Satisficing and Selection in Electoral Competition

Satisficing and Selection in Electoral Competition1
Jonathan Bendor2
Dilip Mookherjee3
Debraj Ray4
1
We thank Alberto Diaz-Cayeros, Jim Fearon, Tim Groseclose, Sunil Kumar, Dave Laitin, Adam Meirowitz, Ken
Shotts, and seminar participants at Berkeley, Stanford, U.C. Irvine, UCLA, the University of Chicago, the 2001 APSA
meetings and the 2002 MPSA meetings for their helpful comments, and our research assistants, past and present—Bill
Hauk, Adam Meirowitz, Dave Siegel, Jon Woon and Muhamet Yildiz—for their work on the computational model.
2
3
4
Graduate School of Business, Stanford University, Stanford CA 94305-5015 (corresponding author).
Professor, Department of Economics, Boston University, Boston MA 02215.
Professor, Department of Economics, NYU, 269 Mercer St., New York City NY 10012.
Abstract
We model political parties as adaptive decision makers who compete in a sequence of elections.
The key assumptions are that winners satisfice (the winning party in period t keeps its platform
in t + 1) while losers search. Under fairly mild assumptions about losers’ search rules, we show
that the sequence of winning platforms is absorbed into the top cycle of the (finite) set of feasible
platforms with probability one. This implies that if there is a majority rule winner then ultimately
the incumbent party must espouse it. However, we also establish, under weak assumptions about
the out-party’s search, that the parties do not stabilize at the majority rule winner (should it exist).
Thus full Downsian convergence is not predicted. We also show, both analytically and computationally, that the adaptive process is robust: in particular, if a majority rule winner “nearly” exists
then the trajectory of winning platforms tends to be “close” to a trajectory of a process which
actually has such a winner.
1
One of the most celebrated and cited5 books in political science is An Economic Theory of
Democracy (Downs 1957). It routinely makes the discipline’s short lists of the most important
works of the post-war era (e.g., Goodin and Klingemann 1996). And it is used as well as cited:
An Economic Theory has spawned a great many papers and books on spatial models of party
competition. This body of work clearly constitutes a research program.
Yet even the most successful research programs have problems. In this paper we focus on three
significant issues that confront the Downsian program. These pertain to its assumptions, robustness/power, and predictions. (1) Almost all Downsian theories assume that decision makers are
fully rational; relatedly, game theoretic ones also make strong common knowledge assumptions (A
knows that B is fully rational, B knows that A knows, ad infinitum). Criticisms of this general
approach once came from outsiders (e.g., Simon 1955)—scholars working in neither the Downsian
program nor its parent (rational choice theories of politics)—and many political scientists and
economists working in this tradition were comfortable with Milton Friedman’s famous “as if” defense of perfect rationality premises. But experimental evidence (Kagel and Roth 1995; Camerer
2003) from the now-flourishing fields of behavioral and experimental economics indicates that in
many choice situations the “as if” defense crumbles: boundedly rational agents do not always behave as if they were cognitively unconstrained. Hence, many economists are now willing to take
objections to full-rationality seriously and to build alternative theories. The time is ripe to do
likewise in political science.
(2) Most Downsian models deliver predictions in a narrow class of situations: when preferences are single-peaked and the issue space is unidimensional. Despite the best efforts of several
generations of talented researchers, getting Downsian models to yield testable predictions for multidimensional issue spaces has been difficult (even if one retains single-peakedness). Because we
have no guarantee that all elections are one-dimensional contests—redistributive issues, e.g., are
notoriously multidimensional—this is troubling.
(3) Lastly, even in the unidimensional setting the accuracy of the best-known Downsian prediction—
the median voter theorem for two-party competition—is questionable. In her review essay on political parties, Stokes says that “Recent research confirms that political parties in two- and multiparty
systems occupy persistently different policy position...[hence] spatial theory’s prediction of programatic convergence is unsustainable” (1999, p.258). Ansolabehere, Snyder and Stewart state that
“Competing candidates in congressional elections almost never converge” (2001, p.152; see also
Chappell and Keech 1986, p.882 and references cited therein, and Levitt 1996, p.438).
The last of these problems is the easiest for the Downsian program to handle. As several scholars
have noted (e.g., Grofman 1993b), some Downsian models do predict divergence, by modifying
several of the canonical premises. This underscores that what is central to a rational choice theory
of electoral competition is not a particular prediction, even one as famous as the median voter
prediction; instead, it is the rationality assumption itself. It is the latter that is central to the
5
“Wattenberg (1991) shows that An Economic Theory of Democracy is now the most widely cited research work
in political science by a living scholar” (Grofman 1993a, p.2). See also the citation data noted by Almond (1993,
p.206).
2
program’s Lakatosian “hard core”.
In this paper we propose a behavioral model of electoral competition which addresses all three
of the above issues. (1) Instead of making restrictive assumptions about rationality and common
knowledge, our model assumes only that candidates are adaptively rational: in the spirit of Cyert
and March (1963), they engage in problem-oriented search for new platforms in the face of electoral
difficulties (losing). (2) The model is not forced to presume either a unidimensional policy space or
single-peaked preferences in order to make testable predictions. (3) Under rather general assumptions about search, the model does not predict that the parties converge to the median position
in a unidimensional policy environment. Finally, by making the parties’ search behavior a central
feature, the model explicitly represents electoral dynamics.
Our work follows that of Kramer (1977) Ferejohn, Fiorina and Packel (1980) and Ferejohn,
McKelvey and Packel (1984; henceforth FFMP), and Kollman, Miller and Page (1992, 1998; henceforth KMP). Although Kramer, FFMP and KMP are rarely regarded as constituting a line of
work, we believe they do: instead of investigating the equilibrium outcomes generated by perfectly
rational decision makers, all studied how a series of electoral outcomes can be generated by myopic
decision makers. Because this body of work is so important to our own, a section of the paper is
devoted to discussing their work.
The paper is organized as follows. The second section analyzes the related work of Kramer,
FFMP and KMP. The third section presents the model, a stochastic analysis of a sequence of elections, and several implications. Proposition 2 shows that if winners satisfice, then experimentation
by losers is necessary for a well-defined type of electoral “progress”. Proposition 3 demonstrates
that if experimentation has certain rather weak properties then it and satisficing-by-winners together are sufficient to ensure that the sequence of winning policies reaches the policy space’s top
cycle set with probability one, and stays there. Proposition 4 establishes that the sequence of
winning policies is absorbed into the top cycle set if winners satisfice and the electoral dynamic
satisfies a general stochastic property (ergodicity). Propositions 3 and 4 imply that if there is a
majority rule winner, then ultimately the incumbent party must espouse it. However, Proposition
5 shows, given a weak assumption about the out-party’s search, that the parties do not stabilize at
the median voter’s bliss point (when it exists). Thus full Downsian convergence is not predicted.
The fourth section investigates alternative specifications of the challenger’s search behavior by
endowing him with different degrees of sophistication and certain kinds of knowledge about the
political terrain, following Kramer (1977), Miller (1980), or FFMP (1980, 1984). The fifth section
analyzes whether our results are sensitive to “small” changes in key assumptions. Proposition
6 shows that they are, in fact, robust: e.g., if a majority rule winner “nearly” exists then the
trajectory of winning platforms tends to be “close” to a trajectory of a process which actually
has a generalized median. We then present a computational model that provides further results
for ill-structured electoral environments. This goes beyond our analytical model by examining
what happens when the profile of voters’ ideal points is not even close to having a generalized
median. Computational results show that if challengers usually search incrementally (i.e., in the
3
neighborhood of their current platform) or blindly then in the steady state winning policies are
centrally located. Moreover, their dispersion is strongly correlated with the size of the uncovered
set. The last section concludes.
RELATED WORK
Despite their common orientation to dynamics and their departure from (Nash) equilibrium
analysis, the work of Kramer, FFMP and KMP move away from standard rational choice models
in different ways. Indeed, the relations among their models are complex. (E.g., although FFMP’s
ideas are applicable to Downsian competition [Ferejohn 1995, p.258], their institutional setting is
committees, not elections.) Hence it is useful to present a table that compares their attributes to
each other and to the current model.
