Solution of Wald's game using loadings and allowed strategies

Abstract

We propose a new interpretation of the strange phenomena that some authors
have observed about the Wald game. This interpretation is possible thanks to
the new language of \emph{loadings} that Morrison and the author have
introduced in a previous work. Using the theory of loadings and allowed
strategies, we are also able to prove that Wald's game admits a \emph{natural}
solution and, as one can expect, the game turns out to be fair for this
solution. As a technical tool, we introduce the notion of \emph{embedding a
game into another game} that could be of interest from a theoretical point of
view. \emph{En passant} we find a very easy example of a game which is loadable
in infinitely many different ways.

Full text

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arXiv:1106.3679v1 [cs.GT] 18 Jun 2011
SOLUTION OF WALD’S GAME USING LOADINGS AND ALLOWED
STRATEGIES
VALERIO CAPRARO
Abstract. We propose a new interpretation of the strange phenomena that some authors have
observed about the Wald game. This interpretation is possible thanks to the new language of
loadings that Morrison and the author have introduced in a previous work. Using the theory
of loadings and allowed strategies, we are also able to prove that Wald’s game admits a natural
solution and, as one can expect, the game turns out to be fair for this solution. As a technical
tool, we introduce the notion of embedding a game into another game that could be of interest
from a theoretical point of view. En passant we find a very easy example of a game which is
loadable in infinitely many different ways.
1. Introduction
Consider Wald’s game pick the bigger integer: the set of pure strategies is the set of non-negative
integers and the payoff function of player 1 (which is the negative of the payoff function of players
2) is
f(s,t) =



1,
0,
−1,
if s > t
if s = t
if s < t
Wald[Wa] proved that this game has no value if just countably additive strategies are allowed.
On the other hand, in [He-Su] it has been shown that this game has a value if one allows finitely
additive probability measures as strategies, but the value depends on the order of integration in
such a way that the internal player has an advantage1. So a game which is naturally symmetric
turns out to be asymmetric.
This note has two goals: first we want to show that this apparently strange situation is perfectly
explained and natural looking at the game from the point of view of the theory of loadings and
allowed strategies; second we want to propose a natural solution of the game using again these new
notions. It is interesting the fact that Wald’s game turns out to be fair for such a natural solution,
that is exactly what one would expect.
In the next section we recall very briefly some of the definitions given in [Ca-Mo] and we introduce
the notion of embedding of games that could be of interest from a theoretical point of view (see
Remark 2). The final section is devoted to the results that motivate this note.
2000 Mathematics Subject Classification. Primary 52A01; Secondary 46L36.
Key words and phrases. game theory, multiplicative game, loaded game, optimal strategy, maxmin strategy,
equilibrium strategy, isomorphic games, approximation of games.
Supported by Swiss SNF Sinergia project CRSI22-130435.
1See [Sc-Se] for more general results and relations with other phenomena, as de Finetti’s non-conglomerability.
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2 VALERIO CAPRARO
2. Basic notions
Let G1 and G2 be two n-player games and let Pi
the set of pure (resp. mixed) strategies of the i-th player of the j-th game. Let us denote by πj
the payoff function of the game Gj. Given a map f : Pi
push-forward strategy f∗µ ∈ Si
j(resp. Si
j), for i ∈ {1,...,n} and j = 1,2, be
1→ Pi
2and µ ∈ Si
1, we can consider the
2.
Definition 1. We say that
(1) G1embeds into G2if there exist n injective maps fi: Pi
S1
none has
1→ Pi
2such that for all (s1
1,...sn
1) ∈
1× ... × S1
π1(s1
1,...sn
1) = π2(f1∗s1
1,...fn∗sn
1))
(2) G1is isomorphic to G2if each fiis bijective.
Remark 2. (Approximation of games) The notion of embedding of games leads very naturally
to some notion of approximation of a game by taking an increasing family of subgames that converges
in some reasonable sense. This could be a useful tool to study complicated games using easier ones.
In order to develop a good theory, one would need some theorem of convergence and this is certainly
the purpose of future research. Here we are just interested in giving a first application of this notion.
Recall that a semigroup S is a set equipped with an associative binary operation S × S → S.
Given x,y ∈ S, the result of the operation is denoted by xy.
Definition 3. Let S be a semigroup and let W be a subset of S. Fix a function h : S → [−1,1]
The Operation Game associated to S, W and h, denoted by G(S,W,h), is the two-person zero-sum
game with S the set of pure strategies for both players: player 1 chooses x ∈ S and player 2 chooses
y ∈ S, then player 1 wins if xy is in W. The payoff to player 1, which is the negative of the payoff
to player 2, is h(xy).2
Remark 4. We can also consider the more general case in which the set of pure strategies of player
i is just a subset Siof S. In this case the Operation Game will be denoted by G(S1,S2,S,W,h). In
particular we are interested in the operation game G(N,−N,Z,N,χN− χZ\N), so that the payoff to
player 1 is



