Department of Social Systems and Management
Discussion Paper Series

No.1287

Critical Comparisons between the Nash Noncooperative
Theory and Rationalizability

byTai-wei HU and Mamoru KANEKO

January 2012 (the latest version: April 2012)

UNIVERSITY OF TSUKUBA
Tsukuba, Ibaraki 305-8573
JAPANCritical Comparisons between
the Nash Noncooperative Theory and Rationalizability∗†
Tai-Wei Hu‡and Mamoru Kaneko§
17 April 2012

Abstract
The theories of Nash noncooperative solutions and of rationalizability intend todescribe the same target problem of ex ante individual decision making, but they
are distinctively diﬀerent. We consider what their essential diﬀerence is by giving
a unified approach and parallel derivations of their resulting outcomes. Our results
show that the only diﬀerence lies in the use of quantifiers for each player’s predictions
about the other’s possible decisions; the universal quantifier for the former and theexistential quantifier for the latter. Based on this unified approach, we discuss thestatuses of those theories from the three points of views: Johansen’s postulates, thefree-will postulate vs. complete determinism, and prediction/decision criteria. One
conclusion we reach is that the Nash theory is coherent with the free-will postulate,
but we would meet various diﬃculties with the rationalizability theory.
JEL Classification Numbers: B40, C70, C72
Key words: Nash equilibrium, Solvability, Rationalizability, Prediction/Decision
Criterion, Infinite Regress, Simultaneous Equations

1. Introduction
We make critical comparisons between the theory of Nash noncooperative solutions due
to Nash [20] and the theory of rationalizable strategies due to Bernheim [3] and Pearce[21]. Each theory is intended to be a theory of ex ante individual decision making ina game, and thus focuses on the decision-making process before the actual play of the∗

The authors are partially supported by Grant-in-Aids for Scientific Research No.21243016 andNo.2312002, Ministry of Education, Science and Culture.
†
The authors thank Geir B. Ascheim and Jean-Jacques Herings for valuable comments and discussions on an earlier draft of this paper.
‡
MEDS Department, Northwestern University, Evanston, IL, (t-hu@kellogg.northwestern.edu)§
IPPS, University of Tsukuba, Ibaraki 305-8573, Japan (kaneko@sk.tsukuba.ac.jp)

game. The diﬀerence in their resulting outcomes has been well analyzed and known.
However, their conceptual diﬀerence has not been much discussed. In this paper, weevaluate these two theories while considering certain conceptual bases of game theory
and addressing the question of logical coherence of these theories with them.
We begin with a brief review of these theories. It is well known that Nash [20] provides
the concept of Nash equilibrium and proves its existence in mixed strategies. However, it
is less known that the main focus of [20] is on ex ante individual decision making. In that
paper, various other concepts are developed, including interchangeability, solvability,
subsolutions, symmetry, and values; those concepts are ingredients of a theory of ex ante
individual decision making, though the aim is not explicitly stated in [20]. This view isdiscussed in Nash’s [19] dissertation (p.23) and a few other papers such as Johansen [11]and Kaneko [12]1 . We call the entire argumentation the Nash noncooperative theory 2 .
On the other hand, in the literature, the theory of rationalizability is typically regarded as a faithful description of ex ante individual decision making in games, expressing the common knowledge of “rationality”. Mas-Colell et al. [15], p.243 wrote “Theset of rationalizable strategies consists precisely of those strategies that may be played ina game where the structure of the game and the player’s rationality are common knowledge among the players.” This view is common in many standard game theory/microeconomics textbooks.
The literature exhibits a puzzling feature: Both theories target ex ante individual
decision making, and both are widely used by many researchers. However, their formal
definitions, predicted outcomes, and explanations diﬀer considerably. This puzzlingfeature raises the following questions: How should we make comparisons between these
theories? Then, what are their main diﬀerences? How would the diﬀerence be evaluated?
What are bases for such an evaluation? This paper attempts to answer these questions.
We formulate the two theories in terms of prediction/decision criteria, which gives aunified framework for comparisons of these theories. For the Nash theory, the criterion
is given by the following requirements:
N1 : player 1 chooses his best strategy against all of his predictions
about player 2’s choice based on N2 ;N2 : player 2 chooses his best strategy against all of his predictions
about player 1’s choice based on N1 .
We may say that player 1 makes a decision if it satisfies N1 ; however, to determine thisdecision, N1 requires a prediction about 2’s possible decisions, which are determined
1

Millham [17] and Jansen [10] study the mathematical structure of the solution and subsolutions,but do not touch the view.
2
The mathematical definition of Nash equilibrium allows diﬀerent interpretations such as a steadystate in a repeated situation (one variant is the “mass-action” interpretation due to Nash [19], pp.21-22),
but we do not touch other interpretaitons. See Kaneko [13] for them.

2

by N2 . The symmetric form N2 determines a decision for 2 if he predicts 1’s decisions.
In this sense, these requirements are circular. Also, they can be regarded as a systemof simultaneous equations with players’ decisions/predictions as unknown. In Section
3, we show that the system N1 -N2 characterizes the Nash noncooperative solution
as the greatest set satisfying them if the game is solvable (the set of Nash equilibria
is interchangeable); and for an unsolvable game, a maximal set satisfying them is a
subsolution.
The rationalizable strategies are characterized by another prediction/decision criterion R1 -R2 :R1 : player 1 chooses his best strategy against some of his predictions
about player 2’s choice based on R2 ;R2 : player 2 chooses his best strategy against some of his predictions
about player 1’s choice based on R1 .
These are obtained from N1 -N2 simply by replacing the quantifier “for all” by “forsome” before predictions about the other player’s decisions. These requirements areclosely related to the BP-property (“best-response property” in Bernheim [3] and Pearce[21]), and the characterization result is given in Section 3.
The above prediction/decision criteria and characterization results unify the Nashnoncooperative theory and rationalizability theory, and pinpoint their diﬀerence: It
is the choice of the universal or existential quantifiers for predictions about the otherplayer’s possible decisions. To evaluate this diﬀerence, we first review the discussion of
ex ante decision making in games given in Johansen [11]. In his argument, a theory of
ex ante decision making in games should describe a player’s active inferences based on
certain axioms about his own and the other’s decision-making. Johansen gives four
postulates for the Nash solution, although his argument there is still informal andcontains some ambiguities.
Our formulation of N1 -N2 may be viewed as an attempt to formalize his postulates
in the language of classical game theory. The pinpointed diﬀerence between the twotheories clarifies the precise requirements in those postulates to obtain the Nash theory.
One of Johansen’s postulates requires that any possible decision be a best response to
the predicted decisions, which is violated by the “for some” requirement in R1 -R2  His
postulates help to clarify N1 -N2 , and vice versa. Nevertheless, his postulates containsome subtle concepts, which go beyond the language of classical game theory.
One such concept is “rationality”. In the theory of rationalizability, “rationality” istypically regarded as equivalent to payoﬀ maximization. In Johansen’s postulates, however, payoﬀ maximization is separated from “rationality”, and is only one componentof “rationality”. We also take this broader view of “rationality”; in our formulation,
it includes, but not limited to, the prediction/decision criterion and logical abilities to3

