The swine flu and its impacts on tourism in Brunei

This paper demonstrates how to disentangle the impacts of the swine flu on tourism in Brunei, which overlaps with the continued effects of the 2008 global financial crisis that occurred earlier using the auto regressive integrated moving average and intervention time series analysis methods. Estimating the impacts of the swine flu for the first 12 months' post-swine flu period, we have predicted the number of tourists by fitting two auto regressive integrated moving average models: one for the swine flu and another for the global financial crisis which occurred and affected the number of tourist arrivals; and one intervention time series analysis model. It is shown that the number of tourists have been reduced significantly due to both the swine flu and the global financial crisis, which is reconfirmed by testing the coefficients of the fitted intervention time series analysis model. It is found that a small country like Brunei lost nearly 30, 000 (15%) tourists and B$15 million dollars due to the swine flu during the first twelve months’ post-swine flu period.

Introduction
This study shows how to disentangle the effects of swine flu (SF) from the effects of earlier global financial crisis (GFC) on tourism and economy in Brunei. This may be a useful contribution to both policy makers, practitioners and to the tourism literature. Brunei is a small country with a population of 400,000 people. It is situated on the north-west side of the Island of Borneo and it has oil and gas reserves. The country is quite safe and tolerant for all types of ethnic people and it attracts many foreign tourists every year from various countries. For example, 226,000 tourists visited Brunei in 2008(Brunei Tourism Development Department, 2010. Brunei lost a significant number of tourists due to the 2008 GFC and this declining trend intensified until mid-2009, which can be seen by comparing tourist arrival numbers: 198,338 and 139,684 respectively during the 10 months pre-and post-GFC periods (Brunei Tourism Development Department, 2010). Then a small window of world economic recovery was observed. But then the world faced another crisis by the news of outbreak of A (H1N1) virus which is popularly known as swine flu (SF). Swine Flu occurs when people are in contact through talking or sitting or walking near the infected person/s and in contact with infected secretions produced through sneezing, coughing, spitting or transfer after touching infected areas. This is a serious infectious disease which people have every reason to fear (Denoon & Hitti, 2010;and Ministry of Health: Brunei Darussalam, 2010). The presence of the A (H1N1) virus affected Brunei sharply and its effect on tourism was serious across the world including Brunei.
Mexico recorded the first known SF case on March 17, 2009. Like the rest of the world, Brunei was not immune from the fear of SF. Mahdini (2010) described how Brunei's government took every possible preventative measure to avoid SF and to boost the confidence of the people. Brunei's government was prepared to face the outbreak as early as May 2, 2009 but Brunei's residents were already panicking by the end of May 2009. These measures failed to protect some of the people of Brunei from SF. A student was identified with positive A (H1N1) virus on June 20, 2009. A 12 year old girl became the first fatality on July 2, 2009. The recorded infected numbers were increasing at a rapid pace and by the reopening time of schools the total number of SF infected cases reached 142 which is a very high rate in a country of less than 400,000 people (Ministry of Health: Brunei Darussalam, 2010;France Presse Agence, 2009;Melvin, 2009;and;Mahdini, 2010). Stephen and Suni (2009) indicated that Brunei was hit very hard due to SF virus and they showed that there was a growth of 547% in H1N1 cases over the last 10 days of July 2009, which was the highest in the world, followed by Bolivia (326%). They further provided statistics and showed that 5.3 people were infected per 10,000 people in Brunei, the second highest in the world, just after Chile's 5.5 people infected per 10,000 people. All this information kept foreign tourists away from the country. Swine flu significantly reduced the number of tourists visiting Brunei, which was already suffering from the earlier GFC (World Tourism Organization, 2010).
The main purpose of this study is to evaluate the impacts of SF on tourism and the economy in Brunei, using two auto regressive integrated moving average (ARIMA) models: one for the SF and another for the GFC which occurred prior to SF, but which continued its effects on tourism during the post-SF period; and one intervention time series analysis (ITSA) method. This is because the ITSA is an important and useful method for analyzing the effects of crisis events on time series (TS) data. It is quasi-experimental in nature and the validity of the model depends on the timing of the intervention and the response of the process to it (see Gilmour, Degenhardt, Hall, & Day, 2006).
The ITSA model considers the effects of SF and the GFC, using two separate transfer functions in one ITSA model. It can also test the effects of the two crisis events separately to show whether the number of tourists has been reduced significantly due to SF and the GFC. The estimated number of lost tourists will then be used to calculate the total amount of money lost due to SF during the first 12 months' post-SF period.
The study is organized as follows. A brief review of the literature is provided in Section 2. Data used in the present study are given in Section 3. Section 4 is concerned with evaluation methods. Empirical illustrations are provided in Section 5, while some important discussions are made in Section 6. Concluding remarks are made in the final Section.

A brief review of literature
Tourism has become an important global industry since 1970 and it is growing rapidly. The World Tourism Organization (2008) mentioned that in 2007 there were nearly 900 million international tourists and it was growing at over 6% per year. Tourism has wide spread implications and this has been the subject of considerable public and academic interest. There is a huge amount of literature on tourism among which Turner (2015), Zhang, Lin, andNewman (2016), Nawijn, Mitas, Lin, andKerstetter (2013), UNWTO (2016) can be mentioned. More importantly, an extensive review of literature can be seen in Witt and Witt (1995), Song and Li (2008), Lee, Lowry, and Delconte (2015), Merinero-Rodriguez and Pulido-Fernandez (2016), Law (2011), Chen andPetrick (2013), Piuchan and Suntikul (2016), and Getz and Page (2016).
There are two main approaches to evaluate the effects of crisis events. These are: (i) the pre-post method, and (ii) the forecasting approach. The pre-post method attempts to evaluate the effects of a crisis by comparing the number of the tourists in the same period of the pre-and post-crisis event (Chen & Chen, 2005). However, it fails to incorporate the pre-trend of the data, which might affect the actual change of the crisis and hence it cannot accurately measure the effects of the crisis event.
