Complexity and Mixed Strategy Equilibria∗
Tai-Wei Hu†
Northwestern University
First Version: Nov 2008
This version: March 2010
Abstract
Unpredictable behavior is central for optimal play in many strategic situations because
a predictable pattern leaves a player vulnerable to exploitation. A theory of unpredictable
behavior is presented in the context of repeated two-person zero-sum games in which the
stage games have no pure strategy equilibrium. We represent each player’s ability to
detect patterns by a set of feasible functions, and the notion of equilibrium can be formally defined. Two dimensions of complexity of these feasible functions are considered:
one concerns the computability relation and the other concerns Kolmogorov complexity.
Existence of a Nash equilibrium is shown under a sufficient condition called mutual complexity. A tight characterization of equilibrium strategies is obtained using a criterion
called stochasticity. In particular, this characterization implies that the failure of some
statistical properties of randomness does not justify the rejection of equilibrium play.
Keywords: Kolmogorov complexity; objective probability; frequency theory of probability;
mixed strategy; zero-sum game; effective randomness
∗
I am grateful to Professor Kalyan Chatterjee for his guidance and encouragement. I also have received
many useful comments from seminar participants in Duke, Rice, Northwestern MEDS, Stanford GSB,
UCLA, Caltech, Iowa, and U Penn. All remaining errors, of course, are my own.
†
E-mail: [email protected]; Corresponding address: 2001 Sheridan road, Jacobs center
548, Evanston, IL 60208-0001, United States.
1
1
Introduction
Unpredictable behavior is central to optimal play in many strategic situations, especially
in social interactions with conflict of interests. The most illustrative examples may come
from competitive sports—think of the direction of tennis serves or soccer penalty kicks.
Indeed, people have found evidence for unpredictable behavior in professional tennis and
soccer (see, for example, Walker and Wooders [34] and Palacios-Huerta [28]). Other relevant examples include secrecy in military affairs, bluffing behavior in Poker, and tax
auditing. But why does a player aim at being unpredictable in these situations? One
suggested answer is that a detectable pattern may leave the player vulnerable to exploitation. This intuition has been around since the beginning of game theory. Von Neumann
and Morgenstern [23], based on a similar idea, points out that even against a moderately
intelligent opponent, a player will concentrate on avoiding his or her intentions being
found out in the context of the matching pennies game. More recently, Dixit and Nalebuff [10] uses this intuition to argue that optimal behavior in these situations has to be
unpredictable.
In game theory, social interactions with most serious conflict of interests are modeled
as zero-sum games, and these are situations where we expect unpredictable behavior to be
most salient. Von Neumann and Morgenstern [23], in their treatment of zero-sum games,
introduces mixed strategies to model unpredictable behavior. The notion of equilibrium
(in mixed strategies) can be formally defined, and it has been shown that equilibrium exists
in any (finite) zero-sum game. However, in a zero-sum game without any equilibrium in
pure-strategies, that theory makes little or no prediction about actions. In recognition
of this fact, there is a literature which identifies mixed strategies with beliefs and which
makes predictions about beliefs in one-shot games (see, for example, Harsanyi [12] and
Aumann and Brandenburger [3]). Indeed, as discussed in Dixit and Nalebuff [10], the
intuition that unpredictable behavior avoids detectable patterns makes little sense in a
single play. But the theory of mixed-strategy equilibrium does not have better predictive
power even in the context of repeated plays. In a game like repeated matching pennies,
any sequence of play is consistent with the (mixed-strategy) equilibrium. In particular, the
2
sequence of plays that alternate between heads and tails is consistent with that theory,
although it has an obvious pattern in it. To the best of our knowledge, a theory of
unpredictable behavior based on the above intuition that has straightforward predictions
about actions is still lacking.
In this paper, we propose a new theory of unpredictable behavior in the context
of (infinitely) repeated two-person zero-sum games in which the stage games have no
equilibria in pure-strategies. Our theory is concerned with unpredictable behavior in
terms of pattern detection. We borrow insights from other fields such as Complexity
theory and Recursion theory to model players’ abilities of pattern detection explicitly.
The notion of equilibrium can be formally defined in our framework, which requires that
each player’s equilibrium strategy best exploit the opponent’s equilibrium strategy with
respect to the patterns detectable by the player.1 We find that for equilibrium to exist,
each player’s ability to detect patterns is necessarily imperfect. Moreover, the nature
of equilibrium is determined by players’ abilities of pattern detection, and our theory
has predictions about the nature of patterns in equilibrium strategies of this repeated
game. In particular, we show a formal correspondence between the relative frequencies of
actions in our equilibrium strategies and the equilibrium stage-game mixed-strategies in
the traditional theory. Thus, in our theory, unpredictable behavior emerges as equilibrium
strategies, and its nature can be characterized to some extent.
The main challenge in our framework is therefore how to represent players’ abilities of
pattern detection. For simplicity, we illustrate our formulation with a particular game—
the repeated matching pennies game played by Ann and Bob. In this game, each player
has two actions at each stage: heads and tails. Ann’s goal in this game is to match Bob’s
actions while Bob’s goal is to mismatch Ann’s actions. The stage game payoff for Ann
is 1 if she wins and -1 otherwise, and both players use the long-run average criterion for
the repeated game payoff specification. We assume that both Ann’s and Bob’s strategies
are sequences of heads and tails.2 This assumption, however, does not affect our main
1
2
For a more detailed discussion on interpretations of equilibrium in our framework, see Section 4.
In the standard formulation of repeated games, players observe past plays and so their strategies are
functions from histories to actions. In this sense, we consider only history-independent strategies.
3
results. Before the game starts, both players have to decide what sequences to use. In
this game, we can represent the pattern in a sequence by a characteristic function. For
example, the sequence 𝜉 𝑒𝑣𝑒𝑛 = (𝜉0𝑒𝑣𝑒𝑛 , 𝜉1𝑒𝑣𝑒𝑛 , 𝜉2𝑒𝑣𝑒𝑛 , ...) defined as
𝜉 𝑒𝑣𝑒𝑛 = (𝐻(ℎ𝑒𝑎𝑑𝑠), 𝑇 (𝑡𝑎𝑖𝑙𝑠), 𝐻, 𝑇, 𝐻, 𝑇, 𝐻, 𝑇, 𝐻, 𝑇, 𝐻, 𝑇, 𝐻, 𝑇, 𝐻, 𝑇, ....)
has a simple pattern: its actions are fully determined by whether the index number is an
even number or not. This pattern can be described by the characteristic function 𝑐𝑒𝑣𝑒𝑛
of even numbers, and a player can perfectly predict the actions in this sequence if this
pattern is detected.
The standard game theory assumes that players can detect the pattern in any sequence
and can use detected patterns to exploit their opponents. For example, if Bob uses the
sequence 𝜉 𝑒𝑣𝑒𝑛 to play the game in equilibrium, then, in the standard game theory, being
able to detect this pattern represented by 𝑐𝑒𝑣𝑒𝑛 , Ann will perfectly predict Bob’s actions
and fully exploit Bob in equilibrium. In general, the pattern in any sequence of heads and
tails corresponds to a characteristic function (i.e., a function over natural numbers with
values 0 and 1) and detecting the pattern corresponding to that function implies a perfect
prediction. Instead of assuming a perfect ability of pattern detection, we give each player
a set of characteristic functions with the following interpretation: if the characteristic
function 𝑐𝐴 for a set 𝐴 ⊆ ℕ is feasible for Ann (Bob), then she (he) can detect the pattern
𝑐𝐴 and perfectly predict the sequence 𝜉 𝐴 defined as 𝜉𝑡𝐴 = 𝐻 if 𝑡 ∈ 𝐴 and 𝜉𝑡𝐴 = 𝑇 otherwise.
We also assume a close relationship between a player’s ability to detect patterns and
his or her ability to generate strategies. To explain this relationship, suppose that the
characteristic function 𝑐𝐴 is feasible for Ann. This implies that Ann has a mechanism
that determines whether a number belongs to 𝐴 or not, and hence, she can generate the
strategy 𝜉 𝐴 with that mechanism. Conversely, if the strategy 𝜉 𝐴 is available for Ann,
then it is natural to assume that Ann can perfectly predict the sequence 𝜉 𝐴 . Thus, the
function 𝑐𝐴 is feasible for Ann if and only if the sequence 𝜉 𝐴 is an available strategy to
her. The set of feasible functions for Ann then describes two aspects of Ann’s strategic
ability: first, it represents Ann’s ability to detect patterns in Bob’s strategies; second, it
determines what strategies are available to Ann. The same relationship holds for Bob as
4
well.
We give a basic assumption on the sets of feasible functions, based on the intuition
that some characteristic function is more complicated than others. For example, it seems
that the task to determine whether a number is an even number or not is simpler than the
task to determine whether a number is a prime number or not. It will be implausible if
a characteristic function for prime numbers is feasible for Ann but that for even numbers
is not. To formalize this intuition, we follow Recursion Theory and define complexity
in terms of computability. We say that a set 𝐴 is simpler than another set 𝐵 if 𝐴 is
computable 3 from 𝐵, that is, if there is a mechanical process that computes 𝑐𝐴 which
allows inquiries about membership of 𝐵 in the computation. We then assume that if the
characteristic function 𝑐𝐵 is feasible for Ann and 𝐴 is computable from 𝐵, then 𝑐𝐴 is
feasible for her as well. This implies, in particular, that any characteristic function that
can be produced by a mechanical process is feasible for both players.4
Given a set of feasible functions for Ann (Bob), we can, as we have discussed earlier,
derive the set of available strategies to Ann (Bob) in the repeated game. In this repeated
game with sets of feasible functions, the notion of equilibrium can be defined as usual.
Our main concern is the existence of equilibrium and the properties of equilibria if they
exist. The first result is a necessary condition for this existence. We show that, if Ann and
Bob have the same ability of pattern detection, or, equivalently, if they share the same
set of feasible functions, there is no equilibrium if the stage game has no equilibrium in
pure strategies. The reasoning is straightforward in the repeated matching pennies game.
Suppose, by contradiction, that Ann has an equilibrium strategy 𝜉. Then the characteristic function corresponding to 𝜉 is feasible for her. This implies that that function is also
feasible for Bob, and Bob can use that function to generate a sequence that mismatches
Ann’s action at each period. This implies that Ann will be fully exploited in equilibrium.
But if we begin with Bob’s equilibrium strategy, the same logic shows that Bob will be
3
For those who know Recursion Theory, the computability relation is also often called Turing re-
ducibility.
4
Equivalently, by Turing’s thesis, this means that any Turing computable function is feasible for either
player.
5
fully exploited by Ann in equilibrium. This leads to a contradiction. By appealing to a
similar argument, we can also show that if Ann’s ability of pattern detection is stronger
than Bob’s in terms of set inclusion, then Ann will fully exploit Bob in any equilibrium if
there is one. Therefore, to obtain an equilibrium where neither player fully exploits the
other, it is necessary that each player has at least one feasible function that is not feasible
for the other player. We call this condition mutual incomputability.
Our second result provides a sufficient condition for the existence of an equilibrium.
First we explain why mutual incomputability may not be sufficient. The problem is
related to partial predictions. As we have seen earlier, Ann can perfectly predict Bob’s
strategy 𝜉 if the characteristic function that has value 1 if and only if 𝜉 selects action
𝐻 is feasible for her. However, Ann does not have to detect this full pattern to obtain
a positive average payoff against 𝜉. If there is a set 𝐵 such that the frequency of 𝐻
in the subsequence of 𝜉 along indices in 𝐵 is, for example, 23 , and if 𝑐𝐵 is feasible for
Ann, then Ann can generate a strategy that gives a payoff at least
1
3
on average against
𝜉. Thus, if both players can partially predict any strategy of the other player in this
sense, there cannot be an equilibrium. As a result, we are not able to show that mutual
incomputability, which does not exclude partial predictions, is sufficient for the existence
of an equilibrium. To resolve this difficulty, we consider the case where each player has a
feasible function (as a sequence) that does not allow even partial predictions.
One approach to obtain such a sequence is proposed by Kolmogorov [14], which suggests a connection between unpredictability and lack of short descriptions. To discuss
complexity in terms of description lengths, we need a language, which is modeled as a
mapping from descriptions to objects. We follow the literature and use finite binary
strings as descriptions. Thus, for a given language, the Kolmogorov complexity of an
object is the length of the shortest description in that language. A natural question arises
as to what language to use. We make another assumption on each player’s set of feasible
functions to settle this question, which requires that there be a function 𝑓 in it such that
any other function in that set is computable from 𝑓 . It can be shown that, under this
assumption, there exists a universal language (for a given set of feasible functions) that
6
is computable from 𝑓 . It is universal in the sense that for any object, it gives the shortest
description asymptotically among all languages computable from 𝑓 .
Under the universal language, it can be shown that sequences that are perfectly predictable have short descriptions: If a characteristic function 𝑐𝐴 (viewed as a sequence
of 0 and 1’s) is feasible for Ann, then the Kolmogorov complexity (with respect to her
universal language) of its initial segment with length 𝑡 is asymptotically log2 𝑡, which vanishes relative to its length. Thus, sequences whose initial segments have high Kolmogorov
complexities are not perfectly predictable and may even be not partially predictable. In
fact, there are sequences whose initial segments with Kolmogorov complexities being essentially the same as their lengths. We call those sequence incompressible sequences, and
we find these sequences relevant to the existence of an equilibrium in our framework.
We use incompressible sequences to formulate our sufficient condition for equilibrium
existence. The properties of incompressible sequences have been well studied in the literature (see, for example, Downey et al. [11]), but most of the studies focus on decision
problems with a single agent. To apply this concept to our framework, where two agents
interact with each other, we formulate a new notion called mutual complexity. It requires
each player to have a feasible function (viewed as a sequence) that is an incompressible
sequence from the other player’s perspective. Mutual complexity is satisfied by a majority of sets of feasible functions. We show that if mutual complexity holds, there is an
equilibrium that has the same value (which represents equilibrium payoffs) as the stage
game in any repeated zero-sum game with sets of feasible functions. Together with our
first result, this implies that each player’s equilibrium strategies thus obtained are not
perfectly predictable by the opponent. In this sense, unpredictable behavior emerges as
equilibrium strategies. However, not every strategy that is not perfectly predictable is
optimal; we have some results regarding the nature of equilibrium strategies in terms of
patterns that do and do not appear in them.
We find a property called stochasticity relevant to describe the nature of equilibrium
strategies. To explain this property, consider again the matching pennies game. Suppose
that the characteristic function 𝑐𝐴 is feasible for Ann. As we have seen earlier, if Bob
7
uses a sequence which takes action 𝐻 more frequently than 𝑇 along the indices in the
set 𝐴 in equilibrium, Ann can obtain a positive average payoff along the indices in 𝐴,
and hence obtain a positive average payoff. However, Ann’s equilibrium payoff is 0 under
mutual complexity. This will not happen if the frequency of 𝐻 in the subsequence of
Bob’s strategy along indices in 𝐴 is 12 . If a sequence satisfies this property for any set 𝐴
such that 𝑐𝐴 is feasible for Ann, we say that it is stochastic relative to Ann.5 Using these
ideas, we show that any stochastic sequence is also an equilibrium sequence. However,
there may be equilibrium strategies that are not stochastic, partly because of possible
multiplicity of mixed-strategy equilibria in the stage game.6 Nonetheless, stochasticity
seems to be a good approximate requirement for optimality in our framework.
Here we discuss the generality of our results. Our previous discussions focus on the
matching pennies game, but the results are general for any repeated game whose stage
game has a finite number of actions. To accommodate an arbitrary number of actions in
the stage game, however, the set of feasible functions is formulated to include functions
with arbitrary ranges within the set of natural numbers. Moreover, we also consider
a model where players can observe past plays, and show that mutual incomputability
is necessary and mutual complexity is sufficient for the existence of an equilibrium. A
notion similar to stochasticity is also relevant to describe equilibrium strategies in that
model. As mentioned earlier, we use long-run average criterion for payoff specifications.
If this assumption is replaced by the discounting criterion, we can show that there is
no equilibrium regardless of the sets of feasible functions, as long as they satisfy our
assumption that any computable function is feasible for both players.
There is also a remarkable connection between equilibrium strategies in our theory and
stage-game equilibrium mixed-strategies: there is an equilibrium strategy in the repeated
game7 that satisfies mutual complexity with a specific relative frequency of actions if and
5
6
Notice that here stochasticity is a property of deterministic sequences instead of random variables.
See Section 3.2 for a more detailed discussion about stochasticity as a necessary requirement for
optimality in repeated games with sets of feasible functions that satisfy mutual complexity.
7
The repeated game refers to the game with sets of feasible functions and the sets of available strategies
are determined by those feasible functions.
8
only if that frequency constitutes a mixed equilibrium in the stage game. In this sense,
we derive probability values from unpredictable behavior in our theory. However, we also
show that there are equilibrium strategies which exhibit patterns that are not expected
from the i.i.d. distribution generated by the stage-game mixed equilibrium. Thus, failure
of certain statistical tests for an i.i.d. distribution does not entail the rejection of the
equilibrium hypothesis in our model.
We close the introduction with some related literature. The idea that players cannot comprehend the full pattern in any sequence has already appeared in Piccione and
Rubinstein [30] and Aragone et al. [1]. However, both papers have quite different formal treatments than ours and both have different purposes. In terms of formal analysis,
the most related paper to ours may be Ben-Porath [4], which studies infinitely repeated
zero-sum games as well. In that model, players use finite automata to generate strategies.
Hence, not every strategy is available to a player. The focus of that paper is to analyze
the effect of strategic complexity on equilibrium behavior. However, because complexity
is measured in terms of number of states in players’ automata, only one player can have
more complicated strategies than the other. As a result, mixed strategies are necessary
to obtain an equilibrium in that model, and hence it does not deal with unpredictable
behavior directly. In contrast, our theory does not assume randomization and shows that
unpredictable behavior can emerge as equilibrium behavior.
The rest of the paper is organized as follows. In Section 2 we formulate horizontal
games and vertical games (two different versions of the repeated game) and present nonexistence results. Section 3 has two parts: we first formulate the notion of incompressible
sequences using Kolmogorov complexity and give an existence result; we then discuss the
unpredictable behavior thus obtained as equilibrium strategies. We give some concluding
remarks in Section 4. The proofs of the main theorems are in Section 5.
9
2
Repeated games with feasible functions
In this section we formulate our framework. We give two alternative formulations of the
repeated game with sets of feasible functions; one is called the horizontal game (𝐻𝐺) and
the other is called vertical game (𝑉 𝐺). Both games consist of infinite repetitions of a
finite zero-sum game, but they differ in their information structures. In both games each
player is given a set of feasible functions to represent his or her ability to detect patterns
and to generate strategies. We begin this section with those feasible functions, and end
with a necessary condition for the existence of an equilibrium in these games.
2.1
Feasible functions
Our approach begins with a representation of players’ abilities to detect patterns. We
model these abilities by giving each player a set of feasible functions over natural numbers.
The intended interpretation is the following. Given a finite set 𝑋 of certain objects, a
player can grasp the full pattern of an infinite sequence 𝜉 over 𝑋 and perfectly predict it
if that sequence (as a function over natural numbers) is feasible for that player. Moreover,
a feasible function may also be used to capture a partial pattern in a sequence and make
partial predictions about it. Partial predictions become more relevant in Section 3. For
our purpose the set 𝑋 will be the set of actions in the stage game. First we define the set
of feasible functions and make an assumption on it, and in the next subsection we define
horizontal and vertical games.
Because we consider only finite stage-games, it is sufficient to consider functions from
natural numbers to natural numbers. However, we include also functions over vectors of
natural numbers. Because there is a constructive method to encode vectors of natural
numbers with natural numbers, this inclusion is not substantial. However, it does simplify
10
many definitions.8 The set of all such functions is denoted by
ℱ=
∞
∪
{𝑓 : ℕ𝑘 → ℕ},
𝑘=1
where ℕ = {0, 1, 2, ...} is the set of nonnegative numbers. We give each player 𝑖 a subset
𝒫 𝑖 of ℱ, which we call the set of feasible functions for 𝑖. 𝒫 𝑖 represents player 𝑖’s ability
to detect patterns.
We give one basic assumption on a player’s feasible functions: if a function is feasible
for a player, so is any function computable from that function. It is natural to assume
that if a function is feasible for a player, so is any function that is simpler. Here we
adopt the computability relation as our complexity measure. This relation measures
complexity qualitatively: a function is more complicated than another if the latter is
computable from the former. To define this relation, we introduce basic functions and
basic operations. The basic functions include (1) the zero function 𝑍, defined by 𝑍(𝑡) = 0;
(2) the successor function 𝑆, defined by 𝑆(𝑡) = 𝑡 + 1; (3) the projection functions 𝑈𝑘𝑖 ,
defined by 𝑈𝑘𝑖 (𝑡1 , ..., 𝑡𝑘 ) = 𝑡𝑖 , with 𝑘 > 0, 𝑖 = 1, .., 𝑘. The basic operations include
(1) composition: from functions 𝑓 (𝑠1 , ..., 𝑠𝑙 ), 𝑔1 (𝑡1 , ..., 𝑡𝑘 ), ..., and 𝑔𝑙 (𝑡1 , ..., 𝑡𝑘 ), obtain
ℎ ≃ 𝑓 (𝑔1 (𝑡1 , ..., 𝑡𝑘 ), ..., 𝑔𝑙 (𝑡1 , ..., 𝑡𝑘 ));
(2) primitive recursion: from functions 𝑓 and 𝑔, obtain ℎ defined as
ℎ(0, 𝑡1 , ..., 𝑡𝑘 ) ≃ 𝑓 (𝑡1 , ..., 𝑡𝑘 ), and
ℎ(𝑠, 𝑡1 , ..., 𝑡𝑘 ) ≃ 𝑔(𝑠 − 1, ℎ(𝑠 − 1, 𝑡1 , ..., 𝑡𝑘 ), 𝑡1 , ..., 𝑡𝑘 ) for 𝑠 > 0.
(3) minimization: from 𝑓 satisfying that for any (𝑡1 , ..., 𝑡𝑘 ) there is some 𝑠 such that
𝑓 (𝑠, 𝑡1 , ..., 𝑡𝑘 ) = 0, obtain 𝑔 defined as
𝑔(𝑡1 , ..., 𝑡𝑘 ) = min{𝑠 : 𝑓 (𝑠, 𝑡1 , ..., 𝑡𝑘 ) = 0 and 𝑓 (𝑟, 𝑡1 , ..., 𝑡𝑘 ) is not zero for all 𝑟 < 𝑠}.