[table 1 about here]
It is especially worth noting that only KMP make bounded rationality a central theme (e.g., 1992,
p.929). Kramer never raises it; FFMP merely touch on it (Ferejohn et al. 1980, p.141, 144). Yet
in several ways all share a family resemblance on this dimension as well.
First, all are what might be called status quo based processes. E.g., in Kramer’s model the winner
in period t retains this winning platform in period t + 1. Although Kramer does not give a cognitive
interpretation for this,6 we believe that there is a natural one, captured by the slogan that winners
satisfice. Obviously, winning is satisfactory for office-oriented candidates, so a winning platform is
the proverbial “needle that is sharp enough” (March and Simon 1958, p.141). Thus satisficing is a
straightforward bounded rationality interpretation of status quo based electoral processes.
Second, agents are assumed to be myopic. This is explicit in KMP and FFMP7 , but it is also
a reasonable interpretation of Kramer’s model in two ways. (a) Satisficing-by-winners, though
sensible, is myopic. (b) Kramer assumes that the out-party selects a platform that maximizes the
vote-share against the status quo policy. This decision rule is an example of what is now called
myopic best response: an alternative is chosen today without any regard for what will happen in
the long run. (Indeed, Kramer’s model is a precursor to this line of theorizing, which is intermediate
between classical full rationality, with its unlimited horizon, and decision rules such as satisficing
that need not be even myopically optimal.)
But these are mostly matters of interpretation. If pushed too far they produce a Procrustean
reading. An inspection of Kramer and FFMP shows that their central concern is not to modify
the rationality postulate of Downsian models; rather, their focus is the so-called “chaos problem”
(Kramer, p.311-315; Ferejohn et al. 1980, p.140-41; Ferejohn et al. 1984, p.45-46). Only KMP
make rationality a central issue.8 We regard this as an important project in its own right, and we
6
About this assumption he says only that “In each period one of the parties is elected, enacts the policy it
advocated, and in the next election must defend this same policy” (p.317).
7
For readers who find this a surprising claim about FFMP, consider the following evidence: “The assumption
that the committee decision process is myopic (3) may not be unreasonable, especially when N [the set of decision
makers] and X [the policy set] are large enough to impede systematic enumeration of preferred alternatives and their
supporting coalitions” (Ferejohn et al. 1980, p.144).
8
In KMP the parties are adaptively rational: they use search heuristics to discover new platforms, and poll data
4
view the current paper as contributing to that effort. That is, we see KMP as part of the larger
bounded rationality research program in political science (Bendor, 2003): the extension of that
program into work on party competition.9 Creating a coherent research program that shares the
rational choice program’s focus on individual agents yet which uses more realistic premises about
cognition is important to the discipline as a whole, not merely to the subfield of electoral politics.
From this perspective, addressing the so-called chaos problem is far from the whole story. The
larger goal is to create a competing research program that can address a wide aray of important
topics on electoral competition—including what happens when there is a majority rule winner.
Where to go from here
Unfortunately, KMP has not had the impact on the field that it deserved. Discipline-wide
factors may be at work—formal models of bounded rationality have diffused slowly in political
science—but we believe that several specific factors also matter. First, in one sense KMP were too
ambitious: they jumped into analyzing races in multidimensional policy spaces without first examining the unidimensional setting.10 Second, in another respect, however, KMP were not ambitious
enough: their major papers (1992, 1998) settled for simulation results rather than analytical ones.
Though this was surely done for pragmatic reasons—closed form results for dynamic processes in
multidimensional spaces are often unattainable—it also made their work differ significantly from
the Downsian mainstream, which emphasizes mathematical models and analytical results. (Even
the substantively unorthodox papers of Kramer (1977) and FFMP (1980, 1984) produced analytical
results.) So KMP differs from the mainstream both in terms of major substantive premises and
the method of analysis. This combination may have marginalized it.
Third, although Simon (1990) and others have listed empirical regularities about informationprocessing that can constrain our assumptions, there are still many plausible ways to model bounded
rationality. Hence, for the sake of robustness it is desirable, ceteris paribus, to model the candidates’
adaptive behavior as generally as possible. This means positing general properties of adaptation
(e.g., that successful strategies are more likely to be used in the future than unsuccessful ones)
rather than a specific form of adaptation. Unfortunately, simulation is ill-suited to the general
approach: a computer must be told exactly what kind of search heuristic to use. This problem can
be alleviated by sensitivity testing, which KMP did, but it remains an issue. Indeed, because there
are many types of bounded rationality (not only many kinds of search heuristics), an efficient form
of sensitivity testing is to establish results analytically.
The current paper thus complements KMP by using a different research strategy. First, we
establish results for the canonical Downsian setting: a unidimensional policy space and voters
is used to evaluate the merits of new platforms.
9
Some might regard the models in KMP and the present paper as belonging to a new research program, behavioral game theory, which merges game theory’s emphasis on strategic intereaction, and behavioral decision theory’s
emphasis on the cognitive constraints of human beings. We have no quarrel with this alternative interpretation.
10
Perhaps the authors thought it was obvious how their search heuristics would perform in unidimensional spaces—
apparently they induce the sequence of winning policies to converge to the median bliss point (Page, personal
communication 1999)—but stating this explicitly in their first paper would have been sensible.
5
with single-peaked preferences. (More precisely, we establish results that include single-peaked
preferences in a unidimensional space as a special case.) Only after we nail down results in that
setting do we move on to the less well understood multidimensional environment. Second, we
push analytical results as far as we can, turning to simulation only when necessary. (We do agree
with KMP that when one cannot derive analytical results, resorting to computation beats giving
up.) Third, in the interests of generality and robustness, we specify a few general properties of
adaptation rather than relying on detailed heuristics. These features are interdependent—e.g., for
modelling general properties of adaptation mathematics is better than simulation—and so they
form a coherent approach with its own distinctive set of tradeoffs.
THE MODEL AND ITS IMPLICATIONS
We study a standard electoral game: a contest between two candidates or parties. A few
modifications have been introduced to enhance analytical tractability.
There are two candidates and a finite set of citizens, N = {1, . . . , n}, with n odd. There is a
finite set of feasible policies or platforms, X = {x1 , . . . , xm }, with m > 1. (X may be huge, but
finiteness is analytically useful for several results.) Each citizen has a strict preference ordering
over the policies. Since n is odd, this ensures that the majority preference relation is also strict:
for any two distinct policies, xi and xj , either a majority of citizens strictly prefer xi to xj (written
xi xj ) or xj to xi .
We do not assume spatial policies or preferences, using instead the above spare assumptions
about individual preference orderings. But we can recover spatial settings (of different dimensionalities) when doing so is necessary or convenient. The following description of the majority preference
relations induced by the above assumptions enables us to make this translation to a spatial framework. We partition the policy set X into z disjoint subsets, {L1 , . . . , Lz }, where 1 ≤ z ≤ m, by
an interative application of the idea of the top cycle set.11 Let L1 denote X’s top cycle set. (It is
known that L1 must be nonempty.) Now consider the reduced set, X/L1 . If this too is nonempty
(it may not be), let L2 denote the top cycle of X/L1 . Proceed in this way until all x ∈ X have
been assigned to an L. This procedure must terminate since X is finite.
We call these subsets of X ‘levels’ to suggest a mental picture: one can see the electoral
environment as a series of levels or plateaus (if z > 1). Each plateau electorally dominates all
those below it in that each policy at a given level is majority-preferred to all policies at lower
levels.12 Further, it follows that each policy at a given level covers (Miller 1980) all policies of lower
elevation plateaus.13 Within a level, however, no policy beats all others, nor does any policy lose
11
The top cycle set can be defined constructively. In our setting (all majority preference relations are strict) a policy
is in the top cycle set if and only if it is reachable from every other policy by a chain of strict majority preference.
12
This follows from two facts. First, no policy outside the top cycle set can beat anything in it. (Austen-Smith
and Banks [1999, p.169] explain why.) Second, given our assumptions there are no ties, so any x in the top cycle
beats any y not in it.
13
This implies that X’s uncovered set is a subset of L1 , the uncovered set of X/L1 is a subset of L2 , and so on.
But some policies in a given Lr may cover other policies in Lr . See Bendor, Mookherjee and Ray 2001 for some facts
about the relations among policies in different L’s.
