In Nite Games

Home , Complete information

In nite Games

Yuri Gurevich

EECS Dept., U. of Michigan, Ann Arb or, MI 48109-2122

In nite games are widely used in mathematical logic [2, 8, 12 ]. In particular, in nite

games proved to b e a useful to ol in dealing with the monadic second-order theories of in nite

strings and in nite trees [3, 4 , 7]. Recently [1, 15 , 13 ], in nite games were used in connection

to concurrent computational pro cesses that do not necessarily terminate. For example, an

op erating system may b e seen as playing a game \against" the disruptive forces of users.

The classical question of the existence of winning strategies turns out to b e of imp ortance

to practice. Here we attempt to explain basics of in nite game theory.

Quisani: What is an in nite game?

Author: In the simplest form, there are two players, Player 1 and Player 2. Player 1

starts bycho osing a binary bit a , then Player 2 cho oses a binary bit a , then Player 1

1 2

cho oses a binary bit a ,thenPlayer 2 cho oses a binary bit a , and so on ad in nitum.

3 4

If the resulting in nite string X (i.e. the function X (i)= a on p ositiveintegers, the

play) b elongs to an a priori xed set W of in nite strings then Player 2 wins the game;

otherwise Player 1 do es.

Q: Do you mean that di erent sets W give di erent games?

A: Yes. The set W de nes the goals of the players. Let W b e the complementof

W and W = W . Then the goal of Player " is to ensure that the play b elongs to his

winning set W .

Q: I guess, your simplest form is a little to o simple. As one of those disruptive users,

Iwant more than 2 keys on mykeyb oard.

A: You are right. Let me describ e a more general setting. We are given an in nite

countable tree A called the arena and a set W of branches of A. No des of A are

p ossible p ositions of the game (A; W ); the ro ot is the initial p osition. Player 1 b egins

bycho osing a child x of the ro ot, then Player 2 cho oses a child x of x , and so on. If

1 2 1

Bulletin of Europ ean Asso c. for Theor. Comp. Science, June 1989, 93{100

Partially supp orted by NSF grant. During the work on this article, the author was with Stanford

University and IBM Almaden ResearchCenter (on a sabbatical leave from the UniversityofMichigan). 1

the resulting branch (the play) b elongs to W ,thenPlayer 2 wins, otherwise Player 1

wins.

Q: Can a branch b e nite? In other words, are there dead-end p ositions?

A: Dead-end p ositions may b e natural in applications and useful in theory [14], but

for simplicity let us stick here to arenas without dead-end no des.

Q: Let me return to the game of an op erating system against the users. If there are n

users then in e ect there are n +1 players.

A: It is easier to analyze 2 player games. We can view all users together as one force

against which the op erating system has to play.

Q: Can you give me an example how the winning set of the op erating system sp eci es

its goals?

A: Supp ose that the users comp ete for some resource, say,aprinter. A computation

is called fair if, whenever a user asks for a printer, she or he eventually gets it. You

maywant to require that the winning set of the op erating system comprises only fair

plays. If the op erating system plays a winning strategy then fairness will b e ensured.

Q: What is a winning strategy exactly? Even simpler, what is a strategy?

A: A (p ossibly nondeterministic) strategy for Player " in a game (A; W ) is a function

F that, given a p osition x where Player " makes a move, pro duces a nonemptyset

F (x)ofchildren of x. F is deterministic if every F (x) is singleton. F is winning if

Player " wins every play consistentwithF .

Q:Ihave forgot the exact de nition of games with complete information, but I remem-

b er that nite games with complete information are always determinate, i.e., one of the

players has a winning strategy. Isn't any(A; W ) a game with complete information?

Is determinacy a problem in the in nite case?

A: Yes, each(A; W ) is a game with complete information, but do not worry ab out the

de nition of games with complete information: we will not use it. And yes, determinacy

is problematic in the in nite case. An example of an indeterminate in nite game

app eared already in the pap er by Gale and Stewart [6] where they intro duced in nite

games of interest to us here. (Actually, Gale and Stewart rediscovered in nite games

in the context of emerging game theory. Similar in nite games were studied by East

Europ ean mathematicians much earlier [11].) Here is their example.

Let A b e the arena of binary strings. Index the deterministic strategies (the notion of

strategy do es not dep end on winning sets) of Player 1 (resp. Player 2) as F (resp. G )

where ranges over ordinals of cardinality less than continuum. By induction on ,

de ne branches X and Y suchthatX is consistent with G and di erent from any

Y with < , and Y is consistentwithF and di erent from any X with .

