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APPLIED MICRO‐ECONOMICS WORKSHOP Scott Ashworth University of Chicago Thursday, March 17, 2016 1:15pm ‐ 2:45pm 333 Sage Hall PRESENTATION Learning About Voter Rationality Learning About Voter Rationality∗ Scott Ashworth† Ethan Bueno de Mesquita‡ Amanda Friedenberg§ March 10, 2016 Preliminary and Incomplete. Please do not circulate. ∗ We are particularly indebted to Anthony Fowler. We have also benefited from comments by John Bullock, Alan Gerber, Greg Huber, Claire Lim, Andrew Little, Neil Malhotra, Mike Ting, Dustin Tingley, Stephane Wolton, seminar audiences at Emory, the Juan March Institute, Princeton, Vanderbilt, Virginia, Yale, and the participants in the EITM Summer Institutes at Princeton (2012) and Mannheim (2014). † Harris School of Public Policy Studies, University of Chicago, email: [email protected] ‡ Harris School of Public Policy Studies, University of Chicago, email: [email protected] § W.P. Carey School of Business, Arizona State University, email: [email protected] Scholars of voter behavior have long been interested in evaluating voters’ competence to fulfill their electoral function. The early literature on voter competence asked whether or not voters were sufficiently informed to make good decisions (Campbell et al., 1960; Fair, 1978; Kinder and Sears, 1985; Popkin, 1991; Sniderman, Brody and Tetlock, 1993; Lupia, 1994; Delli Carpini and Keeter, 1996). A more recent literature focuses on another aspect of the competence of voter decision making—specifically, voter rationality. The question of voter competence is, of course, central to normative debates about the merits of electoral democracy (Downs, 1957; Campbell et al., 1960; Key, 1966; Fiorina, 1981; Fair, 1978; Kinder and Sears, 1985; Popkin, 1991; Sniderman, Brody and Tetlock, 1993; Lupia, 1994; Delli Carpini and Keeter, 1996). But it also raises crucial questions about important positive literatures. As Fearon (1999) highlights, voter learning about incumbent politicians is the key mechanism underlying modern models of electoral accountability. These models form the backbone of an extensive theoretical literature that claims to provide positive insight into a variety of phenomena including the incumbency advantage, term limits, judicial review, delegation, campaign finance, local public good provision, special interest politics, the effects of the media, electoral pandering, federalism, and electoral rules.1 If actual voters process information in a way that is dramatically divergent from the way in which voters process information in these models, then we might worry about the validity of the entire theoretical edifice. This is precisely the conclusion reached by many contributors to an empirical literature that seeks to assess voter rationality by examining how election outcomes respond to various shocks to voter welfare. One strand of that literature examines the electoral consequences of natural disasters and government responses to them (Abney and Hill, 1966; Achen and Bartels, 2004; Healy and Malhotra, 2010; Healy, Malhotra and Mo, 2010; Bechtel and Hainmueller, 2011; Gasper and Reeves, 2011; Cole, Healy and Werker, 2012; Huber, Hill and Lenz, 2012; Chen, 2013). Another strand examines the electoral consequences of economic shocks from outside the local economy (Ebeid and Rodden, 2006; Wolfers, 2002; Leigh, 2009; Kayser and Peress, 2012). Many such studies report that incumbent electoral fortunes suffer following negative events outside the control of policymakers (Achen and Bartels, 2004; Wolfers, 2002; Leigh, 2009; Healy, Malhotra and Mo, 2010). These results are typically interpreted as evidence of voter irrationality. The basic argument is that, if voters are rational, then incumbents’ electoral fortunes should be unaffected by shocks to outcomes outside of the incumbents’ control.2 1 For papers covering these various topics see, among many others, Banks and Sundaram (1993); Lohmann (1998); Persson and Tabellini (2000); Canes-Wrone, Herron and Shotts (2001); Besley and Burgess (2002); Coate (2004); Maskin and Tirole (2004); Ashworth (2005, 2006); Ashworth and Bueno de Mesquita (2006, 2008); Besley (2006); Besley and Prat (2006); Myerson (2006); Gehlbach (2007); Gordon, Huber and Landa (2007); Fox and Van Weelden (2010); Daley and Snowberg (2011); Fox and Jordan (2011); Fox and Van Weelden (2012); Fox and Stephenson (2011, 2015); Almendares and Le Bihan (2015); Eggers (2015). 2 Not all empirical studies find this result. A few studies find that voters do not seem to respond to irrelevant shocks (Abney and Hill, 1966; Ebeid and Rodden, 2006; Kayser and Peress, 2012). Other studies find that the negative electoral consequences of shocks to voter welfare are mitigated once they control for the quality of the government’s response to the crisis (Healy and Malhotra, 2010; Bechtel and Hainmueller, 2011; Gasper and Reeves, 2011; Cole, Healy and Werker, 2012). We address this latter branch of the literature in the Conclusion, where we discuss how adding Incumbent effort would affect our analysis. 1 And, indeed, such findings have sparked a theoretical literature analyzing how specific failures rationality affect the equilibrium predictions in models of elections (Callander and Wilson, 2008; Patty, 2006; Ashworth and Bueno de Mesquita, 2014; Diermeier and Li, 2013; Ortoleva and Snowberg, 2015; Levy and Razin, 2015a,b; Lockwood, 2015). We challenge this interpretation of the empirical literature. This puts on us the burden of answering the following question: Why would incumbent electoral fortunes suffer following a natural disaster or economic downturn, if voters are rational? After all, rational voters certainly do not punish politicians for events outside of the politicians’ control. To get a sense of our answer, think about a simple example of the effect of hurricanes on electoral fortunes. Hurricanes are random shocks, outside the control of the incumbent. But the damage caused by a hurricane depends on the quality of infrastructure maintenance, emergency preparedness, and so on. Suppose there are two types of politicians—high quality and low quality. High quality incumbents do a good job preparing for storms, while low quality incumbents do a poor job. Rational voters are concerned with selecting high quality politicians because they are expected to provide good governance outcomes in the future (Fearon, 1999). If a hurricane does not occur, the voters do not see the difference in preparedness. If a hurricane comes, the voters do see the difference in preparedness. This means that the presence or absence of a hurricane affects the amount of information the voter has about the incumbent—if there is a hurricane the voters learn the incumbent’s quality and if there is no hurricane the voters do not.3 Let’s see that, in this example, disasters can affect the electoral fortunes of the incumbent. Suppose, prior to observing the incumbent govern, the voters believe that the incumbent is more likely to be high quality than is a future electoral challenger. If there is no hurricane, the voters get no new information and, so, continue to believe the incumbent is more likely to be high quality. Thus, they reelect the incumbent. If there is a hurricane, the voters learn the incumbent’s quality. If that quality turns out to indeed by high, the incumbent wins reelection. But if that quality turns out to be low, the incumbent loses. Hence, by providing the voters with new information, the hurricane makes it possible that the incumbent will lose, something that never happens without a hurricane. (Notice, if the voters started with the belief that the incumbent was less likely to be high quality than the challenger, then the effect of the hurricane would be reversed.) Now imagine an empiricist collects the data generated by this example. Some locations have a hurricane and some do not. Since hurricanes happen by chance, the distribution of high and low quality incumbents is the same in each. The empiricist first calculates the reelection rate in locations that did not have a hurricane and finds it to be one. She then calculates the reelection rate in locations that did have a hurricane. It is equal to the share of incumbents who were of high quality—a number less than one. Hence, just as in the literature, she correctly finds that the presence of a hurricane, for which the incumbent bears no responsibility, in fact causes a reduction 3 Our formal model does not make use of such stark differences in information. We do so here for expositional simplicity. 2 in expected incumbent electoral fortunes. Our key point is that this is not evidence of voter irrationality. Indeed, we have shown that, in this example, this must be the result when voters are rational. This example is, of course, contrived. But it raises an important question: In general, if voters are rational, should we expect voter behavior and incumbent electoral fortunes to respond to events outside the incumbent’s control? To answer this question, our main analysis builds on the canonical model of voter learning (Achen, 1992; Bartels, 1993; Gerber and Green, 1999; Wolfers, 2002). In Section 1, we show that under a knife-edge, but common, assumption the answer to that question is no. However, in Sections 3 and 4, we show that moving away from that assumption by any amount changes the answer to yes. Quite generally, rational voter behavior and incumbent electoral fortunes do respond to events outside the incumbent’s control. How exactly they do so is subtle in ways that our analysis elucidates. 