Decentralized One-to-Many Bargaining Chiu Yu Ko National University of Singapore Duozhe Li Chinese University of Hong Kong April 2017 Abstract We study a one-to-many bargaining situation in which one active player seeks to reach an agreement with every passive player on how to share the surplus of a joint project. There is no fixed protocol and the active player decides whom to bargain with in each period. Our model admits a rich set of equilibria and we identify the upper and lower bounds of each player s equilibrium payoff. In particular, there is a class of divideand-conquer equilibria, in which the active player creates an endogenous disparity of bargaining power to her own advantage. We also examine whether two natural bargaining protocols often assumed in existing studies can sustain endogenously. The queuing bargaining protocol may indeed arise in an equilibrium, but it leads to a highly unequal division of the surplus. In contrast, the rotating bargaining protocol yields a plausible equilibrium division, but it is not self-enforcing in generic situations. JEL-codes: C78. Keywords: One-to-many bargaining; queuing protocol; rotating protocol; divideand-conquer equilibrium. Department of Economics, National University of Singapore, Singapore. Department of Economics, Chinese University of Hong Kong, Shatin, N.T., Hong Kong. This project is funded by the Hong Kong Research Grants Council General Research Fund (Project No. CUHK492013). 1
1 Introduction We study the bargaining process that occurs between one active player and multiple passive players regarding how to share the surplus of a joint project, which requires the cooperation of all parties. Relevant real-life situations include a real-estate developer buying pieces of land from multiple owners (Cai 2000, 2003; Xiao 2015), an employer trying to reach deals with several labor unions (Horn and Wolinsky 1988a; Stole and Zwiebel 1996), and a downstream firm acquiring inputs from several upstream firms (Horn and Wolinsky 1988b). A common feature of these situations is that the active player has to reach agreements with all of the passive players, who do not bargain with one another. Such one-to-many bargaining situations have been extensively studied. Existing studies usually assume an exogenously fixed bargaining protocol: (1) bargaining consists of consecutive rounds; (2) in each round the active player bargains with one of the passive players; and (3) the ordering of bargaining rounds and the duration (i.e., offers and counteroffers allowed) of each round are exogenously fixed. Two simple protocols are most commonly assumed in the literature, namely, the queuing protocol (e.g., Stole and Zwiebel 1996) and the rotating protocol (e.g., Horn and Wolinsky 1988a; Cai 2000, 2003). In the queuing protocol, passive players form a queue. The active player starts with the first passive player in the queue and does not switch to the next without reaching an agreement. In the rotating protocol, passive players form a ring. After each round of negotiation, the active player switches to the next regardless of whether an agreement has been reached. In reality, although we may sometimes observe fixed bargaining protocols in centralized negotiations, for example, the accession negotiation of international organizations, such as the World Trade Organization and the European Union, there are also many decentralized one-to-many bargaining situations without fixed protocols. Consider the situation in which the representative of a real-estate developer can visit one landowner each day. If she does not reach a deal with the one she visits, she is not obliged to return the next day, nor is she obliged to visit a different one. It is rather her own choice, which may well be strategic. When analyzing such decentralized bargaining situations, if we were to assume a specific 2
protocol, we must at least ensure that it is self-enforcing. That is, the players have no incentive to deviate from it. Alternatively, we can study a model without a fixed protocol, and then investigate whether any protocol can arise in an equilibrium. We develop a model of decentralized one-to-many bargaining. In each bargaining period, the active player first decides which passive player to bargain with. Then, either the active player or the chosen passive player is randomly selected to make a proposal and the other responds. A binding agreement is reached if the proposal is accepted. If no agreement is reached in the current period, the active player can either continue to bargain with the same passive player or switch to any other player in the following period. Immediately after the active player reaches agreements with all of the passive players, the joint project is implemented and each player receives his or her agreed-upon share of the surplus. 1 In our model, passive players are heterogeneous in terms of bargaining power, which is determined by two factors: the payoff discount factor and the probability of being chosen as the proposer. We aim to achieve two goals in our analysis. The first goal is to identify the upper and lower bounds of each player s equilibrium payoff in the limit in which the players become infinitely patient. It provides the range of outcomes that one can expect from a decentralized one-to-many bargaining situation. The second goal is to investigate whether queuing and rotating bargaining protocols are self-enforcing, that is, whether they can arise as a part of the active player s equilibrium strategy. The conclusion here has important implications for whether an assumption on bargaining protocol for a specific bargaining situation is justifiable. Our bargaining game admits a rich set of equilibria. In her worst equilibrium, the active player adopts a queuing protocol as a part of her strategy, while the first passive player in the queue receives his highest equilibrium payoff. Meanwhile, any ordering of passive players can be sustained in an equilibrium with a queuing protocol. However, a queuing protocol leads to a highly unequal equilibrium division. In the symmetric case in which all players are equally patient and have the same proposing probability, the first passive player receives one half of the surplus, the next one receives one half of the remaining half, and so forth. 1 Henceforth, the active player is referred to as she and a passive player is referred to as he. 3
