- Email: [email protected]
A minimal sufficient set of procedures in a bargaining model
Accepted Manuscript A minimal sufficient set of procedures in a bargaining model Liang Mao, Tianyu Zhang
PII: DOI: Reference:
S0165-1765(17)30016-2 http://dx.doi.org/10.1016/j.econlet.2017.01.006 ECOLET 7475
To appear in:
Economics Letters
Received date : 18 October 2016 Revised date : 17 December 2016 Accepted date : 6 January 2017 Please cite this article as: Mao, L., Zhang, T., A minimal sufficient set of procedures in a bargaining model. Economics Letters (2017), http://dx.doi.org/10.1016/j.econlet.2017.01.006. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
A Minimal Sufficient Set of Procedures in a Bargaining Model Highlights: We study two-player strategic bargaining games with deterministic procedures We define a class of procedures called normalized procedures Each feasible payoff outcome can be implemented by a normalized procedure Different normalized procedures result in different payoff outcomes
*Manuscript Click here to view linked References
A Minimal Sufficient Set of Procedures in a Bargaining Model Liang Mao∗
Tianyu Zhang†
Abstract For a two-player bargaining model, Mao (2016) extends the alternating offers procedure of Rubinstein (1982) to more general procedures and explores which payoff outcomes are feasible, in the sense that they can be supported by some procedures as subgame perfect equilibria. In this paper, we define a special class of procedures called normalized procedures. We show that while the set of normalized procedures can yield all feasible partitions, none of its proper subsets can do so. Keywords: strategic bargaining, subgame perfect equilibrium, normalized procedures, minimal sufficient set JEL code: C72, C78
1
Introduction
Suppose that two players A and B bargain to divide a cake that is perfectly divisible. In a seminal work, Rubinstein (1982) proves that if the bargaining follows an alternating offers procedure, a unique partition of the cake exists; ∗
College of Economics, Shenzhen University, Shenzhen, Guangdong 518060, China. Email: [email protected]. † Economics and Management School, Wuhan University, Wuhan, Hubei 430072, China.
1
this can be supported by a subgame perfect equilibrium (SPE). Numerous studies in the literature have examined extensions or applications of the alternating offers bargaining model. For example, see Shaked and Sutton (1984), Binmore (1985), Binmore et al. (1986), Muthoo (1990), Chatterjee et al. (1993), Krishna and Serrano (1996), Watson (1998), In and Serrano (2004), and Ray (2007), among others.1 Mao (2016) follows this literature to extend the alternating offers procedure to more general procedures, in which a single player can make proposals in several consecutive rounds. The main result of that study is that the game has a unique SPE outcome and thus extends the analogous result of Rubinstein (1982). The paper also studies the payoff outcomes that can be supported as SPE by choosing some appropriate procedures. For an application, suppose a designer chooses the bargaining procedure. Now, the results in Mao (2016) can help us understand which partition is feasible; that is, the designer can implement it as an SPE. These results are listed in Section 2 for the readers’ convenience. Since different procedures may sometimes lead to the same SPE outcome, the designer can actually implement all feasible partitions by some subset of the set of all procedures. In Section 3, we define a set of special procedures called normalized procedures, which can be constructed by an iterative algorithm. We show in Section 4 that while the designer can implement all feasible partitions using the set of all normalized procedures, a smaller set of procedures may fail to implement some feasible partition. Thus, it is always appropriate for the designer to focus on normalized procedures to achieve a desirable bargaining outcome.
2
Preliminaries
In this section, we briefly review the notations, model setup, and some of the conclusions of Mao (2016). We refer the readers to that paper for the proofs of all conclusions arrived at in this section. 1
See Serrano (2008) for a recent survey.
2
The player set is N = {A, B}. Each player i ∈ N has a constant discount factor δi ∈ (0, 1). A procedure consists of a (finite or infinite) sequence of A’s and B’s, and can be denoted by ω = (ω1 , ω2 , . . .) = (A, · · · , A, B, · · · , B , A, · · · , A, · · · ) , [n1 , n2 , n3 , · · · ], | {z } | {z } | {z } n1
n2
n3
where ωi = A or B is the ith element of the sequence. Without loss of generality, we assume that the first element of each procedure is ω1 = A. Let T (ω) be the number of elements ω contains. In particular, T (ω) = ∞ if ω is infinite. Let Ω denote the set of all procedures. Given δA , δB , and ω, the bargaining game G(ω, δA , δB ) proceeds as follows. Time is discrete and can be denoted by period t = 1, 2, · · · , T (ω). Suppose the game has come to period t ≤ T (ω). The proposer in this period is ωt , who makes an offer dt from the agreement set {(dA , dB ) | dA , dB ≥ 0, dA +dB = 1}, where di is i’s share of the cake in the agreement. The other player i 6= ωt decides whether to accept or reject this offer. If dt = (dA , dB ) is accepted, the game ends and player i’s payoff is ui (di , t) = δit−1 di . If t ≤ T (ω) − 1 and the offer is rejected, then the game proceeds to the next period t + 1. If no agreement is ever accepted in all periods t ≤ T (ω), both players receive zero payoff. We solve G(ω, δA , δB ) by subgame perfect equilibrium (SPE). Theorem 1. Given ω = [n1 , n2 , n3 , n4 , · · · ], there exists a unique SPE out come in which players reach agreement θ(ω), 1 − θ(ω) without delay, where θ(ω) = 1 − δBn1 + δBn1 δAn2 − δBn1 +n3 δAn2 + δBn1 +n3 δAn2 +n4 − · · · .
