BARGAINING ON NETWORKS
MIHAI MANEA
Department of Economics, Harvard University, Cambridge, MA 02138
mmanea@fas.harvard.edu
First Version: November 2007
Abstract. We study an infinite horizon game in which pairs of players connected by a
network are randomly matched to bargain over a unit surplus. We prove that for each
discount factor all equilibria are payoff equivalent. The equilibrium payoffs and the set of
equilibrium agreement links converge as players become increasingly patient. We construct
an algorithm that determines the limit equilibrium payoffs by iterating the finding that
players with extreme limit equilibrium payoffs form oligopoly subnetworks. An oligopoly
subnetwork consists of a set of mutually estranged players and their bargaining partners;
in equilibrium, for high discount factors, the partners act as an oligopoly for the mutually estranged players. In the equilibrium limit, surplus within an oligopoly subnetwork
is divided according to the shortage ratio of the mutually estranged players with respect
to their partners, with all players on each side receiving identical payoffs. The algorithm
is used to characterize equitable networks, stable networks, and non-discriminatory buyerseller networks. The results extend to heterogeneous discount factors and general matching
technologies.
1. Introduction
Networks are important in many economic and social interactions. Bilateral connections
in a network facilitate trade, cooperation, investment, information transmission and other
useful activities. The network structure determines the outcomes of such transactions and
relationships. We investigate how the network structure influences the outcomes of decentralized bilateral bargaining. Our bargaining model is tractable and provides insights into
the relative strengths of the positions in the network. The equilibrium payoffs deliver an
index of bargaining power in networks.
Date: May 26, 2008.
I thank Drew Fudenberg for continual guidance in approaching this project.
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MIHAI MANEA
Consider a network where each pair of players connected by a link can jointly produce
a unit surplus. The network generates the following infinite horizon bargaining game in
discrete time. Each period a link is randomly selected, and one of the two matched players
is randomly chosen to make an offer to the other player specifying a division of the unit
surplus between themselves. If the offer is accepted, the two players leave the game with the
shares agreed on. We make the following steady state assumption. In the next period the
two players who reached agreement are replaced in the network by two new players who are
identical copies of the agreeing pair, in the sense that they inherit exactly the same position
and links in the network. If the offer is rejected, the two players stay in the game for the
next period. All players share a common discount factor.
The justification for the steady state assumption is twofold. First, many decentralized
economies are not depleted over time. A series of agents take similar positions in transactions and relationships at different points in time, and the state of many markets does
not change significantly over long time horizons. Second, we are interested in separating
the incentives to reach agreements speedily due to the anxiety of impermanent bargaining
opportunities from the simultaneous determination of bargaining strength in stationary bargaining environments. In numerous circumstances agents face distributions of bargaining
opportunities that are stationary over time, and only have to decide who the most favorable
bargaining partners are. In equilibrium the entire network of connections determines the
agreements that are perpetually feasible.
Examples where our modeling is appropriate include markets for durable goods (e.g., tools
and hardware, used cars), housing and parking spaces in a city, sporadic labor (e.g., moving,
house painting, farming), manufacturing (e.g., subcontracting the production or assembling
of specific items), joint business ventures, and marriage. In these examples the links may be
determined by geographical proximity, social relationships, moral or political principles, joint
business opportunities, lexicographic preferences with a binary first component indicating
whether the bargaining partner is acceptable (depending on the cost of communication or
transportation, brand name, skill level, physical appearance, etc.), and the second component
equal to the benefit from the interaction. The steady state assumption is sensible in instances
of the applications above where the network structure does not change significantly over time.
BARGAINING ON NETWORKS
1
3
1
2
4
5
3
3
2
4
5
Figure 1. Networks G1 , G2
1.1. Outline of the paper. We assume that all players have perfect information about all
the events preceding any of their decision nodes in the game. The equilibrium concept we
adopt is subgame perfect Nash equilibrium. The insensitivity of the results to the information
structure and the equilibrium requirements is addressed in Section 3.
In Section 3, we prove that for every discount factor all subgame perfect equilibria are
(expected) payoff equivalent (Theorem 1). For all but a finite number of discount factors,
there exists a partition of the set of links into equilibrium agreement and disagreement links
(Theorem 2). In every subgame perfect equilibrium, after any history, a pair of players
connected through an equilibrium agreement link reaches an agreement when matched to
bargain, and the division agreed on is uniquely determined by the positions in the network of
the proposer and the responder. Players connected through disagreement links never reach
agreements when matched to bargain.
We prove that there exists a limit equilibrium agreement network that describes the set of
agreement links in every subgame perfect equilibrium for sufficiently high discount factors
(Theorem 3). Also, there is a limit equilibrium payoff vector to which the equilibrium payoffs
converge as the discount factor converges to 1.
Consider the network G1 with 5 players drawn in Figure 1. For every discount factor
there exists a unique equilibrium, with agreement network defined by the entire G1 . In
equilibrium, every match ends in agreement because players 4 and 5 cannot be monopolized
by either player 1 or 2. The limit equilibrium payoff for players 1 and 2 equals 3/5, and for
players 3, 4, and 5 equals 2/5. The limit equilibrium agreement network coincides with G1 .
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MIHAI MANEA
1
3
2
4
5
Figure 2. Network G∗2
Consider next the network G2 , obtained from G1 by removing the link (2, 4). For low
discount factors there exists a unique equilibrium, and the agreement network is the entire
G2 . However, for high discount factors players 1 and 5 do not reach agreement when matched
to bargain in equilibrium. The intuition is that player 1 can extort players 3 and 4, since
these two players do not have other bargaining partners. Player 1 cannot extract as much
surplus from 5, since player 5 has monopoly over the bargaining opportunities of player 2.
The limit equilibrium payoff for player 1 equals 2/3, for players 3 and 4 equals 1/3, and
for players 2 and 5 equals 1/2. The limit equilibrium agreement network G∗2 is depicted in
Figure 2. The equilibria of the bargaining games on the networks G1 and G2 for all discount
factors are described in Example 1 in Section 3.
Our main goal is to determine the limit equilibrium payoff vector for every network. The
following essential observation is presented in Section 4. Consider a set of players for which
no pair is connected by a limit equilibrium agreement link, and the set of players with
whom these players share limit equilibrium agreement links. We refer to the players in
the first group as mutually estranged, and to the ones in the second group as partners.1
Basically, for high discount factors, the partners have control over the relevant bargaining
opportunities of the mutually estranged players, and are able to corner and extort them in
equilibrium. Since for high discount factors the estranged players can only reach bargaining
agreements in pairwise matchings with their partners, the mutually estranged set is weak if
1In order to illustrate the definition,
note that the list of all pairs of mutually estranged sets and corresponding
partner sets in the limit equilibrium agreement network G∗2 for the bargaining game on the network G2 is
({1}, {3, 4}), ({2}, {5}), ({3}, {1}), ({4}, {1}), ({5}, {2}), ({1, 2}, {3, 4, 5}), ({1, 5}, {2, 3, 4}), ({2, 3}, {1, 5}),
({2, 4}, {1, 5}), ({3, 4}, {1}), ({3, 5}, {1, 2}), ({4, 5}, {1, 2}), ({2, 3, 4}, {1, 5}), and ({3, 4, 5}, {1, 2}).
BARGAINING ON NETWORKS
5
the corresponding partners are relatively scarce. The shortage ratio, describing the relative
strength of a mutually estranged set, is defined as the ratio of the number of partners to the
number of estranged players.
For example, in the network G1 the shortage ratios of the mutually estranged sets {3, 4}
and {3, 4, 5} are 1 and 2/3, respectively, since the partner set corresponding to either set
is {1, 2}. In the network G2 the shortage ratios of the mutually estranged sets {3, 4} and
{3, 4, 5} are 1/2 and 2/3, respectively, since the corresponding partner sets are {1} and {1, 2},
respectively. The determination of the partners for the mutually estranged sets considered
here is based on the limit equilibrium agreement subnetworks for the bargaining games on
the networks G1 and G2 described above.
We prove that for every set of mutually estranged players and their partners the ratio of
the limit equilibrium payoffs of the worst-off estranged player and the best-off partner is not
larger than the shortage ratio of the mutually estranged set (Theorem 4). The proof is based
on the following key observation. A player’s equilibrium payoff equals the expected present
value of his stream of first mover advantage. Since the first mover advantage in a bilateral
encounter is symmetric for the two players, the sum of the equilibrium payoffs of every set
of mutually estranged players is not larger than the sum of the equilibrium payoffs of the
corresponding partners. The result yields an upper bound for the limit equilibrium payoff
of the worst-off estranged player and a lower bound for the limit equilibrium payoff of the
best-off partner.
The bounds on limit equilibrium payoffs corresponding to a set of mutually estranged players and their partners need to be binding unless the worst-off estranged player is part of an
even weaker mutually estranged set, and the best-off partner is part of an even stronger partner set. In particular, the bounds are binding for the largest cardinality mutually estranged
set achieving the lowest shortage ratio, and the corresponding partner set. We prove that the
two sets of players have extremal limit equilibrium payoffs, and induce a limit equilibrium
oligopoly subnetwork enclosing all links on which they can reach bargaining agreements in
equilibrium for high discount factors. In the limit, the surplus within the subnetwork is
divided according to the shortage ratio of the mutually estranged players with respect to
their partners, with all players on each side receiving identical payoffs (Theorem 5).
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MIHAI MANEA
The partners of a mutually estranged set with the lowest shortage ratio corner and extort
the estranged players, acting as an oligopoly in the equilibrium limit. The limit equilibrium
payoffs for the networks G1 and G2 are obtained by computing that the lowest shortage ratio
in G1 is 2/3, attained for the mutually estranged set {3, 4, 5} with oligopoly {1, 2}, while in
G2 it is 1/2, attained for the mutually estranged set {3, 4} with oligopoly {1}.
Section 5 defines an algorithm that sequentially determines the limit equilibrium payoffs
of all players based on the ideas above. An important observation, which ensures that the
algorithm identifies and removes all residual players with extremal limit equilibrium payoffs
simultaneously, is that the set of mutually estranged sets with the lowest shortage ratio
is closed with respect to unions, as long as the lowest shortage ratio is less than 1. At
each step, the algorithm determines the union of all the mutually estranged sets with the
lowest shortage ratio, and removes the corresponding estranged players and their partners.
Within the identified extremal oligopoly subnetwork surplus is divided between the two sides
according to the shortage ratio. The algorithm stops when the lowest shortage ratio is larger
than or equal to 1, corresponding to limit equilibrium payoffs for the remaining players of
1/2, or when all players have been removed.
Section 6 employs the algorithm to characterize the class of equitable networks, i.e., networks for which the limit equilibrium payoffs of all players are identical (equal to 1/2). A
network is equitable if and only if it is quasi-regularizable; another equivalent condition is
that the network can be covered by a match and odd cycles disjoint union (Theorem 6).
In Section 7, we apply the algorithm to the analysis of networks where players could
benefit in the equilibrium limit from forming new links or severing existing ones. A network
is unilaterally stable if no player benefits from severing one of his links. A network is pairwise
stable if it is unilaterally stable and no pair of players benefit from forming a new link. We
prove that every network is unilaterally stable, but only equitable networks are pairwise
stable, with respect to the bargaining limit equilibrium payoff (Theorem 7). The same
conclusions hold for approximate stability with respect to the equilibrium payoffs for high
discount factors smaller than 1 (Corollary 2).
In Section 8, we show that restricting attention to buyer-seller networks permits a more
transparent statement of the characterization of limit equilibrium oligopoly subnetworks
and a more straightforward definition of the algorithm used to compute the limit equilibrium
BARGAINING ON NETWORKS
7
payoffs (Theorem 5BS ). Limit equilibrium payoffs of 1/2 play no special role in the results for
buyer-seller networks. We also study non-discriminatory buyer-seller networks, i.e., networks
for which the limit equilibrium payoffs of all buyers are identical. If the ratio of the numbers
of buyers and sellers is an integer, then the network is non-discriminatory if and only if it can
be covered by a disjoint union of clusters formed by one seller connected to a number of buyers
equal to the buyer-seller ratio (Theorem 8). We adapt the definition of pairwise stability
to buyer-seller networks and show that a buyer-seller network is two-sided pairwise stable
with respect to the bargaining limit equilibrium payoff if and only if it is non-discriminatory
(Theorem 7.iiBS ).
Section 9 extends the main results to the cases of heterogeneous discount factors and
general matching technologies. Section 10 discusses the related literature.
2. Framework
Let N denote the set of n players, N = {1, 2, . . . , n}. A network is an undirected
graph H = (V, E) with set of vertices V ⊂ N and set of edges (also called links) E ⊂
{(i, j)|i 6= j ∈ V } such that (j, i) ∈ E whenever (i, j) ∈ E. We identify the pairs (i, j) and
(j, i), and use the shorthand ij or ji instead. We say that player i is connected in H to
player j, or i has an H link to j, if ij ∈ E. We often abuse notation and write ij ∈ H for
ij ∈ E. A network H 0 = (V 0 , E 0 ) is a subnetwork of H if V 0 ⊂ V and E 0 ⊂ E. A network
H 0 = (V 0 , E 0 ) is the subnetwork of H induced by V 0 if E 0 = E ∩ (V 0 × V 0 ). Let G be a fixed
network with vertex set N . A link ij in G is interpreted as the ability of players i and j to
jointly generate a unit surplus.2
Consider the following infinite horizon bargaining game generated by the network G.
Each period t = 0, 1, . . . a link ij in G is selected randomly (with equal probability),3 and
one of the players (the proposer) i and j is chosen randomly (with equal probability) to make
an offer to the other player (the responder) specifying a division of the unit surplus between
themselves. If the responder accepts the offer, the two players exit the game with the shares
agreed on. In period t + 1 players i and j are replaced in the network by two new players
who are identical copies of i and j, in the sense that they inherit exactly the same links as i
2For
3The
simplicity, we assume everywhere except Section 7 that each player has at least one G link.
analysis is not sensitive to the specification of the matching technology. See Subsection 9.2.
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MIHAI MANEA
and j, respectively. If the responder rejects the offer, the two players remain in the game for
the next period. At time t + 1 the game is repeated with the set of n players, consisting of
the ones from time t, with the agreeing pair replaced by a new identical pair in case period
t ends in agreement. All players share a common discount factor δ ∈ (0, 1). The game is
denoted Γδ .
Formally, there exists a sequence i0 , i1 , . . . , iτ , . . . of players of type i ∈ N (a player’s type
is defined by his position in the network). When player iτ exits the game (following an
agreement with another player), player iτ +1 replaces him for the next period.4 We assume
that all players have perfect information of all the events preceding any of their decision
nodes in the game. Possible relaxations of the information structure are discussed in the
next section.
There are three types of histories. We denote by ht a history of the game up to (not
including) time t, which is a sequence of t − 1 pairs of proposers and responders connected in
G, with an offer and a response corresponding to each proposer and responder respectively.
We call such histories, and the subgames that follow them (starting with a move by nature at
time t) complete. For simplicity, we assume that for every history players are only labeled
by their type without reference to the index of their copy. The index τ of the copy of player
i present in the game at time t following the history ht can be recovered by counting the
number of bargaining agreements involving i in pairs of ht . Therefore, a history ht uniquely
determines the copy iτ of player i who is playing the game at time t, and when there is no
risk of confusion we refer to iτ as player i. We denote by H(iτ ) the set of complete histories,
or subgames, where iτ is the copy of player i present in the game at time t. We denote by
(ht ; i → j) the history consisting of the complete history ht followed by nature’s choice of
the link ij at time t, with player i to propose to player j. We denote by (ht ; i → j; x) the
history consisting of (ht ; i → j) followed by player i offering x ∈ [0, 1] to player j at time t.
4The
results translate to an alternative specification of the model in the spirit of Gale (1987). Suppose that
there exists a continuum of players of each type in N . A set of players of type i of measure µi is present in
the game at each period. The matching technology is such that, for each link ij, measure µij of players i
are matched to bargain with oneP
of the players j, through random selection of proposer. It is assumed that
µi > 0, µij = µji > 0 and µi > {j|ij∈G} µij . The probability that a player i is matched to a player j is
µij /µi , and no player is matched to more than one bargaining partner at once. The set of players of each
type who reach agreements is immediately replaced by a set of players of the same type of equal measure.
