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Rev. Econ. Design (2008) 11:295–319DOI 10.1007/s10058-007-0037-0
ORIGINAL PAPER
Bargaining: separately or together?
Alp E. Atakan
Received: 29 May 2007 / Accepted: 4 November 2007 / Published online: 13 December 2007© Springer-Verlag 2007
Abstract This paper explores conditions under which economic agents will wantto bargain collectively instead of individually with a common third party—when, forexample, two firms (or unions), who are bargaining with the same, indispensable,outside party, will want to merge and bargain as one. I use a non-cooperative sequen-tial bargaining model to analyze this question. Previous work has shown that agentsprefer to bargain collectively if they are substitutes for each other in production. Thisresult, however, depends on an exogenously fixed sequence of bargaining. I allow thebargaining sequence to be determined endogenously and investigate how incentivesfor collective bargaining vary with heterogeneity when delay is costly. The previousresults are not robust when the agents are substitutes. In particular, substitute agentsprefer individual to collective bargaining if they are heterogeneous and sufficientlypatient. In the presence of transaction costs, substitutability of agents is no longer thesole determinant of collectivization. Rather, the degree of heterogeneity in produc-tion, in conjunction with the degree of substitutability between agents determine theincentives for collective action.
Keywords Bargaining · Unions · Coalition formation · Hold-up
JEL Classification C73 · C78 · D7
I thank Prajit Dutta, Richard E. Ericson and Levent Koçkesen for patient assistance and advice. This paperhas benefitted from discussions with Atila Abdülkadiroglu, Massimiliano Amarante, Emanuele Gerratana,Sanghoon Lee, Brendan O’Flaherty, Sangwon Park, Michael Riordan, Paolo Siconolfi, and theparticipants of the 2002 SED Conference held at New York University. All remaining errors are my own.
A. E. Atakan (B)MEDS, Kellogg School of Management, Northwestern University,2001 Sheridan Road, Evanston, IL 60208, USAe-mail: [email protected]
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296 A. E. Atakan
1 Introduction
Economic agents frequently develop arrangements, such as unions, syndicates oralliances, that enhance their bargaining positions. These arrangements have long beena subject of economic inquiry. Consider a situation where agents on one-side of themarket (the agents), bargain with a common, indispensable institution (the princi-pal) on the other side of the market. Commonly cited examples of such a situationinclude two unions bargaining with the same firm (Stole and Zwiebel 1996b; Hornand Wolinsky 1988b), or a number of downstream firms bargaining with the sameupstream monopolist (Horn and Wolinsky 1988a; Davidson 1988; Chipty and Snyder1999). Authors have explored conditions under which the agents will want to bargaincollectively with the principal. Results have shown that if the agents are substitutes foreach other in the production process of the principal, then they should bargain collec-tively. Conversely, if they are complements, then they should bargain individually. Thefinding appears in the cooperative game theory literature (see, for example, Gardner1977; Guesnerie 1977; Legros 1987; Segal 2003) as well as the non-cooperative bar-gaining literature (for example, Stole and Zwiebel 1996b in an extensive form gamethat implements the Shapley value and Horn and Wolinsky 1988b in a game that imple-ments the Nucleolus). The main intuition for this finding is as follows: The agents, asa union, bargain over their average contribution to the principal’s production whereasindividually they bargain over their marginal contribution. When the agents are sub-stitutes, i.e., when the principal’s production process exhibits decreasing returns, thenthe agents’ average contribution exceeds the marginal and the result follows. On theother hand, if the agents are complements then their marginal contribution exceedsthe average and consequently they prefer to bargain individually.
This paper adopts a non-cooperative framework, but departs from previous litera-ture in two dimensions: it allows for an endogenous bargaining sequence (or protocol)and explicitly incorporates bargaining (delay) costs. Most previous non-cooperativebargaining models exogenously fix a bargaining sequence (for example, Horn andWolinsky 1988b; Gul 1989) and analyze incentives for collective bargaining as thediscount factor approaches one, or in other words, as bargaining costs disappear. Theabsence of bargaining costs may not seem troubling since the presumption is that theycan only enhance the incentives for collectivization in a straightforward way. How-ever, here I show that bargaining costs can matter due to strategic considerations whenthe bargaining sequence is chosen endogenously. In particular, bargaining costs canactually provide an incentive for substitute agents to bargain separately.
In the case where agents are complements, the results here confirm previous find-ings. Complementary agents should bargain separately if bargaining costs are notprohibitive. Also, with substitutable agents that are homogeneous, allowing for anendogenous bargaining sequence alone does not alter previous findings. However, theprevious result is not robust to all possible bargaining protocols. Specifically, thereare subgame perfect equilibria in the model presented here, in which substitute agentsprefer to bargain individually. Nevertheless, the bargaining sequence used in Horn andWolinsky (1988b) or a sequence that delivers the Shapley value are equilibria of thegame with homogeneous agents. With heterogeneous agents, however, allowing foran endogenous bargaining sequence seriously alters the previous finding. If the agents
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Bargaining: separately or together? 297
are sufficiently heterogeneous in production, then substitute agents prefer to bargainseparately in the unique equilibrium. This finding is in clear contrast to previous resultsand has a clear implication: when delay is costly, substitutability between agents is nolonger the sole determinant for collectivization. Rather, the degree of heterogeneity inproduction, in conjunction with the degree of substitutability between agents jointlydetermine incentives for collective action.
In the game presented here, the principal chooses to meet one of the two agentsin each period, bargains bilaterally with this agent, and if an agreement is reached,the agent begins production, receives a payoff, and leaves the game. The disagree-ment points for the principal and agents, and consequently the division of the surplus,depend crucially on the sequence of bargaining that the principal chooses followinga disagreement. For example, consider the bargaining sequence used by Horn andWolinsky (1988b) in which the principal bargains with agent 2 after a disagreementwith agent 1 and vice versa. If the agents are substitutes, then this bargaining sequencepunishes an agent most severely for a disagreement. The punishment for disagreementis twofold: first, the agent remains idle for one period while the principal bargains withthe second agent; second, given that the principal reaches an agreement with the otheragent in the interim, the agent is left to bargain over her marginal product. The agentsare in a poor bargaining position since the marginal product is small compared to thejoint surplus. If this sequence is used by the principal, then collectivization by theagents would improve their bargaining position as well as allow them to economizeon bargaining costs.
For a contrasting example, consider the bargaining sequence in which the princi-pal bargains with agent 1 until an agreement is reached and only then moves on tobargain with agent 2. In this sequence, agent 1 has considerable bargaining power.After the principal reaches an agreement with agent 1, she will also hire agent 2.Consequently, agent 1 can hold-up the principal for a portion of agent 2’s marginalcontribution to the production process in addition to one-half of her own contribution.If this sequence is used, then although collectivization would allow substitute agentsto economize on bargaining costs, it would actually reduce their bargaining powervis-a-vis the principal.
The previous examples suggest that the division of surplus and consequently theincentives for collective bargaining depend on the choice of bargaining sequence bythe principal. The bargaining sequence that the principal can credibly choose, in turn,depends on the productivity difference between the two agents. If the two agents arehomogenous in terms of productivity, then the principal can choose the bargainingsequence in which she alternates between the two agents. However, with sufficientheterogeneity the principal cannot credibly commit to an alternating sequence. Forexample, if one agent is totally productive while the other is totally unproductive,then the principal must repeatedly bargain with the productive agent. The principalcannot credibly threaten to alternate between the two agents in case of a disagree-ment. Consequently, when there is enough heterogeneity, the high productivity agentcan leverage the principal’s inability to punish disagreements and extract more sur-plus. With the discount factor close to one, this effect is sufficiently strong and out-weighs the cost savings achieved by collective bargaining even when the agents aresubstitutes.
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The paper proceeds as follows: Sect. 2 outlines the bargaining model with one prin-cipal and two agents. Section 3 characterizes the stationary subgame perfect equilibriafor the game and analyzes incentives for collective bargaining. Section 4 generalizesthe results to a model with an arbitrary number of players and explores non-stationaryequilibria. All proof not included in the main text are in the appendix.