In contrast, forecasting methods can predict the tourist numbers for the post-event period as if the crisis event had not occurred. This predicted tourist number is then subtracted from the actual number for a certain post-event period and this is usually interpreted as the effect of the crisis. Most forecasting methods have the advantage of controlling long-term trends. Quantitative forecasting methods are mainly divided into two groups. First, time series (TS) methods (basic, intermediate, and advanced) which have many advantages. It is based on the assumption that the present and past behavior of the data will continue in the future, and observations at different points in time are statistically dependent. The ARIMA model is the most popular TS forecasting model. One can fit an ARIMA model to almost any TS data with the desired precision. However, there are some disadvantages for the ARIMA model such as (i) there is no way to know where to start the ARIMA model; (ii) it cannot be used for non-stationary data; (iii) it requires a large number of data points; and (iv) it is a univariate TS forecasting method and cannot be used for a multivariate TS data analysis. The ITSA model is a further development of the ARIMA model and it can measure and test the effects of intervention/s. It uses both the preand post-intervention data, whereas the ARIMA model uses only the pre-intervention data. It provides better forecasts than the ARIMA model for interrupted TS data. The vector auto-regression (VAR) model is one of the most successful, flexible, and easy to use models for the analysis of multivariate TS data and can provide better forecasts than the ARIMA model. It can be considered as one type of econometric model and it possesses all the advantages of the regression method. Also, the size of the model is not a concern. However, this can be estimated by various ways and hence it can give a wide variety of different forecasts.
On the other hand, causal methods (Regression method and Structural method) are concerned with methodologies for identifying relationships between dependent and independent variables and attempt to incorporate the interdependence of various variables in the real world. However, the difficulty of applying the causal methods is identifying the appropriate independent variables that affect the forecast variable. Structural models are more transparent as they can check if the predicted behavior by the model for each component corresponds to what is expected from the data. The structural forecasting method is relatively straightforward, and missing observations are easier to treat. It can also manipulate multivariate series by direct extension of the univariate structural formulation which is not possible with the ARIMA model. Many researchers forecast with structural models for economic analysis and use the term "econometric model forecasts" and these are routinely used for econometric policy analysis. The main advantages of econometric models are: (i) this method possesses all the advantages of the regression method, (ii) the size of the model is not a problem; and (iii) to estimate and solve the model using software packages is easy. However, the accurate forecasting of these two causal models depends on the quality of independent variables that affect the forecast variable (see Chen, Bloomfield, & Cubbage, 2008 for comparing forecasting models in tourism).
In the past, researchers used various methods as mentioned above to evaluate the impacts of uncertain events such as diseases, earthquake, etc., on tourism, using the ARIMA, ITSA, Structural Time Series, VAR and many other methods. For example, the Severe Acute Respiratory Syndrome (SARS) is a serious disease which affected tourism in most Asian countries. Many authors have analyzed the impacts of various disease outbreaks on travel and tourism for various countries using econometric methods (Chen, Jang, & Kim, 2007;Chien & Law, 2003;Kou, Chen, Tseng, Ju, & Huang, 2008;McAleer, Huang, Kuo, Chen, & Chang, 2010;McKercher & Chon, 2004;WTTC, 2003;Wilder-Smith, 2006;Tung & Chao, 2011;Oukil, Channouf, & Al-Zaidi, 2016;Cuhadar, 2014). By contrast, Cankurt and Subasi (2016), Song, Li, Witt, andAthanasopoulos (2011), Turner (2015) and many others used structural TS analysis to model and forecast tourist demand. On the other hand, many authors such as Aboagye-Sarfo, Cross, and Mueller (2016), Zheng, Farrish, and Kitterlin (2016), Hayashi (2013, 2014), Chung, Ip, and Chan (2009) and many others used the ARIMA and ITSA methods to evaluate the impacts of various crisis events. An excellent survey of the literature on the evaluation of crisis events can be found in Hall (2010) and Goh and Law (2011).
Most advanced TS methods use present and past values and deal with specification, estimation, forecasting and evaluation and they often provide better forecasts than causal methods. Forecasts based on TS methods help in making forward planning, marketing, and resource allocation in the tourism industry, which are now discussed here. Min (2008a) evaluated how Taiwan's inbound tourism was affected by the September 21st, 1999 Earthquake and SARS in 2003, using the ARIMA and ITSA methods. She showed that Taiwan was hit hard by these two calamities. Min (2008b) and Min and Kung (2007) analyzed inbound tourism demand for Taiwan using the ARIMA and ITSA models. Min, Wu, and Wu (2006) also forecasted outbound tourist arrivals using the ITSA model. Bonham and Gangnes (1996) used the ITSA to measure the effect of Hawaii hotel and room tax on tourism. Coshall (2003), Lee, Oh, and O'Leary (2005) and Ismail, Suhartono, Yahaya, and Efendi (2009) evaluated the threat of terrorism on international travel flow using the ITSA method. Goh and Law (2002) also used the ITSA method to forecast tourism demand. Chen (2011), Engin (2015) and many others have evaluated the effects of various crisis events on tourism, using the ARIMA and ITSA techniques. Page, Song, and Wu (2012) Peng, Song, & Crouch, 2014;and Smeral, 2010) have also used the ARIMA and ITSA techniques for modelling and forecasting tourism demand.
In this study, we have used the Box and Jenkins (1970) ARIMA and Box and Tiao (1975) ITSA methods to measure the effect of swine flu in Brunei. This is because the ITSA method is appropriate for explaining the dynamics and the impact of interruptions and changes of TS in a more detailed and accurate manner (Baldigara & Mamula, 2015;Park, Lee, & Song, 2016;Aboagye-Sarfo et al., 2016;Chang & Lin, 1997;Biglan, Ary, & Wagenaar, 2000;Ferrand, Kelton, Guo, Levy, & Yu, 2011;Chung et al., 2009;Wu & Hayashi, 2014). The ARIMA and ITSA methods can provide an appropriate measure of the impacts of crisis events, which advise practical actions to overcome problems associated with tourism, economic and other business matters.