understand their implications. This broader view allows further research on decision criterion (such as the additional principles needed for specific classes of unsolvable games)
and investigations of how players’ logical abilities aﬀect their decisions.
To evaluate the diﬀerence further, we go to deeper methodological assumptions: thefree-will postulate vs. complete determinism. The former, stating that each player has
free will, is automatically associated with decision making. The quantifier “for all” inN1 -N2 is coherent with the application of the free-will postulate between the players.
On the other hand, as will be argued in Section 4, the theory of rationalizability is
better understood from the perspective of complete determinism. Indeed, the epistemic
justification for rationalizability begins with a complete description of players’ actions
as well as mental states, and characterizes classes of those states by certain assumptions.
As a result, our problem is a choice between two methodological assumptions, thefree-will postulate and complete determinism. This choice is discussed in Morgenstern[18] and Heyek [8] in the context of economics and/or social science in general. Basedupon their arguments, we will conclude that the free-will postulate is more coherent
with large part of social science than complete determinism. From this perspective, theNash theory is preferable to rationalizability.
The Nash theory might be less preferred in that it does not recommend definite
decisions for unsolvable games. However, it may not be a defect from the perspectivethat it points out that additional principles, other than the decision criteria given above,
are needed for decision making in unsolvable games. A general study of such additional
principles is beyond the scope of this paper, but we remark that many applied works
appeal to principles such as symmetry (which is already discussed in Nash [20]) and thePareto criterion. As an instance, we will give an argument with the Pareto principle
for the class of games of strategic complementarity in Section 3.1. From a theoreticalperspective, our approach provides a framework to discuss coherence between basic
decision criteria and additional principles.
The paper is written as follows: Section 2 introduces the theories of Nash noncooperative solutions and rationalizable strategies; we restrict ourselves to finite 2-person
games for simplicity. Section 3 formulates N1 -N2 and R1 -R2  and gives two theorems characterizing the Nash noncooperative theory and rationalizability. In Section
4, we discuss implications from them considering foundational issues. Section 5 gives asummary and states continuation to the companion paper.

2. Preliminary Definitions
In this paper, we restrict our analysis to finite 2-person games with pure strategies.
In Section 3.3, we discuss required changes for our formulation to accommodate mixed
strategies.
We begin with basic concepts in a finite 2-person game. Let  = ( { }∈  { }∈ )4

be a finite 2-person game, where  = {1 2} is the set of players,  is the finite set ofpure strategies and  : 1 × 2 →  is the payoﬀ function for player  ∈  We assume
1 ∩ 2 = ∅. When we take one player  ∈  the remaining player is denoted by .
Also, we write  ( ;  ) for  (1  2 ) The property that  is a best-response against
 , i.e.,(2.1)
 ( ;  ) ≥  (0 ;  ) for all 0 ∈  
is denoted by Best( ;  ) Since 1 ∩2 = ∅ the expression Best( ;  ) has no ambiguity.
A pair of strategies (1  2 ) is a Nash equilibrium in  iﬀ Best( ;  ) holds for both
 = 1 2 We use () to denote the set of all Nash equilibria in . The set () may
be empty.
Nash Noncooperative Solutions: A subset  of 1 × 2 is interchangeable iﬀ(1  2 ) (01  02 ) ∈  imply (1  02 ) ∈ 

(2.2)

It is known that this requirement is equivalent for  to have the product form, as statedin the following lemma.
Lemma 2.1. Let  ⊆ 1 × 2 and let  = { : ( ;  ) ∈  for some  ∈  } for = 1 2 Then,  satisfies (2.2) if and only if  = 1 × 2 
Now, let E = { :  ⊆ () and  satisfies (2.2)} We say that  is the Nashsolution iﬀ  is nonempty and is the greatest set in E i.e.,  0 ⊆  for any  0 ∈ E We
say that  is a Nash subsolution iﬀ  is a nonempty maximal set in E i.e., there is no 0 ∈ E such that  (  0 .
Table 21s21
s22
s11 (2 2) (1 1)
s12 (1 1) (0 0)

Table 22s21
s22
s11 (1 1) (1 1)
s12 (1 1) (0 0)

When () 6= ∅ () is the Nash solution if and only if () satisfies (2.2). When
the Nash solution exists for game ,  is called solvable. The game of Table 21 issolvable. On the other hand, a game  may be unsolvable for two reasons: either
() = ∅ or () is nonempty but violates (2.2). For a game  with () 6= ∅ asubsolution exists always; specifically, for any (1  2 ) ∈ () there is a subsolution  containing (1  2 ). This   may not be unique: The game of Table 22 is notsolvable and has two subsolutions: {(s11  s21 ) (s11  s22 )} and {(s11  s21 ) (s12  s21 )}, andboth include (s11  s21 ).
In Section 3, it will be argued that the Nash solution can be regarded as a theoryof ex ante decision making in games. Here we give two comments about this argument.
First, for a solvable game, the theory recommends the set of possible decisions foreach player, i.e., the set of Nash strategies for him; moreover, the recommendation also5includes the set of predicted decisions of the other player. This means that from player
1’s perspective, 1 () = {1 ∈ 1 : (1  2 ) ∈ () for some 2 } describes player 1’s
possible decisions, while 2 () = {2 ∈ 2 : (1  2 ) ∈ () for some 1 } is player
1’s predictions of player 2’s decisions. As shown later, predictions about player 2’s
decisions are crucial to determine player 1’s possible decisions from the perspective of
ex ante decision making in games.
Second, the Nash theory does not provide a definite recommendation for decisions
if the game is unsolvable, even if a subsolution exists. Suppose that  has exactly two
subsolutions, say,  1 = 11 × 21 and  2 = 12 × 22 with 1 6= 2 for  = 1 2. One may
think that the Nash theory would recommend the set  = 1 ∪ 2 for player  as theset of possible decisions to play . However, this is not valid; we cannot find a set 10
or 20 such that 10 × (21 ∪ 22 ) or (11 ∪ 12 ) × 20 satisfies interchangeability.
Rationalizable Strategies: Now, we turn to rationalizability. The pure strategy
version introduced here is known as point-rationalizability due to Bernheim [3]. We
begin with the iterative definition of rationalizability. A sequence of sets of strategies,0{(1 () 2 ())}∞=0 , is inductively defined as follows: for  = 1 2  () =   and () = { : Best( ;  ) holds for some  ∈ −1 ()} for any  ≥ 1(2.3)