BARGAINING ON NETWORKS
9
A strategy σiτ for player iτ specifies, for all players j who are connected to i in G, and
all complete histories ht ∈ H(iτ ) the offer σiτ (ht ; i → j) that i makes to j,5 when nature
matches i to bargain with j, selecting i as the proposer, and the response σiτ (ht ; j → i; x)
that i makes to player j when nature matches i to bargain with j, selecting j as the proposer,
and j offers x to i. We allow for mixed strategies, hence σiτ (ht ; i → j) and σiτ (ht ; j → i; x)
are probability distributions over [0, 1] and {Yes, No}, respectively. We are interested in the
strategy profiles (σiτ )i∈N,τ ≥0 that are subgame perfect Nash equilibria, that is, induce
Nash equilibria in subgames following every history ht , (ht ; i → j), and (ht ; i → j; x).
3. Essential Equilibrium Uniqueness and Discounting Asymptotics
We first prove that in any subgame of the bargaining game the expected payoff of each
active player is uniquely determined by his location in the network for all subgame perfect
equilibria. Hence, the equilibrium payoffs for each player type are unique and stationary over
time, and may be used to describe the possible equilibrium outcomes in every bargaining
encounter.
Theorem 1. For every δ ∈ (0, 1), there exists a payoff vector (vi∗δ )i∈N such that in every
subgame perfect equilibrium of Γδ the expected payoff of player iτ at the beginning of any
H(iτ ) subgame is uniquely given by vi∗δ for all i ∈ N, τ ≥ 0. For every δ ∈ (0, 1), in any
equilibrium of Γδ , in any subgame where iτ is selected to make an offer to jτ 0 , the following
statements are true with probability one: (1) if δ(vi∗δ + vj∗δ ) < 1 then iτ offers δvj∗δ and jτ 0
accepts; (2) if δ(vi∗δ + vj∗δ ) > 1 then iτ makes an offer that jτ 0 rejects.
We can extend the conclusions of Theorem 1 to settings in which players do not have perfect
information about all past bargaining encounters. It may be that players only know their own
history of interactions, or know the history of all pairs matched to bargain but only see the
outcomes in the rounds they bargain. Players who reach an agreement and exit the game may
pass down information to the players who inherit their position in the network. Some players
may be informed of the exact identity of their bargaining partner, or only about his position
in the network. In such settings, players need to form beliefs about the bargaining outcomes
that are not revealed to them. The extension of the definition of sequential equilibrium
5Recall
the convention to refer to the copy iτ (jτ 0 ) of i (j) present in the game following the history ht as i
(j), when ht is clear from the context.
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MIHAI MANEA
(Kreps and Wilson 1982) to the bargaining game, and the generalization of the proof of
Theorem 1 accommodating various information structures are straightforward.
The proof of Theorem 1 can be extended to show uniqueness of the security equilibrium (Binmore and Herrero 1988b) payoffs. The definition of the security equilibrium is
related to the notion of rationalizability introduced by Bernheim (1984) and Pearce (1984).
The requirement that it be common knowledge that a player never uses a strategy that is
not a best response to a rational strategy profile of his opponents is replaced by a similar
requirement concerning security levels. The definition of security equilibrium requires common knowledge of the fact that no player takes an action under any contingency that yields
a payoff smaller than his security level for that contingency. The requirements of security
equilibrium are weaker than the ones of sequential equilibrium.
A steady state strategy σi for player i is defined on {i → j|ij ∈ G} ∪ {(j → i; x)|ij ∈
G, x ∈ [0, 1]}, with σi (i → j) and σi (j → i; x) probability distributions over [0, 1] and
{Yes, No}, respectively. A strategy profile (σiτ )i∈N,τ ≥0 is steady state stationary if each
player iτ ’s strategy at any time t depends exclusively on i and the play of the game in round
t, that is, there exists a steady state strategy profile (σi )i∈N such that σiτ (ht ; i → j) =
σi (i → j) and σiτ (ht ; j → i; x) = σi (j → i; x) for all iτ and ht ∈ H(iτ ). A steady state
stationary subgame perfect equilibrium is a steady state stationary strategy profile
that is a subgame perfect equilibrium.
The following two lemmas are essential to the proof. Lemma 1 is a simple algebraic
observation, and Lemma 2 is the statement of the first part of Theorem 1 restricted to
steady state stationary subgame perfect equilibria. Lemma 1 is used in the proofs of both
Lemma 2 and Theorem 1.
Lemma 1. For all w1 , w2 , w3 , w4 ∈ R,
| max(w1 , w2 ) − max(w3 , w4 )| ≤ max(|w1 − w3 |, |w2 − w4 |).
Proof. Suppose w1 = max(w1 , w2 , w3 , w4 ). Then
| max(w1 , w2 )−max(w3 , w4 )| = w1 −max(w3 , w4 ) ≤ w1 −w3 = |w1 −w3 | ≤ max(|w1 −w3 |, |w2 −w4 |).
The proof is similar for the cases when w2 , w3 , or w4 equals max(w1 , w2 , w3 , w4 ).
BARGAINING ON NETWORKS
11
Lemma 2. For every δ ∈ (0, 1), there exists a payoff vector (ṽi∗δ )i∈N such that in every steady
state stationary subgame perfect equilibrium of Γδ the expected payoff of iτ at the beginning
of any H(iτ ) subgame is uniquely given by ṽi∗δ for all i ∈ N, τ ≥ 0.
The proof of Lemma 2 is provided in the Appendix. The following is a sketch. Let
(σiτ )i∈N,τ ≥0 be a steady state stationary subgame perfect equilibrium of Γδ . For every i,
each player iτ receives the same expected payoff, ṽi , in any subgame ht ∈ H(iτ ) when the
game is played according to (σiτ )i∈N,τ ≥0 . We argue that (ṽi )i∈N is a fixed point of the function
f δ = (f1δ , f2δ , . . . , fnδ ) : [0, 1]n → [0, 1]n defined by
fiδ (v) =
1 X
2e − ei
δvi +
max(1 − δvj , δvi ),
2e
2e
{j|ij∈G}
where e denotes the total number of links in G and ei denotes the number of links player i
has. Next we use Lemma 1 to show that f δ is a contraction with respect to the sup norm
on Rn , hence it has a unique fixed point (ṽi∗δ )i∈N . Therefore, ṽ = ṽ ∗δ .
The set of steady state stationary pure strategy equilibria of Γδ is non-empty. The following
is an element. When ij ∈ G, and player iτ is selected to make a proposal to jτ 0 , if δ(ṽi∗δ +
ṽj∗δ ) ≤ 1, then iτ offers δṽj∗δ to jτ 0 ; if δ(ṽi∗δ + ṽj∗δ ) > 1, then iτ offers 0 to jτ 0 . Player iτ
accepts any offer at least as large as δṽi∗δ , and rejects any offer smaller than δṽi∗δ , from any
jτ 0 . Uniqueness of the steady state stationary subgame perfect equilibrium payoffs does not
imply uniqueness of the equilibrium strategies. The following modification of the equilibrium
description above yields a continuum of (payoff equivalent) steady state stationary subgame
perfect equilibria. When player iτ is selected to make a proposal to jτ 0 and δ(ṽi∗δ + ṽj∗δ ) > 1,
we can replace iτ ’s offer to jτ 0 by any mixed strategy over the entire interval [0, δṽj∗δ ) or
[0, δṽj∗δ ] depending on whether jτ 0 ’s strategy is to accept an offer of δṽj∗δ from iτ with positive
probability.
Proof of Theorem 1. Consider the (non-empty) set of all possible subgame perfect equilibria
of Γδ (including ones that are not steady state stationary). For each i ∈ N , let v δi and v δi be
the infimum and respectively supremum of the expected payoff of iτ in any subgame H(iτ )
over all τ ≥ 0 over all possible subgame perfect equilibria of Γδ .6
6The
approach is similar to the Shaked and Sutton (1984) proof of equilibrium uniqueness in the Rubinstein
(1982) alternating offer bargaining game. For our bargaining game, the steps are complicated by the a priori
12
MIHAI MANEA
Fix a subgame perfect equilibrium. No copy of player j will accept an offer smaller than
δv δj , so i can get a payoff of at most 1−δv δj from an agreement with j when he is the proposer.7
Player i accepts any offer larger than δv δi since he receives at most δv δi in the continuation
subgame after a rejection, so no player j can offer him more than δv δi in equilibrium. If there
exists no agreement involving i in some period, his continuation payoff is at most δv δi . It
follows that for all τ ≥ 0 the expected payoff viτ of iτ in any subgame H(iτ ) in any subgame
perfect equilibrium satisfies
viτ ≤
1 X
2e − ei δ
δv i +
max(1 − δv δj , δv δi ).
2e
2e
{j|ij∈G}
By the definition of v δi , the inequality above implies that
v δi ≤
(3.1)
2e − ei δ
1 X
δv i +
max(1 − δv δj , δv δi ).
2e
2e
{j|ij∈G}
Consider the following deviation for player i from the equilibrium strategy profile. Player
i offers δv δj + ε (ε > 0) to any player j such that δv δi + δv δj + ε ≤ 1. Since each player j
receives at most δv δj in the continuation subgame, the offer is accepted in a subgame perfect
equilibrium. Player i makes unreasonable offers, say offers zero, to all other players, and
rejects any offer he receives. It must be that for all τ ≥ 0 the expected payoff viτ of iτ in any
subgame H(iτ ) in any subgame perfect equilibrium is not smaller than the expected payoff
from the deviation,
viτ ≥
2e − ei δ
1 X
max(1 − δv δj − ε, δv δi ), ∀ε > 0,
δv i +
2e
2e
{j|ij∈G}
and taking the limit ε → 0,
viτ ≥
2e − ei δ
1 X
δv i +
max(1 − δv δj , δv δi ).
2e
2e
{j|ij∈G}
By the definition of v δi , the inequality above implies that
(3.2)
v δi ≥
2e − ei δ
1 X
δv i +
max(1 − δv δj , δv δi ).
2e
2e
{j|ij∈G}
lack of a conjecture about what pairs of players reach agreements in equilibrium when the network structure
determines the possible bargaining partners of each player.
7Throughout this argument i (j) should be read as “copy of i (j).”
BARGAINING ON NETWORKS
13
Let D = maxk∈N v δk − v δk . If i ∈ arg maxk∈N v δk − v δk , then from 3.1, 3.2, and Lemma 1,
D = v δi − v δi ≤
2e − ei
1 X
δ(v δi − v δi ) +
(max(1 − δv δj , δv δi ) − max(1 − δv δj , δv δi ))
2e
2e
{j|ij∈G}
≤
2e − ei
1 X
δD +
max(|1 − δv δj − (1 − δv δj )|, |δv δi − δv δi |)
2e
2e
{j|ij∈G}
≤
1 X
2e − ei
δD +
δ max(v δj − v δj , v δi − v δi )
2e
2e
{j|ij∈G}
≤ δD.
Since D ≥ 0 and δ ∈ (0, 1), it follows that D = 0. Therefore, v δk = v δk for all k ∈ N .
Since v δi = v δi for all i ∈ N , 3.1 and 3.2 imply that
v δi =
2e − ei δ
1 X
δv i +
max(1 − δv δj , δv δi ), ∀i ∈ N,
2e
2e
{j|ij∈G}
which means that v δ is a fixed point of f δ , hence v δ = v δ = ṽ ∗δ . It follows that for all
τ ≥ 0, iτ receives the same expected payoff, equal to ṽi∗δ =: vi∗δ , in any subgame H(iτ ) of
any subgame perfect equilibrium.
We now prove the second half of the theorem. Consider any subgame perfect equilibrium.
The expected payoff for every iτ in any subgame H(iτ ) is vi∗δ . It must be that in any
subgame any player jτ 0 accepts any offer larger than δvj∗δ and rejects any offer smaller than
δvj∗δ . As in the arguments above, if δ(vi∗δ + vj∗δ ) < 1, when iτ is chosen to make an offer
to jτ 0 , in equilibrium it must be that with probability one iτ offers δvj∗δ and jτ 0 accepts. If
δ(vi∗δ + vj∗δ ) > 1, when iτ is chosen to make an offer to jτ 0 , iτ ’s offer is rejected by jτ 0 in
equilibrium with probability one (the rejected offer cannot be pinned down by the equilibrium
requirements, as in the discussion about non-uniqueness of steady state stationary subgame
perfect equilibria).
The following description of the equilibria in two simple networks illustrates the conclusions
of Theorem 1. Equilibria in less trivial networks are analyzed in Example 2.
14
MIHAI MANEA
Example 1. Consider the network G1 illustrated in Figure 1. The equilibrium payoffs in Γδ
are
v1∗δ =
150 − 250δ + 103δ 2
,
5(100 − 220δ + 158δ 2 − 37δ 3 )
v2∗δ =
100 − 160δ + 63δ 2
5(100 − 220δ + 158δ 2 − 37δ 3 )
v3∗δ =
2(25 − 40δ + 16δ 2 )
,
5(100 − 220δ + 158δ 2 − 37δ 3 )
v4∗δ = v5∗δ =
100 − 165δ + 67δ 2
,
5(100 − 220δ + 158δ 2 − 37δ 3 )
converging to v1∗ = v2∗ = 3/5 and v3∗ = v4∗ = v5∗ = 2/5 as δ → 1. There exists a unique
equilibrium in which when i is selected to propose to j he offers δvj∗δ , and when i has to
respond to a proposal he accepts any offer of δvi∗δ or more, and rejects any smaller offers. In
equilibrium, every match ends in agreement.
Consider the network G2 illustrated in Figure 1. The equilibrium payoffs when players
√
have discount factor δ ≤ 10(9 − 2)/79 ≈ 0.9602 =: δ are
v1∗δ =
300 − 520δ + 223δ 2
,
5(200 − 460δ + 346δ 2 − 85δ 3 )
v2∗δ =
100 − 160δ + 63δ 2
5(200 − 460δ + 346δ 2 − 85δ 3 )
v3∗δ = v4∗δ =
2(50 − 85δ + 36δ 2 )
,
5(200 − 460δ + 346δ 2 − 85δ 3 )
v5∗δ =
2(100 − 170δ + 71δ 2 )
.
5(200 − 460δ + 346δ 2 − 85δ 3 )
For δ < δ, there exists a unique equilibrium in which when i is selected to propose to j
he offers δvj∗δ , and when i has to respond to a proposal he accepts any offer of δvi∗δ or more,
and rejects any smaller offers. The inequality δ(vi∗δ + vj∗δ ) < 1 holds for all links ij.
A payoff irrelevant equilibrium multiplicity appears at δ = δ. For δ = δ, it is true that
δ(v1∗δ +v5∗δ ) = 1. The strategy profile described above continues to be an equilibrium. We can
modify the strategy of player 1 for any subgame where he is matched to bargain with player
5 to make an offer that is any probability distribution over [δv5∗δ , 1], and reject any offer
smaller than δv1∗δ , accept with any probability an offer of δv1∗δ , and accept with probability 1
any offer larger than δv1∗δ . We can modify the strategy of player 5 when matched to bargain
with player 1 analogously. We obtain a set of equilibria that are payoff equivalent.
The equilibrium payoffs when players have discount factor δ > δ are
v1∗δ =
2
1
1
, v3∗δ = v4∗δ =
, v2∗δ = v5∗δ =
,
10 − 7δ
10 − 7δ
2(5 − 4δ)
converging to v1∗ = 2/3, v3∗ = v4∗ = 1/3 and v4∗ = v5∗ = 1/2 as δ → 1.
For δ > δ there exists one equilibrium in which when i is chosen to propose to j he offers
δvj∗δ , and when i has to respond to a proposal he accepts any offer of at least δvi∗δ and rejects
BARGAINING ON NETWORKS
15
any smaller offers. The equilibrium is not unique. For example, the strategy of player 1
when matched to bargain with player 5 may be changed to any (mixed) offer smaller than
δv5∗δ , as rejection obtains regardless. However, for δ > δ in any equilibrium the strategies
for the selection of links (1,3), (1,4), and (2,5) have to be as in the equilibrium specification
above.