Data
Brunei Tourism Development Department provided monthly foreign tourist arrivals to Brunei from January 2005 to May 2010. It should be noted that 126,000 tourists arrived in 2005 and it increased to 226,000 in 2008 just before the GFC, and then declined to 157,000 tourists after the occurrence of both the GFC and SF (Brunei Tourism Development Department, 2010). The department only considered entry for overnight transit, government exhibition, business, holidays, visiting friends and relatives as foreign tourists. These are generally considered as the most accurate, complete, and authentic data and are used to estimate the number of tourists lost during the first 12 months post-SF period. At present, there is no information about the activities of tourists in Brunei. In order to get information about the activities of the tourists, we have randomly surveyed a representative sample of 600 (200 and 400 respectively in pre-and post-SF periods) eligible departing adult tourists from Brunei Airport as well as from various shopping centers and restaurants, who did not work and stayed at least one night in Brunei such that each person was selected only once. We then collected information from each of the selected foreign tourists about their length of stay and other costs associated with food, accommodation, transport, shopping and sight-seeing which can be seen from Table 6, and these are used to estimate their spending while they stayed in Brunei. These are then used to estimate the total amount of money lost by the country due to the loss of tourists during the first 12 months post-SF period.

Evaluation methods
In this study, we used the Box and Jenkins (1970); ARIMA and Box and Tiao (1975) ITSA methods to predict the number of tourists which will then be used to estimate the loss of tourists during the 1st 12 months post-SF period. First, we use the ARIMA model to predict the number of tourists using the pre-SF data on the assumption that the past and present behavior of the data will continue. The data series should be stabilized before forecasting. It is a general practice to plot the raw data to see if there is any irregular variation, trend, seasonality, and/or other behavior that can be observed. Most often differencing the data can be used to achieve stationarity. Otherwise more sophisticated techniques of TS data analysis such as the ARIMA model should be applied to predict the number of tourists for the post-SF period.
The ARIMA model is accomplished with the autocorrelation function (acf) and partial autocorrelation function (pacf). By matching the known properties of the model with the observed acf and pacf and undertaking various tests of the stability of the data tentative models are identified, estimated, and checked (see Box & Jenkins, 1970). The Minimum Akaike (1973) Information Criterion Estimate (MAICE) is used to select the best model if several models satisfy the Ljung and Box (1978) Portmanteau test on residuals. Using the estimated parameters of the chosen model, forecasts are made for the post-SF period. In practice TS data are subject to various interventions and hence the ARIMA model alone cannot produce accurate forecasts. Under this situation, the ITSA method can incorporate various interventions with the ARIMA model and can produce a more reliable and accurate forecast. The ITSA method can also control the long-term trend and is more potentially sensitive. More importantly it can test the intervention coefficients to see whether the interventions significantly reduced the tourist numbers by using a one-tailed t-test.
In this study, we should use two ARIMA models. First, for the SF to predict tourist numbers for the post-SF period based on pre-SF data, which can be used to measure the loss of tourists during the post-SF period by comparing the actual and predicted numbers (or ITSA estimate), based on the occurrence of the GFC without SF. Second, we also should use another ARIMA model for the GFC which occurred and affected tourist numbers prior to SF based on the pre-GFC data to predict the number of tourists for the post-SF period due to the GFC.
One ITSA model was then redesigned with the best ARIMA fitted to the SF by incorporating two intervention variables: one for SF and another for the GFC. Now this newly developed ITSA model was estimated using the entire data set from January 2005 to May 2010. This model allows us to test the intervention coefficients for SF and the GFC to investigate whether they significantly reduced tourist numbers. The robustness of the ITSA method can be examined by comparing the predicted tourist numbers with the actual numbers. In this study both the ARIMA and ITSA models are used to estimate the loss of tourists for the post-SF period from June 2009 to May 2010 whose empirical illustrations are given below.

Empirical illustrations
Before fitting the ARIMA and ITSA models we plot the monthly tourist arrival data in Fig. 1 which shows that the data has irregular variation as well as an upward trend until July 2008. It then declined until June 2009 and then it showed a rising trend after the GFC, which can be seen from the tourist arrival numbers: 29,868 (July and August 2009) compared to 24,466 (May and June 2009).
Just at that time SF emerged in Brunei and it further reduced, which can be seen from tourist arrival numbers: 61,047 and 48,959 during SeptembereDecember in 2008 and 2009 respectively (Brunei Tourism Development Department, 2010). This is "just another negative shock when the economy can least afford negative shock" (Wachovia Corporation, 2009). Thus, two interventions disturb our data: the GFC and SF. The behavior of the acf and pacf also showed that the data are non-stationary. To test the non-stationarity of the data we used the Augmented Dickey and Fuller (1979) test for the model with constant and trend: under the null hypothesis that our data has unit root against the alternative hypothesis that data has  Beaulieu and Miron (1993).