In contrast, a rotating protocol leads to a much more plausible equilibrium division. In the symmetric case, it leads to an equal division of the surplus among all of the players. This may explain why the rotating protocol is often assumed in existing studies. However, we find that in generic situations, the rotating protocol is not self-enforcing. More precisely, if and only if all of the passive players are equally patient, our model admits equilibria with rotating protocols. Note that the asymmetry of proposing probabilities plays no role here. Therefore, although the proposing probability and the discount factor together determine the players relative bargaining powers, there are subtle differences between the two factors. Next, we construct a class of divide-and-conquer equilibria in which the active player pits one passive player against the others. The active player obtains her highest payoff in a divide-and-conquer equilibrium, whereas all but one of the passive players receive their lowest equilibrium payoffs, which converge to 0 as the discount factors tend to 1. We also characterize the unique mixed-strategy Markov equilibrium. When all of the passive players are equally patient, in the Markov equilibrium, the active player always randomizes with equal probability when choosing a bargaining opponent. The payoff vector of the Markov equilibrium converges to the same limit as that of any equilibrium with a rotating protocol. It is assumed in our main analysis that after reaching a bilateral agreement, the two players sign a contingent contract, by which the passive player receives his payoff after the project is implemented. There are many real-life situations in which only cash-offer contracts are feasible. Hence, we also consider a modified model in which any bilateral agreement is enforced through an immediate cash payment. We again obtain a class of equilibria with queuing protocols, but in this case, every passive player prefers to be the last to reach an agreement, in contrast to the case with contingent contracts. Again, the rotating protocol is not self-enforcing. A new finding here is that an impasse may arise in an equilibrium when there are at least three passive players. Related Literature. The one-to-many bargaining game is a natural extension of the classic bilateral bargaining game in Rubinstein (1982). Ithasbeenincorporatedintomodelsinvar- 4
ious contexts. Many authors have assumed a fixed bargaining protocol. Horn and Wolinsky (1988a) assume a rotating protocol in the wage negotiation between a firm and two groups of workers. In their seminal paper on intra-firm bargaining, Stole and Zwiebel (1996) specify a queuing protocol and investigate whether the Shapley value of the corresponding coalitional game can be sustained in an equilibrium. Westermark (2003) considers a random-matching protocol, in which the active player chooses a passive player in each round using an exogenously fixed randomization device. He shows that workers receive only a share of marginal products, instead of a share of average marginal products, as specified by the Shapley value. In a recent paper, Brügemann et al. (2015) show that the Shapley value cannot be sustained under the bargaining protocol specified in Stole and Zwiebel (1996) and propose a rotating bargaining protocol to achieve that goal. 2 Cai (2000, 2003) studies a pure one-to-many bargaining problem under a fixed rotating protocol. An important message of these two papers is that inefficient stationary equilibria may arise in a complete information setting. 3 More specifically, although bargaining order is fixed in the models, the order of reaching bilateral agreements is endogenously determined. Hence, delay occurs when the two orders are different. Without a fixed protocol, our model does not admit inefficient stationary equilibria. In other words, the existence of such equilibria is an artifact of the fixed bargaining order. A number of studies have attempted to endogenize bargaining order in various settings. Noe and Wang (2004) consider the negotiation between one buyer and two sellers and in- 2 Two crucial differences between our model and those of Stole and Zwiebel (1996) and Brügemann et al. (2015) should be noted. First, their models consider a more general value structure, in which the firm can settle by reaching agreements with a subset of workers. We study a pure bargaining problem, which requires the active player to reach agreements with all of the passive players. Second, they consider nonbinding bilateral agreements and thus allow renegotiation, whereas we assume binding agreements. In Section 5, we discuss the extension to nonbinding agreements and how our results relate to their studies. 3 Although many other papers have obtained inefficient equilibria in various bargaining games with complete information, their inefficient equilibria are usually sustained by credible punishment schemes as in the repeated game literature, and thus are highly nonstationary (e.g., Haller and Holden 1990; Fernandez and Glazer 1991; Busch and Wen 1995). 5
vestigate the strategic role of the confidentiality of the bargaining order. Notably, the buyer can take advantage of strategic uncertainty by concealing the bargaining order. This model is nicely extended by Krasteva and Yildirim (2012a), who examine the joint effects of confidentiality and offer deadlines. Krasteva and Yildirim (2012b) consider a model with payoff uncertainty and show that the buyer may have strict preferences regarding the order of bilateral negotiations with different sellers. A crucial feature of these models is that the buyer negotiates with each seller only once. Thus, backward induction leads to a unique equilibrium. If, as in our model, the buyer can negotiate with any seller repeatedly until an agreement is reached, multiple equilibria arise. More importantly, the buyer s uncertaintyinduced payoff advantage vanishes. Xiao (2015) also studies a one-to-many bargaining model in which the active buyer chooses which passive seller to bargain with in each period. The key feature of this model is that each passive seller can obtain a constant payoff flow (the inside option) before selling his asset. The heterogeneity of the inside option leads to a unique equilibrium, in which the buyer purchases in order of increasing size. In our model, sellers do not have inside options, but they are heterogeneous in terms of bargaining power. We obtain a vast multiplicity of equilibrium outcomes. The rest of this paper is organized as follows. Section 2 outlines the decentralized one-tomany bargaining model. Section 3 contains the equilibrium analysis. In Section 4, we modify the model using cash-offer contracts to enforce bilateral agreements. Section 5 discusses two interesting extensions. Section 6 concludes the paper. All of the proofs are relegated to a technical appendix. 2 Model 2.1 The Bargaining Game There are +1 players: an active player A and passive players indexed by N {1 2 }. Player A has a project with a commonly known surplus normalized to 1. To undertake the project, she needs the cooperation of all of the passive players. Hence, player 6
