Fully Absorbing Dynamic Compromise Michael Richter∗ Yeshiva University
March 19, 2013
Abstract I consider a repeated divide-the-dollar voting model with rejections leading to the implementation of the previous period’s allocation (see Kalandrakis (2004)). I show that if proposals can be non-exhaustive, then equal division can be achieved as an absorbing steady state from any initial allocation given a large enough discount factor. This result is robust to changes in voting thresholds and persistence in proposal power outside of unanimity or total persistence. Finally, I present a case where equal-division cannot be sustained as an equilibrium in pure strategies if the dollar must be exhaustively divided, for any discount factor. Keywords: Dynamic Legislative Bargaining, Markov Equilibria JEL Codes: C73, D72
1
Introduction
In this paper, I analyze a repeated legislative bargaining game where previous allocations serve as a status-quo for current proposals. I show that an ∗ Corresponding
Author. Email address:
[email protected]
1
equal-division Markov equilibrium is sustainable when discount factors are large enough from any initial status quo proposal. This continues a strand of literature originally started by Kalandrakis (2004) with more recent contributions by Kalandrakis (2010) and Bowen and Zahran (2012) investigating the behavior of agents in the aforementioned setting. The basic motivation for studying equal-division Markov equilibria is the observation that in many real-world settings, budget allocations go far beyond minimal winning coalitions. For example, see the distribution of US Federal Highway funds in 2011.1 In a very related setting, Battaglini and Palfrey (2012) conducts numerical simulations and laboratory experiments to test behavior. In their numerical simulations, they find convergence to a proposer-takes-all equilibrium when agents have a linear period utility function and to a compromise state when the period utility function is a specific CRRA utility function. Additionally, in their laboratory experiment they find “significant evidence of concave utility functions”. There are N players, henceforth referred to as legislators playing a discretetime infinitely repeated game. In each period, each legislator has an equal chance of being selected to be the proposer. The proposer then makes a proposal on how to divide a budget of size one among himself and the other legislators. If a majority of legislators accept this proposal, then the proposed division is implemented for that period. If not, then the division from the previous round is implemented instead. Thus, the division implemented in the previous round acts as an evolving status quo. Per-period utility is a concave function of an individual legislator’s budget allocation and legislators are forward-looking in that they consider not only their present payoff, but also their future possible allocations discounted in a standard exponential fashion. Under these conditions, I will show the existence of an equal-division Markov Perfect equilibrium. Later, I 1 http://www.fhwa.dot.gov/safetealu/fy11comptables.pdf
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provide extensions that show equal-division is sustainable with changes to i). the voting threshold necessary for budget passage and ii). persistence of proposal power. One natural question that may arise relates to the focus to Markov Perfect equilibria. In the current setting, the restriction to Markovian strategies is typically justified on the grounds of simplicity and more importantly, the fact that frequent legislative turnover may lead to a lack of institutional memory.2 In the studied setting, for a strategy to Markovian, the proposal strategy of legislators may only rely upon the status quo proposal, and the accept/reject decision may only rely upon the status quo proposal and the current proposal. The most closely related paper to the current one is Bowen and Zahran (2012), which serves as a primary motivation for this paper, and considers a very similar setting. They find that with sufficient additional conditions on the period utility function and the discount factor, a form of “compromise” is indeed possible. Specifically, the compromise state features all but one legislator equally dividing the dollar and the remaining legislator is frozen out (i.e. this legislator receives 0). Which legislator is frozen out changes over time, because, when the frozen legislator obtains proposal power, he proposes that a different legislator becomes the frozen legislator, and this is accepted. In their model, “compromise” is dependent upon the initial starting status quo. Some initial status quos will lead to this near-equal sharing allocation and other initial status quos will lead to a permanent punishment regime.3 Motivated by real world examples of budgets that go beyond minimal winning coalitions and the above results, I seek an equilibrium that differs from the 2 For
example, if the discount factor comes not only from time discounting, but is also dependent upon the legislator possibly being not re-elected, then the full history dependent equilibria would require his replacement to look at his predecessor’s history in determining his history-dependent action. 3 In this punishment regime, legislators follow the same strategy as in Kalandrakis (2004), namely it is a proposer-takes-all equilibrium.
3
“compromise” equilibrium of Bowen and Zahran (2012) on two fronts: i). I will find an equilibrium that exhibits equal-division, i.e. all legislators will receive an equal 1/N of the budget (there are no frozen legislators) ii). Convergence to the steady state is independent of the initial status quo. Additionally, the restrictions on the utility function are weakened to requiring it to be increasing and strictly concave (which creates a compromise incentive). One could phrase the above finding as “wastefulness may enable cooperation”.4 These stronger results are achieved by allowing budget proposals to be nonexhaustive. Intuitively, this makes it possible for legislators to separate “punishing” from “stealing”. Imagine that a proposer has to divide a dollar and wishes to punish legislators who receive a larger than equal share of the budget. The proposer punishes them by allocating them a smaller than equal share of that dollar. However, if the proposer must make exhaustive proposals, then somebody else will end up with a greater than equal share of the budget. This is undesirable for that legislator because then that legislator will merit future punishment. However, if allocations need not be exhaustive, then it becomes possible to punish some legislators without making it appear as though other legislators are stealing, and thereby meriting future punishment for them. This solves the problem that legislators don’t want to be caught stealing and subsequently punished, and therefore would not want to agree to a punishment of another legislator which would force them to be perceived as stealing. Additionally, it should be mentioned that there are a few other papers related to the current one. The first is Kalandrakis (2004) who originated the study of this type of model and considers a setting with 3 legislators and linear per-period utility in budget allocations. He finds the existence of an equilibrium which eventually degenerates into a proposer-takes-all equilibrium. Specifically, in every round the proposer requests the entire dollar and this proposal is ac4 Alternatively,
“bridges to nowhere” can serve efficiency purposes.