This is p ossible b ecause there are continuum many branches consistentwithany G

(resp. F ) whereas the set fY : < g (resp. fX : g) contains less than

continuum many memb ers.

Let W comprise all branches Y . The game (A; W ) is indeterminate. For, any strategy

F of Player 1 is defeated if Player 2 plays Y , and any strategy G of Player 2 is

defeated if Player 1 plays X .

Q: Is determinacy a big issue in the in nite game theory?

A: Yes. It plays also an imp ortant role in mathematical applications. The so-called

Axiom of Determinacy is an exciting alternative to Axiom of Choice. Much of in nite

game theory is related to determinacy, and it is myintention to sp eak to day mainly

ab out determinacy.Haveyou noticed anything sp ecial ab out the indeterminate game

ab ove?

Q: The arena was natural, but the winning set was weird. I can see an essential

use of Axiom of Choice. Do you wanttosay that games with nice winning sets are

determinate?

A: Yes. And we need top ology to de ne appropriate nice sets. Call a set U of branches

of an arena A open if every branch X 2 U has a nite pre x x such that the cone

[x]= fY : Y isabranch with pre x xg is included into U ; this is the well-known

Cantor top ology.Agame(A; W ) is called open, closed, Borel, etc. if W is so. Tony

Martin proved that every Borel game is determinate [9,10].

Q: Are op en games of interest? Can you prove that op en games are determinate?

A: Op en games are of great interest. In the example with users comp eting for a

printer, consider a particular request R of a printer. It is easy to see that the collection

of branches, i.e. computations, where R is satis ed is op en and gives rise to an op en

game. And I am happytoprove the determinacy of op en games for you.

Theorem 1 [6] Every closedoropen game is determinate.

Q: Isn't enough to sp eak ab out op en games only?

A: Well, there is this little asymmetry: Player 1 b egins. Let me start with a few

de nitions.

A support for an op en set U of branches of an arena A is any subset S of

A such that [x]=U . The collection of all no des x with [x] U is

x2S

the largest supp ort for U .Therank function corresp onding to a subset S of

A and to Player " is the unique function from A to the set of countable

ordinals augmented with a maximal element 1 suchthatforevery no de x: 3

{ (x) = 0 if and only if x has a pre x in S .

{ (x) 6= 0 if and only if either Player " makes a move in p osition x

and (y ) < for some child y of x or else the opp onent makes a move

in p osition x and (y ) < for every child y of x.

If (x)=1, I will saythat (x)isextraordinal.

Nowwe are ready to prove Theorem 1. Let Player " b e the \op en player" in a closed

or op en game (A; W ) (so that W is op en) and let =3 ". The desired winning

strategy of Player " is to minimize the rank function for a supp ort for W , and the

desired winning strategy for Player is to maximize that function.

Q: Will you elab orate?

A: Sure. Let S be any supp ort for W ,and b e the rank function corresp onding to S

and to Player ".Player " wins a play X if and only if X meets S ,i.e.,X has a pre x

that b elongs to S . Let F b e the \Decrease the rank" strategy for Player ":

If 0 <(x) < 1 then F (x) comprises all children y of x such that (y ) <

(x); otherwise F (x) comprises all children of x.

It is easy to see that F is winning if (ro ot ) is ordinal. All the time that the rank is

p ositive, it will decrease with eachmove. Since the order of ordinals is well founded,

the rank will reach 0 in nitely many steps, and therefore Player " will win.

Let G b e the \Keep the rank extra ordinal" strategy for Player :

If (x) is extra ordinal then G(x) comprises all children y of x suchthat(y )

is extra ordinal; otherwise G(x) comprises all children of x if (x) is ordinal.

It is easy to see that G is winning if (ro ot ) is extra ordinal. That nishes the pro of

of Theorem 1.

Q: Whydoyou sp eak ab out countable, rather than nite, ordinals in the de nition of

rank?

A: Consider the case when the ro ot of A has countably manychildren x ;x ;::: .

1 2

Cho ose a set S A in sucha way that the rank of x with resp ect to S and to

Player 2 is n, and consider the game where W is the op en set of branches supp orted

by S . Then the rank of the ro ot is the rst in nite ordinal ! , and the second player

has a winning strategy in the game.

Q: I thought for some reason that we are talking ab out arenas of b ounded branching.

Whydoyou wanttoallow in nite branching?

