loeb measure from the point of view of a coin flipping game

Math. Log. Quart. 42 (1996) 19 - 26

Mathematical Logic Quarterly

@ Johann Ambrosius B a t h 1996

Loeb Measure from the Point of View of a Coin Flipping Game')

Vladimir Kanovei' and Michael Reekenb

a Department of Mathematics, Moscow Transport Engineering Institute, Obraztsova 15, Moscow 101475, Russiaz) Fachbereich 7 Mathematik, Bergische Universitat, GHS Wuppertal, Gauss StraBe 20, Wuppertal42097, Germany3)

Abstract. A hyperfinitely long coin flipping game between the Gambler and the Casino, associated with a given set A, is considered. It turns out that the Gambler has a winning strategy if and only if A has Loeb measure 0. The Casino has a winning strategy if and only if A contains an internal subset of positive Loeb measure.

Mathematics Subject Classiflcation: 03H05, 03E15.

Keywords: Nonstandard andlysis, Loeb measure, Hyperfinite games.

1 Notation

Notation and terminology generally follow KEISLER, KUNEN, MILLER, and LETH [4], and LINDSTR~M [5] on matters of nonstandard andysis.

Let H be a hyperfinite natural number. The hyperfinite set S = (-1, l}H of all internal sequences of -1 and 1 is considered as the measure space with the corresponding Loeb measure L p , constructed in the usual way from the counting measure on S. The latter is defined by p ( X ) = 2-HcardX for internal sets X S, where card X is the (internal) number of elements of X. In particular, p ( S ) = L p ( S ) = 1.

A typical element of S has the form s = ((11, 0 2 , . . . , O H ) , where each ai is -1 or 1. I f n S H , then weset s l n = ( a l , ..., a,); t h e n s l n € S I n = { - l , l ) " . Weput

')The authors are pleased to mention useful conversatiom with M. BRINKMAN, N. CUTLAND,

A. PRESTEL. The authors are in debt to the refem for many corrections and useful suggestions. The first author is indebted to several institutiom and persodties who facilitate his part of work over this paper during his visiting program in 1993/94, in particular Univ&tiea of Bochum and Wuppertal, I.P.M. in Tehran, the organizers of the Oberwolfach (February 1994) and Marseille (July 1994) meetings on Nonstandard Analysis, University of Amsterdam, Caltech, and S. ALBEVERIO, M. J . A. LARIJANI, MARC DIENER, M. VAN LAMBALOEN, A. S. KECHFUS. The work of the first author was partially supported by AMS and ISF grants in 1993 and DFG grant 436 rus 17/215/93 in 1994.

C. W. HENSON, A. S. KECHFUS, H. J . KEISLER, P . LOEB, M. VAN LAMBALGEN, W. A. J . LUXEMBURG,

')e-mail: kanovei0sci.math.mu.su and kanovei0math.d-wuppertal.de %-mail: reekenQmath.uni-wuppertal.de

20 Vladimir Kanovei and Michael Reeken

where A is the empty sequence, and let L(t) denote the internal length of a sequence t E SI<H, - so that L(t) = n iff t E S i n .

For t E SI<H, - we put S(t) = {s E S : 1 c s } (the Baire inierual in S ) .

2 The coin flipping game

We refer to CUTLAND [l] and HENSON [2] for hyperfinite games, KANOVEI and LIN- TON [3] for an exposition of the standard coin flipping game.

HENSON [2] proved that certain simple (in the sense of the place they occupy in the Bore1 hierarchy) games of hyperfinite length are not determined: neither of the two players has a winning strategy. We show in this note that an interesting determined hyperfinite game still exists. This is a nonstandard version of the coin flapping g a m e .

Let A c S be a fixed set, not necessarily internal. Two players, the Gambler and the Casino, participate in the game G(A). The Casino’s aim in this game is to produce a sequence s = ( a l l . . . , aH) E S which should belong to A; the Gambler’s aim is, by betting money, either to force Casino to produce s 4 A or to gain a large amount of money if Casino is willing to reach A by all means.

