Quantal-Response Equilibrium Models
of The Ultimatum Bargaining Game¢
Kang-Oh Yi
November 16, 2001
Department of Economics
Hong Kong University of Science and Technology
Clear Water Bay, Kowloon, Hong Kong
Tel: (852) 2358-7619, Fax: (852) 2358-2084
e-mail: [email protected]
Abstract
This paper investigates the implications of normal- and extensive-form quantal
response equilibrium (QRE) models (McKelvey and Palfrey, 1995, Games and
Economic Behavior; 1998, Experimental Economics) in the ultimatum bargaining
game, assuming that players maximize expected monetary payo¯ s. It is shown that
normal-form QRE can select a non-sequential equilibrium, and that the selection
depends crucially on the noise structure. The normal-form QRE describes the
main qualitative features of experimental subjects' behavior better than extensiveform QRE even in experiments with extensive-form games. Journal of Economic
Literature Classi± cation Number: C79, C92
Key Words: Quantal Response Equilibrium, Ultimatum Bargaining Game, Weakly
Dominated Equilibrium.
I am deeply grateful to Vincent Crawford and Joel Sobel for their advice and encouragement.
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1
Introduction
McKelvey and Palfrey's (1995) notion of quantal response equilibrium or \QRE"
has recently attracted a great deal of attention. In a QRE, players do not always
choose their best responses. Instead their strategy choices are noisy, and strategies
with higher expected payo¯ s are chosen with higher probabilities, with players
taking the noise in each other's strategies rationally into account in equilibrium.
In applications of QRE, the noise in players' strategy choices follows a speci± c
distribution, which allows the degree of noisiness to be represented by as few as
one parameter. The distribution most often used is the logit, and a QRE with
a logit response function is called a logit equilibrium. In some applications the
noise parameter is estimated and the resulting logit equilibrium is compared with
subjects' observed choices period by period. In others, a limiting logit equilibrium,
the limit of logit equilibrium as the noise approaches zero, which is usually an
equilibrium in the game without noise, is compared with limiting behavior in the
experiment.
McKelvey and Palfrey's original notion of QRE is a normal-form concept, and
McKelvey and Palfrey (1995), Anderson et al. (1998, 2001), and others have shown
that the normal-form QRE is surprisingly successful in describing the quantitative
as well as qualitative patters of deviation from equilibrium observed in a variety
of normal-from game experiments.
The normal-from QRE gives an identical prediction for an extensive- and
normal-form representation of the same game. However, experimental subjects
are often sensitive to how a game is presented, and the normal-form QRE has had
less success describing experimental results for some games presented in extensive
form, as in Schotter et al.'s (1994) experiment where the subjects' choice behavior
is systematically di¯ erent in di¯ erent representations of a game.
In response to these di¹ culties, McKelvey and Palfrey (1998) recently extended their notion of QRE to extensive-form games, proposing a notion called
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agent QRE or \AQRE." An AQRE is de± ned like a QRE, but for the agent normal form of an extensive-form game, in which di¯ erent information sets of a given
player are assumed to be played independently by di¯ erent agents, but all of a
given player's agents share the same payo¯ function. Because each agent's noise
is assumed to be independent, for any game with a non-trivial extensive form, an
AQRE di¯ ers from a normal-form QRE, where the noise terms for the agents of a
given player are in e¯ ect assumed to be perfectly correlated.1
In agent normal form, as far as the agents of a subgame are concerned, the
solution of the subgame agrees with the solution of the whole game. This property
brings AQRE much closer to sequential equilibrium in general, and McKelvey and
Palfrey (1998) showed that a limiting AQRE is in fact a sequential equilibrium.
They then used logit-AQRE, AQRE with a logit response function, to analyze
Schotter et al.'s (1994) experimental results, where subjects played a game with
two Nash equilibria, one trembling-hand perfect and one weakly dominated equilibrium, in di¯ erent representations that di¯ er only with respect to inessential
transformation of the extensive form. In one representation, every information
set is a singleton and the weakly dominated equilibrium is not a sequential equilibrium. In the others, subjects made choices simultaneously and both equilibria
are sequential equilibrium. In Schotter et al.'s experiment, the subjects played
mostly the trembling-hand perfect equilibrium strategies but played their dominated equilibrium strategies with signi± cant probabilities in the simultaneous-move
game. McKelvey and Palfrey (1998) showed that logit-AQRE describes subjects'
choice behavior better in the game where the every information set is a singleton
while normal-form logit equilibrium yields a better prediction in the simultaneousmove game. They compared QRE with \noisy Nash model," in which each player's
strategy choice is a convex combination of his Nash equilibrium strategy and random play, and showed that QRE outperforms the noisy Nash model by a large
1
In imperfect-information extensive-form games where every player at move has a single in-
formation set, AQRE and normal-form QRE are identical.
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margin.
McKelvey and Palfrey (1998) also applied logit-AQRE to various extensiveform game experiments on signaling games and centipede games, and AQRE has
had considerable success describing subjects' choice behavior. These results suggest
that AQRE is a promising way to describe subjects' responses to extensive form
games. But, to my knowledge, no one has yet considered the implications of
AQRE or normal-form QRE in ultimatum bargaining games, even though they
are perhaps the extensive-form game that is most often studied in experiments.2
This paper considers whether the notions of QRE can help to explain behavior in
ultimatum bargaining games by characterizing logit-AQRE and normal-form logit
equilibrium in discrete and continuous ultimatum bargaining games whose players
maximize expected monetary payo¯ s.