We obtain rationalizable strategies by taking the intersection of these sets, i.e.,  () =T∞
=0  () for  = 1 2; a pure strategy  ∈  is rationalizable iﬀ  ∈  ()It is shown by induction on  that  () is nonempty for all  and  = 1 2. Also,
each sequence { ()} is monotonically decreasing. Because each  () is finite andnonempty,  () becomes constant after some ; as a result,  () is nonempty. These
facts are more or less known, but we give a proof for completeness.
Lemma 2.2. { ()} is a decreasing sequence of nonempty sets, i.e.,  () ⊇+1 () 6= ∅ for all 

Proof. We show by induction over  that the two sequences { ()}   = 1 2
are decreasing with respect torelation. Once this is shown, since T the set-inclusion () 6= ∅. For the base case of  = 0 we haveis finite, we have  () = ∞
=0 
0 () =  ⊇ 1 () for  = 1 2 Now, suppose the hypothesis that this inclusion holds
up to  and  = 1 2 Let  ∈ +1 () By (2.3), Best ( ;  ) holds for some  ∈  ()Since −1 () ⊇  () by the supposition, Best ( ;  ) holds for some  ∈ −1 ()This means  ∈  ()Criterion for Prediction/Decision Making: Our discussion of ex ante decision
making in games begins with a prediction/decision criterion3 . While comparison between the Nash theory and rationalizability is our concern, some simpler examples of3
A general concept of a prediction/decision criterion is formulated in an epistemic logic of shallow
depths in Kaneko-Suzuki [14].

6decision criteria may be helpful. First, utility maximization can be regarded as a decision
criterion in a non-interactive context, which recommends the set of decisions maximizing a given utility function. In game theory, a classical example of a decision criterion is
the maximin criterion due to von Neumann-Morgenstern [24]: It recommends a player
to choose a strategy maximizing the guarantee level (that is, the minimum payoﬀ for a
strategy). In  = ( { }∈  { }∈ ) let  be a nonempty subset of    = 1 2 The
set  is interpreted as the set of possible decisions for player  based on the maximin
criterion. The criterion is formulated as follows:
NM1: for each 1 ∈ 1  1 maximizes min2 ∈2 1 (1 ; 2 );NM2: for each 2 ∈ 2  2 maximizes min1 ∈1 2 (2 ; 1 )
These are not interactive, since NM  = 1 2 can recommend a decision without
considering NM, and player  needs to know only his own payoﬀ function. Thus,
no prediction is involved for decision making with this criterion.
A more sophisticated criterion may allow one player to consider the other’s criterion.
One possibility is the following:
N1: for each 1 ∈ 1  Best(1  2 ) holds for all 2 ∈ 2 ;NM2: for each 2 ∈ 2  2 maximizes min1 ∈1 2 (2 ; 1 )
The criterion N1 requires player 1 to predict player 2’s decisions and to choose his best
decision against that prediction, while player 2 still adopts the maximin criterion. In
this sense, their interpersonal thinking stops at the second level. In the Nash theory andrationalizability theory, we would meet some circularity and their interpersonal thoughtgoes beyond the second level.
There may be multiple pairs of (1  2 ) that satisfies a given decision criterion.
Without other information than the criterion and components of the game, a player(and we) cannot make a further choice of particular strategies among those satisfying
the criterion. In the case of NM1-NM2,  should consist of all strategies maximizingmin2 ∈2 1 (1 ; 2 ); that is,  is the greatest set satisfying NM. In the case of N1-NM2,
this should also be applied to player 1’s predictions about 2’s choice: 2 in N1 shouldbe the greatest set satisfying NM2. We will impose this greatest-set requirement for in Section 3; this is not a mere mathematical requirement, but is very basic for theconsideration of ex ante decision making, as it will be discussed later.

3. Parallel Derivations of the Nash Noncooperative Solutions and Rationalizable Strategies
In this section we give two parallel decision criteria, and derive the Nash noncooperative
solutions and the rationalizable strategies from those criteria. Our characterization7

results pinpoint the diﬀerence between the two theories. This diﬀerence is used as thebasis for our evaluation of these two theories, which comes in Section 4. We give remarkson the mixed strategy versions of those derivations in Section 3.3.
3.1. The Nash Noncooperative Solutions
The decision criterion for the Nash solution formalizes the statements N1 and N2
in Section 1. This criterion, N1-N2, is formulated as follows: Let  be a subset of   = 1 2 interpreted as the set of possible decisions based on N1-N2,
N1: for each 1 ∈ 1  Best(1 ; 2 ) holds for all 2 ∈ 2 ;N2: for each 2 ∈ 2  Best(2 ; 1 ) holds for all 1 ∈ 1 
These describe how each player makes his decisions; when one player’s viewpoint isfixed, one of N1-N2 is interpreted as decision making, and the other is interpreted as
prediction making. For example, from player 1’s perspective, N1 describes his decision
making, and N2 describes his prediction making.
Mathematically, N1 and N2 can be regarded as a system of simultaneous equations
with unknown 1 and 2 . First we give a lemma showing that (1  2 ) satisfies N1-N2
if and only if it consists only of Nash equilibria.
Lemma 3.1. Let  be a nonempty subset of  for  = 1 2 Then, (1  2 ) satisfies
N1-N2 if and only if any (1  2 ) ∈ 1 × 2 is a Nash equilibrium in 
Proof. (Only-If): Let (1  2 ) be any strategy pair in 1 × 2 . By N1, 1 (1  2 ) is
the largest payoﬀ over 1 (01  2 ) 01 ∈ 1  By the symmetric argument, 2 (1  2 ) is thelargest payoﬀ over 02 ’s. Thus, (1  2 ) is a Nash equilibrium in (If): Let (1  2 ) ∈ 1 × 2 be a Nash equilibrium. Since 1 (1  2 ) ≥ 1 (01  2 ) for all
01 ∈ 1  we have N1. We have N2 similarly.
Regarding N1-N2 as a system of simultaneous equations with unknown 1 and 2 ,there may be multiple solutions; indeed, any Nash equilibrium pair as a singleton set
is a solution for N1-N2. However, the sets 1 and 2 should be based only on theinformation of the game structure  This implies that we should look for the pair ofgreatest sets (1  2 ) that satisfies N1-N24 .
The following theorem dictates that N1-N2 is a characterization of the Nash solution
theory.
Theorem 3.2 (The Nash Noncooperative Solutions): (0):  has a Nash equilibrium if and only if there is a nonempty pair (1  2 ) satisfying N1-N2.
4

If any additional information is available, then we extend N1-N2 to include it and should consider
the pair of greatest sets satisfying the new requirements.

