USC FBE APPLIED ECONOMICS WORKSHOP
presented by Jon Eguia
FRIDAY, Feb. 22, 2013
1:30 pm - 3:00 pm, Room: HOH-706
Endogenous assembly rules, senior agenda power,
and incumbency advantage∗
Jon X. Eguia†
Kenneth Shepsle‡
New York University
Harvard University
February 15, 2013
Abstract
We study repeated legislative bargaining in an assembly that chooses its
bargaining rules endogenously, and whose members face an election after each
legislative term. An agenda protocol or bargaining rule assigns to each legislator
a probability of being recognized to make a policy proposal in the assembly.
We predict that the agenda protocol chosen in equilibrium disproportionately
favors more senior legislators, granting them greater opportunities to make
policy proposals, and it generates an incumbency advantage to all legislators.
Keywords: Seniority, incumbency advantage, endogenous agenda, recognition
rule, legislative bargaining.
∗
We are very grateful to Abhinay Muthoo for his contribution as he collaborated with us in the
early stages of this project, and to Salvatore Nunnari, Santiago Oliveros and Razvan Vlaicu for their
comments.
†
[email protected] 19 West 4th St. 2nd fl. Dept. of Politics, NYU. New York, NY 10012, US.
‡
[email protected] 1737 Cambridge St. CGIS-Knafel Bldg. Harvard University. Cambridge, MA 02138.
1
1
Introduction
Seniority is a characteristic feature of legislative institutions. Senior legislators typically take leadership roles, wield disproportionate clout in the selection of rules of procedure, and are also influential in determining the proposals that ultimately come to a
vote before the assembly. The modern empirical literature on the U.S. Congress, beginning with Abram and Cooper (1968) and Polsby, Gallagher, and Rundquist (1969),
puts considerable emphasis on the importance of seniority as an organizational principle for the conduct of legislative business.
In seeking to explain the legislative
reliance on seniority, however, this literature emphasizes functionalist collective purposes without giving due consideration to the goals of individual legislators — career
or policy.1 An appeal to the functions served by a seniority rule — for example, the
elevation of experienced legislator types to positions of authority, or the economizing
on time and other resources that would otherwise be devoted to the contestation of
authority — is insufficient to explain how these considerations affect the choices of
individual members and decisive coalitions. It fails, that is, to tell us why it is in
the interest of members of a self-governing group to select procedures that bestow
differential advantage on senior members.2
In the more analytical literature on legislative organization, McKelvey and Riez1
For exceptions to functionalist arguments, see the Epstein, Brady, Kawato, and O’Halloran
(1997) and Krehbiel and Wiseman (2001).
2
Functionalist explanations of seniority arrangements also plague the analysis of groups other
than legislatures. On seniority rules in traditional tribal societies, see Simmons (1945). On the
prominence given to seniority in labor contracts, see Mater (1940) and Burda (1990).
2
man (1992) were the first to tackle this issue formally (although a more informal
development anticipating theirs is found in Holcombe 1989).
Instead of offering
functionalist reasons, they provide an explanation of the adoption of an exogenously
given seniority institution based on the benefit it provides each incumbent legislator
in his or her pursuit of reelection. Granting differential power to senior legislators
allows these legislators to obtain in expectation a disproportionate share of resources
for their districts, and it induces voters in every district to prefer reelecting their
incumbent politician rather than electing a newly minted legislator. A seniority rule,
in effect, begets an incumbency advantage.3
Self-interested legislators, caring only
about policies that benefit their constituencies and thus that enhance their prospects
of reelection, support some form of seniority.
McKelvey and Riezman (1992) consider an exogenously given seniority rule that
lumps legislators in two categories: any legislator reelected at least once is in the
favored (senior) category, while newly elected legislators are not. Muthoo and Shepsle
(2012) extend their classic model by allowing for a choice of the number of terms that
a legislator must serve in order to be designated “senior,” but they still work under
the binary restriction of seniority. McKelvey and Riezman, and Muthoo and Shepsle
recognize that, in reality, seniority is ordinal, not categorical, i.e., legislators are
individually ordered from most to least senior.
3
Empirical work on the US Congress confirms that members of the House of Representatives with
greater proposal power obtain more federal funding for their district (Knight 2005), and that greater
federal spending in a district increases the popular vote for the incumbent (Levitt and Snyder 1997).
3
We depart from these models by considering endogenous agenda rules. We study
a game that possesses two legislative stages — a rules-selection stage and a policydetermination stage— followed by an electoral stage in which voters choose whether
to reelect the incumbent that represents their constituency.
The first legislative
stage (rules-selection stage), occurring in a “procedural state of nature” (Cox, 2006),
determines the bargaining rules in operation at the policy-determination stage. A
legislator is selected to propose a procedural rule as a take-it-or-leave-it proposal that
is put to a vote in the assembly. This rule is a distribution of proposal power among
legislators, in the form of a distribution of probabilities of recognition to make the
first proposal at the policy-determination stage. If this procedural rule is rejected, an
exogenous reversion procedure is imposed (equal proposal power for each legislator).
The second stage (policy-determination stage), operating under the procedures just
determined, is the place where actual policy decisions are made.
We let the set of feasible procedural rules be any distribution of proposal power
that conditions each legislator’s proposal power on the legislator’s district of origin,
and on the legislator’s ordinal seniority. Our theory, therefore, allows procedural
rules that treat each legislator differently, unlike previous theories that lump them
in two categories. Furthermore, we allow procedural rules to discriminate among
legislators based on factors other than seniority. In our theory, the procedural rule
and its characteristics are endogenously determined: it is strategically proposed by
a legislator, and strategically voted on by all legislators. We show that equilibrium
4
rules discriminate on the basis of seniority, not on other factors, and favor more senior
legislators. As a result, the expected payoff to each district is strictly increasing in the
seniority of the legislator who represents it. These procedural rules that favor senior
legislators generate an incumbency advantage, even for junior legislators. Because a
reelected incumbent would always be more senior than a newly elected challenger, if
the assembly operates under procedural rules that favor seniority, constituents have
an incentive to reelect their incumbents.
In equilibrium, incumbents choose such
seniority rules to avail themselves of this incumbency advantage.
Additionally, we relax some restrictive assumptions made by McKelvey-Riezman.
In their model, at the third stage in which constituents vote on whether to reelect
their incumbent, citizens can only condition their vote on the policy outcome in the
current period.4 They do not consider feasible any strategy in which voters condition
their actions on their representative’s actions, or their representative’s seniority, or on
any other past action or outcome in previous periods. Given these restrictions, we
interpret their results as partial equilibrium results: the equilibria that they identify
are not shown to be robust against all possible deviations, but only against the very
small set of admissible deviations.
Voters are forced to use very simple reelection
strategies, without it being established that these reelection strategies are best responses among the set of all conceivable strategies. We admit as feasible any voter
strategy, including those that condition on any past action or outcome in every period,
4
See the definition of the voter game at McKelvey-Riezman (1992), p. 956.
5
and while allowing for any feasible strategy, we show that the simple strategies identified in McKelvey and Riezman’s partial equilibrium (representatives institutionalize
seniority rules and voters always reelect their incumbent) are robust against all possible deviations; thus, they constitute true best responses. Additionally, McKelvey
and Riezman assume that the set of legislators is fixed: there is one legislator per
district who serves forever. In their model, not reelecting a legislator only results in a
demotion that causes the legislator to lose her seniority. We let constituents replace
their representative at each election.
We generalize our results in two directions. First, we move beyond simple majority rule, developing our results in a q-majority setting of supermajority rules. Second,
instead of endogenizing proposal power only for the first policy proposal (while preserving equal probabilities of recognition to make subsequent policy proposals if the
first one is rejected), we allow the assembly to consider procedural rules proposals
that specify a distinct distribution of recognition probabilities for each bargaining
round.
In the next section we provide the theoretical context.
In section 3 we derive
some existence results, uncovering equilibria that weren’t feasible under McKelvey
and Riezman’s restrictions on admissible strategies, but can be supported once we
consider the unrestricted game. In section 4 we deal with the multiplicity of equilibria
by proposing and using a selection criterion: we find the equilibrium that maximizes
incumbency advantage, that is, the equilibrium that maximizes the electoral incen6
tive to reelect incumbents. Our main result is to identify this equilibrium. Various
extensions are taken up in section 5, followed by concluding remarks and additional
comparisons to existing literature. All proofs of results are in an Appendix which
follows.
2
The Model
Consider an infinite horizon dynamic game played between a fixed set N of voters,
one voter per district, and a set of politicians that represent voters in a legislative
assembly. Let n denote the number of districts and assume n is odd. An arbitrary
period is denoted by t. Let Γt be the period game played in period t. This period
game is played by 2n agents: the n voters and n politicians, each politician serving as
the representative of a given district in period t. Representatives in any given period
are strictly ordered by their seniority in the assembly: let θt = (θ1t , ..., θnt ) be a state
variable that denotes the position of the representative from district i in the seniority
order in period t, where θit = k ∈ {1, 2, ..., n} means that the representative from
district i is the k − th most senior representative in period t.
The period t game Γt has one non-strategic pre-stage, and 3 stages, which we now
describe.