In an ultimatum bargaining game, one player, called the Proposer, makes an
all-or-nothing o¯ er, which the other, the Responder, can either accept or reject.
When players maximize expected monetary payo¯ s, in any Nash equilibrium, all
o¯ ers made in equilibrium must be accepted. In any sequential equilibrium, the
Proposer o¯ ers 0 to the Responder (or, in the discrete case, the Proposer o¯ ers
either 0 or the minimum positive proposal) and the Responder accepts.
This prediction is chronically violated in experiments. Most o¯ ers are concentrated between 30% and 50%, and smaller positive o¯ ers were often rejected
(see Roth, 1995, Chapter 4). The reasons for these violations have been a source
of controversy for more than a decade. Some studies have sought to explain the
experimental results for ultimatum games by assuming that subjects' preferences
(\social utilities") depend not only on their own monetary payo¯ s but also on others' in various ways, as in the general models proposed by Rabin (1990), Fehr and
2
The game analyzed in McKelvey and Palfrey (1998) can be viewed as a discrete (binary)
version of an ultimatum bargaining game that is studied in Gale et al. (1995). However, it is
not trivial to generalize their results for versions of the ultimatum bargaining game with larger
discrete strategy spaces or with continuous strategy spaces.
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Schmidt (1999), and Bolton and Ockenfels (2000), and the econometric models of
subjects' behavior in ultimatum experiments of Costa-Gomes and Zauner (2001).
Other analyses have sought to explain the results without social utility, assuming
expected monetary payo¯ maximization, but studying adaptive learning dynamics
as in Prasnikar and Roth (1992), evolutionary dynamics as in Gale et al. (1995),
or \limited cognition" as in Johnson et al. (2002).3
This paper takes a di¯ erent approach, assuming expected monetary payo¯
maximization as in the papers just mentioned, but ignoring dynamics, instead using
the normal-form QRE and AQRE as static models of boundedly rational strategic
behavior. Extending results of McKelvey and Palfrey (1995, 1998), the present
analysis gives a complete characterization of both notions of QRE in discrete and
continuous versions of the ultimatum bargaining game. The main result is that
in the discrete versions any o¯ er between 0 and equal split can be supported as
a strict best response in a limiting normal-form logit equilibrium. The limiting
logit-AQRE gives a unique selection of a trembling-hand perfect equilibrium with
the minimum positive o¯ er. In the continuous versions, the limiting normal-form
logit equilibrium and the limiting logit-AQRE coincide with the game's unique
trembling-hand perfect equilibrium.
The key di¯ erence between the two notions of QRE, which allows their implications to di¯ er in the discrete version, is in the noise structure. In an AQRE,
at any information set, the di¯ erence in the expected payo¯ s between accepting
or rejecting the o¯ er is the size of the o¯ er. Therefore, for each agent of the Responder, it is better to accept a positive o¯ er and the limiting AQRE, as the noise
disappears, is a sequential equilibrium. By contrast, in a normal-form QRE, all
3
In QRE, rational players are subject to mistakes in making choices. Prasnikar and Roth
(1992) and Johnson et al. (2002) studied di¯ erent aspects of bounded rationality that players'
game-theoretic reasoning skills are limited, or they do not know how the others behave. Although
their analyses emphasize learning, they acknowledged that social preferences play a signi cant
role in their bargaining experiments.
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agents of the Responder have the same expected payo¯ s. If an information set
is reached with a su¹ ciently small probability, then the agent's strategy choice
should barely a¯ ects the expected payo¯ and the agent should be almost indifferent between accepting and rejecting such o¯ ers. As the Proposer concentrates
the choice probability on a certain o¯ er, the Responder often rejects positive o¯ ers
that are rarely played, and the positive o¯ er can be the Proposer's best response
even without incredible threats. However, in the continuous case, the Proposer's
choice is noisy and the Proposer plays o¯ ers close to the optimal o¯ er as often as
the optimal o¯ er. As the noise vanishes, the Responder accepts those o¯ ers with
higher probability, and the Proposer has no incentive to o¯ er a positive o¯ er in a
limiting QRE.
The rest of the paper is organized as follows. Section 2 introduces the ultimatum bargaining game and the notions of QRE, AQRE, and their limiting
logit counterparts. Section 3 and 4 characterize the QRE's of extensive-form and
normal-form ultimatum bargaining games. Section 5 concludes. Proofs omitted
from the text are in the appendix.
2
QRE of the Ultimatum Bargaining Game
Since the ultimatum bargaining game is two-person perfect-information game whose
players move only once, both game forms can be described using the same notation
without explicitly considering information sets. Let i denote a player, i 2 fp; rg,
where p and r identify the Proposer and the Responder, respectively. the Proposer
chooses a strategy sp 2 Sp . When Sp is discrete, Sp = f0; n1 ; ¸
¸
¸
; 1g for a ± nite
integer n ¡ 1 and Sp = [0; 1] when Sp is continuous.4 In the following, I use (sp ; s0p ]
¸
¸
; s0p g in discrete games. This slight abuse of notation allows
to denote fsp + n1 ; ¸
4
Throughout this paper, the size of pie is normalized to 1. This normalization does not change
the results, and allows me to interpret an outcome as percentage shares of the pie, without
substantively a¯ ecting the results.