1 When Shocks Have No Effect We want to analyze how disasters effect voter behavior and electoral fortunes, using a Bayesian learning framework that is standard in the literature (Achen, 1992; Bartels, 1993; Gerber and Green, 1999). We adapt the specific model in Wolfers (2002), since it speaks directly to the issue of observable shocks and electoral fortunes. In this specific model, the standard interpretation is justified—if voters are rational, electoral fortunes do not depend on observable shocks. But, as we will show in the sequel, this implication is knife-edge: It is driven by a special assumption made in the literature; that assumption is made for reasons of tractability and not verisimilitude. There are two politicians, an incumbent and a challenger. Each is either a good type (θ) or a bad type (θ). The probability that politician p is a good type is πp ∈ (0, 1). The canonical model in the literature instead has a normally distributed prior on type (and so a continuum of types). The two-type structure will only simplify the analysis later. In period t = 1, 2, the voter observes a governance outcome, gt , that depends on the incumbent’s type θt , as well as two shocks: gt = θt − ωt + t . The first shock is an observable disaster which is the realization of a random variable with positive support. We refer to a particular realization of this random variable, ωt , as the disaster intensity in period t. There is also an unobservable shock which is distributed standard normal. Its realization in period t is t . All the random variables are independent from one another. In the election, the voter reelects the incumbent if the voter’s posterior belief that the incumbent is a good type is higher than his prior belief that the challenger is a good type. Using Bayes’ rule, the voter reelects if and only if πI ϕ(g1 − θ + ω1 ) ≥ πC , πI ϕ(g1 − θ + ω1 ) + (1 − πI )ϕ(g1 − θ + ω1 ) 3 (1) where ϕ is the standard normal pdf. It is worth pausing here to see what the rational voter is doing. The voter is trying to learn about θI , but observes g1 . The voter knows that g1 is biased and, in particular, is biased by the disaster intensity ω1 . Hence, the voter “filters out” the bias caused by the disaster, adding ω1 to g1 and then forming his posterior beliefs about the incumbent based on this filtered signal. Rearranging inequality 1, the voter reelects if and only if ϕ(g1 + ω1 − θ) πC (1 − πI ) . ≥ ϕ(g1 + ω1 − θ) 1 − πC πI (2) It can be shown that the left-hand side of this inequality is strictly increasing and continuous in g1 + ω1 . So, there is a g1 + ω1 such that (1) holds with equality. Denote that value of g1 + ω1 by r(πI , πC ) + ω1 . Observe that it depends on the priors πI , πC and the observed disaster intensity ω1 . It is useful to think about r(·) as the benchmark—in terms of governance outcomes—that the voter would adopt if there were no disaster (i.e., ω1 = 0). The voter then ratchets down that benchmark by exactly ω1 if there is a disaster. Now we can see how this model gives rise to the standard intuitions, by showing analogues of Wolfers’s (2002) claim that “events unrelated to a governor’s competence should have no effect on the voting decisions of rational agents” [p. 4, emphasis in original]. First, the voter updates her threshold for reelection to exactly offset the effect of disaster intensity on governance outcomes. To see this, observe that a slight rearrangement gives us the reelection threshold in terms of outcomes: The voter reelects if and only if g1 ≥ r(πI , πC ) − ω1 . So, if a disaster reduces government performance by 1, the voter reduces the reelection threshold by exactly 1. Second, the incumbent’s electoral fortunes do not depend on the presence or magnitude of a disaster. The probability the incumbent is reelected is the probability that g1 ≥ r(πI , πC ) − ω1 . That is, the probability the incumbent is reelection is Pr(θI − ω1 + 1 ≥ r(πI , πC ) − ω1 ) = Pr(θI + 1 ≥ r(πI , πC )). This is a constant function of ω1 , so incumbent electoral fortunes are indeed constant in disaster intensity. The conclusion is consistent with the standard intuition. Given these results and intuitions, the example we described in the introduction is perhaps surprising. In that example, disasters did effect incumbent electoral fortunes. The key is that, there, disaster intensity provided the voter with information about the incumbent’s type: When there was no disaster (ω1 = 0), the voter had no information about preparedness. But when there was a disaster (ω1 > 0), the voter had information about preparedness. By contrast, in this additive model, the intensity of the disaster has no effect on the amount of 4 information available to the voter about the incumbent. Crucially, we will show that, whenever type and disaster intensity are not additively separable in determining governance outcomes, the disaster intensity affects how informative governance outcomes are about type. So, quite generally, if voters are rational, then reelection thresholds will not exactly offset the effect of observable shocks and, so, incumbent electoral fortunes will depend on those shocks. In this sense, the additive benchmark that the literature has focused on is knife-edged. 2 The General Model To formally discuss the effect of shocks on the amount of information available to the voter about the incumbent, we need to generalize the model discussed in Section 1. We do so in this section, where we allow for technologies in which the Incumbent’s type and the disaster intensity are not additively separable. 2.1 Set Up There is an Incumbent (I), a Challenger (C), and a Voter (V ). We refer to each Politician (P ) as “she” and the Voter as “he.” In each of two governance periods, the Voter receives a governance outcome that depends on the type of the Politician in office, an observable level of natural disaster, and an unobservable, idiosyncratic shock. There is an election between the governance periods. The set of types is Θ = {θ, θ}, where θ ≥ 0 is the bad type and θ > θ is the good type. Write πP ∈ (0, 1) for the probability that Politician P is type θ. These probabilities are commonly understood by the players. We will say that the Incumbent enters the first governance period ahead if πI > πC and enters the first governance period behind if πI < πC . The intensity of the natural disaster in period t is ωt ∈ Ω, where Ω is a nonempty, open interval in R. Each ωt is the realization of a random variable that is independent of the Politician’s ability. (The particular distribution will not be relevant.) The governance outcome in a period is a function of a production technology and a random shock. The production function f : Θ × Ω → R is strictly increasing in type (θ) and is strictly decreasing in disaster intensity (ω). We will be interested in whether natural disasters amplify or mute the effect of type on governance outcomes. Definition 2.1. (i) Disasters amplify the effect of type on governance outcomes if, for all ω 0 > ω, f (θ, ω 0 ) − f (θ, ω 0 ) > f (θ, ω) − f (θ, ω). (ii) Disasters mute the effect of type on governance outcomes if, for all ω 0 > ω, f (θ, ω 0 ) − f (θ, ω 0 ) < f (θ, ω) − f (θ, ω). 5 Notice, with additive production, f (θ, ω 0 )−f (θ, ω 0 ) = f (θ, ω)−f (θ, ω). Thus, in that case, disasters neither amplify nor mute the effect of type.4 If, in period t, the type of the Politician in office is θt , the disaster intensity is ωt , and the random shock is t , the governance outcome in that period is f (θt , ωt ) + t . Each t is the realization of a random variable. These random variables are independent of each other, of the Politicians’ abilities, and of the disaster intensities. They are distributed according to an absolutely continuous CDF, Φ, with a continuously differentiable PDF, φ. This distribution satisfies three additional requirements: First, for each x > x0 ≥ 0, the associated likelihood ratio defined by g 7→ φ (g − x) φ (g − x0 ) is onto with non-zero derivatives. Second, the distribution satisfies the (strict) monotone likelihood ratio property (MLRP) relative to all possible realizations of production: If x > x0 , then the associated likelihood ratio φ (g − x) φ (g − x0 ) is strictly increasing in g. Third, the PDF is symmetric: for each x ∈ R, φ(x) = φ(−x). Prior to the game being played, Nature determines the realizations of each Politician’s type and of the random shocks (in all periods). These realizations are not observed by any of the players. Figure 2.1 depicts the timeline: In the initial governance period, the Voter observes the disaster intensity and the governance outcome. This leads to the electoral stage, in which the Voter chooses to reelect the Incumbent or replace her with a Challenger. The winner of the election is the Politician in office in the second governance period. Again, the Voter observes the disaster intensity and the governance outcome, g2 . 