A: Here is one example. In the game of an op erating system against the users, a user

maytyp e a great deal b efore she or he lets the op erating system to makea move. 4

Q:Iwould like to see again the de nition of the Borel hierarchy.

A: Let range over countable ordinals, and let S b e the complement of a subset S of

the given arena A.

{ is the collection of op en sets.

0 0

{ = fS : S 2 g.

0 0

{ Every S 2 is the union of countable many memb ers of .

0 0

Q: Do or winning sets app ear naturally when an op erating system plays against

2 2

the users? If yes, I would like to see the determinacy pro of for games.

A: Recall the fairness condition and consider the game of an op erating system against

the users where the winning set of the op erating system comprises those and only those

plays, i.e. computations, that are fair. That winning set is the intersection of countably

many op en sets U (R) where U (R) is the set of computations where the request R is

satis ed. Thus, the winning set is .

0 0

Theorem 2 [16] Every and every game is determinate.

2 2

0 0 0

To prove Theorem 2, let Player " be the player in a or game (A; W ) (so

2 2 2

that W is ) and let =3 ". Fix op en sets U suchthat W = U , and let

" n " n

S b e the largest supp ort for U . The goal of Player " is to hit every S . All rank

n n n

functions in the pro of of Theorem 2 corresp ond to Player ".

Q: If the ro ot has the extra ordinal rank with resp ect to some S then the strategy

\Keep the rank extra ordinal" of Player is obviously winning. Otherwise, Player "

can hit any S .

A: Being able to hit every S do es not suce: The winning strategy of Player " should

guarantee hitting all sets S . Let P b e the collection of p ositions x suchthatx has an

n 1

ordinal rank with resp ect to every S .From a p osition in P ,Player " can reach some

n 1

p osition in any S , but that p osition mayhave the extra ordinal rank with resp ect to

some other S .

Q: I see your p oint. Let P b e the set of p ositions x such that x has an ordinal rank

with resp ect to every set P \ S .From a p osition in P , Player " can hit any P \ S

1 n 2 1 n

from where he can hit any S ; this is again insucient for a winning strategy. By

induction on k, de ne P to b e the set of p ositions x suchthatx has an ordinal rank

k +1

with resp ect to every P \ S . Finally,let P = P . I think ab out p ositions in P

k n ! k

as those with p otential to hit sets S . It lo oks likePlayer " has a winning strategy

if the initial p osition is in P .

! 5

A: Not quite. Potential ! do es not suce. Given any k and n, where k

and starting in a p osition of p otential ! ,Player " can reach a p osition of p otential k

in S , but it is not guaranteed that he can reach a p osition of p otential ! in S .

n n

Q: But even uncountable p otential do es not guarantee the ability to reach a p osition

of p otential ! in S , then in S , etc. b ecause any descending sequence of ordinals is

1 2

nite. Oh, I see. What we need is that P = P for some . Then p otential

guarantees that Player " can hit a p osition of p otential in S , from where he can hit

a p osition of p otential in S , and so on. Thus, p otential guarantees victory for

Player ". And of course, there is an ordinal with P = P b ecause, for each with

P 6= P , the di erence P P contains at least one p osition and there are only

+1 +1

countable many p ositions.

A: Very go o d. Let me just rep eat your argument a little more formally. By induction

on countable ordinal , de ne sets P :

{ P = A.

{ P is the set of p ositions x such that, for every n, the rank of x with resp ect to

P \ S is ordinal.

{ If is a limit ordinal then P = P :

It is easy to see that the sequence of P decreases; hence indeed P = P for some

. Fix an appropriate and de ne P = P . De ne a strategy F for Player ":

If the set N (x)=fn : x 62 S g is empty,let F (x) comprise all children of

x. Otherwise, let m =minN (x), b e the rank function corresp onding to

P \ S andtoPlayer " and let F (x) comprise all children y of x such that

(y ) <(x).

It is easy to see that F is winning if the initial p osition is in P .

Q: But now it is not obvious that Player has a winning strategy whenever the initial

p osition is not in P .

A: There is only a little work to do. If x 62 P ,let (x) b e the ordinal such that

x 2 P P and n(x)betheleastnumber n suchthatx has the extra ordinal rank

with resp ect to P \ S . De ne a strategy G for Player :

If x 62 P then G(x) comprises children y of x suchthat y has the extra

ordinal rank with resp ect to P \ S ; otherwise it comprises all children

(x) n(x)

of x.