6
to all others at its level. Thus in levels that aren’t singletons, any two policies are connected to
each other by a majority rule cycle. Hence adaptive parties may get “hung up” on the cycles that
pervade non-singleton levels. (Figure 1 gives an example of a policy set with three levels, where
each level has a cycle.)
[figure 1 about here]
Partitioning the policy set into multiple levels turns out to be useful: it allows us to analyze
concisely how the parties hill-climb—or “plateau-climb”—in the electoral landscape.
The partioning easily allows us to describe different majority preference relations. Suppose we
want to examine the classic Downsian context: a unidimensional policy space and citizens with
single-peaked preferences. We obtain this by assuming z = m: each level is a singleton, with the
median voter’s (strict) preference ordering being decisive. At the other extreme, we can represent
the troublesome multidimensional policy space by assuming z = 1: all platforms are connected by
majority rule cycles. Intermediate circumstances are captured by 1 < z < m.14
We assume that if xi xj then xi will defeat xj in an electoral contest. This is consistent
with standard idealized assumptions about Downsian competition: in particular, that turnout is
universal and voting is sincere and error-free. (Proposition 6 shows that the model is robust with
respect to these idealized assumptions.)
There is an indefinitely long sequence of elections, 1, 2, . . ., with one election per period. (At the
start an incumbent party and its platform are randomly picked.) In every election the candidates
simultaneously announce platforms and then citizens vote. The party that gets more votes wins
the election and is the incumbent at the start of the next period. If the candidates adopt the same
policy then a random, nondegenerate tie-breaking procedure picks the winner. (The “coin” need
not be fair; it might, e.g., be biased toward the incumbent.)
The state variables of the associated stochastic process are simply the platforms of the incumbent, It , and the challenger, Ct , in every election. Thus, It is the policy espoused by the winner of
the election in t − 1. We are especially interested in how It —what Kramer (1977) called the trajectory of winning policies—evolves. (Because of It ’s importance we will sometimes abuse terminology
and refer to it as if it were the state variable.)
The parties’ adaptive behavior
The heart of the model is how the candidates respond to winning and losing. Following KMP,
we assume that winners satisfice (Simon 1955).
(A1): the winner of the election in t stands for re-election in t + 1 by keeping the platform that
won him office in t.
In effect, we are assuming that candidates have an implicit aspiration level in-between the
payoffs of winning and losing. Hence winning is satisfying, and incumbents don’t fix what isn’t
broken. (We don’t explicitly model aspirations or their dynamics. For an overview of such models
14
A majority rule winner may exist even if z < m: one exists if and only if X’s top cycle, L1 , is a singleton.
7
see Bendor, Mookherjee and Ray (2001b).)15 Similarly, losing is dissatisfying, and we follow Simon,
and Cyert and March (1963), in assuming that dissatisfaction triggers search. Hence our challengers
(sometimes) search: in at least some elections they will try policies that differ from the one that
they used—and lost with—in the previous election.16
Before examining how challengers search, we investigate an important implication of satisficing.
This will help us better understand which assumptions in our model drive which conclusions.
Proposition 1 refers to a stochastic process, produced by the combined behavior of incumbents and
challengers in the electoral environment of our z levels and (A1). So far, only part of this process
has been defined: incumbents’ (deterministic) behavior.17 The random aspect will be created by
the challenger’s search, as we will see shortly.
Proposition 1:
higher.
Suppose (A1) holds. If It is in Lr , then thereafter it must be at that level or
Proposition 1 says that merely assuming that incumbents satisfice ensures (in the context of
sincere voting, etc.) that electoral outcomes never slip “downhill” to lower plateaus.18 The policy
of the party in power must either stay where it is or climb higher. The proof is simple. Suppose
that It ∈ Lr . The challenger either proposes something from a lower level or not. If he proposes
something lower, then the construction of the L’s plus sincere voting imply that the challenger must
lose. Because winners satisfice, this implies It+1 = It ∈ Lr . If he proposes x ∈ Lr then no matter
who wins today, It+1 ∈ Lr . Finally, if the challenger proposes a platform from a higher level then
he wins and satisficing-by-winners implies It+1 ∈ Lq , where q < r. Induction does the rest.
Note that proposition 1 presumes nothing about the challenger’s behavior. The result follows
just from the combination of the static electoral stage described by the z levels and satisficing by
winners. Recall, however, that we also assume that voters choose without error, or that error is
rare enough so that the underlying majority preference relation is faithfully translated into electoral
outcomes. Thus if we regard the parties as generating options that are collectively “tested” by the
voters (as in Simon’s analysis (1964) of adaptive “generate-and-test” processes), then the “test”
phase is flawless. This flawless collective testing plus satisficing-by-incumbents together imply that
15
Assuming that candidates have exogenously fixed implicit aspirations in-between the payoffs of winning and losing
is equivalent to assuming explicit aspirations that adjust endogenously as follows. Let w be the payoff of winning and
l, losing (w > l). Use the standard assumption (Cyert and March 1963) that tomorrow’s aspirations are a weighted
average of today’s aspiration and today’s payoff: ai,t+1 = λt ai,t + (1 − λt )πi,t , where ai,t is party i’s aspiration in t,
πi,t is its electoral payoff (w or l), and λt , the weighting parameter, is in (0, 1). (Note that the weights can vary over
time.) If initially aspirations are “conventional”—winning is satisfying, losing isn’t—then the endogenous sequence
of aspirations will always be in (l, w). Hence for all possible sequences of elections, winning will be satisfying but
losing won’t.
16
This is such a weak premise that we rarely label it a formal assumption. (We use it explicitly only in proposition 5.)
If it didn’t hold then, given that incumbents satisfice, both parties would always keep the platforms they championed
in period one. This is neither realistic nor interesting.
17
Proposition 1 can therefore say something about specific sample paths; it is not confined to analyzing the probability distribution over a population of sample paths.
18
Proposition 1 has bite only when there are multiple policy levels (z > 1).
8
the process cannot slip downhill.
So one half of the picture of adaptation, how incumbents behave, acts as a brake on the process.
The other half, how challengers behave, is what supplies uphill momentum—not surprisingly, since
only challengers search.
A key aspect of search is discovering new alternatives, as defined below.
Definitions. A party experiments in period t if it selects a platform that it has never used before
t. It innovates in t if it espouses a platform that neither party has ever used.
(A1) implies that incumbents don’t experiment; they satisfice, sticking with what got them
elected. Hence in our model, challengers bear the responsibility for generating novel policies. The
importance of this responsibility is underscored by the next result, which shows that the strong
form of experimentation—innovation—is necessary for upward progress, i.e., for the trajectory of
winning policies to move to higher L’s.19
Proposition 2: Suppose (A1) holds. For all periods s and t where s < t, if Is is in Lr and no
challenger in [s, . . . , t] innovates, then with probability one It is also in Lr .
The proof is simple. Proposition 1 says that no trajectory of winning policies ever slips downhill.
Hence, if Is ∈ Lr , all prior winning platforms (and a fortiori, all losing ones) have come from level
r or lower. Consequently, if no challenger innovates in [s, . . . , t], then all of them must have chosen
platforms from level r or lower. But if that is true and no innovation occurs in [s, . . . , t], then the
set of available policies in that era is in (Lr ∪ · · · ∪ Lz ). So then Is+1 ∈ (Lr ∪ · · · ∪ Lz ), whence by
induction all of the winning policies in [s, . . . , t] must also be in (Lr ∪ · · · ∪ Lz ). Hence innovation
is necessary for progress.20
The electoral dynamic need not grind to a halt if no one innovates: an incumbent can be
beaten even if his opponent does not experiment, much less innovate. For example, suppose that
Lr = {a, b, c}, with a b c a. Then the status quo policy can change endlessly without
experimentation, as the two parties follow the cycle. But this behavior cannot kick the trajectory
of winning policies up to a higher plateau.
When does experimentation suffice for upward progress? Consider this condition.
(A2): There is an > 0 such that for every history and in every election in which the challenger
hasn’t already tried everything, the probability s/he experiments is at least .
The challenger has good reason to try something other than his last platform: it lost in the
last election and the incumbent has stayed put. Further, the challenger has reason to experiment
because everything that he has tried before is electorally flawed.