We call (vi∗δ )i∈N the equilibrium payoff vector at δ. The equilibrium agreement
network at δ, G∗δ , is defined as the subnetwork of G with the link ij included if and
only if δ(vi∗δ + vj∗δ ) ≤ 1. For δ such that δ(vi∗δ + vj∗δ ) 6= 1, ∀ij ∈ G, the agreements and
disagreements in every round of any subgame perfect equilibrium are entirely characterized
as in the statement of Theorem 1. We show that the condition δ(vi∗δ +vj∗δ ) 6= 1, ∀ij ∈ G holds
for all but a finite set of discount factors δ, hence the description of equilibrium agreements
and disagreements is complete generically.
Theorem 2. The condition δ(vi∗δ + vj∗δ ) 6= 1, ∀ij ∈ G holds for all but a finite set of δ.
The proof appears in the Appendix. We outline the approach here since some of the ideas
are used in the proof of the next result as well.
For every δ ∈ (0, 1) and every subnetwork H of G, consider the n × n linear system
2e − eH
1 X
i
(3.3)
vi =
δvi +
(1 − δvj ), ∀i = 1, n,
2e
2e ij∈H
where eH
i denotes the number of H links that player i has (e still denotes the total number of
links in G as in the proof of Theorem 1). We argue that v ∗δ solves the system for H = G∗δ .
∗δ
It is easy to check that the system 3.3 has a unique solution v δ,H . In particular, v ∗δ = v δ,G .
All entries in the augmented matrix of the linear system 3.3 are linear functions of δ.
Then for each i ∈ N the unique solution viδ,H is given by Cramer’s rule, as the ratio of two
determinants that are polynomials in δ of degree at most n,
(3.4)
viδ,H = PiH (δ)/QH
i (δ).
We can then argue that every δ for which there exist i, j with δ(vi∗δ + vj∗δ ) = 1 is a root of
one of a finite number of non-zero polynomials in δ.
Denote by ∆ the finite set of δ for which the condition δ(vi∗δ + vj∗δ ) 6= 1, ∀ij ∈ G does not
hold. As established in Theorem 1, for every δ 6∈ ∆, in every equilibrium of Γδ , after any
16
MIHAI MANEA
history where iτ is chosen to make an offer to jτ 0 , with probability one: (1) if ij ∈ G∗δ then
iτ offers δvj∗δ and jτ 0 accepts; (2) if ij 6∈ G∗δ then iτ makes an offer that jτ 0 rejects.
Theorem 3. (i) There exist δ ∈ (0, 1) and a subnetwork G∗ of G such that the equilibrium
agreement network G∗δ of Γδ equals G∗ for all δ > δ. (ii) The equilibrium payoff vector v ∗δ
of Γδ converges to a payoff vector v ∗ ∈ [0, 1]n as δ tends to 1. The rate of convergence of v ∗δ
to v ∗ is O(1 − δ).
¯ of δ for which there exist a link
Proof. (i) The proof of Theorem 2 shows that the set ∆
¯ =: δ.
ij ∈ G and a subnetwork H of G such that δ(v δ,H + v δ,H ) = 1 is finite. Fix δ0 > max ∆
i
j
Let G∗ = G∗δ0 . We show that G∗δ = G∗ for all δ > δ.
∗
∗
Fix ij ∈ G. The function 1 − δ(viδ,G + vjδ,G ) is continuous in δ for δ ∈ (0, 1) as it is a
∗
rational function (ratio of two polynomials), by 3.4 (QG
i (δ) 6= 0, as we established that
¯ Then the sign
the system is non-singular for all δ ∈ (0, 1)), and it has no roots δ outside ∆.
∗
∗
εδij of 1 − δ(viδ,G + vjδ,G ) is strict and constant for all δ > δ. In particular, εδij = εδij0 for δ > δ.
Since δ0 > δ and G∗ = G∗δ0 , the following conditions hold (1) εδij0 = 1 if and only if ij ∈ G∗ ,
and (2) εδij0 = −1 if and only if ij 6∈ G∗ . For all δ > δ we have εδij = εδij0 , hence the following
conditions must be true: (1) εδij = 1 if and only if ij ∈ G∗ , and (2) εδij = −1 if and only if
ij 6∈ G∗ . For δ > δ, it follows that G∗δ = G∗ .
∗
∗
(ii) Fix i ∈ N . It follows from part (i) that vi∗δ = PiG (δ)/QG
i (δ) for δ > δ. Rewrite
∗
∗
∗
∗
∗
∗
G
G
G
G
PiG (δ)/QG
i (δ) = P̄i (δ)/Q̄i (δ), with P̄i (δ) and Q̄i (δ) relatively prime polynomials.
∗
∗
G
Since vi∗δ ∈ [0, 1] for all δ ∈ (δ, 1), it must be that Q̄G
i (1) 6= 0. For, if Q̄i (1) = 0
∗
∗
∗
then P̄iG (δ) 6= 0 and the rational function P̄iG (δ)/Q̄G
i (δ) diverges at δ = 1. Consequently,
∗
∗
∗
∗
∗
∗
G
G
∗δ
G
G
P̄iG (δ)/Q̄G
i (δ) converges to P̄i (1)/Q̄i (1) as δ tends to 1. Therefore, vi = P̄i (δ)/Q̄i (δ)
∗
∗
has a finite limit, vi∗ := P̄iG (1)/Q̄G
i (1), at δ = 1.
∗
∗
To show that the rate of convergence of v ∗δ to v ∗ is O(1−δ), write vi∗δ −vi∗ = P̄iG (δ)/Q̄G
i (δ)−
∗
∗
∗
∗
∗
∗
∗
∗
G
G
G
G
G
G
P̄iG (1)/Q̄G
i (1). The rational function P̄i (δ)/Q̄i (δ) − P̄i (1)/Q̄i (1) = (Q̄i (1)P̄i (δ) −
∗
∗
∗
∗
G
G
P̄iG (1)Q̄G
i (δ))/(Q̄i (1)Q̄i (δ)) has a vanishing numerator and a non-vanishing denomina∗
∗
tor at δ = 1, hence it can be rewritten as (1 − δ)RiG (δ) where RiG is a rational function
with a finite limit at δ = 1.
BARGAINING ON NETWORKS
17
We call G∗ the limit equilibrium agreement network, and v ∗ the limit equilibrium
payoff vector. Our main goal is to determine the limit equilibrium payoff vector. We first
make two trivial observations.
Proposition 1. If ij ∈ G, then vi∗ + vj∗ ≥ 1. If ij ∈ G∗ , then vi∗ + vj∗ = 1. In particular, if
vi∗ + vj∗ > 1, then ij 6∈ G∗ .
Proof. Let ij ∈ G. If ij ∈ G \ G∗ , then
δ(vi∗δ + vj∗δ ) > 1,
for all δ > δ. Taking the limit as δ tends to 1 we obtain that vi∗ + vj∗ ≥ 1.
If ij ∈ G∗ , then since for i’s links in G∗ to k 6= j, the inequality 1 − δvk∗δ ≥ δvi∗δ holds,
vi∗δ
∗
2e − eG
1 X
2e − 1 ∗δ
1
i
(1 − δvk∗δ ) ≥
=
δvi∗δ +
δvi + (1 − δvj∗δ ),
2e
2e ik∈G∗
2e
2e
for all δ > δ. Taking the limit as δ tends to 1 we obtain that vi∗ + vj∗ ≥ 1.
In conclusion, vi∗ + vj∗ ≥ 1, ∀ij ∈ G. The other claims follow similarly.
Lemma 3. Every player has at least one link in G∗ .
Proof. If i ∈ N had no G∗ link, then vi∗δ = 0 for δ > δ, and since vj∗δ ≤ 1 for all j 6= i it
follows that δ(vi∗δ + vj∗δ ) < 1, and hence ij ∈ G∗ for all ij ∈ G, a contradiction.
4. Bounds for Limit Equilibrium Payoffs
For every network H, and a non-empty subset of players M , let LH (M ) denote the set of
players who have H-links to players in M , LH (M ) = {j|ij ∈ H, i ∈ M }. We say that M is
an H-independent set if there exists no H-link between two players in M , M ∩LH (M ) = ∅.
An H-independent set is mutually estranged in the network H, according to the terminology
of the introduction.
The next result is essential for finding an algorithm that determines the limit equilibrium
payoff vector. For every set M that is mutually estranged in the limit equilibrium agreement
∗
network G∗ , with partner set LG (M ), the ratio of the limit equilibrium payoffs of the worstoff estranged player, mini∈M vi∗ , and the best-off partner, maxj∈LG∗ (M ) vj∗ , is not larger than
∗
the shortage ratio, |LG (M )|/|M |.
18
MIHAI MANEA
Theorem 4. For every G∗ -independent set of players M , the following bounds on limit
equilibrium payoffs hold
∗
min vi∗ ≤
i∈M
max
vj∗ ≥
∗
j∈LG (M )
|LG (M )|
|M | + |LG∗ (M )|
|M |
.
|M | + |LG∗ (M )|
The proof of Theorem 4 is relegated to the Appendix. It uses the following simple fact.
Lemma 4. For every discount factor, the equilibrium payoff of any player is equal to the
expected present value of his stream of first mover advantage, i.e.,
vi∗δ =
X 1
1
max(1 − δvi∗δ − δvj∗δ , 0), ∀i ∈ N, ∀δ ∈ (0, 1).
1−δ
2e
{j|ij∈G}
Proof. Fix a discount factor δ, and a player i. The expected payoff of player i in any
subgame where he is not selected as the proposer is δvi∗δ . The expected payoff of player i in
any subgame where he is selected to make an offer to player j is max(1 − δvj∗δ , δvi∗δ ), which
represents a net gain of max(1 − δvi∗δ − δvj∗δ , 0) over δvi∗δ . Hence, max(1 − δvi∗δ − δvj∗δ , 0)
measures the net first mover advantage that i has when selected to make an offer to j.
Therefore, the expected loss from not being selected as a proposer at some round, measured
by (1 − δ)vi∗δ , equals the sum of the terms max(1 − δvi∗δ − δvj∗δ , 0) weighted by the probability
that i is selected to make a proposal to j at that round.
The intuition for the proof of Theorem 4 is as follows. Suppose that M is a G∗ -independent
set of players, and set L = LG∗ (M ). Fix a discount factor δ > δ, with δ specified as in
Theorem 3. Therefore, G∗δ = G∗ . In every equilibrium, in any subgame, player i only
reaches agreements with players in L with whom he shares G∗ links. Any net first mover
advantage corresponding to a player i in M making an offer to a player j in L is mapped
to an equal net first mover advantage corresponding to player j making an offer to player i
(max(1 − δvi∗δ − δvj∗δ , 0) is symmetric in i and j). It follows that the sum of the expected
present value of the stream of first mover advantage of all players in M is not larger than the
same expression evaluated for the players in L (players in M only gain first mover advantage
from players in L, while players in L may gain first mover advantage from the corresponding
BARGAINING ON NETWORKS
19
P
players in M , and possibly from players that are not in M ). Hence, by Lemma 4, j∈L vj∗δ ≥
P
P
P
∗δ
∗
∗
v
.
Taking
the
limit
δ
→
1
we
obtain
that
v
≥
i
j
i∈M
j∈L
i∈M vi . Then the proof is
finished by repeated use of Proposition 1 in order to obtain that mini∈M vi∗ + maxj∈L vj∗ = 1.
While Theorem 4 invokes knowledge we do not have a priori about the network G∗ , it has
an immediate corollary that involves exclusively properties of G. It is sufficient to note that
since G∗ is a subnetwork of G, for every set of players M , if M is G-independent then M is
also G∗ -independent, and LG∗ (M ) ⊂ LG (M ), so |LG∗ (M )| ≤ |LG (M )|.
Corollary 1. For every G-independent set of players M , the following bounds on limit
equilibrium payoffs hold
min vi∗ ≤
i∈M
max vj∗ ≥
j∈LG (M )
|LG (M )|
|M | + |LG (M )|
|M |
.
|M | + |LG (M )|
However, for the results coming next we need the full strength of Theorem 4.
5. Limit Equilibrium Payoff Computation
Theorem 4 suggests that it may be useful to study the G∗ -independent sets of players
∗
∗
M that minimize |LG (M )|/(|M | + |LG (M )|), or equivalently, minimize the shortage ratio
∗
|LG (M )|/|M |. The next lemma shows that the set of such minimizers is closed with respect
to unions if the attained minimum is less than 1. The proof, which is a conjunction of
combinatorial arguments and brute force algebra, is relegated to the Appendix. For every
network H let I(H) denote the set of non-empty H-independent sets.
Lemma 5. Let H be a network. Suppose that
|LH (M )|
< 1,
M ∈I(H)
|M |
min
and that M 0 and M 00 are two H-independent sets achieving the minimum. Then M 0 ∪ M 00 is
also H-independent, and
|LH (M )|
.
M ∈I(H)
|M |
M 0 ∪ M 00 ∈ arg min
20
MIHAI MANEA
We argue that the bounds on limit equilibrium payoffs corresponding to a set of mutually
estranged players and their partners provided by Theorem 4 need to be binding unless
the worst-off estranged player is part of an even weaker mutually estranged set, and the
best-off partner is part of an even stronger partner set. Nonetheless, each player is part
of a limit equilibrium oligopoly subnetwork where, as the discount factor converges to 1,
some mutually estranged players and their partners share the unit surplus according to the
shortage ratio through the links of the limit equilibrium agreement network. Consequently,
the limit equilibrium payoff of any player cannot be smaller than the upper bound for the
limit equilibrium payoff of the worst-off player from the mutually estranged set with the
lowest shortage ratio. Therefore, the bounds for the limit equilibrium payoffs of the worstoff estranged player and the best-off partner are binding in the case of the mutually estranged
set with the lowest shortage ratio.
Suppose that the lowest shortage ratio r1 = minM ∈I(G) |LG (M )|/|M | is smaller than 1.
Let M1 be the union of all mutually estranged sets M minimizing the shortage ratio. By
Lemma 5, M1 is also a mutually estranged set with minimal shortage ratio. Let L1 = LG (M1 )
be the corresponding set of partners. We argued above that mini∈M1 vi∗ = r1 /(r1 + 1) and
maxj∈L1 vj∗ = 1/(r1 +1). We set out to show that the limit equilibrium payoffs equal r1 /(r1 +1)
for all players in M1 and 1/(r1 + 1) for all players in L1 ; that is, all players in M1 ∪ L1 , not
only the worst-off estranged player and the best-off partner, have extremal limit equilibrium
payoffs. The following algorithm sequentially iterates this hypothesis in order to determine
the limit equilibrium payoffs of all players.
Definition 1 (Algorithm A(G) = (rs , xs , Ms , Ls , Ns , Gs )s=1,2,...,s ). Define the sequence
(rs , xs , Ms , Ls , Ns , Gs )s recursively as follows. Let N1 = N and G1 = G. For s ≥ 1, if
Ns = ∅ then let rs = 1 and stop. Else, let8
(5.1)
8It
rs =
|LGs (M )|
.
M ∈I(G)
|M |
min
M ⊂Ns ,
can be shown that each player in Ns has at least one Gs link if the algorithm does not stop by step s,
hence rs is well-defined and positive.
