Choice against Phantoms
Kazuhiro Hara∗†

Marcel Menezes de Andrade Prado‡
October 13, 2023

Abstract
We study the extent to which individual choice behaviors can be explained as equilibrium strategies in games. We view choice problems as aset of strategies given to the decision-maker, and the chosen alternatives
are considered equilibrium strategies in a game. We assume that players
other than the decision-maker (phantoms) and their set of strategies are unobservable. We will refer to this framework as “choice against phantoms.”
We demonstrate that in a general environment, without any assumptions
about the phantoms, all choice behaviors can be explained as equilibrium
strategies. Under the assumption that the set of phantom’s strategies isfixed, this model is characterized by Sen’s alpha axiom. We also show thata version that uses the strict Nash equilibrium as the equilibrium notion ischaracterized by WWARNI. As an application, we provide an alternative
representation for choice functions satisfying Weak WARP. We also provide
an alternative representation for Rational Shortlist methods.

JEL: C72, D01
Keywords: Rational choice, Nash equilibrium, bounded rationality, α-axiom

1Introduction

We propose and study models of choice that interpret individual choice behaviors
as equilibrium strategies in a game. We characterize choice behaviors that can be
explained under this model. The decision-maker under consideration is a player
in a strategic game. As an outside observer, we observe the set of strategies
available to him and the choices he made. However, we assume that the otherplayers in the game cannot be observed. We refer to these unobserved players
as “phantoms.” In essence, the decision-maker is in a strategic situation against
phantoms where Nash equilibria are played, but we only observe the projection of
the game and equilibria onto the decision-maker. We call this framework “choice∗

Waseda University, School of Commerce, Nishi-Waseda, Shinjuku, Tokyo, 169-8050.
Email: kazuhiro.hara@waseda.jp‡
EPGE Escola Brasileira de Economia e Finanças, Praia de Botafogo 190, Botafogo, Rio deJaneiro, 22250-900.
†1

against phantoms.” Choice against phantoms can arise in different contexts. For
example, an individual in a strategic situation may agree to disclose their choices
while other individuals involved refuse to do so for privacy concerns. Alternatively,
a delusional individual may put him or herself in a hypothetical strategic situationand play a game against phantoms.
We characterize the choice behaviors that can be explained as “choice against
phantoms.” As this is a fairly general framework, various versions can be considered depending on the nature of the phantoms (number of phantoms, set ofstrategies, and payoff functions). Firstly, we consider the situation where multiple phantoms exist and the set of strategies of the phantoms is dependent on the
choice problem:• There is a set N of phantoms. Each phantom has a correspondence Γi that
determines his strategy sets, and a payoff function ui . The decision-makerhas a payoff function ud . Given a choice problem S, the choice is determinedbyYc(S) = {x ∈ S : (x, xN ) ∈ N E(S, (Γi (S))i∈N , ud , (ui )i∈N ) for some xN ∈Γi (S)}
i∈N

where N E stands for the set of Nash equilibrium.
In Proposition 1, we demonstrate that when making almost no assumptions
about the phantoms, anything goes. That is, any pattern of choice can be represented as above.
Seeking positive results, we consider the version where the phantom’s set ofstrategies is fixed:• There is one phantom with a fixed set of strategies Y . Each player i has apayoff function ui . Given a choice problem S, the choice is determined byc(S) = {x ∈ S : (x, y) ∈ N E(S, Y, u1 , u2 ) for some y ∈ Y }.
In Theorem 3, we show that this model is characterized by Sen’s Alpha Axiom.
We show in Theorem 4 that, under suitable conditions, the payoff functions in thismodel can be continuous.
We then explore a version using strict Nash equilibrium as the equilibrium
concept:• There is one phantom with a fixed set of strategies Y . Each player i has apayoff function ui . Given a choice problem S, the choice is determined byc(S) = {x ∈ S : (x, y) ∈ SN E(S, Y, u1 , u2 ) for some y ∈ Y },where SN E stands for the set of strict Nash equilibrium.
In Theorem 6, we show that this model is characterized by WWARNI, a condition introduced by Ribeiro and Riella (2017). This condition is equivalent to2