0. Pre-stage: Assignment of seniority order
7
At the beginning of each period t > 1, we assign seniority ranks θt as follows.5 Reelected representatives, who served in period t−1 and serve again in period t, preserve
their relative position from period t − 1, i.e. for any pair of reelected representatives
serving districts i and j, θit < θjt ⇔ θit−1 < θjt−1 . Each reelected representative j improves as many positions in the seniority order as the number of positions vacated by
representatives with a more senior position than j in period t − 1 who do not serve in
the assembly in period t. Any newly elected politician who serves as a representative
of her district for the first time in period t receives the last position (n) in the seniority order; if there are k ≥ 2 newly elected representatives, some exogenous (possibly
random) rule assigns the k last positions in the seniority ranking among them.6
1. Rules-selection stage
This stage contains three substages. We refer to them as “rounds.” In the first
round, Nature selects a district according to an exogenously given probability distribution p, where p(i, θt ) denotes the probability that district i is selected, as a function
of the seniority ranking.7 Let r(t) be the district selected to propose a rule. In the
5
Representatives in t = 1 and their seniority ranking are given by an exogenous constitutional
process that precedes the rest of the game.
6
This describes how real legislatures, such as the US House of Representatives, operate. Some
measure of previous service weakly orders all legislators; ties are then broken by the application of
second-order criteria (for example, prior service in one in a hierarchy of offices is used in the U.S.
House, e.g., governor, state senator, state representative, etc); if these should fail to break all ties,
randomization is employed (Kellermann and Shepsle, 2009).
7
We assume that the probability that a district is chosen either does not depend on seniority, or
it is weakly increasing in the seniority of the district’s representative. To be precise, we allow the
possibility that a more junior representative is recognized by Nature to make the rule proposal with
higher probability than a senior, but it must be that the junior is recognized by virtue of coming
from her specific district, and not by virtue of her seniority. For example, one recognition practice
consistent with our assumptions allows the representative from district i to be recognized for certain
8
second round, the representative from district r(t) proposes an institutional arrangement at , in effect a recognition rule indicating the probability that each representative
is recognized to make a proposal in the first round of the policy-determination stage
n
X
(see below).8 Formally, at : N −→ [0, 1] is a function such that
at (i) = 1, which
i=1
maps each district i to a probability, so that at (i) denotes the probability that the
representative from district i is recognized to make the first proposal at the policy-
determination stage. In the third round, each representative votes either in favor
of recognition rule at , or against it. If a simple majority of representatives vote in
favor, the outcome of this stage is recognition rule at .9 Otherwise, the outcome is
the reversion rule at = ā which recognizes each representative with equal probability
in the policy-determination stage, that is, ā(i) =
1
n
for each i ∈ N.10
2. Policy-determination stage
At this stage, representatives play the Baron-Ferejohn (1989) legislative bargaining
game in which a unit of wealth is divided.11 This stage has infinitely many rounds.
For each district i, the probability that the representative from i is recognized to
in period t + i for any t that is a multiple of n, regardless of i0 s seniority ranking. This is the case
of recognition rotating among the districts.
8
On the concept of proposal rights as a measure of political power in a democratic assembly, see
Kalandrakis (2006).
9
Our results extend to supermajority acceptance rules as we show in section 5.1
10
In section 5.3 we extend the analysis to multi-round rules-recognition procedures that do not
impose equal recognition probabilities after a failure to determine policy in the first round.
11
The literature on dynamic distributive legislative bargaining that follows Kalandrakis (2004),
(see, for instance, Penn 2009; Bowen and Zahran 2012; and Nunnari 2012), studies models in which
each period contains only a policy-determination stage, with procedural rules fixed across time and
no elections. In a working paper contemporanous to our own, Jeon (2012) merges the rules-selection
and policy-determination stage, so that the policy outcome in period t coincides with the probability
of recognition in period t + 1.
9
make a policy proposal in the first round ρ = 1 of bargaining is at (i); the probability
that the representative from i is recognized to make a policy proposal in round ρ > 1
(if bargaining reaches round ρ) is
1
n
for any i ∈ N.
That is, we assume that the
recognition rule approved at the rules-selection stage can provide only a transitory
advantage to some representatives in the policy-determination stage, an advantage
lasting for only one round of bargaining. If that first round of bargaining leads to
failure, we assume all representatives are recognized to make policy proposals with
equal probability in any subsequent round.
A policy proposal is a partition of the unit of wealth among the n districts. Observing the proposal, representatives vote it up or down by simple majority rule. If
a proposal is accepted in round ρ, the stage ends. If not, the stage moves to round
ρ + 1. We assume there is discounting at the rate δ per round, so the total prize for
each district is discounted by δ ρ−1 if the proposal is accepted in round ρ. At the end
of the policy-determination stage, the assembly ends its session for period t.
3. Election stage
We allow for, but do not require, exogenous turnover (attrition) among representatives: with exogenous probability π ∈ [0, 1], one randomly chosen representative
ends her legislative career and exits the game due to exogenous reasons (e.g., death,
elevation to high executive office, selection for a remunerative private-sector position,
criminal conviction). Each representative then faces probability
π
n
Representatives who do not exit the game enter the election stage.
10
of exogenous exit.
In each district in which the incumbent representative runs for reelection, the
voter in the district chooses whether to reelect her incumbent, or else to elect a new
representative from an infinite pool of identical politicians.
If the voter chooses a
new politician, she enters the assembly at the lowest level of seniority. Incumbents
who are not reelected exit the game.
At the end of the election stage, the period ends, each representative (reelected
or not) keeps a fraction λ of the prize obtained by her district, and the voter in the
district obtains a fraction 1−λ. The game advances to the next period, with discount
π V ∈ (0, 1) for voters. The exogenous turnover suffices to induce time preferences in
representatives if the probability of attrition π is strictly positive, without need for
an additional discount factor of future periods; if π = 0, we assume that politicians
discount future periods with a common discount factor.
The game Γ consists of the infinite sequence of period games Γt . We assume that
all agents maximize the present value of their expected stream of period payoffs.
For each period t, let τ ∈ {1, 2, 3} denote a stage within the period, let τ = 0
denote the pre-stage that sets the seniority order for the period, and let ρ ∈ {1, 2, 3, ...}
denote a round within a stage. A history h(t, τ , ρ) contains all the information about
the actions played by Nature and all agents in all periods through t − 1, in all stages
of period t through stage τ − 1, and in all rounds of stage τ in period t through round
ρ−1. Given h(t, τ , ρ), let h(t, τ , ρ)|=t denote the continuation history of play starting
at the first stage of period t. Let H be the set of all histories.
11
Let θit (h(t, 1, 1)) be the seniority of the representative from district i in period t,
as a function of the history of play up to the end of period t − 1.
A behavioral strategy sj for an agent j is a sequence of mappings, one for each
information set in which player j can be called upon to make a move. Each of these
mappings is a function from the history of play at this information set to the set of
feasible actions for agent j.
We have already specified the set of feasible actions
at each information set: The representative of district r(t) chooses a probability
distribution (a recognition rule); all representatives make a binary choice approving
or rejecting this probability distribution; then representatives engage in the standard
Baron-Ferejohn bargaining game; finally voters make a binary choice. Let s denote a
strategy profile, which maps history of play to an action at each information node in
the game.
We are interested in subgame perfect equilibria of the game Γ that are stationary
as defined by McKelvey and Riezman (1992), so that each period game Γt is solved
independently of the history of play in previous periods. We call this Stationarity I.
That is, we seek equilibria made up of behavioral strategies that describe how to play
each period game conditioning only on information available within the period game,
as if at the end of each period all history were reduced to the state variable of seniority
status and all other details of past play were forgotten. Furthermore, we are interested
in the equilibrium strategies of the policy-determination game that are stationary in
the sense defined by Baron-Ferejohn; without this additional stationarity, the solution
12
to the game is indeterminate, as almost any outcome could then be sustained in
equilibrium (see Baron-Ferejohn). We call this Stationarity II.
Definition 1 A strategy profile s satisfies Stationarity I if for any district i, for any
players (politician or voter) j and j 0 from district i, for any period t, stage τ and
round ρ, and for any two histories h(t, τ , ρ) and h0 (t, τ , ρ) such that θt (h(t, 1, 1)) =
θt (h0 (t, 1, 1)) and h(t, τ , ρ)|=t = h0 (t, τ , ρ)|=t , it follows that sj (h(t, τ , ρ)) = sj 0 (h0 (t, τ , ρ)).12
Given any representative j, a strategy sj satisfies Stationarity II if for any period t, any rounds ρ and ρ0 and any history (h(t, τ , max{ρ, ρ0 })), sj (h(t, 2, ρ)) =
sj (h(t, 2, ρ0 )).
An equilibrium is stationary if every strategy profile satisfies stationarity I and
every representative’s strategy satisfies stationarity II.
The intuition of Stationarity I is that if two histories lead to the same seniority
ranking at the beginning of the period, then in a stationary strategy an agent does not
dwell on details of play in previous periods to decide how to play in the current period.
Stationarity II is the standard stationarity in Baron-Ferejohn bargaining, adapted to
the notation of our framework. It implies that looking only at the bargaining stage
in a given period, given two structurally equivalent subgames (two subgames with
identical continuation extended trees), agents play the same strategies in the two
12
Because we allow for the identity of a district’s representative to vary over time, stationarity of
play by the representative from district i requires symmetry across politicians from the same district:
identical politicians representing the same district must play identical strategies. Our definition of
stationarity I assumes this symmetry.
13
subgames; that is, if probabilities of recognition do not vary, agents play the same
way in the subgame that starts after round 1 of bargaining, or after round k > 1 of
bargaining.
As in most voting games, there exist many implausible equilibria in which all
representatives vote in favor of any proposal: since no representative is pivotal in
this case, representatives are indifferent about the votes they cast.
