by-elections, changing fortunes, uncertainty and the mid-term blues

Public Choice95: 131–148, 1998. 131c 1998Kluwer Academic Publishers. Printed in the Netherlands.

By-elections, changing fortunes, uncertainty and the mid-termblues�

SIMON PRICE1 & DAVID SANDERS21Department of Economics, City University, London, EC1V OHB, U.K.;2Department ofGovernment, University of Essex, Essex, U.K.

Accepted 14 January 1997

Abstract. If voters care about the size of the government’s majority, then by-election votesshould exaggerate national swings. Moreover, if there is uncertainty about the outcome of thegeneral electionand if voters” preferences are skewed in such a way as to give more weightto the “downside” outcome (least favourite party wins) than the “upside” (favoured party winswith a larger than preferred majority), then there will be a systematic tendency for govern-ments to lose by-elections, regardless of any changes in national support. These predictionsgo beyond those generated by conventional explanations. The theory is successfully testedagainst data from 383 post-War elections.

1. Introduction

One of the stylised facts of political life regarding by-elections is that they aremore variable than general election results in the same constituencies; anoth-er is that the government tends to lose them.1 For example, Taylor and Payne(1973) report that in 459 by-elections between 1919 and 1972, the opposi-tion did better than the government in 85% of the cases. One “explanation”of the former phenomenon is that by-elections are complicated by specialfactors that do not hold in general elections. However, this does not increaseour understanding of the problem, and does not explain the systematic pat-tern. The popular common-sense argument is that “the electorate are sendinga message to the government”. It is argued that voters use by-elections toexpress their unhappiness with some or all of the government’s policies. Theadvantage of a by-election, it is argued, is that the message carries very lit-tle risk of actually unseating the government. The by-election result “doesn’treally matter”. The problem with this hypothesis is that it fails to explain whythe messages sent in a by-election differ so markedly from the message sent

� The authors are grateful to Shaun Bowler, Martin Paldam, Hugh Ward and an anonymousreferee for helpful comments on the paper; and to Pippa Norris for supplying her data onpost-War by-elections in the UK.

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via the pollsters. The opinion pollsters ask a completely hypothetical ques-tion about voting in an election which will, in fact,not take place tomorrow.This is a major difficulty with the “message” view. But in the current paperwe are able to explain this dichotomy with a simple theory of voting thatpredicts precisely the observed behaviour and, moreover, generates testablehypotheses which we successfully confront with a large body of UK data.At the core of the answer is the fact that the pollsters ask how voters wouldbehave in ageneral election; but, as we show below, there are good reasonswhy voters may behave quite differently in by-elections, where only one seatis at stake. Our theory, therefore, offers an explanation of by-election lossesthat goes beyond the standard mid-term dip view.2

There are several existing theories of by-election behaviour.3 One charac-teristic of by-elections is a diminished turnout, relative to the general elec-tion. Turnout tends to fall less in marginal constituencies, where votes may beconsidered to count for more (Taylor and Payne (1973)). The anti-governmenteffect has sometimes been ascribed to this phenomenon, although the precisemechanism is unclear. However, Mughan (1986) is able to demonstrate that,although there is a correlation between turnout and the decline in the govern-ment vote, it is spurious, as the effect disappears once a variety of factors arecontrolled for. Mughan also investigates two other hypotheses. The first isthat voting is dominated by “personal votes” for particular MPs. The secondis that by-elections are perceived as referendums on the government’s record.It emerges that this last factor is the only serious contender. This is quite con-sistent with the results below, which condition on national poll results.4

Another feature of Taylor and Payne’s analysis is the “mid-term dip”,which, as we observe above, is a well-defined characteristic of the UK data(and for other countries; see Nannestad and Paldam (1994)). Norris (1990)also clearly demonstrates the co-existence of mid-term dips in both by-elec-tions and poll ratings.5 The mid-term dip is often explained by a a simpleNordhaus (1975) style political business cycle, where governments squeezeinflation after the election, although it should be noted that the empirical con-sensus is now against strong electoral cycles.6 Whatever the explanation ofthe dip,7 a story that rests on economic factors is certainly consistent withMughan’s results, which emphasise economic explanations and governmentapproval.8

What emerges from this literature is that by-election results are certainlyrelated to national poll ratings, but not in a simple way. Similar factors affectboth; but the same news has a differing impact on each. In the next section,we offer an explanation of the relationship.