A has to bargain with every passive player over the payment to be made in exchange for his cooperation. Bargaining takes place over time divided into periods of equal length. In each period {0 1 2}, playerafirst chooses whom to bargain with. Then, either player A or the chosen player is randomly selected to make an offer and the other party responds with acceptance or rejection. The probability that player is selected as the proposer is (0 1) and the probability that player A is selected is 1. The offer is simply the share of the total surplus that player receives. If the offer is accepted, the two parties sign a binding agreement and player A continues to bargain with the other passive players. If it is rejected, bargaining continues in the next period and player A again chooses whom to bargain with. After player A has reached agreements with all of the passive players, the project is immediately implemented and the surplus is realized. A bargaining outcome is denoted by ( ) =1,where [0 1] is player s agreed-upon share and is the period in which this agreement is reached. Let =max { } bethedateonwhichthefinal agreement is reached. As it takes at least periods to reach an agreement with all of the passive players, the bargaining outcome is inefficient if 1. The players discount future payoffs with discount factor (0 1) for player A and (0 1) for passive player. The bilateral agreements between player A and each passive player are contingent contracts under which a passive player receives his share only after agreements have been reached with all of the passive players and the project has been implemented. Therefore, from the outcome ( ) =1,player s payoff is and player A s payoff is,where =1 P =1 is the share for player A. If there is an impasse (i.e., perpetual disagreement), the project is not implemented and everyone gets a payoff of 0. Finally, note that in our model, the passive players are heterogeneous in terms of their bargaining power relative to player A. With a few exceptions (e.g., Xiao 2015), existing studies usually assume that all passive players are identical. Our analysis shows that such an assumption is not entirely innocuous. In particular, with heterogeneous passive players, certain bargaining protocols may not be self-enforcing. 7
2.2 Histories and Strategies The bargaining game just described is an extensive game with perfect information and chance moves. Histories and strategies can be defined as usual. Let N 0 = N and N denote the set of passive players who have not reached agreements by period. Let ( ) denote the outcome of the bargaining period, where N is the identity of the chosen passive player in period, { } is the identity of the randomly selected proposer, [0 1] is the share proposed for player,and {Yes, No} is the response. The history at 0 is =( ) 1 =0 and 0 =. The set of all -histories is H. Each period consists of three stages. In stage 1, player A chooses N to bargain with in the current period. Then, either player A or the chosen passive player is randomly selected as the proposer. The proposer makes an offer in stage 2 and the other responds in stage 3. Hence, player A s strategy can be written as a sequence of functions = 1 2 3 =0, each of which assigns an action from the relevant set to each history. More specifically, in stage 1, we have 1 ( )= N. Depending on the outcome of the chance move, player A moves in either stage 2 or 3. Similarly, each player s strategy can be written as =( 2 3) =0. Only when player is chosen to bargain in period (i.e., 1 ( )=), is he entitled to move in either stage 2 or 3. The actions taken by player A and the chosen passive player in all three stages are specified in the following table: Stage 1 Stage 2 Stage 3 1 ( )= and = 2 ( ) [0 1] 3 ( ) {Yes, No} 1 ( )= and = 2 ( ) [0 1] 3 ( ) {Yes, No} 2.3 Bargaining Protocol Existing studies on one-to-many bargaining often assume a fixed bargaining protocol, which specifies the active player s bargaining opponent for each round and the order of offers and counteroffers in each round. Two most popular and natural bargaining protocols are the 8
queuing and rotating protocols. Under the queuing protocol, the remaining passive players form a queue. The active player always starts by bargaining with the first passive player in the queue. She is not allowed to switch to another passive player before reaching an agreement with the current one. Under the rotating protocol, the remaining passive players form a ring, and the active player moves along the ring. If no agreement is reached in one round,theactiveplayermustmovetothenext passive player on the ring in the following round. In our decentralized bargaining model, a bargaining protocol is not exogenously given. Instead, in each bargaining period, the active player chooses which passive player to bargain with. If no agreement is reached in the current period, the active player can either continue to bargain with the same passive player or switch to any other player in the next period. Nevertheless, as a part of her strategy (i.e., 1 ( )= N ), the active player may still adopt a certain protocol (e.g., queuing or rotating) in choosing her bargaining opponent. In such a case, her choice of bargaining opponent after any history depends only on N and 1, not on other details of the bargaining history, such as previous agreements or previously rejected offers. Acrucialdifference here is that the active player is free to deviate from the protocol. Thus, her continuation strategy after her own deviation must be specified. In our formal definitions, the active player adapts to her own deviation. That is, following the deviation, the active player takes the current passive player as the first one in the original protocol. Without loss of generality, we assume that the active player always follows the ascending order of the indices of passive players. 4 Let (N ) denote the smallest index in N. Definition 1 Player A adopts a queuing bargaining protocol if (i) 1 ( 0 ) = (N 0 ) for =0;(ii) 1 ( )= 1 if N = N 1 and 1 ( )=(N ) if N 6= N 1 for each 0. Definition 2 Player A adopts a rotating bargaining protocol if (i) 1 ( 0 )=(N 0 ) for =0;(ii) 1 ( )= 1 +1 mod N for each 0. 4 As shown in the equilibrium analysis, if there is an equilibrium in which player A adopts a queuing or rotating protocol, any ordering of passive players can be sustained. 9
