proved in [10] :

Theorem 2.4 DGL is complete and its validities are ax-iomatized by the following axioms:

a) all propositional tautologies and inference rules

b) if + $ , % then + (g, i)$ , (g, i)%

75

c) !g, E"! # ¬!g, A"¬!

d) reduction axioms:

!" $ #, E"! % !", E"! & !#, E"!!" $ #, A"! % !", A"! ' !#, A"!!$d, E"! % !$, A"!!$d, A"! % !$, E"!!"; #, i"! % !", i"!#, i"!!%?, E"! % (% ' !)

!%?, A"! % (¬% ' !).

This logic is also decidable. As can be noticed, the truthdefinition of the modal game formulas of the form !$, i"!is given in terms of existence of strategies, without goinginto their structures. In what follows, the strategy structureshave been explicitly dealt together with the game structures.

3. Strategizing DGL

3.1. A logic for strategies

Mentioning strategies explicitly in the dynamic gamelogic framework prompt us to divert from the usual DGLsemantics that takes into consideration ‘generic’ games ongame boards. The whole point is to bring strategies withinthe logical language which till now have their place in giv-ing meaning to the game as well as coalition modalities[14]. Adding explicit strategy terms to DGL, the languageof Strategized DGL (SDGL) is defined by,

Definition 3.1 Given a set of atomic games !, a set ofatomic strategies ", a finite set of atomic actions # and aset of atomic propositions $, game terms $, strategy terms&, action terms ' and formulas ! are defined inductively inthe following way:

$ := g | !? | $d | $; $ | $ $ $& := s | & $ & | &; &' := b | ' $ ' | '!! := ( | p | ¬! | ! & ! | [']! | !'"! | !&, i, $"!

where p ) $, s ) ", g ) !, b ) #, and i ) {A,E}. (

Regarding the intuitive understanding of the strategyterms, ‘$’ corresponds to the choice of strategies, and ‘;’to the composition of strategies. It should be mentionedhere that, the way the semantics is given later, it wouldhave been enough to use just one combination operationof the strategy terms. The use of both of them aids inunderstanding the intuition behind their usage.

Moving away from the ‘generic’ game structures, themodels take the form of extensive game trees with a fewadditional actions. Before going into all these, we need aparent model which is given as follows.

Definition 3.2 A model is a structure M = ! S, {R! : ' ’sare actions}, ref, L, R, V ", where S is a set of states and Vis a valuation assigning truth values to atomic propositionsin states. For each ', R! is a binary relation on S. ref, L,R are all reflexive relations over S, with ! S, {R! : ' ’s areactions}" forming a regular action frame. (

In this model, atomic and composite games from a spec-ified ‘start’-state are defined in the following. It should bementioned that all these game structures are taken to be fi-nite, defined over finite subsets of S.

Definition 3.3 Game(M, s, $) is a structure defined recur-sively as follows:

(i) For atomic games g, Game(M, s, g) is a structuregiven as follows: !W * S, s ) W , {Rb +W : b ) #},V = VM +W , P : W # {E, A, end}".

(ii) For test games !?, Game(M, s, !?) is a structuregiven as follows: !{s}, s, ref +{s}, V = VM +{s}, P :{s}#{ end}".

(iii) Given Game(M, s, $), Game(M, s, $d) is the struc-ture, !W * S, s ) W, {Rb +W : b ) #}, V = VM +W , P :W # {E, A, end}", where all the constituents of the struc-ture are the same as the corresponding ones in Game(M, s,$), except for P"d , which satisfies the property: P"d(w) =E/A, whenever P"(w) = A/E, respectively.

(iv) Given Game(M, s, $) and Game(M, s, $"),Game(M, s, $ $ $") is the structure given by, !W * S, s )W , {Rb +W : b ) #}, L +{s,s}, R +{s,s}, V = VM +W ,P : W # {E,A, end}", where W = W" , W"! , and Pextends both P" and P"! .

(v) Given Game(M, s1, $) and Game(M, s2, $"),Game(M, s, $; $") is defined if for each t ) P#1

" (end),Game(M, t, $") can be defined. Suppose we haveGame(M, t1, $"), . . ., Game(M, tn, $"). In that case,Game(M, s, $; $") is the structure !W * S, s )W, {Rb +W : b ) #}, V = VM +W , P : W #{E,A, end}", where W = W" $W 1

"! $ . . .$Wn"! ; s = s1;

P extends P" , P 1" , . . ., Pn

" , with the restriction that forw ) P#1

" (end) -W , P (w) = P"!(s2).(

Because of some technical reasons regarding satisfiabil-ity, choice games can only be defined for the games withthe same initial state, which is not really a big issue. Thesequential composition game could also be defined undercertain restrictions as mentioned above. It is now time to

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define strategies of the players in a game, which again hasa recursive definition. Note that we will only talk about fullstrategies here and the definition is given likewise.