α-axiom and expansion if there are only finitely many alternatives and the choicefrom every choice set is observed.
The decision-makers described in these models are highly rational in the sensethat they choose equilibrium strategies. In this regard, Theorem 3 suggests that
α-axiom may be an indication of high rationality. On the other hand, however,these models may violate the Weak Axiom of Revealed Preference (WARP) and,
therefore, may not be written as a preference maximization. In this sense, they
can be categorized as models of boundedly rational choice. Indeed, some of theexisting models of boundedly rational choice can be expressed as games.
As an application of the main results, we provide alternative representations
for choice functions that satisfy Weak WARP. We show that Rationalization by
Cherepanov, Feddersen, and Sandroni (2013) and Categorization and choice byManzini and Mariotti (2012) can be viewed as a selection from equilibrium strategies of a strategic game. We also show that the Rational Shortlist Method by
Manzini and Mariotti (2007) can be expressed as a selection from strict equilibrium strategies.

2

Preliminaries

Let X be a nonempty set and A be a nonempty collection of nonempty subsets of
X. A choice correspondence is a map c : A → 2X \ {∅} with c(S) ⊂ S for all S in
A. It is a choice function if c(S) is a singleton for every S in A.
A game (Si , ui )i∈N is a collection of pairs of a nonempty set of strategies Si
and a payoff function ui for each i ∈ N , where N is a nonempty set of players.
A Nash equilibrium of the game is a strategy profile s such that for each i ∈ N ,ui (s) ≥ ui (s0i , s−i ) for every s0i ∈ Si . We denote the set of Nash equilibria of
the game by N E((Si , ui )i∈N ). A strategy profile s is a strict Nash equilibrium if
for each i ∈ N , ui (s) > ui (s0i , s−i ) for every s0i ∈ Si \ {si }. We denote the setof strict Nash equilibria of the game by SN E((Si , ui )i∈N ). A two-player game
(S1 , S2 , u1 , u2 ) is symmetric if S1 = S2 and u1 (x, y) = u2 (y, x) for every (x, y) in
S1 × S2 .

3Models of choice against phantoms

3.1

Anything goes...

A model of choice against phantoms is determined by specifying which game eachdecision problem corresponds to. We consider a model that should be as general
as possible as a starting point. We allow for any number of phantoms, and the setof strategies for each phantom can change depending on the choice problem. To
this end, let N be the set of phantoms, and suppose the set of strategies for eachphantom i is determined by a correspondence Γi . The correspondence Γi matches
each choice problem with the set of strategies available to the phantom. That is,when the choice problem is S, Γi (S) becomes the set of strategies available to thephantom.
3Given the generality of this framework, it is not surprising that every choice
behavior can be explained as equilibrium strategies. In fact, as can be seen fromthe following example, every choice correspondence can be explained using justone phantom.
Example 1. Let c be an arbitrary choice correspondence. For each choice problem
S, let Γ(S) = {S}. Let u2 be an arbitrary constant payoff function for the phantom. Let u1 be the decision maker’s payoff function given by u1 (x, S) = 1c(S) (x).
Clearly, c(S) = arg max{u1 (x, S) : x ∈ S} and hence x ∈ c(S) iff (x, S) is a Nashequilibrium of (S, {S}, u1 , u2 ).
From this example, it is clear that the framework we are considering is too general. In particular, the problem is that the set of phantom’s strategies can change
without any constraints depending on the choice problem. Therefore, it is conceivable to impose some constraints on Γi . Here, we consider a monotonicity condition,
where the set of phantoms’ strategy profiles change monotonically depending on
the choice problem. Formally,Qwe say a collection(Γi )i∈N of correspondences isQmonotone if S ( T implies i∈N Γi (S) ( i∈N Γi (T ). This condition meansthat if the decision-maker’s budget set decreases, the phantoms’ budget sets also
decreases. Specifically, when there is only one phantom, if the decision-maker’s
choice set becomes smaller, the phantom’s choice set will also become smaller. As
the following proposition shows, even in the presence of monotonicity, every choice
behavior can be explained as equilibrium choices.
Proposition 1. There is a set N and a collection of correspondences (Γi )i∈N thatis monotone, and payoff functions (ui )i∈N such that for every choice correspondence c on 2X \ {∅}, there is a payoff function ud with whichYc(S) = {x ∈ S : (x, xN ) ∈ N E(S, (Γi (S))i∈N , ud , (ui )i∈N ) for some xN ∈Γi (S)}i∈N