Part (ii) of each definition specifies how the active player adapts to her own deviation. In the queuing protocol, she always chooses to bargain with the same passive player if no agreement was reached in the previous period. In the rotating protocol, she moves to the next passive player regardless of whether there was an agreement in the previous period. Although the active player may adopt a certain protocol in her strategy, it is not clear whether it can arise in an equilibrium. We say that a bargaining protocol is self-enforcing if it is a part of the active player s equilibrium strategy. 3 Equilibrium Analysis We adopt subgame perfect equilibrium (henceforth, equilibrium) as the solution concept. Following Cai (2000), we restrict our attention to the equilibria satisfying the following condition. Condition 1 If any two bargaining histories, and 0,differ only in the offers made in period 1 and if both offers are rejected, then ( )= ( 0 ) and ( )= ( 0 ). Condition 1 requires the players decisions in each period not to depend on the specific offers that are rejected in the previous period. It rules out the possibility of using rejected offers to coordinate future play and may thus substantially reduce the equilibrium set. For example, although Herrero (1985) and Sutton (1986) obtain a vast multiplicity of equilibria in a multilateral Rubinstein bargaining game, only the unique stationary equilibrium satisfies Condition 1. In contrast, our model admits a rich set of equilibria satisfying Condition 1. Nevertheless, Condition 1 is much weaker than requiring stationary or Markov strategies, as the players can still condition their decisions on other payoff-irrelevant variables, such as the time index and the identities of the remaining passive players. Two straightforward lemmas are shown as follows to begin the equilibrium analysis. Lemma 1 characterizes the unique equilibrium of the continuation bargaining game between player A and the last passive player. Lemma 2 shows that an impasse cannot arise in an equilibrium. 10
Suppose that player A has reached an agreement with every passive player 6= on 0 and P 6= 1. 5 The continuation game between player A and the last passive player is a simple variant of the Rubinstein bargaining game, which has a unique equilibrium with immediate agreement. Let =1 P 6= denote the residual share available in the final bargaining round between players A and. Lemma 1 In the bilateral bargaining between players A and for the last agreement, there is a unique equilibrium in which (i) player A always offers to player and accepts player s demand if it is no greater than ; (ii) player always asks for and accepts player A s offer if it is no less than,where = (1 ) (1 )+(1 )(1 ) and =1 (1 )(1 ) (1 )+(1 )(1 ) Hence, the expected equilibrium shares for players and A are respectively: ( )=(1 ) + = and ( )=(1 ) where = (1 ) (1 )+(1 )(1 ) Note that in the bilateral bargaining game between players A and with randomly selected proposers, player s expected share in the unique equilibrium is simply,which is jointly determined by the discount factors and the proposing probabilities. It may be interpreted intuitively: when player proposes, he is entitled a rent of size (1 ), perceived by player A as the rejection cost, and similarly when player A proposes; hence, player s expected equilibrium share is proportional to (1 ). Lemma 2 Impasse is not an equilibrium outcome of the one-to-many bargaining game. 5 It is imposed as a condition here that the sum of the previously agreed-upon shares, P 6=,isno greater than 1. Given that the active player prefers perpetual disagreement to any outcome that gives her a negative payoff, this condition must be satisfied in any equilibrium. 11
By Lemma 1, if the residual share is positive, player A and the last passive player reach an immediate agreement by which both receive positive payoffs. When there are more than one remaining passive players and a positive residual share, between player A and any passive player, rather than an impasse, it is better for both to agree on a share that does not exhaust the residual share. A simple backward induction argument leads to the conclusion that impasse cannot arise on any equilibrium path. In the follows, we first construct a class of equilibria in which the active player adopts a queuing protocol. It is shown that any ordering of the passive players can be sustained in such an equilibrium. Moreover, in an equilibrium with a queuing protocol, the active player receives her lowest equilibrium payoff, whereasthefirst passive player in the queue receives his highest equilibrium payoff. Next we show that if and only if all of the passive players are equally patient, there exists a class of equilibria in which the active player adopts a rotating protocol. We also construct a class of divide-and-conquer equilibria. The active player obtains her highest equilibrium payoff in a divide-and-conquer equilibrium, in which she pits the weakest passive player against the others. Finally, we characterize the unique Markov equilibrium. For expositional ease, we focus on the case with two passive players. All of the results are proved in the Appendix for general cases with 2 passive players. 3.1 Equilibria with a Queuing Protocol Recall that when the active player adopts a queuing protocol as a part of her strategy, (i) in period 0, she chooses to bargain with the first passive player in the queue; (ii) in each period 0, she stays with the same passive player 1 if no agreement is reached in period 1 and she moves to the next passive player in the queue if there is an agreement. To show that a queuing protocol can arise in an equilibrium, we first assume that the active player commits to it. With this assumption, the analysis is equivalent to that of the bargaining game with a fixed queuing protocol. For a given ordering of the passive players in a queue, we can characterize the unique equilibrium under the fixed queuing protocol and obtain player A s payoff in this equilibrium. Then we relax the protocol commitment 12