(1)
This theorem is an extension of the main theorem of Rubinstein (1982), since, if ω = [1, 1, 1, · · · ] is an infinite alternating offers procedure, it follows 1−δB . from (1) that θ(ω) = 1−δ A δB
3
More formally, given ω = [n1 , n2 , n3 , · · · ], we define 0, r(ω) = m, ∞,
and
p(ω, k) =
(
if ω = [n1 ] if ω = [n1 , n2 , · · · , nm+1 ] , if ω is infinite
1, δBn1 δAn2 δBn3
· · · σknk ,
if k = 0 , if k = 1, 2, . . . , r(ω)
where σk = δA if k is even, and σk = δB if k is odd. We can rewrite (1) as θ(ω) =
r(ω) X
(−1)k p(ω, k).
(2)
k=0
The following two lemmas guarantee that θ(ω) is well defined by (2) even when ω is infinite and that θ(ω) actually defines a partition of the cake. Lemma 1.
P∞
k k=t (−1) p(ω, k)
is absolutely convergent for all t ≥ 0.
Lemma 2. For any ω ∈ Ω, 1 − δB ≤ θ(ω) ≤ 1.
P Furthermore, let zr (ω) = rk=0 (−1)k p(ω, k), r = 0, 1, . . . , r(ω); these can be regarded as the SPE partitions of the corresponding truncated procedures of ω. If ω = [n1 , n2 , · · · ], then z0 (ω) = θ([n1 ]) = 1, z1 (ω) = θ([n1 , n2 ]) = 1 − δBn1 , ..., zr(ω) (ω) = θ(ω). The next lemma implies that the elements in the sequence zr (ω) r=0,1,...,r(ω) are alternately larger and smaller than θ(ω).2 Lemma 3. For any t < s ≤ r(ω), zt (ω) < zs (ω) if t is odd, and zt (ω) > zs (ω) if t is even.
One possible application of Theorem 1 is as a tool to analyze the influence of the procedure on the bargaining outcome. We are particularly interested in which partitions can be implemented in SPE by choosing appropriate procedures. Let Γ(δA , δB ) = {θ(ω) | ω ∈ Ω} collect all partitions that are feasible in the sense that they can be supported in SPE by some procedure. 2
Let s = r(ω) when ω is finite, and let s → r(ω) when ω is infinite.
4
Then, the next two theorems show that all partitions are feasible if the players are patient enough, but almost no partitions are feasible if the players are impatient. Theorem 2. If δA + δB ≥ 1, then Γ(δA , δB ) = [1 − δB , 1].3 Theorem 3. If δA + δB < 1, then the (Lebesgue) measure of Γ(δA , δB ) is 0. Note that we do not consider random procedures in which the proposer is randomly chosen at some period. In fact, when x ∈ Γ(δA , δB ), a designer can achieve expected payoffs (x, 1 − x) by designing a one-period random procedure in which (a) player A (or B) has probability x (or 1 − x) to be the proposer and makes an offer and (b) rejection of this offer leads to the end of the game when both players’ payoffs are zero. However, if the designer is risk averse, this random procedure is not as good for the designer as the deterministic procedure, which can implement the same outcome (x, 1 − x) without uncertainty.
3
Normalized Bargaining Procedures
Given a feasible partition x ∈ Γ(δA , δB ), there may exist multiple procedures that result in the same SPE outcome (x, 1 − x). Let Ω(x) = {ω ∈ Ω | θ(ω) = x} collect all these procedures. For each ω ∈ Ω(x), the elements in the sequence zr (ω) r=0,1,...,r(ω)−1 are alternately larger and smaller than x, and they will either reach or converge to x.4 However, for different ω ∈ Ω(x), zr (ω) may approach x at different speeds. 1−δB = 59 . It is easy to Example 1. Suppose δA = 53 , δB = 32 , and x = 1−δ A δB verify that both the infinite alternating offers procedure ω 1 = [1, 1, . . .] and the finite procedure ω 2 = [2, 1] are in Ω(x), i.e. θ(ω 1 ) = θ(ω 2 ) = x. A designer whose target outcome is (x, 1−x) might well prefer ω 2 to ω 1 , not only because ω 2 is simpler, but also because zr (ω 2 ) approaches x faster than zr (ω 1 ) does, in the sense that |zr (ω 2 ) − x| ≤ |zr (ω 1 ) − x| for all r ≤ min{r(ω 1 ), r(ω 2 )}. 3 4
Note that if Ω also contains the procedures that ω1 = B, then Γ(δA , δB ) = [0, 1]. That is, either zr(ω) (ω) = x when ω is finite, or limr→∞ zr (ω) = x when ω is infinite.