for each S ∈ 2X \ {∅}.
Proof. We take X as the set of phantoms. For each i ∈ X, let Γi (S) = {0, 1} if
i ∈ S and Γi (S) = {0} otherwise. Note (Γi )i∈N is monotone. For each x ∈ X andz ∈ {0, 1}X , let ui (x, z) = zi . Let c be an arbitrary choice correspondence. Define
ud by ud (x, 0) = 0 and ud (x, z) = 1c({i:zi =1}) (x) when z 6= 0. Let S be an arbitrarychoice problem. If x ∈ c(S) then (x, z S ), where ziS = 1 if i ∈ S and ziS = 0
otherwise, is a Nash equilibrium of (S, (Γi (S))i∈X , u, (ui )i∈X ). Conversely, if (x, z)
is an equilibrium of (S, (Γi (S))i∈X , u, (ui )i∈X ) then it must be the case that z = z S .
Therefore, ud (x, z) = 1c(S) (x) = 1 must hold for (x, z) to be an equilibrium. Thisimplies x ∈ c(S).
When the number of alternatives is finite, in Proposition 1, the number ofphantoms can be reduced to one.
Corollary 2. There is a monotone correspondence Γ, a phantom’s payoff functions
u2 such that for every choice correspondence c on 2X \{∅}, there is a payoff function
u1 with whichc(S) = {x ∈ S : (x, y) ∈ N E(S, Γ(S), u1 , u2 ) for some y ∈ Γ(S)}4for each S ∈ 2X \ {∅}.
P
Proof.
Q Construct Γi and ui as in the proof of Proposition 1. Let u2 = i∈X ui andΓ = i∈X Γi . Note Γ is monotone. For any choice correspondence c, define ud asin the proof of Proposition 1 and let u1 = ud . Observe that (x, z) is an equilibriumof (S, Γ(S), u1 , u2 ) iff (x, z) is an equilibrium of (S, (Γi (S))i∈X , u, (ui )i∈X ). Thus,x ∈ c(S) iff (x, z S ) is an equilibrium of (S, Γ(S), u1 , u2 ).
From these “anything goes” type of results, it becomes clear that stronger assumptions are needed to obtain meaningful models of “choice against phantoms”.
In the following sections, we will examine some special cases that have testable
implications.

3.2

Nash Representation

In this model, we consider the decision maker who makes a choice against a player
who has a fixed set of strategies. Specifically, we consider the following representation of a choice correspondence c on an arbitrary collection A of choice problems.
There is a nonempty set Y and two real valued functions u1 and u2 defined on
X × Y such that for each choice problem S in A,c(S) = {x ∈ S : (x, y) ∈ N E(S, Y, u1 , u2 ) for some y ∈ Y }.

(1)

It turns out that this model is characterized by Sen’s α-axiom.
α-axiom. If S ⊂ T then c(T ) ∩ S ⊂ c(S).
This condition states that if an alternative chosen in a larger set is available
in a smaller set, it must be chosen.
Theorem 3. A choice correspondence c on A satisfies α-axiom if and only if it
admits a representation in (1).
Proof. It is easy to see that the representation in (1) implies α-axiom. We focus
on the sufficiency part of the theorem. Let cSbe a choice correspondence thatsatisfies α-axiom. Let Y := A ∪ {∅} and C := S∈A c(S). Define u1 : X × Y → R
by(
1X\S (x) S 6= ∅u1 (x, S) =1C (x)
S=∅
Note 1T stands for the indicator function of the set T . Define u2 : X × Y → R by(
1c(S) (x) S 6= ∅u2 (x, S) =1X\C (x) S = ∅Now, let x ∈ c(S). Then, u1 (x, S) = 0 = u1 (z, S) for all z ∈ S. On the other
hand, u2 (x, S) = 1 ≥ u2 (x, T ) for all T ∈ Y . Thus, (x, S) is a Nash equilibriumof the game (S, Y, u1 , u2 ).
5