At the beginning, Gambler has an amount of money, the init ial balance Bo = $1. The play goes on as follows. First of all, both Gambler and Casino know the set A and have complete information about the course of the play before the current step.

Gambler bets an amount of money b l , a hyperreal satisfying 1611 5 Bo, as to the result of the forthcoming Casino’s move a1 E {-l , l} . Casino knows b1 and moves a1 = -1 or a1 = 1. The next Gambler’s balance B1 is computed by B1 = BO + b l a 1 .

S t e p n + 1. The same way Gambler bets bn+l , lbn+1l 5 Bn, Casino knows bn+l and responds by an+l = -1 or an+l = 1, and the next Gambler’s balance is computed by

(1)

F i r s t s t e p .

Bn+l = Bn + bn+l a n + l . T h e r e s u 1 t o f t h e r u n . If both Gambler and Casino play “internally” , that

is, following certain internal strategies (see below), then this results in the internal sequence SH = ( 0 1 , . . . , aH) E S of Casino’s moves, the internal sequence ( b l , . . . , b ~ ) of Gambler’s bets, and the internal sequence of balances ( E l , . . . , Bn), computed by (l), including the final balance BH. These three sequences will be referred to as a run of the game G(A).4)

The theorems of this paper have the following general form: A has a certain property related to Loeb measure if and only if one among the players guarantees a certain result in G(A). This can be adequately formalized in terms of strategies.

S t r a t e g i e s . A strategy for Gambler is a 5ule” u which tells them how to play, for each n < HI their (n+ 1)th move bn+l = u(a1, . . . , an), given Casino% previous moves a1 , . . . , an. Technically this can be defined as a function ~7 mapping SI<H into hyperreals, so that, for all n < H and all internal sequences s = ( 4 1 , . . . , an) E {-1, l}n,

‘1 It is not an aim of this note to discuss external runs in this game. This may be quite an interesting topic. We refer to CUTLAND 111 and HENSON [2] where externd rum in another hyperfinite game are considered.

Loeb Measure from the Point of View of a Coin Flipping Game 21

Ic(s)I 5 1 + ~ ~ , 1 u(slk-1) ak (in other words, this expresses the request lb,+ll 5 S,). Similarly, a strategy for Casino is a function r mapping internal sequences of hyperreals of length 5 H into { -1,1}. Thus r recommends Casino the move an = r(b1,. . . , b,) given Gambler’s previous moves b l , . . . , b,.

All strategies considered below are assumed to be internal. D e f i n i t i o n 1. Let 0 be a sentence about the result of the game, e. g. “the final

balance BH is unlimited”. We say that Gambler guarantees 0 in G(A) iff there exists a strategy u for Gambler such that in any internal run where Gambler plays by u the following holds: S H 6 A or 0. We say that Casino guarantees 0 in G(A) iff there exists a strategy r for Casino such that in any internal run where Casino plays by r the following holds: S H E A and 0.

Thus, the way that the set A enters the game is the following: Casino musf play S H E A. Therefore Gambler can exploit the unability of Casino to play in an absolutely free way, and make reasonable predictions aiming at increasing the balance. For example if A contains only one sequence then all Casino’s moves are forced’and Gambler obviously can upgrade the initial balance Bo = 1 to BH = 2H.

C o m m e n t . The ideas behind the behaviour of Gambler and Casino in this game may be explained as follows. After n moves have been made, resulting, in particular, in a sequence s, = { a l l . . . , a , } E SI, of Casino’s moves and the Gambler’s balance B,, Gambler has to think about the Casino’s expected response an+l to their next bet b , + l .

In general Casino intends to move to sn--1 or Sn-1’ depending on which interval, S(sn--l) or S(s,-l) , contains a bigger part of A, just to have more possibilities for their play, being aware that the final sequence s should be in A. Thus Gambler has to bet exactly on the predicted direction (a positive hyperreal when Casino is expected to play 1 and a negative one otherwise). After the bet bn+l has been made, Casino moves either in the expected direction, thus, perhaps, gaining something in position, but paying by the increase of Gambler’s balance, or in the opposite direction, decreasing the balance but coming to a worse position.