Source: Brunei Tourism Development Department, Government of Brunei.     no unit root. Our test statistic of À1.98 is greater than the interpolated Dickey and Fuller critical value À3.49 for 66 observations indicating that the null hypothesis cannot be rejected. This means our data has unit root and it is non-stationary. It is also seen from Fig. 1 that our monthly tourist arrival data has seasonal variation which can be deterministic or stochastic (Hylleberg, 1995;Osborn, Heravi, & Birchenhall, 1999and Taylor, 1998. The regression model handles the deterministic process using seasonal dummies. Stochastic process can be stationary or non-stationary which can be investigated by testing the null hypothesis that the data has a seasonal unit root. Thus, the data should be made stationary to make reliable forecasts. In this study, we investigate the seasonal aspects of the data by various tests of seasonality and efforts are made to eliminate the seasonality to make the data stable before making forecasts. Our data suffer from seasonality which can be seen from the estimated acf and pacf and is confirmed by the sinusoidal form of the correlogram of our data. We then used the HEGY test due to Hylleberg, Engle, Granger, and Yoo (1990) to test the null hypothesis of a certain number of seasonal and non-seasonal unit roots in a TS against the alternative hypothesis that the data has no unit roots. The results of the HEGY test are provided in Table 1 and show that all the test statistics at various frequency levels are higher than the corresponding critical values and hence the null hypothesis H 0 cannot be rejected at all frequency levels. This indicates that the data has a non-seasonal unit root corresponding to zero frequency and hence our data is non-stationary. Therefore, its differentiation with filter (1 e B) is required. Moreover, the test accepts the null hypothesis H 0 at all frequency levels showing the presence of unit roots at all seasonal frequencies. Hence, the filter (1 e B 12 ) is required as recommended by Box and Jenkins (1970) depending on the fact that the series is integrated at the seasonal frequency at all levels of frequencies including zero. Because of this the product of the non-seasonal and seasonal filters: (1 À B) (1 À B 12 ) is applied to stablize the seasonal and non-seasonal variation of the data. The Canova and Hansen (1995) test is then employed to test the stability of the data. The Canova-Hansen test results are presented in Table 2 which shows that all calculated statistics are less than the corresponding critical values. Further, this shows that the data has no seasonal unit root and is now stable, although our original data had a stochastic trend and seasonality. To avoid these problems, we differenced our data at non-seasonal and seasonal levels and apply various tests showing that our data became stationary after first differencing at non-seasonal and seasonal levels with lag length one (Ng & Perron, 2001).
The ARIMA and ITSA models are used in the present analyses to evaluate the impacts of SF on tourism and economy. The general ARIMA model is redesigned incorporating two transfer functions for two crises events: the GFC and SF. This model consists of one pulse dummy to consider the effect of the GFC which was long lived and gained a slow recovery. A linear step dummy was used for SF which was a temporary deviation from the slow recovery of the GFC. We use the combination of pulse (for the GFC) and step (for SF) functions as Wei (1990) and add to the general ARIMA (p, d, q) (P, D, Q) s model in formulating our ITSA model, which can be written as follows.
5.1. Fitting the ARIMA models using pre-intervention data We have used the Maximum Likelihood (ML) estimation method to estimate several ARIMA models using the data set from January 2005 to May 2009 (Pre-SF period) and fitted 20 ARIMA models for finding a best ARIMA model for SF using the SAS/ETS (2011) computer software. We also used a second data set from January 2005 to August 2008 (Pre-GFC period) and fitted another 20 ARIMA models for finding a best ARIMA model for the GFC using the same SAS/ETS computer software. The ARIMA model selection procedure for each of the series was based on acf, pacf and inverse autocorrelation (iacf) (see Akaike's, 1973 information criteria (AIC); Schwarz's, 1978 Bayesian criteria (SBC); Chatfield, 1979;Priestley, 1981; for more detailed model selection criteria). The estimated parameters of the best model for each of the data series have satisfied the stationarity and invertibility criteria. The best ARIMA models selected for each of the data series are presented in Table 3. Further diagnostic checks of residuals and autocorrelations for each of the models clearly indicate that the models are acceptable. This can be seen from the Q statistic of Table 3 which shows that the calculated values of Q-statistic for each of the models are less than 1, which is < c 2 .05,8 ¼ 15.50, indicating that we cannot reject the null hypothesis for any one of those selected best models, implying that each of the selected best models adequately fit the data. Also, the standard errors of both the fitted models are very small and hence can be expected to give good forecasts. Estimated parameters for each of the models are also significantly different from zero indicating that both selected models are good.
To undertake sensitivity analyses and predictive performances for each of our best selected models we dropped the last five observations from each of the data series and re-estimated each of those best models. It is seen that the estimated parameters of these newly fitted models are very similar to our original fitted models and certainly none of the parameters of these newly fitted models are significantly different from those of the originally fitted models. Sometimes it is difficult to know the exact point of intervention and as such eliminating the last five points from each of our data series meaning that first we allow the SF to start from January 2009 instead of June 2009. It also allows us to start the GFC from April 2008 instead of September 2008. Eliminating the last five points from each of the original data series means that even if we choose an earlier intervention point for the SF and the GFC even then our original fitted models are still robust and can produce good forecasts.
Re-estimating each of the models without the last five observations from each of the data series allows us to investigate the predictive performance evaluation for each of the best selected models. For the best selected fitted SF model the minimum Mean Absolute Deviation (MAD) and Minimum Mean Sum of Squares of Errors (MSE) are 110.05 and 12,520 respectively. The Percentage of Correct Prediction for this model is 95.2% which shows that the estimated best fitted model for SF is close to the actual number. Similarly, the MAD, MSE and percentage of correct prediction values for the best GFC model are 95; 8504 and 96.4% respectively.
It is thus expected that the pre-intervention model for each of the best ARIMA models given in Table 3 will continue to be adequate for making good post-SF forecasts. The estimated model (a) is used to predict the number of tourist arrivals during the first 12 months post-SF with the knowledge that the GFC affected tourist arrivals but without allowing for the effects of SF. The estimated model (b) is used to forecast the number of tourist arrivals during the first 12 months post-SF as if neither the GFC nor SF occurred.

Intervention model
The best selected ARIMA model for SF is now redesigned to develop an intervention model since our main objective is to estimate the loss of tourists due to SF incorporating two intervention variables: GFC ¼ 0 for pre-GFC months and GFC ¼ 1 for post-GFC months; and SF ¼ 0 for pre-SF months and SF ¼ 1 for post-SF months. It is noted that the impacts of the GFC is long lived or, in other words, the recovery was prolonged and as such a pulse function is appropriate for the GFC. Thus, the SF shock was really a temporary deviation from the slow recovery of the GFC and hence a step function is appropriate for the SF shock. Thus, a combination of pulse and step functions were added to the best selected ARIMA model for the SF and used to estimate an intervention model of type (1), using the entire data set: January 2005eMay 2010. A summary of the estimated ITSA model is presented in Table 4.