Q: I do not understand something. Supp ose that the initial p osition x do es not b elong

to P . The idea of G seems to b e to preventPlayer " from hitting S , but this may

n(x )

b e imp ossible. 6

A: The idea is slightly more subtle. It is true that, if (x ) > 0, then S maybe

n(x )

hit at some p osition x ; but then (x ) < (x ). If (x ) > 0, then S maybehit

1 1 0 1 n(x )

at some x ; but then (x ) < (x ).

2 2 1

Q: I see. Eventually,somex is reached suchthat S is never hit, and therefore

n(x )

Player wins.

A: That's right, and the pro of of Theorem 2 is nished. In connection to sets, the

following observationisofinterest: For every set W of branches of an arena A,

there exists a subset S of A such that an arbitrary branch X b elongs to W if and only

if it hits S in nitely manytimes.To prove this, supp ose that W is the intersection

of op en sets U and let S , n 1, be a support for U . Imagine the tree A growing

n n n

upward. Let T b e the set of those no des x 2 S that there is no y 2 S b elow x; T

1 1 1 1

is still a supp ort for U .LetT b e the set of those no des x 2 S that there is y 2 T

1 2 2 1

below x but there is no y 2 S below x; T supp orts U \ U .LetT b e the set of no des

2 2 1 2 3

x 2 S such that there is y 2 T below x but there is no y 2 S below x; T supp orts

3 2 3 3

U \ U \ U . And so on. The desired S is the union of sets T .

1 2 3 n

Q: How ab out proving Martin's theorem in full?

A: I am reluctant to explain Martin's pro of here. Let me mention however that it uses

a relatively p owerful set theory. Namely, the existence of cardinal BETH is assumed

0 BETH

in the pro of of determinacy. (Recall that BETH = @ ,BETH =2 ,

0 0 +1

and BETH =supfBETH : < g if is limit.) Apparently, the assumption is

essential. Let ZC b e the standard Zermelo-Fraenkel set theory ZFC without Fraenkel's

replacement axiom. ZC has the p ower set axiom and therefore the existence of every

BETH , n< !,isprovable in ZC. However, if ZFC is consistent, then ZC has a mo del

where BETH do es not exist and where a game may b e indeterminate.

One class of Borel games is of sp ecial imp ortance in connection to the monadic second-

order theories of in nite strings and in nite trees. Let B comprise b o olean combinations

of sets.

Theorem 3 Every B game is determinate.

Q: Now the situation is quite symmetric and I wonder whichPlayer will you chose to

be Player ".

A: We'll break symmetry. Let b e a B game (A; W ). Fix sets V ;:::;V such

1 m

that W is a b o olean combination of V ;:::;V . Let Player " b e the player whose

1 m

winning set includes the intersection of all V .

Q: Why should the intersection of all sets V b e included into either winning set?

A: Every b o olean combination E of sets V either includes the intersection of all sets

V or else is disjoint from it. This can b e checked by induction on E .

i 7

Weprove Theorem 3 by induction on m. The case m = 1 is taken care by Theorem 2.

It remains to prove the induction step.

Let D b e the set of p ositions x suchthatPlayer has a winning strategy in the

remainder of the game. If the ro ot of A has an ordinal rank with resp ect to D and

to Player then Player can ensure that a p osition in D is reached and therefore

that he wins the game. Thus, wemay supp ose that the rank of the ro ot with resp ect

to D andtoPlayer is extra ordinal. Obviously,any winning strategy of Player "

is a re nement of the strategy of keeping the D -rank extra ordinal, and therefore we

may restrict attention to the subarena of p ositions reachable when Player " keeps the

D -rank extra ordinal. In other words, without loss of generality,wemay supp ose that

D is empty. In the rest of the pro of, all rank function will corresp ond to Player ".

Q: You were right ab out breaking symmetry. It remains to prove that Player " has a

winning strategy in .

A: For each V , x a no de set S such that an arbitrary branch b elongs to V if and

i i i

only if it hits S in nitely many time. Player " wins if the playhitsevery S in nitely

i i

many times. For example, Player " may adapt the strategy of hitting sets S in the

cyclic order. In the b eginning, he is in mo de 1 of going after S .IfPlayer " is in mo de

i and hits S ,thenhechanges the mo de to i +1 mod m.