Definition 3.4 Given Game(M, s, !), a strategy for aplayer i, given by the relationR!

i is defined by,(i) For Game(M, s, g), Rg

E [RgA] !

!{Rb "Wg : b # !}

satisfying the following conditions:(a) s # Dom(Rg

E)[Dom(RgA)], and Ran(Rg

E)[Ran(Rg

A)] $ P!1g (end) %= &

(b) For each t # P!1g (E, A)' {s}, t # Dom(Rg

E)[Dom(Rg

A)] iff t # Ran(RgE)[Ran(Rg

A)].(c) For each s # P!1(E)[P!1(A)],( unique s" such

that (s, s") # RgE [Rg

A].(d) For each s # P!1(A)[P!1(E)], (s, s") #!{Rb "Wg : b # !} implies (s, s") # Rg

E [RgA].

(e) Nothing else is inRgE [Rg

A].

(ii) For Game(M, s, "?),R"?i = ref "{s}.

(iii) For Game(M, s, !d),R!d

E = R!A, andR!d

A = R!E .

(iv) For Game(M, s, ! ) !"),R!#!!

E = L "{s,s} )R!E or, R "{s,s} )R!!

E ,and,R!#!!

A = L "{s,s} )R "{s,s} )R!A )R!!

A .

(v) For Game(M, s, !; !"),R!;!!

i =R!i )R

!!

ij1) . . .)R!!

ijl,

where the indices correspond to the number of times the‘end’-state is reached inR!

i . #

For an example of the players’ strategies, consider thesimple extensive game tree:

s1, E

!!!!!!!!!!!

"""""""""""

s2, A

######

$$$$$$

s3, A

######

$$$$$$

s4, end s5, end s6, end s7, end

The strategies of E are {(s1, s2), (s2, s4), (s2, s5)},and {(s1, s3), (s3, s6), (s3, s7)}, whereas the strate-gies for A are {(s1, s2), (s1, s3), (s2, s4), (s3, s6)},{(s1, s2), (s1, s3), (s2, s4), (s3, s7)}, {(s1, s2), (s1, s3),(s2, s5), (s3, s6)}, and {(s1, s2), (s1, s3), (s2, s5), (s3, s7)}.

If we consider the choice operations of two such games,the strategies of the players could be easily computed.For the sequential composition, consider the following twogames:

s1, E

%%%%%%

%&&&&

&&&

s2, end G s3, end

t1, A

''''''

'(((

((((

t2, end H t3, end

The strategies of E in G are {(s1, s2)}, and {(s1, s3)},and in H is {(t1, t2), (t1, t3)}, and similarly, that for A in Gis {(t1, t2), (t1, t3)}, and in H are {(t1, t2)}, and {(t1, t3)}.

Suppose, the model is such that G; H could be definedand it is as follows:

s1, E

))!!!!!!!!

**""""""""

s2, A

######

$$$$$$

s3, A

######

$$$$$$

t4, end t5, end t6, end t7, end

The readers can notice that it is just the game givenas example earlier, and hence could easily verify thatthe strategies of the players in this complex game con-form with the definition given to compute the strategiesof the sequential composition games, from the simpler ones.

Before going into the truth-definitions of formulas, let usmention a few words about interpreting the strategy termsof the language. The strategy terms are always interpretedcorresponding to some game structure Game(M, s, !) andplayer i. Let R!

i denote the set of all strategies for player iin Game(M, s, !).

Definition 3.5 Given Game(M, s, !) and player i, a strat-egy functionF!

i is a partial function from the set of all strat-egy terms to R!

i , satisfying the following conditions.

(i) For s # ", F!i (s) is defined, only when ! is an atomic

or a test game.

(ii) For the choice game $ ) %, F##$i is given by,

F##$E (& ) ') = L "W!"" )R#

E iff F#E(&) = R#

E ,

F##$E (& ) ') = R "W!"" )R$

E iff F$E(') = R$

E ,

F##$A (& ) ') = R##$

A iff F#A(&) = R#

A

and, F$A(') = R$

A.

(iii) F!d

E (&) = R!d

E iff F!A(&) = R!