4
cepted. Kalandrakis (2010) extends this result to the setting where the number of legislators is an odd number equal to 5 or greater. Additionally, in a case where legislators have concave per-period utility, which also satisfies some additional restrictions, he finds the existence of Markov Perfect equilibria that feature minimum winning coalitions. The rest of the paper proceeds as follows. In the next section I set up the framework and model. The main result on the achievability of a fully absorbing equal-division equilibrium is presented in the third section. In the fourth section, I analyze two extensions to changes in voting thresholds and proposer persistence. In the fifth section, I exhibit a two player dictatorship game (which fits into the extended model) and show that cooperation is not sustainable in one period dependent pure strategies if budgets must be exhaustive. However, cooperation is sustainable, if budgets are not required to be exhaustive. The final section concludes. Proofs and calculations in general are relegated to the appendix.
2 2.1
The Model Framework
I consider a game with N agents, who I refer to as “legislators.” These legislators bargain in a discrete-time infinitely-repeated game, where in each period, there is a new budget of size 1 to be divided. At the beginning of each period, nature selects a legislator at random to make a proposal. The proposer makes PN t t a proposal on how to share the budget, xt so that i=1 xi ≤ 1, where xi represents the proposed allocation for legislator i in period t. This is the main difference from Bowen and Zahran (2012), there is an inequality requirement PN on budget proposals i=1 xti ≤ 1 as opposed to requiring equality. This is in
5
line with Baron and Ferejohn’s (1989) model, which also allows proposals to be non-exhausting. In that model, legislators would never choose to make an accepted non-exhausting proposal. In addition, in every period t, there is a status quo proposal st−1 . Each legislator votes on whether to accept or reject the proposal. If a majority votes to accept, then the proposal xt is enacted and becomes the new status quo, st := xt . If a majority votes to reject, then the status quo proposal st−1 is implemented and remains the status quo, st := st−1 . Notice that st is both the status quo for period t + 1 and the realized outcome for period t. This game differs from the standard bargaining literature because agreements takes place via majority rule (instead of unanimity), there is a new budget to be shared in every period, and the default option is the previous period’s allocation (as opposed to an assumed fixed default). Legislators’ strategy consists of two functions: • σit (ht ) - the proposal legislator i would make in period t if he happens to be the proposer conditional on the entire history of ht of proposals and voting histories. Note that ht is composed of all proposals, proposer selections, and voting histories in rounds 1, . . . , t − 1. • αit (ht , pt , xt ) - the accept/reject strategy of legislator i in period t where xt is the proposal made and pt is the identity of the proposer. Assumption: Legislators have a strictly concave, strictly increasing, period utility function u which is normalized so that u(0) = 0, u(1) = 1. Utility for a sequence of allocations is geometrically discounted with discount factor δ, so in period t, a legislator’s evaluation of a deterministic sequence of outcomes y t , y t+1 , . . . is:
6
Uit (y t , y t+1 , . . .) =
∞ X
δ k u(yit+k )
k=0
However, due to the stochastic nature of proposer selection, and possibly due to mixed strategies employed by other legislators, a legislator attaches continuation value Vit to the game after history ht and continuation value vit to the game after history ht , pt , xt : " vit (ht , pt , xt , σ, α) = E
∞ X
# δ k u(st+k ) i
: ht , pt , xt , σ, α
k=0
Vit (ht , σ, α) = E vit (ht , pt , xt , σ, α) : ht , σ, α Both expectations are taken over nature choosing the proposer and the (perhaps) random plays of other legislators. I use the above notation because in every period, there are two actions to be taken. The equilibrium notion that I use is Markov Perfect Equilibrium with pivotal voting. As mentioned before, the use of Markov strategies has appeal in a legislative setting due to frequent legislator turnover. For voting, I assume that all legislators vote as if they are pivotal. This restriction avoids many undesirable outcomes, such as all legislators accepting a proposal that none of them prefer. This is an equilibrium because no legislator is pivotal; and therefore no legislator could improve his payoff by switching their vote to reject. The assumption of pivotality removes these “unreasonable” equilibria and stipulates that every voter votes for his preferred outcome. In addition, if there were an > 0 chance of a legislator making a mistake when asked to vote accept/reject, then legislators would always have a non-zero chance of being pivotal and therefore would never vote for unpreferred outcomes. In this situation, they would vote for the alternative that is strictly in their best interest, and no restriction would be imposed in the case of indifference. Note
7
that when legislators are voting, they are taking into account future outcomes and thus consider the expected discounted value of all possible future payoffs when deciding upon accepting or rejecting a budget proposal. In the next definition, when comparing two histories, I use lower and upper case letters to refer to the different histories and their relevant components. Definition: A Markov Perfect Equilibrium (MPE) with pivotal voting is a vector of proposal strategies (σ1 , . . . , σn ) and acceptance strategies (α1 , . . . , αn ) such that: 1. For two histories ht1 and H t2 , if it is the case that st1 −1 = S t2 −1 , then σ t1 (ht1 ) = σ t2 (H t2 ) where st−1 , S t−1 are (resp.) the status quos of ht ,H t . 2. For two histories coupled with a selected proposer and proposal (ht1 , pt1 , xt1 ) and (H t2 , P t2 , X t2 ), if it is the case that (st1 −1 , xt1 ) = (S t2 −1 , X t2 ), then αt1 (ht1 , pt1 , xt1 ) = αt2 (H t2 , P t2 , X t2 ). 3. σi (ht ) ∈ argmaxx vit (ht , i, x, σ, α) A if u(xti ) + δVi (xt , σ, α) ≥ u(st−1 ) + δVi (st−1 , σ, α) i 4. αi (ht , pt , xt ) = R if u(xti ) + δVi (xt , σ, α) < u(st−1 ) + δVi (st−1 , σ, α) i The first two lines of the definition above state the Markov properties. Specifically, i) planned proposals depend only upon the status quo; and ii) accept/reject decisions depend only upon the status quo proposal st and the new proposal xt . The third condition specifies that a legislator is playing an equilibrium - specifically, he maximizes his discounted utility through his proposal choices. Finally, it would be a best response for a legislator to accept/reject proposals whenever he happens to be pivotal, but the fourth condition above is a bit stronger, stating that the legislator acts as if he is always pivotal. He accepts or rejects the new proposal whenever it is best for him to do so. Earlier, I 8
justified this assumption with small mistakes, but this assumption could also be justified via a sequential voting procedure where legislators observably vote in a prespecified order, and one requires subgame perfection. This would not prevent legislators from voting against proposals that benefit them, but it would mean that proposals would pass if they improve the welfare of a majority of legislators and not pass if they hurt a majority of legislators. Then, any equilibrium of this type would be outcome-equivalent to an equilibrium with pivotal voting. Additionally, there is a built-in tiebreaking assumption that agents vote for proposals that do not make them worse off. Since I am constructing an equilibrium, adding in this additional assumption could only make the construction more difficult, but it not an obstacle in the studied setting. Also, note that I changed the inputs for the continuation value function V above by omitting full history dependence and time superscripts due to the Markov assumptions of parts (1) and (2). While I did not, it would have been legitimate to do so for the continuation value function v as well.