8

(1): Suppose that  is solvable. Then the greatest pair (1  2 ) satisfying N1-N2 exists
and  = 1 × 2 is the Nash solution ().
(2): Suppose that  has a Nash equilibrium but is unsolvable. Then  is a Nash subsolution if and only if (1  2 ) is a nonempty maximal pair satisfying N1-N2.
Proof. (0): If (1  2 ) is a Nash equilibrium of  then 1 = {1 } and 2 = {2 } satisfy
N1-N2. Conversely, if a nonempty pair (1  2 ) satisfies N1-N2, then, by Lemma 3.1,
any pair (1  2 ) ∈ 1 × 2 is a Nash equilibrium of (1):(If): Let (1  2 ) be the greatest pair satisfying N1-N2. It suﬃces to show () =1 × 2  By Lemma 3.1, any (1  2 ) ∈ 1 × 2 is a Nash equilibrium. Conversely, let(01  02 ) ∈ () and 0 = {0 } for  = 1 2 Since this pair (10  20 ) satisfies N1-N2, wehave (01  02 ) ∈ 10 × 20 ⊆ 1 × 2  Hence, () = 1 × 2 
(Only-If): Since  is the Nash solution, it satisfies (2.2). Hence,  is expressed as = 1 × 2 by Lemma 2.1. Since it consists of Nash equilibria, (1  2 ) satisfies
N1-N2 by Lemma 3.1. Since () =  = 1 × 2  (1  2 ) is the greatest pair having
N1-N2.
(2): (If): Let (1  2 ) be a maximal pair satisfying N1-N2, i.e., there is no (10  20 )satisfying N1-N2 with 1 × 2 ( 10 × 20 . By Lemma 3.1, 1 × 2 is a set of Nashequilibria. Let  0 be a set of Nash equilibria satisfying (2.2) with 1 × 2 ⊆  0 . Then, 0 is also expressed as 10 × 20 . Since 10 × 20 satisfies N1-N2 by Lemma 3.1, we
have 0 ⊆  for  = 1 2 by maximality for (1  2 ) By the choice of  0  we have
1 × 2 =  0  Thus,  is a maximal set satisfying interchangeability(2.2).
(Only-If): Since  is a subsolution, it satisfies (2.2). Hence,  is expressed as  =1 × 2  Also, by Lemma 3.1, (1  2 ) satisfies N1-N2. Since  = 1 × 2 is a subsolution, (1  2 ) is a maximal set satisfying N1-N2.
When  has a Nash equilibrium but is unsolvable, there are multiple pairs of maximal sets (1  2 ) satisfying N1-N2. We do not have those problems in NM1-NM2 in
Section 2.3, for which the greatest pair always exists and is nonempty. The reason forthis diﬀerence may be the interactive nature of N1-N2, which is lacking in NM1-NM2.
For an unsolvable game  with a Nash equilibrium, there is no single definite recommended set of decisions and predictions based on N1-N2, even though the decision criterion and game structure are commonly understood between the players. Each maximal
pair (1  2 ) satisfying N1-N2 may be a candidate, but it requires further information
for the players to choose among them. Thus, N1-N2 alone is not suﬃcient to provide
a definite recommendation for unsolvable games. Theorem 3.2 gives a demarcation line
between the games with a definite recommendation and those without it.
One possible way to reach a recommendation for an unsolvable game is to impose
an additional criterion, such as the symmetry requirement in Nash [20], so as to selecta certain subset of Nash equilibria. The game of Table 2.2 is unsolvable, but it has a9unique symmetric equilibrium (s11  s21 ). Hence, if we add the symmetry criterion, weconvert an unsolvable game to a solvable game.
Another possible criterion is the Pareto-criterion. It may work to choose one subsolution for some class of games. For example, it is known that a finite game of strategic
complementarity (or super modularity) has a Nash equilibrium in pure strategies, andunder some mild condition, that if it has multiple equilibria, they are Pareto-ranked(see Vives [23] for an extensive survey of this theory and its applications). For thosegames, when there are multiple equilibria, each equilibrium constitutes a subsolution.
However, when we add the Pareto-criterion, the subsolution which Pareto dominates
the other subsolutions is chosen. Since a finite game version of this theory is not wellknown, we give a brief description of this theory in our context.
Assume that the strategy set  is linearly ordered so that  is expressed as {1   }for  = 1 2 Here, 1 ∩ 2 = ∅ is violated but is recovered by a light change. We say thata game  has the SC property iﬀ (1): for  = 1 2  ( ;  ) is concave with respect to  i.e., for all  = 1   − 2 and all  ∈ 
 ( + 1;  ) −  ( ;  ) ≥  ( + 2;  ) −  ( + 1;  );(3.1)

and (2):  ( ;  ) is strategically complement, i.e., for all 1 ∈ 1 \{1 } and 2 ∈ 2 \{2 }
 ( + 1;  ) −  ( ;  ) ≤  ( + 1;  + 1) −  ( ;  + 1)

(3.2)

Then, the following are more or less known results, but we give a proof for selfcontainedness5 .
Lemma 3.3. Let  be a game with the SC property.
(1):  has a Nash equilibrium in pure strategies.
(2): Suppose -single peakedness, i.e., for each  = 1 2 and  ∈    ( ;  ) has aunique maximum over  Then, when  has multiple equilibria, they are linearly ordered with strict Pareto-dominance.
Proof. (1): We will use Tarski’s fixed point theorem: Let ( ≤) be a complete lattice,i.e., any subset of  has both infimum and supremum with respect to ≤. A function :  →  is called increasing iﬀ  ≤  implies () ≤ () Tarski’s theorem statesthat  is an increasing function on a complete lattice ( ≤) to itself then  has a fixed
point. See Vives [23] (the Appendix) and Cousot-Cousot [6] for relevant concepts.
We define the partial order ≤ over 1 × 2 by: (1  2 ) ≤ (01  02 ) ⇐⇒  ≤ 0 for = 1 2 Then, (1 × 2  ≤) is a complete lattice. We also define the (least) best-response
function  : 1 × 2 → 1 × 2 as follows: for  = 1 2 and  ∈  
 ( ) = min{ : Best( ;  ) holds}

(3.3)

5
Intervals of reals are typically adopted for these results. But Tarski’s fixed point theorem is applied
for the existence result in our case, too. In fact we can construct an algorithm to find a Nash equilibrium.