and examine whether the active player can benefit from a one-step deviation from the given protocol. It is crucial to note that after a one-step deviation, the active player simply switches to another fixed queuing protocol with a different ordering of the passive players, which has adifferent unique equilibrium. Hence, the active player has a profitable deviation from a fixed queuing protocol only if her equilibrium payoff under that protocol islowerthanit is under another fixed queuing protocol. Thus, the equilibrium construction is completed by showing that there is no such profitable deviation from any fixed queuing protocol or, equivalently, the active player s equilibrium payoff does not depend on the specific ordering of the passive players in the queue. Given a fixed queuing protocol, it is straightforward to derive the unique equilibrium. We now focus on the case with two passive players. In negotiating the first agreement, the equilibrium offer and demand from players A and and their responses are almost identical to those specified in Lemma 1, except that here the residual share is still 1. More precisely, in the equilibrium, (i) player A always offers to player and accepts player s demand if it is no greater than ; (ii) player always asks for and accepts player A s offer if it is no less than,where = (1 ) (1 )+(1 )(1 ) and =1 (1 )(1 ) (1 )+(1 )(1 ) Hence, under a fixed queuing protocol, player s expected equilibrium share from the first agreement is as follows: ( )=(1 ) + = where the subscript refers to the queuing protocol and the superscript refers to the specific ordering of the passive players in the queue. The first agreement is concluded without any regard to the subsequent negotiations. It is as if player A represents player and herself to bargain with player. After reaching the first agreement, player A and the other passive player immediately reach the second (last) agreement. As specified in Lemma 1, player s expected equilibrium share from the second agreement is ( )= (1 ); therefore, regardless of which 13
passive player reaches the first agreement, the active player s expected payoff is as follows: ( )= (1 1)(1 2) This implies that player A has no incentive to deviate from her strategy, particularly the queuing protocol. We can also conclude that the ordering of the passive players in the queuing protocol does not affect player A s equilibrium payoff. Hence, we have the following proposition. Proposition 1 In the one-to-many bargaining game, any ordering of the passive players can be sustained in an equilibrium with a queuing protocol. Ournextpropositionshowsthatofallpossibleequilibria,theactiveplayerobtainsher lowest expected payoff from the one with a queuing protocol, whereas the passive player who reaches the first agreement in the same equilibrium obtains his highest expected payoff. Proposition 2 Of all possible equilibria of the one-to-many bargaining game, player A s expected equilibrium payoff is no less than Y 1 (1 N ) whereas player s expected equilibrium payoff is no greater than 1. In the case in which all of the players have a common discount factor and equal proposing probability (i.e., = 12), in the equilibria with queuing protocols, the passive player reaching the ( =12 ) agreement receives an expected share of 12 and the active player s expected share is 12, which is less than that of all but the last passive player. 3.2 Equilibria with a Rotating Protocol Next, we investigate whether there also exist equilibria in which the active player adopts a rotating protocol as part of her strategy. As formally specified in Definition 2, under a rotating protocol, all of the passive players are ordered on a ring according to their indices. 14
If the active player bargains with player in period, then she switches to player +1in period +1regardless of whether an agreement is reached in period. Again, we start with the assumption that the active player commits to the rotating protocol. Under the fixed rotating protocol, there also exists a unique equilibrium in which an immediate agreement is reached in every subgame, but it is very tedious to calculate the equilibrium offers. We explain the logic behind the analysis as follows, but skip the details of the calculation. When bargaining for the first agreement, player always asks for and accepts any offer that is no less than, while player A always offers and accepts any demand no greater than. The following indifference conditions hold: = (1 ) 1 + 1 1 (1 )= (1 ) (1 ) 1 + 1 where the first condition says that player is indifferent to accepting and rejecting it, in which case player A reaches an agreement with player in the following period, and then player receives from the residual share an expected share of as specified in Lemma 1; and the second condition says that player A is indifferent to accepting the demand of and rejecting it. From the two pairs of indifference conditions, we can solve for ( 1 1) and ( 2 2). Then, player s expected share from reaching the first agreement can be calculated as follows: and player A s expected equilibrium share is ( )=(1 ) + ( )= 1 ( ) 1 where the subscript refers to the rotating protocol and the superscript refers to the ordering of the passive players on the ring. 15