3 Loeb measure zero case

In this section we are interested mainly in Gambler’s strategies which guarantee an unlimited profit in the game. It is convenient to define that Gambler wins fhe run in G(A) iff either S H 6 A or B H , the final balance, is unlimited (infinitely large). Therefore, a winning sfrategy for Gambler in G(A) is a strategy which guarantees that BH is unlimited in G(A). This definition is used only in this section and Section 5 .

S is a set of Loeb measure 0 if and only if Gambler has a winning sfrafegy in G(A).

T h e o r e m 2. A

P r o o f . (=+). Let u be a Gambler’s strategy which guarantees that BH is unlimited. We

fix a standard real number c > 0 and find an internal set C of measure 5 c-l which covers A .

Let B“ : S ~ S H - nonnegative hyperreals be the balance disfribufion generated by u. In other words, if f = (01,. . ., a,) E Sin, then

SUP) = 1 + Crc’=p(flk-l)ak


is the balance B, after n pairs of moves in the run when Casino plays t = 01, . . . , a,, and Gambler follows Q. Thus B"(A) = BO = 1 and

B'(t-a) = B'(t) + a(t) a, a = - 1 or a = 1.

Obviously B' is internal. Therefore C = {s E S ~ S H : B"(s) 2 c } is also internal. Furthermore the number m = card C of elements of C satisfies m 5 c- l 2 H , because CIES B'(s) = 2 H . Hence L p ( C ) 5 c - l . However A E C by the choice of Q.

(e). Thus we assume that L p ( A ) = 0. There exists a sequence of internal sets I,,, 5 S, m standard, such that card I , = Z H - , - therefore L p ( I m ) = 2-m - for all standard m, and A E &<. I,. In particular, lo = S. By saturation5), the sequence of sets I, can be expanded to an internal sequence of sets Im E S , m 5 H.

We assert that the expanded sequence can be chosen SO that card I , = 2H-m holds for all m 5 H. Indeed, at the beginning, there exists a nonstandard N 5 H such that this property holds for all m < N. Let IN = { x k : 1 5 k 5 2 H - N } . We define anew the sets I , , N < m 5 H, by I , = { 21: : 1 5 k 5 2 H - m } .

Finally one may assume that Im+l C I , for all m.

The strategy for Gambler we are going to define has the aim to multiply the initial balance at least by m - 1 provided Casino plays a sequence s E I,. This is again based on a balance distribution, but in this case we first define the distribution, then obtain the strategy.

Let s E Im-l \I , , m 2 1 . We put B(s) = m - 1 . Separately, B(s) = H + 1 for the unique s E I H . Thus CIES B(s ) = 2H and the function B (defined on S) is internal. We then expand B down by setting

B(t^-l) + B(t -1) 2

B ( t ) = I

so that, for all m, CtESI, B(2) = 2". Finally we obtain B(A) = 1 . The expanded B is internal because it is defined in internal manner from the internal original B (we refer to the internal definition principle in LINDSTR0M [S, p. 781).

The required strategy a is introduced by u(t) = 2- ' [B( t^l ) - B(t - - l ) ] for all t E S / < H . Obviously this strategy has the balance distribution B' equal to the distribution B. We assert that u guarantees that BH is unlimited in G(A) . Indeed, let Casino play an arbitrary sequence s = S H E S. Since A is covered by every I , with standard m, B(s) is unlimited. On the other hand, B(s ) = B'(s) is equal to B H , the

0

R e m a r k . The theorem cannot be strengthened to the form that Gambler guarantees that BH 2 P for some unlimited positive P. Indeed, let A be the collection of all sequences s E S which have an infinitely long "tail" containing only 1's. Thus A has Loeb measure 0. But, given an unlimited real P > 0, Casino guarantees (by an internal strategy, as usual) that BH < P. Therefore, in this case Gambler cannot win any prescribed unlimited amount.

final balance in the run when Casino plays s and Gambler follows Q.