The basic ideas we pursue are very simple. We argue that voters care main-ly aboutpolicies, rather than parties. Although there are good reasons to sup-

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pose that “ideology” should be conceptualised in multidimensional terms,we assume for simplicity of exposition that there is a unidimensional ide-ological spectrum of views, on a continuum from “left” (0) to “right” (1).9

A voter will prefer the party that has a policy position closest to his or herown. But matters are complicated by the government majority. We assumethat the policy position of a party is affected by the majority. The right partybecomes more right as the majority increases, andvice versa.10 A particularvoter, while supporting a unique party, also has a preference over the size ofthe government majority. Thus, conceivably, a Conservative supporter mayvoteagainstthe Conservative candidate, in an attempt to reduce the majority.

Another voter may prefer a larger majority, and vote for the Conservatives.However, this voter may subsequently revise her opinion of the government’sposition (after, for example, new information has arrived about the goven-ment’s stance on a particular policy). She may continue to support the Con-servatives in the sense she prefers a Conservative to a Labour government,but would prefer a lower majority (say). If all voter’s shift their preferences inthis way, then this would affect the marginal constituencies (the ones deter-mining the overall majority) at a general election. But in aby-election, theexistence of marginal constituencies is irrelevant, as their seats are not inplay. The only way to reduce the majority is to oust the candidate in the cur-rent seat. Thus, if support has moved away from the ruling party’s ideologicalposition, the by-election swing is likely to exceed the national swing. We callthis the “excess-swing” hypothesis.

Uncertainty brings an additional effect. If voters have preferences that arestrongly affected by their less preferred party being elected, then they willvote in a way that aims to increase the probability of their preferred partybeing elected - they will “over-vote”. The voter trades off an excessivelyhigh majority against the reduced “downside” risk of the wrong party beingelected. In a general election, there is great uncertainty about the outcome(although less if one party is far ahead in the polls). But in a by-election,the government majority can only change by zero or two, so that the levelof uncertainty is greatly reduced. This means that by-election results shouldtend to show a fall, compared to the general election, and regardless of anynational poll trends, reflecting the fact that the uncertainty surrounding thesize of the majority is largely absent.11

In the next section, we present a formal model of voting under uncertainty.This is extended to encompass uncertainty in Section 3. In Section 4, we testthe model using data from 383 UK by-elections between 1945 and 1995.12

Finally, we draw some conclusions.

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2. A theory of voting under certainty

Our model is designed to be a simplified version of the British electoral sys-tem, where two parties (Conservative and Labour) alternate in office. In theformal model, we ignore the presence of the third party (referred to as the“Liberals” despite the changes of the 1980s and 1990s), although we takeaccount of them in the empirical work.

We make 8 assumptions.A1 There are two political parties.A2 Political preferences may be measured on a uni-dimensional “left-right”

line.A3 Parties have distinct “partisan” characteristics, determining the extrema

of the line.A4 Voting takes place in T constituencies under a first-past-the-post system.13

A5 Policies practised by political parties are modified by the size of theirmajority. A smaller majority moves a right government to the left, andvice versa.

A6 Voters differ in their preferences.A7 Constituencies vary in the composition of left and right voters and

may be ranked by their degree of support for each party.In this section we assumeA8a There is complete information about the outcome of all elections.

AssumptionA3 requires some discussion. We assume parties have fixedideological or partisan positions. This is in contrast to the Downsian (Downs(1957)) view of policy convergence in the face of vote-seeking parties. Amodern expression of this is given by Calvert (1985), who describes thenotion of policy convergence as theoretically “robust”. In fact, the notionis far from robust. Even the smallest ideological bias will lead to a failureto converge, as is pointed out to devastating effect by Alesina and Rosenthal(1995). The argument is a familiar one to economists.14 Since Kydland andPrescott (1977), it has been well known that in the absence of binding pre-commitment governments cannot credibly pursue policies that, while opti-mal ex ante, are against the government’s interestex post. The point is thatthe government always has an incentive to renege (in this case) after winningan election, even if it was clearly in its electoral interest to appease the vot-ers by modifying its policy stance during the campaign. As voters know this,“moderate” policies are not credible.15 There may be reputational equilib-ria (as in Barro and Gordon (1983)), but reputational effects will not ensurefull convergence (Alesina and Rosenthal (1995) pp 30–34). Moreover, thereis a great deal of anecdotal and formal evidence that party policies donotconverge.16

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Figure 1 shows the position of two parties on the left-right continuum,normalised on the range(0;1). The vertical axis shows the “utility” a vot-er derives from a particular policy position. C is the “Conservative”, rightposition at 1, and L the “Labour”, left position at 0.