The parameters of this intervention model were also estimated by the SAS/ETS (2011) computer software which were not significantly different from the pre-SF estimated ARIMA model. Diagnostic checks in terms of residuals, autocorrelations and partial autocorrelations indicate that this model is quite satisfactory. This is reconfirmed by the Q-statistic showing that the null hypothesis cannot be rejected meaning that the model does not exhibit lack of fit since the calculated value of Q is less than 1, which is <c 2 0.05,5 ¼ 11.05. Also, the predictive performance of our ITSA model is excellent which can be seen from Table 5 where our ITSA model  forecasts the tourist number as 170,085 compared to 169,258 who  came to Brunei during June 2009 to May 2010. This means that the percentage of correct prediction of our ITSA model is more than 99.5%. The estimated intervention coefficients W 1 and W 2 are both negative and one-tailed t-tests reconfirm that both the SF and the GFC have reduced the number of tourists visiting Brunei.
The loss of tourists due to SF at a post-SF month is estimated by taking the difference between the actual (or ITSA forecasts) and ARIMA forecasts. Thus, the loss of total number of tourists from June 2009 to May 2010 is estimated by summing the number of tourists lost for those 12 months post-SF period. The ARIMA, ITSA forecasts together with the actual number of tourists lost are presented in Table Haque (2010) has shown that tourists spent B$77 less during post-SF compared to the pre-SF period because of shorter stay and lower accommodation costs due to high hotel vacancies in 2009 and 2010.
Brunei lost nearly 15% of tourists during the first 12 months post-SF period: June 209 to May 2010, which can be estimated from Table 5 and was explained earlier. To recover from this exogenous shock, the Brunei Tourism Development Department undertook several promotional and marketing initiatives by the end of 2009 to establish Brunei as a desirable holiday destination. These initiatives covered acquisition of a new set of iconic images, brochures, information kit, re-branding new websites, promotional materials, media, and TV advertisements, creating search, and booking engines, attending trade events and social networking. Marketing activities are more intense and wider reaching, giving a message of a safe, peaceful, stable, friendly, and hospitable place blessed by nature protected from disasters, healthy clean and free of topical disease, and easily accessible to modern and efficient in fracture: Brunei as 'South-East Asia's best kept secret'.
Group/individual familiarization trips for wholesalers, retailers, organizing sales missions and product updates in all major markets, appointing specialized companies in key markets, financial support for local and overseas industry partners to feature Brunei tourism products in their brochures/websites/fliers; joint marketing activities with local and overseas industry partners for specific technical promotions of tour packages; joint activities with wholesalers to increase destination awareness and sales among retailers; promotion of Brunei tourism initiated thematic packages in short haul markets (such as golf tours and Islamic packages).
The results of all these promotional and marketing initiatives are highly effective which can be seen from Table 7 indicating that there is an overall 20% increase of tourists and hotel accommodations in the first half of 2010 compared to 2009, a similar number to the previous three years' average. These are reconfirmed by the c 2test based on the pre-post method. The actual numbers are well below than the predicted numbers which are supposed to be achieved without crisis events such as the SF, the GFC etc. Increased expected number of tourists can be achieved through increasing promotional and marketing activities in Far-East (China) and Australia/New Zealand marketing areas, providing safety and security information for Brunei, which are currently significantly lower than the three years' average. Only a few tourists (0.4% of the total) are coming from the Middle-East, despite religious and cultural bonds, who can be persuaded to visit Brunei if Islamic tour packages can be arranged.

Discussion
This study shows how the impacts of the SF and the GFC can be separately evaluated. It is important for policy makers and practitioners and is an important contribution to the tourism literature. More importantly, the techniques developed here can be used to disentangle the effects of multiple interventions in a variety of fields including tourism.
In this paper, we investigated how the SF affected tourism in Brunei using the ARIMA and ITSA models. The techniques used here successfully separate the effects of the SF from the GFC on tourism, which occurred earlier and continued after the occurrence of the SF for the period from June 2009 to May 2010. The use of a control group which is free from SF and the GFC may have produced better estimates, but it is difficult to find an appropriate control group. We did not consider other possible interventions that might have affected tourism in Brunei such as the 1997-98 East Asian Economic Crisis (particularly currency crisis in Indonesia and Malaysia), the 2007e2008 Oil shock, the 2010 Arab Spring, etc., (Hamilton, 2011). It seems none of these factors, other than the GFC and SF affected tourism in Brunei during our study period. This is Table 7 Comparison of tourist arrival in Brunei from major countries and market areas in different years: JanuaryeJune a . because a wide range of tourist attractions across the region drew 81 million tourists to ASEAN in 2011, up by 30% compared to 62 million tourists in 2007. In 2010, 47% (34 million) of 73 million tourists in ASEAN countries came from other ASEAN countries (Association of Southeast Asian Nations, 2011). This implies that the reduction of tourists in Brunei may likely to be due to the GFC and SF. In addition to these the effects of cyclical components have also not been considered here which might have affected our estimates of tourists to Brunei.
The ITSA model requires many data points. The model developed here is useful for evaluating planned policy change and/or unexpected events. It is also useful for scenario analysis. The impact of the uncertain events can be tested and measured with great reliability but these models are based on a single parameter throughout the period. More importantly, the techniques used here can easily be extended to disentangle the effects of many interventions which may occur during a certain time-period. On the other hand, structural time series models provide a broader framework and allow the parameters to change over time which is not yet used widely to measure uncertain events. It is also flexible and can be generalized to a multivariate setting without extra effort. Thus, the structural time series model is highly recommended to measure the crisis event such as the SF. More details for the structural time series models can be seen in Durbin and Koopman (2012) among others.