Conversely, let S ∈ A and (x, T ) be a Nash equilibrium of the game (S, Y, u1 , u2 ).
Suppose T = ∅. Then, u1 (x, T ) ≥ u1 (z, T ) = 1 for any z ∈ c(S) and hence x ∈ C.
That is, x ∈ c(A) for some A ∈ A. But then, u2 (x, A) > 0 = u2 (x, T ). This isa contradiction. Thus, T 6= ∅. Suppose x 6∈ C. Then u2 (x, ∅) = 1 > u2 (x, T ),which is a contradiction. Thus, x ∈ C and hence u2 (x, T ) ≥ 1 = u2 (x, V ) for someV ∈ A. This implies x ∈ c(T ). Then, u1 (x, T ) = 0 ≥ u1 (z, T ) for all z ∈ S, which
implies S ⊂ T . By α-axiom, x ∈ c(S) as desired.
An alternative representation for choice correspondences that satisfy α-axiom
appears in Proposition 7 in Cherepanov, Feddersen, and Sandroni (2013). One
may be able to construct preferences for each player in (1) by using their representation.
Under suitable conditions, one can take payoff functions in (1) to be continuous. To this end, let X be a nonempty compact metric space. Let k(X) denote theset of nonempty compact subsets of X endowed with the Hausdorff distance. The
following condition is a standard upper hemicontinuity for choice correspondence.
Upper hemi-continuity. For every convergent sequences (xn ) in X and (Sn ) ink(X), if xn ∈ c(Sn ) for every n, then lim xn ∈ c(lim Sn ).
By adding this condition, we obtain a representation with continuous payoff
functions.
Theorem 4. Let X be a nonempty compact metric space. A choice correspondence c on k(X) satisfies α-axiom and Upper hemi-continuity if and only if there
is a nonempty compact metric space Y and two continuous payoff functions u1
and u2 such that for each S ∈ k(X),c(S) = {x ∈ S : (x, y) ∈ N E(S, Y, u1 , u2 ) for some y ∈ Y }.
Proof. It is easy to see that c satisfies α-axiom and upper hemicontinuity if it admits the representation. Conversely, let c be a choice correspondence that satisfies
α-axiom and upper hemi-continuity. Let Y := k(X) and note that Y is a nonempty
compact metric space. Define u1 : X × Y → R by u1 (x, T ) = miny∈T d(x, y) whered is the metric on X. Observe that u1 is a continuous function. Next, define thebinary relation D2 on X × Y by (x, S) D2 (y, T ) if (x, S) = (y, T ), or x = y andx ∈ c(S). Observe that D2 is a preorder. Moreover, it is continuous by upper hemicontinuity of c. There is a continuous function u2 on X×Y such that (x, S)D2 (y, T )implies u2 (x, S) ≥ u2 (y, T ) and (x, S) B2 (y, T ) implies u2 (x, S) > u2 (y, T ).1
To complete the proof, let x ∈ c(S). Then, u1 (x, S) ≥ u1 (z, S) for all z ∈ S
while u2 (x, S) ≥ u2 (x, T ) for all T ∈ Y . That is, (x, S) is a Nash equilibrium of thegame (S, Y, u1 , u2 ). Conversely, let (x, T ) be a Nash equilibrium of (S, Y, u1 , u2 ).
Then, u1 (x, T ) ≥ u1 (z, T ) for all z ∈ S and u2 (x, T ) ≥ u2 (x, A) for all A ∈Y . In particular, u2 (x, T ) ≥ u2 (x, {x}) while (x, {x}) D2 (x, T ). This implies(x, T ) D2 (x, {x}) and hence x ∈ c(T ). Then by definition of u1 , u1 (x, T ) = 0 and
1

See Levin (1983) and Corollary 1 in Evren and Ok (2011).