5
Inspired by the above example, we are interested in a special class of procedures that keep zr (ω) as close to θ(ω) as possible for each r ≤ r(ω). More specifically, for x ∈ Γ(δA , δB ), we can construct ω ∈ Ω(x) by using the following algorithm. We first assume that δA + δB ≥ 1, and thus Γ(δA , δB ) = [1 − δB , 1] due to Theorem 2. If x = 1, let ω = [1]. Otherwise, we have 1 − δB ≤ x < 1. Let n1 be the integer such that 1 − δBn1 ≤ x < 1 − δBn1 +1 ; that is, n1 is the maximal n such that 1 − δBn ≤ x. If x = 1 − δBn1 , let ω = An1 B; otherwise, let n2 be the integer such that 1 − δBn1 + δAn2 +1 δBn1 < x ≤ 1 − δBn1 + δAn2 δBn1 . Suppose by induction that n1 , . . . , nh have already been defined, satisfying zh (ω) ≤ x < zh−1 (ω) − δB p(ω, h), if h is odd;
(3)
zh−1 (ω) + δA p(ω, h) < x ≤ zh (ω), if h is even.
(4)
If x = zh (ω), then let ω = [n1 , n2 , · · · , nh , 1]. If x 6= zh (ω), we need to define nh+1 . Let nh+1 be the integer such that zh (ω) + δA p(ω, h + 1) < x ≤ zh+1 (ω), if h is odd;
(5)
zh+1 (ω) ≤ x < zh (ω) − δB p(ω, h + 1), if h is even.
(6)
A question is whether nh+1 is well defined, that is, whether there exists exactly one integer nh+1 such that (5) or (6) holds. Mao (2016) provides a positive answer to this question when δA + δB ≥ 1.5 Hence, for each x ∈ [1 − δB , 1], we will either get a finite procedure ω = [n1 , · · · , nm , 1] such that θ(ω) = zm (ω) = x or eventually get an infinite procedure ω = [n1 , n2 , · · · ] such that θ(ω) = limr→∞ zr (ω) = x. Thus, we have constructed ω ∈ Ω(x). We can formally describe the procedures constructed by the above algorithm as follows: Definition 1. ω is called a normalized procedure if it satisfies the following two conditions: (i) When ω = [n1 , . . . , nr(ω)+1 ] is finite, nr(ω)+1 = 1. 5
See the proof of Theorem 2.
6
(ii) When r(ω) ≥ 1, for any q ≥ 1, zq (ω) 6 θ(ω) < zq−1 (ω) − δB p(ω, q), if q is odd;
zq−1 (ω) + δA p(ω, q) < θ(ω) 6 zq (ω), if q is even.
(7) (8)
Note that not all procedures are normalized. For instance, Example 1 shows that the infinite alternating offers procedure may not be normalized. Let Ω∗ denote the set of all normalized procedures. Then, the above analysis establishes the following: Lemma 4. If δA + δB ≥ 1, then for each x ∈ Γ(δA , δB ), there exists a normalized procedure ω ∈ Ω(x). When δA + δB < 1, only a degenerate set of partitions are feasible due to Theorem 3. Under this condition, the above iterative construction algorithm is no longer valid for x ∈ / Γ(δA , δB ), since nh+1 in (5) or (6) need not be well defined. Nevertheless, we still can define normalized procedures by Definition 1 and have an analogous conclusion for each feasible partition. Lemma 5. If δA + δB < 1, then for each x ∈ Γ(δA , δB ), there exists a normalized procedure ω ∈ Ω(x). Proof. Since x ∈ Γ(δA , δB ), there exists ω 0 such that θ(ω 0 ) = x. Let ω = ω 0 if ω 0 is infinite, and let ω = [n1 , . . . , nr(ω0 ) , 1] if ω 0 = [n1 , . . . , nr(ω0 ) , nr(ω0 )+1 ] is finite. It is obvious that θ(ω) = θ(ω 0 ), and thus ω ∈ Ω(x). It remains to prove that ω ∈ Ω∗ . Suppose, for a contradiction, that ω ∈ / Ω∗ . Then, according to Definition 1, there exists an integer s ≥ 1, such that for all integers p = 0, 1, . . . , s − 1, when p is odd, (7) holds, and when p is even, (8) holds, but p = s does not satisfy (7) or (8). Without loss of generality, we assume that s is odd. Then, we either have θ(ω) < zs (ω), which contradicts Lemma 3 or we have θ(ω) > zs−1 (ω) − δB p(ω, s), and thus r(ω) X (−1)k p(ω, k) = θ(ω) − zs−1 (ω) > −δB p(ω, s). k=s
7
(9)
Pr(ω) k On the other hand, if s = r(ω), then k=s (−1) p(ω, k) = −p(ω, s) < −δB p(ω, s), which contradicts (9); if s < r(ω), then according to Lemma 3, Pr(ω) Ps+1 k k k=s (−1) p(ω, k) 6 k=s (−1) p(ω, k) = −p(ω, s)+p(ω, s+1) 6 −p(ω, s)+ δA p(ω, s) < −p(ω, s) + (1 − δB )p(ω, s) = −δB p(ω, s), which also contradicts (9). Therefore, we have proved that ω ∈ Ω∗ . In the proof of Lemma 5, we have shown that when δA +δB < 1, almost all procedures are normalized (except for those finite procedures with nr(ω)+1 6= 1), and thus Ω∗ virtually equals Ω. For each x ∈ [1 − δB , 1], we can check whether x ∈ Γ(δA , δB ) by applying the aforementioned iterative algorithm. If, in the process of this algorithm, we find that for some h, there does not exist nh+1 satisfying (5) or (6), then x ∈ / Γ(δA , δB ); otherwise, we will eventually obtain a normalized procedure ω ∈ Ω(x), which indicates x ∈ Γ(δA , δB ).