2.2
Non-exhaustive Allocations
At first glance, it is unclear whether allowing part of the dollar to be unallocated should make compromise easier or harder. On the one hand, legislators now have access to more proposals, and there are now more paths to the equaldivision allocation. On the other hand, equilibria may be harder to sustain because a proposer now has more possible proposals to deviate to. I find that by allowing for part of the budget to be unallocated, the compromise allocation (1/N, . . . , 1/N ) is supportable as a fully absorbing state of a Markov Perfect equilibrium. The main reason for this is that one can now construct an equilibrium regime where legislators can be punished for taking more than 1/N of the budget.
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Intuitively, when a legislator is to be punished, he must receive less than
1 N
of
the budget. However, if budget allocations must be exhausting, then another legislator will perforce receive more than
1 N
of the budget. If one interprets
receiving a larger-than-equal share of the budget as stealing, then this strategy would stipulate that this legislator will be punished as well. So, by agreeing to punish a legislator, another legislator is induced to be punished, a situation that the legislator who will be punished starting in the next period would not agree to. Allowing for non-exhaustive allocations of the budget makes it possible to punish a legislator without making it appear as if another legislator is stealing and thereby incurring future punishment. Therefore, punishment becomes easier to agree upon in this situation, thus making it possible to sustain compromise as an equilibrium outcome. As mentioned earlier, Baron and Ferejohn (1989) allows for non-exhaustive budget allocations, but in equilibrium no proposer will make an accepted proposal where all of the budget is not allocated. This is because, in their model, agreement is once and for all. Therefore, if a legislator were to make an acceptable proposal that fails to allocate the entire dollar, then the proposer could profitably give more to every other legislator and the excess to himself. All legislators are now better off (including the proposer); thus legislators who agreed to the smaller allocation will also agree to this one. In the model that I present here, an agreed-upon allocation impacts future allocations as well in its role as a status quo. So, there may be (and, in this equilibrium, are) strategic reasons why a legislator would wish to propose a non-exhaustive budget.
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3
Compromise in a Simple Majority Setting
3.1
Construction
In this section, I will present the main equilibrium construction. Specifically a symmetric, Markov Perfect equilibrium that supports the equal-division allocation as a fully absorbing state. I denote legislator i’s proposal strategy by σi and his acceptance strategy by αi . The entire vector of proposal strategies and acceptance strategies are denoted σ and α, respectively. Definition: The budget proposal b is a fully absorbing state of the equilibrium σ ∗ , α∗ if for any initial status quo allocation s, it is the case that lim P r (ω : st (w) = b) = 1, where st denotes the status quo allocations in round
t→∞
t, and the elements ω are drawn from the induced probability distribution over all possible sequences of nature’s proposer selection in rounds 1, . . . , t − 1. Recall that in the model, the status quo allocation in round t is equal to the implemented allocation in round t − 1. Therefore, while the above condition requires the convergence of status quos, this is equivalent to requiring convergence of the implemented allocations. In addition, note that the above definition of convergence is strong in two respects: i). it is stronger than convergence in probability, because it requires the allocation to eventually equal the absorbing allocation, rather than approach it arbitrarily closely as t → ∞, and ii). convergence occurs from any initial status quo allocation. Thus, I call the limit allocation, a “fully absorbing state.” Theorem 1 For every odd N ≥ 1, there exists a symmetric, Markov Perfect equilibrium σ ∗ , α∗ and a δ ∗ < 1 of the repeated divide-the-dollar game with dynamic status quos such that for all δ ≥ δ ∗ , the proposal (1/N, . . . , 1/N ) can be sustained as a fully absorbing state. Proof. See Appendix. 11
While the proof is provided in the appendix, I here provide its construction. First, I define proposal strategies by partitioning all of the possible status quos (denoted by ω) into three cases. In the following, let j denote the proposer and (σj (ω))i denote legislator j’s proposal for legislator i’s allocation when the status quo is ω.
Case 1: ∀i, ωi ≤
1 N.
Then,
(σj (ω))i =
1
N
i=j
ωi
i 6= j
(1)
∀i, αi (ω, σj (ω)) = A Denote by D = {i : ωi >
1 N },
(2)
the set of legislators who are receiving a
greater-than-equal share of the budget - the set of deviators.
Case 2: D 6= ∅ and |D| ≤
N −1 2 .
(σj (ω))i =
Then, 1 N 0 ωi
∀i, αi (ω, σj (ω)) =
Case 3: |D| >
N −1 2 .
i=j i ∈ D\{j} otherwise
A
i 6∈ D or i = j
R
otherwise
(4)
Let E be a set of size
among all such sets. Then,
12
(3)
N −1 2
⊆ D\{j} chosen uniformly
(σj (ω))i =
1 N 0 ωi
i = j or i ∈ D\E (5)
i∈E otherwise
∀i, αj (ω, σj (ω)) =
A
i 6∈ E
R
otherwise
(6)
Note that the acceptance strategy has not yet been completely specified because it has not been defined for off-equilibrium proposals. However, in the studied setting, an acceptance strategy can be simply extended to cover offequilibrium proposals by requiring that legislators accept a proposal if it is weakly in their best interest and not otherwise. This is simply pivotal voting with tie-breaking towards acceptance. In general, in a dynamic game, there is a concern that this type of extension is not well-defined because legislators’ utility depends upon future undefined acceptances. However, this does not prove to be a concern here because on-path acceptance strategies have all been explicitly defined and if a legislator deviates, all future periods are still onpath. Thus, the off-path accept/reject decisions need only take into account future proposals (which are all necessarily on-path) and on-path accept/reject strategies. Therefore, the above extension of accept/reject strategies is not circular and well-defined.