2 The model
The model under consideration is an infinitely repeated sequential bargaining game inwhich the proposer is chosen randomly.1 Player 0 is the principal, and I = {1, ..., N }denotes the set of agents (N = 2 for most of the discussion). Time is discrete, i.e.,n ∈ {1, 2, ...}, and each time period, n, is comprised of four subperiods. In the begin-ning of each period (n0), the principal chooses an agent form the set I as her bargainingpartner and the random proposer bargaining game, that is described below, is played.The principal can choose to continue bargaining with the same agent that she bar-gained with in the previous period, or alternatively, she can dismiss this agent andstart bargaining with another agent still in the game. In stage 1 of the bargaining game(n1), nature designates either the agent or the principal as the proposer. Both playersare equally likely to be chosen as the proposer. In stage 2 (n2), the proposer suggests atransfer, t , that will be paid by the principal to the particular agent i . In stage 3 (n3), theresponder accepts or rejects the proposed transfer. If the proposed transfer is accepted,then the agent and the principal consummate production, the agent receives the agreedtransfer, and leaves the game. If no agreement is reached, then the agent remains inthe game, and the game proceeds to the next period. This procedure is repeated untilall the agents have entered into an agreement with the principal. For an extensive formrepresentation of the stage game, that is repeated in each period, see Fig. 1.
In this bargaining game, the principal is assigned a special role as the residualclaimant: she chooses the bargaining sequence, bargains with the agents, pays themthe agreed upon transfers and receives the residual as profit. In Sect. 3, I briefly discussthe game where the residual claimant is not necessarily the principal but is chosen ran-domly from amongst all players in the game. For example, if agent 1 is chosen as theresidual claimant, she would be in the role of the principal in the stage game describedabove, and the principal would be in the role of any one of the agents.
The game where the agents bargain collectively as a union is exactly analogous. Inthis case, the game has only two participants. Consequently, the game reduces to thestandard two person sequential bargaining model. The principal meets with the unionuntil an agreement is reached. After an agreement is reached, the principal and theagents complete production and the game ends.Production: For clarity, I first outline the production technology for the two agentmodel. The more general set-up with N agents is included in Sect. 4. Production whenthe principal has both agents employed is normalized to equal one. The principal and
1 For the two player version of this model see Binmore (1980) or an overview can be found in Osborneand Rubinstein (1990).
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Bargaining: separately or together? 299
Fig. 1 Extensive form of thestage game
the agents cannot produce when they are alone.2 Production, when the principal hasonly agent i employed, is denoted vi ≤ 1, the production with the principal and agentj , given that agent i has already produced, is 1−vi .3 Let v = v1 and d = v−v2. All theinformation concerning the production process is contained in the pair (v, d). Withoutloss of generality, assume that d ≥ 0. The following formally defines complementarityand substitutability between agents.
Definition 1 If v1 ≥ 1−v2 = 1−v+d, then the agents are substitutes. Alternatively,if v1 ≤ 1 − v2 = 1 − v + d, then the agents are complements.
2 Let g be the characteristic function for the coalitional production game, then g ({0}) = g ({1}) = g ({2}) =g ({1, 2}) = 0, g ({0, 1}) = v1, g ({0, 2}) = v2 and g ({0, 1, 2}) = 1.3 The production figures can also be viewed as infinite streams. For example, suppose that agent 1 is signedin period one and agent 2 is signed in period two and they agree on transfers t1 and t2, respectively. In thiscase, total production is vi in the first period and equal to 1 from period two onwards. Also, the principalpays a transfer equal to t1 in the first period and t1 + t2 from period two onwards.
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300 A. E. Atakan
Preferences: The players have identical preferences and they only care about thetransfer that they receive. Players share a common discount factor δ. Assume thatδ = e−r"t where r > 0 is the instantaneous rate of time preference which is commonamong the players and "t is the time spent negotiating. Bargaining costs fall as "tapproaches zero and consequently δ approaches 1. For the remainder of the paper Iwork only with δ. This should be interpreted as changing "t , while holding constantthe rate of time preference r . "t is the number of periods for which an agent is the“insider” in the bargaining game, using the terminology of Shaked and Sutton (1984).Strategies: Analysis is restricted to equilibria where the players use stationary Markov-ian strategies, that is, to the stationary subgame perfect equilibria of the game (SSPE).This restriction is relaxed in Sect. 4. The strategies are taken as being stationary andMarkovian with respect to the state space S which is given by S = P(I ) × I ∪ {0},where P(I ) denotes the power set of I . A generic state variable s ∈ S specifies A ⊂ I ,the set of agents remaining in the game, and i ∈ I , the principal’s bargaining partner.As stated earlier, the principal has the option of bargaining with i or choosing a newagent as a bargaining partner for the current period. The transitions between statesare as follows: Let sn = (A, i) and assume that the principal chooses to bargain withagent j in period n, then sn+1 = (A\ j, j), if the agent and the principal reach anagreement, and sn+1 = (A, j), otherwise.4 Also, s0 = (I, 0) denotes the initial statewhere all agents remain in the game, but the principal is yet to bargain with any agent.
A stationary history, contains the state, s, and the with-in period play.5 In a stationaryhistory all information concerning play in previous periods is summarized by the statevariable. A stationary strategy for the principal is a function f0 =
(f 10 , f 2
0 , f 30
)from
the set of stationary histories, H , into ((I ) × R × {a, r}. First, the principal choosesf 10 (h) = p ∈ ((I ) which is a probability distribution that determines the princi-
pal’s bargaining partner for the period. Second, the principal makes a transfer offerf 20 (h) = t ∈ R to her current bargaining partner i , when designated as a proposer; and
third, she accepts or rejects the transfer offer, f 30 (h) ∈ {a, r}, when designated as a
responder. Likewise a stationary strategy for agent i is a function fi =(
f 1i , f 2
i
)from
the set of stationary histories, H , into R × {a, r}, where f 1i determines the transfer
offer that agent i will make and f 2i determines whether she will accept a particular
transfer offer made by the principal. A stationary subgame perfect equilibrium (SSPE)is an equilibrium in which all players use stationary strategies.The Question: Let f denote an SSPE profile of the game where the agents bargain indi-vidually. Denote by ui (f) the expected discounted transfer that agent i will receiveunder strategy profile f and let u(f) = ∑N
i=1 ui (f). When the agents bargain col-lectively, the union and the principal play a random proposer sequential bargaininggame over the total surplus which is normalized to equal one. The standard resulthas the two sides receiving one-half of the surplus in the unique SPE of this game.Consequently the question of collective versus individual bargaining then reduces to
4 The state is defined as (A, i) rather than just A in order to allow the principal to change bargainingpartners while using stationary strategies.5 For example, stationary history at time n2, h(n2) = (s, i, i, t), is one in which the state is s, the princi-pal has chosen to meet with agent i , nature has designated agent i as the proposer and she has proposedtransfer t .
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Bargaining: separately or together? 301
checking whether 12 ≥ u(f), i.e., does the expected transfer from bargaining collec-
tively exceed total expected transfers from bargaining individually given a particularSSPE profile f? In what follows I say the agents prefer individual bargaining to col-lective bargaining in a particular equilibrium f, if 1
2 ≤ u(f); and I say that the agentsprefer collective bargaining, if 1
2 ≥ u(f).The model presented here is closely related to the one analyzed in Horn and Wolin-
sky (1988b). However, the models differ in two respects. First, the sequence of bilateralbargaining sessions, between the principal and the agents, is determined endogenouslyby the choice of the principal in this paper. In Horn and Wolinsky (1988b), on the otherhand, the sequence of bargaining is fixed exogenously. The endogenous bargainingsequence is what accounts for the different predictions of the two models. Second, inHorn and Wolinsky (1988b) the principal has the option to delay production until anagreement has been reached with both agents. In this paper production begins imme-diately once the principal reaches an agreement with one of the agents. This seconddifference is less important: the principal only uses the option to delay production, inequilibrium, when the agents are complements; however, the two models’ qualitativepredictions coincide in the case where the agents are complements and the option todelay production is exercised.
3 Main results for the two agent model
This section analyzes the game where the agents bargain individually and the gamewhere they bargain as a union. The SSPE payoffs for the two games are characterizedand the parameter values for which bargaining as a union is preferred to bargainingindividually are determined.
3.1 Characterization of equilibria
Observe that, if an agreement has been reached between agent i and the principal, thenthe subgame, between agent j and the principal, is a standard two player bargaininggame. In this subgame, an agreement is reached immediately, and the principal andagent j each receive 1−vi
2 , that is, one-half of the remaining surplus. Suppose thatboth agents remain in the game and let p denote the probability that the principalchooses agent 1 after state s = ({1, 2} , 1), that is, the probability that she continuesto bargain with player 1, when both agents are still in the game, given that player 1was her partner in the previous period. Likewise, let q denote the probability that shecontinues to bargain with agent 2, given that agent 2 was her bargaining partner in theprevious period. Also, let r denote the probability that the principal begins bargainingwith agent 1 in the first period of the game.6 The following discussion will show thatit suffices to specify the (p, q, r) triple, that can be chosen by the principal, subject tosubgame perfection, to characterize all SSPE of the game.
