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International Game Theory Review, Vol. 1, Nos. 3 & 4 (1999) 241–250
c World Scientific Publishing Company
BOUNDED RATIONALITY AND
ALTERNATING-OFFER BARGAINING
ANA MAULEON
Department of Economic Analysis, University of the Basque Country, Bilbao, Spain
VINCENT J. VANNETELBOSCH∗
CORE and IRES, University of Louvain, Louvain-la-Neuve, Belgium
Institute of Public Economics, University of the Basque Country, Bilbao, Spain
One form of bounded rationality is a breakdown in the commonality of the knowledge
that the players are rational. In Rubinstein’s two-person alternating-offer bargaining
game, assuming time preferences with constant discount factors, common knowledge
of rationality is necessary for an agreement on a subgame perfect equilibrium (SPE)
partition to be reached (if ever). In this note, assuming time preferences with constant
costs of delay, we show that common knowledge of rationality is not necessary to reach
always an agreement on a SPE partition. This result is robust to a generalisation, time
preferences with constant discount factors and costs of delay, if the players are sufficiently
patient.
1. Introduction
A two-person bargaining situation involves two individuals who have the possibility
of concluding a mutually beneficial agreement, but there is a conflict of interests
about which agreement to conclude, and no agreement can be imposed on any individual without her approval. Two approaches are commonly offered to solve a
bargaining situation. In the non-cooperative approach, the outcome is the solution
of a strategic bargaining model which embodies a detailed description of a bargaining procedure (how offers and counter-offers are made, who moves first, how delay
in reaching an agreement imposes costs on the players, and so on). The cooperative approach differs mainly in that the bargaining procedure is left unmodelled. A
survey of the cooperative and non-cooperative bargaining theory can be found in
Thomson (1994) and Binmore et al. (1992), respectively.
One bargaining procedure dominates the literature on non-cooperative bargaining models — the alternating-offer bargaining procedure. The first to investigate
the alternating-offer procedure was Ståhl (1988). He looked at finite-horizon models
in which two players alternate in making proposals until an agreement is reached.
∗ Corresponding
author address: Institute of Public Economics, University of the Basque Country,
Avda. Lehendakari Aguirre 83, 48015 Bilbao, Spain. E-mail: [email protected]
241
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However, the most significant model of alternating-offer bargaining is that of Rubinstein (1982). In Rubinstein’s model, the players alternate in making proposals,
with no exogenous bound on the length of time that they may bargain, and they
have time preferences: among agreements reached at the same time, players prefer
larger shares of the cake, and prefer to obtain any given share of the cake sooner
rather than later. Under mild conditions on the time preferences of the players,
Rubinstein has shown that the alternating-offer bargaining game with complete
information yields a unique SPE. In this SPE, agreement is reached without delay,
the less impatient player obtains a larger share of the cake, and there is a first
mover advantage.a
One form of bounded rationality is a breakdown in the commonality of the
knowledge that the players are rational. See Aumann (1992), Stahl and Wilson
(1994, 1995). A fact is common knowledge if all players know it, all know that
all know it, and so on ad infinitum. A fact is mutual knowledge of order 1 if all
players know it; mutual knowledge of order 2 if all players know that all players
know it, and so on ad finitum. Rationalizability [Bernheim (1984), Pearce (1984)
and Herings and Vannetelbosch (1999)] is a non-equilibrium concept in which the
conjectures of the players about the strategies of their opponents are not assumed
to be correct,b but are constrained by considerations of rationality. All players know
that the strategies chosen by their opponents are best responses to conjectures, and
further, all players know that their opponents know this and hence know that their
opponents know that their strategies are best responses to conjectures, and so on.
In other words, the rationality of the players is common knowledge. Assuming a
breakdown in the commonality of the knowledge reverts to k-step rationalizability
which relies on mutual knowledge up to order k − 1 of rationality.