3.2
Intuition
The above equilibrium construction can be intuitively explained as follows. Any legislator, when she proposes, gives herself the compromise share of the budget 1/N , even if she is receiving a greater then 1/N share in the status quo allocation. If a minority of legislators are receiving more than a 1/N share of the
13
budget (either because of a previous deviation, or because they were just born with a “bad” status quo allocation), then she punishes them all. Agreement is secured as all other legislators are willing to consent to this punishment. However, there can also be situations where a majority of legislators are to be punished. The proposer cannot punish them all simultaneously because each legislator would disagree with being punished and thus, acceptance of such a proposal could not be secured. The above construction sidesteps this problem by having the proposer choose a minority of legislators to punish. Each legislator who is to be punished has an equal chance of being punished. This is sufficient in the simple majority voting scenario, but will not suffice in the case of different voting thresholds. Legislator Status Quo # 1 Proposal # 1 Status Quo # 2 Proposal # 2 Status Quo # 3 Proposal # 3a Proposal # 3b
1 1/4 1/3 1/4 1/3 0 1/3 1/3
2 1/4 1/4 1/2 0 1/2 0 1/3
3 1/4 1/4 1/6 1/6 1/2 1/3 0
Figure 1: Examples of Status Quos and Proposals by Legislator #1 In the figure above, I consider three different possible allocations. In each of them, legislator 1 is the proposer. In the first allocation, no legislator merits punishment, so the resulting proposal is (1/3, 1/4, 1/4). In the second allocation, only legislator 2 merits punishment, and thus the resulting proposal is (1/3, 01/6). Finally, in the third scenario, two legislators warrant punishment, so the proposer punishes each with a 50% probability by randomizing between the two proposals 3a (when legislator 2 is to be punished) and 3b (when legislator 3 is to be punished). Another issue that arises is that it may seem curious that proposers bring
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only themselves to the compromise allocation and do not do so for other legislators who are receiving smaller shares of the budget. Doing so would cause the legislators to reach the equal division allocation faster and should make it easier to garner votes. Moreover, such a proposal is feasible and would waste less of the budget along the way. However, the problem is that this would short circuit others legislators’ punishments. Punishments need to be severe enough to deter deviation, but the worst punishment in a single period is the allocation of 0. Thus, punishments must achieve their strength due to the length of time that they are imposed. This length is stochastic, but in a strategy where each individual j is the only one to “fix” her own allocation, the resultant punishments are of sufficient duration (in expectation) to deter deviations.
4
Two Extensions
In this section, I analyze two different extensions to the model. The first analyzes changes to the number of votes needed to pass a budget and the second considers alterations to the persistence of proposal power. I will show that any voting threshold outside of unanimity and any level of persistence outside of total persistence can admit equal-division as a fully absorbing state, with sufficiently high discount factors.
4.1
Changes to Voting Rules
A natural question regarding the robustness of the previous equilibrium is how dependent is it upon the majoritarian voting rules that I have been considering. For example, in the Senate, 60 votes out of 100 are required to pass a bill over a filibuster. A typical motivation given for this type of restriction is to protect a minority from the “tyranny of the majority” (Tocqueville 1839). Additionally, the analysis performed here extends to the case where voting thresholds are 15
less than 50%. One prominent example of such a setting is the case where the proposer is a dictator. In this case, the voting threshold is just one vote, and the proposer can implement any outcome by proposing it and then accepting it. Another example where there may be an impliciting voting threshold of less than 50% is when votes are taken by roll calls and herding may occur. This could enable a minority of legislators to pass legislation. In this section, let q denote the number of votes required to pass a budget. Otherwise, the model remains as in the previous section. Theorem 2 For any vote threshold q ≤ N −1, there exists a symmetric, Markov Perfect equilibrium σ ∗ , α∗ and a δ ∗ (q) < 1 s.t. for all δ ≥ δ ∗ (q) the proposal (1/N, . . . , 1/N ) can be sustained as a fully absorbing state. The equilibrium construction is very similar to the previous one. Case 1 is the same. The Case 2 proposals are the same as long as |D| ≤ N − q. As for Case 3, the approach from before can be altered to the following: Case 3’: |D| > N − q. Take a distribution over sets E ⊂ D\j such that si −1) (i). |E| ≤ N − q and (ii). if i ∈ D\j, then P r(i ∈ E) = min (N −q)(N , 1 . q−1 Then,
(σj (ω))i =
1 N
i = j or i ∈ D\E (7)
i∈E
0 ωi
otherwise
αj (ω, σj (ω)) =
A
i 6∈ E
R
otherwise
(8)
The above equilibrium construction is very similar to the previous equilibrium construction, except that Case 3 is changed so that a legislator who steals
16
more gets punished with a higher probability. Earlier, punishing legislators with a uniform probability was sufficient, but if q = N − 1, then only one legislator can be punished, and more care needs to be taken as to which legislator should be punished. If all deviators have a uniform chance of being punished, then there may not be sufficient incentive to prevent large deviations. This problem can be overcome by providing larger disincentives to larger deviations by stipulating that the probability of punishment is increasing in the amount of additional budget taken. Off-path acceptance strategies are: A legislator accepts if it is weakly in his best interest and rejects otherwise (taking the future into account). As before, this acceptance strategy is not recursive and well-defined. Notice that in the above construction, the tightness of the voting rules comes into play only during the punishment phase. The problem for the proposer is that, when he wishes to punish, he may only punish up to N − q legislators because no legislator will agree to being punished. Therefore, one arrives at the following corollary. Define δ(q) to be the minimal δ for which cooperation is sustainable in the above equilibrium. Corollary 3 In the above equilibrium, cooperation becomes weakly harder to sustain as q increases. This is in the sense that if q ≤ q 0 , then δ(q) ≤ δ(q 0 ). So, the above corollary differs from the standard “minimum winning coalitions” logic where one expects that the more votes that are necessary, the more equal the distribution. This corollary is in line with Dixit, Grossman, and Gul (2000) who find that higher thresholds can lead to more extreme allocations. Here I find that more votes being necessary for agreement makes achieving the compromise equilibrium more difficult. In fact, in the case of unanimity, achieving compromise is impossible.