10Now,  (1  2 ) = (1 (2 ) 2 (1 )) for each (1  2 ) ∈ 1 × 2  We show that this  isincreasing. Then,  has a fixed point (1  2 ) which is a Nash equilibrium.
Now, suppose   0  Let  ( ) =   By (3.3) and (3.2), we have 0   ( ;  ) − ( − 1;  ) ≤  ( ; 0 ) −  ( − 1; 0 ) By (3.1), we have 0   ( ; 0 ) −  ( − 1; 0 ) ≤ ( ; 0 ) −  ( − 1; 0 ) for all  ≤   Thus,  ( ; 0 ) ≥  ( ; 0 ) for all  ≤   Thisimplies that player ’s best response to 0 is at least as small as   i.e.,  (0 ) = 0 ≥  (2): Let (1  2 ) (01  02 ) be two Nash equilibria with   0  By the monotonicity of 
shown in (1),  =  ( ) ≤  (0 ) = 0  If  = 0  then  (·;  ) takes a maximum at and 0  This is not allowed by -single peakedness. 

When an SC game  with -single peakedness has multiple equilibria,  is unsolvableby (2). However, if we add one criterion for player ’s prediction/decision criterion, thenwe can choose one solution for any SC game with -single peakedness. It may be betterto state the result as a theorem.
Theorem 3.4. Let  be an SC game with -single peakedness. Suppose that (1  2 )and (10  20 ) satisfy N1-N2. Then, if for  = 1 2  () ≥  (0 ) for some  ∈ 1 × 20and 0 ∈ 10 × 2  then 1 × 2 consists of the unique NE Pareto-dominating all other
NE’s.
Thus, one subsolution is chosen by adding the Pareto-criterion to N1-N2.
3.2. Rationalizable Strategies
The decision criterion for rationalizability theory, which formalizes the statements R1
and R2 in Section 1, is given as follows: for 1 and 2 
R1: for each 1 ∈ 1  Best(1 ; 2 ) holds for some 2 ∈ 2 ;R2: for each 2 ∈ 2  Best(2 ; 1 ) holds for some 1 ∈ 1 This criterion diﬀers from N1-N2 only in that the quantifier “for all” before players’
predictions in N1-N2 is replaced by “for some”. In fact, R1-R2 is the pure-strategy
version of the BP-property given by Bernheim [3] and Pearce [21]. The greatest pair(1  2 ) satisfying R1-R2 exists and coincides with the sets of rationalizable strategies(1 () 2 ()). A more general version of the following theorem is reported in Bernheim[3] (Proposition 3.1); we include the proof for self-containment.
Theorem 3.5 (Rationalizability): (1 () 2 ()) is the greatest pair satisfying R1R2.
Proof. Suppose that (1  2 ) satisfies R1-R2. First, we show by induction that 1 ×
2 ⊆ 1 () × 2 () for all  ≥ 0 which implies 1 × 2 ⊆ 1 () × 2 (). Since0 () =  for  = 1 2, 1 × 2 ⊆ 10 () × 20 (). Now, suppose 1 × 2 ⊆111 () × 2 (). Let  ∈  . Due to the R1-R2, there is an  ∈  such that().
Best( ;  ) holds. Because  ⊆  (), we have  ∈  (). Thus,  ∈ +1T
Conversely, we show that (1 () 2 ()) satisfies R1-R2. Let  ∈  () = ∞=0  ().
Then, for each  = 0 1 2 , there exists  ∈  such that Best( ;  ) holds. Since ∗is a finite set, we can take a subsequence {  }∞in { }∞=0 T
=0 such that for some  ∈  
∞

∗
∗

∗
 =  for all    Then,  belongs to  () = =0  () Also, Best ( ;  ) holds.
Thus, (1 () 2 ()) satisfies R1-R2.
Existence of a Theoretical Prediction: Theorem 3.5 and Lemma 2.2 imply thatthe greatest pair satisfying R1-R2 exists and consists of the sets of rationalizable strategies. Interchangeability is automatically satisfied by construction. In this respect, therationalizability theory appears preferable to the Nash theory in that it avoids the issues due to emptiness or multiplicity of subsolutions. We take a diﬀerent perspective
to reverse this preference: Diﬃculties involved in the Nash theory identify situations
where additional requirements other than N1-N2 are required for prediction/decision
making. In this sense, the Nash theory is a more precise and potentially richer theoryof ex ante decision making in an interactive situations.
Set-theoretical Relationship to the Nash Solutions: It follows from Theorem 3.5
that each strategy of a Nash equilibrium is a rationalizable strategy. Hence, the Nashsolution, if it exists, is a subset of the set of rationalizable strategy profiles. However,the converse does not necessarily hold. Indeed, consider the game of Table 34 where
the subgame determined by the 2 and 3 strategies for both players is the “matching
pennies”.

s11
s12
s13

Table 34
s21
s22
(5 5)(−2 −2)(−2 −2) (1 −1)(−2 −2) (−1 1)

s23(−2 −2)(−1 1)(1 −1)This game has a unique Nash equilibrium, (s11  s21 ) Hence, the set consisting of thisequilibrium is the Nash solution.
Both s11 and s21 are rationalizable strategies. Moreover, the other four strategies,s12  s13 and s22  s23 are also rationalizable: Consider s12  It is a best response to s22 which is a best response to s13  and s13 is a best response to s23 , which is a best response
to s12  That is, we have the following relations:
Best(s12 ; s22 ) Best(s22 ; s13 ) Best(s13 ; s23 ) and Best(s23 ; s12 )
By Theorem 3.5, those four strategies are rationalizable. In sum, all the strategies arerationalizable in this game.
12

This example shows that even for solvable games, the Nash solution may diﬀer
from rationalizable strategies6 . As we shall see later, the game of Table 34 becomesunsolvable if mixed strategies are allowed, while the rationalizable strategies remain thesame.
3.3. Mixed Strategy Versions
Theorems 3.2 and 3.5 can be carried out in mixed strategies without much diﬃculty.
The use of mixed strategies may give some merits and demerits to each theory. Here,
we give comments on the mixed strategy versions of the two theories.
The mixed strategy versions can be obtained by extending the strategy sets 1 and2 to the mixed strategy sets ∆(1 ) and ∆(2 ), where ∆( ) is the set of probability
distributions over   The notion of Nash equilibrium is defined in the same manner withthe strategy sets ∆(1 ) and ∆(2 ) : Once the Nash equilibrium is defined, the Nashsolution, subsolution, etc. are defined in the same manner. However, the mixed strategy
version of rationalizability requires some modification: A sequence of sets of strategies,0{(̃1 () ̃2 ())}∞=0 , is inductively defined as follows: for  = 1 2 ̃ () =   andfor any  ≥ 1̃ () = { : Best( ;  ) holds for some  ∈ ∆(̃−1 ())}