')N1-~aturation is assumed throughout the paper.


4 Internal subsets and supersets

This section contains several results which show how the possibilities of the players in G ( A ) depend on the existence of internal subsets or supersets of A of certain counting measure.

T h e o r e m 3. Let r be a posit ive hyperreal. Then A has an internal superset of counting measure r-l if and only if Gambler guarantees BH 2 r in G(A) .

P r o o f . (a). Replace c by r in the proof of the +-part of Theorem 2. (e). Let I C S be an internal set such that A C I and p ( I ) = r - l . Put B(s) = r

0

T h e o r e m 4. Let r be a posit ive hyperreal. T h e n A has a n internal subset of

P r o o f . (3). Let T be a Casino’s (internal) strategy that guarantees BH 5 r in G ( A ) . We

assert that A contains an internal subset X A of counting measure 2 r - l . Since the counting measure takes discrete values, it suffices to prove that for any hyperreal 6 > 0 there exists internal X s A such that cardX 2 r- l 2H (1 - 6).

We introduce, internally, a tree S C SI<H and a function B defined on S and taking values among nonnegative hyperreals- The construction contains an infinitely small parameter E > 0 which is assumed to be very small with respect to both 6 and 2 - H . It will be provided that the condition

(2) B(s -a ) 5 2 B(s)

holds for all s and s-a E S, and for every s E S there exists at least one a = 1 or a = -1 such that s-a E S. The definition goes by the length of sequences in S. First we put A E S and B(A) = 1.

To carry out the step, we assume that s = ( ~ 1 , . . . , an ) E St,,, and it is already known that s E S (then all S k = ( ( 1 1 , . . . , O k ) , 1 5 k < n, also belong to s and the values Bk = B(sk) are defined for k 5 n so that Bk 5 2Bk-1). It will be now determined which among the extensions s--l, s-1 belongs to S (maybe both of them, but at least one). We shall also define the value(s) of B on the new member(s) s-a of s.

The values Bk, k 5 n, can be interpreted as the successive balance values in the run when Casino plays s = ( a l l . . . ,a,) and Gambler bets b k = (Bk - Bk-1) at-’ for k = 1,. . . , n, in accordance with (1). In particular, B,, = B(s ) is the balance after n pairs of moves. The assumption Bk 5 2Bk-1 guarantees the legitimacy of this run: Ibk I 5 Bk-1 for all k = 1, . . . ,n.

We consider the (n + 1)th step. For any hyperreal b such that 161 5 B(s ) = B, (that is, b is a legal Gambler’s bet in this position), let ~ [ b ] be the Casino’s response via T , that is, ~ [ b ] = ~ ( b l , . . . , b,, b ) .

C a s e 1 . ~ [ b ] = -1 for all hyperreals b, Ibl 5 B(s ) . We put s^-1 E S, s-1 $! S, and finally B(s--l) = 2 B(s ) .

for s E I and B(s) = 0 for s $! I , and follow the e - p a r t of Theorem 2 .

counting measure r-l if and only i f Casino guarantees BH 5 r in G ( A ) .


C a s e 2 . ~ [ b ] = 1 for all b. We put 5-1 E S, s--1 $! S, B(s-1) = 2 B(s) . C a s e 3 . T is not a constant on the hyperinterval [ -B(s ) ,+B(s ) ] . There exists

a pair of hyperreals b - , b+ such that ~ [ b - ] = -1, ~ [ b + ] = 1, and lb+ - b-1 < E. We put both s--1 and s-1 in S, and define B(s^-l) = B(s ) - b - , B(s-1) = B(s ) + b+; then B(s-1) + B(s--1) 5 2 B(s ) + 6. This ends the construction. Notice that ( 2 ) is not violated: B(s-a) 5 2 B(s) .