19
As noted earlier, it is trivially true that search by challengers is necessary for upward progress. Proposition 2’s
claim is much stronger: search that yields systemic novelty is required for progress. (By searching its institutional
memory for policies that have worked in the past but which it hasn’t tried recently, a party could search without
experimenting. And it could experiment without innovating.)
20
Neither proposition 1 nor 2 require that the set of policies be finite.
9
Remark 1: Consider the out-party at date t > 1. Every platform that this party has used before
t has lost at least once.
The remark is easily proven by contradiction. If it didn’t hold, then the challenger in t must
have espoused a policy, x, that won in some earlier period s. (A1) implies that that party would
have continued to uphold x in (s + 1, . . . , t − 1), triumphing in every one of those elections. But
then in t that party would be the incumbent, not the challenger.
If we consider platforms that have lost at least once to be electorally damaged goods, then it
makes sense that (A2) would hold.21 However, (A2) does not commit us to a specific assumption
about how much the challenger knows or how sophisticated he is. A challenger might, e.g., know
the entire set X, have beliefs about voters’ preferences about platforms, and sophisticatedly update
those beliefs. Or she might be blindly groping about. (A2) only amounts to a summary statement
about the consequences of the out-party’s knowledge and strategic sophistication: it always has
some chance of experimenting (when that is possible). Moreover, (A2) does not presume any
specific stochastic model, or even any familiar class of stochastic models. For example, (A2) does
not require a Markovian process: what happens today could depend on events in the distant past.22
Although (A2) is a weak assumption about platform-experimentation, it and satificing-byincumbents together ensure that the trajectory of winning policies converges to the policy space’s
top cycle. Satisficing by incumbents and experimentation by challengers work well together: the
former prevents the trajectory of winning policies from slipping downhill and the latter gives the
process upward momentum.
Proposition 3: If (A1) and (A2) hold, then the trajectory of winning policies converges to and
is absorbed by L1 with probability one.
The proof (in the appendix) relies on the fact that any level below L1 must be transitory: if
the process ever goes to such a level, (A2) ensures that eventually it must leave that level and,
by proposition 1, never return. Since all lower levels are transitory, long-run convergence to the
top level is assured. And proposition 1 implies that L1 is absorbing: once a trajectory of winning
policies reaches that level, it stays there.
Note that if L1 is a singleton, then proposition 3 tells us that the trajectory of winning policies
converges to that unique platform. (The challenger may continue to move around; more on this
21
In related work (Bendor, Mookherjee and Ray 2001c) we study adaptive behavior via an aspiration-based model
of reinforcement learning: an agent is more likely to repeat satisfactory actions and less likely to repeat unsatisfactory
ones. In that context, remark 1 is equivalent to saying that every alternative that today’s challenger has tried has
received at least one negative feedback, and (A2) implies that the challenger will, with positive probability, try an
alternative that has never received negative feedback (if one exists).
22
For example, the probability of experimenting could rise the more often the out-party loses, as desperation to
(e.g.) reclaim the White House sets in. And the magnitude of the increases could depend on when the losses occurred:
e.g., the more recent the losses, the more the challenger is willing to experiment. Because the party’s memory is not
itself a state variable in our model, this process is not Markovian.
10
shortly.) Thus when a majority rule winner exists, we have a semi-Downsian result: the government’s policy must converge to the Condorcet winner.23
Convergence to the top cycle can be established without presuming that the electoral dynamic
belongs to a particular class of stochastic processes. In particular, it is not necessary that the
process be ergodic: it could have multiple limiting distributions.24 (For an example of a process—
consistent with assumptions (A1) and (A2)—that has multiple limiting distributions, see Bendor,
Mookherjee and Ray 2003.) However, although convergence to a unique long run distribution is not
necessary for trajectories of winning policies to be absorbed into L1 , it turns out that ergodicity is
sufficient for this convergence. Proposition 4 establishes that assuming that this process converges
to a unique probabilistic steady state substitutes for positing that the challenger experiments.
Proposition 4: If (A1) holds and the (It , Ct ) process is ergodic, then the trajectory of winning
policies converges to and is absorbed by L1 with probability one.
The proof is simple. Suppose that the first incumbent’s policy is in L1 . Then by proposition 1
the It ’s will remain in L1 thereafter, with certainty. But if a process is ergodic then it is independent
of initial conditions, i.e., of the initial state values. Hence, that the It ’s of the above sample path
are always in L1 is consistent with ergodicity only if the process is absorbed into L1 with probability
one.
Proposition 4 implies that if a Condorcet winner exists, then an ergodic process in which
winners satisfice must converge to the majority rule winner. Thus deriving this semi-Downsian
result requires no specific assumption about the challenger’s behavior. Instead, a more abstract
approach—the stochastic properties induced by the parties’ behaviors—suffices. This is useful
knowledge because, given satisficing-by-winners, many kinds of search rules make the electoral
dynamic ergodic.
We have called the convergence of the It ’s into L1 a semi-Downsian result; it is not a prediction
of full convergence (not even when L1 is a singleton). In view of the facts—candidates in two-party
systems rarely adopt identical platforms—that propositions 3 and 4 do not imply the median voter
result is good for the model’s empirical standing.
However, though these results do not imply full convergence, they do not preclude it either.
Proposition 3 is silent on the matter because it uses (A2), which does not say what challengers do
once experimenting becomes impossible (because the parties have tried all feasible platforms). True,
23
Determining the speed of convergence requires assumptions about search that are stronger than (A2). We will
explore this issue in the next section. For now, note that even unsophisticated search can sometimes make the
trajectory of winning platforms move uphill at a good clip. E.g., if the challenger searches completely blindly then
the probability of moving “uphill” in one period equals the fraction of the policies that are in plateaus higher than the
incumbent’s policy. Thus, if It is at a low level then the probability of hillclimbing in one election can be substantial.
24
In one important respect we need not worry about how many limiting distributions the process has. Under the
hypotheses of proposition 3, the electoral dynamic goes into L1 for sure. Then no matter where in L1 it winds up,
we know one significant fact: the government’s policy is ultimately electorally undominated, i.e., it beats all policies
in all lower levels.
11
this is unlikely if X is large, but it is not ruled out. And proposition 4 makes no direct assumptions
about challengers at all. To tie up these loose ends, consider the following weak formalization of
Simon-Cyert-March’s idea that dissatisfaction triggers search. (It also incorporates the domainspecific assumption that losing an election is dissatisfying. For a justification of this via endogenous
aspirations, see footnote 11.)
(A3): There is an > 0 such that for every history, the challenger in t + 1 adopts a platform that
differs from the one it lost with in t with a probability of at least .
This weak specification of problem-driven search ensures that the electoral process will not
settle down into a tweedledum-tweedledee pattern. Note that (A3) is sufficiently weak—e.g., it is
not Markovian—so that the next result cannot pin down the properties of the process’s limiting
distribution, or even if it has only one. But it can say what the process will not do. (Its proof,
being obvious, is omitted.)
Proposition 5: If (A3) holds then no state in which the parties adopt identical platforms is
absorbing, nor is any set of such states.
Proposition 5 implies that if the process does converge to a unique limiting distribution, then
that distribution must put positive weight on states in which the parties adopt different platforms.
And if it has multiple stationary distributions, then all of these must put positive weight on circumstances in which the parties offer distinct policies. This non-convergence of the parties emerges
naturally from two weak behavioral premises: (1) losing is dissatisfying and (2) dissatisfaction
triggers search.25
EXTENSIONS:
MORE INFORMED AND SOPHISTICATED CHALLENGERS
One can extend this model by making the challenger more informed and/or more sophisticated,
hence sharpening his search for winning platforms. Regarding information, the extreme case would
be to assume that the challenger knows the majority preference relation for each pair of platforms.
Then, given varying degrees of strategic sophistication, the challenger might choose platforms that
vote-maximize against the incumbent (Kramer 1977), or those that are in X’s uncovered set (Miller
1980). Or one might assume an intermediate degree of information—e.g., the challenger has some
chance of knowing what vote-maximizes against the status quo—and combine that with a certain
degree of sophistication. Obviously, many different extensions are possible. We can examine only
a few prominent ones. Some are taken directly from the literature; others involve modifications.