6 Formally, p = p (1| {1, 2} , 1), the probability of meeting 1, given that {1, 2} remain in the game and 1was the bargaining partner in the previous period. Likewise, q = p (2| {1, 2} , 2) and r = p (1|s0).
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Fix a SSPE for the game and let πi denote the expected profit of the principal and tithe expected transfer to agent i , when both agents remain in the game and the principalchooses to bargain with agent i . If there is an immediate agreement with agent i , whenthe principal chooses to bargain with agent i , then
πi = vi − ti + δ1 − vi
2. (1)
The following Lemma shows that a disagreement does not occur in any SSPE.
Lemma 1 In any SSPE of the two agent bargaining game, there are no delays on theequilibrium path of play.
Let π = max {π1,π2}. Observe that for the principal to choose to bargain with agenti , it must be the case that πi ≥ π j . Assume that the principal chooses agent 1 asher bargaining partner after a subgame where both agents remain in the game. Thisimplies, π = π1. In any subgame, where agent 1 has been designated as the proposer,the agent will make an offer t that leaves the principal indifferent between acceptingor rejecting the agent’s current offer, that is, t will satisfy v − t + δ
( 1−v2
)= δπ1.
Rearranging gives t = v + δ( 1−v
2
)− δπ1. Likewise, in any subgame where the prin-
cipal is chosen as the proposer, she will make a transfer offer that leaves the agentindifferent between accepting the offer and continuing to bargain. Consequently, theoffer T = δ
(p T +t
2 + (1 − p) δ 1−v+d2
). These two equations imply that, the expected
transfer to player 1, t1 = T +t2 , is given by the following equation:
t1 = 12
(v + δ
1 − v
2+ δpt1 + δ2 (1 − p)
1 − v + d2
− δπ1
). (2)
And the principal’s profit is as follows:
π1 = 12
(v + δ
1 − v
2+ δπ1 −
(δpt1 + δ2 (1 − p)
1 − v + d2
)). (3)
As seen from the above equations, the agent and the principal split the surplus, v+δ 1−v2 ,
according to the Nash Bargaining solution, where the disagreement points for the agentand principal are δ
(pt1 + (1 − p) δ 1−v+d
2
)and δπ1, respectively.7 The disagreement
points for the players are determined by the choice of p (and q). After a rejection,the principal returns to bargain with agent 1 with probability p and agent 1 receivestransfer equal to t1, and with probability 1 − p, the principal bargains with agent 1after reaching an agreement with agent 2 consequently agent 1 receives transfer equalto 1−v+d
2 . Solving for π1 gives the following:
π1 = 1 − δp2 − δp − δ
(v + δ
1 − v
2
)− 1 − p
2 − δp − δδ2 1 − v + d
2. (4)
7 In fact any Pareto optimal, symmetric, and translation invariant solution, e.g., the Kalai–Smordinskysolution or the Egalitarian solution, would also split the surplus equally as in this paper.
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Bargaining: separately or together? 303
The above steps can be repeated to find a similar expression for the expected profit tothe principal from bargaining with agent 2. Observe that, for the principal to be willingto bargain with agent 2, π2 = π . Consequently, if the principal happens to choose tomeet with agent 2 in any subgame, then the two players will split the joint surplus,v − d + δ 1−v+d
2 , according to the Nash bargaining solution, where the disagreementpoints for agent 2 and the principal are δ
(qt2 + (1 − q) δ 1−v
2
)and δπ2, respectively.
Solving for the principal’s profit gives:
π2 = 1 − δq2 − δq − δ
(v − d + δ
1 − v + d2
)− 1 − q
2 − δq − δδ2 1 − v
2. (5)
The above derivation shows that the expression for π1 is independent of q and theexpression for π2 is independent of p.8 Also, the choice of p and q, in conjunctionwith the no delay property uniquely determine, π1, π2, t1 and t2. Consequently, it issufficient to specify p, q and r to pin down SSPE payoffs to the players.
The probability p (or q) can be thought of as a punishment parameter or an endog-enous bargaining sequence. Suppose that the agents are substitutes, or in other words,there are decreasing returns to scale. In an equilibrium where p = 0 (i.e., the principlealternates between the two agents), the principal is punishing agent 1 most severely fora disagreement. The punishment is twofold. First, the agent remains idle for one periodwhile the principal bargains with agent 2. Second, given that the principal reaches anagreement with agent 2 in the interim, agent 1 is left to bargain over her marginalproduct. The marginal product is small compared to the joint surplus available whenboth agents remain in the game since the agents are substitutes.9 In contrast, in anequilibrium where r = p = 1 (i.e., the principle does not alternate), agent 1 hasconsiderable bargaining power. After the principal reaches an agreement with agent1, she will also hire agent 2 in any SSPE. However, since p = 1, the principal willnot move on the bargain with agent 2 before reaching an agreement with agent 1 first.Consequently, agent 1 can hold-up the principal and extract a portion of agent 2’smarginal contribution to the principal’s production in addition to one-half of her owncontribution. The argument above suggests that the principal profit is minimized ifp = 1 and maximized if p is set at a minimum.
The punishment effect of setting p = 0 works in the opposite direction when thetwo agents are sufficiently complementary and discounting is not too severe. By set-ting p = 0, the principal is threatening agent 1 with being signed after agent 2, inthe case of a disagreement. However, since the agents are complementary, agent 1’sproductivity is greatest when she is signed last.10 In a SSPE with p = 1, the principal
8 This statement is not accurate; if p = 1, q = 0 and π1 > π2, then π2 = 12
(v − d + δ 1−v+d
2 + δπ1 (1) −δ2 1−v
2
). However, in this case agent 2 is never visited on equilibrium path.
9 When the bargaining sequence is exogenously set with p = q = 0, for example as in Horn and Wolinsky(1988b), the principal alternates between the agents after each disagreement. This corresponds to the bar-gaining sequence with the most severe punishment. This implies that substitutable agents have the greatestincentive to bargain collectively.10 When the bargaining sequence is exogenously set with p = q = 0 (as in Horn and Wolinsky 1988b)the principal is forgiving of disagreements and complement agents prefer to bargain separately.
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304 A. E. Atakan
faces the same hold-up problem outlined above and agent 1 extracts part of agent 2’smarginal contribution from the principal. Nevertheless, when complementarities aresufficiently strong, profits are maximized in an equilibrium with p = 1 and minimizedin one with p = 0. The characterization of the equilibrium payoffs follows basicallyfrom the insights provided above. The line of reasoning is made exact in the followingLemma. Before proceeding I define expressions, b (δ) = δ
2+δ , a1 (δ) = 2(2−δ)δ(2+δ) and
a2 (δ) = 2δ2−(2−δ)2
δ(2+δ) , which occur repeatedly in the following exposition.
Lemma 2 If b (δ) (d + 1) < v (for example, if the two agents are substitutes), thenπ1 (p) is strictly decreasing in p and otherwise (for example, if there are sufficientlystrong complementarity between the two agents), π1 (p) is increasing in p. Likewise,if b (δ)
(d 2−δ
δ + 1)
< v, then π2 (q), is strictly decreasing in q and otherwise π2 (q)is increasing in q.
In a SSPE in which the principal randomizes or alternates between the agents,more precisely, if p, q or r ∈ (0, 1) or p = q = 0, the principal must be indiffer-ent between her choices of bargaining partners. Consequently, π1 (p) = π2 (q) . Onthe other hand, in a pure strategy SSPE, if r = p = 1 and q = 0 (or r = p = 0and q = 1), then π1 (1) ≥ π2 (0) (or π1 (0) ≤ π2 (1)). Consequently, to character-ize the SSPE, it suffices to find solutions to the equation π1 (p) = π2 (q) and alsocheck for pure strategy SSPE. The following theorem outlines the set of equilibria ofthe game. The theorem partitions the set of v and d pairs into three regions whichare depicted in Fig. 2. In Area 1, where heterogeneity is low and agents are eithersubstitutes or the complementarity between the agents is low, minp∈[0,1] π1 (p) =π1 (1) ≤ maxq∈[0,1] π2 (q) = π2 (0), and so a range of mixed strategy equilibriacan be sustained.11 In Area 3, where complementarity between the agents is strong,minp∈[0,1] π1 (p) = π1 (0) ≤ maxq∈[0,1] π2 (q) = π2 (1), and consequently a rangeof mixed strategy equilibria can again be sustained. On the other hand, in Area 2,where heterogeneity is high, minp∈[0,1] π1 (p) ≥ maxq∈[0,1] π2 (q), and thus the onlySSPE involves repeatedly bargaining with agent 1, i.e., r = p = 1 and q = 0.