A, and,F!d

A (&) = R!d

E iff F!A(&) = R!

A.

(iv) For the composition game $;%, F#;$i satisfies,

F#;$i (' ; () = R#;$

i iff F#i (') = R#

i

and, F$i (() = R$

i .#

Note that the way these partial functions are given, ittakes care of the cases of mismatched syntax (like, *& )', E, $; %+"), which does not have any corresponding struc-ture in the model. For the semantics of our language, we

77

define the truth of a formula ! in M at a state s in theobvious manner, with the action modalities defined in theusual PDL-style and the following key clause for the game-strategy modality:

M, s |= !", i, #"! iff for all s! # Ran(F!i (")) $

P"1(end) in Game(M, s, #), M, s! |= !.

Here are some validities of this logic.

• !", i, #"! % !", i, #"(! & $)

• !", i, #"(! ' $) ( !", i, #"! ' !", i, #"$

3.2. Axioms and completeness

We now provide a complete axiomatization of SDGL.

Theorem 3.6 SDGL is complete and its validities are ax-iomatized by

a) all propositional tautologies and inference rules

b) generalization rule for the action modalities

c) axioms for the action constructs:

[%](! % $) % ([%]! % [%]$)

!%"! ( ¬[%]¬!

!%1 ) %2"! ( !%1"! & !%2"!!%#"! ( (! & !%"!%#"!)

[%#](! % [%]!) % (! % [%#]!)

d) !s, i, g"! % !b1 ) . . .) bn"!(b1 ) . . .) bn)#"!, where! = {b1, . . . , bn}

e) !", i, #"(! % $) % (!", i, #"! % !", i, #"$)

f) if * ! % $ then * !", i, #"! % !", i, #"$

g) reduction axioms:

!" ) &, E, ' ) ("! ( !", E, '"! & !&, E,("!!" ) &, A, ' ) ("! ( !", A, '"! ' !&, A,("!!", E, #d"! ( !", A, #"!!", A, #d"! ( !", E, #"!!& ; ), i, '; ("! ( !&, i, '"!), i, ("!!", E, *?"! ( (* ' !)

!", A, *?"! ( (¬* ' !)

h) strategy rules:

for each X + !, the rule below:

if * ! % !()X)"!()X)#"$ then * ! % !s, i, g"$ .

Proof. Soundness of some of the interesting reduction ax-ioms and the strategy rules for the game-strategy modalityare shown below. The readers can easily verify the validityof the rest.

1. !" ) &, E,' ) ("! ( !", E, '"! & !&, E, ("!Suppose M, s |= !" ) &, E,' ) ("!. Then, for all s! #

Ran(F"$#E (" ) &)) $ P"1

"$#(end) in Game(M, s, ' ) (),M, s! |= !. Now, F"$#

E (" ) &) = L ,W!!" )R"E or,

R ,W!!" )R#E . W.l.o.g. suppose that F"$#

E (" ) &) =L ,W!!" )R"

E . Then, for all s! # Ran(L ,W!!"

)R"E) $ P"1

"$#(end) in Game(M, s,' ) (), M, s! |= !.By definition of strategies in ) games, this implies that,for all s! # Ran(R"

E) $ P"1" (end) in Game(M, s, '),

M, s! |= !. Hence, for all s! # Ran(F"E(")) $ P"1

" (end)in Game(M, s, '), M, s! |= !.So, we have that, M, s |=!", E, '"!. Similarly, if F"$#

E (" ) &) = R ,W!!" )R#E ,

one can show that, M, s |= !&, E,'"!. So, M, s |=!", E, '"! or M, s |= !&, E, '"!. Hence, M, s |=!", E, '"! & !&, E, ("!.

For the converse, suppose that M, s |= !", E, '"!.Then, for all s! # Ran(F"

E(")) $ P"1" (end)

in Game(M, s,'), M, s! |= !. So, for alls! # Ran(R"

E) $ P"1" (end) in Game(M, s, '),

M, s! |= !, which implies that, for all s! #Ran(L ,W!!" )R"

E)$P"1"$#(end) in Game(M, s, ')(),

M, s! |= !. Hence, reasoning as earlier we have that,M, s |= !")&, E, ')("!. The proof for the other disjunctcan be dealt with in a similar manner.

2. !" ) &, A, ' ) ("! ( !", A,'"! ' !&, A, ("!Suppose M, s |= !" ) &, A, ' ) ("!. Then, for all s! #

Ran(F"$#A (" ) &)) $ P"1

"$#(end) in Game(M, s, ' ) (),M, s! |= !. Now, F"$#

A (" ) &) = R"$#A , i.e.