What happens if there is iterated mutual knowledge of rationality up to
any given order, but not common knowledge in Rubinstein (1982) two-person
alternating-offer bargaining game? Assuming time preferences with constant discount factors [as in Rubinstein (1982)], Vannetelbosch (1999a) has shown that
common knowledge of rationality is necessary for an agreement on a SPE partition
to be reached (if ever). Indeed, rationalizability for multi-stage games implies that
either agreement is reached (possibly with delay) on a SPE partition, or perpetual
disagreement occurs.
In this note, assuming time preferences with different constant costs of delay
[as in Rubinstein (1982)], we show that rationalizability or common knowledge of
rationality is not necessary to reach always an agreement on a SPE partition. There
exists an order k ∗ such that for all k ≥ k ∗ , mutual knowledge up to order k − 1 or
a Multiple SPE arise and agreements may be delayed (even with complete information) if at least
one player has the possibility to reduce the value of the cake after her own proposal is rejected
[see Avery and Zemsky (1994)]. Another source for agreements reached with delay is incomplete
information [see e.g. Watson (1998)].
b Equilibrium concepts such as Nash equilibrium or subgame perfect equilibrium assume that all
players hold correct conjectures about the strategies of their opponents.
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k-step rationalizability always leads to an agreement on a SPE partition. Also, the
agreement is reached no later a delay of one period. Moreover, if the players are
sufficiently patient, this result is robust to a generalisation: time preferences with
constant discount factors and costs of delay [as in Avery and Zemsky (1994)].
The paper is organised as follows. Section 2 presents the bargaining game and
gives the definitions of rationalizability and k-step rationalizability for multi-stage
games. Section 3 is devoted to time preferences with constant costs of delay. Section
4 is devoted to time preferences with constant discount factors and costs of delay.
2. The Alternating-Offer Bargaining Game
Two players i (i = 1, 2) are bargaining over the division of a cake of size 1. These
two players must agree on an allocation from the set X. X is called the set of
possible agreements.
X ≡ {(x1 , x2 ) ∈ R2 | x1 , x2 ≥ 0, x1 + x2 ≤ 1} .
We denote by xi player i’s share, for i = 1, 2. Consider Rubinstein’s alternatingoffer bargaining procedure. Player 1 calls (offers/accepts) in even-numbered periods
and player 2 calls in odd-numbered periods. Let n ∈ N be the period at which an
offer is made. The game starts at n = 0 and ends when one of the players accepts
the opponent’s previous offer. Note that an agreement may be reached as early as
in period n = 1.
2.1. Strategies and histories
In each period n, the player on the move chooses an action a(n) ∈ A ≡ X ∪{accept};
except at the very beginning of the game where player 1 cannot accept. Let h0 = ∅
be the history at the start of play. Define a history of the game at the end of period
Qk−1
k − 1 by hk = (a(0), . . . , a(k − 1)) ∈ H k ≡ n=0 X. Histories after which player 1
S∞
has the move are contained in H1 ≡ n=0 H 2n . Histories after which player 2 has
S∞
the move are contained in H2 ≡ n=0 H 2n+1 . Let H ≡ H1 ∪ H2 . A pure strategy
of player i is a function si : Hi → A which maps each possible history after which
player i has the move into an action. Let Si be the set of strategies for player i,
i = 1, 2. −i denotes player i’s opponent. S ≡ S1 × S2 is the set of strategy profiles.
2.2. Time preferences
Payoffs in the bargaining game are given as functions of the players’ strategy profile
according to the utility functions Ui : S → R, i = 1, 2; where Ui (s) ≡ ui (θ(s)). Two
sub-families of time preferences are considered.
(i) Time preferences with a constant cost of delay [Rubinstein (1982)]. For any
outcome θ(s) ≡ ((x1 (s), x2 (s)), n(s)) that specifies an agreement on allocation
(x1 (s), x2 (s)) at period n(s), let ui (θ(s)) ≡ xi (s)−γi ·(n(s)−1), where γi ∈ (0, 1)
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for i = 1, 2. Each player i incurs a fixed cost (cost of delay or bargaining cost)
γi for each unit of period that elapses without an agreement being reached. If
under s the players fail to reach an agreement, let ui (θ(s)) = −∞, for i = 1, 2.