The computability relation is defined as follows.
8
For example, the projection function 𝑈𝑖𝑛 (𝑥1 , ..., 𝑥𝑛 ) = 𝑥𝑖 is very simple to express and understand as
a function over vectors of natural numbers, but it can look complicated if we translate it into a function
over natural numbers.
11
Definition 2.1. A function 𝑓 ∈ ℱ is computable from a function 𝑔 ∈ ℱ if there is a
sequence 𝑓1 , ..., 𝑓𝑛 of functions in ℱ such that 𝑓1 = 𝑔, 𝑓𝑛 = 𝑓 , and for each 𝑖 = 2, ..., 𝑛,
𝑓𝑖 is either a basic function or is obtained from functions in 𝑓1 , ..., 𝑓𝑖−1 through a basic
operation.
Our basic assumption is that any player can implement these basic functions and basic
operations. Thus, if a function can be transformed into another via those basic operations
(possibly with the aid of basic functions), then any player who can implement the former
function is also able to implement the latter. This assumption is stated in the following.
Assumption A1 If 𝑓 ∈ 𝒫 𝑖 and if 𝑔 is computable from 𝑓 , then 𝑔 ∈ 𝒫 𝑖 .
A natural way to compare different players’ abilities of pattern detection is by set
inclusion: player 𝑖 is more competent than player 𝑗 if 𝒫 𝑗 ⊂ 𝒫 𝑖 . Because the computability
relation is transitive, (A1) implies that if any function feasible for player 𝑗 is simpler (in
terms of computability) than some function that is feasible for 𝑖, then 𝑖 is more competent
than 𝑗. Also, it gives a lower bound on players’ abilities: any Turing-computable function
is feasible for both players.9 We remark that the computability relation is not complete,
and hence two players’ abilities may not be comparable. In fact, as we demonstrate later,
this property is crucial for an equilibrium to exist.
To use feasible functions to detect patterns and generate corresponding strategies, one
has to encode both actions and finite sequences of actions (as partial histories of past
actions). Assumption (A1) allows us to focus on a particular encoding. First, any finite
set of actions 𝑋 = {𝑥1 , ..., 𝑥𝑚 } can be identified with the set m = {1, ..., 𝑚}, and hence,
any finite sequences over 𝑋 can be identified with a finite sequence 𝜎 over m. Any such
∑
𝑡
𝜎 can be encoded into ℕ using the mapping 𝜎 7→ ∣𝜎∣−1
𝑡=0 𝜎𝑡 𝑚 . Hence, any function from
ℕ to 𝑋, or equivalently, any infinite sequence over 𝑋 (𝑋 ℕ denotes the set of such infinite
sequences), can be identified with a function in ℱ. Similarly, any function from finite
sequences over 𝑋 (𝑋 <ℕ denotes the set of such finite sequences) to another finite set 𝑌
can be identified with a function in ℱ as well. If a set 𝒫 satisfies (A1), then whether such
9
For a detailed discussion of Turing-computability, see Odifreddi [25].
12
functions belong to 𝒫 or not does not depend on the encoding method. In general, this
would not be the case without (A1).
2.2
Horizontal and vertical games
Given a player’s ability of pattern detection, the set of available strategies to that player
is determined by this ability and the rules of the game. We focus on a finite two-person
zero-sum game without any pure equilibrium that is repeatedly played, and we consider
two different rules to play the repeated game. In a horizontal game, both players simultaneously decide an infinite sequence of actions for all stage games. One example of a
horizontal game is found in Luce and Raiffa [20], which has two aerial strategists deciding
the actions of their pilots in a conflict consisting of many identical aircraft fights. They
use this example to illustrate the meaning of a mixed strategy, which is interpreted as the
distribution of different actions assigned to the pilots. On the other hand, in a vertical
game, players can contingent their actions at each stage on the actions that have been
played in previous stages, which resembles the usual formulation of repeated games.
These two different versions of our model are helpful for several reason. First, horizontal games are easier to analyze and many results on horizontal games hold for vertical
games as well. Second, the two versions help to illustrate a subtle aspect of our theory.
Although in both cases the equilibrium behavior is unpredictable in the sense that there
is no detectable pattern in it that allows a perfect prediction, some patterns that appear
in equilibrium strategies of the horizontal game do not appear in those of the vertical
game, even if we begin with the same abilities of pattern detection. This illustrates how
rules of the game can affect unpredictable behavior that emerges in equilibrium.
Now we formulate our model. The stage game is a finite two-person zero-sum game
𝑔 = ⟨𝑋, 𝑌, ℎ⟩, where 𝑋 = {𝑥1 , ...., 𝑥𝑚 } is the set of player 1’s actions, 𝑌 = {𝑦1 , ..., 𝑦𝑛 } is
the set of player 2’s actions, and ℎ : 𝑋 × 𝑌 → ℚ is the von Neumann-Morgenstern utility
function for player 1, where ℚ denotes the set of rational numbers. We use
Δ(𝑋) = {𝑝 ∈ ([0, 1] ∩ ℚ)𝑚 :
∑
𝑥∈𝑋
13
𝑝𝑥 = 1}
and
Δ(𝑌 ) = {𝑞 ∈ ([0, 1] ∩ ℚ)𝑛 :
∑
𝑞𝑦 = 1}
𝑦∈𝑌
to denote the set of mixed strategies (with rational probability values) of player 1 and
2, respectively.10 Because ℎ is rational-valued, there is always a mixed equilibrium with
rational probability values in 𝑔. Our analysis focuses on games without pure-strategy
equilibria. We say that player 1 fully exploits player 2 if player 1’s equilibrium payoff
in the horizontal or the vertical game is min𝑦∈𝑌 max𝑥∈𝑋 ℎ(𝑥, 𝑦). The horizontal and the
vertical games are defined as follows.
Definition 2.2. Let 𝑔 = ⟨𝑋, 𝑌, ℎ⟩ be a finite zero-sum game and let 𝒫 1 , 𝒫 2 be two sets
of functions satisfying (A1). The horizontal game 𝐻𝐺(𝑔, 𝒫 1 , 𝒫 2 ) is a triple ⟨𝒳 , 𝒴, 𝑢ℎ ⟩
such that
(a) 𝒳 = {𝜉 : ℕ → 𝑋 : 𝜉 ∈ 𝒫 1 } is the set of player 1’s strategies;
(b) 𝒴 = {𝜁 : ℕ → 𝑌 : 𝜁 ∈ 𝒫 2 } is the set of player 2’s strategies;11
(c) 𝑢ℎ : 𝑋 ℕ × 𝑌 ℕ → ℝ is player 1’s payoff function defined as
𝑢ℎ (𝜉, 𝜁) = lim inf
𝑇 →∞
𝑇 −1
∑
ℎ(𝜉𝑡 , 𝜁𝑡 )
𝑡=0
𝑇
,
(1)
while player 2’s payoff function is −𝑢ℎ .
Definition 2.3. Let 𝑔 = ⟨𝑋, 𝑌, ℎ⟩ be a finite zero-sum game and let 𝒫 1 , 𝒫 2 be two sets
of functions satisfying (A1). The vertical game 𝑉 𝐺(𝑔, 𝒫 1 , 𝒫 2 ) is a triple ⟨𝒳 , 𝒴, 𝑢ℎ ⟩ such
that
(a) 𝒳 = {𝑎 : 𝑌 <ℕ → 𝑋 : 𝑎 ∈ 𝒫 1 } is the set of player 1’s strategies;
(b) 𝒴 = {𝑏 : 𝑋 <ℕ → 𝑌 : 𝑏 ∈ 𝒫 2 } is the set of player 2’s strategies;12
10
11
We use rational numbers because of computability issues.
Here, it is implicitly assumed that the sets 𝑋 and 𝑌 are identified with a subset of natural numbers,
and hence each strategy in the horizontal game can be identified with a function in ℱ. As we have
remarked earlier, (A1) ensures that 𝒳 and 𝒴 are well defined regardless of the coding methods.
12
Here the implicit assumption is that any history, as a finite sequence of actions, is identified with a
code number as mentioned in an earlier remark. Again, the sets 𝒳 and 𝒴 are well-defined because of
(A1).
14
(c) 𝑢ℎ : 𝑋 ℕ × 𝑌 ℕ → ℝ is player 1’s payoff function defined as equation (1), while player
2’s payoff function is −𝑢ℎ .
As mentioned in the introduction, we assume a close relationship between pattern
detection and strategy generation in our formulation of horizontal and vertical games: a
strategy is available to a player if and only if that player can detect the full pattern in
it and perfectly predicts it.13
14
As a result, Nash equilibrium is directly applicable in
both horizontal and vertical games, and our analysis focuses on the existence of a Nash
equilibrium and its properties. Roughly speaking, a pair of strategies constitute a Nash
equilibrium if for each player, there is no detectable pattern in the opponent’s strategy
that that player can employ another strategy to exploit it better than the equilibrium
strategy.15
Our first result is a necessary condition for an equilibrium to exist. If the players’
abilities are comparable, then the more competent player can perfectly predict the other
player’s equilibrium strategy and hence will fully exploit the opponent in equilibrium.
However, it cannot be the case that both players fully exploit the other in equilibrium
unless there is a pure equilibrium in the stage game. The following proposition, whose
proof is given in Section 5, formalizes these intuitions.
Proposition 2.1. Let 𝑔 = ⟨𝑋, 𝑌, ℎ⟩ be a two-person zero-sum game without any purestrategy equilibrium. Suppose that 𝒫 1 and 𝒫 2 satisfy (A1).
(a) If 𝒫 1 = 𝒫 2 , then there is no equilibrium in 𝐻𝐺(𝑔, 𝒫 1 , 𝒫 2 ).
13
Notice that 𝒫 1 and 𝒫 2 enter our formulation of the repeated games through 𝒳 and 𝒴 only. An
alternative approach is to define 𝒳 and 𝒴 directly and make assumptions on them. However, that
approach would make our formulation less transparent. Moreover, it is difficult to compare players’
abilities of pattern detection in that approach. As we will see later, this comparison is crucial—in fact,
we are mostly concerned with how different relations between players’ abilities affect the existence of an
equilibrium and its properties.
14
This relationship does not exist, however, in the randomization device interpretation of mixed strategies. In that interpretation, the outcomes of the randomization device are unpredictable to the player
who employ it. However, that interpretation has caused much conceptual discomfort as expressed in
Aumann [2] and Rubinstein [32].
15
For more discussion about the interpretations of Nash equilibrium in our model, see Section 4.
15
(b) If 𝒫 2 ⊂ 𝒫 1 , then the equilibrium payoff (if equilibrium exists) in 𝐻𝐺(𝑔, 𝒫 1 , 𝒫 2 ) for
player 1 is min𝑦∈𝑌 max𝑥∈𝑋 ℎ(𝑥, 𝑦).
This proposition holds for vertical games as well. We will focus on horizontal games
only after this point, because most results hold for both formulations. We return to results
specific to vertical games in section 3.3.
Although we use the long-run average criterion in Proposition 2.1, it is rather clear
from the proof that this proposition does not depend on this specification. It holds as
long as the repeated game is a zero-sum game and the overall payoffs are monotonic with
the payoffs from each stage. In part (b) we give no information about the existence of
an equilibrium, because in this case whether an equilibrium exists or not depends on the
payoff specification, as shown in the following example.
Example 2.1. Consider 𝑔 𝑀 𝑃 = ⟨{𝐻𝑒𝑎𝑑𝑠, 𝑇 𝑎𝑖𝑙𝑠}, {𝐻𝑒𝑎𝑑𝑠, 𝑇 𝑎𝑖𝑙𝑠}, ℎ⟩ with
ℎ(𝐻𝑒𝑎𝑑𝑠, 𝐻𝑒𝑎𝑑𝑠) = 1 = ℎ(𝑇 𝑎𝑖𝑙𝑠, 𝑇 𝑎𝑖𝑙𝑠) and ℎ(𝐻𝑒𝑎𝑑𝑠, 𝑇 𝑎𝑖𝑙𝑠) = 0 = ℎ(𝑇 𝑎𝑖𝑙𝑠, 𝐻𝑒𝑎𝑑𝑠).
In Section 5 (see Lemma 5.1), we show that for any countable 𝒫 1 , there exists a set 𝒫 2
such that 𝒫 1 ⊂ 𝒫 2 and there is an equilibrium in 𝐻𝐺(𝑔 𝑀 𝑃 , 𝒫 1 , 𝒫 2 ).16 Here we show
that if 𝒫 2 ⊂ 𝒫 1 , then there is no equilibrium in 𝐻𝐺(𝑔 𝑀 𝑃 , 𝒫 1 , 𝒫 2 ). Suppose that, by
contradiction, it has an equilibrium. Then the equilibrium payoff to player 1 is 1. Let 𝜉 ∗
be an equilibrium strategy of player 1. Then,
∑𝑇 −1
∑𝑇 −1
∗
∗
𝑡=0 ℎ(𝜉𝑡 , 𝐻𝑒𝑎𝑑𝑠)
𝑡=0 ℎ(𝜉𝑡 , 𝑇 𝑎𝑖𝑙𝑠)
lim inf
= 1 = lim inf
.
𝑇 →∞
𝑇 →∞
𝑇
𝑇
This implies that
1 = lim inf
𝑇 →∞
≥ lim inf
𝑇 →∞
∣{0 ≤ 𝑡 ≤ 𝑇 − 1 : 𝜉𝑡∗ = 𝐻𝑒𝑎𝑑𝑠}∣ + ∣{0 ≤ 𝑡 ≤ 𝑇 − 1 : 𝜉𝑡∗ = 𝑇 𝑎𝑖𝑙𝑠}∣
𝑇
∣{0 ≤ 𝑡 ≤ 𝑇 − 1 : 𝜉𝑡∗ = 𝐻𝑒𝑎𝑑𝑠}∣
∣{0 ≤ 𝑡 ≤ 𝑇 − 1 : 𝜉𝑡∗ = 𝑇 𝑎𝑖𝑙𝑠}∣
+ lim inf
= 2,
𝑇 →∞
𝑇
𝑇
which is a contradiction. Thus, there is no equilibrium in 𝐻𝐺(𝑔 𝑀 𝑃 , 𝒫 1 , 𝒫 2 ).
16
The author is grateful to Eran Shmaya for pointing this out.
16
Proposition 2.1 then gives a necessary condition for the existence of an equilibrium that
describes unpredictable behavior. It shows that if 𝒫 1 ⊂ 𝒫 2 or 𝒫 2 ⊂ 𝒫 1 , in equilibrium (if
an equilibrium exists) the player with better ability perfectly predicts the other’s strategy
and fully exploits the other. Example 2.1 shows that such an equilibrium may not exist.
Thus, to obtain an equilibrium in general, it is necessary that each player has a function
that is not feasible for the other. We call this condition mutual incomputability.
In the next section we give results about the existence of an equilibrium. We close this
section with some comments on the payoff specification and how it affects equilibrium. We
use the long-run average criterion (1) in defining horizontal and vertical games.17 Hu [13]
gives an axiomatization of expected utility from the frequentist perspective that serves as
a foundation for this criterion. However, the next proposition shows that the discounting
criterion is not consistent with the existence of an equilibrium in our model.
Proposition 2.2. Let 𝑔 = ⟨𝑋, 𝑌, ℎ⟩ be a two-person zero-sum game without any pure
strategy equilibrium. If 𝒫 1 and 𝒫 2 satisfy (A1), then there is no equilibrium in
𝐻𝐺′ (𝑔, 𝒫 1 , 𝒫 2 ) = ⟨𝒳 , 𝒴, 𝑣ℎ ⟩,
where 𝑣ℎ is defined by 𝑣ℎ (𝜉, 𝜁) = (1 − 𝛿)
∑∞
𝑡=0
𝛿 𝑡 ℎ(𝜉𝑡 , 𝜁𝑡 ).
The proof of this proposition, which is given in Section 5. The crucial observation is
that (A1) implies that each player can detect the full pattern in any finite initial segment
of the opponent’s strategy and hence can fully exploit that part, and under discounting
criterion, that is sufficient for a player to obtain an overall payoff that is close to full
exploitation. As a result, to obtain an equilibrium with the discounting criterion, (A1)
has to be violated. We leave this difficulty for further research.
We have shown that mutual incomputability is necessary for an equilibrium to exist,
even with the long-run average criterion. However, this condition does not seem sufficient.
Consider the horizontal game based on matching pennies. Suppose that player 1 plays
17
∑𝑇 −1
Because the sequence { 𝑡=0
ℎ(𝜉𝑡 ,𝜁𝑡 ) ∞
}𝑇 =1
𝑇
may not have a limit, it is necessary to introduce limit
inferior, limit superior, or other such notions that extend long-run averages. Our main results are robust
to any such notion between the limit inferior and limit superior.
17
𝐻𝑒𝑎𝑑𝑠 at all places with odd indices and uses an incomputable sequence (from player
2’s perspective) at even ones. The resulting strategy is incomputable from player 2’s
perspective, but player 2 can easily exploit it by playing 𝑇 𝑎𝑖𝑙𝑠 at odd places. As we have
mentioned in the Introduction, this kind of partial predictions is a barrier to the existence
of an equilibrium. In the next section, we introduce another notion of complexity to
resolve this difficulty and obtain the existence of an equilibrium.
3
Complexity and unpredictable behavior
So far we have focused on the complexity of functions in terms of computability. Through
this notion we derive a necessary condition for the existence of an equilibrium, but that
condition does not seem sufficient. In this section we strengthen this condition so that
each player has a function that is not only infeasible for the other player, but is highly
complicated from the other player’s perspective. This condition is formalized with Kolmogorov complexity, which measures how incomputable a function is. We show that if
each player has a function with high Kolmogorov complexity from the other player’s perspective, then there exists an equilibrium. Then we discuss properties of the equilibria
thus obtained.
3.1
Kolmogorov complexity and existence
Kolmogorov complexity [14] measures the complexity of a finite object in terms of its
minimum description lengths. Later we apply it to measure the complexity of functions,
which are our targets. This complexity measure is based on the idea that an object is
more complex if it is harder to describe. To formalize this idea, a language is modeled as a
mapping from words (descriptions) to objects that are to be described. Given a language,
it is then meaningful to discuss the shortest description for a given object. Typically, finite
strings over {0, 1} are used as descriptions (see Li and Vitányi [19]). Here we consider the
complexity of finite binary sequences 𝜎 ∈ {0, 1}<ℕ . A function 𝐿 : {0, 1}<ℕ → {0, 1}<ℕ is
called a language. Its domain consists of descriptions and its range consists of objects to
18
be described. In the literature, another type of languages is also considered: languages
that are partial functions 𝐿 :⊆ {0, 1}<ℕ → {0, 1}<ℕ , which may not be defined over some
strings. If 𝐿 is not defined over a string 𝜎, it is denoted by 𝐿(𝜎) ↑; otherwise, it is denoted
by 𝐿(𝜎) ↓. The set of strings over which 𝐿 is defined is called the domain of 𝐿, denoted
by 𝑑𝑜𝑚(𝐿). A string 𝜏 in 𝑑𝑜𝑚(𝐿) is called a code-word, or simply a word.
The Kolmogorov complexity of a sequence 𝜎 is defined as
𝐾𝐿 (𝜎) = min{∣𝜏 ∣ : 𝜏 ∈ {0, 1}<ℕ , 𝐿(𝜏 ) = 𝜎},
and 𝐾𝐿 (𝜎) = ∞ if there is no word 𝜏 such that 𝐿(𝜏 ) = 𝜎. In this setting, Kolmogorov
complexity measures how much a sequence can be compressed. We may say that a
sequence is incompressible if its minimum description length is equal to its own length.
Notice that this complexity gives different measures if different languages are used. There
is then a natural question about which language to use. Two aspects of this question have
been discussed in the literature: one is concerned about what kind of languages should
be used; the other is concerned with which particular language should be used. Partial
languages play a crucial role in both aspects.
For the first aspect, there is a literature (e.g., Chaitin [7] and Levine [17]) which argues
that only prefix-free languages give meaningful measures of the Kolmogorov complexity.
A languages 𝐿 is prefix-free if its domain has the following property: for any words
𝜎, 𝜏 ∈ 𝑑𝑜𝑚(𝐿), 𝜎 is not an initial segment of 𝜏 . If a language is not prefix-free, a
description from that language may not capture all the information of the object under
consideration. Notice that if a language 𝐿 is prefix-free, then 𝐿 is a partial function, i.e.,
𝑑𝑜𝑚(𝐿) ∕= {0, 1}<ℕ . Following this literature, we consider prefix-free languages only.
The second aspect is related to computability issues. It is well known that if only
Turing-computable languages18 are considered, there is a universal language that gives
shortest descriptions asymptotically and hence is the optimal language to use. For our
purpose, we consider the set of prefix-free languages 𝐷(𝒫 𝑖 ) that is permissible for a given
18
Later on we will see how to identify a language with a function over natural numbers and hence the
notion of Turing-computability is applicable. A function is Turing-computable if it is computable from
any function.
19
𝒫 𝑖 in the sense that any language in 𝐷(𝒫 𝑖 ) is computable from a function in 𝒫 𝑖 . As we
show later, we need one additional assumption to obtain the universal language.
The formal definition of 𝐷(𝒫 𝑖 ) takes some further definitions. First, we encode binary
strings into natural numbers,19 and then any language 𝐿 can be identified with a partial
function 𝑓𝐿 over ℕ. If 𝐿 is total, it is then straightforward to define 𝐷(𝒫 𝑖 ) by requiring
𝐿 ∈ 𝐷(𝒫 𝑖 ) if and only if 𝑓𝐿 ∈ 𝒫 𝑖 . Now, because we consider only prefix-free languages,
which are partial functions, we need to amend our definition of computability. To this
end, we introduce an extended version of the minimization operator:
(3’) minimization: from function 𝑓 obtain 𝑔 defined as
𝑔(𝑡1 , ..., 𝑡𝑘 ) = min{𝑠 : 𝑓 (𝑠, 𝑡1 , ..., 𝑡𝑘 ) = 0 and 𝑓 (𝑟, 𝑡1 , ..., 𝑡𝑘 ) is defined for all 𝑟 < 𝑠};
𝑔(𝑡1 , ..., 𝑡𝑘 ) is undefined otherwise.
Adding this new minimization operator as a basic operator, the definition of computability
in Definition 2.1 is extended to partial functions as well. Thus, a prefix-free language
𝐿 ∈ 𝐷(𝒫 𝑖 ) if and only if 𝑓𝐿 is computable from some 𝑔 ∈ 𝒫 𝑖 with this new definition.