Next, we relax player A s commitment to the rotating protocol and check whether there are any profitable one-step deviations, and in particular, we need to compare 12 ( ) with 21 ( ). The equilibrium with a rotating protocol is valid if and only if 12 ( )= 21 ( ) 1 = 2. In other words, there exists an equilibrium with a rotating protocol if and only if the two passive players are equally patient. If 1 2,then 12 ( ) 21 ( ). This means that if no agreement is reached in period 1 with player 1, then it is strictly better for player A to restart the rotation by choosing player 1 again in period 2, assuming commitment to the rotating protocol from then onward. Hence, the rotating protocol is not self-enforcing in this case, as well as when 1 2.For this negative result, it is sufficient to consider the case with two passive players. Proposition 3 In the case with =2and 1 6= 2, there does not exist an equilibrium with a rotating protocol. As an example, we consider the special case in which = 1 = 2 =. Thetwopairsof indifference conditions become: = (1 ) 1 + 1 (1 )(1 )=(1 ) (1 ) 1 + 1 by which we obtain = (1 ) 1 1 2 and = (1 )+ (1 ) 1 1 2. After reaching the first agreement, player A and the remaining passive player immediately reach the second agreement, as specified in Lemma 1. Being the first one to reach an agreement, player s expected share is ( )= (1 ) 1 1 2, 16
whereas his expected share from reaching the second agreement is ( )= (1 ). 1 1 2 As ( ) ( ), each passive player prefers to be the first one to reach an agreement. However, as tends to 1, the difference between ( ) and ( ) vanishes. It is also easy to verify that player A s expected share is 12 ( )= 21 ( )= (1 1)(1 2 ). 1 1 2 The rotating protocol is self-enforcing in this case. In this special case, it is also easy to verify that in the limit (i.e., 1), the payoff ratio between player and player A goes to (1 ) regardless of whether player is the first or second to reach an agreement. Furthermore, if 1 = 2 =12, then the equilibrium with a rotating protocol induces an equal split across all three players in the limit. The following proposition shows that there exists an equilibrium with a rotating protocol sustaining an arbitrary ordering of passive players on the ring, provided that all of the passive players are equally patient. 6 Proposition 4 If all of the passive players have the same discount factor, then any ordering of the passive players can be sustained in an equilibrium with a rotating protocol. Remark 1 It is shown that the rotating protocol is self-enforcing only under a non-generic symmetry condition. Hence, one must be cautious in assuming this protocol in studying specific one-to-many bargaining situations. It is reasonable to assume the rotating protocol only in situations in which it can be strictly enforced through a centralized mechanism. Remark 2 The rotating protocol is not self-enforcing, as the active player may benefit from deviating to a specific starting point on the rotation. However, as the players become infinitely patient, the deviation gain vanishes. Hence, if we adopt a less stringent solution concept, such as the -perfect equilibrium, the rotating protocol can occur in an equilibrium. 6 Theactiveplayermayhaveadiscountfactordifferent from that of the passive players. It does not affect the existence result. 17
In an earlier version of this paper, we show that when all of the passive players have the same discount factor, there also exist equilibria with general rotating protocols. Here, the active player bargains with player for periods before switching to the next passive player. Equilibrium agreements can be derived in a similar way. For any finite 1 and 2,it is easy to verify that the equilibrium outcome converges to the same limit as that with the basic rotating protocol (i.e., 1 = 2 =1). 3.3 Divide-and-Conquer Equilibria Cai (2003) studies a one-to-many bargaining model with a fixed rotating protocol and contingent contracts. It is shown that the active player can pit one passive player against another in a Markov equilibrium. The key to the equilibrium construction is that the active player creates a disparity of rejection costs by using a strategic delay tactic as a credible threat. As our model does not have a fixed protocol, there is greater room for the active player to exploit the strategic delay tactic. In the following, we construct a divide-and-conquer equilibrium, in which player A pits player 2 against player 1 to force the latter to accept a low share. 7 To simplify the notations, we again focus on the case in which all of the players share a common discount factor. The equilibrium construction can be easily extended to the general case. Let us first describe how player A chooses her bargaining opponent to reach the first agreement: (i) player A bargains with player 1 in period 0; (ii) if player 1 s demand is rejected, player A chooses player 1 again in the next period; (iii) if player A s offer is rejected by player 1, she switches to player 2 in the following ˆ 1 periods; (iv) after ˆ periods of bargaining with player 2 without reaching any agreement, player A switches back to player 1 and continues as specified in (i) to (iv); (v) finally, if player A deviates in any period, she continues with the queuing protocol as specified in Definition 1. 7 Li (2011) studies a bilateral bargaining game in which one party may leave the bargaining table indefinitely after his proposal is rejected. The construction of the divide-and-conquer equilibrium here is similar to that of Li (2011) in the sense that one party can create an endogenous disparity of rejection costs by exploring the strategic delay tactic. 18
Taking (i) to (iv) as a fixed protocol, player 1 s rejection leads to ˆ +1periods of delay, whereas player A s rejection leads to only one period of delay. This creates an endogenous disparity of bargaining power. Hence, in the equilibrium, players A and 1 reach the first agreement in period 0 on either ˆ 1 = if player A is selected to propose or ˆ +1 1 1 +(1 1 ) P ˆ =0 ˆ 1 = 1 (1 1) ˆ +1 1 +(1 1 ) P ˆ =0 if player 1 is selected to propose. After reaching the first agreement with player 1, players A and 2 bargain over the residual share, as specified in Lemma 1. ˆ 1 =(1 1 )(1 ˆ 1 )+ 1 (1 ˆ 1 )= (1 1) P ˆ =0 1 +(1 1 ) P ˆ =0 The ˆ periods of bargaining between players A and 2 serves as player 1 s punishment for rejecting player A s offer, during which no agreement can be reached. To complete the equilibrium construction, we must ensure that it is optimal for players A and 2 to carry out this punishment. By completing the punishment scheme, player A receives an expected payoff ˆ +1 (1 2 )ˆ 1 and player 2 receives ˆ +1 2ˆ 1. If player A deviates during the punishment by switching back to player 1 early, the continuation equilibrium is the one with a queuing protocol, in which player 1 s expected share is 1 and the expected residual share for players Aand2is1 1. It is easy to verify that for any ˆ 2 (1 2 ),when is sufficiently close to 1, players A and 2 have no incentive to deviate during the punishment phase. Proposition 5 For any integer ˆ 2 (1 2 ),when is sufficiently close to 1, thereis a divide-and-conquer equilibrium as described above. When there are more than two passive players, the construction of the divide-and-conquer equilibrium is similar. The active player can always pits one passive player against other 19