Theorem 1 Let $ denote the set of SSPE payoffs to the principal.
1. If π1 (1) ≤ π2 (0) (v ≥ a1 (δ) d + b (δ) or Area 1 in Fig. 2), then any p ∈[
p, 1]
(where p solves π1
(p)
= π2 (0)), the corresponding q that solves π1 (p) =π2 (q), and r ∈ [0, 1] form a SSPE. Also, $ = [π1 (1) ,π2 (0)],
2. If π1 (1) > π2 (0) and π1 (0) > π2 (1) (a2 (δ) d + b (δ) < v < a1 (δ) d + b (δ)or Area 2 in Fig. 2), then there exists a unique SSPE with r = p = 1 and q = 0and $ = {π1 (1)},
3. If π1 (0) ≤ π2 (1) (v ≤ a2 (δ) d + b (δ) or Area 3 in Fig. 2), then r = p = 1and q = 0 is a SSPE. Also, any p ∈ [0, p], (where p solves π1 ( p) = π2 (1)),the corresponding q that solves π1 (p) = π2 (q), and r ∈ [0, 1] form a SSPE.Moreover, $ = [π1 (0) ,π2 (1)] ∪ {π1 (1)}.
11 Lemma 2 implies that π1 (p) and π2 (q) are decreasing in p and q, respectively, in Area 1; and π1 (p) andπ2 (q) are increasing in p and q, respectively, in Area 3. This is because a2 (δ) d + b (δ) ≤ b (δ) (d + 1) ≤b (δ)
(d 2−δ
δ + 1)
≤ a1 (δ) d + b (δ).
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Bargaining: separately or together? 305
Fig. 2 Areas 1, 2 and 3 correspond to Items 1, 2 and 3 in Theorem 1, respectively. Agents are substitutesto the right and complements to the left of the dotted line
In order to make Theorem 1 less opaque the following discussion analyzes the caseswhere the agents are homogeneous (i.e., d = 0) or where they are substitutes butheterogeneous and outlines the implications of the previous Theorem for these twocases.Homogeneous agents: With homogeneous agents, that is if d = 0, the game hastwo pure strategy SSPE, one with p = q = 1 (non-alternating) and another withp = q = 0 (alternating). These two pure equilibria also comprise extreme points forthe SSPE payoffs. In addition to these two SSPE, any p = q ∈ [0, 1] and r ∈ [0, 1]form a SSPE. Consequently, the set of equilibrium profits for the principal is convex,bounded above by π1 (0) = π2 (0) and below by π1 (1) = π2 (1) when the agentsare substitutes. The bounds for the principal’s profit are reversed when the agents aresufficiently complementary.Substitute heterogeneous agents: With sufficient heterogeneity, the principal mayhave difficulties committing to a sequence of play which involves negotiating with thelower productivity agent in a subgame where both agents remain in the game. First,observe adding heterogeneity immediately “kills” the (alternating) pure strategy SSPEwith p = q = 0. This is because v1 > v2 implies π1(0) > π2(0) and consequentlyafter a disagreement with agent 1, the principal cannot choose agent 2 with certainty.The other (non-alternating) pure strategy SSPE with p = 1 remains an equilibriumof the game. When the discount factor δ is low (i.e., bargaining costs are high) orthe degree of heterogeneity, d, is large, this commitment problem is exacerbated. Inparticular, for sufficiently large d, we should expect the principal to bargain with thehigh productivity agent until they reach an agreement and then move on to bargainwith the low productivity agent. To fix intuition take the case with v1 = 1 and v2 = 0.In this case the principal can never commit to bargaining with agent 2 when bothagents remain in the game. Consequently, the bargaining sequence with r = p = 1 isthe unique subgame perfect equilibrium.
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306 A. E. Atakan
Nevertheless, when the inequality, v ≥ a1 (δ) d +b (δ), given in Theorem 1, Item 1is satisfied, a range of mixed strategy SSPE can be sustained. This inequality ensuresthat the principal’s profit in the worst equilibrium when bargaining with agent 1,π1 (1), is less than the profit in the best equilibrium when bargaining with agent 2,π2 (0) . Rewriting the inequality gives d ≤ v−b(δ)
a1(δ), where v−b(δ)
a1(δ)> 0, if the agents are
substitutes. Consequently, if the heterogeneity, d, is less than v−b(δ)a1(δ)
, then a range ofSSPE can be sustained. While if d is large then the unique SSPE involves p = r = 1and q = 0.
Corollary 1 Assume agents are substitutes and let d∗ = v−b(δ)a1(δ)
1. If d ≤ d∗, then any p and q that solve the equation π1 (p) = π2 (q) and anyr ∈ [0, 1] comprise a SSPE.
2. If d > d∗, then the unique SSPE has r = p = 1 and q = 0.
Proof Follows directly from Theorem 1, Items 1 and 2. The inequality given inTheorem 1, Item 3 is never satisfied when the agents are substitutes. This is becausesubstitute agents implies, by Lemma 2, π1 (0) ≥ π2 (0) ≥ π2 (1). *+
3.2 Collective bargaining
It is worthwhile to touch upon the predictions of some cooperative solutions such asthe Shapley value and the Nucleolus, for the production process under consideration.The Nucleolus of the game gives transfers equal to 1−v2
2 and 1−v12 to agent 1 and agent
2, if the complementarity between agents is not too great.12 Consequently, if the agentsare substitutes, then they have a preference for collective bargaining. As demonstratedby Horn and Wolinsky (1988b), the Nucleolus is payoff equivalent to an exogenouslyset bargaining sequence with p = 0 and q = 0, as the discount factor approachesone. However, Theorem 1 showed that the principal cannot choose this sequence ofbargaining if heterogeneity between agents is large. The impact of this commitmentproblem on the incentives for collectivization, is analyzed at length below.
For the production process under consideration, the Shapley values are u1 = 13
v12 +
23
(1−v2
2
), u2 = 1
3v22 + 2
3
(1−v1
2
)for agents 1 and 2, respectively. Since the transfers
that the agents receive are more heavily weighted towards their marginal contribution,i.e., 1−vi
2 , substitute agents again prefer collective bargaining. For this productionprocess, Stole and Zwiebel (1996a) provide an extensive form game that implementsthe Shapley value. In particular, Stole and Zwiebel (1996a) show if bilateral con-tracts between the principal and each agent can be renegotiated until an agreement isreached with all the agents, then the agents receive their Shapley values in equilib-rium. It is also possible to interpret the Shapley value with-in the framework providedin this paper. The procedure that implements the Nucleolus had the principal (or theindispensable party) as the residual claimant. The principal bargains with the agents,pays them the agreed upon transfers and receives the residual as profit. To implement
12 To be precise if v > 23 d + 1
3 .
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Bargaining: separately or together? 307
the Shapley value suppose that one of the three players (the principal, agent 1 oragent 2), is chosen, with equal probability, as the residual claimant. 13 The residualclaimant then plays a game where the bargaining sequence is exogenously set withp (i | {i, j} , i) = p ( j | {i, j} , j) = 0 and p (i |s0) = 1
2 , where {i, j} are the agentswho were not selected as the residual claimant. In other words, agent i or j is chosenin the first period with equal likelihood and the probability of bargaining again withagent i (or j) after a disagreement is zero. For example, if agent 1 is chosen as theresidual claimant, she is in the position of the principal. In the first period, she meets,with equal probability, either the principal or agent 2. If the bargaining results in anagreement, then agent 1 goes on to bargain with the remaining player. In the case ofa disagreement, agent 1 alternates between the two players. For the discount factorclose to one, this game has a unique SSPE, and as δ converges to one, the payoffs tothe players converge to their Shapley value.14
As is the case with the Nucleolus, the exogenously set sequence of contracting thatimplements the Shapley value, cannot always be chosen in a SSPE where the resid-ual claimant decides on her bargaining partner. For example, assume that agent 1 ischosen as the residual claimant. With probability 1
2 , she will meet with the principalin the first period. The productivity of the principal is v1, if she is hired first, and sheproduces 1, if she is hired second. The productivity of agent 2 is zero if she is hiredfirst and 1 − v1, if she is hired second. Consequently, the heterogeneity parameter,that measures the difference between the productivity of agent 2 and the principal, isequal to v1 in this example. If the agents are substitutes, then v1 ≥ 1
2 . This impliesthat the
(v′, d ′) = (v1, v1) pair is in Area 2, Fig. 2 for any δ. This implies that agent
1 would always choose to bargain repeatedly with the principal and then move on tobargain with agent 2 after an agreement has been reached, i.e., the unique equilibriumhas p (0| {0, 2} , 0) = 1.