In [Ca-Mo] it is shown that loadings play a fundamental rule to play in a coherent way and to
solve the Operation Game. Indeed in general it is typical the situation that the game, which is
intrinsically symmetric, loses its symmetry. Here is a sketch of the construction. First we recall the
following
f(x,y) =
1,
0,
−1,
if x + y ∈ N
if x + y = 0
if x + y ∈ −N
Definition 5. A loading on S is any finitely additive probability measure on S which is left and
right invariant with respect to the operation in S. A loading is denoted by ℓ.
The loading is fixed a priori and it is part of the rules of the game, in the sense that it induces
a set of allowed strategies as follows
2The definition in [Ca-Mo] was actually little different, but the reader can easily prove that they are basically
equivalent. Indeed, in [Ca-Mo], we use h = χWand we can pass to function taking values in [−1,1] without modifying
the game with the (usual) affine transformation [0,1] ∋ λ → 2λ−1 ∈ [−1,1]. Finally each proof in [Ca-Mo] uses just
the invariance of the measures and so it could be applied to a a general function h, instead of the particular one χW.

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SOLUTION OF WALD’S GAME USING LOADINGS AND ALLOWED STRATEGIES3
Definition 6. The set of allowed strategies Aℓ is any maximal set of finitely additive probability
measures on S containing the loading ℓ and keeping the symmetry of the game, i.e.
?
S
?
S
χW(xy)dp(x)dq(y) =
?
S
?
S
χW(xy)dq(y)dp(x)
for all p,q ∈ Aℓ(where χ stands for the characteristic function).
In [Ca-Mo] the authors have shown that allowing to play just the strategies in Al, then the
Operation Game in the sense of Definition 3 has the value ℓ(W) and there are at least one optimal
strategies for both the players, which is indeed ℓ. Moreover, it is shown that there are operation
games which are loadable in infinitely many different ways. This basically means that such games
are dramatically not well defined and it is really necessary to fix the loading a priori in order to
play in a coherent way. We are going to show that this is exactly what happens in the case of
Wald’s game. By the way we recall that in many cases (finite games, compact games, other lucky
cases) the choice of a loading is not required, because there is basically a unique way to load the
game and so the game is intrinsically well defined. Unfortunately, this is not the case also for very
natural groups and semigroups, such as (Z,+) and (N,·), as is also shown in Lemma 7.
3. Main results
The following lemma is quite unexpected but probably well-known to the experts. In our case it
is necessary for the proof of the main result and also it provides a very easy example of a game which
is loadable in infinitely many different ways. We recall that at the end of [Ca-Mo], it is proposed
an example of a game which is loadable in infinitely many different ways, but this example is quite
complicated and unnatural from the point of view of a real game, namely a game that can be
proposed to human beings.
Lemma 7. For any real number s ∈ [0,1], there is a finitely additive translation invariant probability
measures m on Z such that m(N) = s. In particular there are uncountably many different ways to
load the Operation Game G(Z,N).
Proof. Let k be a positive integer. If A is a subset of Z, we define its k-density as follows
dk(A) = lim
n→∞
|A ∩ [−kn,n]|
(k + 1)n + 1
where, as usual, the notation [n,m] stands for the set of integers x such that n ≤ x ≤ m. Clearly
there are many subsets A for which dk does not exist, so let Dk= {χA,A s.t. dkexists} and let
Xk⊆ L∞(Z) be the linear span of Dkinside L∞(Z). Clearly dkextends to a linear and bounded
functional on Xk which is dominated by the L∞-norm. Hence we can apply the Hahn-Banach
extension theorem to get a positive linear functional φk : L∞(Z) → R. Now let Xk ⊆ (L∞(Z))∗
be the set of all such extensions φk. It is a convex and compact space with respect to the weak*-
topology. Convexity is indeed trivial and compactness follows from the fact that Xk is weak*
closed in the unit ball of (L∞(Z))∗, which is weak* compact by the Banach-Alouglu theorem. Now
Z acts on Xk via translations as a commutative family of operators and then Markov-Kakutani
fixed point theorem applies. Such a fixed point, say fk, is an element in L∞(Z)∗which is fixed
by every translation and such that fk(1) = 1. It follows that setting mk(A) = f(χA) we get
a translation invariant finitely additive probability measure on Z. Now since fk extends dk, we
have mk(N) = dk(N) =
(k+1). When k goes to infinity, this proves that there exist finitely additive
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4 VALERIO CAPRARO
translation invariant probability measures taking over N values arbitrarily close to 0. Now repeating
the argument with
dk(A) = lim
n→∞
|A ∩ [−n,kn]|
(k + 1)n + 1
we also prove that there exist finitely additive translation invariant probability measures which take
over N values arbitrarily close to 1. Now we know - and this is basically due to Chou [Ch] - that
the set of values which are taken by some finitely additive translation invariant probability measure
on a fixed set W is convex and closed, so, in our case, it has to be the whole interval [0,1].
?
Theorem 8. Wald’s game is equivalent to an Operation Game which is loadable in infinitely many
different ways.
Proof. The idea is easy: consider the Operation Game G(Z,N,h), with h(x) = χN(x) − χZ−N(x).
We want to embed Wald’s game into this game. More precisely we are going to show that Wald’s
game is equivalent to the game G(N,−N,Z,N,χN−χZ\N) of Remark 4 and so in particular, the set
of outcomes3is still Z. This is important in order to apply Lemma 7, since it is certainly false in
general that a subgame of a game which is loadable in infinitely many different ways is still loadable
in infinitely many different ways, but a sufficient condition to pass this property to subgames is
clearly that the set of outcomes S remains the same. Now that we have the idea, the proof is
straightforward: with the notation of Definition 1, define f1(x) = x and f2(y) = −y. So Player
1, after transforming the game, wins if and only if f1(x) + f2(y) ∈ N that happens if and only if
max{x,y} = x. Hence Player 1 wins in the Operation Game G(N,−N,Z,N,χN− χZ\N) if and only
if it wins in the Wald game and the payoff function is clearly preserved. This proves that these
two games are isomorphic and in particular Wald’s game is loadable in infinitely many different
ways.
?
Now we are ready to propose a solution for the Wald game. There is indeed a very natural
approach to solve the game G(N,−N,Z,N,χN− χZ\N), given by the fact that the set of pure
strategies of the first player is the opposite of the set of pure strategies of the second player. This
observation leads to the fact that a loading that can be accepted for playing by both the players is
any loading verifying ℓ(A) = ℓ(−A), for all A ⊆ N. It indeed reflects the point of view of both the
players. It is clear that any loading verifying such equality is such that ℓ(N) =1
2. Hence
Theorem 9. Wald’s game can be solved in a natural way and the value of the game for this solution
is 0, i.e. the game is fair.
Proof. We have already observed that a natural way to solve the game consists in fixing a loading
ℓ verifying ℓ(A) = ℓ(−A), for A ⊆ N, and then playing the allowed strategies induced by ℓ. The
point is that we cannot use directly Definition 6 to construct the set of allowed strategies, since
the mixed strategies of the players are not finitely additive probability measures on Z. Indeed the
strategies of Player 1 (resp. Player 2) are measure on N (resp. −N). But we can take inspiration
from Definition 6 and make the following construction. First of all, observe that given a measure
µ on N, we can construct a probability measure ˆ µ on Z by reflecting µ in the following way: given
A ⊆ Z, we define
ˆ µ(A) = µ(A ∩ N) + µ(−((A − 1) ∩ (−N)))
where A − 1 = {a − 1,a ∈ A}. Analogously we can construct a probability measure ˆ µ on Z by
reflecting a probability measure µ on −N. Let P(X) be the set of probability measures on a set X,
3The set of outcomes in this context is the set of xy, when x ∈ S1 and y ∈ S2.