Time preferences with a constant cost of delay satisfy Osborne and Rubinstein
(1990) assumptions A1 through A5, but violate A6; since the loss to delay
associated with any given share of the cake is constant.c Denote by G(γ1 , γ2 )
the alternating-offer bargaining of complete information with fixed bargaining
costs.
(ii) Time preferences with a constant discount factor and a constant cost of delay
[Avery and Zemsky (1994)]. For any outcome θ(s) ≡ ((x1 (s), x2 (s)), n(s)) that
specifies an agreement on allocation (x1 (s), x2 (s)) at period n(s), let

n−2
X

 (n(s)−1)
δ
· xi (s) −
δ t · γi
for n ≥ 2
ui (θ(s)) ≡
(1)
t=0


for n = 1
xi (s)
where δ ∈ (0, 1), γi ∈ (0, 1) and γi < δ for i = 1, 2. If under s the players fail
to reach an agreement, let ui (θ(s)) = −γi · [1 − δ]−1 , for i = 1, 2. As δ → 1− ,
the disagreement payoffs converge to −∞. Time preferences with a constant
discount factor and a constant cost of delay satisfy Osborne and Rubinstein
(1990) assumptions A1 through A6. Denote by G(δ, γ1 , γ2 ) the alternating-offer
bargaining of complete information with fixed discount factors and bargaining
costs.
2.3. Rationalizability
Without loss of generality (for the results) the players have degenerate conjectures about the behaviour of the opponent. Hence, a conjecture of player i about
the behaviour of player −i is simply a s−i ∈ S−i . Given h ∈ H k , we denote by
Ui (si , s−i | h) the payoff of player i in the game conditional on h describing the
play through period k (or stage k) and (si , s−i ) describing the play thereafter. We
define rationalizability for multi-stage games, as in Vannetelbosch (1999a, 1999b),
by the following iterative process.
Definition 2.1. Consider the alternating-offer bargaining game G. Let R0 ≡ S.
Then Rk ≡ R1k × R2k (k ≥ 1) is inductively defined as follows: for i = 1, 2, si ∈ Rik if:
k−1
(i) si ∈ Rik−1 ; (ii) ∀ h ∈ Hi ∃ s−i ∈ R−i
such that ∀ s0i ∈ Rik−1 , Ui (si , s−i | h) ≥
T
0
k
Ui (si , s−i | h). The set of rationalizable strategy profiles is R∞ ≡ ∞
k=0 R .
c Osborne and Rubinstein (1990, pp. 33–34) assumptions A1 to A6 on players’ preferences over
outcomes are: (A1) disagreement is the worst outcome, (A2) cake is desirable, (A3) time is valuable,
(A4) player i’s preference ordering is continuous, (A5) stationarity of player i’s preference ordering,
and (A6) the loss to delay associated with any given share of the cake is an increasing function of
the share of the cake.
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In Definition 2.1 {Rk ; k ≥ 0} is a weakly decreasing sequence: ∅ 6= Rk+1 ⊆ Rk
∀k ∈ N ∪ {∞}. The set Rk is the set of pure strategy profiles which survive kround of iteration. Each higher step of iteration requires a higher-order of mutual
knowledge of rationality. That is, for all k ≥ 2, Rk relies on the assumption of
mutual knowledge up to order k − 1 of the rationality of the players. Condition (ii)
in Definition 2.1 restricts conjectures to the set of pure strategies that have not
been eliminated at a previous stage. We denote by Rk (γ1 , γ2 )[Rk (δ, γ1 , γ2 )] the set
of k-step rationalizable strategy profiles and by R∞ (γ1 , γ2 )[R∞ (δ, γ1 , γ2 )] the set of
rationalizable strategy profiles for the bargaining game G(γ1 , γ2 ) [G(δ, γ1 , γ2 )].