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me to write the set of o¯ ers less than
1
2
with [0; 12 ) no matter whether n is even or
odd. the Responder's strategy, sr 2 Sr , is a function that maps each possible o¯ er
to fAccept; Rejectg with sr (sp ) 2 fAccept; Rejectg. Let ¼
i
be the probability dis-
tribution over Si and ¼ i (si ) denote the probability of si being played. Each player
is assumed to have risk-neutral preferences which depends only on one's own pecuniary payo¯ . When the o¯ er is accepted, the payo¯ s are up (sp ; sr ) = 1 ∙ sp and
ur (sp ; sr ) = sp . If rejected, ui (sp ; sr ) = 0 for all i. Given a strategy pro± le, player
i's expected payo¯ is ¸i (si ; ¼
arise, ¸i (si ; ¼
¤ i)
¤ i)
R
=
s2S¤
i
¼
¤ i (s)ui (si ; s)ds.
When no confusion will
is denoted by ¸i (si ).
For a given player i, the logit response function maps expected payo¯ s for each
possible pure strategy into a mixed strategy for i, and they are denoted by pi and
fi when the strategy spaces are discrete and continuous, respectively. Pi and Fi
are the associated cumulative probability distributions with pi and fi , respectively.
Letting 0 ½
< 1 be the measure of the amount of noise, or equivalently, the
degree of rationality, the logit responses are determined by
pi (si ) = P
exp ( ¸i (si ))
s2Si exp ( ¸i (s))
and
exp ( ¸i (si ))
:
s2Si exp ( ¸i (s)) ds
fi (si ) = R
(1)
This functional form is called a logit function where the odds are determined
by the exponential transformation of the utility times a given non-negative constant
. The ratio of probabilities of two di¯ erent strategies, si and s0i , are given by
exp[ (¸i (si ) ∙ ¸i (s0i ))]. As
! 1, only the choices having the highest expected
payo¯ can be played with positive probabilities so that the choice behavior becomes
best response; when
= 0 all choices have equal probability. One of the features
that distinguishes QRE from some other noise-based equilibrium notions is that
in a limiting logit equilibrium, a weakly dominated strategy can be played with
positive probability. When ¸i (si ; ¼
holds only for s¤
i
and ¼
¤ i (s¤ i )
¤ i)
¡ ¸i (s0i ; ¼
! 0 as
¤ i)
for all 's, if the strict inequality
increases, then si and s0i should be played
with the same probability in a limiting logit equilibrium. Throughout this paper
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I assume that
is the same for all players and it is common knowledge.5 A logit
equilibrium for
is de± ned by a ± xed point in these probability distributions with
a given .
In games with discrete, ± nite strategy spaces, McKelvey and Palfrey (1995,
1998) showed the existence of logit equilibrium and its convergence to a Nash equilibrium. Although they did not consider games with continuous strategy spaces,
in the ultimatum bargaining game, these properties are preserved in games with
continuous strategies.
Proposition 1. In any versions of ultimatum bargaining game considered, there
exists a logit equilibrium for every
rium.
3
¡ 0 and the limiting QRE is a Nash equilib-
Agent Normal-Form Ultimatum Bargaining Game
In an agent normal-form representation of the ultimatum bargaining game, each
information set is played by a di¯ erent agent, and the agent on move at an information set has the same payo¯ s over terminal nodes as the Responder at the same
information set in the original game. Therefore, A (for Accept) strictly dominates
R (for Reject) at any information set reached by a positive o¯ er, and every responder's agent accepts any positive o¯ er with probability one in a limiting logit
equilibrium. The results for both games with discrete and continuous strategy
spaces are presented for completeness and to discuss some of QRE's interesting
properties.
In an extensive-form game, each agent of the Responder can be identi± ed by
an o¯ er and each agent's choices can be denoted by Ajsp and Rjsp . In both discrete
5
In principle, QRE permits di¯ erent ¼ 's across players, but the common knowledge assumption
is indispensable. In the present analysis, assuming the same ¼ simpli es notation and the proofs
can be directly extended to the case of heterogeneous ¼ 's by choosing ¼ = max ¼ 's.
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and continuous games, the expected payo¯ s can be calculated in the same way,
¸p (sp ) = (1 ∙ sp )pr (Ajsp );
and
¸r (Ajsp ) = sp and ¸r (Rjsp ) = 0:
(2)
Proposition 2. In ultimatum bargaining games, the limiting logit-AQREs are:
1) when n = 1, pp (0) = 1, pr (Aj0) = 12 , and pr (Aj1) = 1.
2) when n = 2, pp (0) = pp ( 12 ) = 21 , pr (Aj0) = 12 , and pr (Aj 12 ) = pr (Aj1) = 1.
3) when n ¡ 3, pp ( n1 ) = 1 and pr (Aj0) =
1
2
and pr (Ajsp ) = 1 for all sp > 0.
4) when the strategy set is continuous, fp converges to a point-mass at 0, and
pr (Aj0) = pr (Rj0) =
1
2
and pr (Ajsp ) = 1 for all sp > 0.