6

hence u1 (z, T ) = 0 for all z ∈ S. Thus, S is a subset of T . By α-axiom, x ∈ c(S)as desired.
Here are some examples of choice models that satisfy α-axiom and their representations of the form in (1).
Example 2. (Rational choice) Every choice correspondence that can be written
as maximization of a preference relation satisfies α-axiom. Such choice correspondences can be written as in (1) by introducing a phantom with only one strategy.
Example 3. (Choice functions) A choice function c on A containing all sets withat most three elements satisfies α-axiom if and only if it is preference maximization. Hence, for such choice functions, the model in (1) coincides with preference
maximization.
Example 4. (Choice against Nature) Let Ω be a state space, Y be a prize space,and X := Y Ω be the set of acts. The decision maker has
R a utility function u andchooses an act to maximize expected utility U (f, p) = Ω u(f ) dp for a given priorp. On the other hand, the nature has a utility function v and chooses a prioron ΩRfrom a set P of priors in order to maximize expected utility V (f, p) = Ω v(f ) dp
for a given act f . The choice is determined byc(S) = {f ∈ S : (f, p) ∈ N E(S, P, U, V ) for some p ∈ P }.
Note that the choice defined above is nonempty, for example, when Ω is finite, Y isa compact convex subset of Rn , P is compact and convex, u and v are affine, and S
is compact and convex. Moreover, if v = −u and S are compact and convex, thenthe model coincides with the maxmin expected utility of Gilboa and Schmeidler(1989).
Example 5. (Path-Independent Choice) A choice correspondence c on 2X \ {∅} ispath-independent if c(S ∪ T ) = c(c(S) ∪ c(T )) for every S and T . It is know thatpath independence implies α-axiom. Models of choosing the two finalists studied
by Eliaz, Richter, and Rubinstein (2011) are examples of path-independent choice.
An important special case of path-independent choice rule is q-acceptance rule.
It is used in school choice to model a school’s preference and is characterized by alinear order  over students and the number q of students the school can accept.
If there are less than or equal to q students applying, the school accepts all ofthem. If there are more than q students applying, the school accepts the top q
students among them according to .
Aizerman and Malishevski (1981) showed that, when X is finite, a choicecorrespondence c on 2X \ {∅} is path-independent if and only if there are linear
orders i for i = 1, . . . , k for some k such thatc(S) = {x ∈ S : x i y for every y ∈ S for some i}.
That is, choices from set S are maximizers of i . Using this representation, one
can easily construct a representation of the form (1). Indeed, let Y := {1, . . . , k}
and u2 be a constant function on X × Y . For each i = 1, . . . , k, let fi be a utility
function representing i . Define u1 on X × Y by ui (x, i) = fi (x).
7Example 6 (Threshold model). A threshold model consists of a utility functionu and a function δ : X × A → R+ . The choice is determined byc(S) = {x ∈ S : max u(S) ≤ u(x) + δ(x, S)}.
It is easy to see that c satisfies α-axiom if S ⊂ T implies δ(x, T ) ≤ δ(x, S).
Under this assumption, c can be written in the form (1) as follows. Let Y := A,
u2 (x, S) = 1c(S) (x), and(max u(S) x ∈ c(S)u1 (x, S) =u(x)otherwise
3.2.1

Strictly Competitive Representation

There are no constraints imposed on games in Nash representation. One canconsider various special cases of Nash representation corresponding to different
classes of games. Here we deal with the case in which the payoffs of the decision
maker and the phantom are in complete conflict in the following sense.
Definition. A game (X, Y, u1 , u2 ) is strictly competitive if for each (x, y) and(x0 , y 0 ) in X × Y , u1 (x, y) ≥ u1 (x0 , y 0 ) iff u2 (x, y) ≤ u2 (x0 , y 0 ).
In short, a strictly competitive game is an ordinal zero-sum game. As thefollowing theorem shows, every choice correspondence that can be expressed as
Nash representation with strictly competitive game is rationalizable in the senseof preference maximization.
Theorem 5. If a choice correspondence c admits a representation in (1) where
the corresponding game is strictly competitive, then it is rationalizable, i.e., there
is a complete and transitive relation % such that c(S) = max(S, %) for all S ∈ A.
Proof. Let (X, Y, u1 , u2 ) be a strictly competitive game and (1) holds for eachS ∈ A. We show that c satisfies the congruence axiom. Let x1 ∈ c(S1 ), x2 ∈S1 ∩ c(S2 ), . . . , xk ∈ Sk−1 ∩ c(Sk ), xk ∈ c(T ), and x1 ∈ T . For each i = 1, . . . , k,
let yi ∈ Y be such that (xi , yi ) ∈ N E(Si , Y, u1 , u2 ) and y ∈ Y be such that
(xk , y) ∈ N E(T, Y, u1 , u2 ). It is enough to show that (x1 , z) ∈ N E(T, Y, u1 , u2 ).
Then, u1 (xi , yi ) ≥ u1 (x, yi ) and u2 (xi , yi ) ≥ u2 (xi , y) for all (x, y) ∈ Si × Y . By
strict competitiveness, therefore, we have u1 (xi , yi ) ≥ u1 (xi+1 , yi ) ≥ u1 (xi+1 , yi+1 )for i = 1, . . . , k − 1. Thus, u1 (x1 , y1 ) ≥ u1 (xk , yk ). On the other hand, strict
competitiveness implies u1 (xk , yk ) ≥ u1 (xk , z) ≥ u1 (x1 , z) ≥ u1 (x1 , y1 ). Therefore, u1 (x1 , z) = u1 (xk , z) = u1 (x1 , y1 ) and by strict competitiveness, u2 (x1 , z) =u2 (xk , z) = u2 (x1 , y1 ). This means x1 is a best response to z and z is a best
response to x1 in (T, Y, u1 , u2 ). Hence, (x1 , z) ∈ N E(T, Y, u1 , u2 ) as desired.