17
Proposition 4 If q = N , then equal-division is not sustainable as a fully absorbing state. Consider a case where one legislator is receiving all of the dollar – i.e. s0 = (1, 0, 0, . . . , 0). In any equilibrium, he will never agree to receive less than all of the dollar and his vote is always needed to change the allocation. Therefore, s0 is a steady state and equal-division is impossible.
4.2
Persistence in Proposer Power
Here, I change the studied setting to allow for persistence (or lack of persistence) in proposal power. Specifically, there is a γ chance that the proposer does not change. Each legislator who is not the proposer has an equal
1−γ N −1
chance of
becoming the proposer in the next period. Note that the case where γ = 0 is also admitted – i.e. the situation where proposal power never persists. Theorem 5 Given γ < 1 , there exists a Markov Perfect equilibrium σ ∗ , α∗ and a δ ∗ (γ) < 1 s.t. for all δ ≥ δ ∗ (γ) the proposal (1/N, . . . , 1/N ) can be sustained as a fully absorbing state. The next proposition illustrates that the requirement γ < 1 is necessary, since, at the boundary, equal division is not possible. This dovetails with the previous section where any voting threshold outside of unanimity admitted the equal-division equilibrium. Proposition 6 If γ = 1, then the proposal (1/N, . . . , 1/N ) is not sustainable as a fully absorbing state in any Markov Perfect Equilibrium. Consider the case where s0 = (1, 0, . . . , 0) and p1 = 1 – i.e. legislator 1 has proposal power and all of the budget. If proposal power is totally persistent, he will always be the proposer in the future, and therefore the only equilibrium
18
proposals that he can make are either proposals that will be rejected or to keep the entire budget for himself. Thus, the state s0 is a steady state. As before, one may wonder how persistence in proposal power affects cooperation. Define δ(γ) to be the minimal δ for which equal-division is sustainable in our specified equilibrium. The following corollary states that in the studied equilibrium, compromise is harder to sustain as proposal power increases. This is interesting because proposer persistence has two different effects. Specifically, as proposer persistence increases, the probability of being punished decreases as a proposer would never punish himself. But, on the other hand, the duration of punishments increases because when a proposer loses proposal power, he loses it for a longer period of time. So, the following corollary can be interpreted as saying that the first of these two effects outweighs the second. Corollary 7 In the above equilibrium, compromise becomes more difficult as proposing power increases, in the sense that if γ < γ 0 , then δ(γ) < δ(γ 0 ).
5
Two Non-Examples
In this section, I present two examples to show when compromise may not be achievable as a fully absorbing steady state in a pure symmetric Markovian Perfect equilibrium when budget proposals must be exhausting. By showing that equal-division is not possible in pure strategies in these settings, one can see the importance of allowing interior budget proposals. The first example is finite and much simpler, but does not strictly fit the model. The second example takes place in a continuous setting that fits the studied model, except for the requirement that proposals must be exhausting. In both of the following examples, I consider rotating dictatorships, i.e. proposal power does not persist (γ = 0) and voting is a dictatorship (q = 1). The previous sections’ results suggest that this may be the setting where compromise 19
is easiest, but I will show its impossibility in pure symmetric Markov Perfect Equilibrium when proposals are exhausting. In fact, in the following examples, equal-division would not be restored if γ or q were to take on different values and proposals are still required to be exhausting.
5.1
A Finite Example
Consider a two-agent setting where budget proposals must be exhaustive and consists of two indivisible 1/2 shares. So, the space of possible proposals is (1, 0), (1/2, 1/2), (0, 1) – i.e. the proposer takes all, the proposer shares equally, or the respondent takes all. Note that cooperation would be clearly unsustainable if u(1)(1 + δ 2 + δ 4 + . . .) > (1 + δ + δ 2 + . . .)u(1/2) because either player would prefer to take the entire budget whenever he proposes, regardless of the other legislators’ proposal/response strategy, rather than compromise. So, I now only consider the case that δ is large enough that cooperation is desirable, i.e. u(1)(1 + δ 2 + δ 4 + . . .) ≤ (1 + δ + δ 2 + . . .)u(1/2). Theorem 8 There is no δ < 1 such that cooperation is sustainable in pure strategies as a Markov Perfect Equilibrium. The previous theorem states that when proposals must be exhaustive, in a random serial dictatorship setting, cooperation is not sustainable. This emphasizes the fact that requiring exhaustive proposals may be harmful for cooperation. In a setting where cooperation should be easiest, none is possible. The above example and following proposition both provide evidence that the difficulty lies in being able to “punish” without “stealing”. Proposition 9 If the proposal space is widened to (1, 0), (1/2, 1/2), (1/2, 0), (0, 1/2), (0, 0), (0, 1), then there exists δ ∗ < 1 s.t. ∀δ > δ ∗ , there is a Markov Perfect Equilibrium that supports equal-division as a fully absorbing state. 20
Note that in the above, with a rotating serial dictatorship with two legislators, equal-division is reached after two rounds. No separate proof is provided for the above proposition because the proposal, acceptance strategies, and proof are exactly as in the first theorem.