3. Fixed Bargaining Costs
Rubinstein (1982) has shown that every alternating-offer bargaining game G(γi , γj )
with γi < γj (i, j = 1, 2) has a unique SPE. In this SPE, player i offers xj = 0
whenever it is his turn to make an offer, and accepts an offer if and only if xi ≥ 1−γi ;
player j offers xi = 1 − γi whenever it is his turn to make an offer, and accepts all
xj ∈ [0, 1]. The SPE outcome is θ = ((1, 0), 1) if γ1 < γ2 and θ = ((γ2 , 1 − γ2 ), 1) if
γ1 > γ2 . So, player 1 gets all the cake if her bargaining cost is smaller than player
2’s bargaining cost, while player 2 gets 1 − γ2 if his bargaining cost is smaller.
When the bargaining costs are the same and less than 1, there is no longer
a unique SPE; every partition in the closed interval [γ1 , 1] is supported by SPE
strategies. Moreover, in some of these equilibria, agreement is not reached in period
n = 1 [see Rubinstein (1982)].
Proposition 3.1. Consider the alternating-offer bargaining game G(γi , γj ) with
γi < γj (i, j = 1, 2). There exists k ∗ ∈ N such that for all k ≥ k ∗ , si ∈ Rik (γi , γj ) if
and only if player i offers xj = 0 whenever it is her turn to make an offer, accepts
all xi > 1 − γi , rejects all xi < 1 − γi ; sj ∈ Rjk (γi , γj ) if and only if player j offers
xi = 1 − γi whenever it is his turn to make an offer, accepts all xj ∈ [0, 1].
Proof. Let γi < γj (i, j = 1, 2). From Definition 2.1, R0 = S. All si ∈ Ri1 are such
that i accepts (after any history h) all xi > 1 − γi , and all sj ∈ Rj1 are such that
j accepts (after any history h) all xj > 1 − γj . Therefore all si which reject (after
any history h) xi > 1 − γi are not included in the set Ri1 and all sj which reject
(after any history h) xj > 1 − γj are not included in the set Rj1 .
All si ∈ Ri2 are such that i (after any history h) never offers xj > 1 − γj and
rejects all xi < γj − γi . Indeed, i never accepts an offer which gives her less than
γj − γi , because she could obtain close to xi = γj next period. All sj ∈ Rj2 are such
that j (after any history h) never offers xi > 1 − γi and rejects all xj < 0. R2 tells
us that the largest share of the cake i [j] could obtain in a continuation game where
she calls first is 1 [1 − (γj − γi )].
All si ∈ Ri3 are such that i (after any history h) never offers xj > 1 − 2γj + γi ,
rejects all xi < 2(γj − γi ), accepts all xi > 1 − γi . All sj ∈ Rj3 are such that j (after
any history h) never offers xi > 1−γi , rejects all xj < 0, accepts all xj > 1−2γj +γi .
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For all k > 3, s ∈ Rk are such that (after any history h) player i never offers
xj > 1 − γj − (k − 2)(γj − γi ) if 1 − γj − (k − 2)(γj − γi ) > 0 and offers xj = 0
otherwise, rejects all xi < (k − 1)(γj − γi) if (k − 1)(γj − γi ) ≤ 1 − γi and rejects all
xi < 1−γi otherwise, accepts all xi > 1−γi ; player j never offers xi > 1−γi , rejects
all xj < 0, accepts all xj > 1 − γj − (k − 2)(γj − γi ) if 1 − γj − (k − 2)(γj − γi ) > 0
and accepts all xj ∈ [0, 1] otherwise.
There exists k ∗ such that for all k ≥ k ∗ , s ∈ Rk if and only if (after any history
h) player i offers xj = 0 since 1 − γj − (k ∗ − 2)(γj − γi ) ≤ 0, rejects all xi < 1 − γi
since (k ∗ − 1)(γj − γi ) ≤ 1 − γi , accepts all xi > 1 − γi ; player j offers xi = 1 − γi ,
accepts all xj ∈ [0, 1] since 1 − γj − (k ∗ − 2)(γj − γi ) ≤ 0.