L ,{s,s} )R ,{s,s} )R!A ) R!"

A . It follows that forall s! # Ran(R"

A) $ P"1" (end) in Game(M, s, '),

M, s! |= !, and for all s! # Ran(R#A) $ P"1

# (end) inGame(M, s,(), M, s! |= !. Then, from the definitions itfollows that M, s |= !", A, '"!' !&, A, ("!. The conversecan be proved by retracing the steps backwards.

3. !", E, #d"! ( !", A, #"!Suppose M, s |= !", E, #d"!. Then, for all

s! # Ran(F!d

E (")) $ P"1!d (end) in Game(M, s, #d),

M, s! |= !. By definition of strategies in the dual game,this implies that, for all s! # Ran(F!

A(")) $ P"1! (end) in

Game(M, s, #), M, s! |= ! and so, M, s |= !", A, #"!.For the converse proof, retrace back.

4. !& ; ), i,'; ("! ( !&, i,'"!), i,("!Suppose M, s |= !& ; ), i, ';("!. Then, for all

78

s! ! Ran(F!;"i (!; ")) " P"1

!;"(end) in Game(M, s,#; $),M, s! |= %. Hence, for all s! ! Ran(R!

i # R"ij1

# . . .# R"ijl

) "P"1!;"(end) in Game(M, s, #; $), M, s! |=

%. Then, for k = 1, . . ., l, for all t ! Ran(R"ijk

) "P"1

"jk(end) in Game(M, tjk ,$), M, t |= %, and hence

for all t! ! Ran(R!i ) " P"1

! (end), in Game(M, s, #),M, t! |= $&, i,$%%. So, M, s |= $", i, #%$&, i, $%%.

For the converse part, suppose M, s |=$", i, #%$&, i, $%%. Then, for all t! ! Ran(R!

i )"P"1! (end),

in Game(M, s, #), M, t! |= $&, i, $%%, whereF!

i (") = R!i . This can be possible, only when, for each

t ! P"1# (end), Game(M, t, '!) can be defined. Hence,

Game(M, s,#;$) is defined, and for all s! ! Ran(R!i

# R"ij1

# . . .# R"ijl

) "P"1!;"(end) in Game(M, s,#; $),

M, s! |= %. So, M, s |= $" ; &, i, #; $%%.

The validity of the strategy rules follows from thefact that, if there is a path from some state s to a statesatisfying some formula %, then the Game(M, s, g) and acorresponding strategy relation Rg

i can be defined in sucha way that $s, i, g%% holds at s.

The completeness of the axiom system is proved byshowing that every consistent formula is satisfiable. Let #be a consistent formula. Let Cl(#) denote the subformulaclosure of #, satisfying the FL-closure conditions for theaction modalities with the following extra conditions:

(i) If $! # ", E,# # $%% ! Cl(#), then $!, E, #%% &$", E,$%% ! Cl(#).(ii) If $! # ", A, # # $%% ! Cl(#), then $!, A, #%% '$", A, $%% ! Cl(#).(iii) If $" ; &, i, #;$%% ! Cl(#), then $", i, #%$&, i, $%% !Cl(#).(iv) Cl(#) is closed under single negations.

Any maximal consistent subset of Cl(#) is said to be anatom. Let A denote the set of all such atoms. For T ! A,let !T denote the conjunction of all the formulas present inT . For C, D ! A, define CR$D if !C ' $(% !D is consistent.The regular canonical model C is defined to be the tuple$A, {R$: (’s are actions}, ref ,L,R,V%, where, ref, L,Rare reflexive relations on A, and V(p) = {T ! A : p ! T},and R$’s satisfy the regularity conditions. The existencelemma for the modalities $(%, can be proved in the usualway, and we have that C, A |= % iff % ! A, for each% ! Cl(#), and each A ! C where % is either an atomic ora boolean or an action modal formula.

It remains to be shown that C, A |= $!, i, '%% iff$!, i, '%% ! A. Because of the reduction axioms, it sufficesto show that for each $s, i, g%% ! Cl(#), and each A ! C,

C, A |= $s, i, g%% iff $s, i, g%% ! A. In other words, wehave to show that $s, i, g%% ! A iff in Game(C, A, g),Ran(Fg

i (s)) " P"1(end) is the set of all atoms T , suchthat % ! T .