To obtain a universal language, we give one additional assumption on the 𝒫 𝑖 ’s.
Assumption A2 There is a function 𝑓 ∗ ∈ 𝒫 𝑖 such that any function 𝑓 ∈ 𝒫 𝑖 is computable
from 𝑓 ∗ .
Assumption (A1) gives a lower bound on what 𝒫 𝑖 has to contain; assumption (A2), on
the other hand, gives an upper bound. In particular, (A2) implies that 𝒫 𝑖 is countable.20
The following proposition shows that, given (A2), there is a universal language in 𝐷(𝒫 𝑖 ).
The argument for the case where 𝒫 𝑖 consists of Turing-computable functions can be found
∑∣𝜎∣−1
One encoding is as follows: for the string 𝜎 ∈ {0, 1}<ℕ , the code for 𝜎 is 𝑡=0 (𝜎𝑡 + 1)2𝑡 .
20
Although our main motivation to introduce (A2) is to avoid language dependence, (A2) may be
19
justified on other grounds. One of them is that it guarantees the set 𝒫 𝑖 to have the following property:
if two functions 𝑓 and 𝑔 are in 𝒫 𝑖 , so are functions that result from 𝑓 and 𝑓 ′ through a series of basic
operations, such as the function 𝑓 ′′ defined by 𝑓 ′′ (2𝑡) = 𝑓 (𝑡) and 𝑓 ′′ (2𝑡+1) = 𝑓 ′ (𝑡) for all 𝑡. This property
is desirable given the motivation behind (A1); however, (A1) alone does not guarantee this property. In
fact, (A2) guarantees that 𝒫 𝑖 includes all the functions derived through basic operations using any finite
number of functions in 𝒫 𝑖 .
20
in Downey et al. [11].21 The general case is an easy extension of the arguments there and
hence we omit its proof.
Proposition 3.1. Suppose that 𝒫 𝑖 satisfies (A1) and (A2). Then a prefix-free language
𝑖
𝐿𝒫 ∈ 𝐷(𝒫 𝑖 ) exists such that for any language 𝐿 ∈ 𝐷(𝒫 𝑖 ) there is a constant 𝑏𝐿 that
satisfies 𝐾𝐿𝒫 𝑖 (𝜎) ≤ 𝐾𝐿 (𝜎) + 𝑏𝐿 for all binary strings 𝜎.
Given Proposition 3.1, it is then meaningful to discuss Kolmogorov complexity from
player 𝑖’s perspective. Now we move on to discuss Kolmogorov complexity of functions.
First we need to translate a function into a binary sequence to use Kolmogorov complexity.
To this end, we identify a function 𝑓 with its graph
{(0, 𝑓 (0)), (1, 𝑓 (1)), ..., (𝑡, 𝑓 (𝑡)), ...},
which can be further identified with an infinite sequence 𝜉 𝑓 over {0, 1}. More precisely,
the infinite binary sequence 𝜉 𝑓 corresponding to 𝑓 is defined as follows: 𝜉𝑡𝑓 = 1 if and only
if 𝑡 = 2𝑠 (2𝑓 (𝑠)+1)−1 for some 𝑠. To measure the complexity of such an infinite sequence,
we follow the literature (see Downey et al. [11]) and consider the complexities of its initial
segments. Intuitively, a function is complex if the complexities of its initial segments are
high relative to their lengths; a function is simple if the opposite is true. Indeed, it can
be shown that if 𝑓 ∈ 𝒫 𝑖 , then lim𝑇 →∞
𝐾
𝐿𝒫
𝑖 (𝜉
𝑓 [𝑇 ])
𝑇
= 0, where 𝜉 𝑓 [𝑇 ] = (𝜉0𝑓 , 𝜉1𝑓 , ..., 𝜉𝑇𝑓 −1 ) is
the initial segment of 𝜉 𝑓 with length 𝑇 .
Thus, in terms of Kolmogorov complexity, mutual incomputability is roughly equivalent to the condition22 that each player has a function 𝑓 𝑖 such that
𝑖
𝐾 𝒫 −𝑖 (𝜉 𝑓 [𝑇 ])
lim sup 𝐿
> 0.
𝑇
𝑇 →∞
We can then see why mutual incomputability may still allow partial predictions: a sequence that is not feasible for a player may still be compressible by that player and hence
21
The notion of relative computability is the standard notion in the literature that is equivalent to what
we refer to as a set of feasible functions. If 𝒫 satisfies (A1) and (A2), then the Kolmogorov complexity
relative to 𝒫 is equivalent to the Kolmogorov complexity relative to the function 𝑓 such that every
function in 𝒫 is computable from 𝑓 . The later notion is what is adopted in Downey et al. [11].
22
To be precise, this condition is slightly stronger than mutual incomputability.
21
some short descriptions may be used to make partial predictions. For this reason, we
consider incompressible sequences.23
Definition 3.1. Let 𝒫 𝑖 be a set of functions. A sequence 𝜉 ∈ {0, 1}ℕ is an incompressible
sequence relative to 𝒫 𝑖 if for any 𝐿 ∈ 𝐷(𝒫 𝑖 ), there is a constant 𝑏𝐿 such that for all 𝑇 > 0,
𝐾𝐿 (𝜉[𝑇 ]) ≥ 𝑇 − 𝑏𝐿 .
By Proposition 3.1, if 𝒫 𝑖 satisfies (A1) and (A2), a sequence is incompressible relative to
𝑖
𝒫 𝑖 if and only if it is incompressible with respect to the universal language 𝐿𝒫 .
We say that two sets 𝒫 1 , 𝒫 2 are mutually complex if there are functions 𝑓 1 ∈ 𝒫 1 , 𝑓 2 ∈
𝑖
𝑖
𝒫 2 such that for both 𝑖 = 1, 2, 𝜉 𝑓 (recall that 𝜉 𝑓 is the binary sequence corresponding
to the graph of the function 𝑓 𝑖 ) is an incompressible sequence relative to 𝒫 −𝑖 . Mutual
complexity, together with (A1) and (A2), guarantees the existence of an equilibrium. The
proof of the following theorem is given in Section 5.
Theorem 3.1. (Existence) Let 𝑔 be a finite zero-sum game. Suppose that 𝒫 1 , 𝒫 2 satisfy both (A1) and (A2) and are mutually complex. Then there exists an equilibrium in
𝐻𝐺(𝑔, 𝒫 1 , 𝒫 2 ) with the same equilibrium payoffs as those in 𝑔.
Theorem 3.1 is meaningful only if mutual complexity is not an empty assumption.
The following proposition, which is a simple implication of what is well known in the
literature, states that it is satisfied by many pairs of sets. Its proof is given in Section 5.
Proposition 3.2. There are uncountably many different pairs of sets 𝒫 1 and 𝒫 2 that
satisfy (A1), (A2), and mutual complexity.
Theorem 3.1 is useful only when 𝑔 has no pure equilibrium.24 For horizontal games
whose stage games have no pure equilibrium, the payoff implication of that theorem also
23
For any prefix-free language 𝐿, we can show that there is a constant 𝑏𝐿 such that 𝐾𝐿 (𝜎) ≤ ∣𝜎∣ +
2 log2 ∣𝜎∣ + 𝑏𝐿 for any binary sequence 𝜎. (The identity function is not a prefix-free language, and the
term 2 log2 ∣𝜎∣ is because of prefix-freeness.) Thus, our definition of incompressibility is less demanding
than what may be achieved otherwise. However, our definition is borrowed from the literature (see Li
and Vitány [19]) and has been proved to be useful in many applications.
24
By this we mean that neither player has a pure equilibrium strategy. It is possible that only one player
22
gives some information about the properties of equilibrium strategies. Together with
Proposition 2.1, it implies that any equilibrium strategy is incomputable from the other
player’s perspective. This shows that an optimal strategy in these games is not perfectly
predictable by the opponent. Although this result sounds intuitive and simple, it cannot
be obtained in the classical framework.
But incomputability can not be the only requirement for optimality. As we have argued
in the end of Section 2, an incomputable strategy may still allow partial predictions and
hence may be exploitable. In the rest of this section we examine what kind of patterns
could and could not exist in an equilibrium strategy.
3.2
Unpredictable behavior in horizontal games
Here we introduce a property, called stochasticity, and show that this property is sufficient
for a sequence to be optimal in a horizontal game satisfying mutual complexity. We
also use this property to give a tight characterization of equilibrium strategies. One
implication of these results is that the relative frequencies of actions in an equilibrium
strategy constitute an equilibrium mixed-strategy in the stage game.
It takes one more concept to define stochasticity. A selection function is a total
function from ℕ to {0, 1} such that ∣{𝑡 : 𝑟(𝑡) = 1}∣ = ∞.25 Given a finite set 𝑋 and an
infinite sequence 𝜉 = (𝜉0 , 𝜉1 , 𝜉2 , ...) ∈ 𝑋 ℕ , a selection function 𝑟 induces a subsequence
𝜉 𝑟 defined as 𝜉𝑡𝑟 = 𝜉𝑔(𝑡) , where 𝑔(0) = min{𝑠 : 𝑟(𝑠) = 1} and 𝑔(𝑡 + 1) = min{𝑠 : 𝑠 >
𝑔(𝑡), 𝑟(𝑠) = 1} for all 𝑡; in other words, 𝜉 𝑟 is the subsequence of 𝜉 selected from indices
where 𝑟 has value 1. Stochasticity is defined as follows.26
has a pure equilibrium strategy but not the other; in this case, only the player without a pure equilibrium
strategy needs an incompressible sequence for an equilibrium to exist. We do not, however, consider this
possibility in the following; moreover, whenever it is said that 𝑔 has no pure strategy equilibrium, it
means that neither player has a pure equilibrium strategy.
25
Selection functions are closely related to the notion of checking rules studied in Lehrer [18] and
Sandroni et al. [33]. The only difference is that we do not have forecasts in our model.
26
In von Mises [22], a sequence 𝜉 is defined to be 𝑝-stochastic if there is no admissible subsequence of
𝜉 that has a relative frequency different from 𝑝. Clearly, without the quantifier “admissible,” there is no
23
Definition 3.2. Let 𝑝 ∈ Δ(𝑋) be a probability distribution and let 𝒫 be a set of functions.
A sequence 𝜉 ∈ 𝑋 ℕ is 𝑝-stochastic relative to 𝒫 if for any selection function 𝑟 ∈ 𝒫 and
for all 𝑥 ∈ 𝑋,
lim
𝑇 →∞
𝑇 −1
∑
𝑐𝑥 (𝜉 𝑟 )
𝑡
𝑡=0
𝑇
= 𝑝𝑥 ,
where 𝑐𝑥 (𝑦) = 1 if 𝑥 = 𝑦, and 𝑐𝑥 (𝑦) = 0 otherwise.
Stochasticity is a sufficient property for a sequence to be optimal in horizontal games.
Before the formal statement, we first give some informal arguments in the context of the
matching pennies game 𝑔 𝑀 𝑃 . We know from Theorem 3.1 that there is an equilibrium in
𝐻𝐺(𝑔 𝑀 𝑃 , 𝒫 1 , 𝒫 2 ) and the equilibrium payoff for player 1 is 21 . Let 𝜉 ∈ {𝐻𝑒𝑎𝑑𝑠, 𝑇 𝑎𝑖𝑙𝑠}ℕ be
an equilibrium strategy of player 1 and consider a selection function 𝑟 ∈ 𝒫 2 . If 𝜉𝑡 = 𝐻𝑒𝑎𝑑𝑠
for all 𝑡 such that 𝑟(𝑡) = 1, then obviously player 2 can generate a strategy 𝜁 (such that
𝜉𝑡 = 𝑇 𝑎𝑖𝑙𝑠 if 𝑟(𝑡) = 1) to fully exploit player 1 along those indices. But we can say more
than that. If the relative frequency of 𝑇 𝑎𝑖𝑙𝑠 along the indices with 𝑟(𝑡) = 1 is greater
than 21 , player 2 can also obtain a payoff higher than − 12 by playing 𝜁.27 A symmetric
argument can be easily imagined for the case where the frequency is less than 21 .
Theorem 3.2. (Stochasticity as a sufficient requirement for optimality) Suppose that
𝒫 1 , 𝒫 2 satisfy (A1), (A2), and mutual complexity. For any player 1’s equilibrium strategy
𝑝 ∈ Δ(𝑋) in 𝑔, the set of 𝑝-stochastic sequences relative to 𝒫 2 in 𝒳 , which is not empty,
is a subset of player 1’s equilibrium strategies in 𝐻𝐺(𝑔, 𝒫 1 , 𝒫 2 ).
A symmetric theorem, of course, holds for player 2 as well. The proof of Theorem 3.2
is given in Section 5. Theorem 3.2 identifies a subset of equilibrium strategies, which is
characterized by stochasticity. In particular, it implies that for any mixed equilibrium
𝑝-stochastic sequence unless 𝑝 is degenerate. Here we define admissibility in terms of feasible functions
in 𝒫. In the literature, stochasticity is defined using selection functions that are history-dependent—i.e.,
𝑟 : 𝑋 <ℕ → {0, 1}. That concept is close to the unpredictability requirement in vertical games. We will
discuss this issue more specifically in Section 4.
27
Here we do not deal with the issue of how player 1 detect such a pattern. Rather, we only consider
what pattern is detectable by the functions in 𝒫 2 , and the notion of equilibrium requires that any
detectable pattern be optimally used.
24
strategy 𝑝 in the stage game, there is an equilibrium strategy in the horizontal game with
the corresponding frequencies of actions. The earlier informal arguments also suggest its
necessity for a strategy to be optimal. Our next theorem, based on those intuitions, gives
some necessary conditions. The formal statement needs one more definition. For any
𝑝 = (𝑝𝑥 )𝑥∈𝑋 ∈ Δ(𝑋), we say that 𝜉 ∈ 𝑋 ℕ is a 𝑝-sequence if for all 𝑥 ∈ 𝑋,
∑𝑇 −1
𝑡=0 𝑐𝑥 (𝜉𝑡 )
lim
= 𝑝𝑥 .
𝑇 →∞
𝑇
Theorem 3.3. (Stochasticity as a necessary requirement for optimality) Suppose that
𝒫 1 , 𝒫 2 satisfy (A1), (A2), and mutual complexity. Suppose that 𝑔 has no pure equilibrium.
Then, for any equilibrium 𝑝-sequence 𝜉 ∈ 𝒳 in 𝐻𝐺(𝑔, 𝒫 1 , 𝒫 2 ) and for any (𝛼, 1 − 𝛼)sequence 𝑟 ∈ 𝒫 2 as a selection function28 with 𝛼 < 1, if 𝜉 𝑟 is a 𝑝′ -sequence, 𝑝′ is an
equilibrium mixed strategy in 𝑔.
Theorem 3.3 gives a stronger condition than incomputability for sequences that have
limit frequencies. It excludes all incomputable sequences with limit frequencies that are
different from any equilibrium mixed-strategy of the stage game. However, it is clearly
weaker than stochasticity. We are not able to characterize equilibrium strategies without
well-defined limit frequencies because they seem to rely on how the long-run average
criterion is extended to sequences of average payoffs without limits. Moreover, different
subsequences may have different limit frequencies because of possible multiplicity of stagegame equilibria. If player 1 has two equilibrium mixed-strategies 𝑝 and 𝑝′ in 𝑔, then playing
a 𝑝-stochastic sequence at odd places and playing a 𝑝′ -stochastic sequences at even places
is an equilibrium strategy in the horizontal game. This naturally follows from the fact that
any convex combination of 𝑝 and 𝑝′ is also an equilibrium strategy in the stage game.29
This issue, of course, does not arise if the stage game has a unique equilibrium, and in
that case we have 𝑝′ = 𝑝 in Theorem 3.3.
Theorem 3.3 implies that some patterns in equilibrium strategies are not expected in
horizontal games. They show that the frequencies (if they exist) of actions have to be
28
Here, we identify a selection function 𝑟 with an infinite sequence in {0, 1}ℕ , and 𝛼 denotes the
frequency of 0.
29
We do not give a formal proof of this fact. However, the proof is available upon request.
25
consistent with the mixed-strategy equilibrium of the stage game. On the other hand,
Theorem 3.2 implies that other patterns may exist. For example, it can be shown that
if 𝜉 ∈ 𝒫 1 is a 𝑝-stochastic sequence relative to 𝒫 2 , the sequence 𝜉 ′ ∈ 𝒫 2 defined as
′
′
𝜉2𝑡
= 𝜉2𝑡+1
= 𝜉𝑡 for all 𝑡 is also 𝑝-stochastic.30 Thus, if 𝜉 is optimal, so is 𝜉 ′ . However,
𝜉 ′ has an obvious pattern in it. Actions played on odd indices are exactly the same as
those on even indices. Nonetheless, the rules in horizontal games prevent players from
exploiting this pattern. As we demonstrate in the next subsection, this pattern does not
exist in equilibrium strategies in vertical games.
3.3
Unpredictable behavior for vertical games
Here we present our results for vertical games. First we remark that both Proposition 2.1
and Proposition 2.2 hold for vertical games without any change. Thus, mutual incomputability is still a necessary condition for the existence of an equilibrium in vertical
games. Theorem 3.1 holds for vertical games as well, and so mutual complexity is also
a sufficient condition to obtain an equilibrium in vertical games. However, as mentioned
above, not every equilibrium strategy in a horizontal game is optimal in a vertical game,
even if 𝒫 𝑖 is the same across these two different games for both 𝑖 = 1 and 2.
To facilitate the comparison between equilibrium strategies in horizontal games and
vertical games, we focus on a particular kind of strategies in vertical games. A strategy of
player 1 𝑎 : 𝑌 <ℕ → 𝑋 is history-independent if there is a sequence 𝜉 ∈ 𝑋 ℕ such that for all
𝜎 ∈ 𝑌 <ℕ , 𝑎(𝜎) = 𝜉∣𝜎∣ . We identify such a strategy 𝑎 with the sequence 𝜉. Similar definition
applies to player 2’s strategies as well. For these strategies, we find that a parallel property
to stochasticity, which we call strong stochasticity, plays a relevant role for optimality in
vertical games. As suggested by its name, this property strengthens stochasticity. It
considers selection functions that can be history-dependent as well. For any given finite
set 𝑋, we define a selection function for 𝑋 as a total function 𝑟 : 𝑋 <ℕ → {0, 1}, which,
as mentioned earlier, can be identified with a total function over natural numbers. Given
a sequence 𝜉 ∈ 𝑋 ℕ , we use 𝑟 to choose a subsequence 𝜉 𝑟 from 𝜉 as follows: 𝜉𝑡𝑟 = 𝜉𝑔(𝑡) ,
30
A proof of this statement is available from the author upon request.
26
where 𝑔(0) = min{𝑡 : 𝑟(𝜉[𝑡]) = 1}, and 𝑔(𝑡) = min{𝑠 : 𝑟(𝜉[𝑠]) = 1, 𝑠 > 𝑔(𝑡 − 1)} for 𝑡 > 0.
Obviously, such a selection function may produce not an infinite subsequence, but only a
finite initial segment. Strong stochasticity is defined formally in the following.
Definition 3.3. Let 𝑝 ∈ Δ(𝑋) be a probability distribution and let 𝒫 be a set of functions.
A sequence 𝜉 ∈ 𝑋 ℕ is strongly 𝑝-stochastic relative to 𝒫 if for any selection function 𝑟 ∈ 𝒫
for 𝑋 such that 𝜉 𝑟 is an infinite sequence,
lim
𝑇 →∞
𝑇 −1
∑
𝑐𝑥 (𝜉 𝑟 )
𝑡
𝑡=0
𝑇
= 𝑝𝑥 for all 𝑥 ∈ 𝑋,
where 𝑐𝑥 (𝑦) = 1 if 𝑥 = 𝑦, and 𝑐𝑥 (𝑦) = 0 otherwise.
When we replace stochasticity with strong stochasticity, Theorem 3.2 continues to
hold, with the understanding that a sequence 𝜉 ∈ 𝑋 ℕ is regarded as a history-independent
strategy in the vertical game. See Theorem 5.1 in Section 5. Thus, strong stochasticity
is a sufficient condition for optimality in vertical games. As for necessity, we have the
following theorem. Its proof is given in Section 5.
Theorem 3.4. Suppose that 𝑔 has no pure equilibrium and that 𝒫 1 , 𝒫 2 satisfy (A1), (A2),
and mutual complexity. Then, for any equilibrium 𝑝-sequence 𝜉 ∈ 𝒳 in 𝑉 𝐺(𝑔, 𝒫 1 , 𝒫 2 ) and
any selection function 𝑟 ∈ 𝒫 2 for 𝑋 such that {𝑟(𝜉[𝑡])}∞
𝑡=0 is a (𝛼, 1 − 𝛼)-sequence with
𝛼 < 1, if 𝜉 𝑟 is a 𝑝′ -sequence, then 𝑝′ is an equilibrium mixed-strategy in 𝑔.
Theorem 3.4 gives a stronger condition than that in Theorem 3.3 because it applies
to history-dependent selection functions as well. Moreover, the following lemma shows
that the obvious pattern that we identify in the last subsection in an optimal strategy in
a horizontal game does not appear in a vertical game. Its proof is given in Section 5.
Lemma 3.1. Suppose that 𝑔 has no pure equilibrium and suppose that 𝒫 1 , 𝒫 2 satisfy
(A1), (A2), and mutual complexity. Let 𝜉 ∈ 𝑋 ℕ . If 𝜉2𝑡 = 𝜉2𝑡+1 for all 𝑡 ∈ ℕ, then 𝜉 is
not a equilibrium strategy in 𝑉 𝐺(𝑔, 𝒫 1 , 𝒫 2 ).
Thus, for any given 𝒫 1 and 𝒫 2 that satisfy (A1), (A2), and mutual complexity, there
are sequences that are equilibrium strategies in horizontal games are not optimal in vertical
27
games. This difference between equilibrium history-independent strategies in vertical
games and equilibrium strategies in horizontal games is the result of different rules instead
of different abilities of pattern detection. On the other hand, in either the horizontal
game or the vertical game, for a fixed 𝒫 1 , the set of player 2’s equilibrium strategies
changes as 𝒫 2 changes (assuming that (A1), (A2), and mutual complexity always hold).
Nonetheless, the nature of equilibrium strategies is roughly captured by stochasticity and
strong stochasticity.