Take γ1 > γ2 . Then, Proposition 3.1 tells us that the rationalizable (or k-step
rationalizable, k ≥ k ∗ ) strategy profiles are not unique. One of them is the SPE,
where player 1 offers 1 − γ2 at period 0, offer which is accepted by player 2 at
period 1. But, player 2 may reject such an offer, knowing that his counter-offer
(x1 , x2 ) = (0, 1) will be accepted by player 1 in period 2. Player 2 is indifferent
between the outcomes ((γ2 , 1 − γ2 ), 1) and ((0, 1), 2). Take now γ1 < γ2 . Then,
rationalizability (or k -step rationalizability, k ≥ k ∗ ) singles out the unique SPE
outcome ((1,0),1).
Corollary 3.1. Consider the alternating-offer bargaining game G(γ1 , γ2 ). If γ1 <
γ2 then θ = ((1, 0), 1) is the unique rationalizable outcome. If γ1 > γ2 then the
rationalizable outcomes are both θ = ((γ2 , 1 − γ2 ), 1) and θ = ((0, 1), 2).
Thus, if the players have time preferences with different constant costs of delay
instead of time preferences with constant discount factors, common knowledge of
rationality is no more necessary for solving Rubinstein’s alternating-offer bargaining
game. There exists a finite order k ∗ up to which, mutual knowledge of rationality
implies that the two players always reach an agreement on a SPE partition no
later than in period 2. It is possible to derive an explicit expression for k ∗ . Let
γ̄ = max{γ1 , γ2 } and γ = min{γ1 , γ2 }. For γ̄ 6= γ,
(
k ∗ (γ1 , γ2 ) =
Φ − mod[Φ, 1] + 1
if mod[Φ, 1] 6= 0
Φ
otherwise
where Φ = (1 + γ̄ − 2γ) · [γ̄ − γ]−1 and mod[Φ, 1] is simply the remainder from
dividing Φ by 1. The largest the cost gap |γ1 − γ2 | the smallest is k ∗ . Hence, an
increase in the cost gap increases the speed of convergence to the SPE partitions
of the set of k-step rationalizable partitions.
If γ1 = γ2 = γ then, for all k ≥ 2, s ∈ Rk are such that (after any history h)
player i never offers x−i > 1 − γ, rejects all xi < 0, and accepts all xi > 1 − γ. It
follows that perpetual disagreement or agreements reached with delay can occur.
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4. Fixed Discount Factor and Costs of Delay
We now consider the case where the players have time preferences with a constant
discount factor and a constant cost of delay [as in Avery and Zemsky (1994)].
Looking at the results on time preferences, either with a constant discount factor
[Vannetelbosch (1999a)] or with a constant cost of delay (Sec. 3), one would expect
a discontinuity in results at δ = 1 when the players have preferences given by (1).
However, the discontinuity occurs at a discount factor smaller than 1 as shown next.
k−1
k
) + γ−i ] − γi and starting
Let {xki }∞
k=0 be sequences with xi ≡ δ[1 − δ(1 − xi
0
k ∞
with xi ≡ δ − γi , i = 1, 2. We have that {xi }k=0 , i = 1, 2, are strict monotonic
decreasing sequences with xki ∈ (0, δ − γi ] that converge to x∞
= [δ(1 − δ) + δγ−i −
i q
γi ] · (1 − δ 2 )−1 if and only if δ ∈ (δ, δ̄), where δ̄ = 12 (1 + γ +
q
δ = 12 (1 + γ − (1 + γ)2 − 4γ̄).
(1 + γ)2 − 4γ̄) and
Lemma 4.1. Consider the alternating-offer bargaining game G(δ, γi , γj ) with γi <
γj (i, j = 1, 2). Then, si ∈ Rik (δ, γi , γj )[k ≥ 2] if and only if player i never offers
xj > xjk−2 if xjk−2 > 0 and offers xj = 0 otherwise, accepts all xi > xik−2 if
xik−2 < δ − γi and accepts all xi > δ − γi otherwise, rejects all xi < δ(1 − xjk−2 ) − γi
if xjk−2 > 0 and rejects all xi < δ −γi otherwise; sj ∈ Rjk (δ, γi , γj )[k ≥ 2] if and only
if player j never offers xi > xik−2 if xik−2 < δ − γi and offers xi = δ − γi otherwise,
accepts all xj > xjk−2 if xjk−2 > 0 and accepts all xj ∈ [0, 1] otherwise, rejects all
xj < δ(1 − xik−2 ) − γj if δ(1 − xik−2 ) − γj > 0 and rejects all xj < 0 otherwise.