Kramer’s 1977 model
Recall that incumbents in Kramer’s model satisfice (in effect) while challengers choose platforms
that vote-maximize against the status quo. His main result, theorem 1, says that the trajectory of
winning policies converges to the minmax set. It need not stay there, but “Theorem 1 does ensure
that a trajectory which jumps outside must immediately return toward the minmax set...” (1977,
25
Note that X’s finiteness plays no essential role in proposition 5.
12
p.324). In contrast, our proposition 5 states that the trajectory converges to the top cycle, where
it stays (by proposition 1).
As noted earlier, our model’s setup bears some significant similarities to Kramer’s. But there
are also some important differences. Most important, perhaps, is the information that the challenger has. Our (A2) presumes only that the challenger knows what he has tried in the past and
has some chance of experimenting. Kramer’s challenger knows what vote-maximizes against the incumbent’s policy. One might object to the latter because it gives challengers an unrealistic amount
of information: to know exactly what policy vote-maximizes against the status quo is asking a lot of
any political party. And to presume that the out-party is so well-organized, so immune to internal
squabbles, that it can always choose a vote-maximizing option may also be questioned. What,
then, happens if one alters our model by assuming that challengers try to vote-maximize but may
occasionally “tremble” and mistakenly pick a platform that is not vote-maximizing? (For simplicity
we make the standard assumption that following a tremble the challenger plays a fixed and “totally
mixed” strategy: anything in X can be picked with positive probability.) The following result,
which follows from proposition 3, gives the answer.
Remark 2: Suppose (A1) holds. With probability 1 − the challenger selects a platform that
vote-maximizes against the incumbent’s policy; with probability > 0 he trembles and plays a
strategy that is totally mixed over X. Then the trajectory of winning policies converges to and is
absorbed by L1 with probability one.
Hence, if the challenger might err in his effort to vote-maximize against the incumbent then
we recover our convergence result, even if the challenger’s chance of erring is arbitrarily small.26
Thus, though the challenger is trying to vote-maximize against the incumbent and usually does so,
the process is led toward the top cycle, not to the minmax set (unless the two happen to coincide).
Why?
The reason is that the challenger’s errors are filtered or tested by the electorate, making the
trajectory of winning policies drift toward higher plateaus (higher L’s) in the short run and the
top cycle in the long run. Proposition 1 tells us that this electoral testing means that no mistake
of the out-party can shove the dynamic down to lower levels. Further, since any kind of mistake
is possible, the challenger has some chance of stumbling onto policies on higher levels. The voters
approve of such mistakes, thus pushing the trajectory uphill—whether or not the minmax set and
the top cycle coincide.
Thus, regardless of the challenger’s intentions, the electoral environment ensures that the dynamic is driven by winning per se rather than by (e.g.) the magnitude of victory. The selection
environment trumps the agent’s intentions, which is consistent with Satz and Ferejohn’s argument
that “[when] we are...interested in explaining...the general regularities that govern the behavior of
26
If one objects to the assumption that the challenger trembles to a strategy that is totally mixed over X, there
are weaker assumptions that still yield convergence to L1 . For example, suppose the challenger vote-maximizes with
probability 1 − and experiments (when this is possible) with some small probability > 0. Then (A2) holds, so
It → L1 with probability one.
13
all agents...it is not the agents’ psychologies that primarily explain their behavior, but the environmental constraints they face” (1994, p.74).
This trumping of the challenger’s intentions by the selection environment holds quite generally;
the Kramerian challenger’s specific objective—to maximize votes against the status quo—played
no essential role in remark 2. As long as the challenger has some chance of trembling and playing a
strategy that is totally mixed over X, the conclusions of remark 2 hold, regardless of the challenger’s
objective function. Thus the parties could have different goals. For example, when the Democrats
challenge they could vote-maximize against the incumbent, but when the Republicans are the outparty they select a policy that maximizes some ideological criterion (as in, e.g., Chappell and Keech
1986, p.884). Or both parties could maximize an objective function based on a mix of office-seeking
and ideology (Wittman 1983). In the long run those goals do not matter. One could even allow for
parties that suffer from Arrovian problems and so lack coherent preferences. All that counts is that
the pattern of error gives the electoral mill enough grist to work on.27 So long as this condition
is met, one can make the challenger as sophisticated and informed as one likes, with any kind of
preferences; the end result is the same.
Miller (et al.) and the uncovered set
Miller (1980, p.93) and others (McKelvey 1986; Cox 1987; Epstein 1998) have argued that if x
covers y then x electorally dominates y. As Cox put it,
If one accepts the extremely mild assumption that candidates will not adopt a spatial
strategy y if there is another available strategy x which is at least as good as y against
any strategy the opponent might take and is better against some of the opponent’s
possible strategies, then one can conclude that candidates will confine themselves to
strategies in the uncovered set (1987, p.420).
As with most solution concepts, in order to use it a candidate must be both well-informed and
reasonably sophisticated. But expecting candidates to invariably pick policies in the uncovered set
may be unrealistic.28 Yet even a bit of information can help the challenger search, as the next
result shows. (The proof is in the appendix.)
Remark 3: If (A1) is satisfied and for every history the challenger alights on X’s uncovered set
with a probability of at least > 0, then the following hold.
• (i) It is in L1 with positive probability for all t > 1.
• (ii) If Pr(It ∈ L1 ) is less than one then Pr(I1 ∈ L1 ) < · · · < Pr(It ∈ L1 ).
• (iii) It → L1 with probability one as t → ∞.
27
Thus, one can regard this as an evolutionary theory: “blind” variation is produced by error; selection is the
electoral environment. We thank John Padgett for this interpretation.
28
If they did then the process would jump to L1 in one period, since X’s uncovered set is a subset of L1 .
14
Thus even fragmentary information about the uncovered set’s location and even crude understanding about the strategic value of uncovered policies can have substantial impacts, both in the
short-run (parts (i) and (ii)) and the long-run (part (iii)).
Consider now a less demanding possibility: the challenger might not know the uncovered set’s
location, but may know policies that cover what he must try to beat today—the incumbent’s platform. (This presumes, of course, that some alternative covers It . If not, then It is in X s uncovered
set and so is already in L1 .)
First we establish the importance of the challenger finding something that covers the incumbent’s platform. It is necessary: electoral hill-climbing cannot occur without it.
Remark 4: Suppose (A1) holds. Consider any r = 1, . . . , z. If It is in Lr and the probability
that Ct covers It is zero, then It+1 must also be in Lr .
The logic is straightforward. Any policy at higher levels, say any xi ∈ L1 ∪ · · · ∪ Lr , covers any
policy at lower ones, i.e., any x ∈ Lr+1 ∪ · · · ∪ Lz . (See Bendor et al. (2001) for the proof.) Hence
if It ∈ Lr and today’s challenger has no chance of finding a platform that covers the incumbent’s,
then he has no chance of finding anything in L1 ∪ · · · ∪ Lr−1 , since anything there would in fact
cover It . Hence hillclimbing cannot occur. Since proposition 1 ensures that the process cannot slip
downhill, it must stay at the same plateau.
Hence, now consider elections in which the challenger does have some chance of finding platforms
that cover the incumbent’s. The following assumption formalizes this idea.
(A4): For every history and in any election in which It is covered by some x ∈ X, with probability
of at least > 0 the challenger finds an option that covers the status quo.
(A4) neither implies nor is implied by (A2), which stipulates the possibility of experimentation.29
But as the next result shows, their long-run effect is the same. (The proof is in the appendix.)
Remark 5: If (A1) and (A4) hold, then the trajectory of winning policies converges to and is
absorbed by L1 with probability one.
Although the challenger is sophisticated enough to have a chance of finding a platform that
covers the incumbent’s, the process is not guaranteed to be absorbed into X’s uncovered set (unless
that set and L1 coincide), though it will visit that set infinitely often. The reason: L1 may have
29
(A2) does not imply (A4) because the challenger might experiment but his set of possible new policies may
not include anything that covers the status quo. The following example shows why (A4) does not imply (A2). Let
Lr = {w, x, y, z}, with r = 1. Suppose w beats x and y, x beats y and z, y beats z, and z beats w. Then a cycle
connects any two policies in Lr , but x covers y since both criteria of the covering relation hold: (1) x beats y, and
(2) x’s winset, {w} ∪ L1 ∪ · · · ∪ Lr−1 , is a subset of y’s winset, which is {w, x} ∪ L1 ∪ · · · ∪ Lr−1 . Suppose the process
has been stuck in level r, and each party has tried all platforms at this level. Yet, if It = y, (A4) could be satisfied
while (A2) isn’t, if the challenger could (re)discover x with probability greater than but with probability one would
not experiment. Thus the possibility of experimenting and of generating a platform that covers the status quo are
independent search assumptions.