In case of discrete strategy spaces, Proposition 1 can be generalized to a game with
an arbitrary set of o¯ ers, Sp = fs0p ; s1p ; ¸
¸
¸
; skp g
[0; 1] where k + 1 is the number
of possible o¯ ers.
Corollary 1. In a limiting logit-AQRE, pr (Ajsp ) = 1 for all sp > 0 and pr (Aj0) =
pr (Rj0) =
1
2
if 0 2 Sp . Letting minsp >0 Sp = sp ,
1) when 0 2 Sp and sp < 21 , or when 0 62 Sp , pp (sp ) = 1.
2) when 0 2 Sp and sp = 12 , pp (0) = pp ( 21 ) = 12 .
3) when 0 2 Sp and sp > 12 , pp (0) = 1.
In discrete games, a limiting logit-AQRE depends only on the size of the
minimum positive o¯ er because ¸p (0) =
1
2
and ¸p (sp ) = 1 ∙ sp . In the continuous
case with 0 2 Sp , for the same reason, the limiting logit equilibrium is not a
logit equilibrium. In the limit, the Responder's logit equilibrium strategy is that
accepting any positive o¯ er with probability one and rejecting o¯ er 0 with a positive
probability. Even though such a strategy is well de± ned, the openness of the
Responder's acceptable set of o¯ ers implies that the Proposer does not have a best
response to it.
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4
Normal-Form Ultimatum Bargaining Game
In a normal-form representation of the ultimatum bargaining game, the Responder's strategy is represented by a complete contingent plan stating whether to
accept or reject each possible o¯ er. Unlike Nash equilibrium, QRE restricts players' behavior in all information sets as in Eq.(1). In the ultimatum bargaining
game, however, even in a limiting logit equilibrium such a restriction is not enough
to rule out the possibility that with positive probability the Responder plays such
a strategy that accepts an o¯ er but rejects bigger o¯ ers so that the Responder accepts all o¯ ers with highest probability but not with probability one. For instance,
the chance that the Proposer o¯ ers the whole pie vanishes so quickly that the Responder is almost indi¯ erent between accepting and rejecting for large 's. Thus,
the Responder rejects the o¯ er of one with probability one half in any limiting
QRE. To rule out such behavior, the present analysis assumes that if the Responder accepts an o¯ er sp then he should accept any o¯ ers larger than sp . That is, if
sr (sp ) = A, then sr (s0p ) = A for all s0p ¡ sp . To distinguish these two normal-form
games, I shall call the game without any restriction on the Responder's choice set
\unabridged normal-form game." After the analysis of the simpli± ed normal-form
game, the limiting logit equilibrium of the unabridged normal-form game is also
presented, which shows that the simplifying assumption does not a¯ ect the set of
limiting logit equilibrium o¯ ers.
Under this simplifying assumption, one can identi± es the Responder's strategy
with his minimum acceptable o¯ ers and can analyze the normal-form representation of a continuous version of the ultimatum bargaining game. This facilitates
the analysis, and allows a direct comparison of the logit equilibrium with limiting
behavior in the evolutionary model of the ultimatum bargaining game in Gale et
al. (1995), where the same restriction is imposed on the Responder's choice set.
Letting sr 2 Sr denote the minimum acceptable o¯ er, the modi± ed rule is that
the Proposer states an o¯ er, sp , and the Responder writes down the minimum
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acceptable o¯ er, sr , simultaneously. Note that Sr 6= Sp because the Responder
could reject the o¯ er of 1 and Sr = Sp [ fsr g with sr > 1. If sp ¡ sr , then
the Proposer receives 1 ∙ sp and the Responder receives sp . Otherwise, both get
nothing. The same rule is applied to both games with discrete and continuous
strategy spaces.
In a game with discrete strategies, the corresponding expected payo¯ s are
given by
¸p (sp ) = (1 ∙ sp )Pr (sp )
and
¸r (sr ) =
sr
X
spp (s):
(3)
s=sr
The simpli± cation makes sr dominate all s0r > sr so that pr (0) = pr ( n1 ) ¡
pr ( n2 ) ¡ ¸
¸
¸¡ pr (1) ¡ pr (sr ). Because the equilibrium selection of logit equilibrium
is based on strategy perturbations, one might expect that pp ( n1 ) = 1 and pr (0) =
pr ( n1 ) =
1
2
in a limiting logit equilibrium. However, this is not necessarily true as
discussed in Section 2. Given a best response, sp , as
grows, the probabilities of
all the o¯ ers other than sp being played become so small that the Responder is
almost indi¯ erent among choices of sr ½sp . Thus the Responder rejects positive
o¯ ers less than sp so often that the Proposer does not have an incentive to make
an o¯ er less than sp .
Lemma 1. When n ¡ 3, in any limiting logit equilibrium, the Proposer has a
unique best response, sp 2 (0; 21 ), and pr is uniformly distributed on [0; sp ].
The Proposer's limiting logit equilibrium strategy should be between 0 and 21 ,
but Lemma 1 does not identify the set of limiting logit equilibrium o¯ ers. Eq.(3)
implies that a su¹ cient condition for a sp to be a unique best response is pr (sp ) ¡
1
.
n(1¤ sp )+1
Since the Responder's strategy is pr (sp ) =
equilibrium with a best response of sp , any sp 2 (0;
1
)
2
1
nsp +1
in a limiting logit
can be a supported as a
unique best response because it can survive su¹ ciently small noise. Therefore, the
conditions in Lemma 1 are not only necessary but also su¹ cient.