1st Governance Period Election 2nd Governance Period Figure 2.1. Timeline The Voter’s payoffs are the sum of governance outcomes in the two periods. 2.2 Some Comments Before turning to the analysis, we reflect on several features of the model. First, notice that we are working with a “pure selection” model. That is, the politician in office cannot influence the governance outcome by taking (potentially costly) actions. That outcome is determined entirely by the politician’s type, the disaster, and the shock. This is clearly an Definition 2.1 is a global requirement, i.e., requiring the function satisfy a condition for all ω 0 > ω. We can relax this requirement to instead consider the following localized definition: Say disasters amplify the effect of type at ω 00 > ω 0 if, for all ω ∈ (ω 0 , ω 00 ), f (θ, ω 00 ) − f (θ, ω 00 ) > f (θ, ω) − f (θ, ω) > f (θ, ω 0 ) − f (θ, ω 0 ). And, analogously, for the mutes case. 4 6 abstraction from reality. And it might seem a consequential one, since an important empirical literature argues that good responses to disasters mitigate the effect of those disasters on incumbent electoral fortunes (Healy and Malhotra, 2010; Bechtel and Hainmueller, 2011; Gasper and Reeves, 2011; Cole, Healy and Werker, 2012). Why do we abstract away from incumbent actions? In this paper, our goal is to explore what the model has to say about what the empirical literature can say about Voter competence. A wellknown feature of agency models of elections—even those where politicians take actions—is that, if voters are uncertain about the politician’s type, then at the time of the election, rational voters are concerned only with “selecting good types,” i.e., with electing the politician who is expected to perform best in the future (Fearon, 1999; Besley, 2006). Allowing politicians to take actions can complicate the analysis of the voters’ inference problem (Ashworth, Bueno de Mesquita and Friedenberg, 2015), but it does not change the logic of voter behavior in a fundamental way. As such, for the questions that are of concern to us, a pure learning model simplifies the analysis with no loss of insight. Section 5 will return to this issue. There we discuss what our analysis has to say about the empirical literature on the relationship between incumbent electoral fortunes and disaster response. Second, when we move away from the additive model, disaster will either amplify or mute the effect of type. As a consequence, it is important to understand when disasters may amplify or mute the effect of type. Substantively, if disasters amplify the effect of type then incumbent competence matters more for governance outcomes, when disasters are large. Suppose, as in the example in the Introduction, that good types are better at maintaining infrastructure. In normal times, it is hard to tell how well infrastructure has been maintained. But severe weather events will reveal weaknesses in the infrastructure—levees will hold or break, for instance. In this case, disasters amplify the effect of type. By contrast, if disasters mute the effect of type, incumbent competence matters less for governance outcomes, when disasters are large. For instance, suppose that good types are better at attracting investment. In normal times, good types will oversee better economic performance. But a natural disaster might stop investment no matter the type of the incumbent. In this case, disasters mute the effect of type. These examples also provide a link to how informative governance outcomes are about the Incumbent’s type. When disasters amplify (resp. mute) the effect of type, the difference in expected performance between competent and incompetent types is larger (resp. smaller) following larger disasters. Thus, during large disasters, it is easier (resp. harder) for voters to infer the incumbent’s competence from noisy governance outcomes. (We formalize this link in Section 3 below.) Third, we describe the Incumbent as ahead if πI > πC and behind if πI < πC . There are two reasons this language makes sense. First, if the Incumbent is ahead then ex ante she is more likely to win than to lose (and the opposite for an Incumbent who is behind). Second, if the Incumbent is ahead (resp. behind), then absent any new information, the Incumbent wins (resp. loses) reelection. 7 2.3 Analysis As in the additive model, the Voter adopts a cutoff rule in the space of governance outcomes. We can characterize this cutoff with a condition analogous to Equation 2. The Voter reelects the Incumbent if the first-period governance outcome, g1 , is greater than or equal to the (unique) ĝ(ω1 ) given by: φ(ĝ(ω1 ) − f (θ, ω1 )) 1 − πI πC . = φ(ĝ(ω1 ) − f (θ, ω1 )) πI 1 − πC Notice, ĝ(ω1 ) corresponds to r(πI , πC ) − ω1 from the additively separable model in Section 1. For the analysis of Voter behavior, it will be convenient to work with the loglikelihood ratio, `(g, ω1 ) ≡ log φ(g − f (θ, ω1 )) . φ(g − f (θ, ω1 )) We can equivalently define ĝ(ω1 ) as `(ĝ(ω1 ), ω1 ) = β(πI , πC ), where 1 − πI πC β(πI , πC ) ≡ log . πI 1 − πC Observe that β(πI , πC ) < 0 if the Incumbent is ahead and β(πI , πC ) > 0 if the Incumbent is behind. Finally, we calculate the ex-ante probability the Incumbent wins reelection, given a disaster intensity ω. The CDF of governance outcomes is given by Γ(g; ω) = πI Φ(g − f (θ, ω)) + (1 − πI )Φ(g − f (θ, ω)). Since the Voter adopts ĝ(ω) as the benchmark for reelection, the probability that the Incumbent is reelected (given ω) is 1 − Γ(ĝ(ω); ω). 3 Voter Behavior Section 1 focused on the case of additively separable production. In that model, the Voter’s reelection threshold was monotonic in disaster intensity: The worse the disaster, the lower the reelection threshold. This is intuitive. Because the production function is decreasing in disaster intensity, an outcome that is sufficiently bad to merit replacement when there is a minor disaster, might well lead to reelection following a larger disaster. Indeed, in the additive model, something even stronger is true: An increase in disaster intensity changes the distribution of outcomes. In a natural sense, the Voter’s reelection threshold changes in a way that exactly offsets the effect of this change in the distribution of outcomes. This fact appears to fit with the intuition motivating much of the empirical literature. As such, it will be useful to be more precise about this idea. For any given level of disaster intensity, ω, there is an outcome that such that the Voter’s posterior belief about the Incumbent’s type is equal to his prior belief about the Incumbent’s type. 8 Refer to this outcome, written ν(ω), as the neutral news outcome. Outcomes that are above (resp. below) the neutral news outcome raise (resp. lower) the Voter’s belief about the Incumbent’s type.5 We refer to such outcomes as good news (resp. bad news) outcomes. Because, at the neutral news outcome, the posterior probability that the Incumbent is a good type equals the prior probability, we have φ(ν(ω) − f (θ, ω)) = φ(ν(ω) − f (θ, ω)). (3) (This follows from Bayes’ rule.) That is, at the neutral news outcome, the likelihood of the outcome given a good type equals the likelihood of the outcome given a bad type. Observe that the symmetry of φ implies that we can write the neutral news outcome as: ν(ω) = f (θ, ω) + f (θ, ω) . 2 An implication of this explicit solution is that the neutral news outcome shifts monotonically with disaster intensity—the more intense the disaster, the lower the neutral news outcome. In the case of the additive production function in Section 1, the neutral news outcome is ν(ω) = θ+θ 2 − ω. And the reelection threshold is ĝ(ω) = r(πI , πC ) − ω. Thus, if the disaster intensity increases from ω to ω 0 , the Voter’s threshold decreases by the same amount as the neutral news signal, namely, by |ω 0 − ω|. Definition 3.1. Let ω 0 > ω. Voter behavior exactly offsets the effect of disasters if: ĝ(ω 0 ) = ĝ(ω) + ν(ω 0 ) − ν(ω). Voter behavior more than offsets the effect of disasters if the left-hand side is less than the righthand side and less than offsets the effect of disasters if the left-hand side is greater than the right-hand side. In the additive model, Voter behavior exactly offsets the effect of disasters. This fact will not hold, quite generally, beyond the additive model. Moreover, without additive separability, the Voter’s reelection threshold need not be monotonic with disaster intensity—larger disasters need not lead the Voter to lower his benchmark for reelection. The key is that, for any production function that is not additively separable, there is a second effect at work. Changes to disaster intensity change how informative the outcome is about the Incumbent’s type (in the sense of Blackwell, 1951). To illustrate these two effects, fix a level of disaster intensity ω. Each type θ generates a distribution on outcomes, i.e., the probability of an outcome g = f (θ, ω) + is the probability that = g − f (θ, ω). The Voter’s challenge is to figure out how likely it is that a particular governance outcome came from the distribution associated with the good type (φ(g − f (θ, ω))) versus the distribution associated with the bad type (φ(g − f (θ, ω))). 