Conclusions
This study investigates the effects of one major crisis event: SF on tourism in Brunei, using forecasting based on the ARIMA and ITSA methods. The effects of SF on tourism in Brunei overlap with the effects of the GFC that occurred earlier and hence we develop a technique to disentangle the effects of SF from the effects of the GFC. This is a useful contribution to the policy makers and to the tourism literature. In fact, the method developed here can be easily extended to disentangle the effects of multiple interventions. It is a useful sophisticated method which could be applied to a wide range of fields. For example, this can be used to evaluate the impacts of other policy/program changes such as to evaluate the impacts of tourism marketing programs. We have successfully used the technique developed here to disentangle the effects of SF and the GFC on tourism and it is shown that Brunei lost 123,737 and 29,919 tourists due to the GFC and SF events respectively during the first 12 months post-SF period.
A survey of tourists in Brunei indicates that on average each tourist spent B$150 per day and stayed 3.4 days in Brunei indicating that each tourist spent B$ 510 while they stayed in Brunei. When this figure is multiplied by the number of tourists lost, it shows that a small country like Brunei lost B$14,882,820 within the first 12 months' post-SF period due to SF alone.
The presence of major international and local events such as the GFC and SF that might produce a shortage of tourists may lead us to believe that the uncertainties caused by the SF are still casting doubts in the minds of tourists. Misunderstanding the situation may impede the flow of tourists since perceptions of risk and safety towards a destination are likely to influence travel decisions (Sonmez & Graefe, 1998).
To recover from these crises, the Brunei Tourism Development Department launched several promotional and marketing initiatives by the end of 2009 to establish Brunei as a desirable holiday destination. The outcome of these activities was promising and there is an overall 20% increase in tourists and hotel bookings in the first half of 2010 compared to 2009, but this is still far short of predicted numbers. Increased expected numbers can be achieved through promotional activities in Far-East (China), Australia/New Zealand and the Middle-East marketing areas.
The propensity for 'family tours' is strong among Muslims, which should be emphasized, encompassing the 'family holidays' by establishing an attractively priced family package and/or offering a package with a variety of highly desirable and unique features to attract them (Kozak, 2016). Muslims travel with family who require special arrangements such as isolated accommodations. Printed advertisement with family-package offerings through travel agents could emphasize the fact that Brunei is one of the best tourism destinations in the world to attract Muslim families particularly from the Middle-East. These are more likely to be responsive to prompt marketing strategies and messages, which could increase higher number of Middle-East tourists.
There are some limitations of this study which could be addressed in future research. This study used only 66 monthly data points and had difficulties in modelling the rebound of foreign tourists. It is recommended to undertake future studies when more data will be available. Other forecasting techniques such as the structural time series, ARMAX, VAR models which consider the dynamic relationship between variables such as gross domestic product, exchange rate and other factors that might influence a rebound such as fading memories, are recommended for future studies.
Despite these limitations, the method developed here is one of the techniques which may be confidently applied to get accurate forecasts even after negative shocks such as the GFC and SF crises. This can be applied to disentangle the effects of two (or more) crisis events when one overlaps the other and this is useful to both policy makers and practitioners and to tourism analysts. Our forecasting models help tourism authorities to make operational, tactical, and strategic decisions (Wang & Lim, 2005;and;Law & Au, 1999), which can be used by practitioners and policy makers to prepare more sophisticated forward planning. This research will potentially benefit researchers of tourism in Brunei and its surrounding countries such as Malaysia, Singapore, Thailand, and other countries which attract many tourists. These countries will benefit by adopting the methods presented in this study to measure the effects of unforeseen events for successful and sustainable tourism development. Marketers of these countries can continuously monitor the changes of attitudes in tourists and can develop appropriate policies accordingly. Thus, this study will help Brunei and other similar countries in South East Asia to improve tourist numbers by various marketing strategies such as advertisements, tour packages etc. To evaluate these strategies for a certain postcampaign period they can use the techniques developed here to measure any associated changes in tourism. Also, it is recommended that the tourism authority should strongly advertise about the safety and security of the environment and accommodation together with recreational facilities and the natural beauty of the country to minimize feelings of uncertainty, which helps tourists to overcome their fears in travelling to Brunei.


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Computer Science
Mathematics

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Proportional Justified Representation

The goal of multi-winner elections is to choose a fixed-size committee based on voters' preferences. An important concern in this setting is representation: large groups of voters with cohesive preferences should be adequately represented by the election winners. Recently, Aziz et al. (2015a;2017) proposed two axioms that aim to capture this idea: justified representation (JR) and its strengthening extended justified representation (EJR). In this paper, we extend the work of Aziz et al. in several directions. First, we answer an open question of Aziz et al., by showing that Reweighted Approval Voting satisfies JR for $k=3, 4, 5$, but fails it for $k\ge 6$. Second, we observe that EJR is incompatible with the Perfect Representation criterion, which is important for many applications of multi-winner voting, and propose a relaxation of EJR, which we call Proportional Justified Representation (PJR). PJR is more demanding than JR, but, unlike EJR, it is compatible with perfect representation, and a committee that provides PJR can be computed in polynomial time if the committee size divides the number of voters. Moreover, just like EJR, PJR can be used to characterize the classic PAV rule in the class of weighted PAV rules. On the other hand, we show that EJR provides stronger guarantees with respect to average voter satisfaction than PJR does.