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Proposition 3. In the simpli± ed normal-form representation of ultimatum bargaining game, limiting logit equilibria are:
1) when n = 1, pp (0) = 1 and pr (0) = pr (1) = pr (sr ) = 31 .
2) when n = 2, pp (0) = pp ( 12 ) =
1
2
and pr (0) = pr ( 12 ) = 12 .
3) when n ¡ 3, the set of limiting logit equilibrium o¯ ers is (0; 21 ). The Proposer
plays only pure strategy sp 2 (0; 21 ), and pr (sp ) =
1
nsp +1
pr (sp ) = 0 otherwise.
for all sp ½ sp and
Unlike Nash equilibrium, a limiting logit equilibrium with a positive o¯ ers
relies on \credible" threats rather than incredible threats. That is, given the consistent beliefs for a su¹ ciently large , since the imperfectly optimizing responder
puts enough weight on the dominated strategies, (0; sp ], the Proposer has a strict
incentive to o¯ er sp . the Responder's imperfectly optimizing behavior e¯ ectively
\threatens" the Proposer. However, this result requires consistent beliefs, and
QRE does not provide any explanation how players can coordinate on a particular
equilibrium.
In the ultimatum bargaining game, the limiting logit equilibrium depends on
redundant strategies. If the Responder's strategies are duplicated, for instance,
1
; 0) while
Sr = f0; 1; 2; ¸
¸
¸
; kg, then the limiting logit equilibrium outcome is ( k+1
that is ( 13 ; 0) when Sr = f0; 1; sr g. This property is of particular interest because
duplicating strategies usually does not a¯ ect the equilibrium outcome. For the
same reason, if the strategy choice of sr = 0 is duplicated enough, zero could be a
unique limiting logit equilibrium o¯ er. On the other hand, the set of logit equilibria
is sensitive to the details of Sp and that makes it hard to generalize Proposition 3 to
1 4 4:5
arbitrary Sp . For instance, when Sp = f0; 10
; 10 ; 10 ; ¸
¸
¸
; 1g,
logit equilibrium o¯ er while
4:5
10
can be. When Sp = f0;
4
cannot be a limiting
10
4:5 6
6
; ;¸
¸
¸
; 1g, 10
can be
10 10
a limiting logit equilibrium o¯ er. Nonetheless, if Sp is ± ne enough, Proposition 3
could serve as a general description of limiting logit equilibrium.
By contrast, when the strategy spaces are continuous, the unique limiting
logit equilibrium is the trembling-hand perfect equilibrium. To get some intuition
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for this result, consider a positive optimal o¯ er in a logit equilibrium for a ± nite
. Since the Proposer's choice probability depends on the expected payo¯ s, in a
continuous case, some sub-optimal strategies adjacent to the optimal strategy are
played almost as often as the optimal o¯ er no matter how large
is. Therefore, as
grows, the Responder accepts those \sub-optimal" o¯ ers with higher probabilities,
and the size of optimal o¯ er gets smaller and becomes zero in the limit.
Proposition 4. When an o¯ er can be made continuously, the unique normal-form
limiting logit equilibrium is the trembling-hand perfect equilibrium.
Now consider the discrete unabridged normal-form game where the Responder
has 2n+1 strategies. In this case, there is no well-de± ned cumulative distribution
function for the Responder's strategy mixture, which makes the analysis a bit more
complicated. Nonetheless, the limiting logit equilibrium can be easily characterized
using the property that if an information set is reached with probability of o(
¤ 1
),
then at the information set the player at move puts equal probability on every
available strategy. If the Proposer's limiting logit equilibrium o¯ er, sp , is strictly
best as in Lemma 1, the Responder plays all choices with sr (sp ) = A with the
same probability and puts zero weight on strategies with sr (sp ) = R. To make the
comparison easier, in Corollary 2 below, I also present the Responder's strategy
using Prob(Accept sp ), the chance that the Responder accepts o¯ er of sp .
Corollary 2. In the unabridged normal-form representation of the ultimatum
bargaining game, limiting logit equilibria are:
1) when n = 1, pp (0) = 1 and pr (sr ) =
Prob(Accept 0) =
1
2
and Prob(Accept 1) = 12 .
1
4
for all sr 2 fAA; AR; RA; RRg.
2) when n = 2, pp (0) = pp ( 12 ) = 21 , and pr (sr ) =
1
4
if sr ( 12 ) = A and pr (sr ) = 0
otherwise. Prob(Accept 12 ) = 1 and Prob(Accept 0) = Prob(Accept 1) = 21 .
3) when n ¡ 3, the set of limiting logit equilibrium o¯ ers is (0; 21 ). The Proposer
plays only pure strategy sp 2 (0; 21 ), and pr (sr ) =
1
2n
if sr (sp ) = A and pr (sr ) = 0
otherwise. Prob(Accept sp ) = 1 and Prob(Accept sp ) =
1
2
for every sp 6= sp .