15
policies that are not in the uncovered set, and because any two policies in the same level are
connected by a cycle, the trajectory of winning policies can leave the uncovered set. ((A4) does
not preclude this possibility.)
FFMP (1980, 1984)
In both papers FFMP assume that the challenges to the status quo, x, come from a uniform
distribution over x’s win set. Assuming that new options come from the status quo’s win set involves
an intermediate degree of information and sophistication: less demanding than assuming that new
options must cover the status quo, but more demanding than assuming experimentation. However,
positing a uniform distribution over the status quo’s win set was for computational reasons: FFMP
(1984) calculated bounds on the limiting distribution of winning platforms. It is unnecessary for
qualitative results, and we shall disregard it here. For our purposes, the key feature in FFMP’s
premise is that when searching the challenger puts positive probability on anything that beats the
incumbent’s platform. As usual, this assumption need not be forced into a Markovian mold.
(A5): Following every history, the challenger’s search has a probability of at least > 0 of finding
any option that beats the status quo.
Because the set of policies that beat any x must include some in L1 , (A5) implies stochastic
hill-climbing in the short-run and convergence to L1 in the long-run, just as remark 3 did. (The
proof is similar to remark 3’s and so is omitted.)
Remark 6:
If (A1) and (A5) are satisfied then the following hold.
• (i) It is in L1 with positive probability for all t > 1.
• (ii) If Pr(It ∈ L1 ) is less than one then Pr(I1 ∈ L1 ) < · · · < Pr(It ∈ L1 ).
• (iii) It → L1 with probability one as t → ∞.
Thus if challengers are as informed and sophisticated as FFMP posit, then long-term convergence to the top level is ensured, as is short-term progress.
In general, this section’s findings show that endowing the challenger with more information
and/or more strategic sophistication has a quantitative effect (speeding up convergence to L1 ); it
does not alter the model’s basic qualitative properties.
ROBUSTNESS ISSUES
A common price for analytical results is that they require stylized assumptions. This can
entail reshaping a vague-but-plausible idea (e.g., incumbents are often content with the platforms
that won them office) into crisp but less plausible ones (incumbents always satisfice). To be sure,
theorizing requires simplification: as Jonathan Swift observed long ago, the most realistic model of
a phenomenon is the phenomenon itself. But it would be troubling if our results turned out to be
knife-edge findings — if changing a premise a little altered the conclusions a lot.
16
Several features of our model might cause concern in this regard. As noted, assuming that
incumbents invariably stick to winning platforms exaggerates the plausible scenario it is meant to
capture. Similarly, that x is majority-preferred to y may not guarantee that x will beat y: e.g.,
variations in turnout or voters’ errors may change the outcome.
A more subtle concern, which is conceptually more serious than the simplifications mentioned
above, is that our model appears to predict little in “ill-structured” environments in which the Plott
conditions fail. In such situations the top cycle may be the entire policy space. (It is well-known
that an n-dimensional (n > 1) spatial voting setting is particularly vulnerable to this problem.)
But then Proposition 3 has no bite at all, and our model apparently loses all predictive power.
Yet in an important sense this conclusion is too hasty: much depends on “how seriously” the
Plott conditions are violated. For instance, consider the most drastic class of perturbations: a y
from the lowest level Lz now beats an x from L1 , causing all levels to collapse into one giant set.
Now the top cycle equals X, so technically our convergence results are vacuous. Yet the substantive
bite of such a perturbation could be minimal: it will matter only when the incumbent’s platform
is that particular x and only if the challenger can find that once-lowly y. Challengers could face a
needle in the haystack problem: in the perturbed electoral environment they might not discover y
very often, if X is large and search is crude.30 Then the probability that the sequence of winning
policies will transit from x to y will also be low, so the process will spend “most” of its time at
the top plateau of policies, even though the top cycle is the entire set of policies in this perturbed
environment.
Below we state a robustness proposition that handles all these (and possibly other perturbations)
in the same general setup. This generality exacts a price—some additional abstraction—but we
will try to link the abstract structure to the substantive robustness questions raised here.
Consider a family of models, indexed by a parameter θ ∈ [0, 1]. In each model θ there is a set
of policies X(θ), partitioned into level sets L1 (θ), L2 (θ), . . . , Lz (θ), where z may also depend on θ.
For each model θ a typical state is defined exactly as before: it is a pair (I, C), called σ here for
notational convenience. Abusing terminology slightly, we say that the state σ = (I, C) lies in some
L if the associated incumbent’s platform I ∈ L. A state σ = (I, C) is in the top set if I ∈ L1 (θ).
For each t ≥ 1, let ht be the t-history at date t: all states that transpired up to t, including the
current state at t. Let σ(ht ) denote the state at date t under the t-history ht .
For each t and each t-history, let π θ (ht , i) denote the probability that the system enters Li (θ)
at the very next date (t + 1), starting from ht . This is information that a particular model would
give us. For instance, Proposition 1 says that given our stylized assumptions, if the state at history
ht lies in some Lj then this transition probability π(ht , i) is precisely zero whenever i > j; i.e., the
process doesn’t move from electorally higher plateaus to lower ones. Here we do want to allow for
these “perverse drifts”, but with low probability.
(A7):
30
There exists a function ψ(θ), with ψ(θ) → 0 as θ → 0, such that whenever σ(ht ) lies in
E.g., if search is completely blind then the chance that the challenger will land on y is
17
1
.
m
Lj (θ) for some j < i, π θ (ht , i) may be positive but is less than ψ(θ).
Because this assumption is the heart of the exercise, it warrants further discussion. It acknowledges that movements to lower plateaus can occur in our family of models. This might happen if,
e.g., voters occasionally punch the wrong spot on the ballot.
But the most subtle interpretation of (A7) is that the multilayered structure of the L’s is
“destroyed” by certain preference cycles. This interpretation is subtle because to accommodate
it, L1 (θ), L2 (θ), . . . , Lz (θ) can no longer be viewed as the true plateaus of the electoral contest,
but as some underlying (counterfactual) structure of plateaus were the “problematic” cycles to be
artificially removed. Now the possible drift from higher to lower levels is interpreted not as a failure
of the model’s assumptions but as a genuine majority-based defeat of some x in one level by some
y in another. Thus, under this last interpretation the true ill-structured environment has only a
few levels (perhaps just one), but our formulation strips away the ill-structure in a minimal way
by placing the burden on “wrong” movements in the state under a well-structured environment.
Return now to (A7). It states that for θ close to zero, the ill-structure “almost” vanishes (or
equivalently, under the other interpretations, that the model’s other assumptions “almost” hold).
This is captured by the bound function ψ(θ) on “wrong” movements, which is assumed to go to
zero as θ → 0.
A second assumption guarantees that the usual upward movements, such as those guaranteed
by (A2), continue to exist throughout:
(A8): The infimum value of π θ (ht , i), over every θ, every t-history, and every i is strictly positive,
provided σ(ht ) ∈ Lj (θ) for some j > i.
Although we need to state (A8) formally for this family of models, we have already seen that
it is an implication of our other assumptions (e.g., experimentation). Since no further comment is
needed regarding (A8), we may now state our robustness result.
Proposition 6:
Assume (A7) and (A8).
• (i) For every > 0, there exist integers T () and θ() such that for every T ≥ T () and
θ ≤ θ(), and every initial state, the system lies in the top set at date T with probability of
at least 1 − .
• (ii) Suppose that the system enters the top set at date t. Pick any date s ≥ t. Then, if
θ ≥ θ(), the system continues to lie in the top set at date s with (conditional) probability of
at least 1 − .
As the formal statement of the result is unavoidably technical, a few informal remarks are
appropriate. Part (i) of the proposition states that if the perturbations (represented by θ) are
small enough, then after “sufficiently” many periods the system must be in the top set L1 (θ) with
very high probability.