4
Main Result
By the definition of Γ(δA , δB ), one can always achieve a feasible partition by choosing some appropriate procedure ω ∈ Ω. Now, we examine the question of whether one can accomplish this by using a smaller set of procedures, and, if the answer is yes, what is the “smallest” set that can accomplish this goal. Definition 2. A set of procedures Ω ⊂ Ω is said to be sufficient when, for each ω ∈ Ω, there exists ω 0 ∈ Ω such that θ(ω 0 ) = θ(ω). In addition, Ω is called a minimal sufficient set if Ω is sufficient, and θ(ω 1 ) 6= θ(ω 2 ) whenever ω 1 , ω 2 ∈ Ω, ω 1 6= ω 2 . In other words, a minimal sufficient set contains enough procedures to implement all feasible outcomes, but none of its proper subsets is sufficient. Note that there could exist more than one minimal sufficient set for a game. For example, suppose Ω is a minimal sufficient set, and θ(ω 1 ) = θ(ω 2 ), where ω 1 ∈ Ω, ω 2 ∈ / Ω, then Ω ∪ {ω 2 } \{ω 1 } is a minimal sufficient set as well. The main finding of this paper is the following proposition. It implies that while the set of normalized procedures can yield all feasible partitions, none of its proper subsets can do so. It also suggests that the normalized 8
procedure ω ∈ Ω(x) that we found in the proof of Lemma 4 or Lemma 5 is actually the only normalized procedure in Ω(x). Proposition 1. Ω∗ is a minimal sufficient set. Proof. According to Lemma 4 and Lemma 5, Ω∗ is a sufficient set. It remains to show that if ω, ω 0 ∈ Ω∗ , ω = [n1 , n2 , · · · ] 6= ω 0 = [n01 , n02 , · · · ], then θ(ω) 6= θ(ω 0 ). Suppose for a contradiction that θ(ω) = θ(ω 0 ). Let l be the least integer such that nl 6= n0l . That is, ni = n0i , i = 1, 2, . . . , l − 1 but nl 6= n0l . Assume without loss of generality that nl < n0l . If l is odd, then we have θ(ω 0 ) = θ(ω) < zl−1 (ω) − δB p(ω, l) ≤ zl−1 (ω) − p(ω 0 , l) = zl (ω 0 ), which contradicts ω 0 ∈ Ω∗ . We can similarly create a contradiction if l is even. This implies that θ(ω) 6= θ(ω 0 ), and this ends the proof.
5
Conclusion
In this study, we define normalized procedures and prove that they form a minimal set that could support all feasible partitions as SPE outcomes. A designer can use the set of normalized procedures as a toolbox for implementing any feasible outcome. Although there may exist other minimal sufficient sets other than Ω∗ , we believe that normalized procedures have some special merits. For each feasible partition, there exists a unique corresponding normalized procedure, which can be easily constructed by using a simple algorithm. This algorithm can also be applied to check whether a partition is feasible in the first place. By contrast, it is usually (to the best of our knowledge) more difficult to find a non-normalized procedure to implement any feasible outcome. Finally, a normalized procedure ω usually makes zr (ω) approach θ(ω) at a relatively fast speed.