Q: I do not see howPlayer " can b e sure that he will eventually hit S when he is in

mo de i.

x x

A: Let S b e the set of no des y 2 S ab ove x. If the rank of x with resp ect to S is

i i

ordinal, then Player " may b e sure to hit S in the remainder of the game. Supp ose

that Player " go es to mo de i in a p osition x such that the S -rank of x is extra ordinal

and that, moreover, Player will play a re nement of the strategy to keep the S -rank

extra ordinal. What will happ en? The strategy of keeping the S -rank extra ordinal

de nes a subarena where all branches avoid V ; it follows that, over the subarena, the

winning sets are b o olean combinations of m 1 sets. By the induction hyp othesis,

one of the players has a winning strategy F in the remainder of the game. Who can

it b e?

Q: Of course, it is Player ". Otherwise x 2 D ,but D is empty.

A: This allows us to complete the description of a winning strategy for Player ".

Supp ose he go es to mo de i in a p osition x.IftheS -rank of x is ordinal, he decreases

the rank until he hits S and go es to a new mo de. Otherwise Player " picks the strategy

F and sticks to it unless Player makes a move that results in a p osition of an ordinal

x x

S -rank. If and when that happ ens, Player " decreases the S -rank until it hits S and

i i

go es to a new mo de. That nishes the pro of of Theorem 3. 8

Q: I like the mathematics of in nite games. And I trust you that in nite games are of

great imp ortance in logic. But what ab out those applications to concurrent pro cessing?

Is it all ab out the game of an op erating system against the user?

A: Op erating systems are but one example. Others are networks and le systems [1].

Q: It seems to me that shear existence of a winning strategy do es not suce. In the

case of, say, op erating systems, the system should b e able to implement a winning

strategy.

A: You are raising a very interesting issue. I hop e we can address it one day.

Acknowledgement I am thankful to Martn Abadi, Egon Borger, Phokion Kolaitis,

Yiannis Moschovakis and Moshe Vardi for useful comments on a draft of this article.

References

[1] Martn Abadi, Leslie Lamp ort and Pierre Wolp er. Realizable and Unrealizable Concur-

rent Program Speci cations. Pro c. of 1989 ICALP conference, Springer Lecture Notes

in Computer Science.

[2] Jon Barwise. Back and Forth Through In nitary Logics. In Studies in Model Theory,

ed.M.D.Morley, 5{34. Mathematical Asso ciation of America, Studies in Mathematics,

vol. 8 (1973).

[3] J. Richard Buc hi and Lawrence H. Landweb er. Solving Sequential Conditions by Finite

State Strategies. Trans. AMS 138 (1969), 295{311.

[4] J. Richard Buc hi. State-strategies for Games in F \ G . Journal of Symb olic Logic,

Vol. 48 (1983), 1171{1198 .

[5] Harvey Friedman. Higher Set Theory and Mathematical Practice. Annals of Math.

Logic 2 (1971), 326{357.

[6] D. Gale and F.M. Stewart. In nite Games with Perfect Information. Ann. of Math.

Studies 28 (Contributions to the Theory of Games I I), 245{266. Princeton, 1953.

[7] Yuri Gurevich and Leo Harrington. Trees, Automata and Games. 14th Symp osium on

Theory of Computations, Asso ciation for Computing Machinery, 1982, 60{65.

[8] Phokion Kolaitis. Game Quanti cation. In Model Theoretic Logics, ed. J. Barwise and

S. Feferman, 365{421, Springer-Verlag, 1985.

[9] Donald A. Martin. Borel Determinacy. Annals of Mathematics 102 (1975), 363{371. 9

[10] Donald A. Martin. Purely Inductive Proof of Borel Determinacy. in Pro c. A.M.S. Sym-

p osia in Pure Math., Vol. 42, Recursion Theory, A. Nero de & R. Shore eds., 1985,

303{308.

[11] R. Daniel Mauldin, editor. The Scottish Book. Birkhauser, Boston, 1981.

[12] Yiannis N. Moschovakis. Descriptive Set Theory. North-Holland, Amsterdam, 1980.

[13] Yiannis N. Moschovakis. A Game Theoretic Modeling of Concurrency. Pro c. of

LICS'89, IEEE, to app ear.

[14] Andre A. Muchnik. Games on In nite Frees and Automata with Dead Ends: A New

Proof of the Decidability of Monadic Theory of two Successor Functions. Semiotics and

Information 24 (1984), 17{42 (Russian).

[15] Amir Pnueli and Roni Rosner. On the Synthesis of a Reactive Module. Proc.of1989

POPL conference, ACM, Jan. 1989.

[16] Philip Wolfe. The Strict Determinateness of Certain In nite Games. Paci c J. Math.

5 (1955), 841{847. 10