In a one-shot
game, such equilibria are discarded assuming that agents never play weakly dominated
strategies, and always vote as if they were pivotal.
The analogous argument for
dynamic games is to refine the set of equilibria by requiring each voter to eliminate
any strategy that is weakly dominated in a given voting stage game considered in
isolation while treating the equilibrium strategies of all players as fixed for all future
stages and periods.
1993).
These are “stage undominated strategies” (Baron and Kalai
Eliminating strategies that violate stage weak dominance is equivalent to
requiring each agent to vote as if she were pivotal in every subgame in which she is
involved (Duggan and Fey 2006). We use this equivalence to define the refinement.
Definition 2 An equilibrium strategy profile s satisfies stage weak dominance if for
any period t, any representative j and any history h(t, τ , ρ) such that a (rule or policy)
proposal p is put to a vote, given s representative j votes for p if the continuation
value for j of passing p is strictly positive and votes against p if the continuation value
for j of passing p is strictly negative.
14
Stage weak dominance merely rules out equilibria in which voters vote against their
strict interest because their votes do not count.
Our solution concept is subgame
perfect, stationary, stage weakly undominated Nash equilibrium. We refer to these
equilibria merely as “equilibria.”
3
Multiplicity of Equilibria
Our game features multiple equilibria. In the next section, we select the equilibrium
that is most favorable for representatives who seek to maximize their electoral incumbency advantage against challengers. First we identify a larger class of equilibria.
In order for all voters to have incentives to reelect their representatives, it must
be that being more senior is not detrimental; otherwise districts would replace their
incumbent and elect a new representative with the lowest seniority ranking. It suffices
that the probability that the representative from district i obtains positive probability
of recognition at the policy-determination stage. This is equivalent to requiring that
the combination of having a positive chance of being chosen by Nature as a rules
proposer and being included in some other representative’s minimal winning rulesselection coalition is weakly increasing in the seniority of the representative from i
(holding fixed the relative seniority order of all other representatives). If so, voters
do not want to replace their incumbent with a new politician who would become the
most junior, and least powerful, representative.
15
Our first result may be stated informally as follows (and is reminiscent of the
standard solution to the Baron-Ferejohn (1989) game): For any period t and any
recognition rule a∗t such that i) under rule a∗t ,
of recognition
1
n
n−1
2
and the rules proposer obtains
representatives obtain probability
n+1
;
2n
and ii) the probability that a
representative is one of those who obtains positive probability of recognition is nondecreasing in seniority, then there exists an equilibrium in which, in each period
t, recognition rule a∗t is approved by the assembly, and all incumbents running for
reelection are reelected.
More formally, for any i ∈ N, let C i (θt ) ⊆ N\{i} be a subset of districts of size
n−1
,
2
chosen as a function of the seniority ranking in period t. This set is composed of
the districts that the representative from i picks for her minimal winning coalition at
the rules stage, if she is the rules proposer in period t. Let C(θt ) = (C 1 (θt ), ..., C n (θt ))
be a list of these subsets, one for each i ∈ N. Let C be the set of all possible such
lists.
Consider any list such that for any district, for any seniority ranking, the probability that its representative is included in the minimal winning coalition of favored
districts, is weakly increasing in the representative’s seniority. That is, consider any
action profile such that representatives are at least as likely to have positive proposal
power as they become more senior. Any such action profile can be sustained in equilibrium. Recall that p(i, θt ) is the probability that Nature selects the representative
from district i to be the rules proposer, given that the seniority ranking is θt , and
16
note that
P
p(j, θt ) is the probability that someone who would include i in the
j∈N:i∈C j
minimal winning coalition at the rules stage, is selected to propose a procedural rule.
Proposition 1 For any C ∈ C such that for any period t and any district i,
p(i, θt ) +
X
(1)
p(j, θt )
j∈N :i∈C j
is weakly increasing in the seniority of the representative from i, there exists an equilibrium in which, in each period t :
i) The rules proposer r(t) proposes recognition rule a∗t such that a∗t (l) =
l ∈ C r(t) and a∗t (r(t)) =
1
n
for any
n+1
.
2n
ii) Recognition rule a∗t is approved by the assembly, and all incumbents running
for reelection are reelected.
If expression (1) is strictly increasing in seniority, each district has a strict incentive
to reelect its incumbent representative. Let N −r(t) = N\{r(t)} denote the set of
districts, excluding the district represented by the rules proposer in period t, and
consider the following example.
Example 1 Consider the following equilibrium rule: If the representative from disto the
n−1
2
, that is, those
n−1
2
trict r(t) is less senior than the median, she offers recognition probability
r(t)− n−1
2
representatives with seniority ranking between θt
r(t)−1
and θt
1
n
legislators immediately more senior than the representative from r(t); if the represen17
tative from r(t) is more senior than the median, she offers recognition probability
to the
n−1
2
1
n
most senior representatives from districts in N −r(t) . In this equilibrium,
expected representative and voter payoffs are strictly increasing in seniority for representatives less senior than the median, and weakly increasing in seniority for those
of at least median seniority (but exceeding the expected payoffs of those below). Thus,
voters in every district have strict incentives to reelect their incumbent.
Example 2 below illustrates that new, qualitatively different, equilibria emerge in
which voters use more sophisticated reelection strategies that condition on the actions
of all representatives.13
Example 2 Suppose there are 3 districts, there is no exogenous turnover (π = 0)
and the probability of recognition to make rules proposals is 1/2 for the most senior
representative and 1/2 for the second most senior. A standard equilibrium (consistent
with Proposition 1 or Example 1) has the rules proposer proposing 2/3 policy-proposal
probability for herself and 1/3 probability for the other senior representative.14 In the
subsequent policy-determination game, the policy proposer — one of the two seniors
— gets 2/3 of the cake and, in expectation, the other two legislators get 1/6 of the
cake (ex post one gets 1/3 the other 0).
Expected payoffs are 5/12 for the two
senior representatives and 2/12 for the junior one. Everyone is reelected, the seniors
13
These novel equilibria could not be imagined or constructed within the restrictions of the
McKelvey-Riezman model — the strategies comprising them are either impermissible or do not exist.
14
For the purpose of this and all other numerical calculations, we take the limit δ −→ 1, that is,
the discount across bargaining rounds is vanishingly small.
18
strictly, the junior just weakly in the sense that the voters of the district are indifferent
between reelecting and replacing.
Now, however, suppose voters use a strategy that reelects their representative if
she is not the least senior, or if she is the least senior and the recognition rule for
bargaining gives her exactly 1/6 of the probability of recognition.
The junior rep-
resentative then is not reelected under an equal policy-proposal recognition rule, for
example, because under this rule her probability of recognition is 1/36=1/6.
So she
only votes in favor of a rule that grants exactly 1/6 recognition probability for herself,
and against all other rules (since any other rule would mean her electoral defeat).
That makes her a cheaper coalition partner at the rules-selection stage.
Thus, the
sequence of stages plays out as follows:
• With probability 1/2 a senior is recognized to propose a rule.
• She proposes 5/6 recognition probability for herself and 1/6 for the junior.
• In the policy-determination game, with probability 5/6 she is recognized and
proposes 2/3 to herself and 1/3 to one of the others; with probability 1/6 the junior
is recognized and proposes 2/3 for herself and 1/3 to one of the others.
•Junior and rules proposer vote in favor of this rule, so it is approved.
The expected payoff for each senior is
1
2
¡5 2
63
for the junior is 28 . Everyone is reelected.15
15
+
111
623
¢
+
11
26
= 38 . The expected payoff
Note that the expected payoff to the district of the junior representative improves with the novel
strategy described above (expected payoff of 2/8 rather than 2/12).
19
4
Equilibrium Selection
Equilibria in which voters use sophisticated reelection rules, as in Example 2 raise
questions of equilibrium selection, since we do not find equilibria such as these very
plausible — in terms of the ability of a constituency either to commit to so exotic a
voting strategy or to communicate this strategy to its representative even if it could
commit.
We select equilibria in which voters do not use such sophisticated rules.
Instead they use reelection rules that condition only on the outcome of the policydetermination stage. In particular, a standard cutoff rule is used according to which a
legislator is reelected if and only if she provides a period payoff at least as high as the
cutoff (assumed fixed across periods).16
We stress that we require the equilibrium
strategies with cutoff reelection rules to be robust against any strategy, including
the more sophisticated ones, so that our equilibrium strategies constitute true best
responses of the full game, and not only a partial equilibrium.
Cutoff reelection strategies rule out unintuitive equilibria such as the one in Example 2, but they still allow for a variety of equilibria, including both equilibria in
which incumbents get reelected, as in Proposition 1 and Example 1, and those in
which legislators serve for one term and are never reelected.
We argue that among all equilibria, representatives have a common incentive to
16
Strategies in which voters do not condition on anything, and either always or never reelect their
incumbents (the ones in the equilibrium identified by McKelvey and Riezman) are special cases of
cutoff rules, with cutoffs set at zero, or above one.
20
coordinate on those in which they are reelected along the equilibrium path, and among
these, on one equilibrium in particular: the one that maximizes their incumbency
advantage.17
Somewhat casually, by “maximize incumbency advantage” we mean that incumbent representatives will seek arrangements which, ex ante, maximize the present
discounted value of their payoff minus that of a newly elected politician.