Acknowledgements We thank the referee and an editor for very helpful comments and suggestions. This work was supported by the National Natural Science Foundation of China (Grant No. 71471119). 9
References Binmore, K., 1985. Bargaining and coalitions, in: Roth, A.E. (Ed.), GameTheoretic Models of Bargaining. Cambridge University Press. Binmore, K., Rubinstein, A., Wolinsky, A., 1986. The Nash bargaining solution in economic modelling. The RAND Journal of Economics 17, 176–188. Chatterjee, K., Dutta, B., Ray, D., Sengupta, K., 1993. A noncooperative theory of coalitional bargaining. Review of Economic Studies 60, 463–477. In, Y., Serrano, R., 2004. Agenda restrictions in multi-issue bargaining. Journal of Economic Behavior and Organization 53, 385–99. Krishna, V., Serrano, R., 1996. Multilateral baigaining. Review of Economic Studies 63, 61–80. Mao, L., 2016. Subgame perfect equilibrium in a bargaining model with deterministic procedures. Theory and Decision (Online first), 1–16. Muthoo, A., 1990. Bargaining without commitment. Games and Economic Behavior 2, 291–297. Ray, D., 2007. A Game-Theoretic Perspective on Coalition Formation. Oxford University Press. Rubinstein, A., 1982. Perfect equilibrium in a bargaining model. Econometrica 50, 97–110. Serrano, R., 2008. Bargaining, in: Durlauf, S.N., Blume, L.E. (Eds.), The New Palgrave Dictionary of Economics. Palgrave Macmillan. Shaked, A., Sutton, J., 1984. Involuntary unemployment as a perfect equilibrium in a bargaining model. Econometrica 52, 1351–1364. Watson, J., 1998. Alternating-offer bargaining with two-sided incomplete information. Review of Economic Studies 65, 573–94.
10
PII: DOI: Reference:
S0165-1765(17)30016-2 http://dx.doi.org/10.1016/j.econlet.2017.01.006 ECOLET 7475
To appear in:
Economics Letters
Received date : 18 October 2016 Revised date : 17 December 2016 Accepted date : 6 January 2017 Please cite this article as: Mao, L., Zhang, T., A minimal sufficient set of procedures in a bargaining model. Economics Letters (2017), http://dx.doi.org/10.1016/j.econlet.2017.01.006. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
A Minimal Sufficient Set of Procedures in a Bargaining Model Highlights: We study two-player strategic bargaining games with deterministic procedures We define a class of procedures called normalized procedures Each feasible payoff outcome can be implemented by a normalized procedure Different normalized procedures result in different payoff outcomes
*Manuscript Click here to view linked References
A Minimal Sufficient Set of Procedures in a Bargaining Model Liang Mao∗
Tianyu Zhang†
Abstract For a two-player bargaining model, Mao (2016) extends the alternating offers procedure of Rubinstein (1982) to more general procedures and explores which payoff outcomes are feasible, in the sense that they can be supported by some procedures as subgame perfect equilibria. In this paper, we define a special class of procedures called normalized procedures. We show that while the set of normalized procedures can yield all feasible partitions, none of its proper subsets can do so. Keywords: strategic bargaining, subgame perfect equilibrium, normalized procedures, minimal sufficient set JEL code: C72, C78
1
Introduction
Suppose that two players A and B bargain to divide a cake that is perfectly divisible. In a seminal work, Rubinstein (1982) proves that if the bargaining follows an alternating offers procedure, a unique partition of the cake exists; ∗
College of Economics, Shenzhen University, Shenzhen, Guangdong 518060, China. Email: [email protected]. † Economics and Management School, Wuhan University, Wuhan, Hubei 430072, China.
1
this can be supported by a subgame perfect equilibrium (SPE). Numerous studies in the literature have examined extensions or applications of the alternating offers bargaining model. For example, see Shaked and Sutton (1984), Binmore (1985), Binmore et al. (1986), Muthoo (1990), Chatterjee et al. (1993), Krishna and Serrano (1996), Watson (1998), In and Serrano (2004), and Ray (2007), among others.1 Mao (2016) follows this literature to extend the alternating offers procedure to more general procedures, in which a single player can make proposals in several consecutive rounds. The main result of that study is that the game has a unique SPE outcome and thus extends the analogous result of Rubinstein (1982). The paper also studies the payoff outcomes that can be supported as SPE by choosing some appropriate procedures. For an application, suppose a designer chooses the bargaining procedure. Now, the results in Mao (2016) can help us understand which partition is feasible; that is, the designer can implement it as an SPE. These results are listed in Section 2 for the readers’ convenience. Since different procedures may sometimes lead to the same SPE outcome, the designer can actually implement all feasible partitions by some subset of the set of all procedures. In Section 3, we define a set of special procedures called normalized procedures, which can be constructed by an iterative algorithm. We show in Section 4 that while the designer can implement all feasible partitions using the set of all normalized procedures, a smaller set of procedures may fail to implement some feasible partition. Thus, it is always appropriate for the designer to focus on normalized procedures to achieve a desirable bargaining outcome.
2
Preliminaries
In this section, we briefly review the notations, model setup, and some of the conclusions of Mao (2016). We refer the readers to that paper for the proofs of all conclusions arrived at in this section. 1
See Serrano (2008) for a recent survey.