5.2
A Continuous Example
The following is a two agent example in which (1/2, 1/2) is not supportable as a fully absorbing state in pure strategies when proposals must be exhausting – i.e. xt1 + xt2 = 1. However, when proposals are allowed to be non-exhaustive, then equal-division becomes possible in a Markov Perfect Equilibrium. This shows that allowing allocations to be non-exhausting allows the equal-division allocation which is otherwise unsupportable in pure strategies. In this example, let u be a twice differentiable increasing continuous concave period-utility function where u(0) = 0, u(1) = 1. Attention here is restricted to pure strategies. Furthermore, proposals must be exhausting, and I consider the case where N = 2, q = 1, γ = 0. Recall from before, that q = 1 and γ = 0 were the parameters that were most favorable for convergence to an equal-division allocation in the studied equilibrium. For simplicity, assume that legislator 1 proposes first. Proposition 10 In the game described above, for any δ < 1, (1/2, 1/2) is not supportable as a fully absorbing state in pure strategies in a symmetric Markov equilibrium. In fact, it is the case that in any equilibrium, if s01 < 1/2 then st 6→ (1/2, 1/2). This stands in contrast to the previous section, which shows that when proposals need not be exhausting, the state (1/2,1/2) can be supported as a fully absorbing state in a pure Markov Perfect Equilibrium. In fact, in such an
21
equilibrium, with q = 1, γ = 0, convergence always occurs after at most two periods.
6
Conclusion
In this paper, I have shown that equal division of a budget is supportable as a fully absorbing steady state in a Markov Perfect Equilibrium. This result is appealing because equitable divisions are often observed in legislative settings and because of the simplicity of Markovian strategies. Previous papers have shown the existence of different possible steady states, but not the case of equal division. To get the above result, I allowed budget proposals to be interior (i.e., notexhausting). The reason that one runs into difficulty when studying the case in which the budge must be exhaustively allocated is: To punish one legislator, another legislator must receive a greater than equitable share of the pie. However, in the next round, because of the Markov setting, it is unclear to the legislators that this legislator received a larger share because he was involved in some punishment or another legislator was stealing and therefore should be punished. By allowing the pie to be non-exhaustively allocated, it becomes possible to punish deviators without giving someone else a larger than 1/N share. Thus, legislators are able to distinguish “punishment” and “stealing”. Put differently, this could lead to an intuition that “wastefulness enables cooperation” in the studied Markov setting. As for this legislative model with dynamic status quos, there are still more open questions left to be answered. One such question would be to characterize the full set of Markov equilibria (or the full set of pure strategy Markov equilibria) in the above setting, but here I have shown a setting where equal-division is possible.
22
References Baron, D. P., and J. A. Ferejohn (1989): “Bargaining in Legislatures,” The American Political Science Review, 83(4), 1181–1206. Battaglini, M., and T. Palfrey (2012): “The dynamics of distributive politics,” Economic Theory, 49(3), 739–777. Bowen, T., and Z. Zahran (2012): “On dynamic compromise,” Games and Economic Behavior, 76(2), 391–419. Budish, E., F. Kojima, Y.-K. Che, and P. Milgrom (2011): “Implementing Random Assignments: A Generalization of the Birkhoff - von Neumann Theorem,” Working Paper. Dixit, A., G. M. Grossman, and F. Gul (2000): “The Dynamics of Political Compromise,” Journal of Political Economy, 108(3), pp. 531–568. Kalandrakis, T. (2004): “A three-player dynamic majoritarian bargaining game,” Journal of Economic Theory, 116(2), 294 – 322. (2010): “Minimum winning coalitions and endogenous status quo,” International Journal of Game Theory, 39, 617–643. Rubinstein, A. (1979): “Equilibrium in supergames with the overtaking criterion,” Journal of Economic Theory, 21(1), 1–9. Tocqueville, A. (1839): Democracy in America, volume 1. Translated by Henry Reeve. New York: Alfred A Knopf. Originally published in 1835.
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Acknowledgements The author would like to thank Renee Bowen, Alessandro Lizzeri, Debraj Ray, Ariel Rubinstein, Lin Zhang, and participants of NYU’s New Research in Economic Theory Seminar for helpful comments and suggestions. The author gratefully acknowledges financial support from ERC grant 269143.
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Appendix Proof of Theorem 1 Consider a legislator j who is making a proposal. For the moment, assume that the proposal is accepted. She can propose according to σ or she can make N an alternate proposal {bi }N i=1 . For her to strictly prefer the proposal {bi }i=1
she must be getting above deliver
1 N
1 N
in at least one period, as equilibrium strategies
in every period for the proposer. But any proposal that has bj ≤ 1 N
delivers less than or equal to
1 N
in every period in equilibrium. Therefore for
there to be any possible strict improvement, it must be that bj >
1 N.
Now, suppose that she makes such a proposal that will result in Case 2 in the next period. For the proposal to be strictly improving, it must be that5
u(bj ) − u
1 N
+
t ∞ X N −1 1 δ t u(0) − u >0 N N t=1
(9)
Rearranging and dividing through by 1 − δ yields u(bj ) − u
1 N
>
δ
N −1 N
u
1−δ
1 N − N −1 N
u(0)
(10)
Simplifying and bounding bj above by 1 yields u(1) − u N1 δ(N − 1) > N − (N − 1)δ u N1 − u(0)
(11)
Concavity of u yields u(1) − u N1 δ(N − 1) N −1> > 1 N − (N − 1)δ u N − u(0)
(12)
As δ → 1, the above equation becomes 5 To check the existence of a sufficient lower bound, a different method of proof is available through (Rubinstein 1979).
25
u(1) − u N1 ≥N −1 N −1> u N1 − u(0)
(13)
This contradiction, along with monotonicity in δ of the RHS in (12), gives ¯ 1] it is the case that that ∃δ¯ such that ∀δ ∈ [δ, u(1) − u N1 δ(N − 1) ≤ N − (N − 1)δ u N1 − u(0)
(14)
So, for high enough δ, a proposer would not like to deviate to a Case 2 1 u(1)−u( N ) proposal. As for an explicit bound, if I let ψn ≡ u 1 −u(0) , then for a lower (N ) bound on δ, I have ψn N = δ¯ (N − 1)(1 + ψn )
(15)
or alternatively
1−
N − 1 − ψn = δ¯ (N − 1)(1 + ψn )
(16)
where 0 < ψn < N − 1. This lower bound is increasing in ψn , and for any fixed N , as ψn converges to 0 (or N − 1 respectively), it is the case that δ¯ converges to 0 (or 1 respectively). Finally, let us check that a legislator j does not wish to deviate to a Case 3 proposal. In this case, let d = |D| for this proposal. He would deviate only if there is a bj , d so that6
u(bj ) − u
1 N
∞
N −1X + 2d t=1
N −1 N
t δ
t
u(0) − u
1 N
>0
(17)
6 In the following equation, there is either a chance of N − 1/2d of being punished or N − 1/2(d − 1) (in the case that the next proposer is a deviator and thus leaves themselves out of the pool of agents to be punished). The following equation just uses the lower bound of being punished, i.e. N − 1/2d.