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Although the Responder's equilibrium strategies of normal-form games appear
quite di¯ erent from those of the simpli± ed games, the implications of logit equilibrium are exactly the same in the sense that those equilibrium strategies in Proposition 3 can be directly generated from those in Corollary 2.
Proposition 5. Given the strategy pro± les in Corollary 2, should the Responder's
strategies such that sr (sp ) = A but sr (s0p ) = R for sp < s0p be deleted and rescaled,
the resulting strategy pro± les are identical to those in Proposition 3.
5
Concluding Remarks
This paper provides a general characterization of logit equilibrium in the ultimatum bargaining game assuming monetary payo¯ maximization. In the discrete
versions, the Responder could reject positive o¯ ers with positive probabilities and
thus the Proposer has a strict incentive to o¯ er a positive share of the pie in a
limiting normal-form logit equilibrium, whereas the Responder receives the minimum positive o¯ er in the limiting logit-AQRE. In the continuous versions of the
game, both notions of limiting QRE coincide with its unique trembling-hand perfect equilibrium.
In the ultimatum bargaining game experiments, most o¯ ers were concentrated
between 30% and 50%, and positive but smaller o¯ ers were often rejected. Clearly,
the predictions of all but the discrete normal-form logit equilibrium do not agree
with the subjects' choice behavior. The normal-form logit equilibrium in the discrete case is compatible with experimental results, but its interpretation is not
compelling. In a normal-form logit equilibrium, the Responder is not willing to reject any positive o¯ ers but he \does" either because he \knows" that the Proposer
should never make those o¯ ers or by mistakes. However, they do not seem the
case in experiments as in Prasnikar and Roth (1992), Binmore et al. (2001), and
Johnson et al. (2002). In particular, Binmore et al. (2001) examined variations of
the ultimatum bargaining game where the size of pie is 100 and the disagreement
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outcome is not (0,0). In the experiment, the subjects were extremely sensitive to
the disagreement outcomes and the result shows clearly that most rejected positive o¯ ers were rejected not by mistakes but intendedly whereas QRE predicts
no di¯ erence in choice behavior.6 Those experiment results question the use of
normal-form logit equilibrium with monetary payo¯ maximization as a description of subjects' choice behavior in ultimatum bargaining game experiments. As
a matter of fact, the Responder having a strict incentive to reject positive o¯ ers is
not compatible with monetary payo¯ maximization in any sensible static models,
and it seems inevitable to incorporate a more elaborate preferences such as social
utilities into analysis.
Appendix
Proof of Proposition 1. This proof deals with only continuous strategy cases.
In the agent normal-form representation, each agent at each information set, sp ,
makes an independent decision. Since accepting an o¯ er sp gives the Responder
( sp )
sp , the agent of the Responder accepts the o¯ er with probability 1+exp
exp( sp ) , which
is independent of pp . Given pr , the expected payo¯ from o¯ ering sp is a function
of pr and the existence follows. For a normal form game, see Anderson, Goeree
and Holt (1998, Appendix A) for the existence proof. The only di¯ erence in the
present analysis is the payo¯ function, but it is still continuous, which is all their
proof requires. Finally, the proofs of convergence in McKelvey and Palfrey (1995,
1998) require only the existence. Q.E.D.
Proof of Proposition 2.
6
( sp )
In any case, since pr (Ajsp ) = 1+exp
exp( sp ) , pr (Aj0) =
When the disagreement outcome is (10,10), 40% (103 of 257) of o¯ ers in the interval [20,30]
were rejected. Only .75% (1 out of 133) are rejected when the disagreement outcome is (70,10).
Those two treatments di¯ er only in the Proposer's disagreement outcomes, and the di¯ erence in
rejection rates are too large to attribute those rejections to mistakes. Instead, for some reasons
the Responder seems to have a strict incentive to reject positive o¯ ers.
15
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1
,
2
pr (Rj0) =
¸p (0) =
pr (Ajsp ) ! 1 as
1
,
2
and ¸p (1) = 0 for all
¡ 0, and for every sp > 0
! 1. Given these, consider a discrete case ± rst. The cases of
n = 1 and n ¡ 3 are trivial. When n = 2, the result follows from that ¸p (1) = 0
and pp (0) = pp ( 21 ),
pp (0)
= exp
pp ( 12 )
" ∙
1 1 exp( 2 )
∙
2 2 1 + exp( 2 )
!#
"
#
1
= exp
! 1 as
2 1 + exp( 2 )
! 1:
With a continuous strategy space, from Eq.(1), we have
Fp (sp ) = R sp
0
R sp
exp ( (¸p (y) ∙ ¸p (sp ))) dy
:
R
exp ( (¸p (y) ∙ ¸p (sp ))) dy + s1p exp ( (¸p (y) ∙ ¸p (sp ))) dy
0
Since (¸p (sp ) ∙ ¸p (s0p )) ! (s0p ∙ sp ) as
Proof of Lemma 1.
! 1, Fp (sp ) ! 1 for all sp > 0. Q.E.D.
After I characterize limiting logit equilibrium, I identify
necessary conditions for a limiting logit equilibrium strategy pro± le.