Because
both constituency payoffs and representative payoffs are proportional to the share
of wealth secured in the equilibrium outcome (no moral hazard), constituents are
provided the sharpest incentives to prefer their incumbent over a challenger when
this difference is maximized. Replacing their incumbent with a newly minted representative when the incumbency advantage is maximal imposes the largest costs on
the constituency. Incumbency advantage in our model is a valence that arises endogenously as a result of the institutional rules chosen in the assembly, and makes
the incumbent more attractive than a potential challenger.18
Previous theories of
elections with endogenous candidate valence assume that politicians can engage in
costly actions to increase their valence (Ashworth and Bueno de Mesquita 2008, 2010;
17
We should emphasize that our more general theoretical framework allows us to move beyond
the binary option (whether to have a seniority system or not) in McKelvey and Riezman. We claim
that, instead of an either-or choice, representatives will select an institution that provides them
maximal advantage, and we provide reasons why. This issue cannot be addressed in a framework
that permits only the binary options of having an exogenous seniority system or not having one.
18
Valence characteristics are those on which there is a constituency consensus that more is preferred
to less, in contrast to positional characteristics (left-wing versus right-wing issue positions) on which
a constituency may be divided. The literature in political science on the importance of valence
in electoral models and voting behavior is large. A sampling includes Ansolabehere and Snyder
(2000), Aragones and Palfrey (2004), Enelow and Hinich (1982), Groseclose (2001), and Stone and
Simas (2010).
21
Serra 2010). We show that incumbent representatives have a unique opportunity to
increase their electoral valence at no cost by approving procedural rules that favor
seniority.
i
More formally, let E denote an equilibrium. Let φi,θ
t (E) be the present value of
the expected stream of future payoffs for district i evaluated at the end of period
t in equilibrium E, and let φi,∅
t (E) be the present value of the expected stream of
future payoffs for the constituents of a newly elected politician from district i with
no seniority at the end of period t, given equilibrium E. Then the incumbency adi
i,∅
vantage of legislator i is φi,θ
t (E)− φt (E), i.e. the value of reelecting the incumbent
from district i given her seniority minus the value of replacing the incumbent with a
challenger. The average incumbency advantage is
i
n
i,∅
X
φi,θ
t (E) − φt (E)
n
i=1
X φi,∅
1
t (E)
=
−
,
(1 − π V )n i=1
n
n
where π V is the voters’ discount factor across periods.19 The average or aggregate
incumbency advantage is maximized if
Pn
i=1
φi,∅
t (E) is minimized.
Let E be any set of equilibria of the game.
Definition 3 We say that equilibrium E ∈ E maximizes incumbency advantage
within E if E = arg min
{E∈E}
Pn
i=1
φi,∅
t (E), that is, if E minimizes the average present
i
P
Note that ni=1 φi,θ
t (E) is the present value of an infinite stream of the total wealth to be
1
distributed each period (one unit). With discount π V , this value is 1−π
.
V
19
22
value of the expected stream of payoffs of a district that replaces its incumbent.
One reason to seek to maximize incumbency advantage is to create the greatest
possible electoral advantage over any challenger. Imagine (outside the model) that
the prospects of an incumbent seeking reelection were affected by stochastic factors.
For a variety of reasons, explicable and inexplicable, a constituency may be “moody,”
possessing an anti-incumbent sentiment in period t.
Perhaps the most compelling
stochastic factor is the nature of the period t challenger. The challenger selected for
period t’s election may possess electorally advantageous valence characteristics (youth,
telegenic looks, reputation for competence or honesty).
If the stochastic effects —
electoral mood, challenger valence — are large and unfavorable to the incumbent,
then she may lose the election. Ex ante, to produce a cushion against adverse draws,
incumbents will want to “stack the procedural deck” maximally in their favor in the
policy-determination game.20
There is a second intuition, without stochastic electoral factors, that rationalizes
the selection of the equilibrium we identify. Imagine (again outside the model) that
the share of the bargaining outcome that a representative keeps for herself, instead
of being exogenously fixed at λ, were endogenously chosen by the representative. If
i
i,∅
some institutional rules generate an incumbency advantage φi,θ
t (E) − φt (E), then
20
We assume an infinite pool of identical politicians who, if elected, become differentiated from
one another by seniority. Some work distinguishes among incumbent “types” on other bases (see
note 18). In these circumstances the “benefits” of seniority may be type-specific. See Fearon (1999),
Mattozzi and Merlo (2008), or Gordon and Landa (2009).
23
the incumbent would be able to appropriate this surplus, instead of passing it on to
her constituents, and still be reelected. Maximizing incumbency advantage through
procedural arrangements thus facilitates “corruption” (personal venality, directing
funds or tax breaks or insider information to friends and family, etc.) and allows
seniors to extract rents that are a function of the difference between senior and junior
bargaining outcomes. Representatives will want to maximize this difference.
Incumbency advantage is maximized by making the payoffs to junior representatives as small as possible in the bargaining game, which maximally punishes any
constituency that replaces its incumbent. In particular, by zeroing out the probability
of recognition of all junior representatives (those with less than median seniority), a
constituency that replaces its incumbent will be required to wait the maximal amount
of time, given exogenous turnover, before their newly minted representative rises sufficiently on the seniority ladder to qualify for positive probability of recognition and a
greater than minimal expectation of payments. In sum, an institutional arrangement
that maximizes incumbency advantage is such that the stream of future payoffs for
the constituency that replaces its representative remains very low for as many periods
as possible, which is achieved by concentrating all the probability of recognition on
senior representatives.
A technical quibble on equilibrium selection is in order. At the policy-determination
stage of any period, the remainder of the stage after the first policy proposer is drawn
is identical to the remainder of the Baron-Ferejohn (1989) bargaining game after
24
the first policy proposer is drawn. An equilibrium is such that the policy proposer
forms a coalition of minimal winning size by offering fraction
actly
n−1
2
1
n
of the cake to ex-
legislators and keeping the rest for herself. Conditional on not being the
proposer in bargaining round ρ, each legislator must be chosen for inclusion in the
round ρ coalition with equal probability.
In the standard, symmetric equilibrium
of the Baron-Ferejohn game, the proposer randomizes over her coalition partners.21
We select equilibria in which, at the policy-determination stage in each period t,
equilibrium play only departs from this standard, symmetric solution if playing the
standard solution and keeping fixed the actions of all agents at all future stages and
periods would not constitute an equilibrium of the continuation game that starts at
the beginning of the stage of period t.22
Let E 0 be the set of equilibria that satisfy the following properties:
Property 1 Voters use cutoff reelection rules, and incumbents are reelected along
the equilibrium path.
Property 2 For any period t, at the policy-determination stage of period t, representatives play the symmetric solution to the Baron-Ferejohn game, unless this play
21
Alternatively, in asymmetric equilibria, different proposers choose different coalitions, but in
such a correlated way that ex-ante all agents are equally likely to be included in a coalition (for
instance, with n = 3, agent 1 may always choose agent 2, who always chooses agent 3, who always
chooses agent 1; see Baron and Ferejohn (1989), footnote 16).
22
In other words, if play according to the standard Baron-Ferejohn solution in period t stage
2, together with the fixed actions of all agents in future stages and periods, would constitute an
equilibrium of the continuation game that starts at the beginning of period t stage 2, the standard
solution should be played at period t stage 2.
25
at stage τ = 2 of period t, keeping fixed equilibrium play in stage τ = 3 of period t
and in any period t0 > t, would be out of equilibrium in the continuation game that
starts at stage τ = 2 of period t.23
Within the class of equilibria satisfying these properties, we find the equilibrium
that maximizes the electoral incentives to reelect incumbents.24 This is our main
result.
Recall that N −r(t) denotes the set of districts excluding the district of the
rules proposer r(t).
Proposition 2 If π > 0, there is an equilibrium that uniquely maximizes incumbency
advantage in E 0 . In this equilibrium, in each period t, the rules-recognition rule a∗t
assigns probability of recognition a∗t (i) =
1
n
to the
n−1
2
most senior representatives
from N −r(t) and leaves the remaining probability a∗t (r(t)) =
n+1
2n
for the rules proposer.
If there is no exogenous turnover (π = 0), this equilibrium still maximizes incumbency
advantage, but not uniquely.
Incumbency advantage is maximized by forming a coalition of minimal winning
size with the proposer and the most senior representatives, and distributing all the
probability of recognition within this senior coalition.
23
Our results hold as well if instead of Property 2, we require the simpler but more restrictive condition that the symmetric solution of the Baron-Ferejohn game be played at every policy-determination
stage.
24
The set E 0 is not empty, so the equilibrium that maximazes seniority advantage within this set
exists. For instance, the class of equilibria identified in Proposition 1 belongs to E 0 . Similarly, all
the equilibria identified in the next section on Extensions also exist, as we prove in the Appendix.
26
5
Generalizations and Extensions
5.1
Supermajority Voting Rules
We have so far assumed that rules and policy proposals are selected by simple majority
rule. However, decisions to adopt or change rules are often subject to supermajority
requirements (Eraslan 2002, Messner and Polborn 2004, Barberà and Jackson 2004).
In the U.S. House of Representatives, for example, the standing rules are adopted at
the beginning of a new Congress by simple majority rule.
If, however, during the
course of considering a specific piece of legislation, proponents wish to cut through
various procedural thickets dictated by the rules and move directly to a vote — that
is, to “suspend the rules” in order to pass the particular bill — then a two-thirds
majority is required. The U.S. Senate is nominally a simple majority rule legislative
chamber.