BARGAINING ON NETWORKS
21
If rs ≥ 1 then stop. Else, set xs = rs /(1 + rs ). Let Ms be the union of all minimizers M in
5.1.9 Denote Ls = LGs (Ms ). Let Ns+1 = Ns \ (Ms ∪ Ls ), and Gs+1 be the subnetwork of G
induced by the players in Ns+1 . Denote by s the finite step at which the algorithm stops.10
At each step, the algorithm A(G) determines the largest cardinality mutually estranged
set achieving the lowest shortage ratio in the subnetwork induced by the remaining players
(Lemma 5), and removes the corresponding estranged players and partners. We are going to
show that for high discount factors the removed players can only reach bargaining agreements
among themselves in equilibrium; the limit equilibrium oligopoly subnetwork they form
encloses all their possible limit agreement links. The definition ensures that A(G) identifies
and removes all residual players with extremal limit equilibrium payoffs simultaneously. The
algorithm stops when every mutually estranged set formed by the remaining players has
shortage ratio larger than or equal to 1 or when all players have been removed.11
The next lemma, which is used in the proofs of Proposition 2 and Theorem 5 below, follows
immediately from the definition of the algorithm A(G).
Lemma 6. A(G) = (rs , xs , Ms , Ls , Ns , Gs )s=1,2,...,s satisfies the following conditions
LGs (Ms ∪ Ms+1 ∪ . . . ∪ Ms0 ) = Ls ∪ Ls+1 ∪ . . . ∪ Ls0 , ∀1 ≤ s ≤ s0 ≤ s
LG (Ns+1 ) ∩ (M1 ∪ M2 ∪ . . . ∪ Ms ) = ∅, ∀1 ≤ s < s.
Also, Ms ∪ Ms+1 ∪ . . . ∪ Ms0 is a G-independent set for all 1 ≤ s ≤ s0 ≤ s.
Proposition 2. The sequences (rs )s=1,2,...,s and (xs )s=1,2,...,s defined by the algorithm A(G)
are strictly increasing.
Proof. By the definition of A(G), it is sufficient to show that (rs )s=1,2,...,s is strictly increasing.
We proceed by contradiction. Suppose rs ≤ rs−1 . It must be that 1 < s < s.
By Lemma 6, LGs−1 (Ms−1 ∪ Ms ) = Ls−1 ∪ Ls and Ms−1 ∪ Ms is a G-independent set. Since
|Ls−1 |
|Ls |
= rs−1 and
= rs ≤ rs−1 ,
|Ms−1 |
|Ms |
9By
Lemma 5, since rs < 1, Ms is also a minimizer in 5.1.
completeness, set xs = 1/2, Ms = Ls = ∅.
11As an illustration, the algorithm A(G ), for the network G introduced in Section 1, ends in s = 2 steps.
2
2
The relevant outcomes in the first step are r1 = 1/2, x1 = 1/3, M1 = {3, 4}, L1 = {1}, and in the second step
are r2 = 1, N2 = {2, 5}.
10For
22
MIHAI MANEA
it follows that
|Ls−1 | + |Ls |
|LGs−1 (Ms−1 ∪ Ms )|
=
≤ rs−1 .
|Ms−1 ∪ Ms |
|Ms−1 | + |Ms |
Therefore,
Ms−1 ∪ Ms ∈ arg
|LGs−1 (M )|
,
M ∈I(G)
|M |
min
M ⊂Ns−1 ,
a contradiction with Ms−1 being the union of all the minimizers of the expression above.
Note that the sets M1 , L1 , . . . , Ms−1 , Ls−1 , Ns partition N . The limit equilibrium payoff of
each player is uniquely determined by the set of the partition that he belongs to.
Theorem 5. Let (rs , xs , Ms , Ls , Ns , Gs )s=1,2,...,s be the outcome of the algorithm A(G). The
limit equilibrium payoffs for Γδ as δ → 1 are given by
vi∗ = xs , ∀i ∈ Ms , ∀s < s
vj∗ = 1 − xs , ∀j ∈ Ls , ∀s < s
vk∗ =
1
, ∀k ∈ Ns .
2
Proof. We prove the theorem by induction on s. Suppose we proved the hypothesis for all
lower values, and we proceed to proving it for s (1 ≤ s ≤ s).12 We treat the case s = s
separately, in the Appendix.
Let s < s and xs = mini∈Ns vi∗ . We first show that xs = xs by arguing that xs ≤ xs and
xs ≥ xs .
Claim 5.1. xs ≤ xs
By Lemma 6, LG (Ms ) ∩ (M1 ∪ M2 ∪ . . . ∪ Ms−1 ) = ∅.
We proceed by contradiction. Suppose that xs > xs . Then vj∗ ≥ 1 − xs−1 > 1 − xs > 1 − xs
for all j in L1 ∪ L2 ∪ . . . ∪ Ls−1 . The first inequality follows from the induction hypothesis
and Proposition 2, and the second from Proposition 2. But vi∗ ≥ xs for all i in Ms . Thus,
vi∗ + vj∗ > 1, ∀i ∈ Ms , ∀j ∈ L1 ∪ L2 ∪ . . . ∪ Ls−1 . By Proposition 1 no agent i ∈ Ms has G∗
∗
links to agents j ∈ L1 ∪ L2 ∪ . . . ∪ Ls−1 , or LG (Ms ) ∩ (L1 ∪ L2 ∪ . . . ∪ Ls−1 ) = ∅.
12The
following technical detail is used in order to avoid analogous arguments proving the base case and the
inductive step. Append step 0 to the algorithm, with (r0 , x0 , M0 , L0 , N0 , G0 ) = (0, 0, ∅, ∅, N, G). Then the
base case s = 0 follows trivially, and the inductive steps, s = 1, 2, . . . , s − 1, involve identical arguments.
BARGAINING ON NETWORKS
23
∗
It follows that LG (Ms ) ⊂ LGs (Ms ) = Ls . Theorem 4 implies that
min vi∗ ≤
i∈Ms
|Ls |
= xs ,
|Ms | + |Ls |
a contradiction with mini∈Ns vi∗ = xs > xs .
Claim 5.2. xs ≥ xs
We proved that xs ≤ xs . Since rs < 1 it follows that xs < 1/2. Let M s = arg mini∈Ns vi∗
be the set of agents in Ns whose limit equilibrium payoffs equal xs , and Ls = LGs (M s ).
By Proposition 1 and Claim 5.1,
vj∗ ≥ 1 − xs ≥ 1 − xs > 1/2,
for all j ∈ Ls . Proposition 1 implies that Ls is a G∗ -independent set.
Fix j ∈ Ls . By Proposition 1 there exist no G∗ links from j to players k ∈ Ns \ M s , since
for these players vk∗ > xs by the definition of M s . Also, there exist no G∗ links from j to
players in M1 ∪ M2 ∪ . . . ∪ Ms−1 by Lemma 6.
By Proposition 1, there exist no G∗ links from j to players k ∈ L1 ∪ L2 ∪ . . . ∪ Ls−1 since
for these players vk∗ ≥ 1 − xs−1 > 1/2, and we need vj∗ > 1/2. Therefore, any j ∈ Ls only
has G∗ links to players in M s . By Proposition 1 and Lemma 3, any player j ∈ Ls must have
limit equilibrium payoff vj∗ = 1 − xs .
Hence Ls is G∗ -independent and LG∗ (Ls ) ⊂ M s , so it follows from Theorem 4 that
xs =
max vi∗ ≥
i∈LG∗ (Ls )
|Ls |
G∗
|L (Ls )| +
|Ls |
≥
|Ls |
.
|M s | + |Ls |
Recall that Ls = LGs (M s ). We can rewrite the inequality above as
(5.2)
xs
|LGs (M s )|
≥
.
1 − xs
|M s |
By the definition of rs and xs ,
(5.3)
|LGs (M s )|
xs
≥ rs =
,
|M s |
1 − xs
and the last two inequalities imply xs ≥ xs .
Claims 5.1 and 5.2 establish that xs = xs . Hence vi∗ = xs , ∀i ∈ M s and vj∗ = 1−xs , ∀j ∈ Ls .
We need to have equalities in the weak inequalities 5.2-5.3, so |Ls |/|M s | = rs .
24
MIHAI MANEA
Claim 5.3. M s ⊂ Ms
Since Ms is the union of all G-independent M ⊂ Ns with |LGs (M )|/|M | = rs and
|LGs (M s )|/|M s | = rs , it follows that M s ⊂ Ms . (M s is G-independent by Proposition
1, as the limit equilibrium payoffs of each player in M s is xs < 1/2.)
Claim 5.4. M s = Ms
We show that M s = Ms by contradiction. Fix i ∈ Ms \ M s . Since i ∈ Ms and LGs (Ms ) =
Ls , i has no G links to players in Ns \Ls . By Lemma 6, i has no G links to M1 ∪M2 ∪. . .∪Ms−1 .
By Proposition 1, i has no G∗ links to players j ∈ L1 ∪ L2 ∪ . . . ∪ Ls−1 ∪ Ls as for such
players vj∗ ≥ 1 − xs , and vi∗ > xs = xs by the definition of M s .
∗
It follows that i only has G∗ links to Ls \ Ls . Therefore, LG (Ms \ M s ) ⊂ Ls \ Ls , implying
∗
that |LG (Ms \ M s )| ≤ |Ls \ Ls | = |Ls | − |Ls |.
Note that
|Ls |
|Ls |
|Ls | − |Ls |
= rs and
= rs imply
= rs .
|Ms |
|M s |
|Ms | − |M s |
Then by Theorem 4,
∗
min
i∈Ms \M s
vi∗
|LG (Ms \ M s )|
|Ls | − |Ls |
rs
≤
=
≤
= xs ,
∗
G
|Ms \ M s | + |L (Ms \ M s )|
|Ms | − |M s | + |Ls | − |Ls |
1 + rs
a contradiction with vi∗ > xs for all i ∈ Ns \ M s .
Therefore, M s = Ms , Ls = Ls , and vi∗ = xs for all i in Ms and vj∗ = 1 − xs for all j in Ls .
Example 2 below shows that the limit equilibrium agreement network does not necessarily
contain all the G links from players in Ms to players in Ls . Although all players in Ms (Ls )
have the same limit equilibrium payoff, their relative bargaining strengths may vary, and
the rates of convergence to the common limit are not identical across Ms (Ls ). Moreover, it
can even be the case that the players in Ms ∪ Ls induce a connected subnetwork in G, but
a disconnected subnetwork in G∗ .
Example 2. Consider the network G3 with 9 players drawn in Figure 3. The algorithm
A(G3 ) ends in one step, with r1 = 1/2, M1 = {4, 5, 6, 7, 8, 9} and L1 = {1, 2, 3}. Therefore,
all players in M1 have limit equilibrium payoff equal to 1/3 and all players in L1 have limit
equilibrium payoff equal to 2/3. However, it is not the case that the limit equilibrium
BARGAINING ON NETWORKS
1
4
3
2
5
6
25
7
9
8
Figure 3. Network G3
1
4
3
2
5
6
7
8
9
Figure 4. Network G∗3
agreement network contains all the links from players in M1 to players in L1 , that is, all the
links of G3 .
Indeed, the limit equilibrium agreement network G∗3 , depicted in Figure 4, excludes the
links (1,8) and (1,9). The intuition is that, although v1∗ = v2∗ = v3∗ = 2/3 and v4∗ = v5∗ = v6∗ =
v7∗ = v8∗ = v9∗ = 1/3, player 1 is relatively stronger than players 2 and 3 as he is connected
to all players that 2 and 3 are connected to, and players 8 and 9 are relatively stronger than
players 4, 5, 6, and 7 as they are connected to all players that 4, 5, 6, and 7 are connected
to. For similar reasons, player 3 is relatively weaker than players 1 and 2, augmenting the
relative strength of 8 and 9 over 4, 5, 6, and 7, and players 4 and 5 are relatively weaker
than players 6, 7, 8, 9, augmenting the relative strength of 1 over 2 and 3. For sufficiently
high δ, in equilibrium the payoffs of player 1, but also of players 8 and 9, will be sufficiently
high such that (1,8) and (1,9) are not equilibrium agreement links.
By the proof of Theorem 1, in order to check that the conjectured G∗3 network is the limit
∗
equilibrium agreement network, we only need to show that v δ,G3 (recall the definition 3.3
26
MIHAI MANEA
1
4
2
3
6
5
7
10
9
8
13
11
14
15
12
16
17
18
Figure 5. Network G4
from the proof of Theorem 2) is a fixed point of f δ for δ sufficiently large. The payoff vector
∗
v δ,G3 solves the n × n linear system
∗
G
2e − ei 3
1 X
vi =
δvi +
(1 − δvj ), ∀i = 1, 9.
2e
2e ij∈G∗
3
∗
A closed form solution for v δ,G3 is immediately obtained, but is omitted for expositional
brevity. For example,
δ,G∗3
v1
=
2(576 − 1068δ + 493δ 2 )
,
3(2304 − 6048δ + 5264δ 2 − 1519δ 3 )
∗
and the other components of v δ,G3 have similar rational function expressions.
√
Simple calculus shows that for all δ > δ =: 6/251(45 − 17) ≈ 0.977 the following
δ,G∗3
conditions hold: δ(vi
δ,G∗3
+ vj
δ,G∗3
) < 1 for all ij ∈ G∗3 , and δ(vi
δ,G∗3
+ vj
) > 1 for all ij ∈
∗
∗
∗
G3 \ G∗3 . Hence v δ,G3 is a fixed point of f δ , and G∗δ
3 = G3 . Therefore, G3 is the limit
equilibrium agreement network. For δ < δ, the equilibrium agreement network is the entire
G3 , and for δ > δ the equilibrium agreement network is G∗3 . The set of equilibria admits a
characterization similar to the one corresponding to the network G2 in Example 1.
An example where the players in Ms ∪ Ls induce a connected subnetwork in G, but a
disconnected subnetwork in G∗ is provided by network G4 from Figure 5. G4 basically
consists of two copies of G3 , with two additional links, (8,10) and (9,10). The corresponding
limit equilibrium network G∗4 is illustrated in Figure 6.13 The links (1,8), (1,9), (8,10), (9,10),
(10,17) and (10,18) are not limit equilibrium agreement links by a logic analogous to the one
suggesting that (1,8) and (1,9) are not limit equilibrium agreement links in G3 .
13The
bipartite nature of G3 and G4 is not critical to the asymptotic results. E.g., the limit equilibrium
payoff vector and agreement network for G3 (G4 ) remain unchanged if the link (2,3) is added to G3 (G4 ).
BARGAINING ON NETWORKS
1
4
2
5
6
3
7
8
27
10
9
13
11
14
15
12
16
17
18
Figure 6. Network G∗4
6. Equitable Networks
We are interested in characterizing the class of equitable networks, that is, networks for
which the limit equilibrium payoffs for all players are identical, equal to 1/2, by Proposition
1. A network G is equitable if and only if r1 ≥ 1, so that the algorithm A(G) stops at
the first step. Therefore, all players have limit equilibrium payoff equal to 1/2 if and only
if |LG (M )| ≥ |M | for every G-independent set M . Networks satisfying the latter property
have been studied in graph theory. The following definitions are necessary.
A graph is an odd cycle if its vertex set has odd cardinality and can be relabeled
ν1 , ν2 , . . . , νk such that the set of edges is {ν1 ν2 , ν2 ν3 , . . . , νk−1 νk , νk ν1 }. A graph is a match
if its vertex set has even cardinality and can be relabeled ν1 , ν2 , . . . , νk such that the set of
edges is {ν1 ν2 , ν3 ν4 , . . . , νk−1 νk }. A graph is a match and odd cycles disjoint union if
it is the union of a match and a number of odd cycles that are pairwise vertex-disjoint. A
subnetwork covers a vertex if the vertex has at least one link in the subnetwork. A subnetwork H 0 of a network H covers (or is a cover of) H if each vertex of H is covered by
H 0 . A perfect match of a network H is a match that is a subnetwork of H covering H. A
graph is regular if all vertices are incident to an identical number of edges.
A graph H is quasi-regularizable (Berge 1981) if there exists d > 0 and non-negative
integer weights ωij associated with each edge ij ∈ H such that the sum of the weights of the
edges incident to any vertex is d,
X
ωij = d, ∀i ∈ N.