3.4
Statistical regularities
Stochasticity or strong stochasticity are the relevant properties for optimality in our framework. However, these properties are not exactly those used in the empirical literature to
test the equilibrium hypothesis in the context of repeated zero-sum games. Instead, tests
based on statistical regularities according to the i.i.d. distribution generated by a mixed
equilibrium in the stage game are generally employed for that purpose.31 Both strong
stochasticity and stochasticity can be regarded as a subclass of those statistical regularities, and we have shown that these regularities do appear in equilibrium strategies.32 As
we have discussed earlier, however, stochasticity is a weak property and there are equilibrium strategies of a horizontal game with patterns that violate other statistical regularities
with respect to any i.i.d. distribution. These patterns, as we have shown, do not exist in
the equilibrium strategies of a vertical game. Nonetheless, the following question remains
unanswered: Are all statistical regularities that are expected from an i.i.d. distribution
satisfied by the equilibrium strategies of a vertical game?
31
In our framework, a statistical regularity can be regarded as a describable event that has probability
1. The term ‘describable’ can be defined with respect to the set of feasible functions in a similar fashion
as we have done for Kolmogorov complexity, and in the Appendix we give a formal definition of statistical
tests in this sense.
32
To be precise, they appear only in some equilibrium strategies. However, we have necessary conditions
for optimality as well. Although they are not as strong as strong stochasticity or stochasticity, the spirit
is the same. Nonetheless, in vertical games this matter is more complicated because there are historydependent equilibrium strategies as well, which we have no characterization at all.
28
In the Appendix (see Theorem 6.2) we show that, under mutual complexity, there
exists an equilibrium strategy that satisfies all the statistical regularities with respect to
the i.i.d. distribution generated by a stage-game mixed equilibrium in both horizontal
and vertical games. Moreover, we show there that mutual complexity is also necessary
for that to happen. Here we show that there are equilibrium strategies that violate a
particular statistical regularity called the Law of the Iterated Logarithm (LIL). Consider
a finite set 𝑋 and a distribution 𝑝 ∈ Δ(𝑋). The i.i.d. distribution 𝜇𝑝 over 𝑋 ℕ generated
∏∣𝜎∣−1
by 𝑝 is defined as follows: for all 𝜎 ∈ 𝑋 <ℕ , 𝜇𝑝 (𝑁𝜎 ) = 𝑡=0
𝑝𝜎𝑡 . LIL states that the
following condition holds for almost all 𝜉 in 𝑋 ℕ (with respect to 𝜇𝑝 ):
∑
∣ 𝑇𝑡=0 𝑐𝑥 (𝜉𝑡 ) − 𝑇 𝑝𝑥 ∣
lim sup √
= 1.
𝑇 →∞
2𝑝𝑥 (1 − 𝑝𝑥 )𝑇 log log 𝑇
(2)
This law gives the exact convergence rate of the frequencies in an i.i.d. sequence. The
following theorem, however, shows that there is always an equilibrium strategy in the
vertical game with mutual complexity that violates this regularity. Its proof is given in
Section 5.
Theorem 3.5. Suppose that 𝒫 1 , 𝒫 2 satisfy (A1), (A2), and mutual complexity. Let 𝑝∗ be
an equilibrium strategy of player 1 in 𝑔 such that 0 < 𝑝∗𝑥 < 1 for some 𝑥 ∈ 𝑋. There exists
an equilibrium strategy 𝜉 in 𝑉 𝐺(𝑔, 𝒫 1 , 𝒫 2 ) that is strongly 𝑝∗ -stochastic and satisfies
∑𝑇
lim √
𝑇 →∞
∗
𝑡=0 𝑐𝑥 (𝜉𝑡 ) − 𝑇 𝑝𝑥
2𝑝∗𝑥 (1 − 𝑝∗𝑥 )𝑇 log log 𝑇
= ∞.
(3)
Theorem 3.5 shows that although certain patterns are detectable, it may not be feasible
to transform them into a strategy to exploit them. This result is similar to what we have
seen earlier when we compare the patterns found in horizontal games with those in vertical
games. Moreover, this result shows that, in vertical games failure of certain statistical
regularities (with respect to the i.i.d. distribution generated by a stage-game mixed
equilibrium) does not entail the rejection of the equilibrium hypothesis. As a result, the
common practice which rejects the equilibrium hypothesis by using statistical tests may
require further reconsideration.
29
4
Discussion
In this section, we discuss the implications of our results for the literature, and give
some interpretations of Nash equilibrium in our model. We end the discussion with two
potential extensions of our framework.
4.1
Unpredictability in game theory
In the literature of game theory, it is commonly assumed that players are able to generate unpredictable strategies. Our nonexistence results suggest, however, that this is
not a trivial assumption. In our model, some minimum complexity requirement—namely,
mutual incomputability—on players’ abilities of strategy generation is necessary for unpredictable behavior to exist in equilibrium. Although there is an extensive literature
that has accumulated evidence of unpredictable behavior and its statistical properties,
little has been said about the complexity of those abilities.
Moreover, the same literature generally reports that players do not generate “random”
strategies in terms of statistical regularities in the context of repeated zero-sum games.
However, our results suggest that not all tests are relevant to the equilibrium hypothesis.
In our model, if an equilibrium exists where unpredictable behavior emerges, then there
always exist equilibrium strategies that fail some regularities with respect to any i.i.d.
distribution. Furthermore, we find that stochasticity roughly corresponds to the property
required for optimality in horizontal games. The matter in vertical games seems more
complicated, but strong stochasticity seems to play a crucial role as well. These results
may help to identify relevant tests for the equilibrium hypothesis.33
33
Tests based on checking rules may be useful to develop tests for (strong) stochasticity. However, the
extant literature on checking rules (see, for example, Lehrer [18] and Sandroni et al. [33]) focuses on
testing experts and hence may not be directly applicable.
30
4.2
Interpretation of Nash equilibrium
Our model adopts Nash equilibrium as the solution concept. Here equilibrium can be
interpreted as a stability condition. Formally this condition says that, given the opponent’s strategy, no player has a profitable deviation. What is conceptually novel in our
framework, however, is about deviations that come from detectable patterns. In games
such as matching pennies, only this kind of deviations is relevant. Equilibrium guarantees
that there be no detectable pattern in the opponent’s strategy that is not used to increase
a player’s payoff, and thus can be interpreted as a stability requirement in the sense that
the system is not stable with a detectable pattern that is not exploited.
Another potential interpretation is inspired by the machine game literature (see Chatterjee and Sabourian [8] for a survey). In that interpretation, players are fully rational,
but are subject to complexity constraints on pattern detection and strategy implementation. Applying this interpretation to our framework, the set of feasible functions gives a
constraint on strategy implementation, which represents patterns that a player can use to
generate a strategy to exploit them. One advantage of this interpretation is that, because
we consider zero-sum games, Nash equilibrium is reduced to the Maximin criterion, and
there is no issue of equilibrium selection.
4.3
Extensions
We propose two potential extensions here. The first considers introducing a learning process, and the second considers finite sequences instead of infinite ones. The first extension
is also related to the interpretation of Nash equilibrium in our model. Equilibrium as a
stability condition presumes a dynamic system in the background, and one may naturally
consider that system as a learning process. Unlike the extant learning literature in game
theory, however, in our framework the main challenge is how to model dynamic pattern
detection in the learning process.
Another extension considers finite sequences. This extension seems natural because
we live in a finite world, and it may give us sharper empirical implications than what we
31
have obtained here. Moreover, the Kolmogorov complexity was originally introduced to
study the complexity of finite sequences, and, in the literature, some results regarding
asymptotical properties of finite sequences with high complexity have been obtained.
Thus, it may be fruitful to consider the asymptotic behavior of finite but long (in terms
of the numbers of repetitions) games.
5
Proofs
In this section we give the proofs. In the Appendix, we review some extant results regarding Kolmogorov complexity and Martin-Löf randomness (which is a notion of randomness
that formalizes the idea that a sequence is random if it satisfies all statistical regularities)
that are necessary in the proofs.
5.1
Proofs of the nonexistence results
Proof of Proposition 2.1: We give the proof for vertical games. The proof for horizontal
games is very similar and is omitted.
(a) We show that this part is implied by part (b). Since there is no pure strategy equilibrium in 𝑔, it follows that 𝑣1 = max𝑥∈𝑋 min𝑦∈𝑌 ℎ(𝑥, 𝑦) < min𝑦∈𝑌 max𝑥∈𝑋 ℎ(𝑥, 𝑦) = 𝑣2 .
Now, 𝒫 1 = 𝒫 2 implies that 𝒫 1 ⊂ 𝒫 2 and 𝒫 2 ⊂ 𝒫 1 . By part (b), this implies that, if
an equilibrium exists, the equilibrium payoff of player 1 is 𝑣1 if 𝒫 1 ⊂ 𝒫 2 while it is 𝑣2 if
𝒫 2 ⊂ 𝒫 1 . Thus, if an equilibrium exists, 𝑣1 = 𝑣2 , which leads to a contradiction.
(b) We first index the actions in 𝑋 as 𝑋 = {𝑥1 , ..., 𝑥𝑚 }. For any 𝑏 ∈ 𝒴, construct the
strategy 𝑎′ such that for all 𝜎 ∈ 𝑌 <ℕ , 𝑎′ (𝜎) = 𝑥𝜁∣𝜎∣ , where 𝜁 is defined as follows:
𝜁0 = min{𝑖 : 𝑥𝑖 ∈ arg max ℎ(𝑥, 𝑏(𝜖))}, and
𝑥∈𝑋
(4)
for 𝑡 > 0, 𝜁𝑡 = min{𝑖 : 𝑥𝑖 ∈ arg max ℎ(𝑥, 𝑏(𝑥𝜁 [𝑡]))},
𝑥∈𝑋
with 𝑥𝜁 [𝑡] = (𝑥𝜁0 , ..., 𝑥𝜁𝑡−1 ). 𝜁 ∈ 𝒫 2 because 𝑏 ∈ 𝒫 2 and 𝒫 2 is closed under composition
32
and primitive recursion. Thus, 𝑎′ ∈ 𝒳 since 𝒫 2 ⊂ 𝒫 1 . By construction, for all 𝑡 ∈ ℕ,
ℎ(𝜁𝑡 , 𝑏(𝑥𝜁 [𝑡])) ≥ min max ℎ(𝑥, 𝑦) = 𝑣2 .
𝑦∈𝑌 𝑥∈𝑋
Then, 𝑢ℎ (𝑎′ , 𝑏) ≥ lim inf 𝑇 →∞ 𝑣2 = 𝑣2 . It follows that sup𝑎∈𝒳 𝑢ℎ (𝑎, 𝑏) ≥ 𝑣2 . Now, let
𝑦 ∗ ∈ arg min𝑦∈𝑌 (max𝑥∈𝑋 ℎ(𝑥, 𝑦)). Let 𝑏 ∈ 𝒫 2 be such that 𝑏(𝜏 ) = 𝑦 ∗ for all 𝜏 ∈ 𝑋 <ℕ .
Then sup𝑎∈𝒳 𝑢ℎ (𝑎, 𝑏) = 𝑣2 . Thus, we have that min𝑏∈𝒴 sup𝑎∈𝒳 𝑢ℎ (𝑎, 𝑏) = 𝑣2 , and hence,
the value of 𝑉 𝐺(𝑔, 𝒫 1 , 𝒫 2 ) is 𝑣2 . □
Proof of Proposition 2.2: Like the previous proof, we give the proof only for vertical
games. Suppose, by contradiction, that (𝑎∗ , 𝑏∗ ) is an equilibrium. Let 𝑐 = max(𝑥,𝑦)∈𝑋×𝑌 ∣ℎ(𝑥, 𝑦)∣.
For each 𝑇 ∈ ℕ, consider the following strategy 𝑏′ such that for all 𝜎 ∈ 𝑋 <ℕ , 𝑏′ (𝜎) =
𝑦𝜁∣𝜎∣ , where 𝜁 is defined as follows:
𝜁0 = min{𝑖 : 𝑦𝑖 ∈ arg min ℎ(𝑎∗ (𝜖), 𝑦)}(𝜖 is the empty string);
𝑦∈𝑌
(5)
for 𝑡 = 1, ...., 𝑇, 𝜁𝑡 = min{𝑖 : 𝑦𝑖 ∈ arg min ℎ(𝑎∗ (𝑦𝜁 [𝑡]), 𝑦)};
𝑦∈𝑌
for 𝑡 > 𝑇 + 1, 𝜁𝑡 = 1.
Since 𝜁𝑡 is constant for all 𝑡 > 𝑇 , 𝜁 ∈ 𝒫 2 and so 𝑏′ ∈ 𝒴. We have 𝑣(𝑎∗ , 𝑏′ ) ≤ (1 −
∑
∑
𝑇 +1
(𝑐 − 𝑣1 ). Since (𝑎∗ , 𝑏∗ ) is an equilibrium, it
𝛿) 𝑇𝑡=0 𝛿 𝑡 𝑣1 + (1 − 𝛿) ∞
𝑡=𝑇 +1 𝑐 = 𝑣1 + 𝛿
follows that 𝑣(𝑎∗ , 𝑏∗ ) ≤ 𝑣(𝑎∗ , 𝑏′ ) ≤ 𝑣1 + 𝛿 𝑇 +1 (𝑐 − 𝑣1 ). Since this holds for all 𝑇 , we have
𝑣(𝑎∗ , 𝑏∗ ) ≤ lim𝑇 →∞ 𝑣1 + 𝛿 𝑇 +1 (𝑐 − 𝑣1 ) = 𝑣1 . Similarly, we can show that 𝑣(𝑎∗ , 𝑏∗ ) ≥ 𝑣2 . It
then follows that, 𝑣1 < 𝑣2 ≤ 𝑣(𝑎∗ , 𝑏∗ ) ≤ 𝑣1 , which is a contradiction. □
Here we give the proof for the assertion in Example 2.1 regarding the existence of an
equilibrium when 𝒫 1 ⊂ 𝒫 2 .
Lemma 5.1. If 𝒫 1 is countable, then there is a set 𝒫 2 that includes 𝒫 1 such that there
is an equilibrium in 𝐻𝐺(𝑔 𝑀 𝑃 , 𝒫 1 , 𝒫 2 ).
Proof. Let 𝒳 = {𝜉 0 , 𝜉 1 , ...}. For each 𝑛, define 𝑈 𝑛 = {𝜁 ∈ {𝐻𝑒𝑎𝑑𝑠, 𝑇 𝑎𝑖𝑙𝑠}ℕ : 𝑢ℎ (𝜉 𝑛 , 𝜁) =
0}. We show that for all 𝑛, 𝑈 𝑛 is a co-meager set, by showing that 𝑈 𝑛 can be expresses
as a countable intersection of dense open subsets. ({𝐻𝑒𝑎𝑑𝑠, 𝑇 𝑎𝑖𝑙𝑠}ℕ has the product
33
topology.) For each 𝑘, 𝑇 > 0, define
𝑈
𝑛,𝑘,𝑇
′
= {𝜁 ∈ {𝐻𝑒𝑎𝑑𝑠, 𝑇 𝑎𝑖𝑙𝑠} : (∃𝑇 > 𝑇 )
ℕ
′ −1
𝑇∑
ℎ(𝜉𝑡𝑛 , 𝜁𝑡 )/𝑇 ′ <
𝑡=0
1
}.
𝑘
𝑈 𝑛,𝑘,𝑇 is clearly an open set. We now show that it is dense. Consider any 𝜀 > 0 and
any 𝜁 ∈ {𝐻𝑒𝑎𝑑𝑠, 𝑇 𝑎𝑖𝑙𝑠}ℕ . We show that there is 𝜁 ′ ∈ 𝑈 𝑛,𝑘,𝑇 such that the distance
∑ 𝑖(𝜁𝑡 ,𝜁𝑡′ )
between 𝜁 ′ and 𝜁 is less than 𝜀 (we use the metric defined by 𝑑(𝜁, 𝜁 ′ ) = ∞
, where
𝑡=0
2𝑡
𝑖(𝐻𝑒𝑎𝑑𝑠, 𝐻𝑒𝑎𝑑𝑠) = 0 = 𝑖(𝑇 𝑎𝑖𝑙𝑠, 𝑇 𝑎𝑖𝑙𝑠) and 𝑖(𝑇 𝑎𝑖𝑙𝑠, 𝐻𝑒𝑎𝑑𝑠) = 1 = 𝑖(𝐻𝑒𝑎𝑑𝑠, 𝑇 𝑎𝑖𝑙𝑠)). Let
∑
1
′
′
′
𝑇1 be so large that ∞
𝑡=𝑇1 2𝑡 < 𝜀. Take 𝜁 be such that 𝜁𝑡 = 𝜁𝑡 for 𝑡 ≤ 𝑇1 and 𝜁𝑡 is different
∑ 1 1
from 𝜉𝑡𝑛 for 𝑡 > 𝑇1 . Now, let 𝑇 ′ > max{𝑇1 , 𝑇 } be such that 𝑇𝑡=0
< 𝑘1 . Such 𝑇 ′ exists
𝑇′
∩ ∩∞
𝑛,𝑘,𝑇
,
and this shows that 𝜁 ′ ∈ 𝑈 𝑛,𝑘,𝑇 , and hence, 𝑈 𝑛,𝑘,𝑇 is dense. Now, 𝑈 𝑛 = ∞
𝑇 =1 𝑈
𝑘=1
∩∞
and hence, it is co-meager. It follows that 𝑈 = 𝑛=0 𝑈 𝑛 is also co-meager and by the
Baire category theorem, it is dense and hence nonempty. Take any 𝜁 ∈ 𝑈 . Then any 𝒫 2
that includes 𝜁 will work.
5.2
Proofs of results in Sections 3
This section gives proofs for Sections 3. We begin with the proof of Proposition 3.2.
Then, we proceed to Theorem 3.2, which implies Theorem 3.1 as a corollary. We then
use a similar argument to to prove an analogous result in vertical games, Theorem 5.1.
After them, we proceed to the necessary conditions, which include Theorem 3.3 and
Theorem 3.4. Then we prove Lemma 3.1. We end with Theorem 3.5.
Proof of Proposition 3.2: (This proof needs some background knowledge that is presented in the Appendix.) By Theorem 6.1, for any 𝜉, 𝜁 ∈ {0, 1}ℕ , if 𝜉 is 𝜇( 1 , 1 ) -random
2 2
relative to 𝐶(𝜁) (𝐶(𝜁) denotes the set of functions computable from 𝜁), then 𝜉 is incompressible relative to 𝐶(𝜁). Moreover, by Theorem 12.17 in Downey et al. [11], if 𝜉 ⊕ 𝜁
(where 𝜉 ⊕ 𝜁2𝑡 = 𝜉𝑡 and 𝜉 ⊕ 𝜁2𝑡+1 = 𝜁𝑡 for all 𝑡) is 𝜇( 1 , 1 ) -random, then 𝜉 is 𝜇( 1 , 1 ) -random
2 2
2 2
relative to 𝐶(𝜁) and vice versa. Hence, 𝐶(𝜉) and 𝐶(𝜁) are mutually complex. Now, by
Proposition 6.1, the set
𝐴 = {𝜉 ⊕ 𝜁 : 𝜉 ⊕ 𝜁 is 𝜇( 1 , 1 ) -random} ⊂ {𝜉 ⊕ 𝜁 : 𝐶(𝜉) and 𝐶(𝜁) are mutually complex}
2 2
34
has measure 1. Therefore, the set A is uncountable. Since for any 𝜉, the set of sequences
𝜉 ′ such that 𝐶(𝜉) = 𝐶(𝜉 ′ ) is countable, we then conclude that there are uncountably many
different pairs of sets satisfying (A1) and (A2) that are mutually complex. □
Before the other proofs, we give a lemma concerning expected values that is used
repeatedly in what follows. For (strongly) stochastic sequences, the expected values correspond to long-run averages. We present the proof for strongly stochastic sequences; the
arguments for stochastic sequences are almost the same.
Lemma 5.2. Let 𝑋 be a finite set. Suppose that 𝒫 satisfies (A1) and (A2). Let 𝑝 ∈ Δ(𝑋)
be a distribution and let ℎ : 𝑋 → ℚ be a function. If 𝜉 is a strongly 𝑝-stochastic sequence
relative to 𝒫, then, for any selection function 𝑟 for 𝑋 in 𝒫 such that 𝜉 𝑟 ∈ 𝑋 ℕ ,
lim
𝑇 −1
∑
ℎ(𝜉 𝑟 )
𝑡
𝑇 →∞
𝑡=0
𝑇
∑
=
𝑝𝑥 ℎ(𝑥).
𝑥∈𝑋
Proof. Let 𝑟 be a selection function in 𝒫 such that 𝜉 𝑟 ∈ 𝑋 ℕ . Then, for any 𝑥 ∈ 𝑋,
lim
𝑇 →∞
𝑇 −1
∑
𝑐𝑥 (𝜉 𝑟 )
𝑡
𝑇
𝑡=0
= 𝑝𝑥 .
Therefore,
lim
𝑇 →∞
𝑇 −1
∑
ℎ(𝜉 𝑟 )
𝑡
𝑡=0
𝑇
= lim
𝑇 →∞
𝑇 −1
∑∑
𝑐𝑥 (𝜉 𝑟 )ℎ(𝑥)
𝑡
𝑥∈𝑋 𝑡=0
𝑇
=
∑
𝑝𝑥 ℎ(𝑥).
𝑥∈𝑋
Now we give the proof of Theorem 3.2.
Proof of Theorem 3.2: Suppose that (𝑝∗ , 𝑞 ∗ ) ∈ Δ(𝑋) × Δ(𝑌 ) is an equilibrium in
𝑔. Such an equilibrium exists because 𝑔 has rational payoff functions. By Theorem 6.2
and Theorem 6.5 in the appendix, mutual complexity implies that there is a 𝑝∗ -stochastic
sequence 𝜉 ∗ relative to 𝒫 2 in 𝒳 and there is a 𝑞 ∗ -stochastic sequence 𝜁 ∗ relative to 𝒫 1 in
𝒴. We show that (𝜉 ∗ , 𝜁 ∗ ) is an equilibrium in 𝐻𝐺(𝑔, 𝒫 1 , 𝒫 2 ).
Because (𝑝∗ , 𝑞 ∗ ) is an equilibrium of the game 𝑔, ℎ(𝑥, 𝑞 ∗ ) ≤ ℎ(𝑝∗ , 𝑞 ∗ ) for all 𝑥 ∈ 𝑋.