Returning to the original game where the principal is the residual claimant, thefollowing proposition outlines the set of transfers received by the agents in SSPE ofgame when the two agents bargain separately. It follows directly from Theorem 1. Ifthe agents are substitutes, then transfers are maximized in a SSPE in which agent 1has the greatest bargaining power, i.e., when p = 1. Alternatively, if the agents aresufficiently complementary, then transfers are minimized in a SSPE with p = 1.
Proposition 1 Let u (f) = u1 (f)+u2 (f), i.e., the total expected payoffs to the agentsin a SSPE where the players use strategy profile f . Also, let u = minf∈Es u (f) andu = maxf∈Es u (f) where Es denotes the set of SSPE strategy profiles.
1. If v and d lie in Area 1 in Fig. 2 (v ≥ a1 (δ) d + b (δ)), then u = v − d +δ (1 − v + d) − π2 (0) and u = v + δ (1 − v) − π1 (1) = v
2 + δ 3(1−v)4 ,
2. If v and d lie in Area 2 in Fig. 2 (a2 (δ) d + b (δ) < v < a1 (δ) d + b (δ)), thenu = u = v
2 + δ 3(1−v)4 ,
13 This is in contrast to the game that implements the Nucleolus, where the principal is chosen as theresidual claimant with certainty.14 This extensive form game is a variant of the one presented in Gul (1989).
123
308 A. E. Atakan
3. If v and d lie in Area 3 in Fig. 2 (v ≤ a2 (δ) d + b (δ)), then u = v2 + δ 3(1−v)
4 ifδ ≥ 2
3 , and u = v−d2 + δ 3(1−v+d)
4 otherwise. Also, u = v + δ (1 − v) − π1 (0) .15
Substitute agents: In the SSPE of the game with p = 1 (i.e., the SSPE that maximizestransfers to the agents), the agents prefer to bargain individually if δ > 2
3 while theprincipal prefers for the agents to bargain collectively for any range of parameter val-ues. This follows since this equilibrium gives agent 1 considerable bargaining power.Ignoring discounting, agent 1 gets half of her contribution to the principal’s productionv2 as well as one-half of the principal’s share of agent 2’s contribution (that is, one-halfof 1−v
2 ). Consequently, the only benefit of collective bargaining in this SSPE is thenegation of delay costs. When the discount factor is sufficiently large, the cost savingsof collective bargaining are overwhelmed by the bargaining power gained throughindividual bargaining. Observe that with any heterogeneity, i.e., for any d > 0, theSSPE with p = 1 is also the unique pure strategy SSPE of the game. The followingtheorem and its corollary show, even when mixed equilibria are considered, with suf-ficient heterogeneity, the unique pure SSPE is also the unique SSPE of the game andbargaining individually is preferred to bargaining collectively for δ > 2
3 .
Theorem 2 Assume that the agents are substitutes.
1. In the worst equilibrium for the agents (a SSPE with q = 0), collective bargain-ing is preferred by the agents for all δ ∈ (0, 1). Also in this SSPE, there exists aδ′ ∈ (0, 1) such that the principal prefers the agents to bargain individually forall δ > δ′.
2. In the best equilibrium for the agents (a SSPE with p = 1), for all δ ≥ δ∗ = 23 , the
agents prefer to bargain individually and for δ < δ∗ = 23 , they prefer collective
bargaining. Also, in this SSPE, collective bargaining is preferred by the principalfor all δ ∈ [0, 1).
As was demonstrated earlier in Corollary 1, if the heterogeneity between the twoagents is large, then the game has a unique SSPE with p = 1. Combining the finding ofCorollary 1 with the previous theorem delivers the following corollary. The corollarystates that if heterogeneity is large and transaction cost are not prohibitive, then theagents prefer to bargain separately. The principal, in contrast, prefers for the agents tobargain as a collective.
Corollary 2 Assume that the agents are substitutes and δ > 23 . If d > d∗, where d∗ is
defined as in Corollary 1, then the agents prefer bargaining individually in the uniqueSSPE of the game.
Proof By Theorem 1, Item 2, d > d∗ implies that the game has a unique SSPE withr = p = 1 and q = 0. In this SSPE bargaining individually is preferred when δ > 2
3 ,by Theorem 2, Item 2. *+
15 The explicit expression for u in items 1 and 3 have been suppressed since they are cumbersome. Seethe appendix for details.
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Bargaining: separately or together? 309
Complement agents: For complementary agents, the only incentive for collectivebargaining is to economize on bargaining costs. The principal however, always pre-fers the agents to bargain collectively. This is because the agents always have leveragein the bargaining game: in the SSPE with p = 0 this strength stems from the comple-mentarity between the two agents and in the SSPE with p = 1, is due to the hold-uppower of agent 1. The following theorem summarizes these findings.
Theorem 3 Assume that the agents are complements. The principal prefers the agentsto bargain collectively in any SSPE of the game.
1. Assume v ≥ a1 (δ) d + b (δ). In the worst equilibrium for the agents, there existsa δ′ ∈
( 23 , 1
)such that if δ ≥ δ′, then the agents prefer to bargain individually.
2. Assume v < a1 (δ) d + b (δ). In the worst equilibrium for the agents, if δ ≥ 23 ,
then the agents prefer to bargain individually.
3.3 Summary of results
Fix the discount factor δ to be larger than 2/3.16 The analysis in this section showedthat if heterogeneity is large (d > d∗), then the game has a unique SSPE with p = 1;and, in this SSPE, agents prefer to bargain individually. In contrast, if heterogeneityis small, then a range of SSPE can be sustained. If the agents are substitutes, then theyprefer collective bargaining in the SSPE that minimizes transfers to the agents (i.e., aSSPE with q = 0 and p chosen to solve π1 (p) = π2 (0)) and they prefer individualbargaining in the SSPE that maximizes transfers (i.e., the pure SSPE with p = 1).Complementary agents, on the other hand, always prefer to bargain individually. Thesefindings are summarized in the table below.
Homogeneous Small heterogeneity Large heterogeneitySubstitutes Individual Individual in unique pure SSPE Individual
or collective Collective in mixed SSPE in unique SSPEComplements Individual Individual Individual
4 More than two agents and non-stationary equilibria
This section extends some of the findings for the model with two agents to one withan arbitrary number of agents and non-stationary strategies. The goal is not to give afull characterization of the set of equilibria but to show that the main insights from thetwo agent model extend to the more general set-up.
Given the set I of agents, vi (A) denotes the additional value added by agent i ∈ Ato the production process which already has the first {I \ A} agents employed. 17 Theagents are homogeneous if vi (A) = v j (A) for all i, j ∈ A and all A ⊂ I . The
16 Observe that for δ < 2/3 agents always prefer to bargain collectively in order to minimize the time costof bargaining.17 If g represents the characteristic function of a standard cooperative game in coalitional form, thenvi (A) = g ((I \ A) ∪ {i}) − g (I \ A) .
123
310 A. E. Atakan
agents are substitutes, when the following condition is satisfied: If |B| ≥ |A|, thenvi (B) ≥ vi (A) for all A, B and i ∈ A ∩ B. Throughout this section, I assume thatthe agents are substitutes.
Suppose that the agents are ranked according to decreasing productivity, i.e., agent1 is the most productive agent when all N agents remain in the game. The follow-ing assumption, which is retained throughout the section, states that if a subset B ofagents that excludes agents 1 and 2 have produced, then agents 1 and 2 are still thefirst and second most productive agents remaining in the game. In other words, theranking of the agents’ productivity is independent of the set of agents that have alreadyproduced.18
Assumption 1 Pick i and j with vi (I ) ≥ v j (I ) . If i, j ∈ A, then vi (A) ≥ v j (A)for any A ⊂ I .