5 This follows from the MLRP. 9 f (θ , ω ) −4 ν (ω ) f (θ , ω ) 4 2ι(ω ) −4 f (θ , ω ʹ) ν (ω ʹ) f (θ , ω ʹ) 4 2ι(ω ʹ) Figure 3.1. Amplifies: Effect of Increase in Disaster on Conditional Distributions Refer to Figure 3.1. The top picture depicts the densities of outcomes associated with the good type and the densities of outcomes associated with the bad type when the level of disaster intensity is ω and the bottom picture depicts the densities of outcomes when the disaster intensity is ω 0 > ω. In each picture, the density to the left corresponds to the bad type and the density to the right corresponds to the good type. Note, because the level of disaster intensity is higher under ω 0 versus ω, the mean of each density is lower under ω 0 versus ω. As a consequence, the neutral news outcome, is lower under ω 0 versus ω. (Equation (3) says that the neutral news outcome is found at the intersection of these two densities.) Now let’s turn to the informational effect. Figure 3.1 was drawn for the case when disasters amplify the effect of type. As a consequence, the distance between the mean of the two densities is larger for ω 0 versus ω. This increased distance between the two densities corresponds to an improvement in Voter information—informally, when the means of the densities are further apart, it is easier for the Voter to determine which density a given signal came from. It will be convenient to measure this informativeness effect as: ι(ω) = f (θ, ω) − f (θ, ω) . 2 10 LHgL {Hg, ΩL 0 Β HΠI ,ΠC L g g̀HΩL ΝHΩ'L ΝHΩL g Figure 3.2. How Voter behavior responds to a shift in expected performance that ignores the effect on informativeness. (Details on parameters used are in Appendix B.1.) By Theorem 3.1 of Ashworth, Bueno de Mesquita and Friedenberg (2015), if ω 0 > ω and ι(ω 0 ) > ι(ω), then outcomes are more informative (in the sense of Blackwell, 1951) about the incumbent’s type when the disaster intensity is ω 0 then when the disaster intensity is ω. It’s easy to see that disasters amplify (resp. mute) the effect of type if and only if ι(·) is increasing (resp. decreasing) in disaster intensity. That is, governance outcomes are more informative (resp. less informative) to the Voter following larger disasters if disasters amplify (resp. mute) the effect of type. Observe that: f (θ, ω) = ν(ω) + ι(ω) and f (θ, ω) = ν(ω) − ι(ω). As a consequence, for each ω ∈ Ω, we can express the loglikelihood ratio in terms of the variables (g, ν(ω), ι(ω)), `(g, ω) = log φ(g − ν(ω) − ι(ω)) . φ(g − ν(ω) + ι(ω)) This will allow us to separately analyze how changing the disaster intensity changes the Voter’s behavior via a change in the neutral news outcome (ν(ω)) and a change in informativeness (ι(ω)). Neutral News Increasing disaster intensity from ω to ω 0 decreases the neutral news outcome from ν(ω) to ν(ω 0 ). To understand how this effect impacts the reelection rule, let’s consider a thought experiment in which there is a change to the neutral news outcome with no corresponding change in information. To do so, define a new function: L̃(g) = log φ(g − ν(ω 0 ) − ι(ω)) . φ(g − ν(ω 0 ) + ι(ω)) This function is obtained from `(·, ω) by decreasing the neutral news outcome from ν(ω) to ν(ω 0 ), but leaving ι(ω) fixed. That is, this function is obtained by shifting `(·, ω) up. (See Lemma B.1, which shows that, for any given g, L̃(g) < `(g, ω).) 11 {Hg, Ω'L LHgL {Hg, ΩL 0 Β HΠI ,ΠC L g g̀HΩ'L g̀HΩL ΝHΩ'L ΝHΩL g Figure 3.3. How Voter behavior responds to a shift in expected performance and informativeness when disasters amplify the effect of type. (Details on parameters used are in Appendix B.1.) This upward shift is depicted in Figure 3.2. The dark solid line is the function `(·, ω); the fact that it is increasing follows from the MLRP. The light solid line is the function L̃(·).6 Let us point to several additional features: First, the neutral news outcome, ν(ω), is the outcome where the loglikelihoood ratio `(·, ω) equals zero. Second, the reelection threshold, ĝ(ω), is the outcome where the loglikelihood ratio `(·, ω) equals β(πI , πC ). This figure is drawn for a case where the Incumbent is ahead—i.e., ex-ante the Incumbent is more likely than the Challenger to be a good type. In this case β(πI , πC ) < 0 and, as a consequence, the reelection threshold, ĝ(ω), is smaller than the neutral news outcome, ν(ω). Third, because L̃(·) lies above `(·, ω), L̃(·) intersects β(πI , πC ) at a governance outcome g̃ that is lower than ĝ(ω). This last fact reflects the intuition with which we began: The Voter wants to “filter out” the mechanical effect of the shock on the expected outcome—i.e., he doesn’t blame the Incumbent for events outside of her control. Because the neutral news outcome is worse (and there is no change in informativness) the Voter is willing to reelect an Incumbent at a lower level of performance—i.e., use a more lax reelection standard. Figure 3.2 corresponds to this thought experiment of changing the neutral news outcome with no corresponding information change. The fact that g̃ < ĝ(ω) corresponds to the the standard intuition. And, in fact, the difference between ĝ(ω) and g̃ is exactly the change in he neutral news outcome, i.e., ĝ(ω) − g̃ = ν(ω) − ν(ω 0 ). (See Lemma B.1.7 ) Informativeness Now let’s add the informativeness effect. To do so, begin with the function L̃ and allow the level of informativeness to change, going from ι(ω) to ι(ω 0 ). Figure 3.3 depicts the case where disasters amplify the effect of type, so that more intense disasters are associated with increased informativeness. As above, the neutral news outcome ν(ω 0 ) is the outcome where the loglikilihood ratio `(·, ω 0 ) equals 0. That is, `(·, ω 0 ) intersects 0 at ν(ω 0 ). Notice, the loglikelihood ratio `(·, ω 0 ) 6 In the figure, the two functions are parellel lines. This is true when φ is a normal density, but not true more generally. Our argument will not rely on this feature of the graphs. 7 Observe that ν(ω 0 ) is the point at which L̃(ν(ω 0 )) = 0: By symmetry of φ, L̃(ν(ω 0 )) = log φ(−ι(ω)) = log 1. φ(ι(ω)) 12 intersects the function L̃(·) at this neutral news outcome. This is because the two functions differ only in their level of informativeness—holding the neutral news outcome fixed. Increasing informativeness is equivalent to a given increase in the governance outcome inducing a larger increase in the Voter’s loglikelihood ratio. Geometrically this means that the graph of the loglikelihood ratio becomes steeper—specifically the graph of `(·, ω 0 ) is steeper than the graphs of both `(·, ω) and L̃(·). As a consequence, `(·, ω 0 ) is a rotation of L̃ around the point ν(ω 0 ). How does this rotation affect the reelection threshold? Without the informativeness effect, the Voter would have been indifferent between the Incumbent and Challenger at the governance outcome g̃. In Figure 3.3, the Incumbent is ahead and so g̃ is less than ν(ω 0 ); so, g̃ is a bad news outcome. But, an increase in informativeness makes bad news worse. So, increasing informativeness lowers the Voter’s expectation of the Incumbent’s type, given an observation of g̃. Putting the two effects together, the Voter strictly prefers the Challenger to the Incumbent when the governance outcome is g̃. This is why, in Figure 3.3, `(g̃, ω 0 ) < β(πI , πC ). Since `(ĝ(ω 0 ), ω 0 ) = β(πI , πC ), it follows from the MLRP that ĝ(ω 0 ) > g̃. Substantively, when the Incumbent is ahead, the Voter is indifferent between the Incumbent and the Challenger when the news about the Incumbent is somewhat bad. When the governance outcome becomes more informative, this bad news is worse, so the Voter is no longer indifferent—he strictly prefers the Challenger. Hence, the Voter shifts to a more stringent reelection rule. Overall Effect In sum, when the Incumbent is ahead and disasters amplify the effect of type, the informativeness effect and neutral news effect work in opposite directions: the latter serves to lower the Voter’s benchmark for reelection and the former serves to increase it. The overall effect is that ĝ(ω 0 ) > g̃ and so Voter behavior less than offsets the effect of a disaster. More surprisingly, it is possible for the informativeness effect to swamp the neutral news effect. That is, sometimes the Voter holds the Incumbent to a more stringent reelection threshold following a more intense disaster—i.e., ĝ(ω 0 ) > ĝ(ω). Such a case is illustrated in Figure 3.4. (See Appendix B.1 for numerical examples that generate the pictures in Figures 3.3-3.4.) This counterintuitive result causes problems relative to standard intuitions about rational voter behavior: Not only is it incorrect to assert that rational Voter behavior exactly offsets the effect of disasters, a rational voter’s reelection threshold need not even move in the opposite direction from disaster intensity. The situation is reversed when the Incumbent is behind. In that case, g̃ > ν(ω 0 ) and so g̃ is a good news outcome. An increase in informativeness makes good news better news. As a consequence, at g̃, an increase in informativeness makes it more likely that the Incumbent is a good type and so further lowers the Voter’s benchmark. That is, both the neutral news and the informativeness effects work in the same direction. As a consequence, Voter behavior more than offsets the effect of disasters. Now consider the case where disasters mute the effect of type on outcomes. Now, when the level 13 {Hg, Ω'L LHgL {Hg, ΩL 0 Β HΠI ,ΠC L g g̀HΩL g̀HΩ'L ΝHΩ'L g ΝHΩL Figure 3.4. A case where disasters amplify the effect of type and the informational effect dominates the expected performance effects, so the Voter uses a more stringent reelection rule following a more intense disaster. (Details on parameters used are in Appendix B.1.) LHgL {Hg, Ω'L 0 {Hg, ΩL Β HΠI ,ΠC L g̀HΩ'L g g̀HΩL ΝHΩ'L ΝHΩL g Figure 3.5. How Voter behavior responds to a shift in expected performance and informativeness when disasters mute the effect of type. (Details on parameters used are in Appendix B.1.) of disaster intensity rises, there is a decrease in informativeness. This implies that the likelihood ratio `(·, ω 0 ) is a rotation of L̃ at ν(ω 0 ) that is shallower than L̃. This is illustrated in Figure 3.5, which shows that the implications are reversed: If the Incumbent is ahead, g̃ is a bad news outcome—an increase in disaster intensity lowers informativeness and, in turn, tempers bad news. Hence, both the neutral news and informational effects serve to lower the Voter’s benchmark. The overall result is that Voter behavior more than offsets the effect of disasters. By contrast, if the Incumbent is behind, g̃ is good news and the two effects work in opposite directions. In that case, Voter behavior less than offsets the effect of disasters. (Again, the counterintuitive shift in the reelection threshold is possible.) Table 3.1 summarizes these results, which are formalized as Proposition B.1 in the Appendix. When πI = πC , Voter behavior exactly offsets the effect of disaster. However, when πI 6= πC , Voter behavior either more than or less than offsets the effect of disaster. 14 Amplifies Mutes Ahead Less than offsets More than offsets Behind More than offsets Less than offsets Table 3.1. How Voter Behavior Responds to Disasters 4 Electoral Fortunes In this section, we investigate how a change in the level of disaster intensity effects the Incumbent’s electoral fortunes. In the additive model, we saw that an increase in the level of disaster intensity did not alter the probability of reelection. We will now see that, when disasters provide information about the Incumbent’s type, the Incumbent’s electoral fortunes may well be altered by the level of disaster intensity. In particular, the Incumbent may well be hurt by (or benefit from) the disaster. The probability of reelection is determined by two things: the reelection rule the voter uses (ĝ(·)) and the distribution of governance outcomes Φ(·). Recall that, for a given disaster intensity ω, the CDF of governance outcomes is Γ(g; ω) = πI Φ(g − ν(ω) − ι(ω)) + (1 − πI )Φ(g − ν(ω) + ι(ω)). If the Voter adopts ĝ(ω) as the benchmark for reelection, the probability that the Incumbent is reelected (given ω) is 1 − Γ(ĝ(ω); ω). Changes to disaster intensity affect the probability of reelection by changing both the reelection rule and the distribution of governance outcomes. As we saw in the previous section, a change in disaster intensity has two effects on the reelection rule. The first is a neutral news effect—the reelection rule changes to offset the change in productivity (holding fixed the level of informativeness). The second is an informativeness effect—the reelection rule changes to incorporate the change in information about the Incumbent’s type. A change in disaster intensity also affects the distribution of outcomes through two analogous effects. Separating these, along with the two effects on the reelection rule already discussed, will allow us to clarify the overall effect of disaster intensity on electoral fortunes. As we did in the previous section, we will start by studying the neutral news effect. We will see that the effect of a change in neutral news (absent any information effect) on the reelection rule exactly offsets the change in the distribution of outcomes associated with the neutral news shift. That is, if all a change in disaster intensity did was shift the neutral news outcome, there would be no effect on electoral fortunes. We will then turn to the effect of disasters on information. An increase in informativeness is associated with a shift of probability mass toward the tails of the distribution of governance outcomes. (We will be formal about what this means below.) That is, an increase in disaster 15 intensity that increases informativeness makes both very good and very bad outcomes more likely (though not necessarily symmetrically). Even if the reelection rule did not change, this change in the distribution of outcomes would already have an effect of electoral fortunes. Of course, as we’ve already seen, the reelection rule also changes. To understand the overall impact of disaster intensity on electoral fortunes, we put all of these effects together. Neutral News Let’s start with the offsetting neutral news effects. Toward that end, as we did above, consider a hypothetical situation in which the Voter ignores the change in informativeness—holding it fixed at ι(ω)—and instead simply filters out the effect of a disaster on the likelihood of observing any given level of public goods. In this case, the relevant CDF of governance outcomes is Γ̃(g) = πI Φ(g − ν(ω 0 ) − ι(ω)) + (1 − πI )Φ(g − ν(ω 0 ) + ι(ω)). Notice that, for any g, Γ(g; ω) > Γ̃(g), so that this new CDF can be seen as a downward shift of the original CDF on governance outcomes Γ(·, ω). That is, the distribution of outcomes gets worse when disasters are more intense. (Formally, Γ(·, ω) first-order stochastically dominates Γ̃.) Because the Voter filters out the effect of a disaster on the likelihood of observing any given level of public goods (and ignores the informational effect), he adopts a benchmark for reelection that exactly offsets the effect of a disaster. We found this benchmark in the previous section, it is g̃ = ĝ(ω) − ν(ω) + ν(ω 0 ) (instead of the actual benchmark ĝ(ω 0 )). Overall, these two changes—in the distribution of outcomes and the reelection rule—leave the probability of reelection unchanged, i.e., 1 − Γ̃(g̃) = 1 − Γ(ĝ(ω); ω). Notice that the analysis of the neutral news effect corresponds exactly to the additive model. There, changing the level of disaster intensity leaves the level of informativeness unchanged. So, the new distribution of governance outcomes, Γ(g; ω 0 ), corresponds exactly to the distribution Γ̃. Moreover, as we have seen, in that case, the Voter’s actual new benchmark ĝ(ω 0 ) exactly offsets the effect of disaster (i.e., ĝ(ω 0 ) = g̃). Thus, 1 − Γ̃(g̃) = 1 − Γ(ĝ(ω 0 ); ω 0 ). That said, when changing the level of disaster changes the level of informativeness, these two properties no longer hold: The new actual distribution Γ(·, ω 0 ) is typically different from the ‘shifted down’ distribution Γ̃ and the new benchmark for reelection ĝ(ω 0 ) typically does not exactly offset the effect of disaster. Informativeness Given the analysis above, any effect of disasters intensity on electoral fortunes must work through informativeness. We’ve already seen how the informativeness effect changes the reelection rule. So, let’s now consider the effect of informativeness on the distribution of governance outcomes. To do so, we begin with the hypothetical distribution Γ̃ and change only the level of informativeness from ι(ω) to ι(ω 0 ); this leads to the new distribution Γ(·; ω 0 ). Figure 4.1 depicts the relationship between the new distribution Γ(·; ω 0 ) and the neutral news distribution Γ̃(·). The new 16 G GH × ; Ω'L G GH × ; Ω'L g g g (a) Disasters Amplify the Effect of Type g (b) Disasters Mute the Effect of Type Figure 4.1. How Informativeness Changes the Distribution of Outcomes distribution is a rotation of Γ̃ around some point, namely g. (The particular point depends on both ω and ω 0 .) The nature of the rotation depends on whether disasters amplify vs. mute the effect of type. Figure 4.1a focuses on the case of amplifies. There, when the governance outcome g is below g, the new true distribution lies above the neutral news distribution—that is, if g > g, then Γ(g, ω 0 ) > Γ̃(g). Figure 4.1b focuses on the case of mutes. There, when the governance outcome g is below g, the new true distribution lies below the neutral news distribution—that is, if g > g, then Γ̃(g) > Γ(g, ω 0 ). Matters are reversed for g > g. This single-crossing relationship between these CDFs formalizes the idea, discussed at the beginning of this section, that an increase in informativeness is associated with a shift of probability mass toward the tails of the distribution of governance outcomes. (Because g need not equal 1/2, this shift need not be symmetric.) This change in the distribution of outcomes has a direct effect on the probability of reelection. That is, it would directly influence the probability of reelection, even if the Voter were to adopt a threshold for reelection that exactly offsets the effect of disaster. To see this, focus on the case where disasters amplify the effect of type and recall that g̃ is the outcome level that exactly offsets the effect of disaster. If g̃ were below g, the change in the distribution of outcomes would result in a lower the probability of reelection (i.e., 1 − Γ(g̃; ω 0 ) < 1 − Γ̃(g̃) = 1 − Γ(g; ω)) and if g̃ were above g the change in the distribution of outcomes would raise the probability of reelection (i.e., 1 − Γ(g̃; ω 0 ) > 1 − Γ̃(g̃) = 1 − Γ(g; ω)). This fact captures an important intuition. When the Incumbent is way ahead, g̃ is below g. Such an incumbent will only be replaced if there is a very bad governance outcome. In this case, an increase in informativeness lowers the probability of reelection because moving probability mass to the tails increases the likelihood of such an outcome. (It also increases the likelihood of a very good outcome, but that doesn’t matter for such an incumbent.) By contrast, when the Incumbent is way behind, g̃ is above g. In this case an increase in informativeness raises the probability of reelection because such an Incumbent will only be reelected if the governance outcome is very good. Of course, the analysis above does not tell the whole story. Disaster intensity has a second effect on the probability of reelection: the Voter’s benchmark for reelection does not exactly offset 17 G GH × ; Ω'L GHg̀HΩ'L;Ω'L GHg ;Ω'L GHg L g g̀HΩ'L g g Figure 4.2. Amplify and g̃ < ĝ(ω 0 ) < g: Probability of Reelection Falls the effect of disaster. That is, ĝ(ω 0 ) 6= g̃. We will see that whether this effect serves to increase or decrease the probability of reelection depends on whether the benchmark more than or less than offsets the effect of disaster. To illustrate this indirect effect, focus on the case where disasters amplify the effect of type and g > g̃. Figure 4.2 depicts the case where the benchmark less than offsets the effect of disaster, i.e., ĝ(ω 0 ) > g̃. In this case, Γ(ĝ(ω 0 ); ω 0 ) > Γ(g̃; ω 0 ) > Γ̃(g̃). Here, the direct and indirect effects work in the same direction and so the probability that the Incumbent is reelected falls. Overall Effect Figure 4.3 depicts the case where the benchmark more than offsets the effect of disaster, i.e., g̃ > ĝ(ω 0 ). In this case, the indirect effect causes the Voter to lower his benchmark for reelection and thereby increase the probability that the Incumbent is reelection. Thus, the direct and indirect effect work in opposite directions. Figure 4.3a depicts the case where the direct effect dominates the indirect effect (causing the probability of reelection to fall) and Figure 4.3b depicts the case where the indirect effect dominates the direct effect (causing the probability of reelection to rise). Table 4.1 summarizes when the Incumbent’s fortunes will increase versus decrease, based on whether (a) disasters amplify or mute the effect of type, (b) Voter behavior more than or less than offsets the effect of disasters, and (c) the outcome that exactly offsets the effect of disaster lies above or below the rotation point. In boxes labeled with [?], the direct and indirect effects compete, and so the Incumbent’s electoral fortunates can either increase or decrease. [NOTE: Comment from Amanda to Ethan and Scott: This next discussion seems very economicsy to me. What we really want to express is ”and this is important from the perspective of getting the 18 G G GH × ; Ω'L GH × ; Ω'L GHg ;Ω'L GHg L GHg ;Ω'L GHg̀HΩ'L;Ω'L GHg L GHg̀HΩ'L;Ω'L g̀HΩ'L g g g g g̀HΩ'L (a) Probability of Reelection Falls g g (b) Probability of Reelection Rises Figure 4.3. Amplifies and ĝ(ω 0 ) < g̃ < g Amplifies Mutes g̃ > g ? Reelec Prob ↓ g̃ < g Reelec Prob ↓ ? Amplifies Mutes g̃ > g Reelec Prob ↑ ? g̃ < g ? Reelec Prob ↑ ĝ(ω 0 ) > g̃ g̃ > ĝ(ω 0 ) Table 4.1. Change in Electoral Fortunes empirical identification correct.” This is not coming across.] While the first of these factors, i.e., (a), is a primitive of the model, the latter two, i.e., (b)-(c), are endogenous aspects of the model. We have already seen that (b) can be determined based on (a) and whether the Incumbent is ahead or behind. (Refer to Table 3.1.) In fact, we can likewise determine whether (c) holds based on primitives of the model. Refer to Figure 4.4, which depicts the situation where disasters amplify the effect of type. The shaded region indicates the the pairs (πI , πC ) for which the reelection probability rises and the nonshaded regions correspond to the pairs for which the reelection probability falls. When (πI , πC ) = ( 12 , 12 ), the neutral news outcome g̃ correspond exactly to the rotation point g and the Voter exactly offsets the effect of disaster. Thus, that case acts as the additive model and the Incumbent neither benefits from nor is hurt by the effect of disaster. But, for any pair (πI , πC ) 6= ( 12 , 12 ), the Voter either benefits from or is hurt by the effect of disaster. Notice that, when πC ≥ max{ 12 , πI }, the Incumbent’s electoral fortunes necessarily benefit from the effect of disaster. In that case, disasters increase the Incumbent’s probability of reelection through both the direct and indirect effects. Likewise, when min{ 12 , πI } ≥ πC disasters decrease the Incumbent’s probability of reelection through both the direct and indirect effects. Proposition C.1 in the Appendix shows these claims. Proposition C.7 in the Appendix shows analogous claims for the case when disasters mute the effect of type. The Appendix also explains why these primitives 19 ΠC 1 0.5 0.5 1 ΠI Figure 4.4. Amplifies: Shaded Reelection Probability Rises correspond to the boxes in Table 4.1. 5 Conclusion In their seminal paper, Achen and Bartels (2004, pp.7–8) write “To the extent that voters engage in sophisticated attributions of responsibility they should be entirely unresponsive to natural disasters, at least on average; to the extent that they engage in blind retrospection, they should exhibit ‘systematic attribution errors’.” This conclusion is reflective of a large empirical literature. That literature argues that, because natural disasters are outside the control of incumbents, their electoral fortunes will systematically suffer following natural disasters only if voters are irrational. This argument has, further, motivated a variety of new models of elections with non-rational voters (Callander and Wilson, 2008; Patty, 2006; Ashworth and Bueno de Mesquita, 2014; Diermeier and Li, 2013; Ortoleva and Snowberg, 2015; Levy and Razin, 2015a,b; Lockwood, 2015). Our results suggest that this interpretation is not warranted. We study a canonical model of voter learning. We have shown that, in all but knife-edged cases, rational voters behavior implies that disasters—and other shocks outside the control of policy makers—will affect incumbent electoral fortunes. There is a strand of empirical work which argues that incumbent electoral fortunes are helped by effective disaster response (Bechtel and Hainmueller, 2011; Healy and Malhotra, 2009, 2010). These findings are typically interpreted as evidence for voter rationality.8 In this paper, we abstract 8 Because disaster response is post-treatment, it may be endogenous to anticipated electoral competitiveness and so identification is often difficult. 20 away from Incumbent behavior (assuming that Incumbent’s cannot take any actions), so our work does not directly speak to this literature. Nonetheless, in light of several prominent models in the political agency literature, such conclusions should also be treated with caution. To understand why it will be useful to distinguish between two models. First, suppose all incumbents are known to have the same quality, but are differentiated based whether or not they are intrinsically motivated to work on behalf of voters (this is as in Myerson, 2006). In such a model, effort toward disaster response is direct evidence that a politician is more likely to be a good type and so, higher effort leads to a greater probability of reelection. In that case, the relationship between the disaster response and incumbent electoral fortunes is, in fact, consistent with the interpretation of the empirical literature. Now, instead, suppose all incumbents are known to have identical motivations, but are differentiated by how competent they are believed to be. In such a model, an incumbent who expects to win handily has weaker incentives to work hard than does an incumbent who expects a close election. Thus an increase in the prior probability that the incumbent is competent leads to both a greater probability of reelection and lower effort. Such a model can easily predict a negative relationship between incumbent effort and expected incumbent electoral fortunes. (An earlier version of this paper [CITE working paper], we showed this formally, in a model where governance outcomes are the sum of competence, effort, an observable shock, and an unobservable shock.) The comparison of these models highlights two points. First, assessing the rationality of voter responses to incumbent effort requires understanding which aspects of incumbent type voters are learning about, in their attempt to predict future performance. Second, regardless of the source of uncertainty or the predicted relationship between effort and incumbent electoral fortunes, rational voters do not follow a logic of reward and punishment. Instead, they vote based on their observations of past behavior and outcomes solely because such observations are informative about future performance. Taken together with the results in this paper, this raises challenges for identifying voter rationality. The empiricist would need to identify aspects of the learning environment that are currently ignored by empirical work. An alternate approach may be to identify voter irrationality instead of directly identifying voter rationality. As we’ven noted, a recent literature explores the implications of specific failures of rationality for equilibrium predictions in models of elections (Callander and Wilson, 2008; Patty, 2006; Ashworth and Bueno de Mesquita, 2014; Diermeier and Li, 2013; Ortoleva and Snowberg, 2015; Levy and Razin, 2015a,b; Lockwood, 2015). At this point, much of the literature has been concerned with normative questions—demonstrating that certain types of voter irrationality can improve democratic performance, at least sometimes. As the literature develops, it may produce predictions that distinguish equilibria with fully rational voters from equilibria with less rational voters. These can then be used to further guide the empirical literature. 