First we show that for all 𝜉 ∈ 𝒳 ,
lim sup
𝑇 →∞
𝑇 −1
∑
ℎ(𝜉𝑡 , 𝜁 ∗ )
𝑡
𝑡=0
𝑇
35
≤ ℎ(𝑝∗ , 𝑞 ∗ ).
(6)
Suppose that 𝜉 ∈ 𝒳 , and so 𝜉 ∈ 𝒫 1 . For each 𝑥 ∈ 𝑋, let 𝑟𝑥 : ℕ → {0, 1} be the
selection function such that 𝑟𝑥 (𝑡) = 1 if 𝜉𝑡 = 𝑥, and 𝑟𝑥 (𝑡) = 0 otherwise. Define
𝑥
𝐿𝑥 (𝑇 ) = ∣{𝑡 ∈ ℕ : 0 ≤ 𝑡 ≤ 𝑇 − 1, 𝑟𝑥 (𝑡) = 1}∣ and 𝜁 𝑥 = (𝜁 ∗ )𝑟 .
(7)
It is easy to see that 𝑟𝑥 is in 𝒫 1 since 𝜉 is. Let
ℰ 1 = {𝑥 ∈ 𝑋 : lim 𝐿𝑥 (𝑇 ) = ∞} and ℰ 2 = {𝑥 ∈ 𝑋 : lim 𝐿𝑥 (𝑇 ) < ∞}.
𝑇 →∞
𝑇 →∞
∑𝐵𝑥
𝑥
For each 𝑥 ∈ ℰ 2 , let 𝐵𝑥 = lim𝑇 →∞ 𝐿𝑥 (𝑇 ) and let 𝐶𝑥 =
𝑡=0 ℎ(𝑥, 𝜁𝑡 ). Then, for any
∑ −1 ℎ(𝑥,𝜁𝑡𝑥 )
= ℎ(𝑥, 𝑞 ∗ ) ≤ ℎ(𝑝∗ , 𝑞 ∗ ).
𝑥 ∈ ℰ 1 , by Lemma 5.2, lim𝑇 →∞ 𝑇𝑡=0
𝑇
We claim that for any 𝜀 > 0, there is some 𝑇 ′ such that 𝑇 > 𝑇 ′ implies that
𝑇 −1
∑
ℎ(𝜉𝑡 , 𝜁 ∗ )
𝑡
𝑇
𝑡=0
≤ ℎ(𝑝∗ , 𝑞 ∗ ) + 𝜀.
(8)
Fix some 𝜀 > 0. Let 𝑇1 be so large that 𝑇 > 𝑇1 implies that, for all 𝑥 ∈ ℰ 1 ,
𝑇 −1
∑
ℎ(𝑥, 𝜁 𝑥 )
𝑡
𝑡=0
and, for all 𝑥 ∈ ℰ 2 ,
𝐶𝑥
𝑇
<
𝑇
≤ ℎ(𝑝∗ , 𝑞 ∗ ) +
𝜀
,
∣𝑋∣
(9)
𝜀
.
∣𝑋∣
Let 𝑇 ′ be so large that, for all 𝑥 ∈ ℰ1 , 𝐿𝑥 (𝑇 ′ ) > 𝑇1 . If 𝑇 > 𝑇 ′ , then
(𝑇 )−1
𝐿𝑥 (𝑇 )−1
∑ 𝐿𝑥 (𝑇 ) 𝐿𝑥∑
ℎ(𝑥, 𝜁𝑡𝑥 ) ∑ ∑ ℎ(𝑥, 𝜁𝑡𝑥 )
=
+
𝑇
𝑇
𝐿
𝑇
𝑥 (𝑇 )
𝑡=0
𝑡=0
𝑥∈ℰ1
𝑥∈ℰ2
∑ 𝐿𝑥 (𝑇 )
∑ 𝜀
𝜀
(ℎ(𝑝∗ , 𝑞 ∗ ) +
)+
≤ ℎ(𝑝∗ , 𝑞 ∗ ) + 𝜀.
≤
𝑇
∣𝑋∣
∣𝑋∣
𝑥∈ℰ
𝑖∈ℰ
𝑇 −1
∑
ℎ(𝜉𝑡 , 𝜁 ∗ )
𝑡
𝑡=0
1
(10)
2
Notice that 𝐿𝑥 is weakly increasing, and 𝐿𝑥 (𝑇 ) ≤ 𝑇 for all 𝑇 . Thus, 𝑇 > 𝑇 ′ implies that
𝐿𝑥 (𝑇 ) ≥ 𝐿𝑥 (𝑇 ′ ) > 𝑇1 , and so 𝑇 > 𝑇1 .
This proves the inequality (8), and it in turn implies that, for any 𝜀 > 0, there is some
𝑇 such that
𝛼𝑇 = sup
′ −1
𝑇∑
𝑇 ′ >𝑇 𝑡=0
ℎ(𝜉𝑡 , 𝜁𝑡∗ )
≤ ℎ(𝑝∗ , 𝑞 ∗ ) + 𝜀.
𝑇′
Now, the sequence {𝛼𝑇 }∞
𝑇 =0 is a decreasing sequence, and the above inequality shows that
for any 𝜀 > 0, lim𝑇 →∞ 𝛼𝑇 ≤ ℎ(𝑝∗ , 𝑞 ∗ ) + 𝜀. Thus, we have lim𝑇 →∞ 𝛼𝑇 ≤ ℎ(𝑝∗ , 𝑞 ∗ ). This
proves (6).
36
Now, (6) implies that for all 𝜉 ∈ 𝒳 ,
lim inf
𝑇 →∞
𝑇 −1
∑
ℎ(𝜉𝑡 , 𝜁 ∗ )
𝑡
𝑡=0
𝑇
≤ ℎ(𝑝∗ , 𝑞 ∗ ).
With a symmetric argument, we can show that for all 𝜁 ∈ 𝒴, lim sup𝑇 →∞
(11)
∑𝑇 −1
𝑡=0
−ℎ(𝜉𝑡∗ ,𝜁𝑡 )
𝑇
−ℎ(𝑝∗ , 𝑞 ∗ ). This implies that for all 𝜁 ∈ 𝒴,
lim inf
𝑇 →∞
𝑇 −1
∑
ℎ(𝜉 ∗ , 𝜁𝑡 )
𝑡
𝑡=0
𝑇
≥ ℎ(𝑝∗ , 𝑞 ∗ ).
(12)
By (11), we have for all 𝜉 ∈ 𝒳 , 𝑢ℎ (𝜉, 𝜁 ∗ ) ≤ ℎ(𝑝∗ , 𝑞 ∗ ). By (12), we have for all 𝜁 ∈ 𝒴,
𝑢ℎ (𝜉 ∗ , 𝜁) ≥ ℎ(𝑝∗ , 𝑞 ∗ ). Therefore, we have
𝑢ℎ (𝜉 ∗ , 𝜁 ∗ ) ≤ ℎ(𝑝∗ , 𝑞 ∗ ) ≤ 𝑢ℎ (𝜉 ∗ , 𝜁 ∗ ) and hence 𝑢ℎ (𝜉 ∗ , 𝜁 ∗ ) = ℎ(𝑝∗ , 𝑞 ∗ ).
This, together with (11) and (12), shows that (𝜉 ∗ , 𝜁 ∗ ) is an equilibrium.
On the other hand, if 𝜉 ′ ∈ 𝒳 is a 𝑝′ -stochastic sequence for some equilibrium strategy
𝑝′ in 𝑔, then ℎ(𝑝′ , 𝑞 ∗ ) = ℎ(𝑝∗ , 𝑞 ∗ ). Moreover, all the above argument holds for (𝜉 ′ , 𝜁 ∗ ) and
so 𝜉 ′ is also an equilibrium strategy. □
As we mentioned in Section 3.3, an analogous result holds for vertical games as well.
The following is the formal statement. Its proof is very similar to that of Theorem 3.2.
Theorem 5.1. Suppose that 𝒫 1 and 𝒫 2 satisfy (A1) and (A2) and mutual complexity.
Let (𝑝∗ , 𝑞 ∗ ) ∈ Δ(𝑋) × Δ(𝑌 ) be an equilibrium of 𝑔. Then, the set of strongly 𝑝∗ -stochastic
sequences relative to 𝒫 2 in 𝒳 , which is not empty, is a subset of player 1’s equilibrium
strategies in 𝑉 𝐺(𝑔, 𝒫 1 , 𝒫 2 ).
Proof. For any strategy profile (𝑎, 𝑏) ∈ 𝒳 × 𝒴, let 𝜃𝑎,𝑏 be the sequence of actions obtained by playing this profile, i.e., 𝜃𝑡𝑎,𝑏 = (𝜃𝑡𝑎,𝑏,𝑋 , 𝜃𝑡𝑎,𝑏,𝑌 ) with 𝜃𝑡𝑎,𝑏,𝑋 = 𝑎(𝜃𝑎,𝑏,𝑌 [𝑡]) and
𝜃𝑡𝑎,𝑏,𝑌 = 𝑏(𝜃𝑎,𝑏,𝑋 [𝑡]). By Theorem 6.2 and Theorem 6.5, there exists a strongly 𝑝∗ -stochastic
sequences 𝜉 ∗ in 𝒳 and a strongly 𝑞 ∗ -stochastic sequence 𝜁 ∗ in 𝒴. We show that (𝜉 ∗ , 𝜁 ∗ ) is
an equilibrium (as history-independent strategies).
37
≤
Because (𝑝∗ , 𝑞 ∗ ) is an equilibrium of the game 𝑔, ℎ(𝑥, 𝑞 ∗ ) ≤ ℎ(𝑝∗ , 𝑞 ∗ ) for all 𝑥 ∈ 𝑋.
First we show that for all 𝑎 ∈ 𝒳 ,
lim sup
𝑇 →∞
∗
𝑇 −1
∑
ℎ(𝜃𝑡𝑎,𝑏 )
𝑡=0
𝑇
≤ ℎ(𝑝∗ , 𝑞 ∗ ),
(13)
∗
where 𝑏∗ is such that 𝑏∗ (𝜎) = 𝜁∣𝜎∣
for all 𝜎 ∈ 𝑋 <ℕ .
Suppose that 𝑎 ∈ 𝒳 , and so 𝑎 ∈ 𝒫 1 . For each 𝑥 ∈ 𝑋, let 𝑟𝑥 : 𝑌 <ℕ → {0, 1} be the
selection function such that 𝑟𝑥 (𝜎) = 1 if 𝑎(𝜎) = 𝑥, and 𝑟𝑥 (𝜎) = 0 otherwise. Define
𝑥
𝐿𝑥 (𝑇 ) = ∣{𝑡 ∈ ℕ : 0 ≤ 𝑡 ≤ 𝑇 − 1, 𝑟𝑥 (𝜁 ∗ [𝑡]) = 1}∣ and 𝜁 𝑥 = (𝜁 ∗ )𝑟 .
It is easy to see that 𝑟𝑥 is in 𝒫 1 since 𝑎 is. Let
ℰ 1 = {𝑥 ∈ 𝑋 : lim 𝐿𝑥 (𝑇 ) = ∞} and ℰ 2 = {𝑥 ∈ 𝑋 : lim 𝐿𝑥 (𝑇 ) < ∞}.
𝑇 →∞
𝑇 →∞
∑𝐵𝑥
𝑥
For each 𝑥 ∈ ℰ 2 , let 𝐵𝑥 = lim𝑇 →∞ 𝐿𝑥 (𝑇 ) and let 𝐶𝑥 =
𝑡=0 ℎ(𝑥, 𝜁𝑡 ). Then, for any
∑ −1 ℎ(𝑥,𝜁𝑡𝑥 )
= ℎ(𝑥, 𝑞 ∗ ) ≤ ℎ(𝑝∗ , 𝑞 ∗ ).
𝑥 ∈ ℰ 1 , by Lemma 5.2, lim𝑇 →∞ 𝑇𝑡=0
𝑇
We claim that for any 𝜀 > 0, there is some 𝑇 ′ such that 𝑇 > 𝑇 ′ implies that
𝑇 −1
∑
ℎ(𝑎(𝜁 ∗ [𝑡]), 𝜁 ∗ )
𝑡
𝑇
𝑡=0
≤ ℎ(𝑝∗ , 𝑞 ∗ ) + 𝜀.
(14)
Fix some 𝜀 > 0. Let 𝑇1 be so large that 𝑇 > 𝑇1 implies that, for all 𝑥 ∈ ℰ 1 ,
𝑇 −1
∑
ℎ(𝑥, 𝜁 𝑥 )
𝑡
𝑡=0
and, for all 𝑥 ∈ ℰ 2 ,
𝐶𝑥
𝑇
<
𝜀
.
∣𝑋∣
𝑇
≤ ℎ(𝑝∗ , 𝑞 ∗ ) +
𝜀
,
∣𝑋∣
(15)
Let 𝑇 ′ be so large that, for all 𝑥 ∈ ℰ1 , 𝐿𝑥 (𝑇 ′ ) > 𝑇1 . If
𝑇 > 𝑇 ′ , then
(𝑇 )−1
𝐿𝑥 (𝑇 )−1
∑ 𝐿𝑥 (𝑇 ) 𝐿𝑥∑
ℎ(𝑥, 𝜁𝑡𝑥 ) ∑ ∑ ℎ(𝑥, 𝜁𝑡𝑥 )
=
+
𝑇
𝑇
𝐿
(𝑇
)
𝑇
𝑥
𝑡=0
𝑡=0
𝑥∈ℰ1
𝑥∈ℰ2
∑ 𝐿𝑥 (𝑇 )
∑ 𝜀
𝜀
(ℎ(𝑝∗ , 𝑞 ∗ ) +
)+
≤ ℎ(𝑝∗ , 𝑞 ∗ ) + 𝜀.
≤
𝑇
∣𝑋∣
∣𝑋∣
𝑥∈ℰ
𝑖∈ℰ
𝑇 −1
∑
ℎ(𝑎(𝜁 ∗ [𝑡]), 𝜁 ∗ )
𝑡
𝑡=0
1
(16)
2
Notice that 𝐿𝑥 is weakly increasing, and 𝐿𝑥 (𝑇 ) ≤ 𝑇 for all 𝑇 . Thus, 𝑇 > 𝑇 ′ implies that
𝐿𝑥 (𝑇 ) ≥ 𝐿𝑥 (𝑇 ′ ) > 𝑇1 , and so 𝑇 > 𝑇1 . This proves the inequality (14), and it in turn
implies (13).
38
Now, (13) implies that for all 𝑎 ∈ 𝒳 ,
lim inf
𝑇 →∞
∗
𝑇 −1
∑
ℎ(𝜃𝑡𝑎,𝑏 )
𝑡=0
𝑇
≤ ℎ(𝑝∗ , 𝑞 ∗ ).
Similarly, we can show that for all 𝑏 ∈ 𝒴, lim sup𝑇 →∞
(17)
∗ ,𝑏
−ℎ(𝜃𝑡𝑎
𝑡=0
𝑇
∑𝑇 −1
)
≤ −ℎ(𝑝∗ , 𝑞 ∗ ), where
𝑎∗ is such that 𝑎∗ (𝜎) = 𝜉∣𝜎∣ for all 𝜎 ∈ 𝑌 <ℕ . This implies that for all 𝑏 ∈ 𝒴,
lim inf
𝑇 →∞
∗
𝑇 −1
∑
ℎ(𝜃𝑡𝑎 ,𝑏 )
𝑡=0
𝑇
≥ ℎ(𝑝∗ , 𝑞 ∗ ).
(18)
By (17), we have for all 𝑎 ∈ 𝒳 , 𝑢ℎ (𝑎, 𝑏∗ ) ≤ ℎ(𝑝∗ , 𝑞 ∗ ). By (18), we have for all 𝑏 ∈ 𝒴,
𝑢ℎ (𝑎∗ , 𝑏) ≥ ℎ(𝑝∗ , 𝑞 ∗ ). Therefore, we have
𝑢ℎ (𝑎∗ , 𝑏∗ ) ≤ ℎ(𝑝∗ , 𝑞 ∗ ) ≤ 𝑢ℎ (𝑎∗ , 𝑏∗ ).
This, together with (17) and (18), implies that (𝑎∗ , 𝑏∗ ) is an equilibrium.
On the other hand, if 𝜉 ′ ∈ 𝒳 is a strongly 𝑝′ -stochastic sequence for some equilibrium
strategy 𝑝′ in 𝑔, then ℎ(𝑝′ , 𝑞 ∗ ) = ℎ(𝑝∗ , 𝑞 ∗ ). Moreover, all the above argument holds for
(𝜉 ′ , 𝜁 ∗ ) and so 𝜉 ′ is also an equilibrium strategy.
Now we proceed to necessary conditions.
Proof of Theorem 3.3:
We consider two cases (recall that 𝛼 < 1 and so 1 − 𝛼 > 0):
Case 1: lim𝑇 →∞
∑𝑇 −1
𝑡=0
𝑟(𝑡)/𝑇 = 1−𝛼 = 1. Suppose that 𝑝′ is not an equilibrium strategy
in 𝑔. Then, there is an action 𝑦1 ∈ 𝑌 such that ℎ(𝑝′ , 𝑦) < 𝑣, where 𝑣 is the equilibrium
payoff for player 1 in 𝑔. Define y1 as y1 𝑡 = 𝑦1 for all 𝑡 ∈ ℕ. Let 1 − 𝑟 be the selection
∑ −1
function such that (1 − 𝑟)(𝑡) = 1 − 𝑟(𝑡) for all 𝑡 ∈ ℕ, and let 𝑆(𝑇 ) = 𝑇𝑡=0
𝑟(𝑡) for all
𝑇 ∈ ℕ.
∑ )−1 ℎ(𝜉𝑡𝑟 ,𝑦1 )
∑ −𝑆(𝑇 )−1 ℎ(𝜉𝑡1−𝑟 ,𝑦1 )
∑𝑇 −1 ℎ(𝜉𝑡 ,y1 𝑡 )
= lim𝑇 →∞ 𝑆(𝑇
+ lim𝑇 →∞ 𝑇𝑡=0
.
Then, lim𝑇 →∞ 𝑡=0
𝑡=0
𝑇
𝑇
𝑇
∑𝑆(𝑇 )−1 ℎ(𝜉𝑡𝑟 ,𝑦1 )
∑𝑆(𝑇 )−1 ℎ(𝜉𝑡𝑟 ,𝑦1 )
)
Clearly, lim𝑇 →∞ 𝑡=0
= lim𝑇 →∞ 𝑡=0
lim𝑇 →∞ 𝑆(𝑇
= ℎ(𝑝′ , 𝑦1 ) < 𝑣.
𝑇
𝑆(𝑇 )
𝑇
∑𝑇 −𝑆(𝑇 )−1 ∣ℎ(𝜉𝑡1−𝑟 ,𝑦1 )∣
))𝐶
Let 𝐶 = max𝑥∈𝑋,𝑦∈𝑌 ∣ℎ(𝑥, 𝑦)∣. Then,
≤ (𝑇 −𝑆(𝑇
→ 0, and so
𝑡=0
𝑇
𝑇
∑𝑇 −𝑆(𝑇 )−1 ℎ(𝜉𝑡1−𝑟 ,𝑦1 )
lim𝑇 →∞ 𝑡=0
= 0.
𝑇
39
Therefore, 𝑢ℎ (𝜉, y1 ) < 𝑣, and so 𝜉 is not an equilibrium strategy.
∑𝑇 −1
Case 2:: lim𝑇 →∞
𝑡=0
𝑟(𝑡)/𝑇 = 1 − 𝛼 < 1.
First we show that the sequence 𝜉 1−𝑟 has limit relative frequency for all 𝑥 ∈ 𝑋. Indeed,
for each 𝑥 ∈ 𝑋,
lim
𝑇 −1
∑
𝑐𝑥 (𝜉𝑡1−𝑟 )
𝑇 →∞
= lim
= ( lim
𝑇 →∞
𝑇
𝑡=0
∑𝑇 −1
where 𝑝𝑥 = lim𝑇 →∞
𝑡=0
= lim
𝑇 →∞
𝑡=0
𝑇 −1
∑
𝑐𝑥 (𝜉𝑡 )(1 − 𝑟(𝑡))
𝑇 →∞
𝑇 −1
∑
𝑐𝑥 (𝜉𝑡 )
𝑇
𝑡=0
𝑇 −1
∑
𝑐𝑥 (𝜉𝑡 )(1 − 𝑟(𝑡))
𝑇
𝑡=0
− lim
𝑇 −1
∑
𝑐𝑥 (𝜉𝑡 )𝑟(𝑡)
𝑇 →∞
𝑐𝑥 (𝜉𝑡 )
𝑇
𝑡=0
lim
𝑇 →∞
𝑆(𝑇 )
𝑇 − 𝑆(𝑇 )
𝑇 −1
∑
𝑡=0
𝑇
𝑇 − 𝑆(𝑇 )
𝑇 −1
∑
𝑆(𝑇 )
𝑇
lim
) lim
𝑇 →∞ 𝑇
𝑇 →∞
𝑇 − 𝑆(𝑇 )
𝑡=0
1
= (𝑝𝑥 − (1 − 𝛼)𝑝′𝑥 ) ,
𝛼
∑𝑇 −1
and 𝑆(𝑇 ) = 𝑡=0 𝑟(𝑡) for all 𝑇 ∈ ℕ.
Suppose that 𝑝′ is not an equilibrium strategy in 𝑔. Then, there is an action 𝑦1 ∈ 𝑌
such that ℎ(𝑝′ , 𝑦1 ) < 𝑣, where 𝑣 is the equilibrium payoff for player 1 in 𝑔. Moreover,
there is an action 𝑦2 ∈ 𝑌 such that ℎ( 𝛼1 (𝑝 − (1 − 𝛼)𝑝′ ), 𝑦2 ) ≤ 𝑣. Define 𝜁 as follows: 𝜁𝑡 = 𝑦1
if 𝑟(𝑡) = 1 and 𝜁𝑡 = 𝑦2 if 𝑟(𝑡) = 0.