The discussion here focuses on two bargaining procedures: a non-alternating strat-egy (or SSPE), which corresponds to the SSPE in the two agent game with p = 1,and an alternating strategy (or SSPE) which corresponds to the SSPE in the two agentgame with p = 0. In a non-alternating strategy, the principal begins bargainingwith the highest productivity agent (agent 1). If there is an agreement in the currentperiod, then the principal chooses the highest productivity agent remaining in the gameand if there is a disagreement with the highest productivity agent, then the principalagain chooses this agent. Consequently, the principal bargains with agent 1 until theyreach an agreement, and then with agent 2 and so on. In an alternating strategy, theagents are organized around a circle in order of increasing productivity. The principalbegins bargaining with the lowest productivity agent. After either an agreement or adisagreement the principal moves to the next agent in line and proceeds around thecircle. Consequently, after a disagreement, the principal bargains with the agent thatcaused the disagreement last. The following proposition shows that non-alternatingstrategy is always a SSPE and the alternating strategy is a SSPE of the game if theagents are substitutes and homogeneous. Also, the proposition shows that the insightfrom the two agent model, concerning incentives for collectivization, extends to thismore general model. In particular, it shows that if the discount factor is greater than a
threshold, δ (N ) =(
2N−1
2N −1
) 1N−1 , then the agents prefer individual bargaining to col-
lective bargaining when the principal uses a non-alternating strategy. Also, substituteagents always prefer collective bargaining if the principal uses an alternating strategy.
Proposition 2 1. There exists a SSPE in which the principal uses the non-alternatingstrategy. In this SSPE, if δ > δ (N ), then the agents prefer to bargain individually.Also, the principal prefers for the agents to bargain collectively for all δ ∈ [0, 1).
2. If the agents are substitutes and homogeneous, then there exists an SSPE in whichthe principal uses the alternating strategy. In this SSPE, the agents prefer col-lective bargaining for all δ ∈ [0, 1). Also if δ < δ∗, then the principal preferscollective bargaining.
18 I do not believe that the assumption is essential for the results presented, nevertheless it greatly simplifiesthe proofs.
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Bargaining: separately or together? 311
In the two agent case with sufficient heterogeneity, I found a range of δ such thatthe non-alternating SSPE was the unique equilibrium and also individual bargainingwas preferred by the agents. The following is an N -agent analog and shows that ifheterogeneity is greater than a threshold level and discounting is not prohibitive, thenthe agents prefer to bargain separately, i.e., the result for the two agents extends tothe N agent model and arbitrary subgame perfect equilibria. In the following theoremthe heterogeneity parameter, d(A), denotes the productivity difference between thefirst and the second most productive agents in the set A; and d∗ (A), defined in theappendix, denotes the threshold level of heterogeneity.
Theorem 4 Assume the agents are substitutes. If d (A) > d∗ (A), for all A ⊆ I ,then the game has a unique SPE in which the principal uses the non-alternating strat-egy and the principal prefers for the agents to bargain collectively. In addition, ifd (A) > d∗ (A), for all A ⊆ I , and δ > δ (N ), then the agents prefer individualbargaining.
Appendix
Proof of Lemma 1 Let f = (f0, f1, f2) be a SSPE strategy profile and let π(s, f) andui (s, f) denote equilibrium payoffs under f , to the principal and agent i , given that thestate is s = (A, j). A disagreement occurs between the principal and agent i , if andonly if, the total surplus available for division, is less than the sum of the disagreementpayoffs, that is, vi + δπ((A\i, i), f) < δπ((A, i), f) + δui ((A, i), f). Assume A ={i, j}, i.e., both agents are yet in the game. As stated in the text, a disagreement cannotoccur when there is only one agent remaining in the game and π((A\i, i), f) = 1−vi
2 .
Also, it is immediate that a disagreement cannot occur with both players in a SSPEsince this would imply π (f) = ui (f) = u j (f) = 0, and vi + δπ((A\i, i), f) > 0.
Assume that a disagreement occurs on equilibrium path with agent 2. Because agent 2is visited on the equilibrium path, the equality, π1 = π2, must hold. Disagreement withagent 2 in any meeting implies v2 + δ 1−v2
2 < δπ(({1, 2}, 2), f) + δu2(({1, 2, }, 2), f)and u2(({1, 2, }, 2), f) ≤ δ 1−v1
2 (which is the transfer that agent 2 gets when she issigned after agent 1). This implies that v2 + δ 1−v2
2 − δ 1−v12 < δπ(({1, 2}, 2), f), thus
π(({1, 2}, 2), f) > 0. Again since agent 2 is visited on the equilibrium path, π1 = π2.However, there is a disagreement with agent 2 so π2 = δπ1 = π1 and thus π1 = 0,which gives a contradiction. *+
Proof of Lemma 2 Taking the derivative of π1 with respect to p gives the following
dπ1
dp= −1 + δ
(2 − δp − δ)2 δ
(v + δ
1 − v
2
)+ 2 − 2δ
(2 − δp − δ)2 δ2 1 − v + d2
. (6)
The expression is positive if
(1 − δ) δ (1 − v + d) − (1 − δ)
(v + δ
1 − v
2
)> 0. (7)
123
312 A. E. Atakan
Simplifying shows dπ1dp < 0, if δ
2+δ + δd2+δ < v (e.g., when the agents are substitutes)
and dπ1dp ≥ 0 otherwise (e.g. when the agents are sufficiently strong complements).
Also,
dπ2
dq= −1 + δ
(2 − δq − δ)2 δ
(v − d + δ
1 − v + d2
)+ 2 − 2δ
(2 − δq − δ)2 δ2 1 − v
2. (8)
The expression is positive if
δ (1 − v) −(
v − d + δ1 − v + d
2
)> 0
Consequently, if δ2+δ +d 2−δ
2+δ < v, then π2 (q), is strictly decreasing in q and otherwiseπ2 (q) is increasing in q. *+
Proof of Theorem 1 Observe that π1 (1) ≥ π2 (1) and π1 (0) ≥ π2 (0) for all v, d andδ. For π1 (1) ≥ π2 (1), the following inequality,
12
(v + δ
1 − v
2
)≥ 1
2
(v − d + δ
1 − v + d2
), (9)
which simplifies as 0 ≥ − (2−δ)2 d, needs to be satisfies. Consequently, 0 ≥ − (2−δ)
2 d(which is always true) implies that π1 (1) ≥ π2 (1) . Also, for π1 (0) ≥ π2 (0), thefollowing inequality,
12 − δ
(v + δ
1 − v
2− δ2 1 − v + d
2
)≥ 1
2 − δ
(v − d + δ
1 − v + d2
− δ2 1 − v
2
),
(10)
which simplifies as δ2d ≤ (2 − δ) d, needs to be satisfied. Consequently, δ2d ≤(2 − δ) d (which is again always true) implies that π1 (0) ≥ π2 (0) .
Any p and q such that π1 (p) = π2 (q) and r ∈ [0, 1] is a SSPE. After any his-tory, in which the principal has to choose an agent as a bargaining partner, she has noincentive to deviate from the prescribed randomization since bargaining with eitheragent results in an immediate agreement and either continuation gives equal profit tothe principal. Consequently, the one-shot deviation property implies that the abovep, q and r comprise a SSPE.
Item 1. The condition implies π2 (1) ≤ π1 (1) ≤ π2 (0) ≤ π1 (0) consequently, byLemma 2, π1 (p) and π2 (q) are decreasing in p and q, respectively. Also, workingthe algebra shows that π1 (1) ≤ π2 (0) implies v ≥ a1 (δ) d + b (δ) .
I show for any SSPE p and q, π1 (p) = π2 (q) . Assume to the contrary, in anSSPE profile f π1 (f) > π2 (f) . This implies that r = p = 1, q = 0 in this SSPE and
123
Bargaining: separately or together? 313
π1 (f) = π1 (1). Consequently,
π2 (f) = 12
(v − d + δ
1 − v + d2
+ δπ1 (1) − δ2 1 − v
2
)< π1 (1) . (11)
Simplifying gives
12 − δ
(v − d + δ
1 − v + d2
− δ2 1 − v
2
)< π1 (1) . (12)
However, 12−δ
(v − d + δ 1−v+d
2 − δ2 1−v2
)= π2 (0) < π1 (1), which is a contradic-
tion.Consequently, the set of SSPE is defined by the p and q that solve π1 (p) = π2 (q) .
The characterization of the SSPE payoffs follows from the fact that π1 (p) and π2 (q)are decreasing in p and q, respectively.
Item 2. π1 (1) > π2 (0) and π1 (0) > π2 (1) implies that π1 (p) > π2 (q) for all pand q. Consequently, the unique SSPE must involve r = p = 1 and q = 0.