3.3

Stric Nash Representation

Next, we consider a version of Nash representation in which strict Nash equilibrium
is used rather than Nash equilibrium. In this model, the decision maker, taking
8the choice set as the set of strategies, plays a game against a phantom as inNash representation. His or her choices are strict Nash equilibria of the game.
Specifically, we study the following representation for choice correspondence c.
There is a nonempty set Y and two payoff functions u1 and u2 defined on X × Y
such that for each choice problem S in A,c(S) = {x ∈ S : (x, y) ∈ SN E(S, Y, u1 , u2 ) for some y ∈ Y }.

(2)

It is easy to see that this representation implies α-axiom. In fact, it satisfies
the following slightly stronger condition .
WWARNI: If x ∈ S and for every z ∈ S there is T such that {x, z} ⊂ T andx ∈ c(T ) then x ∈ c(S).
This condition is employed by Ribeiro and Riella (2017) as part of conditions
that characterize the maximalization of a preorder. It is a weaker version of
the Weak Axiom of Revealed Inferiority (See Eliaz and Ok (2006)) and implies αaxiom. The following theorem shows that the representation in (2) is characterized
by this condition.
Theorem 6. A choice correspondence c on A satisfies WWARNI if and only if it
admits a representation in (2).
Proof. Suppose c(S) = {x ∈ S : (x, y) ∈ SN E(S, Y, u1 , u2 )} for all S ∈ A. Fix
S ∈ A and x ∈ S such that for each z ∈ S, there is Tz with {x, z} ⊂ Tz and x ∈c(Tz ). For each z ∈ S \ {x}, there is yz ∈ Y such that (x, yz ) ∈ SN E(Tz , Y, u1 , u2 ).
As (x, yz ) is a strict equilibrium, yz = yz0 for each z, z 0 ∈ S \ {x}. Put y := yz
for any z ∈ S \ {x}. Then, for each z ∈ S \ {x}, u1 (x, y) > u1 (z, y) as (x, y) ∈SN E(Tz , Y, u1 , u2 ). Hence, (x, y) ∈ SN E(S, Y, u1 , u2 ) and thus x ∈ c(S).
To see the other direction, let c be a choice correspondence that satisfies
WWRNI. For each x ∈ X, define Sx as follows: Sx = {x} if there is no T ∈ A
with x ∈ c(T ), otherwise[
Sx = {T ∈ A : x ∈ c(T )}.
Note, WWARNI implies x ∈ c(S) iff x ∈ S ⊂ Sx . We use X as the phantom’s setof strategies and define u1 : X × X → R and u2 : X × X → R as follows:(2 x 6∈ Sy
1 x=y
u1 (x, y) = 1 x = yu2 (x, y) =