Thus the parties” positions are at the extrema of the line. We are effectivelyassuming all voters” ideal or “bliss-point” preferences, such as the maximumof the curve illustrated, lie between 0 and 1. This is less restrictive than itmay seem at first sight. If some right voters are more right than the Con-servatives, then they will always vote Conservative. The point of this paperis that if a voter’s preferences are to the left of 1 but still greater than 0:5(so that she is in some sense a Conservative supporter) she may still votefor Labour. But for the hard-core of extreme right or left voters, none ofthis applies. Effectively, we are normalising the bedrock party support (vot-ers who will never change their vote) on zero. A voter’s decision to supporta party depends on her position relative to C or L. Were there unique poli-cy positions, the voter would choose the party closest to her ideal position,markedP on the figure.17 AssumptionA5, modification of the policy posi-tion by size of majority, complicates the decision. We rule out the possibilitythat voters can achieve the same point on the policy line with either a Labouror Conservative administration, with suitable choice of majorities. In otherwords, a Conservative government with a majority of one is always to theright of any Labour government. This leads to Figure 2, which illustratesthe choice open to a Conservative supporter.18 The figure shows the voter’spreferences defined over the number of Conservative seats.(T=2) (strictly,T=2+ 0:5) defines the point at which the Conservatives form a government.We assume voters have preferences that are symmetric around the maximum,but because the policy position of Labour is strictly to the left of the Conserv-atives at the point where the majority switches, there is a discontinuity at T=2.To make the argument a little more concrete, we work an example assumingquadratic preferences19 over policy position, P. For a representative voter,utility U (normalised to take a maximum of zero) is given by

U = �(P� P)2

2(1)

whereP is the voter’s bliss point. There is a simple linear relationship betweenthe number of Conservative seats M and P, given by

P= M=T for Conservative victory, M> T=2 (2)

andP= M=T� � for Labour victory, M< T=2;� > 0 (3)

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Figure 1. Symmetric policy preferences.

The higher is the number of Conservative seats, the more to the right are poli-cies (for both Labour and Conservative administrations). However, a positivevalue of� ensures that Labour is always to the left of the Conservatives.The existence of non-zero� implies that even if preferences are symmetricwith regard to policies, they will not be so over majorities. The figure showsthe preferences of a Conservative voter whose ideal number of Conservativeseats isM.

The choice open to a voter in the Nth ranked20 constituency is whether tovote Conservative or Labour. We adopt a Nash equilibrium solution concept;namely, the outcome is an equilibrium if no voter has an incentive to deviatefrom her strategy given other voters do not deviate from theirs.21 Voters mustfirst determine their optimal choice of partyand majority. They must thendecide which party to vote for. Some Conservative supporters (Conservative,in the sense they prefer Conservative to Labour governments) will decide tovote for Labour. Consider the case where a Conservative supporting voter isin a constituency where N> M. This means that if all constituencies up toand including her own fell to the Conservatives, the number of seats wouldexceedM.

In this case, as her desired number of seats is less than N, she will vote forLabour.22 If all voters act in this way, it is a Nash equilibrium. Furthermore,the decision satisfies Alesina and Rosenthal’s “conditional sincerity” condi-

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Figure 2. Asymmetric majority preferences.

tion: “no voter would prefer a decrease in the expected vote for the party[s]he has voted for in an electoral contest” (Alesina and Rosenthal (1995), p50).23

This strategy (under certainty) will result in the expected outcome. Butover time circumstances and preferences change. Suppose that at a by-election,support for the government has risen. If so, voters previously desiring a Con-servative win and a numberM0 of seats will now preferM1, whereM1 > M0.SupposeM1 � M0 > 1. Then even if the byelection takes place in a seatthat would be won by Labour in a general electiongiven the current state ofnational support,24 there may still be enough Conservative-supporting but (ingeneral elections) Labour-voting voters who nevertheless vote Conservativein the by-election, in order to increase the Conservative majority. A similarargument holds for a fall in national support. If the desired majority fallsby more than one, some Conservative-supporting, Conservative-voting vot-ers will switch to Labour in an attempt to reduce the Conservative majority.Thus the theory predicts the proposition thatP1 By-election votes will tend to exaggerate changes in national support.