However, in order to proceed to a vote, debate must be brought to a
close (cloture), and this requires the support of sixty out of the hundred members.
Even more restrictive, a motion to end debate and proceed to vote on a rules change
requires two-thirds of those present and voting — that is, 67 votes when all senators
participate.
For any integer q ∈
£ n+1
2
¤
, n − 1 , let Γq be the generalized game in which in any
period, q votes are needed to approve at at the rules-selection stage; otherwise, the
equal recognition rule ā is the default rule in the subsequent policy-determination
stage. Let Eq be the set of all equilibria of Γq that satisfy properties 1 and 2. The
27
supermajority requirement forces the rules proposer to grant recognition probability
to more representatives.
Proposition 3 If π > 0, the equilibrium that maximizes incumbency advantage in Eq
is such that, in each period t, the equilibrium recognition rule a∗t assigns probability of
recognition a∗t (i) =
1
n
to each of the q − 1 most senior representatives other than the
rules proposer, and leaves the remainder probability 1 −
q−1
n
to the rules proposer.
Comparing the simple majority result of Propositions 2 to its q-majority generalization Propositions 3, notice that r(t)’s payoff declines with q: the representative
from r(t) must distribute recognition probability to a greater number of colleagues.
Given two rules q and q 0 such that q < q 0 , in expectation (before the uncertainty over
the identity of r(t) is resolved), any representative at least as senior as q and any
representative less senior than q0 is strictly better off with rule q than with rule q 0 .25
As in previous results, if there is no turnover, this equilibrium maximizes incumbency
advantage, but not uniquely.
25
In particular, “any representative at least as senior as q” is the set A = {1, 2, ..., q} in the
seniority ranking, and “any representative less senior than q 0 ” is the set B = {q 0 + 1, q 0 + 2, ..., n}.
Consider an i ∈ A. He would clearly prefer q to q 0 if he were the proposer (because he would
have to pay off q − 1 rather than q 0 − 1 others), and he would be indifferent between q and q 0 if he
were not the proposer (because in either case being in A means he will be included in the winning
rules-selection coalition and receive 1/n of recognition probability). In expectation (over whether or
not he is the proposer), then, he prefers q to q 0 .
What about an i ∈ B? If she is not the proposer, she is indifferent between q and q 0 (because under
neither rule would she be included in the winning rules-selection coalition). If she is the proposer,
she is better off under q (because she has to include fewer people in her coalition). So, as above,
she prefers q to q 0 .
28
5.2
Constraints on Admissible Rules
We next consider a more restricted environment, in which any proposed recognition
rule governing the policy-determination stage must satisfy a weak monotonicity constraint based on seniority. For any two legislators from districts i and j with seniority
ranks such that θit < θjt , any admissible recognition at must be such that representatives with greater seniority (lower θit ) are at least as likely to be recognized as less
senior representatives in the first round of policy bargaining.
Definition 4 A recognition rule at satisfies the Weak Seniority condition if θit < θjt
implies at (i) ≥ at (j) for any districts i and j.26
Let ΓW S denote the constrained game in which any rules proposal must satisfy
Weak Seniority and decisions are taken by simple majority rule, and let EW S be the
set of all equilibria of ΓW S that satisfy properties 1 and 2.
Proposition 4 Assume that either Nature’s exogenous probability of recognition to
make a rules proposal pt (i, θt ) depends only on i (seniority does not matter) or it
depends only on θit (only seniority, and not district’s identity, matter). If π > 0,
the equilibrium that uniquely maximizes incumbency advantage in EW S is such that
in each period t, the recognition rule a∗t assigns probability of recognition a∗t (i) =
26
The binary rules considered by McKelvey and Riezman [40] and by Muthoo and Shepsle [42] are
special cases of rules that satisfy Weak Seniority, so this subsection can be seen as an intermediate
step between their more restrictive setup, and our model with a larger class of rules in the rest of
our paper.
29
r(t)
1
1− n
max{0, n+1
−θt
2
r(t)
θt
}
to any representative at least as senior as the rules proposer (who
represents r(t)), probability of recognition a∗t (i) = n1 to each representative (if there is
³
i
r(t)
any) with seniority rank θit ∈ θt , n+1
, and zero probability of recognition to any
2
representative with less than median seniority.
If pt (i, θt ) > 0 for each i ∈ N and θt ∈ Θ, the expected period payoff for each
district in this equilibrium is strictly increasing in the seniority of the district’s representative. If π = 0 (no turnover), this equilibrium maximizes incumbency advantage,
but once again, not uniquely.
The rules proposer gives: (i) the highest recognition probability to all those with
at least as much seniority as her — including herself — as Weak Seniority requires;
(ii) lower recognition probability to those below her but necessary to make up a
majority (if any are required), and (iii) zero to those unnecessary to a majority and
not of higher seniority than her.
This means that any vector θt maps into three
seniority classes — seniors, semi-seniors, and juniors (the first two combine when the
representative from r(t) has less seniority than the median).
Finally, we underscore the finding that if the probability of selection to make a
rules proposal in the “procedural state of nature” is strictly positive for all representatives, then it follows that expected payoff is strictly monotonic in θit — the more
experienced a representative, the greater his or her expected payoff.
30
5.3
Multi-Round Recognition Rules
The institutional arrangement at that we have considered so far involves the selection in the period t rules stage of a distribution of recognition probabilities for the
first round of bargaining in that period’s policy-determination stage. If the policydetermination stage moves on to a second (or subsequent) bargaining round, we have
assumed that all representatives are recognized with equal probability
1
n
to make a
proposal. That is, our results thus far have assumed that there is an exogenously
imposed default distribution of recognition probabilities (equal recognition) for each
bargaining round beyond the first if it is required. We now relax this assumption.
The representative selected to propose a distribution of recognition probabilities at
the rules-selection stage now is empowered to propose a distribution for each possible
round of policy bargaining, not just the first. There now is no exogenously imposed
default distribution of recognition probabilities in rounds after the first.
Formally, once selected in the first round of the rules-selection stage, r(t) proposes
an institutional arrangement at (ρ, i), which consists of a proposal for recognition rules
in the policy-determination stage. A recognition rule indicates the probability that
each representative from district i is recognized to make a proposal in each round of
bargaining ρ; thus, the recognition rule contains a countable infinity of probability
distributions, one distribution for each of the countable infinity of possible bargaining
n
X
rounds. Specifically, at : N×[1, ..., N] −→ [0, 1] is a function such that
at (ρ, i) = 1
i=1
31
for each positive integer ρ.27
As in the benchmark model, representatives vote on at (ρ, i) and approve it or
reject it by simple majority, and if they reject it, the outcome is the status quo rule
at = ā that assigns equal probability of recognition in all rounds of bargaining. Thus,
there is still an exogenously imposed default if a rules proposal is rejected: the default
option gives an equal probability of recognition in each required round of bargaining
in period t.
The definition of Stationarity II must also be adjusted.
Stationarity II re-
quires that representatives play two structurally equivalent games in the same manner.
With heterogeneous probability distributions in various rounds, two policy-
determination games are only structurally equivalent if the probability distributions
in subsequent rounds are the same in the two games. Only in this case must agents
play the same behavioral strategies in the two games.28
Let ΓMR be the modified game in which at assigns a probability distribution in
each round of bargaining (and decisions are made by simple majority rule), and let
EMR be the set of all equilibria of ΓMR that satisfy properties 1 and 2. If multi-round
27
To keep the model tractable, we require that at (ρ, i) be such that there exists some K such that
at (ρ, i) = at (K, i) for any ρ > K.
Eraslan (2002) shows that under stationarity II, there is a unique equilibrium of the policydetermination game, even if agents have asymmetric probabilities of recognition, as long as these
probabilities are constant. Our analysis allows asymmetric probabilities as in Eraslan, but does not
require them to be constant for the first K rounds.
28
Formally: Given any politician j, a strategy sj satisfies stationarity II if for any period t in
which j serves as a representative, for any rounds ρ and ρ0 , and for any history (h(t, τ , max{ρ, ρ0 }))
such that at (ρ + k, i) = at (ρ0 + k, i) for any district i and any k ∈ N, sj (h(t, 2, ρ)) = sj (h(t, 2, ρ0 )).
32
recognition rules are feasible, the rules proposer r(t) proposes a rule that distributes
all the probability of recognition and all the expected payoff to a minimal winning
majority of agents.
Proposition 5 An equilibrium that maximizes incumbency advantage in EMR is such
that, in each period t, the equilibrium recognition rule a∗t assigns, in each round ρ,
probability of recognition a∗t (ρ, i) =
1
n
for each of the
from districts other than r(t), and a∗t (ρ, r(t)) =
n+1
2n
n−1
2
most senior representatives
to the rules proposer, while all
other representatives are assigned zero probability of recognition in every round. In
this equilibrium, the period utility for representatives less senior than the median
representative is zero.
Multi-round recognition rule proposals allow rules proposers to completely exclude
n−1
2
representatives from any share in the wealth to be distributed.
Incumbency
advantage is maximized by excluding the most junior representatives, uniquely so if
π > 0.
We have considered two restrictions on recognition rules: equal probability of
recognition after the first round, and the Weak Seniority condition. We illustrate the
relevance of each restriction on the ability of the proposer to extract additional surplus
by the following numerical example.
The top of the table (rows 1-3) reports the
probability of recognition in the first round of bargaining in the policy-determination
stage, and the bottom of the table (rows 4-6) the expected payoff in the equilibria
33
described in propositions 2, 4 and 5.