Then,
lim
𝑇 −1
∑
ℎ(𝜉𝑡 , 𝜁𝑡 )
𝑇 →∞
𝑇
𝑡=0
𝑇 −𝑆(𝑇 )−1
∑ ℎ(𝜉 𝑟 , 𝜁 𝑟 )
∑ ℎ(𝜉𝑡1−𝑟 , 𝜁𝑡1−𝑟 )
𝑡
𝑡
= lim
+ lim
𝑇 →∞
𝑇 →∞
𝑇
𝑇
𝑡=0
𝑡=0
𝑆(𝑇 )−1
𝑇 −𝑆(𝑇 )−1
∑ ℎ(𝜉 𝑟 , 𝑦1 )
∑ ℎ(𝜉𝑡1−𝑟 , 𝑦2 )
𝑆(𝑇 )
𝑇 − 𝑆(𝑇 )
𝑡
= lim
lim
+ lim
lim
𝑇 →∞
𝑇 →∞
𝑆(𝑇 ) 𝑇 →∞ 𝑇
𝑇 − 𝑆(𝑇 ) 𝑇 →∞
𝑇
𝑡=0
𝑡=0
𝑆(𝑇 )−1
1
= (1 − 𝛼)ℎ(𝑝′ , 𝑦1 ) + 𝛼ℎ( (𝑝 − (1 − 𝛼)𝑝′ ), 𝑦2 ) < 𝑣.
𝛼
Therefore, 𝑢ℎ (𝜉, 𝜁) < 𝑣 and so 𝜉 is not an equilibrium strategy. □
We give the proof of Theorem 3.4. The arguments are very similar to those in the
proof of Theorem 3.3.
Proof of Theorem 3.4: We consider two cases:
40
Case 1: lim𝑇 →∞
∑𝑇 −1
𝑡=0
𝑟(𝜉[𝑡])/𝑇 = 1. Suppose that 𝑝 is not an equilibrium strategy in
𝑔. Then, there is an action 𝑦1 ∈ 𝑌 such that ℎ(𝑝, 𝑦1 ) < 𝑣, where 𝑣 is the equilibrium
payoff for player 1 in 𝑔. Define 𝑏 as 𝑏(𝜎) = 𝑦1 for all 𝜎 ∈ 𝑋 <ℕ . Let 1 − 𝑟 be the selection
∑ −1
function such that (1 − 𝑟)(𝜎) = 1 − 𝑟(𝜎) for all 𝜎 ∈ 𝑋 <ℕ , and let 𝑆(𝑇 ) = 𝑇𝑡=0
𝑟(𝜉[𝑡]) for
all 𝑇 ∈ ℕ.
∑ −1 ℎ(𝜉𝑡 ,𝜁𝑡 )
∑ )−1 ℎ(𝜉𝑡𝑟 ,𝜁𝑡𝑟 )
∑ −𝑆(𝑇 )−1 ℎ(𝜉𝑡1−𝑟 ,𝜁𝑡1−𝑟 )
Then, lim𝑇 →∞ 𝑇𝑡=0
= lim𝑇 →∞ 𝑆(𝑇
+lim𝑇 →∞ 𝑇𝑡=0
.
𝑡=0
𝑇
𝑇
𝑇
∑ )−1 ℎ(𝜉𝑡𝑟 ,𝑦1 )
∑ )−1 ℎ(𝜉𝑡𝑟 ,𝜁𝑡𝑟 )
)
= lim𝑇 →∞ 𝑆(𝑇
lim𝑇 →∞ 𝑆(𝑇
= ℎ(𝑝, 𝑦1 ) < 𝑣.
Clearly, lim𝑇 →∞ 𝑆(𝑇
𝑡=0
𝑡=0
𝑇
𝑆(𝑇 )
𝑇
∑𝑇 −𝑆(𝑇 )−1 ∣ℎ(𝜉𝑡1−𝑟 ,𝜁𝑡1−𝑟 )∣
))𝐶
Let 𝐶 = max𝑥∈𝑋,𝑦∈𝑌 ∣ℎ(𝑥, 𝑦)∣. Then,
≤ (𝑇 −𝑆(𝑇
→ 0, and so
𝑡=0
𝑇
𝑇
∑𝑇 −𝑆(𝑇 )−1 ℎ(𝜉𝑡1−𝑟 ,𝜁𝑡1−𝑟 )
lim𝑇 →∞ 𝑡=0
= 0.
𝑇
Therefore, 𝜉 is not an equilibrium strategy.
Case 2:: lim𝑇 →∞
∑𝑇 −1
𝑡=0
𝑟(𝜉[𝑡])/𝑇 = 1 − 𝛼 < 1.
First we show that the sequence 𝜉 1−𝑟 has limit relative frequency for all 𝑥 ∈ 𝑋. Indeed,
for each 𝑥 ∈ 𝑋,
lim
𝑇 −1
∑
𝑐𝑥 (𝜉𝑡1−𝑟 )
𝑇 →∞
𝑇 →∞
= ( lim
𝑇 →∞
𝑡=0
where 𝑞𝑥 = lim𝑇 →∞
𝑇
∑𝑇 −1
𝑡=0
= lim
𝑇 →∞
𝑇
𝑡=0
− lim
𝑇 →∞
𝑐𝑥 (𝜉𝑡 )
𝑇
lim
𝑇 →∞
𝑇 −1
∑
𝑐𝑥 (𝜉𝑡 )𝑟(𝜉[𝑡])
𝑡=0
𝑇 − 𝑆(𝑇 )
𝑡=0
𝑇 −1
∑
𝑐𝑥 (𝜉𝑡 )(1 − 𝑟(𝜉[𝑡]))
= lim
𝑇 −1
∑
𝑐𝑥 (𝜉𝑡 )
𝑇
𝑡=0
𝑇 −1
∑
𝑐𝑥 (𝜉𝑡 )(1 − 𝑟(𝜉[𝑡]))
𝑆(𝑇 )
𝑇 −1
∑
𝑡=0
𝑇
𝑇 − 𝑆(𝑇 )
𝑇 −1
∑
𝑆(𝑇 )
𝑇
lim
) lim
𝑇 →∞ 𝑇
𝑇 →∞
𝑇 − 𝑆(𝑇 )
𝑡=0
1
= (𝑞𝑥 − (1 − 𝛼)𝑝𝑥 ) ,
𝛼
∑𝑇 −1
and 𝑆(𝑇 ) = 𝑡=0 𝑟(𝜉[𝑡]) for all 𝑇 ∈ ℕ.
Suppose that 𝑝 is not an equilibrium strategy in 𝑔. Then, there is an action 𝑦1 ∈ 𝑌
such that ℎ(𝑝, 𝑦) < 𝑣, where 𝑣 is the equilibrium payoff for player 1 in 𝑔. Moreover, there
is an action 𝑦2 ∈ 𝑌 such that ℎ( 𝛼1 (𝑞 −(1−𝛼)𝑝), 𝑦2 ) ≤ 𝑣. Define 𝑏 ∈ 𝒴 as follows: 𝑏(𝜎) = 𝑦1
if 𝑟(𝜎) = 1 and 𝑏(𝜎) = 𝑦2 if 𝑟(𝜎) = 0.
Then,
lim
𝑇 →∞
𝑇 −1
∑
ℎ(𝜃𝑡𝜉,𝑏 )
𝑡=0
𝑇
𝑇 −𝑆(𝑇 )−1
∑ ℎ(𝜉 𝑟 , 𝑦1 )
∑ ℎ(𝜉𝑡1−𝑟 , 𝑦2 )
𝑡
= lim
+ lim
𝑇 →∞
𝑇 →∞
𝑇
𝑇
𝑡=0
𝑡=0
𝑆(𝑇 )−1
41
𝑇 −𝑆(𝑇 )−1
∑ ℎ(𝜉 𝑟 , 𝑦1 )
∑ ℎ(𝜉𝑡1−𝑟 , 𝑦2 )
𝑆(𝑇 )
𝑇 − 𝑆(𝑇 )
𝑡
= lim
lim
+ lim
lim
𝑇 →∞
𝑇 →∞
𝑆(𝑇 ) 𝑇 →∞ 𝑇
𝑇 − 𝑆(𝑇 ) 𝑇 →∞
𝑇
𝑡=0
𝑡=0
𝑆(𝑇 )−1
1
= (1 − 𝛼)ℎ(𝑝, 𝑦1 ) + 𝛼ℎ( (𝑞 − (1 − 𝛼)𝑝), 𝑦2 ) < 𝑣.
𝛼
Therefore, 𝜉 is not an equilibrium strategy. □
Now we give the proof of Lemma 3.1
Proof of Lemma 3.1: Suppose that 𝜉 ∈ 𝑋 ℕ satisfies that 𝜉2𝑡 = 𝜉2𝑡+1 for all 𝑡. Let
𝑞 ∗ ∈ Δ(𝑌 ) be an equilibrium mixed-strategy of player 2 in 𝑔. Because 𝑔 has no pure
equilibrium, max𝑥∈𝑋 min𝑦∈𝑌 ℎ(𝑥, 𝑦) =𝑑𝑒𝑓 𝑣1 < 𝑣, where 𝑣 is the equilibrium payoff for
player 1 in 𝑔.
Because 𝒫 1 and 𝒫 2 satisfy mutual complexity, there is a 𝑞 ∗ -stochastic sequence 𝜁 ∈ 𝑌 ℕ .
Let 𝑏 : 𝑋 ℕ → 𝑌 be a strategy such that for any 𝜎 ∈ 𝑋 <ℕ ,
(1) 𝑏(𝜎) = 𝜁𝑡 if ∣𝜎∣ = 2𝑡 for some 𝑡;
(2) 𝑏(𝜎) ∈ arg min{𝑦 ∈ 𝑌 : ℎ(𝜎∣𝜎∣−1 , 𝑦)} if ∣𝜎∣ = 2𝑡 + 1 for some 𝑡.
Clearly, we can make 𝑏 be a function in 𝒫 2 because 𝜁 belongs to 𝒫 2 . By a similar argument
to what is used in the proof of Theorem 3.2, we can show that
lim sup
𝑇 →∞
𝑇 −1
∑
ℎ(𝜉2𝑡 , 𝑏(𝜉[2𝑡]))
𝑇
𝑡=0
≤ max ℎ(𝑥, 𝑞 ∗ ) = 𝑣.
𝑥∈𝑋
(19)
Moreover, for any 𝑇 > 0,
𝑇 −1
∑
ℎ(𝜉2𝑡+1 , 𝑏(𝜉[2𝑡 + 1]))
𝑡=0
𝑇
𝑇 −1
∑
max𝑥∈𝑋 min𝑦∈𝑌 ℎ(𝑥, 𝑦)
≤
= 𝑣1 .
𝑇
𝑡=0
We now show that for any 𝜀 > 0, there is a number 𝑇0 such that 𝑇 ≥ 𝑇0 implies that
𝑇 −1
∑
ℎ(𝜉𝑡 , 𝑏(𝜉[𝑡]))
𝑡=0
𝑇
<
𝑣 + 𝑣1
+ 𝜀.
2
𝜀
Let 𝜖′ = min{{ ∣1+𝑣∣
, 𝜀′ }. By (19), there is a number 𝑇1 so that for all 𝑇 > 𝑇1 ,
𝑇 −1
∑
ℎ(𝜉2𝑡 , 𝑏(𝜉[2𝑡]))
𝑡=0
𝑇
42
≤ 𝑣 + 𝜀′ .
Let 𝑇0 = max{( 2𝜀1 ′ − 1), 2𝑇1 }. Then, for all 𝑇 such that 2𝑇 > 𝑇0 , we have
2𝑇
−1
∑
𝑡=0
𝑇 −1
𝑇 −1
1 ∑ ℎ(𝜉2𝑡 , 𝑏(𝜉[2𝑡])) 1 ∑ ℎ(𝜉2𝑡+1 , 𝑏(𝜉[2𝑡 + 1]))
ℎ(𝜉𝑡 , 𝑏(𝜉[𝑡]))
=
+
2𝑇
2 𝑡=0
𝑇
2 𝑡=0
𝑇
1
1
𝑣 + 𝑣1
< (𝑣 + 𝜀′ ) + 𝑣1 <
+ 𝜀;
2
2
2
and
2𝑇
∑
ℎ(𝜉𝑡 , 𝑏(𝜉[𝑡]))
𝑡=0
2𝑇 + 1
𝑇 −1
𝑇
=
𝑇 + 1 ∑ ℎ(𝜉2𝑡 , 𝑏(𝜉[2𝑡]))
𝑇 ∑ ℎ(𝜉2𝑡+1 , 𝑏(𝜉[2𝑡 + 1]))
+
2𝑇 + 1 𝑡=0
𝑇
2𝑇 + 1 𝑡=0
𝑇
1
1
𝑣 + 𝑣1
< ( + 𝜀′ )(𝑣 + 𝜀′ ) + 𝑣1 <
+ 𝜀.
2
2
2
Therefore, 𝑢ℎ (𝜉, 𝑏) < 𝑣, and hence 𝜉 cannot be an equilibrium strategy of player 1. □
Finally, we give the proof of Theorem 3.5.
Proof of Theorem 3.5: Recall that 𝑝∗ is an equilibrium strategy with 0 < 𝑝∗𝑥 < 1. Let
𝑥′ ∕= 𝑥 be such that 𝑝∗𝑥′ > 0. Such 𝑥′ exists because 𝑝∗𝑥 < 1.
For any real number 𝑠, let └𝑠┘ be the largest integer less than or equal to 𝑠. Construct
the sequence p = (𝑝0 , 𝑝1 , ...) as follows (𝑡 is the smallest 𝑡 such that └𝑡0.4 ┘ >
1
):
𝑝∗𝑥
(a) 𝑝𝑡𝑦 = 𝑝∗𝑦 if 𝑦 ∕= 𝑥 and 𝑦 ∕= 𝑥′ ;
(b) 𝑝𝑡𝑥 = 𝑝∗𝑥 if 𝑡 ≤ 𝑡 and 𝑝𝑡𝑥 = 𝑝∗𝑥 −
1
└𝑡0.4 ┘
(c) 𝑝𝑡𝑥′ = 𝑝∗𝑥′ if 𝑡 ≤ 𝑡 and 𝑝𝑡𝑥′ = 𝑝∗𝑥′ +
otherwise;
1
└𝑡0.4 ┘
otherwise.
By construction, 𝑝𝑡𝑥 = 0 if and only if 𝑝∗𝑥 = 0, and lim𝑡→∞ 𝑝𝑡 = 𝑝∗ . Clearly, p is
computable. By Theorem 6.2, there is a 𝜇p -random sequence 𝜉 relative to 𝒫 2 in 𝒫 1 .
Now, let 𝑋0 = {𝑦 ∈ 𝑋 : 𝑝𝑦 > 0}, then the sequence 𝜉 is 𝜇p -random can be regarded as a
sequence in 𝑋0ℕ . Therefore, Theorem 6.5 is applicable and so 𝜉 is strongly 𝑝∗ -stochastic.
Thus, by Theorem 3.2 and Theorem 5.1, 𝜉 is an equilibrium strategy for player 1 in both
𝐻𝐺(𝑔, 𝒫 1 , 𝒫 2 ) and 𝑉 𝐺(𝑔, 𝒫 1 , 𝒫 2 ).
By Theorem 6.6, we have
∑𝑇 −1
𝑡
𝑡=0 (𝑐𝑥 (𝜉𝑡 ) − 𝑝𝑥 )∣
= 1.
lim sup √ ∑
√ ∑ −1
𝑇 −1 𝑡
𝑇 →∞
2( 𝑡=0
𝑝𝑥 (1 − 𝑝𝑡𝑥 )) log log ( 𝑇𝑡=0
𝑝𝑡𝑥 (1 − 𝑝𝑡𝑥 ))
∣
43
(20)
For any 𝑇 large enough,
∑𝑇 −1
∑𝑇 −1
∑𝑇 −1
1
∗
𝑡
𝑡=𝑡+1 └𝑡0.4 ┘
𝑡=0 𝑐𝑥 (𝜉𝑡 ) − 𝑇 𝑝𝑥
𝑡=0 (𝑐𝑥 (𝜉𝑡 ) − 𝑝𝑥 )
√
√
√
=
+
2𝑇 𝑝∗𝑥 (1 − 𝑝∗𝑥 ) log log 𝑇
2𝑇 𝑝∗𝑥 (1 − 𝑝∗𝑥 ) log log 𝑇
2𝑇 𝑝∗𝑥 (1 − 𝑝∗𝑥 ) log log 𝑇
We claim that
∑𝑇 −1
lim √
𝑇 →∞
1
𝑡=𝑡+1 └𝑡0.4 ┘
2𝑇 𝑝∗𝑥 (1 − 𝑝∗𝑥 ) log log 𝑇
= ∞;
(21)
and there exists some 𝐵 > 0 such that for all 𝑇 large enough,
∑𝑇 −1
∣ 𝑡=0
(𝑐𝑥 (𝜉𝑡 ) − 𝑝𝑡𝑥 )∣
√
< 𝐵.
2𝑇 𝑝∗𝑥 (1 − 𝑝∗𝑥 ) log log 𝑇
(22)
The conclusion follows directly from (21) and (22).
For all 𝑡, we have └𝑡0.4 ┘ ≤ 𝑡0.4 and so
𝑇 −1
∑
𝑡=1
1
𝑡0.4
≤
1
.
└𝑡0.4 ┘
It is easy to check that
∫ 𝑇 −1
𝑇 −1
∑
1
≥
≥
𝑥−0.4 𝑑𝑥 − 1 ≥ (𝑇 − 1)0.6 − 2.
0.4
└𝑡0.4 ┘
𝑡
𝑥=1
𝑡=1
1
Therefore, for 𝑇 large enough,
∑𝑇 −1
1
0.5𝑇 0.6
𝑇 0.1
𝑡=𝑡+1 └𝑡0.4 ┘
√
≥√
= 𝐴0 √
log log 𝑇
2𝑇 𝑝∗𝑥 (1 − 𝑝∗𝑥 ) log log 𝑇
2𝑇 𝑝∗𝑥 (1 − 𝑝∗𝑥 ) log log 𝑇
for some constant 𝐴0 > 0. Because lim𝑇 →∞
0.1
√ 𝑇
log log 𝑇
(23)
= ∞, (23) implies that (21).
Because of (20), to prove (22), it suffices to show that for 𝑇 large enough,
√ ∑
√ ∑ −1
−1 𝑡
𝑝𝑥 (1 − 𝑝𝑡𝑥 )) log log ( 𝑇𝑡=0
𝑝𝑡𝑥 (1 − 𝑝𝑡𝑥 ))
2( 𝑇𝑡=0
√
2𝑇 𝑝∗𝑥 (1 − 𝑝∗𝑥 ) log log 𝑇
(24)
is bounded. Now, for 𝑇 large enough,
𝑇 −1
∑
𝑝𝑡𝑥 (1
−
𝑝𝑡𝑥 ))
=
𝑇 𝑝∗𝑥 (1
−
𝑝∗𝑥 )
+
(2𝑝∗𝑥
− 1)
𝑡=0
𝑇 −1
∑
𝑡=𝑡+1
1
└𝑡0.4 ┘
−
𝑇 −1
∑
(
𝑡=𝑡+1
1
└𝑡0.4 ┘
)2 .
Because for 𝑡 large enough, 12 𝑡0.4 < └𝑡0.4 ┘, there is a constant 𝐴 > 0 such that
𝑇 −1
∑
𝑡=𝑡+1
1
└𝑡0.4 ┘
<
𝑇 −1
∑
2
+ 𝐴 < 2𝑇 0.6 + 𝐴.
0.4
𝑡
𝑡=𝑡+1
Similarly, there is a constant 𝐴′ > 0 such that
𝑇 −1
∑
𝑇 −1
∑
4
( 0.4 ) <
+ 𝐴′ < 4𝑇 0.2 + 𝐴′ .
0.8
└𝑡 ┘
𝑡
𝑡=𝑡+1
1
2
𝑡=𝑡+1
44
(25)
Hence,
∑𝑇 −1
𝑡
𝑡=0 𝑝𝑥 (1
𝑇 𝑝∗𝑥 (1 −
− 𝑝𝑡𝑥 )
2∣2𝑝∗𝑥 − 1∣
∣2𝑝∗𝑥 − 1∣𝐴 + 𝐴′
4
<
1
+
+
+ 0.8 ∗
,
∗
0.4
∗
∗
∗
∗
𝑝𝑥 )
𝑇 𝑝𝑥 (1 − 𝑝𝑥 )
𝑇 𝑝𝑥 (1 − 𝑝𝑥 )
𝑇 𝑝𝑥 (1 − 𝑝∗𝑥 )
and so, for 𝑇 large enough,
∑𝑇 −1
𝑝𝑡𝑥 (1 − 𝑝𝑡𝑥 ))
< 2.
𝑇 𝑝∗𝑥 (1 − 𝑝∗𝑥 )
𝑡=0
Equation (25) also implies that, for 𝑇 large enough,
𝑇 −1
∑
𝑝𝑡𝑥 (1 − 𝑝𝑡𝑥 ) ≤ (2∣2𝑝∗𝑥 − 1∣ + 5 + 𝑝∗𝑥 (1 − 𝑝∗𝑥 ))𝑇 = 𝐴′′ 𝑇,
𝑡=0
and hence
√ ∑𝑇 −1 𝑡
( 𝑡=0 𝑝𝑥 (1 − 𝑝𝑡𝑥 ))
log(log 𝑇 + log 𝐴′′ )
log 2 + log log 𝑇
≤
≤
.
log log 𝑇
log log 𝑇
log log 𝑇
√ ∑ −1 𝑡
𝑡
log log ( 𝑇
𝑡=0 𝑝𝑥 (1−𝑝𝑥 ))
So for 𝑇 large enough,
≤ 2. Thus, the expression in (24) is bounded
log log 𝑇
log log
by 2, and we have proved (22). □
6
Appendix
6.1
Recursive functions
6.2
Martin-Löf randomness
Here we discuss a property called Martin-Löf randomness [21] (henceforth ML randomness), which defines random sequences in terms of statistical regularities. Its definition
begins with a formulation of idealized statistical tests. Some notation is needed to define
these tests and the criterion. Let 𝑋 be a finite set. We endow the set of infinite sequences
𝑋 ℕ with the product topology. Any open set can be written as a union of basic sets,
where a basic set has the form 𝑁𝜎 = {𝜁 ∈ 𝑋 ℕ : 𝜎 = 𝜁[∣𝜎∣]} for some 𝜎 ∈ 𝑋 <ℕ . We give a
formal definition of ML-randomness in the following.
45
Definition 6.1. Let 𝑋 be a finite set and let 𝒫 be a set of functions satisfying (A1) and
(A2). Suppose that 𝜇 is a computable probability measure over 𝑋 ℕ ,34 i.e., the mapping
𝜎 7→ 𝜇(𝑁𝜎 ) is computable.35 A sequence of open sets {𝑉𝑡 }∞
𝑡=0 is a 𝜇-test relative to 𝒫 if it
satisfies the following conditions:
(1) There is a function 𝑓 : ℕ → ℕ × 𝑋 <ℕ in 𝒫 such that for all 𝑡 ∈ ℕ,
𝑉𝑡 =
∪
{𝑁𝜎 : (∃𝑛)(𝑓 (𝑛) = (𝑡, 𝜎))}.