21 A Preliminary Results In this Appendix, we report a preliminary result that will be useful in the subsequent analysis. Lemma A.1. Consider a function L : R × R+ → R+ given by L(x, δ) = φ(x − δ) . φ(x + δ) (i) L is increasing in δ if x > 0. (ii) L is constant in δ if x = 0. (iii) L is decreasing in δ if x < 0. The proof can be found in Ashworth, Bueno de Mesquita and Friedenberg (2015) (Lemma B2). B Supporting Material for Section 3 The following Proposition formalizes the results reported in Table 3.1: Proposition B.1. Fix ω 0 > ω. (i) Suppose disasters amplify the effect of type. Then πI ≥ πC if and only if ν(ω) − ν(ω 0 ) ≥ ĝ(ω) − ĝ(ω 0 ) and πC ≥ πI if and only if ĝ(ω) − ĝ(ω 0 ) ≥ ν(ω) − ν(ω 0 ). (ii) Suppose disasters neither amplify nor mute the effect of type. Then, for any πI and πC , ν(ω) − ν(ω 0 ) = ĝ(ω) − ĝ(ω 0 ). (iii) Suppose disasters mute the effect of type. Then πI ≥ πC if and only if ĝ(ω) − ĝ(ω 0 ) ≥ ν(ω) − ν(ω 0 ) and πC ≥ πI if and only if ν(ω) − ν(ω 0 ) ≥ ĝ(ω) − ĝ(ω 0 ). Observe that the statement of the proposition has implications for strict inequalities: Suppose disasters amplify the effect of type and πI > πC . Then we can apply the fact that “πI ≥ πC ” and “not πC ≥ πI ” to obtain that ν(ω) − ν(ω 0 ) > ĝ(ω) − ĝ(ω 0 ). This Proposition follows from two Lemmata. The first formalizes the discussion surrounding Figure 3.2. Lemma B.1. Fix ω 0 > ω. (i) For each g, L̃(g) > `(g, ω). (ii) If L̃(g̃) = β(πI , πC ), then ĝ(ω) − g̃ = ν(ω) − ν(ω 0 ). Proof. Observe that, for each g, there exists some g 0 > g so that g = g 0 + (ν(ω 0 ) − ν(ω)). Since L̃(g) = log φ(g 0 − ν(ω) − ι(ω)) φ(g − ν(ω 0 ) − ι(ω)) = log = `(g 0 , ω) φ(g − ν(ω 0 ) + ι(ω)) φ(g 0 − ν(ω) + ι(ω)) 22 and `(g 0 , ω) > `(g, ω), it follows that L̃(g) > `(g, ω). If L̃(g̃) = β(πI , πC ), then log φ(g̃ − ν(ω 0 ) − ι(ω)) φ(ĝ(ω) − ν(ω) − ι(ω)) = log . φ(g̃ − ν(ω 0 ) + ι(ω)) φ(ĝ(ω) − ν(ω) + ι(ω)) By the strict MLRP, this implies that g̃ − ν(ω 0 ) = ĝ(ω) − ν(ω). The second lemma formalizes the “rotation” effect discussed surrounding Figures 3.3, 3.4, and 3.5. Lemma B.2. Fix ω 0 > ω. (i) Suppose disasters intensity the effect of type. Then g ≥ ν(ω 0 ) if and only if `(g, ω 0 ) ≥ L̃(g) and g ≤ ν(ω 0 ) if and only if `(g, ω 0 ) ≤ L̃(g). (ii) Suppose type and disaster mute the effect of type. Then g ≥ ν(ω 0 ) if and only if `(g, ω 0 ) ≤ L̃(g) and g ≤ ν(ω 0 ) if and only if `(g, ω 0 ) ≥ L̃(g). [NOTE: I changed the omegas to omega’s, since i think that was an error. Also, i turned it to iff statements about weak inequaliies since that implies the stronger statements.] Proof. Immediate from Lemma A.1. Note, again, that while these statements involve weak inequalities they have implications about strict inequalities. For instance, if disasters amplify the effect of type and g > ν(ω 0 ) then, using the fact that “not g ≤ ν(ω 0 )” it follows that `(g, ω 0 ) > L̃(g). Proof of Proposition B.1. We show the result when disasters amplify the effect of type and πI ≥ πC ; the other cases are analogous. Since πI ≥ πC , β(πI , πC ) ≤ 0. From this, ĝ(ω) < ν(ω) and Lemma B.2 implies that β = `(ĝ(ω), ω) < L̃(ĝ(ω)). This implies that the g̃ that solves L̃(g̃) = β satisfies g̃ < ĝ(ω). Thus ν(ω 0 ) − ν(ω) = ĝ(ω 0 ) − g̃ > ĝ(ω 0 ) − ĝ(ω). B.1 Computational Examples Figures 3.2–3.5 were drawn based on numerical examples. We now give details on the numerical examples, to illustrate that the effects depicted can indeed arise. [NOTE: Elaborate?] The examples have several features in common: In each, φ is the pdf of a standard normal random variable: φ(x) = √12π exp −x2 /2 . Also, in each πC = 21 , ω 0 = 21 , and ω 00 = 1. The 23 examples vary in the production function and the choice of πI . These choices are described in Table B.1. Figure 3.4 depicted the case where, when disaster intensity increased, the benchmark for reelection rose, i.e, ĝ(ω 00 ) > ĝ(ω 0 ). We show this fact analytically. Using the formula for the normal pdf, `(g, ω) = 2ι(ω) (g − ν(ω)) . So we can solve explicitly for ĝ: ĝ(ω) = β(πI , πC ) + ν(ω). 2ι(ω) Using the parameters in the middle column of Table B.1, we calculate ν(ω) = and β(πI , πC ) = −2. Then ĝ(ω 0 ) = − 2−3ω 2 , ι(ω) = ω 2, 10 15 > − = ĝ(ω 00 ). 4 4 Thus, as the disaster becomes more intense, the reelection threshold becomes more stringent. I [NOTE: Comment from Amanda: Is β correct? Since πC = .5, β = log( 1−π πI ). Using the table above, I get something different.] exp(2) From the File’s computation: 1− 1+exp(2) exp(2) 1+exp(2) 1 + exp(2) −1 exp(2) log(1 + exp(2)) − 2 − 1) If we want β = −2 then I log 1−π πI = −2 1−πI πI = exp(−2) 1 − πI = πI ∗ exp(−2) πI = 1 1+exp(−2) On the other hand, its not obvious to me from the calculations above that ĝ(ω) involves β = −2. ĝ(ω) = − f (θ, ω) f (θ, ω) πI 1 + ν(ω). ι(ω) 3.2 & 3.3 1−ω 1 − 4ω Figure 3.4 1−ω 1 − 2ω 3.5 4 − 2ω 1−ω 1 1+exp(−2.5) exp(2) 1+exp(2) 1 1+exp(−2.5) Table B.1. Parameters for the numerical examples. 24 ĝ(ω 0 ) = − 2 .5 + = −4 + (1/4) .5 2 ĝ(ω 00 ) = − C 1 − .5 = −1.5. .5 Proofs for Section 4 [NOTE: Can we find a better term than ”the election is not random”?] Say that the election is random if πI = πC = 12 . In what follows, we will restrict attention to elections that are not random. This Appendix will show the following results. Proposition C.1. Suppose disasters amplify the effect of type and the election is not random. (i) If πC ≥ max{ 12 , πI }, then 1 − Γ(ĝ(ω 0 ); ω 0 ) > 1 − Γ(ĝ(ω); ω). (ii) If min{ 21 , πI } ≥ πC , then 1 − Γ(ĝ(ω); ω) > 1 − Γ(ĝ(ω 0 ); ω 0 ). Proposition C.2. Suppose disasters mute the effect of type and the election is not random. (i) If min{πI , 21 } ≥ πC , then 1 − Γ(ĝ(ω 0 ); ω 0 ) > 1 − Γ(ĝ(ω); ω). (ii) If πC ≥ max{ 21 , πI }, then 1 − Γ(ĝ(ω); ω) > 1 − Γ(ĝ(ω 0 ); ω 0 ). We begin by analyzing how the change in informativeness—going form Γ̃ to Γ(·, ω 0 )—rotates the distribution of public goods. We then use this fact to conclude whether the reelection probability would increase or decrease, depending on where g̃ lies relative to the rotation point and whether the incumbent is ahead versus behind. Finally, we conclude by providing parameters of the model that pin down where g̃ lies relative to the rotation point. The Effect of Informativeness: Rotating the Distribution Γ̃ Write G(x; δ) = πI Φ(x − δ) + (1 − πI )Φ(x + δ). Observe that ∂G (x; δ) = −πI φ(x − δ) + (1 − πI )φ(x + δ). ∂δ Notice that Γ̃(g) = G(g − ν(ω 0 ), ι(ω)) and Γ(g; ω 0 ) = G(g − ν(ω 0 ), ι(ω 0 )). Thus, we will be interested in how the function G(x; δ) responds to changes in δ. Lemma C.1. Fix δ 0 6= δ. There is exactly one x such that G(x; δ 0 ) = G(x; δ). Moreover: (i) For δ 0 > δ: If x > x then G(x; δ 0 ) > G(x; δ) and if x > x then G(x; δ) > G(x; δ 0 ). 25 (ii) For δ > δ 0 : If x > x then G(x; δ) > G(x; δ 0 ) and if x > x then G(x; δ 0 ) > G(x; δ). The level x that solves G(x; δ 0 ) = G(x; δ) will typically depend on both δ and δ 0 . To show Lemma C.1, it will be useful to fix δ 0 6= δ and define auxiliary functions ∆(·) : R → R and Ξ(·) : R → R so that ∆(x) = G(x; δ 0 ) − G(x; δ) = πI Φ(x − δ 0 ) − Φ(x − δ) + (1 − πI ) Φ(x + δ 0 ) − Φ(x + δ) and Ξ(x) = − Φ(x − δ 0 ) − Φ(x − δ) . Φ(x + δ 0 ) − Φ(x + δ) We begin with the following observation, which follows directly from calculations: Lemma C.2. Suppose δ 0 > δ. (i) ∆(x) ≥ 0 if and only if 1−πI πI ≥ Ξ(x). (ii) ∆(x) ≤ 0 if and only if 1−πI πI ≤ Ξ(x). Lemma C.3. Suppose δ > δ 0 . (i) ∆(x) ≥ 0 if and only if 1−πI πI ≤ Ξ(x). (ii) ∆(x) ≤ 0 if and only if 1−πI πI ≥ Ξ(x). Lemma C.4. (i) If δ 0 > δ, the function Ξ(·) is strictly increasing in x. (ii) If δ > δ 0 , the function Ξ(·) is strictly decreasing in x. Proof. Observe that Ξ0 (x) = − (φ(x − δ 0 ) − φ(x − δ))(Φ(x + δ 0 ) − Φ(x + δ)) − (Φ(x − δ 0 ) − Φ(x − δ))(φ(x + δ 0 ) − φ(x + δ)) . (Φ(x + δ 0 ) − Φ(x + δ))2 We will show: δ 0 > δ implies that Ξ0 (x) > 0 for all x. Reversing the inequalities below gives the desired result when δ > δ 0 . Observe that Ξ0 (x) > 0 if and only if −(φ(x − δ 0 ) − φ(x − δ))(Φ(x + δ 0 ) − Φ(x + δ)) + (Φ(x − δ 0 ) − Φ(x − δ))(φ(x + δ 0 ) − φ(x + δ)) > 0. Since δ 0 > δ, Ξ0 (x) > 0 if and only if or if and only if φ(x + δ) − φ(x + δ 0 ) φ(x − δ 0 ) − φ(x − δ) > . Φ(x − δ 0 ) − Φ(x − δ) Φ(x + δ) − Φ(x + δ 0 ) 26 Applying Cauchy’s Mean Value Theorem, there exists c ∈ (x − δ 0 , x − δ) and c ∈ (x + δ, x + δ 0 ) such that φ(x − δ 0 ) − φ(x − δ) φ0 (c) = Φ(x − δ 0 ) − Φ(x − δ) φ(c) and φ(x + δ) − φ(x + δ 0 ) φ0 (c) = . Φ(x + δ) − Φ(x + δ 0 ) φ(c) Observe that c < c and φ is