0 x 6= y
0 x ∈ Sy and x 6= y
We show that c(S) = {x ∈ S : (x, y) ∈ SN E(S, Y, u1 , u2 )} for all S ∈ A.
Let x ∈ c(S). Then, S ⊂ Sx and hence u1 (x, x) = 1 > 0 = u1 (z, x) for allz ∈ S \ {x}. Similarly, u2 (x, x) = 1 > 0 = u2 (x, y) for all y ∈ X \ {x}. Hence,(x, x) ∈ SN E(S, Y, u1 , u2 ).
Conversely, let (x, y) ∈ SN E(S, Y, u1 , u2 ). It is clear from the definition of u2
that x = y. Suppose there is no T ∈ A with x ∈ c(T ). Then, there is z ∈ S \ {x}9and u1 (z, x) = 2 > 1 = u1 (x, x). This is a contradiction. Hence, x ∈ c(T ) forsome T ∈ A. If there is z ∈ S \ Sx , then u1 (z, x) = 2 > 1 = u1 (x, x). This is acontradiction. Hence, S ⊂ Sx and thus x ∈ c(S) by WWARNI.
When X is finite, and A contains all choice problems, WWARNI can be decomposed into α-axiom and the following condition.
Expansion: If x ∈ c(S) and x ∈ c(T ) then x ∈ c(S ∪ T ).
Corollary 7. Let X be a nonempty finite set. A choice correspondence on 2X \{∅}
satisfies α-axiom and Expansion if and only if it admits a representation in (2).
Proof. By the previous Theorem, it is enough to show that WWARNI is equivalent to α-axiom and Expansion. Clearly, WWARNI implies α-axiom and Expansion. To see the converse, suppose c satisfies α-axiom and Expansion. Let S be achoice problem such thatSx ∈ S, and for every y ∈ S, there is Ty with {x, y} ⊂ Ty
and x ∈ c(Ty ). Let T := y∈S Ty and note that S ⊂ T . Since X is finite, x ∈ c(T )by Expansion. By α-axiom, x ∈ c(S).
Here are some examples of choice models that satisfy WWARNI.
Example 7 (Maximization of an acyclic relation). Sen (1971) showed that, when
X is finite, a choice correspondence c on 2X \ {∅} satisfies α-axiom and Expansion if and only if there is a complete and acyclic relation % on X such thatc(S) = max(S, %) for each S. The above corollary states that such a choicecorrespondence can be expressed as a game in the sense of (2).
Example 8 (Maximalization of an preorder). Let % be a preorder on a finite set
X and c be a choice correspondence defined by c(S) = MAX(S, %). It is easy toshow that c satisfies WWARNI.
Example 9 (Maximization of an interval order). A special case of threshold models (See Example 6) in which the threshold function δ depends only on x corresponds to the maximization of an interval order. The choice is determined byc(S) = {x ∈ S : max u(S) ≤ u(x) + δ(x)}.
This model satisfies WWARNI.

4

Application

4.1

Weak WARPThe following weaker version of the weak axiom of revealed preference for choicefunction has been studied recently by Manzini and Mariotti (2007), Manzini andMariotti (2012), and Cherepanov, Feddersen, and Sandroni (2013).

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Weak WARP: If x 6= y, {x, y} ⊂ S ⊂ T and c({x, y}) = c(T ) = x then c(S) 6= y.
We provide an alternative representation for models consistent with Weak
WARP. The following theorem shows that all choice functions that satisfy Weak
WARP can be expressed as a selection from equilibrium strategies.
Theorem 8. Let X be a finite set and c a choice function on 2X \ {∅}. Then,c satisfies Weak WARP if and only if there is a nonempty set Y and functions
u1 : X × Y → R, u2 : X × Y → R, and an irreflexive binary relation  on X suchthatc(S) = max ({x ∈ S : (x, z) ∈ N E (S, Y, u1 , u2 ) for some z ∈ Y }, )for each S ∈ 2X \ {∅}.
Proof. First, assume a choice function c admits a representation in the statement. Let x 6= y, {x, y} ⊂ S ⊂ T , and c({x, y}) = c(T ) = x. If (y, z) 6∈
N E (S, Y, u1 , u2 ) for all z ∈ Y then, c(S) 6= y. If (y, z) ∈ N E (S, Y, u1 , u2 ) for some z ∈Y , then x  y as c({x, y}) = x. Since c(T ) = x, (x, z 0 ) ∈ N E (T, Y, u1 , u2 ) andhence (x, z 0 ) ∈ N E (S, Y, u1 , u2 ) for some z 0 ∈ Y . Therefore, c(S) 6= y.
Conversely, let c be a choice function that satisfies Weak WARP. Define acorrespondence α by α(S) = {y ∈ S : y = c(T ) for some T ⊃ S} for eachS ∈ 2X \ {∅}. Define  by x  y if x 6= y and there are S ⊂ T such that{x, y} ⊂ S, x = c(S) and y = c(T ). Then,  is asymmetric by Weak WARP andc(S) = max (α(S), ). Notice that α satisfies α-axiom. By Theorem 3, there is anonempty set Y and functions u1 : X × Y → R and u2 : X × Y → R such thatα(S) = {x ∈ S : (x, y) ∈ N E(S, Y, u1 , u2 ) for some y ∈ Y }.
Thus, a desired representation is obtained.