(2) For all 𝑡 ∈ ℕ, 𝜇(𝑉𝑡 ) ≤ 2−𝑡 .
A sequence 𝜉 ∈ 𝑋 ℕ is 𝜇-random relative to 𝒫 if it passes all 𝜇-tests relative to 𝒫, i.e., for
∩
/ ∞
any 𝜇-test {𝑉𝑡 }∞
𝑡=0 relative to 𝒫, 𝜉 ∈
𝑡=0 𝑉𝑡 .
The crucial part of this definition is to define a “describable” event with probability
1.36 Martin-Löf [21] defines “describable” by saying that an event with probability 1 is
describable if there is a constructive proof for that fact. A test {𝑉𝑡 }∞
𝑡=0 is used to establish
∩∞
the statistical regularity corresponding to the complement of 𝑡=0 𝑉𝑡 .37 Conditions (1)
and (2) require that this test, as a sequence of open sets, can be generated by functions
in 𝒫, and hence is constructive with respect to 𝒫. A sequence is random if it passes all
such tests. We remark that if 𝒫 satisfies both (A1) and (A2), then for any computable
probability measure 𝜇, the set of 𝜇-random sequences relative to 𝒫 has probability 1 with
respect to 𝜇 (see Downey et al. [11]), as stated in the following proposition. Its proof
for a special case can be found in Martin-Löf [21] (with some minor modifications to
accommodate general computable measures). See also Nies [24], p.123. The proof can be
easily extended to cover the general case.
34
In the definition, we implicitly assume that 𝜇(𝑁𝜎 ) is always a rational number for 𝜇 to be computable.
In the literature, the computability of a measure is defined more generally, but this definition is sufficient
for our purpose.
35
Here a function is computable if it is Turing-computable, i.e., if it belongs to the smallest set of
functions that satisfy both (A1) and (A2).
36
The quantifier “describable” is necessary (because there exists no sequence belonging to all events
with probability 1), but hard to define.
∩∞
37
Notice that 𝜇( 𝑡=0 𝑉𝑡 ) = 0.
46
Proposition 6.1. Suppose that 𝑋 is a finite set and 𝜇 is a computable measure over 𝑋 ℕ .
Let 𝒫 satisfy (A1) and (A2). Then
𝜇({𝜉 ∈ 𝑋 ℕ : 𝜉 is 𝜇-random relative to 𝒫}) = 1.
We are mostly interested in Bernoulli measures—i.e., measures generated by a distribution over 𝑋 in an i.i.d. manner. For any 𝑝 ∈ Δ(𝑋), we use 𝜇𝑝 to denote the Bernoulli
∏
measure over 𝑋 ℕ generated by 𝑝: for all 𝜎 ∈ 𝑋 <ℕ , 𝜇𝑝 (𝑁𝜎 ) = ∣𝜎∣−1
𝑡=0 𝑝𝜎𝑡 . In what follows, if
ML-randomness is mentioned without reference to any probability measure, it means ML
randomness with respect to an i.i.d. measure 𝜇𝑝 . ML-randomness is a stronger property
than strong stochasticity. See Theorem 6.5 in the next subsection.
6.2.1
Generating random sequences
First we give a theorem that directly connects incompressible sequence to random sequence: in general, a sequence is incompressible if and only if it is random w.r.t. the
uniform distribution. The proof of the following theorem can be found in Nies [24], p.
122.
Theorem 6.1. Suppose that 𝒫 satisfies (A1) and (A2). Then, 𝜉 ∈ {0, 1}ℕ is an incompressible sequence relative to 𝒫 if and only if 𝜉 is a 𝜆-random sequence relative to 𝒫,
where 𝜆(𝑁𝜎 ) = 2−∣𝜎∣ for all 𝜎 ∈ {0, 1}<ℕ .
The following theorem shows that, if mutual complexity holds, each player can generate
a 𝜇p -random sequence relative to the other player for any p, where 𝜇p is defined as
∏
𝑡
0 1
𝜇p (𝜎) = ∣𝜎∣−1
𝑡=0 𝑝 (𝜎𝑡 ), with p = (𝑝 , 𝑝 , ...) being a sequence of probability measures over
𝑋. The proof for the case ∣𝑋∣ = 2 can be found in Zvonkin and Levin [35]. We follow
a similar logic. Although the proof is lengthy, it is based on a computable version of the
well-known result that one can generate a random variable with an arbitrary distribution
function from the uniform distribution. In the following theorem and the subsequent
analysis, we assume that p is computable and 𝑝𝑡𝑥 > 0 for all 𝑥 ∈ 𝑋 and all 𝑡 ∈ ℕ
throughout the following unless stated otherwise.
47
To prove this theorem, some additional tools from Recursion theory is necessary. For
our purpose here, it is more convenient to consider Turing machines with oracle inquiries
(c.f. Odifreddi [25]). Given a finite set 𝑋, an element 𝜉 in 𝑋 ℕ is regarded a Turing oracle.
We then allow a set of feasible functions to include partial functions as well. Thus, if 𝒫
satisfies (A1) and (A2), then 𝒫 = {𝑓 : ℕ𝑘 → ℕ∣ 𝑓 is computable from 𝜉} for some oracle
𝜉.38 Here, however, we consider also partial functions. We define 𝐶(𝜉), which consists of
partial functions computable from 𝜉, as the following:
𝐶(𝜉) = {𝑓 :⊆ ℕ𝑘 → ℕ∣ 𝑓 is a partial function and is computable from 𝜉}
It is well known that any function in 𝐶(𝜉) can be computed by a Turing machine with
(𝑘),𝜉
oracle 𝜉. As a result, we can use 𝜑𝑒
to denote the function that is computed by
the machine with Gödel number 𝑒 and with oracle 𝜉, and the superscript 𝑘 indicates
that we put 𝑘 numbers as input in the computation. Moreover, it is known that the
(𝑘),𝜉[𝑠]
predicate 𝜑𝑒,𝑠
(𝑡1 , ..., 𝑡𝑘 ) = 𝑟, which indicates whether the machine with index 𝑒 halts
(using (𝑡1 , .., 𝑡𝑘 ) as inputs and 𝜉 as the oracle) within 𝑠 steps and produce output 𝑟, is
computable. This result is usually called the Enumeration theorem in the literature (see
Downey et al. [11]). This also allows us to define recursively enumerable sets, which will
ℕ
be useful in what to come. Consider a sequence {𝑉𝑡 }∞
𝑡=0 of subsets of 𝑋 . Such a sequence
<ℕ
is recursively enumerable in 𝜉, or is of Σ0,𝜉
1 , if there is a total function 𝑓 : ℕ → ℕ × 𝑋
in 𝐶(𝜉) such that for all 𝑡 ∈ ℕ and for all 𝜁 ∈ 𝑋 ℕ ,
𝜁 ∈ 𝑉𝑡 ⇔ (∃𝑛)(𝑓 (𝑛) = (𝑡, 𝜎) ∧ 𝜎 = 𝜁[∣𝜎∣]).
(26)
In this case, we can find a total function ℎ ∈ 𝒯 such that39
(1),𝜉⊕𝜁[𝑠]
𝜁 ∈ 𝑉𝑡 ⇔ (∃𝑠)𝜑ℎ(𝑡)
(0) ↓,
(27)
This result transform the relation in (26) into a parameterized form in (27).
38
By (A2) we know that this is true for some total function 𝑓 ; we can then use the graph of 𝑓 , which
is also computable from 𝑓 , to construct the oracle 𝜉. For more details, see Odifreddi [25].
39
This follows directly from the Parametrization Theorem for relative computability. Please see Downey
et al. [11] for a more detailed discussion.
48
Theorem 6.2. Suppose that 𝒫 satisfies (A1) and (A2). Let 𝑋 be a finite set. Suppose
that 𝜉 ∈ {0, 1}ℕ is an incompressible sequence relative to 𝒫. Then, there is a 𝜇p -sequence
𝜁 ∈ 𝑋 ℕ relative to 𝒫 in 𝐶(𝜉).
Proof. First we remark a known fact: there is a 𝜆𝑋 -random sequence 𝜉 ′ ∈ 𝑋 ℕ relative to
𝒫 that is in 𝐶(𝜉), where 𝜆𝑋 (𝑁𝜎 ) = ∣𝑋∣−∣𝜎∣ for all 𝜎 ∈ 𝑋 <ℕ (i.e., the uniform distribution
over 𝑋 ℕ ). For a proof, see Calude [6], Theorem 7.18.
We will construct a partial computable functional Φ : 𝑋 ℕ → 𝑋 ℕ such that if 𝜉 ′ is a
𝜆𝑋 -random sequence, then Φ(𝜉 ′ ) is a 𝜇p -random sequence. Instead of constructing that
functional directly, we construct a computable function 𝜙 : 𝑋 <ℕ → 𝑋 <ℕ such that
for any 𝜎, 𝜏 ∈ 𝑋 <ℕ , 𝜎 ⊂ 𝜏 implies 𝜙(𝜎) ⊂ 𝜙(𝜏 ).
(28)
Then we define Φ as Φ(𝜉)𝑡 = 𝜙(𝜉[min{𝑘 : ∣𝜙(𝜉[𝑘])∣ ≥ 𝑡}])𝑡 . Notice that Φ is well defined
if lim𝑡→∞ 𝜙(𝜉[𝑡]) = ∞ and it outputs a finite sequence otherwise (i.e., for large 𝑡’s, Φ(𝜉)𝑡
is not defined). We will show that Φ satisfies the following properties:
1. Φ is well-defined over any sequence in 𝑋 ℕ that is not computable.
2. 𝜆𝑋 (Φ−1 (𝐴)) = 𝜇p (𝐴) for any measurable 𝐴.
3. If 𝜉 ′ is a 𝜆𝑋 -random sequence, then Φ(𝜉 ′ ) is a 𝜇p -random sequence.
(Construction of 𝜙) 𝜙 is constructed through the distribution function of 𝜇p , which is
introduced as follows. There is a natural mapping Γ between 𝑋 ℕ and [0, 1]:
Γ(𝜁) =
∞
∑
𝜄(𝜁𝑡 )
𝑡=0
1
𝑛𝑡+1
,
where 𝑋 = {𝑥1 , ..., 𝑥𝑛 } and 𝜄(𝑥) = 𝑖 − 1 if and only if 𝑥 = 𝑥𝑖 . Γ is onto but not one-to-one.
However, the set {𝜁 ∈ 𝑋 ℕ : Γ(𝜁) = Γ(𝜁 ′ ) for some 𝜁 ′ ∕= 𝜁} is countable, since for any such
∑∣𝜎∣−1 𝜄(𝜎𝑡 )
𝜁, Γ(𝜁) is a rational number. Γ can be extended to 𝑋 <ℕ by defining Γ(𝜎) = 𝑡=0
.
𝑛𝑡+1
Given Γ, there distribution function of 𝜇p over [0, 1] is defined by 𝑔 : [0, 1] → [0, 1] as
𝑔(𝑟) = 𝜇p ({𝜁 : Γ(𝜁) ≤ 𝑟}).
Therefore, 𝑟 ≤ 𝑔(𝑠) if and only if ℎ(𝑟) ≤ 𝑠. Hence,
𝜇p (Γ−1 ([0, 𝑟])) = 𝑔(𝑟) = 𝜆𝑋 (Γ−1 ([0, 𝑔(𝑟)])) = 𝜆𝑋 (Γ−1 (ℎ−1 ([0, 𝑟]))).
49
Now we construct the function 𝜙 using the distribution function 𝑔. The idea of construction is the following: Because 𝑔 is continuous, any open interval has a converse that
is also open. Now, each finite sequence in 𝑋 <ℕ can be regarded as an open interval, and
so it can be mapped into another via 𝑔. As the length of the finite sequence increases,
the interval shrinks and finally the functional Φ obtains.
Let 𝜖 be the empty string. Define 𝑔 0 (𝜖) = 0 and 𝑔 1 (𝜖) = 1, and, for 𝜏 ∈ 𝑋 <ℕ − {𝜖},
∑
∑
define 𝑔 0 (𝜏 ) = {𝜇p (𝑁𝜎 ) : Γ(𝜎) ≤ Γ(𝜏 ) − 𝑛1∣𝜏 ∣ , ∣𝜎∣ = ∣𝜏 ∣}, where
∅ = 0, and 𝑔 1 (𝜏 ) =
∑
{𝜇p (𝑁𝜎 ) : Γ(𝜎) ≤ Γ(𝜏 ), ∣𝜎∣ = ∣𝜏 ∣}. For any 𝜁 ∈ 𝑋 ℕ , Γ(𝜁) ≤ Γ(𝜏 ) ⇔ Γ(𝜁[∣𝜏 ∣]) ≤
Γ(𝜏 ) − 𝑛1∣𝜏 ∣ ∨ Γ(𝜁) = Γ(𝜏 ), and Γ(𝜁) ≤ Γ(𝜏 ) + 𝑛1∣𝜏 ∣ ⇔ Γ(𝜁[∣𝜏 ∣]) ≤ Γ(𝜏 ) ∨ Γ(𝜁) = Γ(𝜏 ) + 𝑛1∣𝜏 ∣ .
Since 𝜇p has no atoms, we have that 𝑔 0 (𝜏 ) = 𝑔(Γ(𝜏 )) and 𝑔 1 (𝜏 ) = 𝑔(Γ(𝜏 )+ 𝑛1∣𝜏 ∣ ). Therefore,
for each 𝑡 > 0, the class of intervals
{[𝑔 0 (𝜏 ), 𝑔 1 (𝜏 )] : 𝜏 ∈ 𝑋 <ℕ , ∣𝜏 ∣ = 𝑡}
(29)
forms a partition of [0, 1].
Construct 𝜙 as follows: Given a string 𝜎 ∈ 𝑋 <ℕ , let
𝑎𝜎 = Γ(𝜎) and 𝑏𝜎 = Γ(𝜎) +
1
.
𝑛∣𝜎∣
Let 𝜙(𝜎) be the longest 𝜏 with ∣𝜏 ∣ ≤ ∣𝜎∣ such that [𝑎𝜎 , 𝑏𝜎 ] ⊂ [𝑔 0 (𝜏 ), 𝑔 1 (𝜏 )]. Now, for
any 𝜎 ∈ 𝑋 <ℕ , 𝜙(𝜎) is well-defined, because the intervals in (29) forms a partition and
[𝑔 0 (𝜖), 𝑔 1 (𝜖)] = [0, 1]. Clearly 𝜙 is computable.
(Φ satisfies property 1.) First we show that 𝜙 satisfies (28) and so Φ is can be defined
from 𝜙. Suppose that 𝜎 ⊂ 𝜎 ′ and 𝜏 = 𝜙(𝜎), 𝜏 ′ = 𝜙(𝜎 ′ ). It is easy to check that 𝑎𝜎 ≤ 𝑎𝜎′
and 𝑏𝜎′ ≤ 𝑏𝜎 . Now, if Γ(𝜏 ′ ) ≥ Γ(𝜏 ) +
1
,
𝑛∣𝜏 ∣
then
𝑎𝜎′ ≥ 𝑔 0 (𝜏 ′ ) = 𝑔(Γ(𝜏 ′ )) ≥ 𝑔(Γ(𝜏 ) +
1
) = 𝑔 1 (𝜏 ) ≥ 𝑏𝜎 ≥ 𝑏𝜎′ ,
∣𝜏
∣
𝑛
a contradiction. On the other hand, we have ∣𝜏 ′ ∣ ≥ ∣𝜏 ∣. If Γ(𝜏 ′ ) < Γ(𝜏 ), then Γ(𝜏 ′ ) ≤
Γ(𝜏 ) −
1
,
𝑛∣𝜏 ∣
and hence,
𝑏𝜎 ≤ 𝑏𝜎′ ≤ 𝑔 1 (𝜏 ′ ) = 𝑔(Γ(𝜏 ′ ) +
1
0
′ ∣ ) ≤ 𝑔(Γ(𝜏 )) = 𝑔 (𝜏 ) ≤ 𝑎𝜎 ,
∣𝜏
𝑛
a contradiction. Thus, we have 𝜏 ⊂ 𝜏 ′ .
50
Then we show that, for any sequence 𝜉 not in the set
{𝜁 ∈ 𝑋 ℕ : ℎ(Γ(𝜁)) =
𝑚
for some 𝑚, 𝑛, 𝑡 ∈ ℕ},
𝑛𝑡
𝑚
𝑛𝑡
lim𝑡→∞ 𝜙(𝜉[𝑡]) = ∞. Suppose that ℎ(Γ(𝜁)) ∕=
for any 𝑚, 𝑛, 𝑡 ∈ ℕ. Let 𝐾 be given.
There exists some 𝑙 ∈ ℕ such that ℎ(Γ(𝜁)) ∈ ( 𝑛𝑙𝐾 , 𝑙+1
). Let
𝑛𝐾
𝜀 = min{ℎ(Γ(𝜁)) −
𝑙 𝑙+1
,
− ℎ(Γ(𝜁))}.
𝑛𝐾 𝑛𝐾
Since ℎ is continuous, there is some 𝑇 such that 𝑡 ≥ 𝑇 implies that
min{∣ℎ(𝑏𝜁[𝑡] ) − ℎ(Γ(𝜁))∣, ∣ℎ(Γ(𝜁) − ℎ(𝑎𝜁[𝑡] )∣} ≤
𝑙 𝑙+1
𝜀
and so [ℎ(𝑎𝜁[𝑡] ), ℎ(𝑏𝜁[𝑡] )] ⊆ ( 𝐾 , 𝐾 ).
2
𝑛
𝑛
Thus, if 𝑡 ≥ max{𝑇, 𝐾}, then
[𝑎𝜁[𝑡] , 𝑏𝜁[𝑡] ] ⊂ [𝑔(
𝑙+1
𝑙
𝑙
𝑙
), 𝑔( 𝐾 )] = [𝑔 0 ( 𝐾 ), 𝑔 1 ( 𝐾 )],
𝐾
𝑛
𝑛
𝑛
𝑛
and so ∣𝜙(𝜁[𝑡])∣ ≥ 𝐾. Clearly, any sequence 𝜁 that satisfies
ℎ(Γ(𝜁)) =
𝑚
for some 𝑚, 𝑛, 𝑡 ∈ ℕ
𝑛𝑡
is computable, and so if Φ(𝜉) is not well-defined, 𝜉 is computable.
(Φ satisfies property 2.) We first claim that if Φ is well-defined over 𝜁 (the set of
such 𝜁’s is denoted by 𝐷(𝜙)), then Γ(Φ(𝜁)) = ℎ(Γ(𝜁)). Let 𝜀 be given, and let 𝐾 be
so large that 𝜀 <
1
.
𝑛𝐾−1
Since 𝜁 ∈ 𝐷(𝜙), there exists 𝑇 such that 𝑡 ≥ 𝑇 implies that
∣𝜙(𝜁[𝑡])∣ ≥ 𝐾. Then, for all 𝑡 ≥ 𝑇 , ℎ(Γ(𝜁)) ∈ [ℎ(𝑎𝜁[𝑡] ), ℎ(𝑏𝜁[𝑡] )] ⊆ [𝑎𝜙(𝜁[𝑡]) , 𝑏𝜙(𝜁[𝑡]) ], and so
ℎ(Γ(𝜁)) − Γ(𝜙(𝜁[𝑡])) ≤
1
𝑛𝐾
≤ 𝜀. Thus, Γ(Φ(𝜁)) = lim𝑡→∞ Γ(𝜙(𝜁[𝑡])) = ℎ(Γ(𝜁)). Moreover,
for almost all 𝑟 ∈ [0, 1] (except for countably many of them), there is a sequence 𝜁 ∈ 𝑋 ℕ
such that Γ(Φ(𝜁)) = 𝑟, since ℎ is strictly increasing and is continuous. Also, we have that
Γ(Φ(𝜁)) ≥ Γ(Φ(𝜁 ′ )) ⇔ Γ(𝜁) ≥ Γ(𝜁 ′ ).
(30)
We show that 𝜆𝑋
Φ = 𝜇p by demonstrating that they share the same distribution func𝑋
−1
∗
tion 𝑔, where 𝜆𝑋
Φ (𝐴) = 𝜆 (Φ (𝐴)): for any 𝜁 ,
∗
𝑋
∗
𝜆𝑋
Φ ({𝜁 : Γ(𝜁) ≤ Γ(Φ(𝜁 ))}) = 𝜆 ({𝜁 : Γ(Φ(𝜁)) ≤ Γ(Φ(𝜁 ))})
51
= 𝜆𝑋 ({𝜁 : Γ(𝜁) ≤ Γ(𝜁 ∗ )}) = Γ(𝜁 ∗ ) = 𝑔(Γ(Φ(𝜁 ∗ ))).
(Recall that, for all but a countable set of numbers 𝑟 ∈ [0, 1], there is a 𝜁 ∗ such that
Γ(Φ(𝜁 ∗ )) = 𝑟. The gaps may be filled by assigning arbitrary values on Φ when it is not
well-defined. The first equality comes from the definition of 𝜆𝑋
Φ and the second comes
from equation (30).)
(Φ satisfies property 3.) Recall that if Γ(𝜁) = 𝑔( 𝑛𝑚𝑡 ) ∈ ℚ, then 𝜁 is computable. Thus,
if 𝜉 ′ is 𝜆𝑋 -random, 𝜉 ′ ∈ 𝐷(𝜙). Let 𝜁 ′ = Φ(𝜉 ′ ). Now we show that 𝜁 ′ is 𝜇p -random relative
∩∞
′
to 𝒫. Suppose not. Then there is a 𝜇p -test {𝑉𝑡 }∞
𝑡=0 relative to 𝒫 such that 𝜁 ∈
𝑡=0 𝑉𝑡 .
Specifically, suppose that for some 𝜉-computable total function ℎ,
(1),𝜉⊕𝜁[𝑠]
𝑉𝑡 = {𝜁 : (∃𝑠)𝜑ℎ(𝑡)
(0) ↓}.
Define 𝑈𝑡 as
(1),(𝜉⊕𝜙(𝜁[𝑠′ ]))[𝑠]
𝑈𝑡 = {𝜁 : (∃𝑠)(∃𝑠′ )𝜑ℎ(𝑡)
(0) ↓}.