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3. The effect of uncertainty

Now we change assumption A8 toA8b Voters are uncertain about the outcome of elections other than in their

own constituency.The implication of this is that under uncertainty and with asymmetric pref-erences over majorities, voters will not select their general-election votingbehaviour to achieveM. Instead, if there is a “downside”25 risk, they willtend to choose~M > M. This cannot be demonstrated using calculus in ourexample as there is a discontinuity at T=2. But the principle can be illustratedwith a continuous example. In Figure 3, Voter A has symmetric (in this casequadratic) preferences. For simplicity we assume the outcome in the voter’sconstituency is known26 by the voters in that constituency, but there is uncer-tainty about other constituencies, which leads to uncertainty about the totalvotes cast for the two parties.

To fix ideas, assume there is an equal chance of the national majority result-ing from a general election being over or under the expected outcome by a.27

For Voter A, this set up does not affect her optimal decision,~M = M. Utilityis given by

U = �(M �M)2

2(4)

There is an equal chance that the expected vote E(M) will be realised as M+aor M� a. Formally, she wishes to

maxM

E(U) = :5

"�(M �M + a)2

2�

(M �M � a)2

2

#(5)

The first order condition is evidently

:5[(M �M + a) + (M �M � a)] = 0 (6)

so that the optimal~M is simply M.But with Voter B, illustrated in Figure 4, with asymmetric preferences,

U = ��(M �M)3

3�

(M �M)2

2where� > 0 (7)

In this case the first order condition is

:5[�((M �M � a)2 + (M �M + a)2) + 2(M �M)] (8)

For� = 0 (quadratic preferences)~M = M. For� > 028 we have a quadraticform for the first order conditions. There may be two solutions but in all

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Figure 3. Symmetric majority preferences under uncertainty:~M = M.

Figure 4. Asymmetric majority preferences under uncertainty:~M > M.

cases~M > M.29 The point is that the loss from a downside risk exceeds thegain from an upside risk, so the voter insures against these possibilities by“over-voting” for the preferred party. In other words, uncertainty will imply

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that voter’s vote in such a way as to select a larger majority than they wouldhave done under certainty. This solution remains consistent with the Nashand conditional sincerity conditions.

However, this analysis applies only to a general election. In a by-election,there can only be a marginal change in the majority. So (given the simplifyingassumption of certainty regarding the voter’s own constituency), there is nouncertainty regarding the outcome M and~M = M; there is, in other words,no incentive to “over-vote” in by-elections. ThusP2 By-election results will show systematically lower support for the gov-

ernment than was exhibited at the previous general election.This argument holds for a unique by-election. However, by-elections are

necessarily random events.Given this, even if the outcome of the by-election in the voter’s own con-

stituency is certain, there is some uncertainty about the government’s major-ity remaining as the results of potentialfuturecontests are unknown. Howev-er, with locally symmetric preferences over M this will only bite if there is anon-negligible probability of the government losing its majority. This impliesthat voters in by-elections will be more cautious the smaller is the nationalmajority. ThusP3 The size of by-election swings against the government will be negatively

related to the size of the government majority.

4. Econometric specification and results

4.1. Specification and data

We collected data on 383 by-elections between 1945 and 1995 where bothmajor parties fielded a candidate.30 We had data on the Conservative andLabour vote at the by-election, the corresponding votes at the general elec-tion, the level of national opinion-poll support for the Conservatives, Labourand the Liberals at the time of the by-election and at the previous generalelection.31 In addition we had data on turnout, on the government majority atthe time of the last election and 12 regional dummies, rj . We also constructeda series measuring the time in months since the general election, s. The basicmeasures of popularity and support were handled with a logit transform. Thelogit or “log odds ratio” transform is the standard specification for examiningpopularity functions, for good statistical reasons.