2
The player set is N = {A, B}. Each player i ∈ N has a constant discount factor δi ∈ (0, 1). A procedure consists of a (finite or infinite) sequence of A’s and B’s, and can be denoted by ω = (ω1 , ω2 , . . .) = (A, · · · , A, B, · · · , B , A, · · · , A, · · · ) , [n1 , n2 , n3 , · · · ], | {z } | {z } | {z } n1
n2
n3
where ωi = A or B is the ith element of the sequence. Without loss of generality, we assume that the first element of each procedure is ω1 = A. Let T (ω) be the number of elements ω contains. In particular, T (ω) = ∞ if ω is infinite. Let Ω denote the set of all procedures. Given δA , δB , and ω, the bargaining game G(ω, δA , δB ) proceeds as follows. Time is discrete and can be denoted by period t = 1, 2, · · · , T (ω). Suppose the game has come to period t ≤ T (ω). The proposer in this period is ωt , who makes an offer dt from the agreement set {(dA , dB ) | dA , dB ≥ 0, dA +dB = 1}, where di is i’s share of the cake in the agreement. The other player i 6= ωt decides whether to accept or reject this offer. If dt = (dA , dB ) is accepted, the game ends and player i’s payoff is ui (di , t) = δit−1 di . If t ≤ T (ω) − 1 and the offer is rejected, then the game proceeds to the next period t + 1. If no agreement is ever accepted in all periods t ≤ T (ω), both players receive zero payoff. We solve G(ω, δA , δB ) by subgame perfect equilibrium (SPE). Theorem 1. Given ω = [n1 , n2 , n3 , n4 , · · · ], there exists a unique SPE out come in which players reach agreement θ(ω), 1 − θ(ω) without delay, where θ(ω) = 1 − δBn1 + δBn1 δAn2 − δBn1 +n3 δAn2 + δBn1 +n3 δAn2 +n4 − · · · .
(1)
This theorem is an extension of the main theorem of Rubinstein (1982), since, if ω = [1, 1, 1, · · · ] is an infinite alternating offers procedure, it follows 1−δB . from (1) that θ(ω) = 1−δ A δB
3
More formally, given ω = [n1 , n2 , n3 , · · · ], we define 0, r(ω) = m, ∞,
and
p(ω, k) =
(
if ω = [n1 ] if ω = [n1 , n2 , · · · , nm+1 ] , if ω is infinite
1, δBn1 δAn2 δBn3
· · · σknk ,
if k = 0 , if k = 1, 2, . . . , r(ω)
where σk = δA if k is even, and σk = δB if k is odd. We can rewrite (1) as θ(ω) =
r(ω) X
(−1)k p(ω, k).
(2)
k=0
The following two lemmas guarantee that θ(ω) is well defined by (2) even when ω is infinite and that θ(ω) actually defines a partition of the cake. Lemma 1.
P∞
k k=t (−1) p(ω, k)
is absolutely convergent for all t ≥ 0.
Lemma 2. For any ω ∈ Ω, 1 − δB ≤ θ(ω) ≤ 1.
P Furthermore, let zr (ω) = rk=0 (−1)k p(ω, k), r = 0, 1, . . . , r(ω); these can be regarded as the SPE partitions of the corresponding truncated procedures of ω. If ω = [n1 , n2 , · · · ], then z0 (ω) = θ([n1 ]) = 1, z1 (ω) = θ([n1 , n2 ]) = 1 − δBn1 , ..., zr(ω) (ω) = θ(ω). The next lemma implies that the elements in the sequence zr (ω) r=0,1,...,r(ω) are alternately larger and smaller than θ(ω).2 Lemma 3. For any t < s ≤ r(ω), zt (ω) < zs (ω) if t is odd, and zt (ω) > zs (ω) if t is even.
One possible application of Theorem 1 is as a tool to analyze the influence of the procedure on the bargaining outcome. We are particularly interested in which partitions can be implemented in SPE by choosing appropriate procedures. Let Γ(δA , δB ) = {θ(ω) | ω ∈ Ω} collect all partitions that are feasible in the sense that they can be supported in SPE by some procedure. 2
Let s = r(ω) when ω is finite, and let s → r(ω) when ω is infinite.
4
Then, the next two theorems show that all partitions are feasible if the players are patient enough, but almost no partitions are feasible if the players are impatient. Theorem 2. If δA + δB ≥ 1, then Γ(δA , δB ) = [1 − δB , 1].3 Theorem 3. If δA + δB < 1, then the (Lebesgue) measure of Γ(δA , δB ) is 0. Note that we do not consider random procedures in which the proposer is randomly chosen at some period. In fact, when x ∈ Γ(δA , δB ), a designer can achieve expected payoffs (x, 1 − x) by designing a one-period random procedure in which (a) player A (or B) has probability x (or 1 − x) to be the proposer and makes an offer and (b) rejection of this offer leads to the end of the game when both players’ payoffs are zero. However, if the designer is risk averse, this random procedure is not as good for the designer as the deterministic procedure, which can implement the same outcome (x, 1 − x) without uncertainty.