We assert that the (internal) set X = SnS of all elements of S having the maximal length H satisfies X C A and has the counting measure 2 r - l ( l -a), provided E > 0 was chosen sufficiently small.

1. Let s = ( a l l . . . , a H ) E X; we prove that s E A. Indeed, consider the run when Casino plays the sequence s while Gambler responds by

B( (a1 ,'. . . , a,)) - B( (a1 , . . . , an-1)) an

b, = I

for all n , 1 5 n 5 H. By definition, this run is in accordance with T . Therefore, by the choice of T , first, s E A, and second, the value B(s ) , which is precisely the final balance BH in this run, satisfies B(s ) 5 r.

L(,)=n B(s) . Then B[O] = B ( A ) = 1. Furthermore, we have 2 . Let B[n] =

B(s--l] = 2 B ( s ) - in Case 1,

B(s-1) = 2B(s ) - in Case 2 ,

B(s--l)+B(s-l) 2 2 B ( s ) - - E - i n c a s e 3

by the construction. This implies B[n + 11 2 2B[n] - ~ 2 " for all n. Therefore, assuming that -E > 0 is small enough, we obtain

B I H 1 = CaES, L ( S ) = H ~ ( ~ ) 2 2H (l - '1 '

Finally, since B(s ) 5 r for all s E X = {s E S : L(s) = H}, the internal number card X of elements of X is 2 r-l 2H (1 - a), as required. (e). Let X C A be an internal set of counting measure r - l . We define a strategy

T for Casino which guarantees BH 5 r in G(A) . Let n 5 HI 1 E 5'1,. We put d ( t ) = 2"-H card ( X n S(t ) ) , the density of X on

S(t) = { s E S : t s}. In particular, d ( A ) is precisely the internal counting measure of X, that is, d ( h ) = r-l. On the other hand, d(s--1) + d ( s - 1 ) = 2 d ( s ) .

We let Casino play as follows. Let B, be the balance after n steps, and let s, be the sequence (01,. . . , an) of initial n moves of Casino. Let, finally, b = b,+l be the next of Gambler's moves. Thus, if Casino responds by an+l = 1, the next balance is B$+l = B, + b; if the (n + 1)th Casino's move is an+l = -1, then the next balance is Bz+l = B, - b. (Take notice that b itself can be negative.) To respond, Casino compares the values

Q = d ( s n ) B,' , Q+ = d(sn-1) (B;+i)-', Q- = d(sn--l) (B;+i)-' .

Obviously at least one among Q+, Q- is not less than Q. Casino moves in this direction, that is, plays an+l = 1 provided Q 5 Q+, and an+l = -1 otherwise.


This can be formalized in a certain (internal, since it is defined in an internal manner) strategy r for Casino having the following property: the value Q n = d(s,) B,’ does not decrease when n increases from 0 to H.

We claim that r is the required strategy. Let, indeed, s = SH E S be the complete sequence of Casino’s moves in a run when Casino follows r. I t is asserted that, first, s E X , second, BH 5 r in this run.

Indeed, d ( s ) = 1 provided s E XI and d ( s ) = 0 otherwise. Since the values of Qn do not decrease and Qo = r- l B- l 0 = r-l > 0, we obtain d ( s ) = 1, that is, s E X. Moreover, QH = d(s) BH1 = BH1 2 Qo = r-l, which implies BH 5 r, as required. 0

C o r o 11 a r y 5. A has an iniernal subsei of a nonzero Loeb measure if and only if Casino guarantees ihat BH is limiied in G ( A ) .

P r o o f . (a). First of all, by saturation, the statement “BH is limited in any internal run

in which Casino follows r” implies the existence of a standard real number r such that the final balance BH does not exceed r whenever Casino follows r in G ( A ) . It remains to apply the *-part of Theorem 4.

0 (e). This is an immediate corollary of the +part of Theorem 3.