There is an underlying model which relates the transform zi to a set ofrelevant variables, xi, as in the following expression (where pi is support fora party in the ith constituency and�i is an error term).

zi = ln(pi=(1� pi)) = f(xi) + �i (9)

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Note that zi is unbounded, while pi is restricted to the range (0,1). The pointis that a given change in xi will have less effect the closer is pi to either 0 or 1.The largest effect is when p= 0:5. Put simply, for low and high values of pi,most voters have made their minds up and are unlikely to change behaviourfollowing a shift in xi.32 Note that if the function f(xi) is common to allconstituencies, varying only in the xi, a given change (say�) to one of thearguments in any two constituencies will shift zi by the same amount in both,ceteris paribus.33 This means that if our model is wrong and by-electionsare affected by the same variables as national poll supportand to the sameextent, then (subject to the standard variation in the error term) aggregate zand constituency zi should move together.34 Thus the model we estimate maybe written as

dvi = �c + �l + �1dp

i + �2dli + �3m dp

i +11X

j=0

j rji (10)

where�c and�l are the constants for Conservative and Labour administra-tions (which may differ), dv and dp are the changes in the logit transform ofthe government constituency vote and national popularity since the generalelection, dl is the change in the logit transform of the Liberal national popu-larity since the general election, m is a dummy for governments with smallmajorities35 at the general election, and the rj are regional dummies wherethe coefficients are constrained to sum to zero. We allow the constant to alterbetween types of government to allow for different perceptions of “down-side” risk; we expect�h � 0 for h = c; l. We expect�1 > 1. To reiteratethe argument, this follows from prediction that by-election votes exhibit an“excess swing” - by-electionsexaggeratenational opinion poll swings. Thisis more than saying that by-elections offer a chance to register a “protestvote”, as we predict thatpositiveswings should be exaggerated as well asnegative ones. It should also be recalled that the effects we are looking forgo beyond the well established mid-term dip effect, which is captured inthe national opinion poll data. The Liberal term is introduced to allow fora third-party dilution effect. The effect of the increased Liberal vote will beto reduce the credibility of the opposition threat and make it more difficultto see off a government candidate. So we expect�2 > 0. The governmentmajority dummy allows voting to revert towards general-election behaviour,as it were. For small majorities, we expect caution to reassert itself, so that theswing becomes less pronounced. Thus we expect that�3 < 0. One interestinghypothesis is that�1 + �3 = 1. In this case by-election voting exactly minicsgeneral election behaviour. Finally, the regional dummies are designed topick up idiosyncratic regional effects.

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Table 1. Explaining changes in electionvotes – Dependent variable: dv

Parameter Estimate T-ratio

�c –0.278 –5.31�l –0.131 –2.08�1 1.265 12.10�2 0.176 4.14�3 –0.681 –2.09

Regional dummies not reportedAbsolute t-ratios based on Newey-Westadjusted standard errors with 50 periodParzen windown = 383R2 = 0.455Serial Correlation: LM(1)�2

1 = 8.349Heteroscedasticity: LM(1)�2

1 = 1.501

The data set has unusual characteristics. It is ordered temporally but atirregular intervals; some by-elections are held simultaneously. Although weare conditioning on national support, we know other factors matter, whichwill tend to induce serial correlation.36 There may also be other dynamics;by-elections may exhibit bandwagon effects. However, it would be inappro-priate to use parametic methods to model these, given the irregularity ofthe frequency. We therefore use the Newey-West (1987) correction to thevariance-covariance matrix, which requires no prior information about theerror structure.

4.2. Results

Our theory offers a fairly tight specification embedded in one equation. Theresults of estimating this equation are given in the table, which reports theOLS estimates (with Newey-West adjusted t ratios).37 The key coefficientsare the�h, �1 and�3. Taking the coefficients in the order they appear, thetwo �h parameters are both significantly less than zero, in line with the pre-dictions of the theory. Moreover, the Conservative effect is larger than theLabour effect, suggesting Conservative voters perceive the “downside” con-sequences of a Labour administration to be greater than the correspondingperception of Labour voters. However, the difference is on the border of sig-nificance:�2

1 = 3:428, which has a marginal significance of 0.065. The para-meter�1 is expected to be greater than one. This is indeed the case. The coef-

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ficient is significantly greater than unity (�21 = 6:431, marginal significance

0.011) and very precisely determined.This is strong evidence for the theory. The excess swing hypothesis cannot

be rejected. The other important coefficient is�3, which tells us how muchthe excess swing falls when there is a narrow majority. The estimated para-meter is significantly negative, as we expect.