{j|ij∈H}
Examples of quasi-regularizable graphs are regular graphs (set all the weights equal to 1),
and match and odd cycles disjoint unions (set the weights of the edges of the odd cycles
28
MIHAI MANEA
1
2
3
4
5
6
Figure 7. Network G5
equal to 1 and the weights of the edges of the match equal to 2). If a network H 0 is quasiregularizable then so is any network H for which H 0 is a subnetwork that covers H. Hence
networks that are covered by regular or match and odd cycles disjoint unions subnetworks
are also quasi-regularizable. Berge (1981) shows that [G satisfies |LG (M )| ≥ |M | for every
G-independent set M ] if and only if [G is quasi-regularizable] if and only if [G can be covered
by a match and odd cycles disjoint union].
Theorem 6. (i) G is equitable if and only if G is quasi-regularizable. (ii) G is equitable if
and only if G can be covered by a match and odd cycles disjoint union.
Proof. By Theorem 5, [G is equitable] iff [r1 ≥ 1] iff [G satisfies |LG (M )| ≥ |M | for every Gindependent set M ]. By Berge’s alternative characterizations of quasi-regularizable graphs,
[G satisfies |LG (M )| ≥ |M | for every G-independent set M ] iff [G is quasi-regularizable] iff
[G can be covered by a match and odd cycles disjoint union]. The result is established by
sequencing all the iff statements above.
One important corollary is that every network that is regular or a match and odd cycles
disjoint union, and any network that can be covered by such subnetworks, is equitable.
Example 3. Consider the network G5 with 6 players drawn in Figure 7. By Theorem 6, G5
is equitable since it has the perfect match {(1, 4), (2, 5), (3, 6)}. By methods similar to the
ones used in Example 2 we can prove that the limit equilibrium agreement network is G∗5 ,
illustrated in Figure 8.
Limit equilibrium payoffs depend on the network structure in a more complicated fashion
than through the relative number of bargaining partners. In the network G5 the number
BARGAINING ON NETWORKS
1
2
3
4
5
6
29
Figure 8. Network G∗5
of links of each player ranges from 1 to 3, but all players receive limit equilibrium payoff
1/2. Therefore, while regular networks are equitable, regularity is far from being a necessary
condition for equitability. Quasi-regularizability is a necessary and sufficient condition.
7. Stable Networks
In our framework the network structure is exogenously given, and we do not study network formation. Nonetheless, it is important to characterize the networks where players
could benefit in the equilibrium limit from forming new links or severing existing ones. The
algorithm A(G) enables us to address this problem. Fix the set of players N , and let G be
the set of networks G with set of vertices equal to N . A payoff function u assigns each
player i ∈ N a payoff, denoted by ui (G), for every network G ∈ G. If vi∗ (G) denotes the
limit equilibrium payoff of player i for the bargaining game on network G then the profile
(vi∗ (G))i∈N,G∈G defines a payoff function, which we call the bargaining limit equilibrium
payoff. For every network G̃ and any i 6= j ∈ N , let G̃ + ij (G̃ − ij) denote the network
obtained by adding (deleting) link ij to (from) G̃.
Definition 2 (Stability). Network G̃ is unilaterally stable with respect to payoff function
(ui (G))i∈N,G∈G if ui (G̃) ≥ ui (G̃ − ij) for any pair of players i 6= j ∈ N . Network G̃ is
pairwise stable with respect to payoff function (ui (G))i∈N,G∈G if it is unilaterally stable
with respect to (ui (G))i∈N,G∈G and ui (G̃ + ij) > ui (G̃) ⇒ uj (G̃ + ij) < uj (G̃) for any pair of
players i 6= j ∈ N .
To rephrase, a network is unilaterally stable if no player benefits from severing one of his
links. Pairwise stability requires additionally that no pair of players benefit from forming
30
MIHAI MANEA
a new link. Jackson and Wolinsky (1996) motivate the definitions by the fact that the
formation of link ij necessitates the consent of both players i and j, but its severance can
be done unilaterally by either player i or j.
Theorem 7. (i) Every network is unilaterally stable with respect to the bargaining limit
equilibrium payoff. (ii) A network is pairwise stable with respect to the bargaining limit
equilibrium payoff if and only if it is equitable.
Part (i) of the statement is not surprising, but its proof is involved. All proofs are in the
Appendix.
While unilateral stability with respect to the bargaining equilibrium payoffs holds for every
network in the limit as players become patient, the conclusion does not apply before taking
the limit. Let vi∗δ (G) be the equilibrium payoff of player i for the bargaining game on network
G when all players have discount factor δ < 1. Not every network is unilaterally stable with
respect to (vi∗δ (G))i∈N,G∈G . Consider the networks G2 and G∗2 from Figures 1 and 2. G∗2 is
obtained from G2 by removing the link between players 1 and 5. For every discount factor
√
δ ∈ (10(9 − 2)/79, 1), players 1 and 5 receive higher equilibrium payoffs in the bargaining
game on network G∗2 than in the one on network G2 ,
v1∗δ (G2 ) =
2
2
1
1
<
= v1∗δ (G∗2 ) and v5∗δ (G2 ) =
<
= v5∗δ (G∗2 ).
10 − 7δ
8 − 5δ
2(5 − 4δ)
2(4 − 3δ)
Thus both players 1 and 5 could benefit from removing the link connecting them and play
the game on network G∗2 rather than on network G2 .
The intuition for this observation is simple. For the range of discount factors considered,
the link between players 1 and 5 is not part of the equilibrium agreement network, whence
it becomes a source of delay for the possible agreements and deflates the equilibrium payoffs
of all players. However, the gain to player 1 (5) from severing the link (1,5) vanishes as
the discount factor approaches 1. If players only consider deleting or adding links when the
resulting gains are significant, we need to focus on approximate stability.
Definition 3 (ε-Stability). Network G̃ is unilaterally ε-stable with respect to payoff
function (ui (G))i∈N,G∈G if ui (G̃) + ε ≥ ui (G̃ − ij) for any pair of players i 6= j ∈ N . Network
G̃ is pairwise ε-stable with respect to payoff function (ui (G))i∈N,G∈G if it is unilaterally
BARGAINING ON NETWORKS
31
ε-stable with respect to (ui (G))i∈N,G∈G and ui (G̃ + ij) > ui (G̃) + ε ⇒ uj (G̃ + ij) < uj (G̃) + ε
for any pair of players i 6= j ∈ N .
For any sufficiently low ε > 0, there exists a discount factor threshold δ < 1 such that the
two statements of Theorem 7 also hold for ε-stability with respect to the equilibrium payoffs
in Γδ for any δ > δ. The next result is based on ideas from the proofs of Theorems 3 and 7.
Corollary 2. There exists ε̄ > 0 such that for all ε < ε̄ there exists δ < 1 such that for all
δ > δ the following statements are true. (i) Every network is unilaterally ε-stable with respect
to the bargaining equilibrium payoff (vi∗δ (G))i∈N,G∈G . (ii) A network is pairwise ε-stable with
respect to the bargaining equilibrium payoff (vi∗δ (G))i∈N,G∈G if and only if it is equitable.
8. Buyer and Seller Networks
Let N = B ∪ S, where B and S denote the sets of buyers and sellers, respectively. Each
seller owns one unit of a homogeneous indivisible good. Each buyer demands one unit of
the good. The utilities of the buyers and sellers for the good are normalized to 1 and 0,
respectively. Some buyer-seller pairs are connected and can trade. G denotes the network
of connections. With this interpretation, only pairs of buyers and sellers who are connected
in G can generate a unit surplus, as in our benchmark model. The buyers and sellers have
common discount factor δ, and play the game Γδ on network G described in Section 2. Since
buyer-seller networks form a special class of networks, all the results of the paper apply, and
some refinements are possible.
The bipartite nature of buyer-seller networks permits a more straightforward description
of the bounds and the accompanying algorithm for computing the limit equilibrium payoffs.
The amendments to Theorems 4 and 5 for the special case of buyer-seller networks are
based on the following observations. Fix a buyer-seller network H. For every subset of
players M = B 0 ∪ S 0 (B 0 ⊂ B, S 0 ⊂ S), LH (M ) = LH (B 0 ) ∪ LH (S 0 ) with LH (B 0 ) ⊂ S and
LH (S 0 ) ⊂ B. The set M is H-independent if there exists no H-link between buyers in B 0
and sellers in S 0 , or LH (B 0 ) ∩ S 0 = ∅ (or LH (S 0 ) ∩ B 0 = ∅). B 0 and S 0 are H-independent.
32
MIHAI MANEA
Theorem 4BS . For every set of buyers B 0 , the following bounds on limit equilibrium payoffs
hold
∗
min0 vi∗
i∈B
max
∗
j∈LG (B 0 )
|LG (B 0 )|
≤
|B 0 | + |LG∗ (B 0 )|
vj∗ ≥
|B 0 |
.
|B 0 | + |LG∗ (B 0 )|
If we restrict attention to the sets of buyers M that minimize |LH (M )|/|M |, Lemma 5 can
be extended to show that the set of such minimizers is closed with respect to unions without
the assumption that the attained minimum is strictly less than 1.
Lemma 5BS . Let H be a buyer-seller network. Suppose that
|LH (M )|
.
M ⊂B
|M |
B 0 , B 00 ∈ arg min
Then
|LH (M )|
B ∪ B ∈ arg min
.
M ⊂B
|M |
0
00
Remark 1. The only new observation we need to add to the proof of Lemma 5 in order to
prove Lemma 5BS is that M 0 , M 00 ⊂ B implies that A4 , A5 = ∅, hence a4 + a5 = 0. Then the
display equation A.13, (r − 1)(a4 + a5 ) ≥ 0 holds with equality, and thereafter the proof can
be completed as in Lemma 5. The inequality r < 1 becomes unnecessary for the conclusion.
By Theorem 4BS and Lemma 5BS , the algorithm computing limit equilibrium payoffs does
not have to treat rs ≥ 1 as a stopping condition at step s if the focus is on sets of buyers
M that minimize |LGs (M )|/|M |. While the algorithm A(G) is effective for buyer-seller
networks, the revision ABS (G) adapted to buyer-seller networks offers a simplified solution.
Definition 4 (Algorithm ABS (G) = (rl , xl , Ml , Ll , Nl , Gl )l=1,2,...,l ). Define the sequence
(rl , xl , Ml , Ll , Nl , Gl )l recursively as follows. Set N1 = N and G1 = G. For l ≥ 1, let
(8.1)
rl =
|LGl (B 0 )|
.
B ⊂Nl ∩B
|B 0 |
min
0
Set xl = rl /(1 + rl ). Let Bl be the union of all minimizers B 0 in 8.1. By Lemma 5BS , Bl
l
is also a minimizer in 8.1. Denote Sl = LG (Bl ). If Nl = Bl ∪ Sl then stop. Else, set
Nl+1 = Nl \ (Bl ∪ Sl ), and let Gl+1 be the subnetwork of G induced by the players in Nl+1 .
Denote by l the finite step at which the algorithm stops.
BARGAINING ON NETWORKS
33
Note that the sets B1 , S1 , . . . , Bl , Sl partition N .
Proposition 2BS . The sequences (rl )l=1,2,...,l and (xl )l=1,2,...,l defined by the algorithm ABS (G)
are strictly increasing.
Theorem 5BS . Let (rl , xl , Ml , Ll , Nl , Gl )l=1,2,...,l be the outcome of the algorithm ABS (G) for
the buyer-seller network G. The limit equilibrium payoffs for Γδ as δ → 1 are given by
vi∗ = xl , ∀i ∈ Bl , ∀l ≤ l
vj∗ = 1 − xl , ∀j ∈ Sl , ∀l ≤ l.
Similarly to the study of equitable networks for the general model, we are interested in
characterizing the class of non-discriminatory networks in the bipartite case. A buyerseller network is non-discriminatory if the limit equilibrium payoffs of all buyers, hence of all
sellers as well, are identical. By Proposition 2BS , a network G is non-discriminatory if and
only if the algorithm ABS (G) stops at the first step, hence B1 = B, S1 = S, r1 = |S|/|B|.
Assume that β := |B|/|S| is an integer. In a non-discriminatory network the common
limit equilibrium payoffs of all buyers and sellers are 1/(β + 1) and β/(β + 1), respectively.
Therefore, the limit equilibrium price is defined by β/(β + 1).
A β-buyer-seller cluster is a subnetwork formed by a seller connected to β buyers.
A β-buyer-seller cluster disjoint union is a network that is a union of vertex-disjoint
β-buyer-seller clusters. Set B = {4, 5, 6, 7, 8, 9} and S = {1, 2, 3} for the network G3 from
Example 2. The ratio of buyers to sellers is 2 to 1, and the network can be covered by the
disjoint union of three 2-buyer-seller clusters, {(1, 4), (1, 5)} ∪ {(2, 6), (2, 7)} ∪ {(3, 8), (3, 9)}.
The network G3 is non-discriminatory, with a limit equilibrium price of 2/3. The observation
can be generalized.
Theorem 8. Suppose that β = |B|/|S| is an integer. A buyer-seller network is nondiscriminatory if and only if it can be covered by a β-buyer-seller cluster disjoint union.
Proof. Let G be a buyer-seller network. G is non-discriminatory if and only if B1 = B, S1 =
S, r1 = 1/β if and only if |LG (B 0 )|/|B 0 | ≥ 1/β for all subsets of buyers B 0 .
Let H be the graph obtained from G by replacing each vertex corresponding to a seller
with β identical copies (each connected to all the buyers the initial seller had links to). Note
34
MIHAI MANEA
that |LG (B 0 )|/|B 0 | ≥ 1/β for all subsets of buyers B 0 is equivalent to |LH (B 0 )|/|B 0 | ≥ 1 for
all subsets of buyers B 0 , which is equivalent to H having a perfect match by Hall (1935)’s
theorem. By construction, H has a perfect match if and only if G contains a subnetwork
cover that is a β-buyer-seller cluster disjoint union, which finishes the proof.
Corollary 3. A buyer-seller network is equitable if and only if it has a perfect match.
Remark 2. Corollary 3 also follows from Theorem 6 since a bipartite network contains no
odd cycles, so any cover by a match and odd cycle disjoint union is a perfect match.
Remark 3. A symmetric set of results obtains if we interchange the roles of buyers and
sellers in Theorems 4BS , 5BS and 8.
In the context of buyer-seller networks the definition of pairwise stability should account
for the fact that only buyer-seller pairs may consider forming new links.
Definition 5 (Buyer-seller stability). Buyer-seller network G̃ is two-sided pairwise stable
with respect to payoff function (ui (G))i∈N,G∈G if for all (i, j) ∈ (B ×S)∪(S ×B) the following
conditions hold: (1) ui (G̃) ≥ ui (G̃ − ij), and (2) ui (G̃ + ij) > ui (G̃) ⇒ uj (G̃ + ij) < uj (G̃).
The next result is the analogue of Theorem 7.
Theorem 7.iiBS . A buyer-seller network is two-sided pairwise stable with respect to the
bargaining limit equilibrium payoff if and only if it is non-discriminatory.
9. Robustness of the Results
9.1. Heterogeneous discount factors. All players are assumed to share a common discount factor hitherto. We can extend the results of the paper to the case of heterogeneous
discount factors, where all copies of player i have the same discount factor, denoted δi . The
accumulation points of the equilibrium payoff vectors and agreement networks along a sequence (δ1 , δ2 , . . . , δn ) converging to (1, 1, . . . , 1) depend on the choice of the sequence.14 One
14For
example, in the game corresponding to a two player network, if the discount factors are given by
the pair (δ a , δ b ) (a, b > 0), then as δ → 1 the limit equilibrium payoffs are (b/(a + b), a/(a + b)). For
different choices of (a, b) the limit equilibrium payoffs for the corresponding sequence of discount factors
vary accordingly. For the sequence of discount factor vectors indexed by n given by (1 − 1/n, 1 − 1/n)
BARGAINING ON NETWORKS
35
constraint that guarantees convergence of the equilibrium payoffs and agreement network is
that the relative rates of convergence of δi and δj to 1 be constant along the sequence of
discount factor vectors. Let α = (α1 , α2 , . . . , αn ) be the vector of (strictly positive) rates of
convergence of the sequence of discount factor vectors to (1, 1, . . . , 1). Denote by Γδ,α the
game as described in Section 2 modified such that the discount factor of player iτ is δ αi for
all i ∈ N and τ ≥ 0.