Example 3 Suppose n = 15, all decisions are taken by simple majority rule, and
the representative with seniority rank 5 is selected as rules proposer. Representatives
with seniority ranks 1 − 4 are senior; those with ranks 6 − 8 are semi-senior; and
those with ranks 9 − 15 are junior.
Prob. of 1st round recognition
Cases
Proposer Senior Semi-Senior Junior
1
Multi-round (Prop 5)
0.533
0.067
0.067
0
2
1-Round (Prop 2)
0.533
0.067
0.067
0
0.160
0.160
0.067
0
3 1-Round, Weak Sen. (Prop 4)
Expected payoff
Cases
Proposer Senior Semi-Senior Junior
4
Multi-round (Prop 5)
0.533
0.067
0.067
0
5
1-Round (Prop 2)
0.300
0.067
0.067
0.033
0.113
0.113
0.067
0.033
6 1-Round, Weak Sen. (Prop 4)
The takeaway points from the tables in Example 3 are the following. First, the
completely unconstrained proposer who need not observe Weak Seniority and whose
proposal contains recognition probability distributions for each round of bargaining
(rows 1 and 4) obtains the highest payoff, zeroing out junior legislators, and keeping
the payoffs to senior legislators low. The introduction of constraints reduces the rules
proposer’s advantage, redistributing some expected payoff to junior legislators when
34
unequal recognition rules are restricted to one round (rows 2 and 5) and to juniors
and seniors, leading to a more equal distribution of expected payoffs, when unequal
recognition rules are restricted to one round and constrained by Weak Seniority (rows
3 and 6).
6
Discussion
Formal research on the origin of institutional rules that favor seniority in legislatures
begins with the seminal paper by McKelvey and Riezman (1992).
They establish
the endogenous choice of a legislative seniority system, one that gives agenda recognition advantage to senior legislators, as an equilibrium feature of Baron-Ferejohn
(1989)-style bargaining games in which participants make organizational choices at a
prior stage of the game. This formalizes an informal argument of Holcombe (1989).
Muthoo and Shepsle (2012) generalize their results, allowing for the endogenous determination of who is senior and who is junior.
The political imperative that drives these organizational choices is an incumbency
advantage. Rather than taking the bargaining game as one in which each legislator
has a fixed likelihood of being recognized to make a proposal (as in Baron-Ferejohn),
legislative incumbents use their control of the rules and other organizational features
to assign recognition likelihoods differentially with the aim of inducing constituencies to reelect their incumbents. They do this by selecting rules that make it very
35
disadvantageous for a constituency to replace its incumbent representative with a
newly minted legislator. However, these papers take a restrictive view of the menu
of organizational options available (and in this sense are partial equilibrium results).
For McKelvey and Riezman, the choice is between having a seniority system or not
(where seniority is defined exogenously). For Muthoo and Shepsle, the choice is between having no seniority system or of adopting one and endogenously determining
who qualifies as senior. In both of these, once the class of seniors is established, their
members share recognition probability in a pre-assigned way.
The results of the present paper allow for a more nuanced set of organizational
options.
We characterize the organizational choice as one of selecting the rule by
which an agenda setter is chosen. An individual is randomly chosen to propose such
a rule — a vector of probabilities giving for each legislator the likelihood of being
selected to make a first policy proposal or, more generally, a sequence of probability
vectors for making a proposal in each of the rounds of bargaining required to arrive
at a decision. This rule proposer is not constrained in the rule proposal he or she
makes (as is the case in the papers we identified above). We establish in our first two
propositions that a “seniority equilibrium” exists, one in which the randomly selected
proposer and other senior legislators share agenda power. Among the equilibria that
exist in our formulation, after excluding those that fail to satisfy either stationarity
or stage undominance, we defend the selection of the seniority equilibrium we identify
because it is the one that uniquely maximizes the incentives of constituents to reelect
36
incumbent legislators.
It is worth noting that this equilibrium does not privilege other characteristics
of a legislator or the district she represents.
Suppose, for example, district i were
privileged by the rules to make proposals in the policy-determination stage. While
constituents in that district would be delighted with this advantage, it provides no
incentive for them to reelect their representative, since her replacement would inherit this advantage. Her seniority, on the other hand, would not be inherited by a
victorious replacement.
We extend our results to supermajority voting rules and to recognition rules
that allow for differential recognition probabilities in each of the required bargaining rounds. Finally, an example illustrates the skewness of payoffs of the equilibria
we identify.
In this concluding discussion we note other research papers that identify agenda
power and the selection of voting rules as key aspects of majoritarian decision making
in legislative bodies. Early papers by Romer and Rosenthal (1978), Banks and Gasmi
(1987), and Harrington (1990) (along with the aforementioned Holcombe (1989) and
Baron and Ferejohn (1989)) make clear that agenda power affects equilibrium outcomes, showing that payoffs are skewed toward those possessing the power to propose. They identify specific institutional features of agenda power that reduce the
skew. Harrington, for example, shows that the skew is monotonically decreasing in
the voting majority needed to make final decisions. Baron and Ferejohn demonstrate
37
that allowing amendments to a proposal also reduces the skew. But in each of these
papers the agenda institutions are taken as given rather than chosen.
A number of more recent papers provide insights about specific features of agenda
institutions. Breitmoser (2011) considers a variation of the Baron-Ferejohn bargaining game in which the identity of the policy proposer is deterministic for a finite
number of rounds, and it varies across rounds. He shows that there are multiple
stationary equilibria, and all these equilibria favor the first proposer at least as much
as the equilibrium of the standard Baron-Ferejohn game.
In a similar spirit Fan,
Ali, and Bernheim (2010) examine a dynamic Baron-Ferejohn bargaining setting in
which there is an informational connection between Γt and Γt+k — viz., at time t the
agents know the subset of agents who will have positive recognition probability at
time t + k. In this circumstance they find highly skewed expected payoffs — indeed,
skewed in the extreme. However, in both of these papers, as in the earlier papers,
the details of the agenda-power arrangement are given exogenously. Yildirim (2007)
assumes that each legislator’s probability of recognition is proportional to the effort
that the legislator invests into gaining agenda power; since becoming a proposer is
costly, he finds that expected payoffs are less unequal. While these papers show that
various agenda institutions can exaggerate or diminish the skew in expected payoffs,
they provide no sense of whether such arrangements would ever have been chosen by
the legislature in a “procedural state of nature.”29
29
A number of other contributions have this same flavor, focusing attention on some specific
38
Agenda setting, and the endogeneity of agenda institutions, is central to a recent
paper by Diermeier, Prato and Vlaicu (2012).
They consider a “procedural state
of nature” in which a self-governing group selects the procedures by which it will
conduct its business.
They note two stylized facts and regard them as puzzles
to be explained: (1) Why are procedures restrictive, granting asymmetric agenda
advantage to some agents? and (2) Why are these procedures persistent, that is,
not (often) revoked by a majority? To address these questions, they build a model
combining features of Baron-Ferejohn bargaining and single-peaked legislator policy
preferences based on a one-dimensional spatial model.
They account for the first
question with an appeal to risk aversion — symmetric recognition rules provide the
expectation of centrist policy outcomes preferred by majorities (i.e., the median voter),
but are accompanied by undesirable variance. For the second question they appeal to
opportunity costs — time spent on revisiting procedural decisions is time taken away
from substantive policy making. These conceptual developments require additional
theoretical machinery (one-dimensional policy space, concave utility functions, costly
decision making) that specializes the argument, but nevertheless is well within the
spirit of the legislative modeling literature. While bargaining in a legislative assembly
is a possible application of Diermeier, Prato and Vlaicu (2012), their theory is static:
features of agenda-setting rules. These include Gersbach (2004) on voting rules that depend on the
motion on the floor, Cotton (2010) on proposal power that is retained across bargaining periods,
and Diermeier and Fong (2011) that allows for reconsideration of an approved policy. The latter
paper has an extensive bibliography of related papers.
39
it considers only one session of the assembly, and at the end of this legislative session,
the game ends.
Since there is no future, legislators need not take into account
reelection pressures, and thus there is no scope to study the incentives to institute
seniority rules.
To sum up, we present a dynamic theory of institutions, legislative bargaining, and
elections, which advances an understanding of the relationship between procedural
choice and seniority, and their connection to an incumbency advantage. McKelvey
and Riezman (1992) allow consideration of a specific procedural arrangement before
divide-the-dollar bargaining commences. Muthoo and Shepsle (2012) extend their
argument by allowing senior status to be determined endogenously.
But, as in
McKelvey and Riezman, the procedural options are limited and pre-determined. Our
model allows the procedure itself to be endogenously determined.
We identify an
equilibrium in which the proposer and senior colleagues share proposal power and
juniors are excluded. We demonstrate that this seniority rule is chosen in equilibrium
— it is proposed, passes the assembly, and produces a subsequent electoral result in
which constituencies prefer to reelect their incumbent representatives.
While it is
not the only equilibrium rule, it uniquely maximizes the incentive for a constituency
to maintain its representative’s place on the seniority ladder — the pressure to reelect
is at a maximum.
40
7
Appendix
We begin with two lemmas.
Lemma 1 An equilibrium in which voters use cutoff reelection rules and incumbents
are reelected along the equilibrium path is such that incumbents are reelected on and
off the equilibrium path.
Proof. Stationarity II implies that at the bargaining stage,
tain a zero share of the pie.
n−1
2
representatives ob-
If these representatives are reelected, and voters use
cutoff reelection strategies, it must be that the cutoff is set at zero, thus reelecting
incumbents on and off the equilibrium path.