18
In particular, the proposition says that if the perturbations are due to ill-structured majoritarian
preferences but the “destructive” cycles are sparse enough, then by artificially stripping away the
destructive cycles one can significantly boost the framework’s predictive power. One interpretation,
consistent with the family of models studied here, is that as θ becomes small the number of feasible
policies grows without bound, while the policies that cause the ill-structure increase more slowly.
If the challenger’s search is crude, the chances of alighting on the latter will become very small;
then (A7) holds. Proposition 6 follows.
Part (ii) states that for any sample path that enters the top set, the conditional probability
of staying there thereafter is also high. Informally, the system enters the top set and stays there
with high probability. This rules out certain perverse dynamics (e.g., entry into the top set being
positively correlated with swift departure from that set).
Note that this result used no Markovian assumptions. This is striking: it shows that the model’s
robustness does not rest on any special stochastic features.
Ill-structured majoritarian preferences: a computational model
Proposition 6 studies only “small” changes in the model’s key assumptions. Hence it does
not tell us what happens to the results when there are big changes in, e.g., preference-profiles. In
particular, it does not say what happens to the trajectory of winning platforms when the preferenceprofile is far from having a generalized median. This is a difficult question; analytical results are
hard to come by (though see proposition 7). Hence we resort to a computational model. (It can
address other issues, but due to space constraints we confine ourselves here to the above question.
Subsequent work will tap the computational model’s potential more fully.)
We now briefly describe the computational model. (For a description of the computer program,
see https://faculty-gsb.stanford.edu/bendor/.) Because the simulation is a special case of our
mathematical model—the policy space is finite, (A1) and (A2) hold, etc.—we focus on its distinctive
properties. The central ones are that policies are embedded in a two-dimensional space and voters
have quadratic loss functions in this space.31
To ensure that the simulation results are meaningful and interpretable we make the search
Markovian, which implies that the probability distribution of (It+1 , Ct+1 ) depends only the parties’
current platforms and the transition rules created by the incumbent’s satisficing and the challenger’s
search. By stipulating time-homogeneous search rules, we ensure that the (It , Ct ) process is stationary. Hence we can invoke powerful theorems for finite-state stationary Markov chains (e.g.,
Kemeny and Snell 1960) which tell us when such processes are ergodic.32 All of our computational
31
The following ensures that, as the analytical model requires, each citizen has a strict preference ordering over
the policies: if policies x and y are equidistant from voter i’s ideal point, then x and y are given randomly drawn
distinct “valence” (nonspatial) values. (The draws are independent across voters.) Thus, the voters’ preferences are
lexicographic, with the spatial component being primary.
32
Models of trial-and-error learning that are not ergodic may lack empirical content. See Bendor, Diermeier and
Ting (2002) for several “folk theorems” for such models.
19
results arise from ergodic processes.33 Thus we will be scrutinizing the steady state distributions
of the winning platforms. (More precisely, the output—for every set of parametric values, 1,000
different sample paths run for 1,000 periods—will closely approximate such steady states.)
Results
We examine two types of results: (1) how different profiles of voters’ ideal points affect the
steady state distribution of winning platforms (holding the challenger’s search rule constant) and
(2) how different search rules affect the long run distribution.
(1) To measure the symmetry of a profile of ideal points, we use a standard metric: the relative
size of the uncovered set. (At one extreme, if the uncovered set is a singleton then a generalized
median exists; at the other extreme, the uncovered set is the entire policy space. So the measure
1
ranges from m
to 1.) The challenger’s search rule is represented by a probability distribution over
the policy space; here the distribution is single-peaked (a truncated normal). Thus in period t the
challenger is more likely to choose a platform close to the one he espoused in t − 1 than something
far away.
Figure 2 reveals a strong relation between the size of the uncovered set and the steady-state
dispersion of winning platforms. This pattern reflects how the strength of the centripetal forces
varies across electoral environments. When the uncovered set is small these centripetal forces are
strong, so in the steady state winning platforms are centrally located; when this set is big the
centripetal forces are weak, so winning platforms are scattered throughout the policy space.
[figure 2 about here]
This pattern complements the analytical result of proposition 6, which showed that the process
is well-behaved for “small” perturbations to voter profiles. The computational results of figure 2
suggest that the process is well-behaved globally: the dispersion of the winning platforms seems to
increase steadily as the uncovered set expands.
However, our explanation for this pattern was too terse: the electoral environment can mold
the steady-state distribution of winning platforms only if the out-party’s search generates enough
variety for the selective forces of the electoral environment to work on. So we now briefly examine
the effect of different search rules.
(2) The output in the first part of table 2 is based on naive search: challengers search blindly,
1
putting probability of m
on every platform. Yet the size of the uncovered set and the long-run
dispersion of winning platforms remain highly correlated: the main pattern—centrist platforms
tend to win when the uncovered set is small—continues to hold.
[table 2 about here]
Hence, this main pattern does not require search to be prospectively attuned to winning. Instead,
what suffices is that challengers generate enough grist (variety) for the electorate’s mill, per the
evolutionary arguments proposed earlier.
33
Establishing that these processes are ergodic is straightforward, so the proofs are omitted.
20
The second part of table 2 shows that the relation between the uncovered set’s size and the
steady-state dispersion of winning platforms is attenuated if the out-party’s search is keyed to its
policy preferences. This makes sense: that search rule creates a centrifugal force that operates
independently of the size of the uncovered set—the challenger always reverts to this distribution.
However, although the basic pattern is weakened, it is still present, even in this extreme case
when the challenging party is, in effect, completely dominated by ideologues. (We have also studied
cases where the challenger’s search is a mix of the above pure types, i.e., search centers on a policy
that is a weighted average of the party’s ideal point and its last platform. Results [unreported
here due to space constraints] show that the system’s long run tendencies are then intermediate
between those produced by the two pure search rules whose outcomes are reported by table 2.)
Even ideological parties cannot completely ignore the strong electoral forces that are present when
the uncovered set is small.
The shaping power of the electoral environment: analytics once more
Our computational results are consistent with the argument that even when the electorate’s
preference-profile is far from having a generalized median, all hell does not break loose because
the electoral environment still has centripetal forces. But computational models necessarily use
quite specific assumptions: here, a two-dimensional policy space, quadratic utility, and a small
electorate. So, because it would be helpful to supplement these findings with analytical ones, we
return to deductive methods for a final result.
Naturally, since we resorted to computation because we couldn’t derive general results analytically when the Plott conditions are far from holding, we must simplify our analytical model in
some way. But the simplification should be consistent with our substantive objective, which is to
examine the electoral environment’s centripetal forces. Therefore we should not impose simplifying
assumptions on voters’ preferences. Instead, we make our mathematical model tractable by simplifying the challenger’s search: we assume that it is blind—uniform over X. This assumption not
only helps to ensure tractability; it is also substantively useful: since challengers don’t learn, we
know that conclusions about the steady-state probabilities of winning platforms depend only on
the selective forces inherent in the electorate (and on satisficing-by-incumbents).
Our last result shows that even when the top cycle is the entire policy space and even if no
assumption is made about whether the Plott conditions “nearly” hold, the electoral environment
plus satisficing-by-incumbents can still impart some stochastic order to outcomes. (The proof is in
the appendix.)
Proposition 7: Assume (A1) and blind search by challengers. If L1 = X and x covers y, then
the steady-state probability that x is the incumbent’s platform exceeds y’s steady-state probability.
Thus, whether or not a generalized median almost exists, if the electoral environment is characterized by long strings of covering relations (e.g., a covers b which covers c which...), then the
steady state probabilities of winning policies will be nicely structured by monotonicity (e.g., in the
21
long-run a is more likely than b which is more likely than c which...).34 Proposition 7 also implies
that if a platform’s steady-state probability of being the government’s policy is maximal, then it
must be in the uncovered set. These properties buttress the claims of Miller et al. that if x covers
y then x electorally dominates y.
CONCLUSIONS
This paper has examined the electoral dynamic generated by adaptively rational incumbents
and challengers in two-party competition. The results emphasize how powerfully certain electoral
landscapes shape the dynamic. In highly structured electoral environments—those with many
levels—electoral competition constrains the process greatly, even if challengers are both ignorant
and unsophisticated. Thus our paper is consistent with work in economics on zero-intelligence
agents (e.g., Gode and Sunder 1993), which tries to ascertain how much of market performance
is due to the market environment rather than to the intelligence of agents in those environments.