For a fixed α, we are interested in the asymptotic equilibrium behavior in Γδ,α as δ →
1. Theorems 1-3 generalize verbatim. The notation for Γδ , v ∗δ , δ, G∗δ , G∗ , v ∗ , . . . needs to
be replaced by Γδ,α , v ∗δ (α), δ(α), G∗δ (α), G∗ (α), v ∗ (α), . . . to reflect the dependence of the
variables on α.
Remark 4. For a subnetwork H of G, the analogue of the linear system 3.3 used in the
proofs of Theorems 2 and 3 is
vi =
1 X
2e − eH
i
δ αi vi +
(1 − δ αj vj ), ∀i = 1, n.
2e
2e ij∈H
As in the proof of Theorem 2 the unique solution viδ,H (α) is given by Cramer’s rule, as
the ratio of two determinants that are finite sums of positive real powers of δ. In order to
show that for fixed i, j, H there exists a finite number of δ solving the equation δ(viδ,H (α) +
vjδ,H (α)) = 1 we invoke a result due to Laguerre (1883). The result extends Descartes’ rule
of signs, which provides a bound for the number of positive real roots of polynomials in δ, to
the case of linear combinations of powers of δ. A corollary of Laguerre’s result is that every
finite linear combination of positive powers of δ that does not vanish everywhere has a finite
number of solutions.15
for odd n and (1 − 1/n, (1 − 1/n)2 ) for even n, the set of equilibrium payoff vectors has two accumulation
points, (1/2, 1/2) and (2/3, 1/3). Similarly, for more complicated network structures, the limit equilibrium
agreement network depends on the sequence of discount factor vectors, and convergence does not always
obtain.
15If all components of α are rational numbers we can avoid non-polynomial functions by using the substation
δ → δ c where c is the least common multiple of the denominators of α1 , α2 , . . . , αn .
36
MIHAI MANEA
Theorem 4α . For every G∗ (α)-independent set of players M , the following bounds on limit
equilibrium payoffs hold
P
min vi∗ (α) ≤ P
∗ (α)
j∈LG
(M )
αj
P
αi + j∈LG∗ (α) (M ) αj
P
i∈M αi
∗
P
P
.
max
v
(α)
≥
j
∗
∗
j∈LG (α) (M )
i∈M αi +
j∈LG (α) (M ) αj
i∈M
i∈M
Proof. The proof of Theorem 4α follows analogous steps to the proof of Theorem 4. The
only innovation is that the analogue of Lemma 4 is
X 1
1 − δ αi ∗δ
1
vi (α) =
max(1 − δvi∗δ (α) − δvj∗δ (α), 0), ∀i ∈ N, ∀δ ∈ (0, 1).
1−δ
1−δ
2e
{j|ij∈G}
Then
X 1 − δ αj
j∈L
1−δ
vj∗δ (α)
≥
X 1 − δ αi
i∈M
1−δ
vi∗δ (α),
which after taking the limit δ → 1 becomes
X
X
αj vj∗ (α) ≥
αi vi∗ (α).
j∈L
i∈M
The conclusion is reached as in the proof of Theorem 4.
If we replace the definition 5.1 in the algorithm A(G) with
P
rs (α) =
min
M ⊂Ns , M ∈I(G)
∗ (α)
j∈LG
(M )
P
αi
i∈M
αj
,
and leave the rest of the specification of the algorithm and the definition of the other variables
unchanged, the characterization of the limit equilibrium payoffs provided by Theorem 5
carries through.16
9.2. General matching technologies. The assumption in the benchmark model of Section
2 is that each pair of connected players is equally likely to be selected to bargain at each
round, and conditional on the link selection each of the two matched players is equally
likely to be the proposer. Let (p(i → j) > 0)ij∈G be a probability distribution over the
oriented links of G. Consider the game Γδ,p defined identically to the game Γδ except for the
probability distribution defining nature’s moves. At each round of Γδ,p , player i is chosen
to make an offer to player j with probability p(i → j). Hence, each link ij is selected for
16The
extension of the proofs of Lemma 5 and Theorem 5 virtually consists inPreplacing everywhere the
cardinality set operator | · | by the α-weight set operator | · |α , defined by |M |α = i∈M αi for every M ⊂ N .
BARGAINING ON NETWORKS
37
bargaining with probability p(i → j) + p(j → i), and conditional on the selection, i is the
proposer with probability p(i → j)/(p(i → j) + p(j → i)). The uniform distribution pu
defining the benchmark game Γδ is given by pu (i → j) = 1/(2e), ∀ij ∈ G.
Fix p. We are interested in the asymptotic equilibrium behavior in Γδ,p as δ → 1. As in
the previous subsection, Theorems 1-3 generalize with Γδ , v ∗δ , δ, G∗δ , G∗ , v ∗ , . . . replaced by
Γδ,p , v ∗δ (p), δ(p), G∗δ (p), G∗ (p), v ∗ (p), . . .
The observation that for every discount factor the equilibrium payoff of any player equals
the expected present value of his stream of first mover advantage, which is the first ingredient
for the proof of Theorem 4, generalizes to Γδ,p . Next suppose that conditional on the selection
of any link, each of the two players is equally likely to become the proposer, i.e., p(i → j) =
p(j → i) for all ij ∈ G. Then the second ingredient for the proof of Theorem 4 also
generalizes to Γδ,p . Any net first mover advantage corresponding to a player i making an
offer to a player j is mapped to an equal net first mover advantage corresponding to player j
making an offer to player i, and both gains are weighted by probability p(i → j) = p(j → i)
in the expected payoffs of i and j. Therefore, the sum of the expected present value of the
stream of first mover advantage of all players in LG
∗ (p)
(M ) is at least as large as the same
expression evaluated for the players in M .
Proposition 3. If p(i → j) = p(j → i) for all ij ∈ G then Theorems 4 and 5 extend
without change. The limit equilibrium payoff vector for Γδ,p as δ → 1 is identical to the one
for Γδ,pu = Γδ as δ → 1.
Remark 5. Suppose that G is the complete network, and p(i → j) = πi πj , ∀i 6= j ∈ N , for
some vector π describing relative matching frequencies (πi > 0, ∀i ∈ N ). The consequence
of Proposition 3 for this setting is that the limit equilibrium payoff of each player is 1/2,
independently of π. While players with higher matching frequencies obtain larger equilibrium
payoffs for any discount factor smaller than 1, all equilibrium payoffs converge to 1/2 as the
discount factor goes to 1. The intuition is that patient players can postpone agreement until
they are matched to bargain with the player who has the lowest matching frequency (the
match occurs with positive probability each period).
Another case where the results extend easily is the setting of buyer-seller networks where
there exists q > 0 such that p(i → j) = qp(j → i) for all ij ∈ G, i ∈ B, j ∈ S (B
38
MIHAI MANEA
1
4
2
5
6
3
7
8
9
Figure 9. Network G∗3 (p)
and S denote the sets of buyers and sellers, respectively). In any pairwise matching, the
buyer is q times more likely than the seller to be the proposer. Consider the sequence
(rl , xl , Bl , Sl , Nl , Gl )l=1,2,...,l generated by the algorithm ABS (G). The same sequence can be
used to characterized the limit equilibrium payoff of each player, with the unique change in
definition, xl = qrl /(1 + qrl ).
For more general matching technologies it is not possible to extend the results. The
following example illustrates some of the difficulties.
Example 4. Consider again the network G3 with 9 players drawn in Figure 3. Let p be the
probability distribution determined as follows. Each link is selected with equal probability,
and for each selection except (2, 6) and (2, 7), one of the two matched players is chosen with
equal probability to be the proposer. If the selected link is (2, 6) ((2, 7)) then player 2 is
twice as likely to be chosen as the proposer than player 6 (7). Mathematically, p(2 → 6) =
p(2 → 7) = 1/18, p(6 → 2) = p(7 → 2) = 1/36 and p(i → j) = 1/24 for all other links
ij in G3 . We define the probability distribution p0 similarly to give player 2 asymmetric
bargaining power in pairwise matchings to players 8 and 9, by p0 (2 → 8) = p0 (2 → 9) =
1/18, p0 (8 → 2) = p0 (9 → 2) = 1/36 and p0 (i → j) = 1/24 for all other links ij in G3 .
Consider first the game Γδ,p . By ideas and arguments similar to the ones used in Example
2, we obtain that the limit equilibrium agreement network in Γδ,p is given by the subnetwork
G∗3 (p) illustrated in Figure 9. The limit equilibrium payoffs are v1∗ (p) = v2∗ (p) = 8/11, v3∗ (p) =
2/3, v4∗ (p) = v5∗ (p) = v6∗ (p) = v7∗ (p) = 3/11 and v8∗ (p) = v9∗ (p) = 1/3. The reasoning for these
results is that player 2 can extort players 6 and 7 for more than 2/3 since in pairwise
interactions with these players he has increased bargaining power. Players 6 and 7 will have
BARGAINING ON NETWORKS
39
to reach agreements with 1 when matched since they could not receive limit equilibrium
payoffs larger than 1/5 if they only reached agreements with 2. Then player 1 will be able to
take advantage of the weakness of 6 and 7, and also reach equally favorable agreements with
4 and 5 in the limit since 4 and 5 do not have other possible bargaining partners. In the
limit, since players 1 and 2 obtain high payoffs from players 4, 5, 6 and 7, they will not be
attractive bargaining partners for 8 and 9. Players 8 and 9 are the only bargaining partners
of player 3, and can secure limit equilibrium payoffs of 1/3. Hence the links from 1 and 2 to
8 and 9 cannot be part of equilibrium agreements when players are sufficiently patient.
0
0
Consider next the game Γδ,p . The limit equilibrium agreement network in Γδ,p is identical
to G∗3 , illustrated in Figure 4. The limit equilibrium payoffs are v1∗ = v2∗ = v3∗ = 2/3 and
v4∗ = v5∗ = v6∗ = v7∗ = v8∗ = v9∗ = 1/3. Player 2 cannot use his stronger bargaining position to
secure a limit equilibrium payoff larger than 2/3. The intuition is that 8 and 9 can secure
limit equilibrium payoffs of 1/3 in pairwise agreements to 3, who has no other bargaining
partner. Hence 8 and 9 cannot be compelled to surrender more than 2/3 to player 2 in the
limit. The links from 1 to 8 and 9 are not part of equilibrium agreements when players are
sufficiently patient for the reasons outlined in Example 2.
9.2.1. More than one match per period. Suppose that nature selects a set of disjoint links for
bargaining every period. At every round, there may be multiple pairs of players selected, and
all pairs bargain simultaneously. We assume that the distribution over the sets of disjoint
links is stationary, and each link is selected with positive probability. The preliminary
results on uniqueness of the equilibrium payoffs and convergence of the equilibrium payoffs
and agreement links as players become patient extend to this setting. Suppose further that
each of any two players matched to bargain is equally likely to be the proposer. Then, by an
argument similar to the one for Proposition 3, the limit equilibrium payoff of the bargaining
game with the general matching technology is identical to the one in the benchmark game.
10. Related Literature
One important study of decentralized trading is Rubinstein and Wolinsky (1985). The
paper considers a market where each seller owns one unit of a homogeneous indivisible good,
and each buyer demands one unit of the good. The utilities of the buyers and sellers for the
good are normalized to 1 and 0, respectively. At each round buyer-seller pairs are matched
40
MIHAI MANEA
by the following stochastic process. Each buyer is matched to a new seller with probability
αb and each seller is matched to a new buyer with probability αs . For every buyer-seller
match, each of the two agents is chosen with equal probability to make a price offer. If
the offer is accepted then the two agents leave the market. Every pair of matched agents
continues to bargain until one of them is matched to a new bargaining partner.
Rubinstein and Wolinsky focus on a market at a steady state, where the numbers of buyers
and sellers are constant over time. The probabilities αb and αs determine the rate at which
matches are formed, and also determine the rate at which agreements are reached and agents
leave the market in equilibrium. At a steady state there exits an exogenous flow of agents
into the market equal to the endogenous equilibrium flow of post-agreement departures. The
numbers of buyers and sellers are left unspecified, and the matching technology determining
the probabilities αb and αs is not explicitly described. Rubinstein and Wolinsky provide the
following example of a matching technology. Suppose there are nb buyers and ns sellers, and
each period a set of m new matches is randomly chosen among the sets of m disjoint buyerseller pairs. If nb and ns are large, so that the probability that an agent is rematched with
his current partner is small, then the matching technology yields approximately αb = m/nb
and αs = m/ns . For m = 1, the example is a special case of our model, whereby the network
is a bipartite complete graph with nb buyers and ns sellers.
Rubinstein and Wolinsky (1985) restrict attention to semi-stationary strategies, in which
each agent uses the same rule of behavior during every match, and all agents of the same type
(buyer/seller) use identical strategies. There is a unique semi-stationary equilibrium.17 One
implication of the analysis is that as the common discount factor of the players approaches
1, the equilibrium payoffs of the buyers and sellers converge to αb /(αs +αb ) and αs /(αs +αb ),
respectively. For the matching technology that approximates our model when the network
is a bipartite complete graph with nb buyers and ns sellers, the limit equilibrium payoffs
can be rewritten as ns /(nb + ns ) and nb /(nb + ns ), respectively. Hence, as players become
increasingly patient, the unit surplus is split between a buyer and a seller according to the
shortage ratio defined by our algorithm, ns /nb , describing the relative strength of the group
of buyers.
17Binmore
and Herrero (1988b) show that the stationarity assumption is not necessary in order to obtain
equilibrium uniqueness. The Rubinstein and Wolinsky bargaining model has a unique security equilibrium.
BARGAINING ON NETWORKS
41
Gale (1987) considers a generalization of the Rubinstein and Wolinsky (1985) model in
which agents on both sides of the market are distinguished by their reservation prices. There
exists a continuum of agents for each reservation price. Gale first studies the case in which
a finite measure of players participate in the game, entering the economy according to some
fixed pattern. The finite measure assumption implies that the set of all agents constitutes
a well-defined economy, with a generically unique competitive equilibrium allocation. As
players become increasingly patient the limit point of any sequence of equilibria is the Walras
outcome of the underlying exchange economy. The result is driven by the fact that in
a sequential bargaining model with a finite measure of agents all possible agreements are
eventually depleted.
Gale recognizes that in real life markets the number of active participants may fluctuate,
but there is no long-run tendency for the population to disappear. Therefore, a more appropriate model of decentralized markets has an infinite measure of agents entering the game
over time, with entry flows specified so that the economy is in a steady state. As players
become increasingly patient the limit point of any sequence of stationary equilibria is the
Walras outcome of the exchange economy induced by the distribution of agent types in the
steady state inflow.
Three observations in Gale’s argument parallel the preliminary results of the present paper
stated in Lemma 2, Proposition 1, and respectively Theorem 4. First, the discounting
assumption facilitates a contraction argument showing that there exists a unique stationary
equilibrium. Second, patient players can wait for the agreement that is asymptotically most
favorable. In Gale’s setting this remark implies that there exists a unique limit price for all
equilibrium agreements. In our setting the consequence is that the limit equilibrium payoffs
of any two players who share a link sum to at least 1. Third, the equilibrium payoff of a
player equals the present value of a stream of payoffs corresponding to the net first mover
advantage weighted by the probability distribution of agreements involving the player as
the proposer. Furthermore, the first mover advantage in a bilateral encounter is symmetric
for the two players. In Gale’s framework the observation implies that the sums of expected
utilities on each side of the market are equal, hence the limit equilibrium price must be the
competitive price of the inflow economy. In our framework the conclusion is that the sum of
42
MIHAI MANEA
the limit equilibrium payoffs of any mutually estranged set of players is not larger than the
sum of the limit equilibrium payoffs of the corresponding partners.
Binmore and Herrero (1988a) and Rubinstein and Wolinsky (1990) further study the
relationship between the equilibrium outcomes of various decentralized bargaining procedures
and the competitive equilibrium allocations as the costs of search and bargaining become
negligible. The findings are that the relationship is sensitive to the assumptions on the flow
of agents entering the market, the amount of information available to agents, the matching
technology, and the discounting. Osborne and Rubinstein (1990) and Binmore et al. (1992)
are excellent surveys of the literature on bargaining in markets.