Let sp be the largest o¯ er played with positive probabilities in a limiting logit
equilibrium. Since in a limiting logit equilibrium the o¯ er should be accepted
with probability one, if pp (sp ) > 0, Pr (sp ) = 1 and ¸p (sp ) ¡ ¸p (sp ) +
sp > sp . On the other hand, from Eq.(3), ¸p (sp ) ∙ ¸p (sp ∙
1
)
n
Pr (sp )
p)
¡ 0. Since pr (sp ) ∙ n(1¤Pr (s
is strictly
n(1¤ sp )+1
sp )+1
Pr (sp )
implies ¸p (sp ) ¡ ¸p (sp ∙ n1 ) > ¸p (sp ) for all
n(1¤ sp )+1
if pr (sp ) ∙
1
n
for all
¡ 0 if and only
decreasing in sp ,
1
,
n
and
there exists a " > 0 such that ¸p (sp ) ∙ ¸p (sp ) > " > 0 for all sp 2 Sp nfsp ; sp ∙
1
g.
n
pr (sp ) ¡
sp < sp ∙
Therefore, if any, the possible multiple best responses are sp and sp ∙
pp (sp ∙
1
)
n
1
n
but with
= 0.
Since the best response in a limiting logit equilibrium is best along the asso-
ciated converging sequence of logit equilibria, for all sp 6= sp ∙
" > 0 such that ¸p (sp ∙
1
)∙
n
¸p (sp ) > ", and
h
1
,
n
there exists a
pp (sp ) ! 0 for any ± nite h because
exp( ¸p (sp ))=exp( ¸p (sp ))
exp[ (¸p (sp ) ∙ ¸p (sp ))]
=
:(¢ )
P
1 + j6=sp exp[ (¸p (j) ∙ ¸p (sp ))]
j=0 exp( ¸p (j))=exp( ¸p (sp ))
pp (sp ) = P1
This implies that pr (sp ) = pr (s0p ) + o(
h
) for all sp ; s0p ½sp ∙
1
,
n
and pr (sp ) = o(
h
)
for all sp > sp . For a su¹ ciently large , that allows us to write pp (sp ) and pr (sp )
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as
pp (sp ) = P
pr (sp ) = P
exp[ (1 ∙ sp )]
1
+ o(
=
1
1 + exp[ ( n ∙ (1 ∙ sp + n1 )pr (sp )]
sp 2Sp exp[ (1 ∙ sp )Pr (sp )]
exp[
sr 2Sr
Since pp (sp ∙
pr (sp ) =
1
)
n
P sr
sr pp (sr )]
sr =sp
Psr
exp[
s=sr spp (s)]
= 1 ∙ pp (sp ) + o(
1 + nsp exp
¸
=
h
1
1 + nsp exp[ (sp ∙
1
)p (s
n p p
1
pr (sp ¤ n
)
pr (sp )
1
)]
n
+ o(
h
);
):
), substituting pp (sp ) into pr (sp ) yields
1
µ + o(
exp[ ( n1 ¤ (1¤ sp + n1 )pr (sp )]
1
(sp ∙ n ) 1+exp[ ( 1 ¤ (1¤ s + 1 )pr (s )]
p n
p
n
Since pp (sp ∙ n1 ) ! 0 and
∙
h
h
):
= exp[ (sp ∙ n1 )pp (sp ∙ n1 )] converges to a positive
constant, the above equation could hold only when pr (sp ) >
" > 0. Therefore, pr (sp ) converges to
1
nsp +1
1
1
n(1¤ sp )+ n
+ " for some
and sp is a unique best response in
a limiting logit equilibrium. This also characterizes the limiting logit equilibrium
strategy pro± le given sp .
Next, let's ± nd boundaries. Since Pr ( n1 ) = 2Pr (0) ¡
Pr (0) ∙ 2 n¤n 1 Pr (0) >
2
3n
2
n
for all , ¸p ( n1 )∙ ¸p (0) =
and a logit equilibrium o¯ er is positive. For the upper
bound, from
¸p (sp ) ∙ ¸p
¶
1
sp ∙
n
¸p (sp ) ∙ ¸p (sp ∙
1
)
n
¶
1
1
= (1 ∙ sp )pr (sp ) ∙ Pr sp ∙
n
n
nsp
½ (1 ∙ sp )pr (sp ) ∙
pr (sp ) = (1 ∙ 2sp )pr (sp );
n
½ 0 for any sp ¡
1
.
2
1
2
Thus, sp ¡
cannot be played with
probability one and the result follows. Q.E.D.
Proof of Proposition 3.
This proof considers only the games with n = 1; 2.
When n = 1, pr (0) = pr (1) = pr (sr ) = 13 , and ¸p (0) ¡
1
2
> ¸p (1) for all . When
n = 2, since pr (0) = pr ( 12 ) ¡ pr (1) ¡ pr (sr ), ¸p (0) = ¸p ( 21 ) > ¸p (1) or pp (0) =
pp ( 21 ) > pp (1). Then the result follows from ¸r (0) = ¸r ( 21 ) ¡
1
2
>
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1
3
¡ ¸r (1).