Lemma 2 Any equilibrium in which any incumbent running for reelection is reelected
on and off the equilibrium path, is such that for any period t, at stage τ = 2, round
ρ = 1, the policy proposer keeps
2n−δ(n−1)
2n
to herself and offers
δ
n
to a set of
n−1
2
other
representatives, and this policy is approved.
Proof. In an equilibrium in which incumbents are reelected on and off the equilibrium
path, the actions in period t have no effect over actions or expected payoffs in future
periods, hence each representative seeks to maximize her period payoff, which means
that each representative plays the policy-determination stage myopically as if there
were no continuation game. For any ρ ≥ 2, the probability of recognition in bargaining
round ρ is symmetric across representatives and constant over ρ. Hence, for any ρ ≥ 1,
41
the subgame of the stage τ = 2 bargaining game that starts with a representative
recognized to make a proposal in round ρ is identical to the subgame that starts with
a legislator being recognized in the standard Baron-Ferejohn bargaining game, and
the unique stationary II class of equilibria of that game is such that the proposer
keeps
2n−δ(n−1)
2n
and offers
δ
n
to a set of
n−1
2
other agents.
Proof of Proposition 1
Proof. Assume first that each representative running for reelection is reelected in
every period (we later show that this holds in equilibrium). By lemma 2, the unique
equilibrium of the bargaining game in this case is such that the policy proposer obtains
2n−δ(n−1)
2n
of the cake, and
n−1
2
other representatives obtain nδ ; let these representatives
be randomly chosen. Then, the expected period payoff for each representative from
district i is
λat (i)
2n − δ(n − 1)
δ
+ λ[1 − at (i)] .
2n
2n
(2)
Given that the equal recognition default rule ā(i) grants an expected period payoff of
λ n1 to each representative, in order for rule at (i) to be approved, the expected payoff
(2) must be at least λ n1 for at least
n+1
2
representatives, hence the rules proposer
solves
λa∗t (i)
1
2n − δ(n − 1)
δ
+ λ[1 − a∗t (i)]
= λ
2n
2n
n
1
a∗t (i) =
n
42
and offers a∗t (i) =
1
n
to
n−1
2
agents, keeping the rest of the probability of recognition to
herself. This is the optimal rule for the representative from district r(t), among those
that can be approved in equilibrium. Thus, the representative from r(t) best responds
by proposing it, and the
n−1
2
representatives who get probability of recognition
1
n
best
respond by voting to approve it.
Given these representatives’ strategies, if for any i ∈ N, the expected payoff for a
voter i is weakly increasing in the seniority of the representative from i, then voters
best respond by reelecting their representatives. The expected period payoff for the
voter in district i is:
p(i, θt )
2n − δ(n − 1)
+
2n
X
j∈N:i∈C j
p(j, θt )
δ
2n
Given that, by assumption, p(i, θt ) is weakly increasing in θit , and
(3)
2n−δ(n−1)
2n
>
δ
,
2n
it
follows that a sufficient condition for expression (3) to be weakly increasing in θit is
that p(i, θt ) +
P
p(j, θt ) be weakly increasing in θit , as stated in the proposition.
j∈N:i∈C j
We complete the characterization of the equilibrium by describing the actions
dictated by the equilibrium strategies off the equilibrium path. First, any deviations
in any period prior to t is ignored at period t; play returns to equilibrium play as if
the play had stayed along the equilibrium path. Suppose the rules proposer deviates
to propose recognition rule a0t 6= a∗t . At τ = 1, ρ = 3 (the voting round of the
proposal stage) any representative from district i ∈ N for whom a∗t (i) >
43
1
n
votes in
favor of the proposal and any legislator such that a∗t (i) ≤
1
n
votes against it. At the
policy-determination stage, the policy proposer ignores deviations at the rules stage
and plays as if along the equilibrium path, while following a deviation by the policy
proposer, any representative from district i ∈ N who obtains at least
of the proposal and any representative who obtains less than
δ
n
δ
n
votes in favor
votes against it. At
the election stage, voters reelect their representatives, even off the equilibrium path.
Lemma 3 Let σ be an equilibrium strategy profile in which incumbents are reelected
on and off the equilibrium path in every period. Then, σ is such that, for any period t,
the following strategies constitute an equilibrium of the continuation game that starts
at the beginning of stage τ = 2 in period t :
(i) In stage τ = 2 of period t, let the policy proposer in each round ρ offer
a randomly selected set of
n−1
2
δ
n
to
other representatives, and let representatives vote in
favor of a policy in any round if and only if it allocates them at least nδ .
(ii) In all future stages and periods after t and τ = 2, play according to σ.
Proof. Since incumbents are always reelected, and the equilibrium is stationary, their
actions during period t have no consequences outside period t, and hence representatives maximize period t payoff myopically. The solution to myopic maximization of
period payoffs in the continuation game that starts at the beginning of τ = 2 in period
t coincides with the solution to a standard Baron-Ferejohn bargaining game. Play
44
according to (i) is a solution to the standard Baron-Ferejohn game. Since voters are
forward looking, their reelection strategy is unaffected by the past actions, and since
the equilibrium is stationary, strategies in future periods are unaffected by actions in
period t. Thus (i) and (ii) constitute an equilibrium of the continuation game.
Lemma 4 Consider game Γq . Let σ be an equilibrium strategy profile in which incumbents are reelected on and off the equilibrium path in every period, and the policydetermination stage is solved as in Lemma 3 (i). Then at the rules stage, the representative from district r(t) proposes rule a∗t such that a∗t (r(t)) =
n+1
2n
and a∗t (i) =
1
n
for a set of representatives of size q − 1.
Proof. Since incumbents are always reelected, and the equilibrium is stationary, their
actions during period t have no consequences outside period t, and hence representatives maximize period t payoff myopically. Given that the policy-determination stage
is solved as in the standard, symmetric solution to the Baron-Ferejohn bargaining
game, the expected period payoff for each legislator is given by expression (2) in the
proof of Proposition 1, and thus, by the same argument as in the proof of Proposition
1, the rules proposer, who represents district r(t), offers probability of recognition
at (i) =
1
n
to q − 1 agents and keeps all the rest to herself.
Proof of Proposition 2 and Proposition 3
Proof. We prove Proposition 3.
q=
Proposition 2 is obtained as a special case for
n+1
.
2
45
Suppose an equilibrium in Eq exists. By Lemma 1, incumbents are reelected on
and off the equilibrium path. Given that the equilibrium is stationary, and incumbents are always reelected, every representative plays each period myopically, aiming
to maximize the period payoff. Furthermore, playing any given policy-determination
stage according to the standard, symmetric solution to the Baron-Ferejohn game,
together with equilibrium play in the rest of the game, constitutes an equilibrium
of the continuation game that starts at the given bargaining stage (Lemma 3). By
definition of Eq , policy-determination stages are then played according to this symmetric solution to the Baron-Ferejohn bargaining game (Property 2). Then, at the
rules stage, the rules proposer, who represents district r(t), proposes rule a∗t such that
a∗t (r(t)) =
n+1
2n
and a∗t (i) =
1
n
for a set of representatives of size q − 1 (Lemma 4).
It remains to be shown which choice of coalition of representatives at the rules
stage maximizes incumbency advantage. Incumbency advantage is maximized by
minimizing the present value of the future stream of payoffs to a district that replaces
its incumbent. The minimization is over the identity of the coalition partners chosen
by each rule proposer.
The representative of a district i that replaces its incumbent enters the assembly
with low seniority ranking: if only district i has a new representative, her rank is
n (the lowest); if there are two new representatives, the new representative from i
enters with equal probability with rank n − 1 or n.
For any rank k ∈ {1, ..., n},
given the exogenous turnover, a representative from district i who has seniority rank
46
k in period t moves up to rank k − 1 in period t − 1 with probability π k−1
, exits the
n
assembly with probability π n1 and stays at rank k with probability
that the probability of ever reaching ranking k − 1 is
k−1
k
n−πk
.
n
It follows
and the expected number
of periods that the representative stays at rank k conditional on ever reaching k is
πk X
γ
n γ∈N
µ
n − πk
n
¶γ−1
=
πk 1
n
.
¡ πk ¢2 =
n
πk
n
Thus, for any k0 ∈ {2, 3, ..., n}, conditional on initial entry at rank k0 , the expected
number of periods spent at rank k0 − 1 is
k0 −1
n
k0 π(k0 −1)
=
n
.
πk0
Iterating, it follows that conditional on initial entry at rank k0 , the expected number of periods at any rank k ≤ k 0 is
n
.
πk0
If π > 0, the seniority of the representative
of a district that replaces its incumbent enters a cycle, with equal expected number
of periods at each seniority ranking. Since the periods with highest seniority ranking
occur later in the future, given that the district discounts future payoffs, the present
value of the future stream of payoffs for the district are minimized by allocating higher
probability of recognition (and thus higher expected payoffs) to the q − 1 most senior
representatives, so as to maximally defer into the future the periods of higher period
payoffs. Therefore, to minimize the payoff of a district that replaces its incumbent
(the equilibrium that maximizes incumbency advantage), if π > 0, in each period t
the rules proposer chooses the q − 1 most senior representatives other than herself for
her minimal winning q−majority of representatives who approve rules proposal a∗t .
47
Finally, note that the constructed strategy profile is indeed part of an equilibrium:
voters prefer to always reelect their incumbents, since the expected payoff for a district
is weakly increasing in the seniority of its representative.