26
Rearrange as before to get u(bj ) − u N1 (N − 1)2 δ > 1 2d (N − (N − 1)δ) u N − u(0) Note that bj < 1 −
d−1 N ,
(18)
so
u 1 − d−1 −u N 1 u N − u(0)
1 N
>
(N − 1)2 δ 2d (N − (N − 1)δ)
(19)
Concavity of u and rearranging yields
(N − d)2d > The LHS is maximized at d =
(N − 1)2 δ N − (N − 1)δ
N −1 2
(20)
since d is an integer and 0 ≤ d ≤ N .
Substituting and rearranging yields N (N + 1) >δ (N + 3)(N − 1)
(21)
So, if
δ ∗ = max
N (N + 1) ψn N , (N + 3)(N − 1) (N − 1)(1 + ψn )
(22)
then as long as δ ≥ δ ∗ , it is the case that assuming that other legislators play their equilibrium proposal and acceptance strategies, σ is an equilibrium proposal strategy. Now, there is one important point of caution. It is not always the case that as
2
N +N N 2 +N +(N −3) ,
N (N +1) (N +3)(N −1)
< 1. Actually, that fraction can be rewritten
so it is less than 1 if N > 3. Since N is odd, there are two
additional cases to check, specifically N = 1 and N = 3. However, if N = 1, then there is a unique equilibrium where the legislator proposes to have the entire budget for himself and accepts this for himself. This case trivially fits the definition of equal-division. If N = 3 and d = 1, then the deviator is always punished, and actually the above case fits in Case 2. 27
So, the only additional case to consider is if N = 3 and d = 2. In this case, the essential inequality becomes
u(2/3)−u(1/3) u(1/3)−u(0)
≤
δ 3−2δ .
By the concavity of u,
the LHS is less than 1 and as δ ↑ 1, the RHS ↑ 1. Thus, for large enough δ, agents would not want to make Case 3 deviations in the case N = 3, d = 2. From the above argument, one can see that whenever a legislator is proposed to receive 1/N , it is weakly in his best interest to accept, because that is optimal for him to propose if he could dictate the outcome. As for acceptance strategies, in Case 1, it is the case that all legislators perperiod payoffs are unaffected except for the proposer’s when comparing ω and σj (ω). Therefore acceptance by every legislator is equilibrium behavior. In Case 2, legislators in D have a worse flow of payoffs (specifically, they receive 0 now and the same payoffs thereafter) and reject the proposal. The proposer is weakly better off (obtaining the best possible allocation conditional on δ above) and all other legislators are indifferent, hence their acceptance is equilibrium behavior as well. In Case 3, legislators in E have worse per-period payoffs (they receive 0 now and until they propose, this is the worst possible allocation) and hence reject the proposal. All other legislators are weakly better off receiving either the best possible allocation 1/N or being unaffected. As for pairs of proposals where the proposal xt is not an equilibrium proposal, I let legislators accept according to whichever alternative is best for them, and therefore, this portion of the acceptance strategy trivially satisfies the equilibrium definition. Note this is not important for establishing the Nash equilibrium because even a dictator has no strict incentive to deviate in his proposals. Finally, note that in the above equilibrium, once each legislator has been the proposer, the allocation, in equilibrium will be (1/N, 1/N, . . . , 1/N ). The probability that this has not happened by period t is bounded above by
28
N·
N −1 t N
→ 0 as t → ∞.
Proof of Theorem 2: The acceptance strategies in this equilibrium strategy are identical to the ones outlined in the previous theorem and therefore constitute an equilibrium via the same argument. Proposals that lead to the proposer being punished with certainty are prevented as before by taking a δ ≥ δ ∗ , as discovered from the previous theorem. In addition, however, I need to prevent deviations to claiming a share of the dollar in a situation in which only some of the deviators may be punishable. For notation’s sake, let r = N − q, where r is the maximum number of legislators that the proposer may punish. In this setting, there must be at least r + 1 cheaters. Therefore, the max that is stealable is 1 − (r + 1) N1 =
N −r−1 . N
In this case, one would like to punish a legislator who receives an allocation 1 ) r(w− N w−1) w > N1 with probability min N −r−1 , 1 = min( r(N N −r−1 , 1). Note that it is N
possible to do this because of Budish, Kojima, Che, and Milgrom (2011).7 Let D = {j : j 6= i, wj > 1/N }. A legislator j ∈ D receiving an allocation r(N bj −1) r(N bj −1) bj is punished with probability min N −r−1 , 1 ≤ N −r−1 . Since at most r P r(N bj −1) ≤ r. Since, at legislators can be punished at a time, it must be j∈D N −r−1 least r+1 legislators must be punished overall, the sum is ≤
r(r+1) rN N −r−1 − N −r−1
= r.
As before, legislators have no incentive to claim a share of the dollar of size <
1 N.
So, all I need to show is the existence of a δ ∗ such that they have no
incentive to claim a share greater than 1/N . Moreover, I know that by using a δ ∗ at least as large as the one found in the previous theorem, I have prevented deviations that result in a sure punishment. 7 The argument for how to apply Budish, Kojima, Che, and Milgrom (2011) is as follows: Let pj be the probability with which legislator j is punished and qj be the probability with P which legislator j is not punished. Our constraint structure is ∀j, Pj + Qj = 1, N i=1 Pj ≤ r, PN Q ≤ N , and ∀J, 0 ≤ P , Q ≤ 1. This forms a bihierarchy. Due to the simple structure j j j i=1 of the punishment mechanism, more primitive arguments are also available.
29
1 N
Let us consider a deviation where a legislator requests
+ z and it is ac-
cepted. This legislator receives u( N1 + z) + δ (p(z)P + (1 − p(z))V ) where z is the continuation value when being punished and V is the continuation value when receiving the equal-division continuation value – i.e. that p(z) =
N −1 rN z N N −r−1
1 1−δ u(1/N ).