Moreover, although the point estimate of the sum of�1 and�3 is only0.584, this is insignificantly different from one (�2

1 = 2:068, marginal sig-nificance 0.150). So when the government is in danger of losing its over-all majority from a run of adverse by-election results, we cannot reject thehypothesis that voters behave as if they were voting at a general election.Finally,�2 shows that an increase in Liberal support does benefit the govern-ment, as we expected.

Given that we cannot reject the hypothesis that�1 + �3 = 1, we re-estimated the relationship imposing this restriction. The results are largelyunchanged; the estimate of�3 rises slightly to 1.310. We also explored thepossibility that the excess swing we have identified applies only to down-turns. This helps us to distinguish our model from the “message” story (whichis usually interpreted as the sending of disapproving, rather than approving,signals). To test this, we divided dp

i into two series - one (dp+i ) where dpi tookpositive values, and one ( dp�

i ) where dpi took negative values. The point esti-mates of the effect on dp+

i and dp�i respectively were 1.145 (1.79) and 1.275(9.85), with t-ratios in parentheses.

Evidently the numbers are numerically close; they are also insignificantlydifferent (�2

1 = 0:037). We were also concerned there might be evidenceof a mid-term dip beyond that included in the change in popularity, dp. Wetherefore defined s as the time in months elapsed since the general election,and added s, s2 and s3 to the regression. This had no explanatory power;�2

3 = 2:652.This is a short results section. We have shown that the predictions of the

theory stand up remarkably well to the data. All the results are consistent withthe theory. Conservative voters appear to fear Labour governments more thanLabour voters fear Conservative ones.

5. Conclusions

By-elections are notoriously idiosyncratic. They do not reflect national pollratings in any simple way. One consequence of this is that by-election resultsare a poor indicator of general election success. Some prominent recent by-election defeats have been quietly negated at the next election. It seems thatelectors use by-elections to “send a message” to the government. And yet

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they do not choose to do this when responding to the standard poll questionasking for voting intentions “if there were to be a general election tomor-row”. On the face of it, this discrepancy is irrational. However, in this paperwe argue that rational by-election votersshouldact in a way that is systemat-ically different from the way the same voters will choose to act in the generalelection, if they care about the size of the government’s majority. In particu-lar, by-election swings should exaggerate national swingsin both directions(for both positive and negative swings). This is strongly supported by thedata. Moreover, if we assume a particular type of preference over parties,where the “downside” risk of the less preferred party being elected outweighsthe “upside” risk of having your preferred party elected with a majority aboveyour ideal level, then by-elections should, on average, show a fall in sup-port from the election (over and above the effect expected from the nationalpoll rating). This turns out to be the case for both Conservative and Labourgovernments, although the effects is slightly greater for the Conservatives,implying Labour voters are more sanguine about Conservative policies thanConservatives are about Labour’s.

The final conclusion we draw is simply that there does seem to be some-thing in this model. The next step in the empirical project is to deepen the dataset, in order to condition on a wider range of variables, and thus to model theunexplained component currently reducing the efficiency of our estimates.

Notes

1. A related issue is the evident loss of support that governments suffer as time passes after anelection; see Paldam and Skott (1995). However, as reported below, we find no evidencefor this effect in our sample over and above the contribution from the national poll.

2. It also provides a grain of comfort to unpopular governments suffering by-election losses.The scale of defeat is unlikely to be matched in general elections. Politicians know this, ofcourse; we simply have a theory about why this occurs.

3. See Norris (1990) for a comprehensive description and analysis of all by-elections between1945 and 1987.

4. Although there are a multitude of relevant factors, economic considerations and percep-tions of competence at economic management are important parts of the explanation ofthe level of national support. See Price and Sanders (1993) and Sanders and Price (1996)for some recent analysis of both these issues.