log-concave (see Lemma B.1 in Ashworth, Bueno de Mesquita and Friedenberg, 2015). So φ(x − δ 0 ) − φ(x − δ) φ0 (c) φ0 (c) φ(x + δ) − φ(x + δ 0 ) = > = , 0 Φ(x − δ ) − Φ(x − δ) φ(c) φ(c) Φ(x + δ) − Φ(x + δ 0 ) implying that Ξ0 (x) > 0. Lemma C.5. There exist some x so that ∆(x) = 0. Proof. Observe that ∆(x) = 0 if and only if Ξ(x) = 1−πI πI . So it suffices to show that Ξ(·) is onto. To do so, observe that Ξ(·) is continuous. We will show it is not bounded on R, from which it follows that the function is onto. We show this when δ 0 > δ; a symmetric argument applies for the reverse inequality. Observe that, by the Cauchy Mean Value Theorem, for any given x, there exists some δ̃(x) ∈ (δ, δ 0 ) so that Ξ(x) = φ(x − δ̃(x)) . φ(x + δ̃(x)) It follows from Lemma A.1 that, when x ≥ 0 (resp. x ≤ 0), Ξ(x) is greater than or equal to (resp. less than or equal to) φ(x − δ) . φ(x + δ) Now, using the fact that the likelihood ratio is unbounded, for any given K > 0, there exists some x(K) so that Ξ(x(K)) ≥ φ(x(K) − δ) ≥ K. φ(x(K) + δ) Likewise, for any given K < 0. Thus, Ξ(·) is unbounded. Proof of Lemma C.1. Immediate from Lemmata C.2-C.3-C.4-C.5. We can apply Lemma C.1 to the functions Γ̃(g) = G(g − ν(ω 0 ), ι(ω)) and Γ(g; ω 0 ) = G(g − ν(ω 0 ), ι(ω 0 )). Corollary C.1. Suppose disasters amplify the effect of type. If ω 0 > ω, then there exists some g(ω, ω 0 ) so that the following hold: (i) If g ≥ g(ω, ω 0 ), then Γ̃(g) ≥ Γ(g; ω 0 ). 27 (ii) If g ≤ g(ω, ω 0 ), then Γ̃(g) ≤ Γ(g; ω 0 ). Corollary C.2. Suppose disasters mute the effect of type. If ω 0 > ω, then there exists some g(ω, ω 0 ) so that the following hold: (i) If g ≥ g(ω, ω 0 ), then Γ̃(g) ≤ Γ(g; ω 0 ). (ii) If g ≤ g(ω, ω 0 ), then Γ̃(g) ≥ Γ(g; ω 0 ). Implications for Reelection Probabilities The following Lemmata follow immediately from Corollaries C.1-C.2, the analysis in the main text, and Proposition TK. Lemma C.6. Suppose that disasters amplify the effect of type. (i) If g̃ ≥ g(ω, ω 0 ) and πC ≥ πI with at least one strict inequality, then Γ(ĝ(ω), ω) > Γ(ĝ(ω 0 ), ω 0 ). (ii) If g(ω, ω 0 ) ≥ g̃ and πI ≥ πC with at least one strict inequality, then Γ(ĝ(ω 0 ), ω 0 ) > Γ(ĝ(ω), ω). Lemma C.7. Suppose that disasters mute the effect of type. (i) If g̃ ≥ g(ω, ω 0 ) and πC ≥ πI with at least one strict inequality, then Γ(ĝ(ω 0 ), ω 0 ) > Γ(ĝ(ω), ω). (ii) If g(ω, ω 0 ) ≥ g̃ and πI ≥ πC with at least one strict inequality, then Γ(ĝ(ω), ω) > Γ(ĝ(ω 0 ), ω 0 ). Where Does g̃ Lie Relative to g(ω, ω 0 ) We begin with a preliminary result that will be of use. Lemma C.8. There is an ω̃ ∈ (ω, ω 0 ) such that φ(g(ω, ω 0 ) − ν(ω 0 ) − ι(ω̃)) 1 − πI = . 0 0 φ(g(ω, ω ) − ν(ω ) + ι(ω̃)) πI Proof. By Lemmata C.2-C.3, − Φ(g(ω, ω 0 ) − ν(ω 0 ) − ι(ω 0 )) − Φ(g(ω, ω 0 ) − ν(ω 0 ) − ι(ω)) 1 − πI = . 0 0 0 0 0 Φ(g(ω, ω ) − ν(ω ) + ι(ω )) − Φ(g(ω, ω ) − ν(ω ) + ι(ω)) πI Applying the Cauchy Mean Value Theorem to the functions given by h− (x) ≡ Φ(g(ω, ω 0 ) − ν(ω 0 ) − ι(x)) and h+ (x) ≡ Φ(g(ω, ω 0 ) − ν(ω 0 ) + ι(x)), we see that − φ(g(ω, ω 0 ) − ν(ω 0 ) − ι(ω̃)) Φ(g(ω, ω 0 ) − ν(ω 0 ) − ι(ω 0 )) − Φ(g(ω, ω 0 ) − ν(ω 0 ) − ι(ω)) = . Φ(g(ω, ω 0 ) − ν(ω 0 ) + ι(ω 0 )) − Φ(g(ω, ω 0 ) − ν(ω 0 ) + ι(ω)) φ(g(ω, ω 0 ) − ν(ω 0 ) + ι(ω̃)) From this the claim follows. In light of Lemma C.8, in all subsequent results, we fix ω̃ so that φ(g(ω, ω 0 ) − ν(ω 0 ) − ι(ω̃)) 1 − πI = . 0 0 φ(g(ω, ω ) − ν(ω ) + ι(ω̃)) πI 28 Lemma C.9. Fix ω 0 > ω. (i) If πI ≥ 12 , g(ω, ω 0 ) ≤ ν(ω 0 ). (ii) If πI ≤ 12 , g(ω, ω 0 ) ≥ ν(ω 0 ). Proof. We show that g(ω, ω 0 ) ≤ ν(ω 0 ) if and only if πI ≥ 12 . Reversing the inequalities gives that g(ω, ω 0 ) ≥ ν(ω 0 ) if and only if πI ≤ 12 , and so gives the desired result. Observe that g(ω, ω 0 ) ≤ ν(ω 0 ) if and only if 1= φ(ν(ω 0 ) − ν(ω 0 ) − ι(ω̃)) φ(g(ω, ω 0 ) − ν(ω 0 ) − ι(ω̃)) 1 − πI , ≥ = 0 0 0 0 φ(ν(ω ) − ν(ω ) + ι(ω̃)) φ(g(ω, ω ) − ν(ω ) + ι(ω̃)) πI where the first equality follows from symmetry, the inequality follows from the MLRP, and the last equality is by definition of ω̃. It follows that g(ω, ω 0 ) ≤ ν(ω 0 ) if and only if πI ≥ 21 . Lemma C.10. Fix ω 0 > ω and let 1 2 ≥ πI . (i) Suppose that disasters amplify the effect of type and πC > 12 . Then g̃ > g(ω, ω 0 ). (ii) Suppose that disasters mute the effect of type and 1 2 > πC . Then g(ω, ω 0 ) > g̃. Proof. We begin with Part (i). Observe that φ(g(ω, ω 0 ) − ν(ω 0 ) − ι(ω̃)) 1 − πI = 0 0 φ(g(ω, ω ) − ν(ω ) + ι(ω̃)) πI 1 − πI πC < πI 1 − πC φ(ĝ(ω) − ν(ω) − ι(ω)) = φ(ĝ(ω) − ν(ω) + ι(ω)) φ(g̃ − ν(ω 0 ) − ι(ω)) = . φ(g̃ − ν(ω 0 ) + ι(ω)) where the first line follows from Lemma C.8, second line follows from the fact that πC > 21 , the third line follows from the definition of ĝ(ω), and the last line follows from the definition of g̃. With this, to show that g̃ > g(ω, ω 0 ), it suffices to show that φ(g(ω, ω 0 ) − ν(ω 0 ) − ι(ω̃)) φ(g(ω, ω 0 ) − ν(ω 0 ) − ι(ω)) ≥ . φ(g(ω, ω 0 ) − ν(ω 0 ) + ι(ω̃)) φ(g(ω, ω 0 ) − ν(ω 0 ) + ι(ω)) In that case, the inequalities above imply that φ(g̃ − ν(ω 0 ) − ι(ω)) φ(g(ω, ω 0 ) − ν(ω 0 ) − ι(ω)) > φ(g̃ − ν(ω 0 ) + ι(ω)) φ(g(ω, ω 0 ) − ν(ω 0 ) + ι(ω)) From this, the MLRP implies that g̃ > g(ω, ω 0 ), as desired. By Lemma C.9 and the fact that πI ≤ 12 , g(ω, ω 0 ) ≥ ν(ω 0 ). With this, the function δ 7→ φ(g(ω, ω 0 ) − ν(ω 0 ) − δ)) φ(g(ω, ω 0 ) − ν(ω 0 ) + δ)) 29 (4) is (weakly) increasing in δ. (See Lemma A.1.) Now using the fact that disaster amplifies the effect of type, it follows that ι(ω̃) > ι(ω). From this, Equation (4) follows. Part (ii) involves reserving certain inequalities in the above argument. First, since 1 2 > πC , φ(g(ω, ω 0 ) − ν(ω 0 ) − ι(ω̃)) 1 − πI πC > 0 0 φ(g(ω, ω ) − ν(ω ) + ι(ω̃)) πI 1 − πC φ(g̃ − ν(ω 0 ) − ι(ω)) = . φ(g̃ − ν(ω 0 ) + ι(ω)) Thus, it suffices to show that φ(g(ω, ω 0 ) − ν(ω 0 ) − ι(ω)) φ(g(ω, ω 0 ) − ν(ω 0 ) − ι(ω̃)) ≥ . φ(g(ω, ω 0 ) − ν(ω 0 ) + ι(ω)) φ(g(ω, ω 0 ) − ν(ω 0 ) + ι(ω̃)) (5) If so, the MLRP analogously implies that g(ω, ω 0 ) > g̃. Now observe that ω̃ > ω and disaster mutes the effect of type; so ι(ω) > ι(ω̃). Recall, since πI ≤ 21 , g(ω, ω 0 ) ≥ ν(ω 0 ). With this, the function δ 7→ φ(g(ω, ω 0 ) − ν(ω 0 ) − δ)) φ(g(ω, ω 0 ) − ν(ω 0 ) + δ)) is (weakly) increasing in δ and so Equation (5) holds, as desired. Several remarks are in order. First, observe that the proof of Lemma C.10 also gives the following: Remark C.1. Fix ω 0 > ω and let 1 2 > πI . (i) Suppose that disasters amplify the effect of type and πC ≥ 12 . Then g̃ > g(ω, ω 0 ). (ii) Suppose that disasters mute the effect of type and 1 2 ≥ πC . Then g(ω, ω 0 ) > g̃. Moreover, reversing the argument in the proof of Lemma C.10, we get the following: Lemma C.11. Fix ω 0 > ω and let πI > 12 . (i) Suppose that disasters amplify the effect of type and 1 2 ≥ πC . Then g(ω, ω 0 ) > g̃. (ii) Suppose that disasters mute the effect of type and πC ≥ 21 . Then g̃ > g(ω, ω 0 ). We will not need to make use of part (ii), as it will be redundant. We only include it for completeness. Remark C.2. Fix ω 0 > ω and let πI ≥ 12 . (i) Suppose that disasters amplify the effect of type and 1 2 > πC . Then g(ω, ω 0 ) > g̃. (ii) Suppose that disasters mute the effect of type and πC > 21 . Then g̃ > g(ω, ω 0 ). We will give a further condition under which g(ω, ω 0 ) > g̃ (resp. g(ω, ω 0 ) > g̃). 30 Lemma C.12. (i) g̃ ≤ ν(ω 0 ) if and only if πI ≥ πC . (ii) g̃ ≥ ν(ω 0 ) if and only if πC ≥ πI Proof. Note that g̃ ≤ ν(ω 0 ) if and only if 1= φ(ν(ω 0 ) − ν(ω 0 ) − ι(ω)) φ(g̃ − ν(ω 0 ) − ι(ω)) ≥ , φ(ν(ω 0 ) − ν(ω 0 ) + ι(ω)) φ(g̃ − ν(ω 0 ) + ι(ω)) where the inequality follows from the MLRP and the equality follows from symmetry. Using the fact that φ(g̃ − ν(ω 0 ) − ι(ω)) φ(ĝ(ω) − ν(ω) − ι(ω)) 1 − πI πC = = 0 φ(g̃ − ν(ω ) + ι(ω)) φ(ĝ(ω) − ν(ω) + ι(ω)) πI 1 − πC it follows that g̃ ≤ ν(ω 0 ) if and only if 1≥ 1 − πI πC πI 1 − πC or if and only if πI ≥ πC . Reversing the inequalities gives part (ii). Lemma C.13. (i) If πC ≥ πI > 12 , then g̃ > g(ω, ω 0 ). (ii) If 1 2 > πI ≥ πC , then g(ω, ω 0 ) > g̃. Proof. We show part (i); part (i) follows by reversing inequalities: Since πC ≥ πI , g̃ ≥ ν(ω 0 ). (See Lemma C.12(i).) Since, πI ≥ 21 , g̃ > ν(ω 0 ) > g(ω, ω 0 ). (See Lemma C.9.) Conclusion of Proof Proposition C.1 follows from Lemmata C.6, Lemma C.10-Remark C.1, Lemma C.11-Remark C.2, and Lemma C.13. Proposition C.2 follows from Lemmata C.7, Lemma C.10-Remark C.1, Lemma C.13 and the following additional result. Lemma C.14. Suppose disasters mute the effect of type and πI > 1 2 ≥ πC . Then Γ(ĝ(ω), ω) > Γ(ĝ(ω 0 ), ω 0 ). Proof. If g(ω, ω 0 ) ≥ g̃, then Lemma C.7 says that Γ(ĝ(ω), ω) > Γ(ĝ(ω 0 ), ω 0 ). Suppose that g̃ > g(ω, ω 0 ). We will show that g(ω, ω 0 ) > ĝ(ω 0 ). With this, Γ(ĝ(ω), ω) = Γ̃(g̃) > Γ̃(g(ω, ω 0 )) = Γ(g(ω, ω 0 ), ω 0 ) > Γ(ĝ(ω 0 ), ω 0 ), where the first inequality follows from the assumption that g̃ > g(ω, ω 0 ) and the MLRP, the second equality follows by Corollary C.2, and the last inequality follows from the fact that g(ω, ω 0 ) > ĝ(ω 0 ) (that we will show) and the MLRP. 31 We now turn to show that g(ω, ω 0 ) > ĝ(ω 0 ): By Lemma C.8, there exists some ω̃ ∈ (ω, ω 0 ) so that φ(g(ω, ω 0 ) − ν(ω 0 ) − ι(ω̃)) 1 − πI . = φ(g(ω, ω 0 ) − ν(ω 0 ) + ι(ω̃)) πI Since disaster mutes the effect of type, ι(ω) > ι(ω̃) > ι(ω 0 ). Moreover, since πI > 12 , Lemma ?? implies that g(ω, ω 0 ) − ν(ω 0 ) < 0. 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