Proof of Proposition 4
Proof. By Lemma 1, incumbents are reelected on and off the equilibrium path (we
later show that indeed it is a best response for voters to reelect their incumbents).
Given that the equilibrium is stationary, and incumbents are always reelected, every
representative solves each period myopically, aiming to maximize the period payoff.
Furthermore, playing any given policy-determination stage according to the standard,
symmetric solution to the Baron-Ferejohn game, together with equilibrium play in
the rest of the game, constitutes an equilibrium of the continuation game that starts
at the given policy-dtermination stage (the statement and proof of Lemma 3 apply
unchanged to game ΓW S ).
By definition of EW S , policy-determination stages are
then played according to this symmetric solution to the Baron-Ferejohn bargaining
game (Property 2). Thus, the expected period payoff for the representative from each
district i is given by expression (2) in the proof of Proposition 1 above. Equating (2)
to the expected period payoff under the default equal recognition rule ā, and solving
for at (i) we obtain
at (i) =
48
1
.
n
At the rules stage, the representative from district r(t) maximizes a∗t (r(t)) subject
r(t)−1
to two restrictions: (i) a∗t (i) ≥ a∗t (r(t)) for the θt
(ii) a∗t (i) ≥
1
n
r(t)
binds; if θt
for at least
≤
n+1
,
2
n+1
2
most senior representatives, and
r(t)
representatives. If θt
>
n+1
,
2
only the first constraint
the two constraints are binding. The unique solution is to assign
probability of recognition a∗t (i) ≥
1
n
to a set A ⊆ N of size |A| =
³
i
r(t)
i
representatives with seniority ranking θt ∈ θt + 1, n and
a∗t (i) =
1−
1
n
n+1
2
r(t)
− θt
of
³
´
r(t)
max{0, n+1
−
θ
}
t
2
to representatives with seniority ranking
r(t)
θt
θit
∈
h
r(t)
1, θt
i
. It remains to be shown which
choice of coalition of representatives at the rules stage maximizes incumbency advantage within the class of equilibria in which incumbents are reelected. Incumbency
advantage is maximized by minimizing the present value of the future stream of payoffs to a district that replaces its incumbent. The minimization is over the identity
of the coalition partners (the set A) chosen by each rule proposer who is more senior
than the median..
Period expected payoffs are increasing in the probability of being recognized as
rules proposer. The present value of the stream of payoffs for a district depends on
the expected seniority of its representative, and on the expected period payoffs at
each seniority level. As argued in the proof of Proposition 3, if π > 0, the seniority of
the representative of a district that replaces its incumbent enters a cycle, with equal
49
expected number of periods at each seniority ranking. Since the periods with highest
seniority ranking occur later in the future, given that the district discounts future
payoffs, the present value of the future stream of payoffs for the district are minimized
by allocating higher probability of recognition (and thus higher expected payoffs) to
the
n+1
2
most senior legislators, so as to maximally defer into the future the periods
of higher period payoffs. Therefore, in the equilibrium that minimizes the payoff of
a district that replaces its incumbent (the equilibrium that maximizes incumbency
advantage), in each period t, a rules proposer more senior than the median chooses the
most senior representatives among those that are less senior than herself to complete
her minimal winning majority of representatives who approve rules proposal a∗t .
For any i ∈ N, let pt (r, θt ) be the probability that Nature recognizes r to be the
rules proposer in period t, given seniority ranking θt and let at (i; r) be the probability
of recognition at the bargaining stage for the representative from district i, given that
the representative from district r is the rules proposer, and let E[at (i)] be the expected
probability of recognition at the policy-determination stage for the representative
from i, evaluated before the identity of the rules proposer is known. Then, E[at (i)] =
50
P
pt (r, θt )at (i; r). The period t expected payoff for district i is
r∈N
(1 − λ)
X
r∈N
⎛
pt (r, θt ) ⎝at (i; r)
2n − δ(n − 1)
+
2n
X
j∈N\{i}
⎞
δ
at (j; r)⎠
2n
2n − δ(n − 1)
δ
E[at (i)] + (1 − λ) (1 − E[at (i)])
2n
2n
¶
µ
δ
(2 − δ)n
E[at (i)] +
,
= (1 − λ)
2n
2n
= (1 − λ)
which is increasing in E[at (i)] =
P
pt (r, θt )at (i; r). In equilibrium, for any i with
r∈N
θit ≥ 12 ,
X
pt (r, θt )a∗t (i; r) =
r∈N
X
pt (r, θt )
r∈N:θr ≥θi
1
θrt
(4)
which, if pt (r, θt ) depends only on r, is weakly increasing in the seniority rank of the
representative from i. Similarly, if pt (r, θt ) depends only on θrt and is weakly increasing
in θrt , expression (4) is weakly increasing in the seniority rank of the representative
from i. Furthermore, if pt (r, θt ) > 0 for each r, then the expression is strictly increasing
in θit . Similarly, for any i with θit ≤ 12 ,
E[at (i)] =
X
r∈N:θr <θi
1
pt (r, θt ) +
n
X
pt (r, θt )
r∈N:θr ∈[θi , N 2−1 ]
1−
¡ n+1
2
θrt
− θrt
¢1
n
+
X
pt (r, θt )
r∈N:θr ∈[ N2+1 ,N]
which again is strictly increasing in seniority rank if if pt (r, θt ) > 0 and pt (r, θt ) for
r 6= i depends only on r or only on θrt , but not on θit . Therefore, voters best respond
by reelecting their incumbents: under equilibrium strategies, senior representatives
51
1
,
θrt
have a strictly greater expected period payoffs than junior ones at every period, so
voters have strict incentives to reelect their representatives.
Proof of Proposition 5
Proof. Assume EMR is not empty (we prove this at the end of proof). Since any
equilibrium in EMR is stationary, and such that voters use cutoff reelection strategies
at each round of the bargaining stage, it follows that each representative also uses a
cutoff strategy, accepting a policy proposal if and only if the proposal assigns at least
some fixed quantity to the representative. Since a policy proposer makes a proposal
that maximizes her own allocation among those that would be approved, she offers a
positive allocation only to
n−1
2
other representatives, in particular, to those with the
lowest cutoff for approval of a policy. It follows that
n−1
2
districts receive an allocation
of zero. Since all incumbents are reelected, and voters use cutoff reelection rules, it
must be that the cutoff is set at zero. Thus, incumbents are reelected on an off the
equilibrium path.
Since incumbents are reelected on and off the equilibrium path, and the equilibrium is stationary, representatives solve each period game independently, myopically
maximizing their expected period-payoff. At the recognition rules stage, the default
rule ā grants an expected payoff in the bargaining game of
1
n
to each legislator. Thus,
in order to have recognition rule a∗t approved, the rules proposer must grant an expected period payoff of at least
n+1
2n
1
n
to at least
n−1
2
other legislators, and thus at most
to the rules proposer. It follows that if there exists a rule that can pass and gives
52
an expected period payoff of
n+1
2n
to the rules proposer, then the equilibrium rule must
offer an expected period payoff of
n+1
2n
to the rules proposer, and
1
n
to at least
n−1
2
other representatives. Rule a∗t achieves this: at the policy-determination stage, the
policy proposer offers an allocation that gives her the whole unit, and this allocation
is approved with her vote and the votes of the
n−1
2
junior representatives who are
never recognized to make a policy proposal. Other rules also yield the desired payoffs
and form part of other equilibria in EMR , but among all rules that assign expected
period payoff of
n+1
2n
to the rules proposer, and
1
n
to
n−1
2
other representatives, one
that minimizes the present value of the stream of payoffs of a district that replaces
its incumbent is one in which the districts receiving a positive payoff are the most
senior ones, so as to maximally defer into the future the stream of positive payoffs of
a district that replaces its incumbent.
It follows that an equilibrium with rule a∗t maximizes incumbency advantage.
We finish the proof noting that voters have incentives to reelect their incumbent
on and off the equilibrium path, and thus these behavioral strategies at each stage
for representatives and voters constitute an equilibrium in EMR .
Calculations for Example 3
Calculations for the example. Values are for the limit as δ converges to 1.
53
Rows 1-4: The rules proposer is recognized as policy proposer with probability
n+1
2n
=
16
30
= 0.533. Her expected period payoff is 0.533(1) + (1 − 0.533)(0) = 0.533.
8 n+1
7 11
8 8
7
+
=
+
= 0.3.
15 2n
15 2 n
15 15 450
Senior and semi-senior representatives are recognized with probability
is
1
1
15
+
14
0
15
1
15
=
1
.
15
Their payoff
= 0.067.
Junior representatives are never recognized. Their payoff is 0.
Rows 2-5: The rules proposer is recognized with probability
Her payoff is
16 16
30 30
+
14 1
30 30
=
270
900
n+1
2n
=
16
30
= 0.533.
= 0.300. Seniors and semi-seniors have the same
recognition probabilities and payoffs as in rows 1-4. Juniors are never recognized.
Their payoff is
1
2n
= 0.033.
Rows 3-6: Semi-seniors are recognized with probability
their payoff is
probability
1 8
15 15
1
1−3 15
5
=
+
12
75
14 1
15 30
=
1
15
1
n
=
1
15
= 0.067 and
= 0.067. Proposer and seniors are recognized with
= 0.160. Their expected payoff is
Juniors are never recognized. Their payoff is
1
2n
4 8
25 15
1
+ 21
=
25 30
17
150
= : 0.113.
= 0.033.
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