Gode and Sunder concluded, “Adam Smith’s invisible hand may be more powerful than some may
have thought: when embodied in market mechanisms such as a double auction, it may generate
aggregate rationality not only from individual rationality but also from individual irrationality”
(p.136). The median voter’s invisible hand, when it exists, is similarly powerful, in guiding the
trajectory of winning platforms to what that pivotal citizen wants.35
At the other end of the spectrum, one can investigate the effects of endowing candidates (especially challengers) with more information and/or sophistication. The results in the fourth section
show that incorporating such extensions is straightforward. In general, their effect is quantitative,
not qualitative: they tend to increase the likelihood of hill-climbing in the short-run, making It
converge faster to the top level.
A concern regarding our basic model is that slight perturbations of voters’ preferences can create
an ill-structured electoral environment—the top cycle is everything—making many of our results
vacuous. But proposition 6 shows that small perturbations (of preferences and other parameters)
have only a minor effect on winning platforms in the steady-state. And our computational results
and proposition 7 indicate that the process is well-behaved even when the voters’ preference profile
is far from having a Condorcet winner.
34
Proposition 7 cannot be generalized by replacing “x covers y” with “x beats more policies than y does”. (If x
covers y then it beats more policies than y does, but the converse doesn’t hold.) Though the resulting conjecture—
platforms that beat more rivals should be more likely to be the government’s policy in the steady state—is intuitively
plausible, it is not true in general. This is so even if one rules out what would be, from a spatial perspective, bizarre
“preferred-to” relations, e.g., x beats y even though the former loses to thousands of other platforms while the latter
loses only to a handful. (We will provide, upon request, a counterexample to the conjecture. This counterexample is
not bizarre: if x beats y then x loses to at most only one more policy than y does.)
35
The parallel with models of zero-intelligence agents is incomplete because throughout our paper winners satisfice,
which is sensible. For a true zero-intelligence benchmark, one should posit that both candidates choose platforms
blindly. Because we are interested in more realistic models of bounded rationality, we have not pursued the pure
benchmark case. But even a cursory investigation of that benchmark would reveal that the electoral environment
strongly shapes the trajectory of winning policies.
22
Appendix
Proof of Proposition 3
If I1 ∈ L1 with certainty then the result is immediate (given proposition 1), so assume that
Pr(I1 ∈ L1 ) < 1. Define Ut (i) as the number of platforms untried by party i (i = D, R) at the start of
period t. Ut (i) is a weakly decreasing process, with U0 (i) = m. Note that if min(Ut (D), Ut (R)) = 0
then It ∈ L1 . (This holds because if one of the parties has tried all platforms then it must have tried
those in L1 , and once something in L1 was tried, proposition 1 implies that the process never leaves
L1 thereafter.) So we need to show only that the probability of the event that min(Ut (D), Ut (R)) > 0
goes to zero as t → ∞.
Since there is a challenger in every period, at least one party (say D) must be a challenger
infinitely often. Suppose, for date t, Ut (D) = n, where 0 < n. Consider the event that Ut (D)
stays at n. Suppose D’s experimentation probability were exactly , whenever D challenges and
Ut (D) > 0. Then the counting process Ut (D) is just a geometric process with probability , whence
limt→∞ [Pr(Ut (D) = n)] = 0. Since D experiments with a probability of at least , this must
continue to hold. Hence with probability one Ut (D) will hit n − 1 eventually. If n − 1 = 0 we
are done; if not, just repeat the above argument. So with probability one Ut (D) = 0 eventually,
implying that the probability of min(Ut (D), Ut (R)) > 0 goes to zero as t → ∞.
QED.
Proof of Remark 3
(i) Proposition 1 implies that if a sample path of the winning policies is in L1 in t then it stays
in L1 thereafter. Hence if Pr(It ∈ L1 ) > 0 then Pr(Is ∈ L1 ) > 0, ∀s > t, and in particular if
Pr(I1 ∈ L1 ) > 0 then Pr(Is ∈ L1 ) > 0, ∀s > 1. Alternatively, suppose that Pr(It ∈ L1 ) = 0. Let
U C(X) denote X’s uncovered set. Since U C(X) ⊆ L1 it follows immediately that Pr(I2 ∈ L1 ) > 0,
and we can then use the preceding argument.
(ii) If Pr(It ∈ L1 ) < 1 then by proposition 1, Pr(Is ∈ L1 ) < 1, ∀s < t. Because U C(X) ⊆ L1 ,
it follows that Pr(Is+1 ∈ L1 |Is ∈
/ L1 ) ≥ Pr(Is+1 ∈ U C(X)|Is ∈
/ L1 ). Since Pr(Is+1 ∈ U C(X)|Is ∈
/
L1 ) = Pr(Cs ∈ U C(X)|Is ∈
/ L1 ), and the latter term is bounded away from zero, the result follows.
(iii) Let the bound on hitting U C(X) be some > 0. If the challenger’s probability of hitting
U C(X) were exactly , then the process would have a geometrically distributed waiting time,
whence the probability that it stays out of L1 would go to 0 as t → ∞. Since the process’s
probability hitting U C(X) is at least , the result follows.
QED.
Proof of Remark 5
Consider the event that the challenger fails to espouse an alternative that covers It . Since the
probability of this event is at most 1 − < 1, the probability that it will recur infinitely often
is zero. Hence with probability one the challenger will eventually find something that covers It .
Because It is arbitrary here, this holds for every status quo platform. Hence, since the covering
relation is transitive, the trajectory of winning policies must land in U C(X) with probability one.
Because U C(X) ⊆ L1 , the result is established.
QED.
23
As the proof of proposition 6 is rather technical, it is omitted. (It is available upon request,
and has been provided to the referees.)
Proof of Proposition 7
Given the proposition’s hypotheses, it is easily established that the process is ergodic. Hence
the proposition’s conclusion is a meaningful claim.
Given blind search, the steady state probability of policy i (denoted πi ) equals j∈l(i) πj +
πi (1 − |w(i)|), where l(i) denotes the set of policies that lose to i and w(i) is the set that wins
against i. Thus
|w(i)| · πi =
As
1
m
1 πj .
m j∈l(i)
is a constant in this set of simultaneous equations, it drops out; hence
πi =
1
πj .
|w(i)| j∈l(i)
Because either x j or j x for all i = j, it follows that |l(i)| + |w(i)| + 1 = m. So
w(i) ⊂ w(j) ⇒ l(j) ⊂ l(i)
Hence, if x covers y then l(y) ⊂ l(x). And since πx =
the result follows by simple algebra.
24
1
|w(x)|
j∈l(x) πj
whereas πy =
1
|w(y)|
j∈l(y) πj ,
QED.
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27
Figure 1:
An electoral landscape
with three plateaus.
a
c
L 1 : top cycle of X
b
d
f
L 2 : top cycle of
reduced set,
X/L 1 .
e
x
z
L 3 : top cycle of
reduced set,
X/(L 1 U L 2 ).
y
Key:
a Æ b : a beats b.
a
b
c
f
d
e
:
everything in {a,b,c} beats everything in {d, e, f}.
Variance of Limiting Distribution
Mean = 13.3335 St. Dev. = 12.1236
60
50
40
30
20
10
0
-10
0
0.05
0.15
0.2
Figure 2: Normal search rule around last policy
0.1
Ratio of Size of the Uncovered Set to Size of Top Cycle
Mean = 0.1000 St. Dev. = 0.0594
2
Regression Statistics: Slope = 187.6 T-statistic = 16.17 R = 0.84
0.25
0.3
stochastic?
Table 1
dynamic?
no
is model…
parties
myopic?
yes
key foci
the “chaos”
problem?
yes
bounded
rationality?
yes
nonequilibrium
solution
concepts used?
analytical
solutions?
yes
no
yes
yes
Kramer
(1977)
yes
yes
yes
yes
no
yes
yes
(Markovian)
FFMP
(1980, 1984)
yes
no
yes
yes
yes
yes
yes
yes
(Markovian)
KMP
(1992, 1998)
no
yes
yes
yes
BMR
(2004)
yes
(not
restricted to
Markovian)
Table
bl 2