3
Normalized Bargaining Procedures
Given a feasible partition x ∈ Γ(δA , δB ), there may exist multiple procedures that result in the same SPE outcome (x, 1 − x). Let Ω(x) = {ω ∈ Ω | θ(ω) = x} collect all these procedures. For each ω ∈ Ω(x), the elements in the sequence zr (ω) r=0,1,...,r(ω)−1 are alternately larger and smaller than x, and they will either reach or converge to x.4 However, for different ω ∈ Ω(x), zr (ω) may approach x at different speeds. 1−δB = 59 . It is easy to Example 1. Suppose δA = 53 , δB = 32 , and x = 1−δ A δB verify that both the infinite alternating offers procedure ω 1 = [1, 1, . . .] and the finite procedure ω 2 = [2, 1] are in Ω(x), i.e. θ(ω 1 ) = θ(ω 2 ) = x. A designer whose target outcome is (x, 1−x) might well prefer ω 2 to ω 1 , not only because ω 2 is simpler, but also because zr (ω 2 ) approaches x faster than zr (ω 1 ) does, in the sense that |zr (ω 2 ) − x| ≤ |zr (ω 1 ) − x| for all r ≤ min{r(ω 1 ), r(ω 2 )}. 3 4
Note that if Ω also contains the procedures that ω1 = B, then Γ(δA , δB ) = [0, 1]. That is, either zr(ω) (ω) = x when ω is finite, or limr→∞ zr (ω) = x when ω is infinite.
5
Inspired by the above example, we are interested in a special class of procedures that keep zr (ω) as close to θ(ω) as possible for each r ≤ r(ω). More specifically, for x ∈ Γ(δA , δB ), we can construct ω ∈ Ω(x) by using the following algorithm. We first assume that δA + δB ≥ 1, and thus Γ(δA , δB ) = [1 − δB , 1] due to Theorem 2. If x = 1, let ω = [1]. Otherwise, we have 1 − δB ≤ x < 1. Let n1 be the integer such that 1 − δBn1 ≤ x < 1 − δBn1 +1 ; that is, n1 is the maximal n such that 1 − δBn ≤ x. If x = 1 − δBn1 , let ω = An1 B; otherwise, let n2 be the integer such that 1 − δBn1 + δAn2 +1 δBn1 < x ≤ 1 − δBn1 + δAn2 δBn1 . Suppose by induction that n1 , . . . , nh have already been defined, satisfying zh (ω) ≤ x < zh−1 (ω) − δB p(ω, h), if h is odd;
(3)
zh−1 (ω) + δA p(ω, h) < x ≤ zh (ω), if h is even.
(4)
If x = zh (ω), then let ω = [n1 , n2 , · · · , nh , 1]. If x 6= zh (ω), we need to define nh+1 . Let nh+1 be the integer such that zh (ω) + δA p(ω, h + 1) < x ≤ zh+1 (ω), if h is odd;
(5)
zh+1 (ω) ≤ x < zh (ω) − δB p(ω, h + 1), if h is even.
(6)
A question is whether nh+1 is well defined, that is, whether there exists exactly one integer nh+1 such that (5) or (6) holds. Mao (2016) provides a positive answer to this question when δA + δB ≥ 1.5 Hence, for each x ∈ [1 − δB , 1], we will either get a finite procedure ω = [n1 , · · · , nm , 1] such that θ(ω) = zm (ω) = x or eventually get an infinite procedure ω = [n1 , n2 , · · · ] such that θ(ω) = limr→∞ zr (ω) = x. Thus, we have constructed ω ∈ Ω(x). We can formally describe the procedures constructed by the above algorithm as follows: Definition 1. ω is called a normalized procedure if it satisfies the following two conditions: (i) When ω = [n1 , . . . , nr(ω)+1 ] is finite, nr(ω)+1 = 1. 5
See the proof of Theorem 2.
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(ii) When r(ω) ≥ 1, for any q ≥ 1, zq (ω) 6 θ(ω) < zq−1 (ω) − δB p(ω, q), if q is odd;
zq−1 (ω) + δA p(ω, q) < θ(ω) 6 zq (ω), if q is even.