0,𝜉
Because 𝜙 is computable, {𝑈𝑡 }∞
𝑡=0 is of Σ1 . It is easy to check that
Φ−1 (𝑉𝑡 ) ∩ 𝐷(𝜙) ⊂ 𝑈𝑡 ⊂ Φ−1 (𝑉𝑡 ) ∪ (𝑋 ℕ − 𝐷(𝜙)).
Because 𝜆𝑋 (𝐷(𝜙)) = 1 by property 1, for all 𝑡, 𝜆𝑋 (𝑈𝑡 ) = 𝜇p (𝑉𝑡 ) ≤
𝑋
′
Therefore, {𝑈𝑡 }∞
𝑡=0 is a 𝜆 -test relative to 𝒫. But 𝜉 ∈
∩∞
𝑡=0
1
.
2𝑡
𝑈𝑡 since 𝜁 ′ ∈
∩∞
𝑡=0
𝑉𝑡 , a
contradiction. Since 𝜙 is computable, 𝜁 ′ ∈ 𝐶(𝜉 ′ ) ⊂ 𝐶(𝜉).
Here we show that there is always an equilibrium strategy in both the vertical game
and the horizontal game that satisfies all statistical regularities with respect to the i.i.d.
distribution generated by the stage-game mixed equilibrium.
Theorem 6.3. Suppose that 𝒫 1 , 𝒫 2 satisfy (A1), (A2), and mutual complexity. Let 𝑝∗
be an equilibrium strategy of player 1 in 𝑔. There exists an equilibrium strategy 𝜉 ∈ 𝒳 for
player 1 that is 𝜇𝑝∗ -random relative to 𝒫 2 in 𝑉 𝐺(𝑔, 𝒫 1 , 𝒫 2 ).
Proof. Let 𝑝 be an equilibrium strategy of player 1 in 𝑔. By Theorem 6.2, there exists
a 𝜇𝑝 -random sequence relative to 𝒫 2 in 𝒳 . By Theorem 6.5, such a sequence is strongly
𝑝-stochastic and hence is an equilibrium strategy in both the horizontal game and the
vertical game.
52
6.2.2
Martingales and stochastic sequences
Here we prove that any 𝜇p -random sequence is strongly 𝑝-stochastic if lim𝑡→∞ 𝑝𝑡 = 𝑝. To
this end, we need a new concept called martingales. First we give a definition.
Definition 6.2. Let 𝑋 be a finite set and let p = (𝑝0 , 𝑝1 , ...) be a computable sequence of distributions over 𝑋 such that 𝑝𝑡𝑥 > 0 for all 𝑥 ∈ 𝑋 and for all 𝑡 ∈ ℕ.
A function 𝑀 : 𝑋 <ℕ → ℝ+ is a martingale with respect to 𝜇p if for all 𝜎 ∈ 𝑋 <ℕ ,
∑
∣𝜎∣
𝑀 (𝜎) = 𝑥∈𝑋 𝑝𝑥 𝑀 (𝜎⟨𝑥⟩).
Let 𝒫 ⊂ ℱ satisfy (A1) and (A2). A martingale 𝑀 is 𝒫-effective if there is a sequence of
martingales {𝑀𝑡 }∞
𝑡=0 that satisfies the following properties:
(a) 𝑀𝑡 (𝜎) ∈ ℚ+ for all 𝑡 ∈ ℕ and for all 𝜎 ∈ 𝑋 <ℕ ;
(b) the function 𝑔 defined as 𝑔(𝑡, 𝜎) =𝑑𝑒𝑓 𝑀𝑡 (𝜎) belongs to 𝒫;
(c) lim𝑡→∞ 𝑀𝑡 (𝜎) ↑ 𝑀 (𝜎) for all 𝜎 ∈ 𝑋 <ℕ .
In this case, we say that the sequence {𝑀𝑡 }∞
𝑡=0 supports 𝑀 .
The following theorem characterizes randomness in terms of martingales. The proof
for a special case of this theorem can be found in Downey et al. [11]. We provide a short
sketch of the proof for self-containment.
Theorem 6.4. Suppose that 𝒫 satisfy (A1) and (A2). Let p be a computable sequence
such that 𝑝𝑡𝑥 > 0 for all 𝑥 ∈ 𝑋 and for all 𝑡 ∈ ℕ. A sequence 𝜉 ∈ 𝑋 ℕ is 𝜇p -random relative
to 𝒫 if and only if for any 𝒫-effective martingale 𝑀 w.r.t. 𝜇p , lim sup𝑇 →∞ 𝑀 (𝜉[𝑡]) < ∞.
Proof. (⇒) Suppose that lim sup𝑇 →∞ 𝑀 (𝜉[𝑇 ]) = ∞ and 𝑀 is a 𝒫-effective martingale.
Let 𝑉𝑡 = {𝜉 : (∃𝑠)(𝑀 (𝜉[𝑠]) > 2𝑡 )}. It is easy to show that for any 𝜎 and any prefix-free
∑
set 𝐴 ⊂ {𝜏 : 𝜏 ⊆ 𝜎}, 𝜏 ∈𝐴 𝜇p (𝑁𝜏 )𝑀 (𝜏 ) ≤ 𝜇p (𝑁𝜎 )𝑀 (𝜎). Thus, 𝜇p (𝑉𝑡 ) ≤ 21𝑡 . It is routine
∩
to check that {𝑉𝑡 } is a 𝜇P -test relative to 𝒫 because 𝑀 is 𝒫 effective. Then 𝜉 ∈ ∞
𝑡=0 𝑉𝑡
and hence is not 𝜇p -random.
∩∞
𝑉𝑡 for a 𝜇P -test {𝑉𝑡 } relative to 𝒫. Let 𝑀 𝑡 (𝜎) = 𝜇p (𝑉𝑡 ∩
∑∞
𝑡
𝑡
𝑁𝜎 )/𝜇p (𝑁𝜎 ) and let 𝑀 =
𝑡=0 𝑀 . It is routine to check that 𝑀 is a martingale.
(⇐) Suppose that 𝜉 ∈
𝑡=0
53
∑∞
𝜇p (𝑉𝑡 ) = 1 and for any 𝜎, 𝑀 𝑡 (𝜎) ≤ 𝑀 (𝜖)/𝜇p (𝑁𝜎 ) and so 𝑀 is well-defined.
∩
It is easy to show that 𝑀 is 𝒫-effective because {𝑉𝑡 } is a 𝜇P -test relative to 𝒫. 𝜉 ∈ ∞
𝑡=0 𝑉𝑡
𝑀 (𝜖) =
𝑡=0
implies that lim sup𝑇 →∞ 𝑀 (𝜉[𝑇 ]) = ∞.
The following theorem shows that any 𝜇p -random sequence is also strongly stochastic.
As a corollary, any 𝜇𝑝 -random sequence is also strongly 𝑝-stochastic. Moreover, by Theorem 6.2, if mutual complexity holds, then each player can generate a 𝜇𝑝 -random sequence
for any 𝑝 relative to the other player and hance can generate a strongly 𝑝-random sequence
relative to the other player.
Theorem 6.5. Suppose that 𝒫 satisfies (A1) and (A2). Suppose that 𝜉 is 𝜇p -random
relative to 𝒫 with 𝑝𝑡𝑥 > 0 for all 𝑡 ∈ ℕ and for all 𝑥 ∈ 𝑋 and lim𝑡→∞ 𝑝𝑡 = 𝑝. Then, 𝜉 is a
strongly 𝑝-stochastic sequence relative to 𝒫.
Proof. By Theorem 6.4, for any 𝒫-effective martingale 𝑀 w.r.t. 𝜇p , lim sup𝑇 →∞ 𝑀 (𝜉[𝑡]) <
∞. We prove by contradiction. Suppose that the theorem does not hold, and so suppose
that 𝑟 is a selection function for 𝑋 in 𝒫 such that 𝜉 𝑟 is a total function and assume that
there exists some 𝜀 > 0 and a sequence {𝑇𝑘 }∞
𝑘=0 such that for all 𝑘 ∈ ℕ,
𝑇∑
𝑘 −1
𝑡=0
𝑐𝑥 (𝜉𝑡𝑟 )
≥ 𝑝𝑥 + 𝜀.
𝑇𝑘
We construct a martingale (w.r.t. 𝜇p ) such that lim sup𝑇 →∞ 𝑀 (𝜉[𝑡]) = ∞.
(Construction) We define 𝑀 as follows:
(a) 𝑀 (𝜖) = 1;
∣𝜎∣
∣𝜎∣
(b) 𝑀 (𝜎⟨𝑥⟩) = (1 + 𝜅(1 − 𝑝𝑥 ))𝑀 (𝜎) and 𝑀 (𝜎⟨𝑦⟩) = (1 − 𝜅𝑝𝑥 )𝑀 (𝜎) for all 𝑦 ∕= 𝑥 if
𝑟(𝜎) = 1;
(c) 𝑀 (𝜎⟨𝑦⟩) = 𝑀 (𝜎) for all 𝑦 ∈ 𝑋 if 𝑟(𝜎) = 0.
Here, 𝜅 is a fixed rational number whose value will be determined latter. Clearly, by
construction, 𝑀 ∈ 𝒫 because 𝑟 ∈ 𝒫 and p is computable.
54
(Verify 𝑀 is a martingale) To check that 𝑀 is a martingale, note that if 𝑟(𝜎) = 1,
then
∑
∣𝜎∣
∣𝜎∣
𝑝∣𝜎∣
𝑦 𝑀 (𝜎⟨𝑦⟩) = 𝑝𝑥 (1 + 𝜅(1 − 𝑝𝑥 ))𝑀 (𝜎) +
𝑦∈𝑋
∑
𝑝𝑦∣𝜎∣ (1 − 𝜅𝑝𝑥∣𝜎∣ )𝑀 (𝜎)
𝑦∕=𝑥
∣𝜎∣
∣𝜎∣ ∣𝜎∣
= 𝑀 (𝜎) + 𝜅𝑀 (𝜎)(𝑝∣𝜎∣
𝑥 (1 − 𝑝𝑥 ) − (1 − 𝑝𝑥 )𝑝𝑥 ) = 𝑀 (𝜎);
if 𝑟(𝜎) = 0, then
∣𝜎∣
∑
𝑦∈𝑋
𝑝𝑦 𝑀 (𝜎⟨𝑦⟩) =
∣𝜎∣
∑
𝑦∈𝑋
𝑝𝑦 𝑀 (𝜎) = 𝑀 (𝜎).
(𝑀 satisfies lim sup𝑇 →∞ 𝑀 (𝜉[𝑡]) = ∞) For 𝑘 ≥ 1, define
𝐷𝑘 = {𝑡 ≤ 𝑘 − 1 : 𝑟(𝜉[𝑡]) = 1, 𝜉𝑡+1 = 𝑥} and 𝐸𝑘 = {𝑡 ≤ 𝑘 − 1 : 𝑟(𝜉[𝑡]) = 1, 𝜉𝑡+1 ∕= 𝑥}.
Then, 𝑀 (𝜉[𝑘]) =
∏
𝑡∈𝐷𝑘 (1
+ 𝜅(1 − 𝑝𝑡+1
𝑥 ))
∏
𝑡∈𝐸𝑘 (1
𝑟
𝜉 −1
− 𝜅𝑝𝑡+1
𝑥 ). Let 𝑙𝑘 = (𝐿𝑟 ) (𝑇𝑘 ). Since 𝜉
is total, 𝑙𝑘 is well defined for all 𝑘 ∈ ℕ.
Let 𝛿 = min{𝑝𝑥 , 1 − 𝑝𝑥 , 2𝜀 }. Since lim𝑡→∞ 𝑝𝑡 = 𝑝, let 𝑇 be so large that 𝑡 ≥ 𝑇 implies
that ∣𝑝𝑡𝑥 − 𝑝𝑥 ∣ < 𝛿. Let 𝐾 be the first 𝑘 such that 𝑇𝑘 > 𝑇 . Then, for all 𝑘 > 𝐾,
𝑀 (𝜉[𝑙𝑘 ]) =
∏
(1 + 𝜅(1 − 𝑝𝑡+1
𝑥 ))
𝑡∈𝐷𝑙𝑘
≥
∏
(1 + 𝜅(1 − 𝑝𝑡+1
𝑥 )
𝑡∈𝐷𝑙𝐾
∏
∏
(1 − 𝜅𝑝𝑡+1
𝑥 )
𝑡∈𝐸𝑙𝑘
∣𝐷𝑙𝑘 −𝐷𝑙𝐾 ∣
(1 − 𝜅𝑝𝑡+1
(1 − 𝜅𝑝𝑥 − 𝜅𝛿)∣𝐸𝑙𝑘 −𝐸𝑙𝐾 ∣ .
𝑥 ))(1 + 𝜅(1 − 𝑝𝑥 − 𝛿))
𝑡∈𝐸𝑙𝐾
∏
Let 𝐴 =
𝑡∈𝐷𝑙
𝐾
(1+𝜅(1−𝑝𝑡+1
𝑥 )
∏
𝑡∈𝐸𝑙
𝐾
(1−𝜅𝑝𝑡+1
𝑥 ))
(1+𝜅(1−𝑝𝑥 −𝛿))𝐿1 (1−𝜅𝑝𝑥 −𝜅𝛿)𝐿2
, where 𝐿1 = ∣𝐷𝑙𝐾 ∣ and 𝐿2 = ∣𝐸𝑙𝐾 ∣. Since for
each 𝑘, ∣𝐷𝑙𝑘 ∣ ≥ 𝑇𝑘 𝑝𝑥 + 𝑇𝑘 𝜀,
𝑀 (𝜉[𝑙𝑘 ]) ≥ 𝐴((1 + 𝜅(1 − 𝑝𝑥 − 𝛿))𝑝𝑥 +𝜀 (1 − 𝜅𝑝𝑥 − 𝜅𝛿)1−𝑝𝑥 −𝜀 )𝑇𝑘 .
Define 𝐹 (𝜅) = (1 + 𝜅(1 − 𝑝𝑥 − 𝛿))𝑝𝑥 +𝜀 (1 − 𝜅𝑝𝑥 − 𝜅𝛿)1−𝑝𝑥 −𝜀 . We have ln 𝐹 (0) = 1 and
(ln 𝐹 )′ (0) = (𝑝𝑥 + 𝜀)(1 − 𝑝𝑥 − 𝛿) − (1 − 𝑝𝑥 − 𝜀)(𝑝𝑥 + 𝛿) = 𝜀 − 𝛿 > 0. Thus, for 𝜅 small
enough, 𝐹 (𝜅) > 1, and so lim sup𝑇 →∞ 𝑀 (𝜉[𝑇 ]) = ∞, a contradiction.
6.2.3
Law of the Iterated Logarithm
Here we give a general Law of the Iterated Logarithm that is satisfied by any 𝜇p -random
sequence.
55
Theorem 6.6. Suppose that 𝜉 is a 𝜇p -random sequence relative to 𝒯 with p = (𝑝0 , 𝑝1 , ..., 𝑝𝑡 , ...).
Then, for any 𝑥 ∈ 𝑋,
𝑇 →∞
∑𝑇 −1
(𝑐𝑥 (𝜉𝑡 ) − 𝑝𝑡𝑥 )∣
= 1,
√∑
∑𝑇 −1 𝑡
𝑇 −1 𝑡
𝑡
𝑡
2( 𝑡=0 𝑝𝑥 (1 − 𝑝𝑥 )) log log ( 𝑡=0 𝑝𝑥 (1 − 𝑝𝑥 ))
lim sup √
∣
𝑡=0
(31)
Proof. The positive part of equation (31) is equivalent to the following two conditions:
(a) for all rational 𝜀 > 0,
v
v
u
u 𝑇 −1
u
𝑇 −1
𝑇 −1
u∑
∑
∑
u
(∃𝑆)(∀𝑇 ≥ 𝑆)
(𝑐𝑥 (𝜉𝑡 ) − 𝑝𝑡𝑥 ) ≤ ⎷2(1 + 𝜀)(
𝑝𝑡𝑥 (1 − 𝑝𝑡𝑥 )) log log ⎷(
𝑝𝑡𝑥 (1 − 𝑝𝑡𝑥 )).
𝑡=0
𝑡=0
𝑡=0
(b) for all rational 𝜀 > 0,
v
v
u
u 𝑇 −1
u
𝑇 −1
𝑇 −1
u∑
∑
∑
u
(𝑐𝑥 (𝜉𝑡 ) − 𝑝𝑡𝑥 ) ≥ ⎷2(1 − 𝜀)(
(∀𝑆)(∃𝑇 ≥ 𝑆)
𝑝𝑡𝑥 (1 − 𝑝𝑡𝑥 )) log log ⎷(
𝑝𝑡𝑥 (1 − 𝑝𝑡𝑥 )).
𝑡=0
𝑡=0
𝑡=0
Let
v
v
u
u 𝑇 −1
u
𝑇 −1
𝑇 −1
u∑
∑
∑
u
(𝑐𝑥 (𝜁𝑡 ) − 𝑝𝑡𝑥 ) > ⎷2(1 + 𝜀)(
𝐸𝑇𝜀 = {𝜁 :
𝑝𝑡𝑥 (1 − 𝑝𝑡𝑥 )) log log ⎷(
𝑝𝑡𝑥 (1 − 𝑝𝑡𝑥 ))},
𝑡=0
𝑡=0
𝑡=0
and
v
v
u
u 𝑇 −1
u
𝑇 −1
𝑇 −1
u∑
∑
∑
u
𝜀
𝑡
𝑡
𝑡
𝐹𝑇 = {𝜁 :
(𝑐𝑥 (𝜁𝑡 ) − 𝑝𝑥 ) < ⎷2(1 − 𝜀)(
𝑝𝑥 (1 − 𝑝𝑥 )) log log ⎷(
𝑝𝑡𝑥 (1 − 𝑝𝑡𝑥 ))}.
𝑡=0
𝑡=0
𝑡=0
∩
∪∞
𝜀
Clearly, condition (a) is equivalent to 𝜉 ∈
/ ∞
𝑆=0
𝑇 =𝑆 𝐸𝑇 and condition (b) is equiva∪
∩∞
𝜀
lent to 𝜉 ∈
/ ∞
𝑆=0
𝑇 =𝑆 𝐹𝑇 . By Theorem 7.5.1 in Chung [9],
𝜇p (
∞ ∪
∞
∩
𝐸𝑇𝜀 ) = 0
𝑆=0 𝑇 =𝑆
and
𝜇p (
∞ ∩
∞
∪
𝑆=0 𝑇 =𝑆
56
𝐹𝑇𝜀 ) = 0.
∩
𝜀
𝜀
It then follows that 𝜇p ( ∞
𝑇 =𝑆 𝐹𝑇 ) = 0 for any 𝑆 ∈ ℕ. Because 𝐹𝑇 is computable (uniformly
𝜀
in 𝑇 ), {𝐹𝑇𝜀 }∞
𝑇 =𝑆 is a 𝜇p -test for any 𝑆 (notice that 𝜇p (𝐹𝑇 ) is also computable). Therefore,
∪
∩∞
𝜀
𝜉∈
/ ∞
𝑆=0
𝑇 =𝑆 𝐹𝑇 .
∪
𝜀
On the other hand, the set 𝐸𝑇𝜀 is computable (uniformly in 𝑇 ) and so the set ∞
𝑇 =𝑆 𝐸𝑇 is
∪
∪∞
𝜀 ∞
𝜀
of Σ01 (uniformly in 𝑆). For { ∞
𝑇 =𝑆 𝐸𝑇 }𝑆=0 to be a test, we need to show that 𝜇p ( 𝑇 =𝑆 𝐸𝑇 )
has a computable upper bound for all 𝑆. From the proof in Theorem 7.5.1 in Chung [9],
we know that there exists a constant 𝐴 > 0 and a number 𝑘 > 0 such that for all 𝑘 ≥ 𝑘
(with the provision that 𝑐2 (1 + 2𝜀 ) < 1 + 𝜀), c.f. p. 216),
𝑇𝑘+1 −1
𝜇p (
∪
𝐸𝑇𝜀 ) <
𝑇 =𝑇𝑘
where 𝑇𝑘 = max{𝑇 :
√∑
𝑇
𝑡=0
𝐴
𝜀 ,
(𝑘 log 𝑐)1+ 2
𝑝𝑡𝑦 (1 − 𝑝𝑡𝑦 ) ≤ 𝑐𝑘 } and 𝑐 = 1 +
𝜀
10
(for 𝜀 small enough,
𝑐2 (1 + 2𝜀 ) < 1 + 𝜀).
Let’s define 𝐺0 =
∪𝑇1 −1
𝑇 =0
𝐸𝑇𝜀 and 𝐺𝑘 =
∞
∞ ∪
∩
∪𝑇𝑘+1 −1
𝑇 =𝑇𝑘
𝐸𝑇𝜀 for 𝑘 > 0. Clearly,
∞
∞ ∪
∩
𝐺𝑘 =
𝐸𝑇𝜀 .
𝑆=0 𝑇 =𝑆
𝑆=0 𝑘=𝑆
Now, because 𝑇𝑘 is a computable function of 𝑘, {
∪∞
𝑘=𝑆
𝐺𝑘 }∞
𝑆=0 is also a sequence of
uniformly Σ01 sets. We now show that there is a computable mapping 𝑖 7→ 𝑆𝑖 so that
∪
1
𝜇p ( ∞
𝑘=𝑆𝑖 𝐺𝑘 ) ≤ 2𝑖 . We know that
∫ ∞
∞
∑
𝜀
𝜀
𝐴
𝐴
≤
= (𝑆 − 1)− 2 (log 𝑐)−1− 2 .
1+ 2𝜀
1+ 2𝜀
(𝑘 log 𝑐)
𝑥=𝑆−1 (𝑥 log 𝑐)
𝑘=𝑆
𝜀
2
Let 𝐵 ∈ ℕ be such that 𝐵 > (𝐴(log 𝑐)−1− 2 ) 𝜀 and let 𝑁 ∈ ℕ be such that 𝑁 > 2𝜀 . Take
∪
1
𝑆𝑖 = 𝐵2𝑁 𝑖 + 1, and it follows that 𝜇p ( ∞
𝑘=𝑆𝑖 𝐺𝑘 ) ≤ 2𝑖 .
∪
∩
∪∞
∩∞ ∪∞
∞
𝜀
This shows that { ∞
/ ∞
𝑘=𝑆 𝐺𝑘 }𝑆=0 is a 𝜇p -test, and so 𝜉 ∈
𝑆=0
𝑘=𝑆 𝐺𝑘 =
𝑆=0
𝑇 =𝑆 𝐸𝑇 .
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