Introduction
Decision-making based on the aggregation of possibly conflicting preferences is a central problem in the field of social choice, which has received a considerable amount of attention from the artificial intelligence researchers (Conitzer 2010;Brandt et al. 2016). The most common preference aggregation scenario is the one where a single candidate has to be selected. However, there are also many applications where the goal is to select a fixed-size set of alternatives: example range from choosing a parliament or a committee to identifying a set of plans, allocating resources, shortlisting candidates for a job or an award, picking movies to be shown on a plane or creating a conference program (Barberà and Coelho 2008;Monroe 1995;Elkind et al. 2017). Recently, the complexity of multi-winner voting rules (Betzler et al. 2013;Aziz et al. 2015b) and their social choice properties (Elkind Multi-winner voting rules are often applied in scenarios in which the set of winners needs to represent the different opinions or preferences of the agents involved in the election. Thus, it is important to formulate axioms that capture our intuition about what it means for a set of winners to provide a faithful representation of voters' preferences (Monroe 1995;Dummet 1984;Black 1958). Aziz et al. (2015a; have recently proposed two such axioms for approvalbased multi-winner voting, namely justified representation (JR) and extended justified representation (EJR). Intuitively, JR requires that a large enough group of agents with similar preferences is allocated at least one representative; EJR says that if this group is large enough and cohesive enough, it deserves not just one, but several representatives (see Section 2 for formal definitions). Similar axioms have been proposed for multi-winner voting rules with ranked ballots (Dummet 1984;Elkind et al. 2017). Aziz et al. show that for every collection of ballots there is a winning set that provides EJR; they then explore a number of popular multi-winner voting rules and show that several of them satisfy JR, but only one rule satisfies EJR.
Our first contribution in this paper is to answer a question left open by Aziz et al. Specifically, Aziz et al. prove that Reweighted Approval Voting (RAV) 1 satisfies JR if the desired number of winners k is 2 but it fails JR if k ≥ 10. We close this gap and prove that RAV satisfies JR if k ≤ 5 and fails JR if k ≥ 6. Our proof proceeds by constructing and solving a linear program that establishes bounds on RAV scores.
We then formulate an axiom that we call Perfect Representation (PR), which says that if a given instance admits a 'perfect solution' (all voters are represented, and each winner represents the same number of voters), then we expect a voting rule to output such a solution. This axiom is very appealing in parliamentary elections and similar applications of multi-winner voting. However, it turns out to be incompatible with EJR: there is an election where these two axioms correspond to disjoint sets of winning committees.
Motivated by this result, we propose a relaxation of EJR, which we call Proportional Justified Representation (PJR). PJR is more demanding than JR, but, unlike EJR, it is compatible with perfect representation, and a committee that provides PJR can be computed in polynomial time; in particular, we show that a well-studied efficiently computable voting rule satisfies PJR if the committee size k divides the number of voters n. In contrast, it is conjectured that finding committees that provide EJR is computationally hard. Moreover, just like EJR, PJR can be used to characterize the classic Proportional Approval Voting (PAV) in the class of weighted PAV rules. However, we then show that the additional flexibility supplied by PJR comes at a cost: we define a measure of average voter satisfaction and show that EJR provides much stronger guarantees with respect to this measure than PJR does. We conclude the paper by discussing our results and indicating directions for future work.

Preliminaries
Given a positive integer s, we denote the set {1, . . . , s} by [s]. We consider elections with a set of voters N = {1, . . . , n} and a set of candidates C = {c 1 , . . . , c m }. Each voter i ∈ N submits an approval ballot A i ⊆ C, which represents the subset of candidates that she approves of. We refer to the list A = (A 1 , . . . , A n ) as the ballot profile. An approval-based multi-winner voting rule takes as input a tuple (N, C, A, k), where k is a positive integer that satisfies k ≤ |C|, and returns a subset W ⊆ C of size k, which we call the winning set, or committee. We omit N and C from the notation when they are clear from the context.
The following voting rules have received a considerable amount of attention in the literature (Kilgour 2010;Elkind et al. 2017;Aziz et al. 2015a;Aziz et al. 2017): Proportional Approval Voting (PAV) Under PAV, an agent is assumed to derive an utility of 1 + 1 2 + 1 3 + · · · + 1 j from a committee that contains exactly j of her approved candidates, and the goal is to maximize the sum of the agents' utilities. Formally, the PAV-score of a set W ⊆ C is defined 1 j , and PAV outputs a set W ⊆ C of size k with the highest PAV-score.
We can generalize the definition of PAV by using an arbitrary non-increasing score vector in place of (1, 1 2 , 1 3 , · · · ): for every vector w = (w 1 , w 2 , . . .), where w 1 , w 2 , . . . are non-negative reals, w 1 = 1 and w 1 ≥ w 2 ≥ . . ., we define a voting rule w-PAV that, given a ballot profile (A 1 , . . . , A n ) and a target number of winners k, returns a set W of size k with the highest w-PAV score, defined as Reweighted Approval Voting (RAV) RAV is a multi-round rule that in each round selects a candidate and then reweighs the approvals for the subsequent rounds. Specifically, it starts by setting W = ∅. Then in round j, j ∈ [k], it computes the approval weight of each candidate c as selects a candidate with the highest approval weight, and adds her to W . Just as for PAV, we can extend the definition of RAV to score vectors other than (1, 1 2 , 1 3 , · · · ): every vector w = (w 1 , w 2 , . . .), where w 1 , w 2 , . . . are non-negative reals, w 1 = 1 and w 1 ≥ w 2 ≥ . . . defines a sequential voting rule w-RAV, which proceeds as RAV, except that it computes the approval weight of a candidate c in round j as i:c∈Ai w |W ∩Ai|+1 , where W is the winning set after the first j − 1 rounds. The Monroe rule For each voter i ∈ N and each candidate c ∈ C we write u i (c) = 1 if c ∈ A i and u i (c) = 0 if c ∈ A i . Given a committee W ⊆ C of size k, we say that a mapping π : N → W is valid if it satisfies |π −1 (c)| ∈ n k , n k for each c ∈ W . The Monroe score of a valid mapping π is given by i∈N u i (π(i)), and the Monroe score of W is the maximum score of a valid mapping from N to W . The Monroe rule returns a committee of size k with the maximum Monroe score. The Greedy Monroe rule Given a ballot profile A = (A 1 , . . . , A n ) over a candidate set C and a target committee size k, the Greedy Monroe rule proceeds in k rounds. It maintains the set of available candidates C and the set of unsatisfied voters N ; initially C = C and N = N . It starts by setting W = ∅. In round t, t = 1, . . . , k, it selects a candidate c t from C and a group of voters N t from N of size approximately n k (specifically, n k if t ≤ n − k n k , and n k if t > n − k n k ) so as to maximize the quantity |{i ∈ N t : c t ∈ A i }| over all possible choices of (N t , c t ). The candidate c t is then added to W , and we set C = C \ {c t }, N = N \ N t . We say that the candidates in N t are assigned to c t . After k rounds, the rule outputs W .