Suppose $s, i, g%% ! A. Then because of axiom (d),Game(C, A, g), and Fg

i can be defined in such a way, thatthe implication holds. The converse follows from the factthat if < bi1 > . . . < bim > % is consistent, then so is< bi1 > . . . < bim > % ' $s, i, g%%, which holds becauseof the strategy rules. QED

4. DGL and SDGL - a comparison

As mentioned earlier, DGL talks about generic gamesplayed on game boards, and the meaning of the gamemodalities is given by existence of strategies. SDGLbrings out these strategies to the fore. Strategy combi-nations for playing composite games are talked about inthis framework which brings out the extensional nature ofstrategies, though according to certain views, strategiesare inherently intensional. As mentioned by van Benthem[4, 5], strategies of the players in the game tree can betalked about using the program constructs of the dynamicmodal logic. Some proposals for combining strategies toachieve a certain goal are also made there.

The task was to combine the strategy constructs togetherwith the game constructs. SDGL proposes a way to doit. As evident from the previous section, one has to resortto the PDL-style action constructs. To make strategiesexplicit, one can no longer talk about generic games.Extensive game trees come into the treatise - games aredefined as tree structures, and strategies are defined assubtrees.

In the tradition of DGL semantics, the so-called forcingrelations satisfy the conditions of upward-monotonicityand consistency (determinacy also, in case of Parikh’sand Pauly’s DGL). The sets of states forced by theserelations have an inherent ‘disjunctive’ interpretation.A ‘conjunctive’ interpretation of these sets which isneeded when parallel game constructs are introduced,has been taken in [10]. It is interesting to note that, theway strategies are defined as relations between states, itcorroborates with the ‘conjunctive’ interpretation of theset of ‘end’-states reached. Hence, this language rathersuggests ‘downward monotonicity’ at this conjunctive level.

It is clear that there are some sentences which could beexpressed in SDGL, but not in DGL. But it is also thecase that there are certain statements that can be expressedin DGL, but not in SDGL : for example, ‘player i does

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not have any strategy in the game g to achieve !’ can beexpressed in DGL as ¬!g, i"!. Under these circumstancesit would be ideal to have a logic that could express both.This gives rise to the following issue:

Question What would be the complete axiomatizationof a logic that has both Parikh’s original game modalitiesas well as the game-strategy modalities presented in theearlier section of this paper?

In fact, for the set of strategy relations R!i for player i in

Game(M, s, "), one can easily define #i! (cf.§2), as follows:

s#i!X iff X = Ran(R!

i ) # P!1! (end), for someR!

i $ R!i .

It remains to be seen what conditions have to be im-posed on #i

! to maintain compatibility. This is precisely thesame issue as finding joint logics of proofs and provabilityin arithmetic, on which a lot of effort has been made in therecent past. For a detailed overview, one can have a lookat [2]. The most natural analogy that one can think of hav-ing both such existential criterion, as well as the witnessesconforming to it could be found in first order logic - %x!together with term substitutions like ![$/x].

5. Conclusions and intentions

This paper proposes a logic which makes strategies ex-plicit in the dynamic game logic framework. The needfor the dynamic modal logic syntax for achieving such tar-gets becomes apparent. An interesting issue of getting ajoint logic of complex game modalities together with game-strategy modalities emerges. Some possible areas for futureinvestigations are given below.

Explicit strategies for other logics Several other lan-guages talking about game structures and coalition struc-tures like Alternating-time temporal logic and Coalitionlogic could be investigated so as to add an explicit notionof strategies, which merely occur as an existential notion inthe semantics of these logics. This could very well aid inthe social choice mechanism designs.

Adding knowledge and preference notions To comecloser to the real game scenario which are played by the ra-tional players, one has to incorporate the knowledge/beliefas well as preference modalities in the existing framework,i.e. epistemic versions of these game logics with explicitstrategies need to be explored.

Games with imperfect information It is evident that theuniform strategies in the imperfect information games donot conform with the compositional analysis that has been

done here. That study is inherently different taking into ac-count the knowledge level of the players, which provides avery interesting challenge.

Acknowledgements

I thank Johan van Benthem for introducing me to the log-ics of games and strategies and providing me with constantsupport and invaluable suggestions throughout this effort. Ihave been greatly enriched with the many discussions I hadwith Fenrong Liu and Cedric Degremont during my stay inAmsterdam in the year 2006-2007. R. Ramanujam, SunilSimon and Fernando R. Velazquez-Quesada went througha preliminary draft of this paper and provided me withthoughtful comments.

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