5. This emerges in our sample too. In the 383 by-elections that we examine, the averagechange in the logit-transformed vote share from the general election is –0.462 (standarddeviation 0.435). The corresponding figure for national support relative to the last electionis also negative, but smaller and less variable: –0.252 (standard deviation 0.226). So gov-ernments tend to lose support relative to the election, but relatively more in by-elections.However, there is one additional feature of the data. Even though by-election results areworse for the government than national polls, thenumberof positive or negative swings ispractically the same (331 negative swings in the polls; 334 negative by-election swings).As we show below, our theory explains all these features of the data.

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6. See Alesina (1988, 1990) and Alesina, Cohen and Roubini (1993), for example.7. Paldam and Skott (1995) have an intriguing explanation of the dip, based on a rational

desire for change. Some voters prefer a combination of policies from different parties. Aspolicies take time to implement, this outcome can partly be achieved by alternating periodsof office. The model is not tested (and it is hard to see how it could be), but Paldam andSkott give an example where, with plausible parameters, there is a dip roughly in linewith that observed in reality. The regression to the mean argument sometimes advanced(statistically, the party elected is likely to have its support overstated in the election) cannotgenerate either the magnitudes or the particular pattern required (Paldam (1986)).

8. Mughan (1988) compares the factors affecting the national poll and by-elections, and findsthey differ in magnitude and some times qualitatively (for instance, he suggests that eco-nomic crises tend to rally the by-election vote while weakening poll ratings). Probably, nottoo much should be read into the detail of his comparisions. Nevertheless, it seems clearthat perceptions of economic competence affect both; and that the effects are different.This is consistent with our results.

9. This is the simplification adopted by Alesina and Rosenthal (1995) among others.10. While plausible, this need not be the case. In the UK, there is a concept of “a working

majority”, above which the government can do what it likes. This is not fatal to our argu-ment; effectively it introduces a step-change in the trade-off between policy and majority.However, if the government has a small majority “rebels” may be able to exert dispro-portionate power; witness the “Euro-sceptics” revolt in the Major government in 1995.Against this, it is likely that parliamentary candidates tend to be more extreme in the moremarginal constituencies, on the grounds that “moderate” politicians are more electable,and are therefore selected for the safest seats first. If so, higher majorities are associatedwith a more extreme sample of MPs.

11. A referee suggested that the idea of “expressive voting” introduced by Brennan and Lomasky(1993) may be relevant to our analysis. Like many others, Brennan and Lomasky are puz-zled by voters’ motives. Why should a individual, who has a vanishingly small chance ofbeing the decisive voter, bother to make even a minimal effort to turn out in the polls? Theysuggest there is a category of actions, which they call “expressive”, that have an intrinsicutility quite apart from any “instrumental” effect they may have. Thus a football supportermay cheer his or her team on in the privacy of his own room; a voter expresses supportfor a party in the polling booth. Brennan and Lomasky argue these expressive motivesallow conflict with the standard individualistic calculus. For example, a voter (Brennanand Lomasky (1993) pp 41–49) may express a preference for higher, redistributive taxa-tion in conflict to the instrumental or selfish desire for lower taxes. Voters feel good abouttheir gesture, knowing it is unlikely their vote will be decisive. This may, therefore, pro-vide a rationale for the “message” view of by-election behaviour. Voters may send strongerexpressive signals in by-elections as the cost is lower than in a general election. Howev-er, none of the arguments developed below - about uncertainty and voter behaviour - areaffected by this notion.

12. We concentrate on by-elections in this paper. But the theory has applications beyond this.Alesina and Rosenthal’s model of institutional balancing can be used to explain anti-government voting in local and European elections.

13. To rule out hung parliaments we assume T is an odd number and ignore the existence of aSpeaker.

14. While the arguments have been developed largely by economists and are generally in thecontext of macroeconmic policy, the implications go beyond economics to other socialand public policies.

15. There is a large literature on this “policy game” question. See Alesina (1988), Price (1995)and Perrson and Tabellini (1990, 1994) for surveys. The seminal articles in addition toKydland and Prescott are Barro and Gordon (1983) and Backus and Driffill (1985). Notethat we implicitly assume that both parties are more extreme than the pivotal voter, so that

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“reneging” implies a move toward extremism; this is not an exceptional assumption; seebelow.

16. This begs the question ofwhy parties are polarised in this way. One story is that partiespolicies are determined by activists, and that activism is a costly activity (in terms oftime and effort). Plausibly, people with extreme views care more about which policiesare implemented and are therefore more willing to be activists. So the activists are over-sampled from among the extremists in the population. See Feddersen (1992) for a formalmodel: Alesina and Rosenthal (1995) discuss some other possibilities.