T
A pure strategy  ∈  is rationalizable iﬀ  ∈ ̃ () = ∞=0 ̃ ()Requirements N1-N2 are modified by replacing  by ∆( )  = 1 2;for  ⊆ ∆( )
 = 1 2
N1 : for each 1 ∈ 1  Best(1 ; 2 ) holds for all 2 ∈ 2 

N2 : for each 2 ∈ 2  Best(2 ; 1 ) holds for all 1 ∈ 1 

Notice that N1 -N2 is the same as N1-N2 with diﬀerent strategy sets. Moreover,
Theorem 3.2 still holds without any substantive changes.
In a parallel manner, the mixed strategy version of rationalizability can also be
obtained: for  ⊆ ∆( )  = 1 2
R1 : for each 1 ∈ 1  Best(1 ; 2 ) holds for some 2 ∈ 2 
R2 : for each 2 ∈ 2  Best(2 ; 1 ) holds for some 1 ∈ 1 This is a direct counterpart of R1-R2 in a game with mixed strategies. In this case,
6

When a 2-person game has no Nash equilibria, each player has at least two rationalizable strategies.
If a player has a unique rationalizable strategy, it is a Nash strategy. Moreover, when each player has aunique rationalizable strategy, then the pair of them is a unique Nash equilibrium. Example 3.4 statesthat the converse does not necessarily hold.

13

a player is allowed to play mixed strategies. However, in the original version of rationalizability in Bernheim [3] and Pearce [21], the players are allowed to use purestrategies only; indeed, mixed strategies are interpreted as a player’s beliefs about theother player’s decisions. We can reformulate R1 -R2 based on this interpretation of
mixed strategies: In R1  the first occurrence of 1 is replaced by a pure strategy inthe support of 1  and R2 is modified in a parallel manner. This reformulation turns
out to be mathematically equivalent to R1 -R2 
With the replacement of R1-R2 by R1 -R2 in Theorem 3.5, the following statement
holds:
Theorem 3.50  (∆(̃1 ()) ∆(̃2 ())) is the greatest pair satisfying R1 -R2 .
A simple observation is that a rationalizable strategy in the pure strategy version
is also a rationalizable strategy in the mixed strategy version. Similarly, since a Nashequilibrium in pure strategies is also a Nash equilibrium in mixed strategies, it may beconjectured that if a game  has the Nash solution  in the pure strategies, it might bea subset of the Nash solution in mixed strategies. In fact, this conjecture is answered
negatively.
Consider the game of Table 3.4. This game has seven Nash equilibria in mixed
strategies:
4 7 7
4 7 7
 18  18 ) ( 18
 18  18 ))((1 0 0) (1 0 0)) ((0 12  12 ) (0 12  12 )) (( 18
3 7
3
7
3 7
3
7
(( 18  78  0) ( 10
 10  0)) (( 18  0 78 ) ( 10
 0 10)) (( 10 10  0) ( 18  0 78 )) (( 10 0 10) ( 18  78  0))This set does not satisfy interchangeability (2.2). For example, ((1 0 0) (1 0 0)) and((0 12  12 ) (0 12  12 )) are Nash equilibria, but ((0 12  12 ) (1 0 0)) is not a Nash equilibrium.
Thus, (2.2) is violated, and the set of all mixed strategy Nash equilibria is not the Nashsolution. This result depends upon the choice of payoﬀs: In Table 3.5, (11  21 ) isa unique Nash equilibrium even in mixed strategies, while all pure strategies are stillrationalizable.

s11
s12
s13

s21
(5 5)( 12  12 )( 12  12 )Table 35
s22
s23
1 1
1 1
(2 2)(2 2)(1 −1) (−1 1)(−1 1) (1 −1)

4. Evaluations of N1-N2 and R1-R2 as Prediction/Decision Criteria
Our unified approach pinpoints the diﬀerence between the Nash and rationalizability
theories: the choice of quantifier “for all” or “for some” for each player’s predictions.
14Here we evaluate this diﬀerence reflecting upon on the conceptual bases of game theory.
We take Johansen’s [11] argument on the Nash theory as our starting point. Then, we
make comparisons between the two theories by considering two methodological principles: the free-will postulate and complete determinism. We also consider multiplicity
in prediction/decision criteria and how we should take it in our research activities.
4.1. Johansen’s Argument
Johansen [11] gives the following four postulates for prediction/decision making in games
and asserts that the Nash noncooperative solution is derived from those postulates
for solvable games. For this, he assumes (p.435) that the game has the unique Nashequilibrium, but notes (p.437) that interchangeability is suﬃcient for his assertion.
Postulate J1. A player makes his decision  ∈  on the basis of, and only on thebasis of information concerning the action possibility sets of two players 1  2 and their
payoﬀ functions 1  2 
Postulate J2. In choosing his own decision, a player assumes that the other is rationalin the same way as he himself is rational.
Postulate J3. If any7 decision is a rational decision to make for an individual player,then this decision can be correctly predicted by the other player.
Postulate J4. Being able to predict the actions to be taken by the other player,
a player’s own decision maximizes his payoﬀ function corresponding to the predictedactions of the other player.
Notice that the term “rational” occurs in J2 and J3, and “payoﬀ maximization” in J4.
The term “rational” in Johansen’s argumentation is broader than its typical meaning in
the game theory literature referring to “payoﬀ maximization.” Indeed, He regards thesefour postulates together as an attempt to define “rationality”; “payoﬀ maximization” isonly one component of “rationality”. We may further disentangle it using the conceptof prediction/decision criterion, which includes J4 as its component, and the concept oflogical abilities. Then, the above four postulates will be well understood, which is now
discussed.
Postulate J1 is the starting point for his consideration of ex ante decision making.
Postulate J2 requires the decision criterion be symmetric between the decision makerand the other player in his mind. Postulate J3 requires each player’s prediction about theother’s decision be correctly made. Postulate J4 corresponds to the payoﬀ maximization
requirement In the following, we first elaborate Postulates J2 and J3, and then use J1-J4
as a reference point for our critical comparisons between N1-N2 and R1-R2.
7This “any” was “some” in Johansen’s orginal Posutlate 3. According to logic, this should be “any”.
However, this is expressed as “some” by many scientists (even mathematicians).