Studying the relationship between our model and the Walrasian setting is only justifiable
for buyer-seller networks in which each buyer is connected to each seller. As argued above,
for such networks our conclusions coincide with the ones of Rubinstein and Wolinsky (1985).
Other buyer-seller networks exhibit frictions precluded in competitive environments–some
pairs of buyers and sellers are not connected, and can never be matched to bargain. The
Walrasian analysis is ineffective in decentralized markets where the impact of excess supply
or demand in the far future on current prices is properly discounted (as discussed by Binmore
and Herrero 1988a), and agent encounters are restricted by network connections determined
by geographical proximity, social relationships, moral or political principles, sympathy, etc.
Our model is related to the one of Polanski (2007). We differ from Polanski regarding the
choice of the matching technology and the steady state assumption. In Polanski’s model a
maximum number of pairs of connected players are selected to bargain every period, and all
pairs that reach agreements are removed from the network without replacement. Polanski
claims that his game has a unique subgame perfect equilibrium, but his proof is flawed, and
the validity of the result is unclear.18 Only uniqueness of the Markov perfect equilibrium is
proven.
18A
correct proof showing subgame perfect equilibrium uniqueness in the case of a network with one seller
and a number of buyers is provided by Rubinstein and Wolinsky (1990) (Proposition 4(i)). The proof is
lengthy and the authors explain that their approach cannot be easily extended to more general networks.
Rubinstein and Wolinsky (1990) also point out that an alternative matching technology whereby sellers
choose their bargaining partners at each round yields multiple equilibria. The conclusion extends to our
model if the game is modified such that at each round one player is randomly selected to make a proposal
to a player of his choice.
BARGAINING ON NETWORKS
43
Corominas-Bosch (2004) considers a model in which buyers and sellers connected through
a network alternate in making public offers that may be accepted by any of the responders
connected to a specific proposer. As in Polanski (2007), players who reach agreements are
removed from the network without replacement and the matching technology is chosen such
that when there are multiple possibilities to match buyers and sellers (that is, there are
multiple agents proposing or accepting identical prices) the maximum number of transactions occur. Corominas-Bosch notes that the model features equilibrium multiplicity, and
provides a characterization of the networks that implement the Walrasian outcome as a subgame perfect equilibrium. We find the assumption of the efficient matching technology in
both Corominas-Bosch (2004) and Polanski (2007) inconsistent with the goal of modeling
a decentralized market mechanism based on bilateral bargaining. The market forces that
would organize the matches in order to achieve maximal total surplus are not explicitly
modeled by way of decentralized self-interested strategic choices by the agents.
Centralized trading mechanisms may be employed in order to implement efficient matching
outcomes. Kranton and Minehart (2001) study a model similar to the one of CorominasBosch (2004). The valuations of the buyers are heterogeneous, and sellers are non-strategic.
Prices are determined by the ascending-bid auction mechanism designed by Demange, Gale
and Sotomayor (1986). Kranton and Minehart show that for a given link pattern the unique
equilibrium of the auction mechanism using strategies that are not weakly dominated leads to
an efficient allocation of the goods. The result enables the study of the relationship between
overall economic welfare and the incentives of buyers to form the network when links are
costly to maintain. For small link costs only efficient link patterns emerge in equilibrium.
Calvo-Armengol (2001, 2003) introduces the following model of sequential bargaining on
a network. In the first round a proposer is randomly chosen, and a responder is randomly
selected among his neighbors in the network. Every round that ends in disagreement is followed by a new round where the disagreeing responder makes an offer to a randomly chosen
neighbor. The game ends when one agreement is reached by two bargaining partners. The
unique stationary subgame perfect equilibrium specifies offer and acceptance behavior identical to the one in the two-person Rubinstein (1982) game. Therefore, the network structure
has no effect on the ex-post equilibrium partition, and only influences the bargaining power
index through the probabilities that an agent is selected to be the proposer or the responder
44
MIHAI MANEA
in the first round of the game, which are defined ad hoc in the model. Furthermore, the
assumptions that a proposer forfeits to the responder his bargaining ability in the case of
disagreement and that the game ends as soon as one pair reaches an agreement are unrealistic
in applications.
The formal models of non-cooperative two player bargaining were introduced in the pioneering work of Stahl (1972), Rubinstein (1982) and Binmore (1987). Pairwise stability was
introduced by Jackson and Wolinsky (1996). Jackson (2001) surveys the extensive related
literature.
Appendix A. Proofs
Proof of Lemma 2. Fix the common discount factor for all players δ. Let (σiτ )i∈N,τ ≥0 be a
steady state stationary subgame perfect Nash equilibrium of Γδ . For every i, each player
iτ receives the same expected payoff, ṽi , in any subgame ht ∈ H(iτ ) when the game is
played according to (σiτ )i∈N,τ ≥0 . It must be that the equilibrium specifies that any player j
accept any offer larger than δṽj , and reject any offer smaller than δṽj . Then, if ij ∈ G and
δ(ṽi + ṽj ) < 1, when i is chosen to propose to j, it must be that in equilibrium i offers δṽj
to j and j agrees with probability 1. Similarly, if ij ∈ G and δ(ṽi + ṽj ) > 1 and i is chosen
to propose to j, in equilibrium i makes an offer that is rejected by j with probability 1.
By the analysis above, the payoff vector (ṽi )i∈N solves the system
ṽi =
1 X
2e − ei
δṽi +
max(1 − δṽj , δṽi ), ∀i ∈ N
2e
2e
{j|ij∈G}
(e and ei are defined in Section 3).
Therefore, ṽ is a fixed point of the function f δ = (f1δ , f2δ , . . . , fnδ ) : Rn → Rn defined by
fiδ (v) =
2e − ei
1 X
δvi +
max(1 − δvj , δvi ).
2e
2e
{j|ij∈G}
Note that f δ maps [0, 1]n into itself.
We show that f δ is a contraction with respect to the sup norm || · ||∞ on Rn , defined by
||z||∞ = maxi=1,n |zi |. Specifically,
||f δ (v) − f δ (u)||∞ ≤ δ||v − u||∞ , ∀v, u ∈ [0, 1]n .
We need to prove that for each i, |fiδ (v) − fiδ (u)| ≤ δ||v − u||∞ .
BARGAINING ON NETWORKS
45
By f δ ’s definition,
|fiδ (v) − fiδ (u)| ≤
2e − ei
1 X
δ|vi − ui | +
| max(1 − δvj , δvi ) − max(1 − δuj , δui )|
2e
2e
{j|ij∈G}
≤
2e − ei
1 X
δ|vi − ui | +
max(|1 − δvj − (1 − δuj )|, |δvi − δui |)
2e
2e
{j|ij∈G}
=
1 X
2e − ei
δ|vi − ui | +
δ max(|vj − uj |, |vi − ui |)
2e
2e
{j|ij∈G}
≤
2e − ei
1 X
δ||v − u||∞ +
δ||v − u||∞
2e
2e
{j|ij∈G}
= δ||v − u||∞ ,
where the second inequality follows from Lemma 1, and the others from the definition of the
sup norm and algebraic manipulation.
Because f δ is a contraction, it has exactly one fixed point; denote it ṽ ∗δ . Since f δ ([0, 1]n ) ⊂
[0, 1]n , it follows that ṽ ∗δ ∈ [0, 1]n . Steady state stationary subgame perfect equilibrium
payoffs of Γδ need to be fixed points for f δ , hence they are uniquely given by ṽ ∗δ .
Proof of Theorem 2. By the definition of G∗δ , ij ∈ G∗δ ⇐⇒ ij ∈ G & max(1−δvj∗δ , δvi∗δ ) =
1 − δvj∗δ . Since v ∗δ is a fixed point of f δ , v ∗δ solves the n × n linear system
∗δ
2e − eG
1 X
i
vi =
δvi +
(1 − δvj ), ∀i = 1, n.
2e
2e
∗δ
ij∈G
For every δ ∈ (0, 1) and every subnetwork H of G, consider more generally the n × n linear
system
(A.1)
vi =
2e − eH
1 X
i
δvi +
(1 − δvj ), ∀i = 1, n,
2e
2e ij∈H
where eH
i denotes the number of H links that player i has. It is easy to check that the system
∗δ
A.1 has a unique solution v δ,H , and the solution belongs to [0, 1]n . In particular, v ∗δ = v δ,G .
The simplest path to show uniqueness of the solution to A.1 is analytical rather than
linear algebraic, by showing that the function hδ,H : Rn → Rn defined by
hδ,H
i (v) =
2e − eH
1 X
i
δvi +
(1 − δvj ), ∀i = 1, n
2e
2e ij∈H
46
MIHAI MANEA
is a contraction with respect to the sup norm on Rn . The proof is omitted as it is very
similar (but simpler, as it does not involve Lemma 1) to the proof that f δ is a contraction.
Therefore, for all δ ∈ (0, 1) and all non-empty subnetworks H of G the corresponding system
A.1 is non-singular.
All entries in the augmented matrix of the linear system A.1 are linear functions of δ.
Then for each i ∈ N the unique solution viδ,H is given by Cramer’s rule, as the ratio of two
determinants that are polynomials in δ of degree at most n,
viδ,H = PiH (δ)/QH
i (δ).
(A.2)
QH
i (δ) 6= 0 for all δ ∈ (0, 1) and all non-empty subnetworks H of G as the corresponding
system A.1 is non-singular.
¯ be the set of δ for which there exist i, j, H with δ(v δ,H + v δ,H ) = 1. Fix i, j, H. The
Let ∆
i
j
equation δ(viδ,H + vjδ,H ) = 1 is equivalent to
H
H
1 = δ(viδ,H + vjδ,H ) = δ(PiH (δ)/QH
i (δ) + Pj (δ)/Qj (δ)).
If the equation above has an infinite number of solutions δ, it follows that
H
H
H
H
δ(PiH (δ)QH
j (δ) + Pj (δ)Qi (δ)) = Qi (δ)Qj (δ)
is a polynomial identity, i.e., it holds for all δ. In particular, equality needs to hold in the
1/3,H
two display equations above for δ = 1/3, implying that 1/3(vi
1/3,H
vi
1/3,H
+ vj
1/3,H
+ vj
) = 1, hence
= 3, a contradiction with v 1/3,H ∈ [0, 1]n .
Since for fixed (i, j, H) the equation δ(viδ,H + vjδ,H ) = 1 has a finite number of solutions δ,
¯ is finite. The equality
and the number of triplets (i, j, H) is finite, it follows that the set ∆
∗δ
v ∗δ = v δ,G implies that the set of δ for which there exist i, j s.t. δ(vi∗δ + vj∗δ ) = 1 is included
¯
in ∆.
Proof of Theorem 4. Let M be a G∗ -independent set of players, and set L = LG∗ (M ). Fix
δ > δ, with δ specified as in Theorem 3. A pair of players connected in G agree in equilibrium
in Γδ when matched to bargain if and only if they are connected in G∗ .
BARGAINING ON NETWORKS
47
By Lemma 4, for all i in M ,
vi∗δ =
(A.3)
X 1
1
max(1 − δvi∗δ − δvj∗δ , 0)
1−δ
2e
{j|ij∈G}
=
(A.4)
1
1−δ
X
{j|ij∈G, j∈L}
1
max(1 − δvi∗δ − δvj∗δ , 0),
2e
/ L.
since i only has G∗ links to players in L, so max(1 − δvi∗δ − δvj∗δ , 0) = 0 if ij ∈ G, j ∈
By Lemma 4, for all j in L,
vj∗δ =
(A.5)
≥
(A.6)
1
1−δ
1
1−δ
X
{k|kj∈G}
1
max(1 − δvk∗δ − δvj∗δ , 0)
2e
X
{i|ij∈G, i∈M }
1
max(1 − δvi∗δ − δvj∗δ , 0).
2e
Adding up A.3-A.4 for all i ∈ M and A.5-A.6 for all j ∈ L we obtain
X
vi∗δ =
i∈M
X
vj∗δ ≥
j∈L
1
1−δ
1
1−δ
X
{(i,j)|ij∈G, i∈M, j∈L}
X
{(i,j)|ij∈G, i∈M, j∈L}
1
max(1 − δvi∗δ − δvj∗δ , 0)
2e
1
max(1 − δvi∗δ − δvj∗δ , 0),
2e
hence
X
(A.7)
vj∗δ ≥
X
vi∗δ .
i∈M
j∈L
Taking the limit δ → 1,
X
(A.8)
vj∗ ≥
j∈L
X
vi∗ .
i∈M
We can now manipulate the inequality A.8,
|L| max vj∗ ≥ |M | min vi∗ .
j∈L
i∈M
Player i ∈ arg mini∈M vi∗ is connected in G∗ to a player j̃ ∈ L (Lemma 3), hence by
Proposition 1, vi∗ + vj̃∗ = 1. Therefore, mini∈M vi∗ = 1 − vj̃∗ ≥ 1 − maxj∈L vj∗ .
Also, any j ∈ arg maxj∈L vj∗ is connected in G∗ to a player ĩ ∈ M (it is crucial to note here
∗
that we defined L to be LG (M ), rather than LG (M ); by definition any player in LG∗ (M )
has a G∗ link to some player in M , but this is not necessarily true for players in LG (M )),
and vĩ∗ + vj∗ = 1 by Proposition 1. Hence maxj∈L vj∗ = 1 − vĩ∗ ≤ 1 − mini∈M vi∗ .
48
MIHAI MANEA
We proved that mini∈M vi∗ = 1 − maxj∈L vj∗ . It follows that
|L| max vj∗ ≥ |M |(1 − max vj∗ ),
j∈L
j∈L
which is equivalent to
max vj∗ ≥
j∈L
|M |
.
|M | + |L|
Moreover,
min vi∗ = 1 − max vj∗ ≤ 1 −
i∈M
j∈L
|M |
|L|
=
.
|M | + |L|
|M | + |L|
Recalling that L = LG∗ (M ), the last two display equations yield the desired conclusion. Proof of Lemma 5. Let 1 > r := minM ∈I(H) |LH (M )|/|M |, and M 0 , M 00 be two H-independent
sets achieving the minimum. Decompose the set M 0 as the union of the sets A2 = M 0 ∩ M 00 ,
A1 = (M 0 \M 00 )\LH (M 00 ), that is, the set of players in M 0 \M 00 who do not have any H links to
M 00 , and A4 = (M 0 \M 00 )∩LH (M 00 ), that is, the set of players in M 0 \M 00 who do have H links
to M 00 . Similarly, decompose the set M 00 as the union of the sets A2 , A3 = (M 00 \M 0 )\LH (M 0 )
and A5 = (M 00 \ M 0 ) ∩ LH (M 0 ). Let B2 = LH (A2 ), B1 = LH (A1 ) \ B2 , B3 = LH (A3 ) \ B2 .
Denote |Ai | = ai , |Bj | = bj for i = 1, 5, j = 1, 3.
Note that LH (A4 ) ⊃ A5 and LH (A5 ) ⊃ A4 . We have
LH (A1 ∪ A2 ∪ A3 ) = B1 ∪ B2 ∪ B3
LH (M 0 ) = LH (A1 ∪ A2 ∪ A4 ) ⊃ B1 ∪ B2 ∪ A5
LH (M 00 ) = LH (A2 ∪ A3 ∪ A5 ) ⊃ B2 ∪ B3 ∪ A4
LH (A2 ) = B2.
The middle two expressions may be strict inclusions as the players in A4 (A5 ) may have H
links to players not in B1 ∪ B2 ∪ A5 (B2 ∪ B3 ∪ A4 ).
Since A1 ∪ A2 does not have H links to M 00 it follows that (B1 ∪ B2 ) ∩ A5 ⊂ (B1 ∪ B2 ) ∩
M 00 = ∅; analogously (B2 ∪ B3 ) ∩ A4 = ∅. By definition, B1 ∩ B2 = ∅ and B2 ∩ B3 = ∅.