17. We have assumed symmetric preferences, which is needed to justify the “closeness” argu-ment. We work with symmetric preferences partly for simplicity, but also because,ceterisparibus, symmetry will imply that uncertainty over policy outcomes after a vote leavesthe optimal choice unaffected. This does not imply voters are risk neutral. In all cases weconsider, concavity of preferences ensures risk aversion. As we will see, even with thisassumption that loads the results against uncertainty having an effect on voting, there willstill be asymmetry in preferences over majorities.

18. The analysis is exactly symmetrical for a Labour supporter.19. Quadratic preferences have two advantages. Firstly, they possess the “right” shape, and

secondly, they make the solution very tractable as the first-order conditions are linear.20. That is, constituencies are ordered from 1 (with highest Conservative support) to T (with

lowest Conservative support).21. This is, of course, not to be confused with the Nash bargaining solution concept.22. It may be easier to understand this if we imagine all constituencies have a unique voter; in

our set up, all voters know each others preferences.23. In Alesina and Rosenthal there are two equilibria, corresponding to Democratic or Repub-

lican presidential victories. At first blush, there are there are two potential equilibria in ourmodel too, where the outcome is conditioned onexpectationsof the overall outcome. Ifthe voter expects a Conservative victory, then under the conditions set out above (N> M),she will vote Labour. By contrast, if she expects a Labour victory, she will vote Conser-vative, as she will always wish to minimise the Labour majority. But in each case sheexhibits conditional sincerity. As in Alesina and Rosenthal, this is not a simple medianvoter model, as the choice of vote is conditioned on the expectation of a Conservative orLabour victory. However, under certainty the notion of an “expectation” is trivial. Theseperfectly informed voters know the precise rankings of the constituencies. But if we allowuncertaintyabout the outcome of contests in other constituencies, then it makes sense tocondition on the expected victory. Nevertheless, the menu of possibilities is inevitably lessrich than in Alesina and Rosenthal, given the US electoral system of two houses, separatePresidential elections and fixed mid-term elections.

24. Note that “national support”, as measured by the response to the standard polling question,refers explicitly to a (admittedly hypothetical) general election.

25. That is to say, if they are particularly fearful of their non-preferred party forming a gov-ernment.

26. Taken literally, this seems to remove any motive for voting. If the outcome is “known”,then an individual’s vote is redundant. The issue of why people vote at all is a well troddenone, which we do not propose to rexamine here, although we note that the strategy ofchoosing to vote is a Nash equilibrium, whereas the strategy “don’t vote” cannot be so, asif a voter believed other voters were not voting, she would.

27. As in footnote 21, one way of thinking about this is to imagine there is a unique constituen-cy for each voter. Thus each voter knows with certainty the results in her own constituency,but not in others.

28. We assume a suitable choice of� to ensure concavity (single peakedness) of U.

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29. We require a� < 1 for real solutions. The second order condition for a maximum isM = M + 2(1��)

�2 where� = 2p(1� a2�2). These conditions place no major restrictions

on the solution.30. The core data set, which we subsequently supplemented, was supplied by Pippa Norris.31. This is defined as the “poll of polls” average poll rating at the time of the by-election, and

the actual votes cast in the general election.32. See Price and Sanders (1993) for a fuller exposition.33. Strictly, this assumes linearity of f(xi). This is probably not an unreasonable assumption;

see Price and Sanders (1993). If we relax it, we have a non-trivial aggregation problem.34. In terms of the empirical specification set out below this means our null hypothesis is

formulated as H0: �1 = 1, with alternative H1: �1 > 1.35. “Small” is defined as less than 20. The relevant periods are April 1950 to June 1951;

February 1952 to February 1955; January 1965 to January 1966; June 1974; and June1975 to March 1979.

36. The idiosyncratic nature of by-elections inevitably implies that the error tem will be largerelative to the main explanatory variable, the national poll rating. However, this in itselfleads to no special econometric problems, as long as the excluded idiosyncratic factors areorthogonal to the explanatory variables, as seems likely.

37. The results are for a 50 period Parzen window. Other lag lengths and windows give similarresults.

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