Note
is linear in z, the probability that a legislator other
than our deviator is selected as the proposer times the probability that he will then be subsequently punished. So, a legislator has no incentive to deviate if u0 (1/N )+δ 0. Note that the continuation value P = u(0) + δ
N −1 N P
r(N −1) N −r−1 (P
+
1 NV
−V) ≤
⇒ P (N −
(δ−1) −u(1/N ) δ(N − 1)) = δV ⇒ P − V = N N −δ(N −1) V = N −δ(N −1) . The above inequality u(1/n) −1) 1 becomes 1 ≤ δ r(N N −r−1 N −δ(N −1) u0 (1/n) . The RHS is increasing in δ
and r and equals
u(1/n) u0 (1/n)
when δ = 1, r = 1. The inequality holds strictly here
because of the concavity of u and the normalization u(0) = 0. Call the minimal δ that satisfies the above inequality δ ∗∗ (q). δ ∗∗ (q) is well-defined because the RHS is monotonically increasing in δ. So, this shows the theorem with δ ∗ (q) = min(δ ∗ , δ ∗∗ (q)), where δ ∗ is the bound found in the previous theorem. Proof of Corollary 3: From the above corollary, one can see that δ(q) = min(δ ∗ , δ ∗∗ (q)) and that δ ∗∗ (q) is increasing in r, hence decreasing in q = N − r. Proof of Proposition 4: As mentioned in the text, consider the case where s0 = (1, 0, 0, . . . , 0). The legislator receiving the entire dollar can guarantee himself the first-best payoff of u(1) ∗ (1 + δ + δ 2 + . . .) by only accepting the status-quo proposal and rejecting all others. Hence, in any Markov Perfect Equilibrium, if the above s0 is the initial status quo, then the allocation cannot change.
30
Proof of Proposition 5: First, notice as before that there are no profitable deviations for a proposer to Case 1 proposals. In the case where the persistence parameter is γ, Eq. (12) becomes
N −1>
u(bj ) − u(1/N ) δ(1 − γ)(N − 1) ≥ u(1/N ) N − 1 − δ(N − 2 + γ)
(23)
and the lower bound on the RHS converges to N-1 as δ → ∞. Moreover, the right hand side is monotonic in δ. Thus, there are no profitable deviations to Case 2 proposals when δ ≥ δ¯ where
δ¯ =
ψN (N − 1) (1 − γ)(N − 1) + ψN (N − 2 + γ)
(24)
Finally, to prevent Case 3 deviations, one needs to consider equation 20 which becomes:
(N − d)(2d) >
(1 − γ)(N − 1)2 δ (N − 1) − δ(N − 2 + γ)
As before, the LHS is maximized at d =
N −1 2
(25)
and rearranging yields:
(N + 1)(N − 1) >δ (N + 1)(N − 1) + (1 − γ)(N − 3)
(26)
So, a sufficient lower bound for delta is now
∗
δ = max
(N + 1)(N − 1) ψN (N − 1) , (N + 1)(N − 1) + (1 − γ)(N − 3) (1 − γ)(N − 1) + ψN (N − 2 + γ) (27)
Again, as before, the right term is always less than one, and the left term is as well, if N > 3. Again, equal-division is trivial if N = 1 and follows from the right hand bound if d = 1. So, the only case to check is N = 3, d = 2. In this case, the essential equality becomes 31
u(2/3)−u(1/3) u(1/3)−u(0)
≤
(1−γ)δ 2−δ(1+γ) .
As before,
the LHS is strictly less than 1 and the RHS ↑
(1−γ) 2−(1+γ)
= 1 as δ ↑ 1. Thus, for
large enough δ, agents would not want to make Case 3 deviations in the case that N = 3, d = 2. Finally, as before, acceptance strategies are that agents accept if they weakly prefer the proposal to the status quo (taking future payoffs into account). Proof of Corollary 7: In the above proof, it is clear that
(N +1)(N −1) (N +1)(N −1)+(1−γ)(N −3)
γ as its denominator decreases in γ. This is true for
is increasing in
ψN (N −1) (1−γ)(N −1)+ψN (N −2+γ)
as
well, and this can be seen by rewriting the denominator as (1 − γ)(N − 1 − ψN ) + ψN (N − 1) and noting that N − 1 − ψN > 0. Thus, both denominators are decreasing in γ. Proof of Theorem 8: Note that it must be the case that σ1 (0, 1) = (1, 0). Each legislator must totally punish the other legislator for taking the entire dollar; otherwise, each legislator would wish to request the entire dollar when it is her turn to propose. But, then, cooperation is never achieved because legislators are stuck in an infinite punishment loop depending upon the initial status quo of the system. In fact, the curious reader should note that the only MPE for the above game in pure strategies is that each legislator takes the entire dollar when it is her turn. Also, cooperation is sustainable in mixed equilibria for appropriate δ. Proof of Proposition 10: Suppose legislator 1 proposes to receive x for himself and 1 − x for the other legislator. First, it must be the case that x > 1/2 (because otherwise, legislator 1 yields over half of the pie when he receives less than 1/2 and legislator 2 can
32
do significantly better than 1/2 in every period.) Then, in equilibrium, legislator 2 will respond by asking r(x) > 1/2 for herself and 1 − r(x) for legislator 1. If (1/2, 1/2) is a fully absorbing state, then it must be that r(1/2) = 1/2. Therefore for any other proposal x, it must be true that u(x) + δu(1 − r(x)) ≤ u(1/2) + δu(1/2). But, rearranging terms yields:
u(x) − u(1/2) ≤ δ(u(1/2) − u(1 − r(x))) < u(1/2) − u(1 − r(x))
Concavity of u then yields:
x − 1/2 < 1/2 − (1 − r(x)) = r(x) − 1/2
Thus, it must be the case that r(x) > x. The same argument applies to legislator 1’s responses. Thus, legislator 1 must request more than 1/2 in the initial round, and in every subsequent round, the proposer requests an everlarger share of the budget.
33