passive players one after another and force them to accept very low shares, provided that is sufficiently close to 1. Finally, the active player and the last passive player split the residual share as specified in Lemma 1. As tends to 1, themaximum ˆ in a divide-and-conquer equilibrium tends to infinity. Thus, player 1 s minimum expected share tends to 0 and player A s maximum expected share tends to 1 2. Similarly, player A can pit player 1 against player 2, such that her maximum expected share tends to 1 1 in the limit. Hence, when the players are extremely patient, the active player s greatest equilibrium payoff is arbitrarily close to what she may obtain from a bilateral bargaining with only the weakest passive player on a surplus of size 1. By the same token, a passive player s lowest equilibrium payoff tends to 0. Corollary 1 As tends to 1, the upper bound of the active player s expected equilibrium payoff tends to 1 min { }, while the lower bound of each passive player s expected equilibrium payoff tends to 0. 3.4 Markov Equilibrium Having obtained a vast multiplicity of equilibria, it is natural to consider Markov equilibrium as a refinement. A Markov strategy is a strategy that only depends on payoff-relevant variables. In the current model, these include the residual share and the set of passive players who have not reached agreements. Thus, the active player s strategy is Markovian if in all subgames with the same set of remaining passive players and the same residual share, (i) she always chooses the same one to bargain with or randomizes using the same probabilities; (ii) she always makes the same offer and accepts the same set of demands when bargaining with a specific passive player. A Markov equilibrium is a subgame perfect equilibrium in Markovian strategies. An immediate observation is that in our model a Markov equilibrium does not involve any delay. In Cai (2003), when there are two passive players, the game has three Markov equilibria and one of them is inefficient. The inefficient Markov equilibrium is an artifact of the fixed rotating protocol. More precisely, in Cai (2003), although the order of reaching 20
agreements is endogenously determined, the order of bargaining is exogenously given. Delay occurs when the two orders are different. Next, there does not exist any Markov equilibrium in pure strategies. To see this, let us again focus on the case where all of the players share a common discount factor. In a pure-strategy Markov equilibrium, player A always bargains with the same passive player (e.g., player 1) for the first agreement. The outcome should be the same as in an equilibrium with a fixed queuing protocol, in which player 1 s expected payoff is 1 and player 2 s is (1 1 ) 2. If player A makes a one-step deviation to choose player 2, the latter would be willing to accept any offer greater than 2 (1 1 ) 2. This makes the deviation profitable forplayerawhen is sufficiently close to 1. This leads to the conclusion that any equilibrium with either a queuing or rotating protocol is not Markovian. In an equilibrium with a queuing protocol, once player A deviates and bargains with another passive player, it is crucial that player A does not switch again until an agreement is reached. However a pure-strategy Markov equilibrium (in pure strategies) requires player A to choose the same passive player to bargain with after any history. When player A adopts a rotating protocol, her strategy is obviously not Markovian. In a mixed-strategy Markov equilibrium, in each period player A randomizes with the same probabilities ( 1 2) in choosing a passive player to bargain with. An immediate agreement is reached by either the chosen passive player accepting player A s offer or player A accepting player s demand. The following indifference conditions must hold in such an equilibrium: = [(1 ) + ]+ (1 ) 1 = 1 1 ( )+ 2 2 ( ) (1 ) 1 + 1 1 ( )= 2 ( ) 21
where 1 ( )=[(1 1 )(1 1)+ 1 (1 1)] (1 2) 2 ( )=[(1 2 )(1 2)+ 2 (1 2)] (1 1) The first condition states that player must be indifferent to accepting player A s offer and rejecting it. If player rejects the offer, with probability, he is chosen again in the following period and reaches an agreement on either or. With probability, player A reaches an agreement with player on either or, after which player receives ashareof from the residual share. Similarly, the second condition says that player A must be indifferent to accepting and rejecting player s demand. The last condition states that player A must be indifferenttochoosingplayers1or2ineachperiod,where ( ) denotes her expected payoff from choosing player. Hence, there are six unknowns, ( 1 1 2 2 1 2), five indifference conditions, and the condition of 1 + 2 =1. After some tedious algebra, we find the unique Markov equilibrium in mixed strategies. In this Markov equilibrium, before the first agreement is reached, in each period player A randomizes in choosing whom to bargain with according to the following probabilities: 1 = 1 (1 2 ) 1 (1 2 )+ 2 (1 1 ) and 2 = 2 (1 1 ) 1 (1 2 )+ 2 (1 1 ) When the two passive players are equally patient, each of them is chosen with probability 12 regardless of their different proposing probabilities. Again, the two factors determining relative bargaining power play different roles in one-to-many bargaining. In the general case, there also exists a unique Markov equilibrium in mixed strategies. In this equilibrium, player A s randomization probabilities depend on their discount factors in an extremely complicated way. However, when all of the passive players are equally patient, they are always chosen with equal probabilities. Proposition 6 There is a unique Markov equilibrium in mixed strategies. If all of the passive players have the same discount factor, then player A chooses each passive player with probability 1 N in every subgame, where N is the set of remaining passive players. 22