Proof of Proposition 4. When the strategy space is continuous, the expected
payo¯ s are
¸p (sp ; fr ) = (1 ∙ sp )Fr (sp );
Since for any ± nite
and
¸r (sr ; fp ) =
Z 1
sfp (s)ds:
sr
the logit equilibrium densities, fi (si )'s, are di¯ erentiable with
respect to si , we have fi0 (si ) = fi (si )Dx ¸i (si ), or
7
fp0 (sp )
=
¸
µ
(1 ∙ sp )fr (sp ) ∙ Fr (sp ) fp (sp );
fr0 (sr ) = ∙ sr fp (sr )fr (sr ):
By integrating these from 0 to x,
fp (sp ) = fp (0) +
fr (sr ) = fr (0) ∙
Z sp ¸
Z0sr
0
µ
(1 ∙ s)fr (s) ∙ Fr (s) fp (s)ds
sfr (s)fp (s)ds:
In a logit equilibrium for a ± nite , since fr0 (sr ) ½0 and Dx2 ¸p (sp ) = (1∙ sp )fr0 (sp )∙
2fr (sp ) < 0, the Proposer's best choice is unique for every
¡ 0.
Let sp = argmaxsp ¸p (sp ; fr ). To obtain a contradiction, suppose that there
exists a su¹ ciently large
such that sp ¡ " > 0 for all
>
. Then from the
equation for fp0 , (1 ∙ sp )fr (sp ) = Fr (sp ), or
Z s
Fr (sp )
p
1 ∙ sp =
=
exp[
fr (sp )
0
"
sp
=
exp
2
"
sp
exp
¡
2
∙
(¸r (s) ∙ ¸r (sp )]ds ¡
∙
Z
0
sp
2
!
exp
" Z
sp
sp
2
#
sfp (s)ds
!#
Z s
p
sp
sp
sp Fp (sp ) ∙
Fp
∙ sp Fp (s)ds
2
2
2
∙
∙
!!#
"
#
sp
sp
sp
sp
Fp (sp ) ∙ Fp
¡
exp
Fp (sp ) :
2
2
2
4
The last inequality holds because fp (sp ) is increasing on [0; sp ]. Since sp > ",
sp Fp (sp ) should be bounded and this implies that fr (0) is bounded because
fr (0)
= exp
fr (sp )
7
i (si )'s
" Z
sp
0
#
sfp (s)ds ½exp[ sp Fp (sp )]:
are di¯ erentiable with respect to si , and in Eq.(1), given strictly positive denominator,
fi (si )'s are the ratio of di¯ erentiable functions of
i (si ).
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Finally,
fp (sp ) = fp (0) +
Z s ¸
p
0
µ
(1 ∙ s)fr (s) ∙ Fr (s) fp (s)ds ½fp (0) + fr (0)Fp (sp )
where the inequality follows from that fr (sr ) is non-increasing in sr . This implies
that fp (sp ) is bounded, which is a contradiction. Q.E.D.
Proof of Corollary 2.
When n = 1, Sp = f0; 1g and pr (AA) ¡
1
4
be-
cause AA is dominant. Thus ¸p (0) > ¸p (1) for all , and the result follows that
pr (AA)=pr (RR) = exp[ pp (1)] and ¸r (AA) ¡ ¸r (RA) ¡ ¸r (AR) = ¸r (RR).
all
When n = 2, since AAA is dominant, pr (AAA) ¡
1
8
and ¸p (0) > ¸p (1) for
. Since sr (0) does not a¯ ect the Responder's payo¯ , the Responder puts
positive weights only on sr 's with sr ( 12 ) = A. the Proposer's equilibrium strategy
is determined by the fact that pr (sr )'s for sr 's with sr ( 12 ) = R vanish at the rate
of o(
¤ 1
).
When n ¡ 3, one can show that the Proposer's limiting logit equilibrium o¯ er
is unique and strictly positive using the similar argument in the proof of Lemma 1.
Only di¯ erence is that the probability of sp being accepted is the sum of ¼ r (sr ) over
all sr 2 fsr jsr (sp ) = Ag. Given this, it is straightforward to ± nd the Responder's
equilibrium strategy using the fact that pp (sp ) ! 0 for all sp 6= sp and Eq.(*).
Finally, since sr (0) does not a¯ ect the Responder's expected payo¯ , the set of
equilibrium o¯ ers is determined by the fact that the Responder accepts any o¯ er
other than the equilibrium o¯ er with probability one half so that ¸p (0) =
1
2
for all
. Q.E.D.
Proof of Proposition 5.
When n = 1, after deleting AR, rescaling pr 's
gives pr (AA) = pr (RA) = pr (RR) =
Prob(Accept 1) =
2
3
1
,
3
which is Prob(Accept 0) =
1
3
and
as in Proposition 3.
When n = 2, fAAA; AAR; RAA; RARg are played with positive probability
of 41 . Deleting AAR and RAR and rescaling yields pr (AAA) = pr (RAA) = 21 , with
which Prob(Accept 0) = 12 , Prob(Accept 21 ) = 1, and Prob(Accept 1) = 1.
19
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When n ¡ 3, let sp be the limiting logit equilibrium o¯ er. First, delete the
Responder's strategies such that sr (sp ) = A but sr (s0p ) = R for sp < s0p . Then
there remains nsp + 1 strategies that are played with equal probabilities of
1
nsp +1
after rescaling. Then Prob(Accept sp ) = Prob(Accept sp ∙ n1 )+ ns 1+1 for all sp ½sp
p
and Prob(Accept sp ) = 1, and the result follows. Q.E.D.
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