15

Postulate J2 implies that from player 1’s perspective, the decision criterion has to besymmetric between the two players. In our context, this is interpreted as applied to the
choice of prediction/decision criterion. Both N1-N2 and R1-R2 satisfy this symmetricrequirement. The combination N1-NM2 discussed in Section 2 violates symmetry, andso does N1-R2, which will be further discussed in Section 4.3.
Postulate J3 is interpreted in the following manner: First, player 1 thinks about
the whole situation, taking player 2’s criterion as given, and makes inferences from thisthinking. Based on such inferences, player 1 makes a prediction about 2’s decisions.
This prediction is correct in the sense that player 1’s prediction criterion is the same as
2’ decision criterion and 1 has the same logical ability as player 2’s. In this sense, predictability in J3 is a result of a player’s contemplation of the whole interactive situation8 .
In this reasoning, “rationality” in J3 emphasizes symmetry in players’ interpersonal logical abilities, while that in J2 emphasizes symmetry in his prediction/decision criterion.
Postulates J1-J3 are compatible with N1-N2 and R1-R2. Only Postulate J4 makes
a distinction between the Nash theory and rationalizability theory. If we read Postulate
J4 in light of his assertion that interchangeability is a suﬃcient condition for J1-J4 tolead to the Nash solution, we can interpret J4 as adopting “for all” predicted actions of
the other player’s possible decisions.
Johansen [11] does not give a formal analysis of his postulates. Our N1-N2 may be
regarded as a formulation of these postulates in the language of classical game theory.
In this sense, Theorem 3.2 formalizes Johansen’s assertion that the Nash solution ischaracterized by J1-J4. If we modify Postulate J4 so that the “for all” requirement is
replaced by the “for some” requirement, Theorem 3.5 for R1-R2 would be a result. We
still need to discuss what are bases for the choice of “for all” or “for some”.
4.2. The Free-will Postulate vs. Complete Determinism
Here, we evaluate the diﬀerence between N1-N2 and R1-R2, based on two conflicting
meta-theoretical principles: the free-will postulate and complete determinism.
The Free-will Postulate: This states that players have freedom to make choices
following their own will. Whenever the social science involves value judgements for individual beings and/or the society, they rely on the free-will postulate as a foundation9 .
In a single person decision problem, utility maximization may eﬀectively void this postulate10 . However, in an interactive situation, even if both players are very smart, it
8Bernheim’s [4], p.486, interpretation of J3 in his criticism against these postulates is quite diﬀerent
from our reasoning. In his framework, predictability simply means that the belief about the otherplayer’s action, which is exogenously given, coincides with the actual action.
9
The free-will postulate is needed for deontic concepts such as responsibility for individual choice
and also for individual and social eﬀorts for future developments.
10This does not imply that utility maximization even for 1-person problem violates the free-will
postualte; he has still freedom to ignore his utility.

16

is still possible that individual decision making, based on utility maximization alone,
may not result in a unique decision. This is first argued in Morgenstern [18], using theparadox of Moriarty chasing Holmes. This is still a central problem in game theory; thefree-will postulate constitutes an important part of this problem. In this respect, thefree-will postulate still remains relevant to game theory.
Consider applications of the postulate at two diﬀerent layers in terms of interpersonal thinking:(): It is applied by the outside observer to the (inside) players;(): It is applied by an inside player to the other player.
In application () the outside theorist respects the free will of each player; the theorist
can make no further refinement than the inside player. This corresponds to the greatestness requirement for (1  2 ) in Theorems 3.2.(1) and Theorem 3.5. In (), when
one player has multiple predictions about the other’s decisions, the free-will postulate,applied to interpersonal decision making, requires the player take all possible predictions into account. N1-N2 is consistent with this requirement in that it requires eachplayer’s decision be optimal against all predictions11 .
Criterion R1-R2 involves some subtlety in judging whether it is consistent with application () The main diﬃculty is related to the interpretation of “for some” before
the prediction about the other’s decision. This leads us to another view, “complete
determinism.”Complete Determinism: The quantifier “for some” in R1-R2 has two diﬀerent interpretations:(): it requires only the mere existence of a rationalizing strategy;(): it suggests a specific rationalizing strategy predetermined for some other reason.
Interpretation () is more faithful to the mathematical formulation of R1-R2 as a decision criterion. If we accept (), then arbitrariness of the rationalizing strategy shows norespect to the other player’s free will, but we would not find a serious diﬃculty in R1-R2
with the free-will postulate in that R1-R2 is a prediction/decision criterion adopted by aplayer. However, this reminds us Aesops’ sour grapes that the fox finds one convenient
reason to persuade himself: For R1-R2, it suﬃces to find any rationalizing strategy.
This interpretation of “rationalization” is at odds with the purpose of a theory of ex
ante decision-making for games, since such a theory is supposed to provide a rationale
for players’ decisions as well as predictions. Interpretation () requires no rationale foreach specific rationalizing strategy.
Interpretation () resolves the arbitrariness in (): According to (), there are somefurther components, not explicitly included in the game description  and R1-R2, that
11
There are many other criteria consistent with the requirement. For example, player 1 uses the
maximin criterion to choose his action against 2  Another possibility is to put equal probability oneach action in 2 and to apply expected utility maximization.

17

determine a specific rationalizing strategy. However, a specific rationalizing strategy
for each step has to be uniquely determined, for otherwise the player would have toarbitrarily choose among diﬀerent strategies or to look for a further reason to choosesome of them. Thus, interpretation () violates Johansen’s postulate J1.
Interpretation () deserves a further analysis, since it is related to complete determinism, which has been regarded as very foundational in natural sciences. To determine a
specific rationalizing strategy, one possibility is to refer to a full description of the world
including players’ mental states; this presumes some form of determinism. We consider
only complete determinism for simplicity. Such a full description in a situation withtwo persons may require an infinite hierarchy of beliefs. Indeed, there is a literature,
beginning from Aumann [2]12 , to justify the rationalizability theory or alike along thisline (see Tan-Werlang [22]).
Complete determinism is incompatible with the free-will postulate in that it contains
no room for decision; ex ante decision making is an empty concept from this perspective.
From this view, R1-R2 is regarded as a partial description of a law of causation.
Except for conflicting against the free-will postulate, complete determinism may not
be very fruitful as a methodology for social science in general, which is aptly describedby Hayek [8], Section 8.93:“Even though we may know the general principle by which
all human action is causally determined by physical processes, this would not mean thatto us a particular human action can ever been recognizable as the necessary result of aparticular set of physical circumstances.”
Complete determinism is justified only because of its non-refutability by withdrawing from concrete problems into its own abstract world. In fact, neither complete
determinism nor the free-will postulate can be justified by its own basis. Either shouldbe evaluated with coherency of the entire scope and the scientific and/or theoretical
discourse.
Our conclusion is that the free-will postulate is needed for the perspective of social
sciences, and complete determinism has no such a status in social sciences. The Nashnoncooperative theory is constructed coherently with the free-will postulate, but therationalizability theory meets a great diﬃculty to reconcile with it.
4.3. Prediction/Decision Criteria
The characterizations of the Nash and rationalizability theories in terms of prediction/decision criteria are helpful to find their diﬀerences as well as to understand Jo12
In the problem of common knowledge in the information partition model due to Robert Aumann,
the information partitions themselves are assumed to be common knowledge. He wrote in [1], p.1237:“Included in the full description of a state  of the world is the manner in which information is imparted to the two persons”. This can be interpreted as meaning that the primitive state  includes
every information. A person receives some partial information about , but behind this, everything ispredetermined. This view is shared with Harsanyi [7] and Aumann [2].