Proof. Let γi < γj (i, j = 1, 2). From Definition 2.1, R0 = S. All si ∈ Ri1 are such
that i accepts (after any history h) all xi > δ − γi , and all sj ∈ Rj1 are such that j
accepts (after any history h) all xj > δ − γj .
All si ∈ Ri2 are such that i (after any history h) never offers xj > δ − γj and
rejects all xi < δ[1 − (δ − γj )] − γi . Indeed, i never accepts an offer which gives her
less than δ[1−(δ −γj )]−γi , because she could obtain close to xi = [1−(δ −γj )] next
period. All sj ∈ Rj2 are such that j (after any history h) never offers xi > δ − γi ,
rejects all xj < δ[1 − (δ − γi )] − γj if δ[1 − (δ − γi )] − γj > 0 and rejects all xj < 0
otherwise. The expression δ[1 − (δ − γi )] − γj is positive if and only if δ ∈ (δ, δ̄).
R2 tells us that the largest share of the cake i [j] could obtain in a continuation
game where he calls first is 1 − δ[1 − (δ − γi )] + γj if δ[1 − (δ − γi )] − γj > 0 and 1
otherwise [1 − δ[1 − (δ − γj )] + γi )].
All si ∈ Ri3 are such that i (after any history h) never offers xj > δ[1 − δ[1 − (δ −
γj )]+γi ]−γj if δ[1−δ[1−(δ −γj )]+γi ]−γj > 0 and offers xj = 0 otherwise, accepts
all xi > δ[1−δ[1−(δ−γi)]+γj ]−γi if δ[1−δ[1−(δ−γi)]+γj ]−γi < δ−γi and accepts
all xi > δ − γi otherwise, rejects all xi < δ(1 − δ[1 − δ[1 − (δ − γj )] + γi ] − γj ) − γi if
δ[1 − δ[1 − (δ − γj )] + γi ] − γj > 0 and rejects all xi < δ − γi otherwise; sj ∈ Rj3 are
such that j (after any history h) never offers xi > δ[1 − δ[1 − (δ − γi )] + γj ] − γi if
δ[1 − δ[1 − (δ − γi )] + γj ] − γi < δ − γi and offers xi = δ − γi otherwise, accepts all
xj > δ[1 − δ[1 − (δ − γj )] + γi ] − γj if δ[1 − δ[1 − (δ − γj )] + γi ] − γj > 0 and accepts
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all xj ∈ [0, 1] otherwise, rejects all xj < δ(1 − δ[1 − δ[1 − (δ − γi )] + γj ] − γi ) − γj if
δ(1 − δ[1 − δ[1 − (δ − γi )] + γj ] − γi ) − γj > 0 and rejects all xj < 0 otherwise. And
so on.
We can rewrite xki ≡ δ[1 − δ(1 − xik−1 ) + γ−i ] − γi as xki = [(δ(1 − δ) + δγi −
γ−i )δ 2k+1 + δ(1 − δ) + δγ−i − γi ] · (1 − δ 2 )−1 (which converges to x∞
i = [δ(1 − δ) +
δγ−i − γi ] · (1 − δ 2 )−1 ) if and only if δ ∈ (δ, δ̄), i = 1, 2. Finally, notice that for all
δ∈
/ (δ, δ̄) there exist k ∈ N such that xkj < 0.