5 A proof of countable additivity of the Loeb measure

To demonstrate how the connection between the Loeb measure and the game can be exploited, we outline the proof of countable additivity in the easiest form: a union of countably many sets Ak, 1 5 k E w , each of Loeb measure 0, has Loeb measure 0. Thus it is assumed that Gambler wins (that is, guarantees that BH is unlimited) every game G(Ak) , and prove that he wins also G ( A ) , where A = U l l k E w A,.

Let Gk(X) denote G ( X ) started with the initial balance Bo = B; = 2-‘. Then Gambler evidently wins every game G t ( A t ) . To win G ( A ) , Gambler divides their initial balance $1 into infinitely many pieces BI, 1 5 k E w , and plays against Casino in the corresponding games Gk(Ak) using the winning strategies. Then Casino has to play s = SH E S, so that s E Ak for a t least one k. In this case, Gambler guarantees that the final balance is unlimited in GI(&) , therefore has an unlimited income even in this particular game.

We now demonstrate that this idea can be formalized in a concrete Gambler’s strategy in G(A) . This strategy requests that Gambler generates, together with the current balance B,, 0 5 n 5 H, its internal partition B , = Cf=’=, I$ onto H nonnegative hyperreals, each Bk being interpreted as the current balance in kth imaginary game, 1 5 k S H .

Let uk, for a standard k, denote a winning (internal) Gambler’s strategy in Gk(Ak). We expand this sequence of strategies to an internal sequence (oh : 1 5 k 5 H) SO that , for every nonstandard k, ck is a “legal” strategy which recommends moves satisfying lbn+ll 5 B, for Gambler. (No winning property is assumed for uk, k nonstandard.) Such an expansion can be done by the same reasons as the expansion of the sequence of sets I,,, in the proof of Theorem 2.


At the beginning, we put B; = 2'k for 1 5 k < H and Bf = 2 - H + 1 , to keep the equality cf=, B; = 1.

To describe the step, assume that 0 5 n < H and Bn = Cf=, B i has been defined in the course of the play; every Bfi being a nonnegative hyperreal. Then, for every k, Gambler plays a hyperreal 6:+1 satisfying Ib;+, I 5 B; following the strategy Uk in the position determined by moves b:, 01, 6$, a2,. . . , b;, a,,. The moves 6;+1 are summed up to bn+, = cf=, 6:+1, which is formally the Gambler's move in G(A) .

Casino responds by some an+, = -1 or an+l = 1. Obviously the updated balance B n + l = Bn+bn+lan+l in G(A) is equal to the sum zf=, Bi+, of the partial updated balances Bi+, = B; + Ef=, b;+,a ,+l .

The described behaviour of Gambler can be converted to a certain strategy u which is asserted to be a winning strategy in G(A). To prove the winning property, let Casino play the sequence S H = ( a l , ~ , . . . , aH) E A. Then S H E Ak for a standard index k. Since Gambler follows U k , a winning strategy in Gk(Ak), the kth partial balance B$ is unlimited. However the common balance B H is B$ plus the nonnegative sum cl$k @f.

References

CUTLAND, N. , A question of Borel hyperdeterminacy. Zeitschrift Math. Logik Grund- lagen Math. 30 (1984), 313 - 316. HENSON, C. W., Strong counterexamples to Borel hyperdeterminacy. Archive Math. Logic 31 (1992), 215 - 220. KANOVEI, V., and T. LINTON, Lebesgue measure and gambling. Submitted. KEISLER, H. J., K. KUNEN, A. MILLER, and S. LETH, Descriptive set theory over hypefinite sets. J. Symbolic Logic 54 (1989), 1167 - 1180. LINDSTRBM, T., An invitation to nonstandard analysis. In: Nonstandard Analysis and its Applications (N . CUTLAND, ed.), London Math. SOC. Student Texts 10, Cambridge Univ. Press, Cambridge 1988, pp. 1 - 105.

(Received: October 19, 1994; Revised: March 16, 1995)

loeb measure from the point of view of a coin flipping game

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