Item 3. Working through the algebra shows that π1 (0) ≤ π2 (1) implies v ≤a2 (δ) d + b (δ) . The condition implies that π2 (0) ≤ π1 (0) ≤ π2 (1) ≤ π1 (1). Con-sequently, a variety of p and q solve π1 (p) = π2 (q) and are SSPE. Also, r = p = 1and q = 0 is a SSPE since π2 (0) ≤ π1 (1).
By Lemma 2, π1 (p) and π2 (q) are increasing in p and q, respectively and thecharacterization of SSPE profits follows. *+
Proof of Proposition 1 Item 1. The condition for Item 1 implies that π2 (1) ≤ π1 (1) ≤π2 (0) ≤ π1 (0). To specify a particular SSPE profile f, it suffices to specify (p, q, r) .
Consequently,
u = min(p,q,r)∈Es
r (v + δ (1 − v) − π1 (p)) + (1 − r)
× (v − d + δ (1 − v + d) − π2 (q)) (13)
= min{(p, q, r) s.t.
π1 (p) = π2 (q)
} r (v + δ (1 − v) − π1 (p)) + (1 − r)
× (v − d + δ (1 − v + d) − π2 (q)) (14)
= min(q,r)
r (v + δ (1 − v)) + (1 − r) (v − d + δ (1 − v + d)) − π2 (q) (15)
u = v − d + δ (1 − v + d) − π2 (0) . (16)
Plugging in for π2 (0) and simplifying gives the following:
u = v − d + δ (1 − v + d)− 12 − δ
(v − d+δ
1 − v + d2
)+ δ
2 − δδ
1 − v
2(17)
= (v − d)(2 − 5δ + δ2) − δ2d + δ (3 − δ)
2 (2 − δ). (18)
123
314 A. E. Atakan
Also
u = max(p,q,r)∈Es
r (v + δ (1 − v) − π1 (p)) + (1 − r)
× (v − d + δ (1 − v + d) − π2 (q)) (19)
= max
(p, q, r) s.t.
π1 (p) = π2 (q)
r (v + δ (1 − v) − π1 (p)) + (1 − r)
× (v − d + δ (1 − v + d) − π2 (q)) (20)
= max(p,r)
r (v + δ (1 − v)) + (1 − r) (v − d + δ (1 − v + d)) − π1 (p) (21)
= v + δ (1 − v) − π1 (1) . (22)
Substituting in for π1 (1) and simplifying delivers the result.Item 2. The statement follows directly from Theorem 1, Item 2, since the game has
a unique SSPE with r = p = 1 and q = 0.
Item 3. The condition on v and d implies that π2 (0) ≤ π1 (0) ≤ π2 (1) ≤ π1 (1).
u = max(p,q,r)∈Es
r (v + δ (1 − v) − π1 (p)) + (1 − r)
× (v − d + δ (1 − v + d) − π2 (q)) (23)
= max
max(p,r) r (v + δ (1 − v)) + (1 − r) (v − d + δ (1 − v + d))−π1 (p) ,
v + δ (1 − v) − π1 (1)
(24)
= max {v + δ (1 − v) − π2 (0) , v + δ (1 − v) − π1 (1)} (25)
u = v + δ (1 − v) − π2 (0) . (26)
Substituting for π2 (0) delivers the following:
u = v + δ (1 − v) − 12 − δ
(v + δ
1 − v
2
)+ δ
2 − δδ
1 − v + d2
(27)
= v(2 − 5δ + δ2) + dδ2 + δ (3 − δ)
2 (2 − δ). (28)
Also,
u = min(p,q,r)∈Es
r (v + δ (1 − v) − π1 (p)) + (1 − r)
× (v − d + δ (1 − v + d) − π2 (q)) (29)
= min
min(q,r) r (v + δ (1 − v)) + (1 − r) (v − d + δ (1 − v + d))−π2 (q) ,
v + δ (1 − v) − π1 (1)
(30)
= min {v − d + δ (1 − v + d) − π2 (1) , v + δ (1 − v) − π1 (1)} (31)
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Bargaining: separately or together? 315
= 12
min{v − d + 3
2δ (1 − v + d) , v + 3
2δ (1 − v)
}. (32)
Hence u = v2 + δ 3(1−v)
4 , if δ ≥ 23 , and u = v−d
2 + δ 3(1−v+d)4 otherwise. *+
Proof of Theorem 2 Item 1. For collective bargaining to be preferred the followinginequality must hold
u − 12
= (v − d)(2 − 5δ + δ2) − δ2d + δ (3 − δ) − (2 − δ)
2 (2 − δ)≤ 0. (33)
Consequently, the numerator of the above equation must be negative. Let f (v, d, δ)=(v − d)(2 − 5δ + δ2) − δ2d + δ(3 − δ) − (2 − δ). Rewrite f (v, d, δ) = v(2 − 5δ +δ2)−d(2−5δ +2δ2)+4δ −2− δ2 and observe that this linear in v and d. The agentsbeing substitutes implies that v ≥ 1+d
2 , d ≥ 0, and v ≤ 1. This defines a triangle withvertices at (1, 1),
( 12 , 0
), (1, 0), (see Fig. 1). Since f is affine for fixed δ, it will reach a
maximum at one of the three vertices and to sign the numerator it suffices to check thevalue at these three points. If (v, d) = (1, 1), then f (1, 1, δ) = 4δ − 2 − 2δ2 ≤ 0, forall δ ∈ [0, 1] . If (v, d) =
( 12 , 0
), then f
( 12 , 0, δ
)= 3δ−2−δ2
2 ≤ 0, for all δ ∈ [0, 1] .
If (v, d) = (1, 0), then f (1, 0, δ) = −δ ≤ 0.
For the principal’s incentives:
π2 (0) − 12
= 12 − δ
(v + δ (1 − δ)
1 − v
2
)− d
2− 1
2(34)
= 2v + δ (1 − δ) (1 − v) − 2 + δ
2 − δ− d
2. (35)
This expression is negative when δ = 0, positive when δ = 1 and increasing in δ.
Item 2. In the best equilibrium for the agents u − 12 = v
2 + δ3 1−v4 − 1
2 =1−v
2
(δ 3
2 − 1)
which is positive if δ ≥ 23 and negative otherwise.
For the principal’s incentives, the following inequality delivers the result.
π1 (1) − 12
= 12
(v + δ
1 − v
2
)− 1
2= 1
2
(δ
1 − v
2− (1 − v)
)≤ 0. (36)
*+Proof of Theorem 3 In order to prove that the principal always prefers for the agentsto bargain collectively, I show that π2 (1) ≤ π1 (1) ≤ 1
2 and π2 (0) ≤ π1 (0) ≤ 12 . For
π1 (1) ≤ 12 see the proof of Theorem 2.