(7) (8)
Note that not all procedures are normalized. For instance, Example 1 shows that the infinite alternating offers procedure may not be normalized. Let Ω∗ denote the set of all normalized procedures. Then, the above analysis establishes the following: Lemma 4. If δA + δB ≥ 1, then for each x ∈ Γ(δA , δB ), there exists a normalized procedure ω ∈ Ω(x). When δA + δB < 1, only a degenerate set of partitions are feasible due to Theorem 3. Under this condition, the above iterative construction algorithm is no longer valid for x ∈ / Γ(δA , δB ), since nh+1 in (5) or (6) need not be well defined. Nevertheless, we still can define normalized procedures by Definition 1 and have an analogous conclusion for each feasible partition. Lemma 5. If δA + δB < 1, then for each x ∈ Γ(δA , δB ), there exists a normalized procedure ω ∈ Ω(x). Proof. Since x ∈ Γ(δA , δB ), there exists ω 0 such that θ(ω 0 ) = x. Let ω = ω 0 if ω 0 is infinite, and let ω = [n1 , . . . , nr(ω0 ) , 1] if ω 0 = [n1 , . . . , nr(ω0 ) , nr(ω0 )+1 ] is finite. It is obvious that θ(ω) = θ(ω 0 ), and thus ω ∈ Ω(x). It remains to prove that ω ∈ Ω∗ . Suppose, for a contradiction, that ω ∈ / Ω∗ . Then, according to Definition 1, there exists an integer s ≥ 1, such that for all integers p = 0, 1, . . . , s − 1, when p is odd, (7) holds, and when p is even, (8) holds, but p = s does not satisfy (7) or (8). Without loss of generality, we assume that s is odd. Then, we either have θ(ω) < zs (ω), which contradicts Lemma 3 or we have θ(ω) > zs−1 (ω) − δB p(ω, s), and thus r(ω) X (−1)k p(ω, k) = θ(ω) − zs−1 (ω) > −δB p(ω, s). k=s
7
(9)
Pr(ω) k On the other hand, if s = r(ω), then k=s (−1) p(ω, k) = −p(ω, s) < −δB p(ω, s), which contradicts (9); if s < r(ω), then according to Lemma 3, Pr(ω) Ps+1 k k k=s (−1) p(ω, k) 6 k=s (−1) p(ω, k) = −p(ω, s)+p(ω, s+1) 6 −p(ω, s)+ δA p(ω, s) < −p(ω, s) + (1 − δB )p(ω, s) = −δB p(ω, s), which also contradicts (9). Therefore, we have proved that ω ∈ Ω∗ . In the proof of Lemma 5, we have shown that when δA +δB < 1, almost all procedures are normalized (except for those finite procedures with nr(ω)+1 6= 1), and thus Ω∗ virtually equals Ω. For each x ∈ [1 − δB , 1], we can check whether x ∈ Γ(δA , δB ) by applying the aforementioned iterative algorithm. If, in the process of this algorithm, we find that for some h, there does not exist nh+1 satisfying (5) or (6), then x ∈ / Γ(δA , δB ); otherwise, we will eventually obtain a normalized procedure ω ∈ Ω(x), which indicates x ∈ Γ(δA , δB ).
4
Main Result
By the definition of Γ(δA , δB ), one can always achieve a feasible partition by choosing some appropriate procedure ω ∈ Ω. Now, we examine the question of whether one can accomplish this by using a smaller set of procedures, and, if the answer is yes, what is the “smallest” set that can accomplish this goal. Definition 2. A set of procedures Ω ⊂ Ω is said to be sufficient when, for each ω ∈ Ω, there exists ω 0 ∈ Ω such that θ(ω 0 ) = θ(ω). In addition, Ω is called a minimal sufficient set if Ω is sufficient, and θ(ω 1 ) 6= θ(ω 2 ) whenever ω 1 , ω 2 ∈ Ω, ω 1 6= ω 2 . In other words, a minimal sufficient set contains enough procedures to implement all feasible outcomes, but none of its proper subsets is sufficient. Note that there could exist more than one minimal sufficient set for a game. For example, suppose Ω is a minimal sufficient set, and θ(ω 1 ) = θ(ω 2 ), where ω 1 ∈ Ω, ω 2 ∈ / Ω, then Ω ∪ {ω 2 } \{ω 1 } is a minimal sufficient set as well. The main finding of this paper is the following proposition. It implies that while the set of normalized procedures can yield all feasible partitions, none of its proper subsets can do so. It also suggests that the normalized 8
procedure ω ∈ Ω(x) that we found in the proof of Lemma 4 or Lemma 5 is actually the only normalized procedure in Ω(x). Proposition 1. Ω∗ is a minimal sufficient set. Proof. According to Lemma 4 and Lemma 5, Ω∗ is a sufficient set. It remains to show that if ω, ω 0 ∈ Ω∗ , ω = [n1 , n2 , · · · ] 6= ω 0 = [n01 , n02 , · · · ], then θ(ω) 6= θ(ω 0 ). Suppose for a contradiction that θ(ω) = θ(ω 0 ). Let l be the least integer such that nl 6= n0l . That is, ni = n0i , i = 1, 2, . . . , l − 1 but nl 6= n0l . Assume without loss of generality that nl < n0l . If l is odd, then we have θ(ω 0 ) = θ(ω) < zl−1 (ω) − δB p(ω, l) ≤ zl−1 (ω) − p(ω 0 , l) = zl (ω 0 ), which contradicts ω 0 ∈ Ω∗ . We can similarly create a contradiction if l is even. This implies that θ(ω) 6= θ(ω 0 ), and this ends the proof.
5
Conclusion
In this study, we define normalized procedures and prove that they form a minimal set that could support all feasible partitions as SPE outcomes. A designer can use the set of normalized procedures as a toolbox for implementing any feasible outcome. Although there may exist other minimal sufficient sets other than Ω∗ , we believe that normalized procedures have some special merits. For each feasible partition, there exists a unique corresponding normalized procedure, which can be easily constructed by using a simple algorithm. This algorithm can also be applied to check whether a partition is feasible in the first place. By contrast, it is usually (to the best of our knowledge) more difficult to find a non-normalized procedure to implement any feasible outcome. Finally, a normalized procedure ω usually makes zr (ω) approach θ(ω) at a relatively fast speed.
Acknowledgements We thank the referee and an editor for very helpful comments and suggestions. This work was supported by the National Natural Science Foundation of China (Grant No. 71471119). 9
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