From Lemma 4.1, in the alternating-offer bargaining game G(δ, γ1 , γ2 ) with δ ∈
(δ, δ̄), a strategy si ∈ Ri∞ (δ, γ1 , γ2 ) if and only if (after any history h) player i offers
x−i = [δ(1−δ)+δγi −γ−i ]·(1−δ 2 )−1 , accepts all xi > [δ(1−δ)+δγ−i −γi ]·(1−δ 2 )−1 ,
rejects all xi < [δ(1 − δ) + δγ−i − γi ] · (1 − δ 2 )−1 , i = 1, 2. Hence, the rationalizable
strategy profiles are not unique. One of them is Avery and Zemsky (1994) SPE,
where player 1 offers x2 = [δ(1−δ)+δγ1 −γ2 ]·(1−δ 2 )−1 at period n = 0, offer which
is accepted by player 2 at n = 1. But as in the case of time preferences with constant discount factors [see Vannetelbosch (1999a)], players may even play a strategy
profile leading to an agreement reached with delay or to perpetual disagreement.d
For δ ∈ [δ̄, 1), the SPE outcome is ((1,0),1) if γ1 < γ2 and ((1 − δ + γ2 , δ − γ2 ), 1)
if γ1 > γ2 .
However, from Lemma 4.1, if the players are sufficiently patient (δ ∈ [δ̄, 1)) then
there exists a finite order k ∗∗ up to which, mutual knowledge of rationality implies
that the two players always reach an agreement on a SPE partition and no later
than in period n = 2.
Proposition 4.1. Consider the alternating-offer bargaining game G(δ, γi , γj ) with
γi < γj (i, j = 1, 2) and δ ∈ [δ̄, 1). There exists k ∗∗ ∈ N such that for all k ≥ k ∗∗ ,
si ∈ Rik (δ, γi , γj ) if and only if player i offers xj = 0, accepts all xi > δ − γi , rejects
all xi < δ − γi ; sj ∈ Rjk (δ, γi , γj ) if and only if player j offers xi = δ − γi , accepts
all xj ∈ [0, 1].
Corollary 4.1. Consider the alternating-offer bargaining game G(δ, γ1 , γ2 ) with
γ1 6= γ2 and δ ∈ [δ̄, 1). If γ1 < γ2 then θ = ((1, 0), 1) is the unique rationalizable
outcome. If γ1 > γ2 then the rationalizable outcomes are both θ = ((1 − δ + γ2 , δ −
γ2 ), 1) and θ = ((0, 1), 2).
Thus, if the players are sufficiently patient and have time preferences given by
(1), common knowledge of rationality is not necessary to reach always an agreement on a SPE partition. It is possible to derive an explicit expression for k ∗∗ .
d Player 2’s expected payoff is equal to [δ(1 − δ) + δγ − γ ](1 − δ 2 )−1 . Hence, player 2 may
1
2
reject an offer x2 = [δ(1 − δ) + δγ1 − γ2 ](1 − δ2 )−1 , hoping that his counter-offer (x1 , x2 ) =
[δ(1 − δ) + δγ2 − γ1 , 1 − δ − δγ2 + γ1 ](1 − δ2 )−1 will be accepted next period. A similar reasoning
can be made for player 1.
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For γ̄ 6= γ,
(
∗∗
k (δ, γ1 , γ2 ) =
Ψ − mod[Ψ, 1] + 1
if mod[Ψ, 1] 6= 0
Ψ
otherwise
where Ψ = (ln[−δ(1 − δ) − δγ + γ̄] − ln[δ(1 − δ) + δγ̄ − γ] + 3 ln δ) · (2 ln δ)−1 . As
δ → 1− , we have Ψ → Φ (using l’Hopital’s rule), k ∗∗ (δ, γ1 , γ2 ) → k ∗ (γ1 , γ2 ) and
Rk (δ, γ1 , γ2 ) → Rk (γ1 , γ2 ).
Acknowledgment
We wish to thank an associate editor and two anonymous referees for helpful
comments. The research of Vincent Vannetelbosch has been made possible by
a fellowship of the Basque Country government. Financial support from the research projects PI-1998-48 (Basque Country government) and G17-99 (UPV-EHU)
is gratefully acknowledged.
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