It follows that the triplets (B1 , B2 , A5 ) and (B2 , B3 , A4 ) are made of disjoint sets, hence
|B1 ∪ B2 ∪ A5 | = b1 + b2 + a5 and |B2 ∪ B3 ∪ A4 | = b2 + b3 + a4 . The intersection of B1 and
B3 may be non-empty, hence |B1 ∪ B2 ∪ B3 | ≤ b1 + b2 + b3 .
BARGAINING ON NETWORKS
49
By the definitions for r, M 0 , M 00 , the identities and inclusions above imply19
|LH (A1 ∪ A2 ∪ A3 )|
b1 + b2 + b3
≥
≥ r
a1 + a2 + a3
|A1 ∪ A2 ∪ A3 |
r =
|LH (M 0 )|
b 1 + b 2 + a5
≥
|M 0 |
a1 + a2 + a4
r =
|LH (M 00 )|
b 2 + b 3 + a4
≥
00
|M |
a2 + a3 + a5
b2
|LH (A2 )|
=
≥ r,
a2
|A2 |
which can be rewritten
(A.9)
b1 + b2 + b3 ≥ ra1 + ra2 + ra3
(A.10)
ra1 + ra2 + ra4 ≥ b1 + b2 + a5
(A.11)
ra2 + ra3 + ra5 ≥ b2 + b3 + a4
(A.12)
b2 ≥ ra2 ,
or adding up all the inequalities and canceling terms,
(A.13)
(r − 1)(a4 + a5 ) ≥ 0.
Since r < 1, it follows that a4 + a5 = 0, hence there are no H links joining players in M 0
with players in M 00 , so the set M 0 ∪ M 00 is H-independent.
Since the sum of all weak inequalities A.9-A.12 leads to an equality, it follows that they
should all hold with equality. In particular,
|LH (A1 ∪ A2 ∪ A3 )|
b1 + b2 + b3
=
= r.
a1 + a2 + a3
|A1 ∪ A2 ∪ A3 |
Since A4 = A5 = ∅, it follows that A1 ∪ A2 ∪ A3 = M 0 ∪ M 00 , hence
|LH (M 0 ∪ M 00 )|
= r,
|M 0 ∪ M 00 |
which finishes the proof.
19The
case a1 + a2 + a3 = 0 is not possible, as it would imply LH (M 0 ) ⊃ M 00 and LH (M 00 ) ⊃ M 0 , leading
to r ≥ 1. If a2 = 0, the bottom inequality becomes irrelevant for the proof.
50
MIHAI MANEA
Proof of Theorem 5, case s = s. This case is only relevant when Ns 6= ∅, which is assumed
in the claims below. Note that rs ≥ 1.
Claim 5.5. vk∗ ≥ 1/2, ∀k ∈ Ns
Again, let xs = mini∈Ns vi∗ , M s = arg mink∈Ns vk∗ and Ls = LGs (M s ). We show that
xs ≥ 1/2 by contradiction. Suppose that xs < 1/2.
By arguments parallel to the ones in Claim 5.2, it can be shown that any j ∈ Ls only has
G∗ links to players in M s , and must have limit equilibrium payoff vj∗ = 1 − xs > 1/2. Then
Ls is G∗ -independent by Proposition 1.
Since LG∗ (Ls ) ⊂ M s and Ls is G∗ -independent, by Theorem 4,
xs =
max vi∗ ≥
i∈LG∗ (Ls )
|Ls |
.
|M s | + |Ls |
Recalling that Ls = LGs (M s ), we can rewrite the inequality above as
|LGs (M s )|
xs
≥
.
1 − xs
|M s |
If xs < 1/2, then xs /(1 − xs ) < 1, hence
1>
|LGs (M s )|
,
|M s |
leading to a contradiction with rs ≥ 1.
Claim 5.6. vk∗ ≤ 1/2, ∀k ∈ Ns
We showed that for all k ∈ Ns , vk∗ ≥ 1/2. Fix k ∈ Ns . By Lemma 6 there exist no G∗
links from k to players in M1 ∪ M2 ∪ . . . ∪ Ms−1 . By Proposition 1, there exist no G∗ links
from k to players j ∈ L1 ∪ L2 ∪ . . . ∪ Ls−1 since for these players vj∗ ≥ 1 − xs−1 > 1/2,
and we need vk∗ ≥ 1/2. Therefore, k only has G∗ links to players in Ns . By Claim 5.5 the
limit equilibrium payoff of any player in Ns is at least 1/2, and Proposition 1 implies that
vk∗ ≤ 1/2.
By Claims 5.5 and 5.6, vk∗ = 1/2 for all k ∈ Ns .
Proof of Theorem 7. (i) Let G̃ be a network with ij ∈ G̃ and let G = G̃ − ij be the network
obtained by deleting link ij from G̃. Denote by A(G) = (rs , xs , Ms , Ls , Ns , Gs )s=1,2,...,s and
BARGAINING ON NETWORKS
51
A(G̃) = (r̃s̃ , M̃s̃ , L̃s̃ , Ñs̃ , G̃s̃ )s̃=1,2,...,s̃ the outcomes of the algorithm for computing the limit
equilibrium payoffs for the bargaining games on the networks G and G̃ respectively. Let
s(i) and s(j) be the steps at which players i and j are removed in the algorithm A(G), i.e.,
s(i) = max{s|i ∈ Ns }, s(j) = max{s|j ∈ Ns }.
Without loss of generality, we may assume that s(i) ≤ s(j) and set out to prove both of
the following inequalities,
vi∗ (G̃) ≥ vi∗ (G) and vj∗ (G̃) ≥ vj∗ (G).
If i ∈ Ls(i) or s(i) = s(j) then the algorithms A(G) and A(G̃) lead to identical outcomes.
Therefore, we can assume that i ∈ Ms(i) and s(i) < s(j). In particular, s(i) < s and rs(i) < 1.
Note that the outcomes of the algorithms A(G) and A(G̃) are identical for steps 1, . . . , s(i)−
1 and r̃s(i) ≥ rs(i) . Therefore r̃s̃ ≥ rs(i) for s̃ ≥ s(i) and vk∗ (G̃) ≥ rs(i) /(1 + rs(i) ) = vi∗ (G) for
all k ∈ Ns(i) = Ñs(i) . Hence vi∗ (G̃) ≥ vi∗ (G).
We next show that vj∗ (G̃) ≥ vj∗ (G). There are three cases to consider: j ∈ Ls(j) , j ∈ Ms(j) ,
and j ∈ Ns . We only solve the former case; the other two can be handled by similar methods.
Henceforth we focus on the case s(i) < s(j), i ∈ Ms(i) , j ∈ Ls(j) . The following lemma will
be used repeatedly.
Lemma 7. Suppose that (rs , xs , Ms , Ls , Ns , Gs )s=1,2,...,s is the outcome of the algorithm A(G).
For any s < s, and any non-empty L0 ⊂ Ls ,
|L0 |
≤ rs .
|LGs (L0 ) ∩ Ms |
Proof. Let M 0 = LGs (L0 ) ∩ Ms . Then Ls = LGs (Ms ) and (Ms \ M 0 ) ∩ LGs (L0 ) = ∅ imply that
LGs (Ms \ M 0 ) ⊂ Ls \ L0 . Since
rs =
min
M ⊂Ns , M ∈I(G)
|LGs (M )|/|M |
it follows that |Ls \L0 |/|Ms \M 0 | ≥ rs . But rs = Ls /Ms , hence |Ls \L0 |/|Ms \M 0 | ≥ |Ls |/|Ms |,
which implies that |L0 |/|M 0 | ≤ |Ls |/|Ms | = rs .
Let s̃(j) be the step where player j is removed in the algorithm A(G̃), i.e., s̃(j) =
s(j)
max{s̃|j ∈ Ñs̃ }. For all s̃0 = s(i)−1, s(i), . . . , s̃(j) we show that r̃s̃0 ≤ rs(j) , M̃s̃0 ∩(∪s=s(i) Ls ) =
52
MIHAI MANEA
s(j)
∅ and L̃s̃0 ∩ (∪s=s(i) Ms ) = ∅ by induction on s̃0 . The base induction case s̃0 = s(i) − 1 is
obviously true. Suppose the induction hypothesis holds for s̃0 = s(i) − 1, s(i), . . . , s̃ − 1. We
show that it holds for s̃0 = s̃.
Part 7.1. We prove the first part of the induction step, r̃s̃ ≤ rs(j) . Let M̃ = M̃s(i) ∪. . .∪ M̃s̃−1
and L̃ = LG̃s(i) (M̃ ) = L̃s(i) ∪ . . . ∪ L̃s̃−1 . Note that L̃ = LG̃s(i) (M̃ ) = LGs(i) (M̃ ) since i, j 6∈ M̃
(i ∈ M̃ ⇒ j ∈ L̃, a contradiction with s̃ − 1 < s̃(j) and the definition of s̃(j); j ∈ M̃ leads
to a similar contradiction).
Fix s ∈ s(i), s(j). Let L0 = Ls \ LGs (Ms ∩ M̃ ). Note that LGs (L0 ) ∩ Ms ⊂ Ms \ M̃ . Lemma
7 applied to step s of A(G) with L0 defined above implies that
|Ls | − |LGs (Ms ∩ M̃ )|
≤ rs .
|Ms \ M̃ |
(The argument below is only necessary and relevant for s with Ms ∩ M̃ 6= ∅, M̃ .)
Note that LGs (Ms ∩ M̃ ) ⊂ LGs(i) (M̃ ) ∩ Ls , hence
|Ls | − |LGs(i) (M̃ ) ∩ Ls |
≤ rs .
|Ms \ M̃ |
Since rs ≤ rs(j) for all s ∈ s(i), s(j), the set of inequalities above imply
Ps(j)
Gs(i)
(M̃ )
s=s(i) (|Ls | − |L
Ps(j)
s=s(i) |Ms \ M̃ |
∩ Ls |)
≤ rs(j) ,
or equivalently,
s(j)
s(j)
| ∪s=s(i) Ls | − |LGs(i) (M̃ ) ∩ (∪s=s(i) Ls )|
s(j)
| ∪s=s(i) Ms \ M̃ |
≤ rs(j) ,
which can be rewritten
s(j)
| ∪s=s(i) Ls \ LGs(i) (M̃ )|
s(j)
| ∪s=s(i) Ms \ M̃ |
s(j)
≤ rs(j) .
s(j)
s(j)
s(j)
But LG̃s̃ (∪s=s(i) Ms \ M̃ ) ⊂ LG̃s(i) (∪s=s(i) Ms ) = LGs(i) (∪s=s(i) Ms ) = ∪s=s(i) Ls (the first
equality follows from the fact that G̃ = G + ij), and G̃s̃ does not contain any players in
L̃ = LGs(i) (M̃ ), hence the inequality above implies that
s(j)
|LG̃s̃ (∪s=s(i) Ms \ M̃ )|
s(j)
| ∪s=s(i) Ms \ M̃ |
≤ rs(j) .
BARGAINING ON NETWORKS
53
s(j)
s(j)
Since ∪s=s(i) Ms \ M̃ is G̃-independent, it follows that r̃s̃ ≤ rs(j) < 1. Note that ∪s=s(i) Ms \ M̃
s(j)
is non-empty as it contains i, and it is contained in Ñs̃ since for s̃0 < s̃, L̃s̃0 ∩ (∪s=s(i) Ms ) = ∅
by the induction hypothesis.
s(j)
Part 7.2. We prove the second part of the induction step, M̃s̃ ∩ (∪s=s(i) Ls ) = ∅, by contras(j)
diction. Suppose that M̃s̃ ∩ (∪s=s(i) Ls ) 6= ∅, and let s0 be the smallest index s ∈ s(i), s(j)
for which M̃s̃ ∩ Ls 6= ∅. Define B = M̃s̃ ∩ Ls0 and A = LGs0 (B) ∩ Ms0 .
We argue that A ⊂ Ñs̃ . Fix l ∈ A. Player l has a G-link to a player k ∈ B. If l is
removed at step s̃0 < s̃ in the algorithm A(G̃) then l ∈ M̃s̃0 by the induction hypothesis
s(j)
(L̃s̃0 ∩ (∪s=s(i) Ms ) = ∅). Then k ∈ L̃s̃0 , a contradiction with k ∈ M̃s̃ . Therefore, l ∈ Ñs̃ .
Note that LG̃s̃ (A) ∩ M̃s̃ ⊂ B ∪ {i} since players in Ms0 may only have G-links to players
in L1 ∪ L2 ∪ . . . ∪ Ls0 , and M̃s̃ ∩ (L1 ∪ L2 ∪ . . . ∪ Ls0 ) = B by the definition of s0 . If i ∈ A
then we could have j ∈ LG̃s̃ (A) ∩ M̃s̃ . Lemma 7 applied for step s̃ of algorithm A(G̃) with
L0 = A and Part 7.1 imply that
|A|
≤ r̃s̃ < 1,
|B| + 1
hence |A| < |B| + 1, or |A| ≤ |B|.
Since A = LGs0 (B) ∩ Ms0 , Lemma 7 applied to step s0 of algorithm A(G) with L0 = B
implies that
|B|
≤ rs0 < 1,
|A|
s(j)
hence |A| > |B|, a contradiction with |A| ≤ |B|. Therefore M̃s̃ ∩ (∪s=s(i) Ls ) = ∅.
s(j)
Part 7.3. To establish the third part of the induction hypothesis, L̃s̃ ∩ (∪s=s(i) Ms ) = ∅, we
s(j)
proceed by contradiction. Suppose that k ∈ L̃s̃ ∩ (∪s=s(i) Ms ). It should be that there exists a
s(j)
player l ∈ M̃s̃ sharing a G̃s̃ link with k. But k ∈ ∪s=s(i) Ms can only have G̃s̃ links to players
s(j)
s(j)
s(j)
in ∪s=s(i) Ls , hence l ∈ ∪s=s(i) Ls . Therefore, l ∈ M̃s̃ ∩ (∪s=s(i) Ls ), a contradiction with Part
7.2.
In particular, for s̃ = s̃(j) the induction hypothesis implies that j ∈ L̃s̃(j) and r̃s̃(j) ≤ rs(j) .
Then vj∗ (G̃) = 1/(1 + r̃s̃(j) ) ≥ 1/(1 + rs(j) ) = vj∗ (G).
(ii) To prove the “if” part of the statement, let G̃ be an equitable network. Part (i) shows
that G̃ is unilaterally stable with respect to the bargaining limit equilibrium payoff. Note
that by Theorem 6 when a link is added to an equitable network another equitable network
54
MIHAI MANEA
obtains. Hence G̃ + ij is equitable, and vi∗ (G̃ + ij) = vi∗ (G̃) = 1/2 for any pair of players
i 6= j ∈ N . Therefore, G̃ is pairwise stable with respect to the bargaining limit equilibrium
payoff.
To prove the “only if” part of the statement, let G̃ be a network that is pairwise stable
with respect to the bargaining limit equilibrium payoff. Suppose that G̃ is not equitable.
Then, if (r̃s̃ , M̃s̃ , L̃s̃ , Ñs̃ , G̃s̃ )s̃=1,2,...,s̃ denotes the outcome of the algorithm A(G̃), there exist
i, j ∈ M̃1 such that vi∗ (G̃) = vj∗ (G̃) < 1/2 (|M̃1 | ≥ 2). The limit equilibrium payoffs of
players i and j in the game on network G̃ + ij satisfy vi∗ (G̃ + ij) ≥ vi∗ (G̃) and vj∗ (G̃ + ij) ≥
vj∗ (G̃) by part (i) of the theorem. By Proposition 1, vi∗ (G̃ + ij) + vj∗ (G̃ + ij) ≥ 1. Hence,
vi∗ (G̃ + ij) + vj∗ (G̃ + ij) > vi∗ (G̃) + vj∗ (G̃), which together with vi∗ (G̃ + ij) ≥ vi∗ (G̃) and
vj∗ (G̃ + ij) ≥ vj∗ (G̃), leads to a violation of the pairwise stability of G̃.20 The contradiction
proves that G̃ is equitable.
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