Recall that when all of the players have the same discount factor, there also exist multiple equilibria with rotating protocols. As tends to 1, the Markov equilibrium outcome converges to the same limit as that of the equilibria with rotating protocols. Remark 3 Westermark (2003) considers a random-matching protocol in which the active player randomly chooses a passive player in each round and the randomization probabilities may depend on the number and identities of the remaining passive players. Our result suggests that there exists one and only one self-enforcing randomization probability. Remark 4 The absence of a Markov equilibrium in pure strategies raises the question of whether the notion of a Markov equilibrium is too restrictive in the current setting. Alternatively, we can also include the identity of the bargaining opponent in the previous period as a state variable. Then, whether to switch to another opponent and which one to switch to become the decisions of the current period. The equilibria with either queuing or rotating protocols become Markovian after the introduction of this extra state variable. Remark 5 The mixed-strategy Markov equilibrium is robust in the following sense. Suppose that the active player deviates and commits to a randomization probability that is slightly greater than.player would demand a greater share compared with that under, whereas the other passive player s demand would decrease. This gives the active player the incentive to reduce to. 4 Cash-Offer Contract In some real-life situations, such as land acquisitions, only cash-offer contracts are feasible. That is, when the active player reaches an agreement with a passive player, the latter immediately receives an unconditional cash payment from the former. For example, in Coase s (1960) well-known railroad example, it is reasonable to assume that the negotiating parties are limited to cash-offer contracts. Obvious differences exist between contingent contracts and cash-offer contracts. Thus, it is desirable to investigate how the equilibrium set of the 23
one-to-many bargaining game is affected by the contract form adopted in the model. The following analysis is in the same vein as that in the previous section, except that we defer the discussion of an impasse being a possible equilibrium outcome to the end. To emphasize the key points, we assume that all of the players have a common discount factor. With cash-offer contracts, when the active player and a passive player reachanagreement on in period,player immediately receives a cash payment.thushispayoff is. This cash payment also becomes player A s sunk cost. From the bargaining outcome ³ ( ) =1, player A obtains a payoff of P =1, as she receives the entire surplus at =max { } after reaching agreements with all of the passive players. An immediate observation is that passive player, who reaches the last agreement, obtains an expected equilibrium payoff of. The last agreement is not affected by previous agreements (or payments), as they have become player A s sunk costs. This is in contrast to the model with contingent contracts, in which each agreement is reached without any regard to future bargaining, whereas the last agreement is negotiated over the residual share. In the following analysis, we again focus on the equilibria satisfying Condition 1 as stated in Section 3. In the case with two passive players, let us assume that player A commits to a queuing protocol (as specified in Definition 1). When players A and 1 bargain for the first agreement, they both anticipate a future payment of expected size 2 to player 2. Thus, they are effectively bargaining over a surplus of (1 2 ). Hence, in the equilibrium, player 1 s expected payment is 1 (1 2 ), whereas player A s expected payoff is (1 1 )(1 2 ). This leads to the following proposition. Proposition 7 With cash-offer contracts, any ordering of the passive players can be sustained in an equilibrium with a queuing protocol, in which player A s expected payoff is 1 Y N (1 ) Suppose that the passive players are ordered by their indices in the queue. Player s expected payoff is 1 Y =+1 (1 ) 24
Recall that with contingent contracts, in an equilibrium with a queuing protocol, every passive player prefers to reach an early agreement. Here, it is the opposite: every passive player prefers to be the last one to reach an agreement. In the special case with identical proposing probabilities (i.e., =12), the passive player who reaches the last agreement receives an expected payment of 12, the second-to-last passive player receives 4, andthe first one receives 1 2, which is also player A s expected equilibrium payoff. With contingent contracts, the active player obtains the lowest payoff in the equilibria with queuing protocols. This is also different here. The active player obtains a lower payoff in any equilibrium with a rotating protocol when it exists, and an impasse may occur when there are three or more passive players. With cash-offer contracts, equilibria with rotating protocols require more restrictive conditions than they do for contingent contracts. More precisely, when =2, a rotating protocol is self-enforcing only when the two passive players have the same proposing probability that is no greater than 12. Proposition 8 ( =2) With cash-offer contracts, an equilibrium with a rotating protocol does not exist if either (i) 1 6= 2 or (ii) 1 + 2 1 and is sufficiently close to 1. In an equilibrium with a rotating protocol, each passive player can secure a payment of by refusing to reach the first agreement. Hence, when tends to 1, the expected equilibrium payoff of player tends to.if 1 + 2 1 and is sufficiently close to 1, such an equilibrium is obviously not viable. When 1 + 2 1 but 1 6= 2, a rotating protocol is not self-enforcing, as the active player always prefers to start the rotation with the passive player with the larger. This negative result is almost identical to Proposition 3. However, with contingent contracts, nonexistence results solely from asymmetry on discount factors, whereas with cash-offer contracts, asymmetry on proposing probabilities causes the same negative result. It is also straightforward to characterize divide-and-conquer equilibria with cash-offer contracts. The key step is to ensure that it is optimal for the active player to carry out the 25