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SOLUTION OF WALD’S GAME USING LOADINGS AND ALLOWED STRATEGIES5
we define the set of allowed strategies for G(N,−N,Z,N,χN− χZ\N) induced by ℓ to be any subset
Aℓ⊆ P(N) × P(−N) which is maximal with respect to the following two properties:
(1)
?
N
?
−N
for all (p,q) ∈ Aℓ.
(2) ℓ ∈ {ˆ µ,µ allowed strategy}
Observe that this maximal set exists and contains the pair (ˇℓ1,ˇℓ2), whereˇℓ1∈ P(N) andˇℓ2∈ P(−N)
are defined by setting
ˇℓ1(A) = ℓ(A) + ℓ(−A)
for all A ⊆ N (ˇℓ2∈ P(−N) is defined in an analogue way). It is now easy to check, being the details
basically the same as the main theorem in [Ca-Mo], that (ˇℓ1,ˇℓ2) is a profile of optimal strategies
and that the value of the game is indeed 2ℓ(N) − 1 = 2 ·1
(χN(x + y) − χZ\N(x + y))dp(x)dq(y) =
?
−N
?
N
(χN(x + y) − χZ\N(x + y))dq(y)dp(x)
2− 1 = 0.
?
References
[Ca-Mo] V. Capraro and K.Morrison, Existence of optimal strategies for the operation game on amenable semigroup,
Preprint (2011)
[Ch]C. Chou, On topologically invariant means on a locally compact group, Trans. Amer. Math. Soc. 151 (1970)
443–456.
[He-Su] D. Heath and W. Sudderth, On a theorem of de Finetti, oddsmaking, and game theory, Ann. of Math. Stat.
43 (1972) 2072–2077.
[Sc-Se] M. J. Schervish and T. Seidenfeld, A fair minimax theorem for two-person (zero-sum) games involving finitely
additive strategies, in Rethinking the Foundations of Statistics, ed: J. B. Kadane, M. J. Schervish, and T.
Seidenfeld, Cambridge: Cambridge Univ. Press (1999) 267–291.
[Mo] K. E. Morrison, The multiplication game, Math. Mag. 83 (2010) 100–110.
[Wa]A. Wald, Statistical decision functions, New York: Wiley (1950).
University of Neuchatel, Switzerland
E-mail address: valerio.capraro@unine.ch