4.2

Rational Shortlist Method

Manzini and Mariotti (2007) proposed the Rational Shortlist Method, a model
for choice function, and showed that it is characterized by Weak WARP andExpansion. Here, we propose an alternative model characterized by the same
axioms. In this model, the decision maker makes a selection from equilibrium
strategies in symmetric equilibria.
Theorem 9. Let X be a finite set and c a choice function on 2X \ {∅}. Then, csatisfies Weak WARP and Expansion if and only if there is a nonempty set Y andfunctions u1 : X × Y → R, u2 : X × Y → R, and an irreflexive binary relation 
on X such thatc(S) = max ({x : (x, y) ∈ SN E (S, Y, u1 , u2 ) for some y ∈ Y }, )for each S ∈ 2X \ {∅}.

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Proof. First, assume c admits a representation in the statement. Define a correspondence α by α(S) = {x : (x, y) ∈ SN E (S, Y, u1 , u2 ) for some y ∈ Y }. By
Theorem 6, α satisfies α-axiom and Expansion. Hence, c satisfies Weak WARP.
To see that c satisfies Expansion, let x = c(S) = c(T ). Then, as α satisfies Expansion, x ∈ α(S ∪ T ). On the other hand, for each y ∈ α(S ∪ T ), y ∈ α(S) ory ∈ α(T ). Without loss of generality, let y ∈ α(S). Then, x  y as x = c(S).
Hence, x = c(S ∪ T ).
Conversely, let c be a choice correspondence that satisfies Weak WARP andExpansion. Define α(S) = {y ∈ S : y = c(T ) for some T ⊃ S} for each S ∈2X \ {∅}. Then, α satisfies α-axiom. Define  by x  y if x 6= y and there are
S ⊂ T such that {x, y} ⊂ S, x = c(S) and y = c(T ). Then,  is asymmetricby Weak WARP and c(S) = max (α(S), ). Notice that α satisfies Expansion.
Indeed, if x ∈ α(S) and x ∈ α(T ) then, there is S 0 ⊃ S such that x = c(S 0 )and T 0 ⊃ T such that x = c(T 0 ). As c satisfies Expansion, x = c(S 0 ∪ T 0 )and thus x ∈ α(S ∪ T ). Therefore, by Theorem 6, there is a nonempty set Y
and functions u1 : X × Y → R, u2 : X × Y → R such that α(S) = {x :(x, y) ∈ SN E (S, Y, u1 , u2 ) for some y ∈ Y }. Hence, the desired representation is
obtained.

5

Related LiteratureThis paper is related not only to the theory of boundedly rational choice but also
to the emerging literature on the revealed preference approach to game theory.
The primary focus of this literature is on the identification of testable restrictionsof game-theoretic solution concepts. The literature can be classified into two parts
depending on whether or not a game form is given.
Sprumont (2000) and Galambos (2004) studied the rationalization of choicecorrespondence in terms of Nash equilibria when there are given game forms andthe observed choices are strategy profiles. In the same setup, Lee (2012) studied
the rationalization by Nash equilibria of a zero-sum game. Ray and Zhou (2001)
studied similar problems when extensive game forms with complete information
are given. In contrast, we do not assume game forms as given. In addition, we
assume that the choices of other players are unobservable.
The other approach takes choice correspondence as primitive and does notassume game forms to be observed. For example, Bossert and Sprumont (2013)
showed that every choice function is rationalizable as an outcome of backward
induction.2 Rehbeck (2014) and Xiong (2014) extended the result to choice correspondences. Li and Tang (2017) showed a similar result for random choice rules. In
these papers, each alternative corresponds to an outcome of a game, and the chosenalternatives are interpreted as equilibrium outcomes. In contrast, this paper considers the rationalization of choice correspondence in terms of strategic games. We
interpret each alternative as a decision maker’s strategy in an unobserved strategic
environment, and chosen alternatives are understood as equilibrium strategies.
2

See also Xu and Zhou (2007), and Horan (2012).

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