π1 (0) − 12
= 12 − δ
(v + δ (1 − δ)
1 − v
2− d (2 − δ)
2
)− 1
2(37)
= δv −(δ2 + 2 (1 − δ)
)(1 − v) − (2 − δ) d
2 (2 − δ)(38)
123
316 A. E. Atakan
≤ δv −(δ2 + 2 (1 − δ)
)(1 − v + d)
2 (2 − δ)
(since δ2 + 2 (1 − δ) < 2 − δ) (39)
≤ (1 − v + d) (δ (1 − δ) − 2 (1 − δ))
2 (2 − δ)
(since 1 − v + d ≥ v) ≤ 0. (40)
If v < a1 (δ) d + b (δ), then u = 12
(v + 3δ 1−v
2
)or 1
2
(v − d + 3δ 1−v+d
2
). Conse-
quently, if δ ≥ 23 , then the agents would like to bargain individually. If v ≥ a1 (δ) d +
b (δ), then u = (v−d)(2−5δ+δ2)−δ2d+δ(3−δ)
2(2−δ) . Hence for individual bargaining to be
preferred (v−d)(2−5δ+δ2)−δ2d+δ(3−δ)
2(2−δ) − 12 > 0 or
(v − d)(2 − 5δ + δ2) − δ2d + δ (3 − δ) − (2 − δ)
2 (2 − δ)> 0. (41)
Let f (δ) = (v − d)(2 − 5δ + δ2) − δ2d + δ (3 − δ) − (2 − δ), i.e., the numera-
tor of the above equation. f (1) = 1 − 2v + d ≥ 0 because the inequality impliesv ≤ 1 − v + d and the agents are complements. Also, f (0) = 2 (v − d) − 2 < 0and ∂ f (δ)
∂δ = (v − d) (2δ − 5) − 2δd + 4 − 2δ > 0 because the agents are comple-ments. Consequently, the expression is positive when δ = 1, negative when δ = 0and increasing in δ. This implies that there exists a δ′ > 2
3 such that for all δ > δ′ theagents prefer to bargain individually. *+
Proof of Proposition 2 Item 1. The proof proceeds by induction. Assume that for allproper subsets A ⊂ I , there is a SSPE that involves the principal using the non-alternating strategy (an non-alternating SSPE). Assume that after an agreement withany agent i , the game proceeds according to a non-alternating SSPE. Let pi denotethe probability that the principal returns to bargain with agent i after a disagreementwhen all agents remain in the game. If the principal does not bargain with agent iafter a disagreement, she bargains with agent 1, i.e., she bargains with agent 1 withprobability 1 − pi . Let π1 (I |p1 = 1) denote the expected profit to the principal ofbargaining with agent 1 and returning to bargain with agent 1 after any disagreement.It is straight forward to show π1 (I |p1 = 1) = 1
2 (v1 (I ) + δπ (I\1)) . Also, assuminga non-alternating SSPE for the continuation π (I\1), implies
π1 (I |p1 = 1) = 12
N−1∑
k=0
(δ
2
)k
vk+1, (42)
where vk+1 = vk+1 (I\ {1, ..., k}). Likewise let πi (I |pi ) be the profit to the prin-cipal of bargaining with agent i and returning to bargain with player i after anydisagreement with probability pi and going to bargain with agent 1 with probability1 − pi . To show that a non-alternating SSPE exists it suffices to show that there exists
123
Bargaining: separately or together? 317
{pi }i∈I such that p1 = 1, and if pi ∈ (0, 1], then πi (I |pi ) = π1 (I |p1 = 1) andπi (I |pi ) < π1 (I |p1 = 1) otherwise.
Pick a particular agent i /= 1.
πi (I |pi ) = 1 − δpi
2 − δpi − δ(vi (I ) + δπ (I\i)) − 1 − pi
2 − δpi − δδui (I, 1) , (43)
where ui (I, 1) denotes the expected payoff to agent i , given that the principal is bar-gaining with agent 1. Taking the derivative with respect to pi delivers the following:
∂πi (I |pi )
∂pi= (2ui (I, 1) − vi (I ) − δπ (I\i))
δ (1 − δ)
(2 − piδ − δ)2 . (44)
Hence, if 2ui (I, 1) > vi (I ) + δπ (I\i), then πi (I |pi ) is maximized when pi = 1.
However, πi (I |pi = 1) = 12 (vi (I ) + δπ (I\i)) ≤ π1 (I |p1 = 1) . This implies
that, πi (I |pi = 0) ≤ πi (I |pi = 1) ≤ π1 (I |p1 = 1), showing that setting pi = 0would work to support the non-alternating SSPE. If 2ui (I, 1) ≤ vi (I ) + δπ (I\i),then πi (I |pi ) is maximized when pi = 0. If πi (I |pi = 0) ≤ π1 (I |p1 = 1), thenagain pi = 0 would work to support the non-alternating SSPE. On the other hand,if πi (I |pi = 0) > π1 (I |p1 = 1), then there exists a p∗
i such that πi(I |p∗
i
)=
π1 (I |p1 = 1) and this p∗i supports the non-alternating SSPE. This is because,
πi (I |pi = 0) > π1 (I |p1 = 1) ≥ πi (I |pi = 1) and πi (I |pi ) is decreasing in pi .
The comparison with collective bargaining. Let f denote a non-alternating SSPEequilibrium profile. This implies that the transfers to agents are as follows:
u (f) =∑
i
ui (f) =N−1∑
k=0
δkvk+1 − 12
N−1∑
k=0
(δ
2
)k
vk+1 (45)
=N−1∑
k=0
δkvk+12k+1 − 1
2k+1 . (46)
Hence, the agents prefer individual bargaining if the following holds
u (f) − 12
=N−1∑
k=0
δkvk+12k+1 − 1
2k+1 − 12
> 0 (47)
=N−1∑
k=0
δk (2k+1 − 1
)− 2k
2k+1 vk+1 > 0. (48)
Each expression in this sum is positive if δk(2k+1−1)−2k
2k+1 > 0 for each k. Rewriting,
δ > δ (k) =(
2k
2k+1−1
) 1k . δ (k) is increasing in k. Consequently, the expression is
positive if δ > δ (N ) =(
2N−1
2N −1
) 1N−1
. It is straight forward to show that the principalalways prefers collective bargaining.
123
318 A. E. Atakan
Item 2. Given either a disagreement or a disagreement with agent i, the principalmoves to agent i + 1 with probability one. This is indeed an equilibrium since theagents are homogeneous and the profit from going to any of the agents is the same.The equation for the profit is as follows:
π(I ) = 12
(v1 (I ) + δπ (I\1) + δπ(I ) − δN v1(1)
2
)(49)
= 12 − δ
(v1 (I ) + δπ (I\1) − δN v1(1)
2
), (50)
which can be solved using the boundary condition π(0) = 0. Solving with the bound-ary condition gives the following equation for profits:
π(I ) =N−1∑
k=0
δkvN−k1
(2 − δ)k+1 − δN 1(2 − δ)k+1
vN
2, (51)
where vk+1 = vk+1 (I\ {1, ..., k}) . Also,
u =N−1∑
k=0
δkvN−k − π(I ) =N−1∑
k=0
δkvN−k(2 − δ)k+1 − 1
(2 − δ)k+1 + δN 1(2 − δ)k+1
vN
2.
It is cumbersome but straight forward to show u < 12 . Observe that when δ = 1,
u = NvN2 which is less than 1
2 because the agents are substitutes. *+
Proof of Theorem 4 Let d (A) = v1A (A) − v2A (A) where 1A denotes the most pro-ductive and 2A is the second most productive agent in the set A. If d (A) > d∗ (A) :=1+δ
2 v1A (A) for all A ⊆ I , then the unique SPE of the game is the non-alternatingequilibrium. Assume that d (A) > d∗ (A) for all A ⊆ I. The proof will proceedby induction. Assume, for all proper subsets A ⊂ I , the unique SPE is the non-alternating equilibrium. Suppose that π1 (I ) = 1
2 (v1 (I ) + δπ (I\1)) (this will beestablished at the end of the proof). Also, πi (I ) ≤ vi (I ) + δπ (I\1) = vi (I ) +δ( 1
2v1 (I\i) + δ2π (I\ {i, 1})
), i.e., the principal extracts all the surplus if she bar-
gains with agent i. The condition on d (A) ensures that π1 (I ) > πi (I ) for all i.Observe,
π1 (I ) − πi (I ) = 12
(v1 (I ) + δπ (I\1)) − vi (I ) − δ
(12v1 (I\i) + δ
2π (I\ {i, 1})
)
(52)
≥ 1 − δ
2v1 (I ) − vi (I ) ≥ 1 − δ
2v1 (I ) − v1 (I ) + d (I ) . (53)
This follows since v1 (I ) ≥ v1 (I\i) and π (I\1) ≥ π (I\ {i, 1}) by the substi-tutes assumption. Consequently, the principal must always choose to bargain withagent 1, if this agent remains in the game. To complete the proof I must show
123
Bargaining: separately or together? 319
that π1 (I ) = 12 (v1 (I ) + δπ (I\1)). First observe, u1 (I ) = t1 (I ), i.e., the best
continuation payoff for agent 1 occurs when they are bargained with first and reachan agreement with the principal. This follows since the payoff that agent 1 receivesafter any agent i is signed is unique and equals 1
2 (v1 (I\i) + δπ (I\ {1, i})) by theinduction hypothesis. However, we know that a non-alternating SSPE exists by Prop-osition 2 and the transfer in this equilibrium t1 = 1
2 (v1 (I ) + δπ (I\1)) exceedsany transfer that agent 1 gets when she is signed after an agent i which equals12 (v1 (I\i) + δπ (I\ {1, i})) . Also, t1 ≤ 1
2
(v1 (I ) + δπ (I\1) + δt1 − δπ1 (I )
)and
π1 (I ) ≥ 12
(v1 (I ) + δπ (I\1) − δt1 + δπ1 (I )
). Combining the two shows that
π1 (I ) ≥ 12 (v1 (I ) + δπ (I\1)) . However, we know that the non-alternating SSPE
delivers exactly this profit consequently π1 (I ) = 12 (v1 (I ) + δπ (I\1)) . *+
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