18hansen’s argument and vice versa. However, these characterization results also introduce
a new problem: Among all possible prediction/decision criteria, why should we focus
particularly on the Nash theory or the rationalizability theory? Here we consider a few
exmaples of prediction/decision criteria and their resulting outcomes.
Relativistic View: It may be the case that people adopt diﬀerent prediction/decision
criteria. In addition to N1-N2 and R1-R2, as already indicated, NM1-NM2, N1-NM2
and N1-R2 are also possible candidates, among others. Even restricting our focus toN1-N2 and R1-R2, it is natural to ask why we avoid a mixture, such as N1-R2, of thosecriteria. Moreover, this combination actually generates a diﬀerent outcome either from
N1-N2 or R1-R2. Consider the game which is obtained from Table 3.4 by changing thepayoﬀs in the first row and first column.

s11
s12
s13

Table 41
s21
s22

(1 1)(1,1)
(1,0)(1 −1)(0,0)(−1 1)s23(0,0)(−1 1)(1 −1)For this game, we can calculate the greatest pairs (1  2 ) satisfying N1-N2, R1-R2 andN1-R2 as follows:
N1-N2: ({s11 } {s21  s22 })R1-R2: ({s11  s12  s13 } {s21  s22  s23 })N1-R2: ({s11  s12 } {s21  s22 })As soon as we start considering diﬀerent combinations, they could provide actuallydiﬀerent recommendations.
This relativistic view may turn our target problem into an empirical study of such
criteria in real societies. However, a prediction/decision criterion itself is still an analytic concept that serves as a benchmark to understand the prediction/decision-making
process in interactive situations. From this perspective, the focus should rather be a
study of the underlying structures and rationales for those criteria; if a criterion is incoherent with other bases, people will eventually avoid it. The goal of such study is thento separate some criteria from others, even if we take the relativistic view that people
follow diverse ways of prediction/decision making. For example, Johansen’s postulate
J2 accepts N1-N2 and R1-R2 but rejects N1-R2 as a legitimate criterion.
In this paper we analyze two specific criteria, N1-N2 and R1-R2, taking Johansen’s
postulates and the current literature of game theory as given. However, once we enter
the relativistic world of prediction/decision criterion, we may require rationales for the
postulates such as J2. A full analysis, which would involve broader conceptual bases forprediction/decision criterion and more explicit study of the underlying thought processes

19for prediction/decision making, is way beyond the current research. Nevertheless, Section 5 mentions a further research possibility on these problems as a continuation of thepresent paper.

5. Conclusions
5.1. Summary: the Unified Framework and Parallel Derivations
We presented the unified framework and parallel derivations of the Nash noncooperative
solutions and rationalizable strategies. The diﬀerence between them is pinpointed to be
the choice of the quantifier “for all” or “for some” for predictions about the other player’s
possible decisions. In Section 4, we discussed various conceptual issues by viewing thequantifier “for all” and “for some” from the perspectives of Johansen’s postulates, thefree-will postulate vs. complete determinism, and prediction/decision criteria.
Comparisons with Johansen’s postulates help us well understand our unified framework and derivations. The argument from the perspective of the free-will postulate vs.
complete determinism concludes that the Nash theory is more coherent to social sciences
as a whole than the rationalizability theory. Nevertheless, as a descriptive concept, it
would be possible for some people to use a criterion with “for some” for their decision
making. Reflections upon our approach in terms of prediction/decision criteria manifest
that vast aspects of prediction/decision making in social context are still hidden.
One such hidden problem is the treatment of the assumption of common knowledge.
We started this paper with the quotation to Mas-Colell et al. [15] about the standard
interpretation of rationalizability theory in terms of common knowledge. It is also a common interpretation that the Nash theory requires the common knowledge of the gamestructure. In this paper, the notion of common knowledge or even knowledge/beliefs
remains interpretational. To study the thought process for prediction/decision making
explicitly, we meet new issues and additional framework is necessary. The following
section discusses these issues.
5.2. Continuation: Thought Process for Prediction/Decision Making
The present paper employs the standard game theory language. In this language,
many essential elements remain informal and hidden, including a player’s beliefs or
knowledge. Those elements are essential for understanding the thought process forprediction/decision-making. Here we consider only N1-N2, but the argument is also
applicable to R1-R2.
Prediction Making (Putting Oneself in the Other’s Shoes): N1-N2 is understood as describing both prediction making and decision making: From player 1’s perspective, 1 in N1 is his decision variable, while 2 in N1 is his prediction variable.
20

Here, player 1 puts himself into player 2’s shoes to make predictions. In fact, this argument could not stop here; by putting himself in 2’s shoes, 1 needs to think about 2’s
predictions about 1’s decisions. Continuing this argument ad infinitum, we meet theinfinite regress described in Diagram 5.1, which is made from the viewpoint of player 1.
A symmetric argument from player 2’s viewpoint can be constructed.
Double Uses of N1-N2: In the infinite regress, N1 is a decision criterion for 1 andis a prediction criterion for 2, while N2 is a decision criterion for 2 and a prediction
criterion for 1. Thus, N1 and N2 are used both as decision and prediction criteria. This
double makes the infinite regress in Diagram 4.1 collapse into a system of simultaneous
equations described by Diagram 5.2. Theorem 3.2 solves this system of equations.

N1
↓
N2

%

Diagram 5.1
N1
N1
↓
% ↓
N2
N2

%

···

=⇒

Diagram 5.2
N1
↓ ↑
N2

The language of classical game theory is incapable to explicitly distinguish between
player 1’s and 2’s perspectives; as a result, many foundational problems can only be
discussed at interpretational levels. One way to formalize those issues is to reformulate
the above problem in the epistemic logic framework. Then, we can avoid the collapses
from Diagram 5.1 into Diagram 5.2, and explicitly discuss the relationship between theabove infinite regress and the common knowledge of N1-N2. In doing so, we will beable to evaluate the standard interpretations, such as the quotation from Mas-Colell etal. [15] in Section 1, of the rationalizability theory as well as the Nash theory. Also, we
can more explicitly discuss Johansen’s [11] argument. The research on these problems
will be undertaken in the companion paper [9].

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