PAV and RAV were defined by Thiele (1895). The Monroe rule was proposed by Monroe (1995) and Greedy Monroe is due to Skowron et al. Under each of the rules we consider, there may be more than one winning committee. In what follows, we assume that all ties are broken in some deterministic way; none of our results depends on the tie-breaking rule.
We will now define the key concepts in the work of Aziz et al. (2015a;: justified representation and extended justified representation. Definition 1. (Extended) justified representation ((E)JR) Consider a ballot profile A = (A 1 , . . . , A n ) over a candidate set C and a target committee size k, k ≤ |C|. Given a positive integer ∈ [k], we say that a set of voters N * ⊆ N is -cohesive if |N * | ≥ n k and | i∈N * A i | ≥ . A set of candidates W is said to provide -justified representation ( -JR) for (A, k) if there does not exist an -cohesive set of voters N * such that |A i ∩ W | < for each i ∈ N * . We say that W provides justified representation (JR) for (A, k) if it provides 1-JR for (A, k); it provides extended justified representation (EJR) for (A, k) if it provides -JR for (A, k) for all ∈ [k]. An approval-based voting rule satisfies -JR if for every ballot profile A and every target committee size k it outputs a committee that provides -JR for (A, k). A rule satisfies JR (respectively, EJR) if it satisfies -JR for = 1 (respectively, for all ∈ [k]).
By definition, EJR implies JR. Aziz et al. (2015a; show that PAV satisfies EJR (and hence JR), Monroe satisfies JR, but fails EJR, and RAV fails JR for sufficiently large values of k; they do not consider Greedy Monroe in their work.
3 Justified Representation and RAV Aziz et al. (2015a; prove that RAV satisfies JR for k = 2, but fails it for k ≥ 10. Whether RAV satisfies JR for k = 3, . . . , 9 was left an open problem. The following theorem provides a complete answer.
Theorem 1. RAV satisfies JR for k ≤ 5 but fails it for k ≥ 6.
Proof. For each k ∈ N, we construct a linear program LP k whose value is the maximum possible 'relative' approval weight of a yet unelected candidate (i.e., the ratio between her approval weight and the total number of voters) after k − 1 steps of RAV. Fix a k ∈ N. We can assume without loss of generality that RAV elects a committee {c 1 , . . . , c k }, where for i ∈ [k] candidate c i is added to the committee in round i; our linear program includes constraints that impose this order. Moreover, as non-elected candidates do not have any influence on the approval weight under RAV, we may assume that LP k has a variable x A for each nonempty candidate subset A ⊆ C k ; this variable corresponds to the fraction of voters that submit the approval ballot A. The objective function of LP k is the ratio of the approval weight of candidate c k and the total number of agents n after candidates {c 1 , . . . , c k−1 } have already been elected. The constraints say that all variables should be non-negative and sum up to 1, and that RAV can select for i = 1, . . . , k − 1 and for j = i + 1, . . . , k.
The number of variables in LP k grows exponentially with k, but this is not an issue, because we only have to solve this linear program for small values of k. Solving LP k for k = 3, 4, 5, 6, we obtain the following result.
Lemma 1. For k = 6 the value of LP k is 0.204 > 1 k−1 . For k = 3, 4, 5 the value of LP k is smaller than 1 k−1 .
Consider an optimal solution (x A ) A⊆C6 of LP 6 . We can find a positive integer n such all values n A = x A · n for A ⊆ C 6 are integer, and construct an n-voter ballot profile A = (A 1 , . . . , A n ) where each ballot A ⊆ C 6 occurs exactly n A times; moreover, we can pick n so that n/5 is an integer. Lemma 1 implies that when we execute RAV on (A, k), in each round RAV selects a candidate whose approval weight is at least 0.204n. Now, consider the ballot A = (A 1 , . . . , A 6n/5 ) over and each of the the additional n/5 voters has A i = {c 7 }. Suppose that we run RAV on A with k = 6. As we have 1 6 · (6n/5) = n/5, the JR axiom requires that c 7 is elected. However, this does not happen: at each point the approval weight of c 7 is n/5, whereas by Lemma 1 in each round RAV can find a candidate whose approval weight is at least 0.204n > n/5. In the appendix, we provide a concrete implementation of this idea: we describe an election with 5992 candidates on which RAV fails JR for k = 6. We also explain how to extend this example to k > 6.
We will now show that RAV satisfies JR for k = 3, 4, 5. Suppose for the sake of contradiction that RAV violates JR for some n-voter ballot profile A and some k ∈ {3, 4, 5}. That is, for some way of breaking ties RAV outputs a winning set W and there exists a set of voters N * of size at least n k such that all voters in N * approve some candidate c, yet no voter in N * approves any candidate in W . Let w be the candidate that is added to W during the k-th round.
Consider a ballot profile A obtained from A by removing all voters in N * , and suppose that we execute RAV on (A , k). It is possible to break intermediate ties in the execution of RAV so that RAV outputs W on (A , k), and, moreover, candidates are added to W in the same order on both inputs. Indeed, as none of the candidates in W is approved by the voters in N * , removing these voters does not change the approval weights of the candidates in W in each round of RAV, and can only lower the scores of the other candidates. Thus, we can assume that RAV selects W , and, moreover, adds w to W at the k-th step.
We can now apply Lemma 1 to A , which contains at most n = n − n k = n k−1 k voters; by the lemma, when w is added to W , its approval weight is strictly less than n k−1 = n k . Since none of the removed voters approved w, when RAV is executed on A, candidate w's approval weight in the k-th round is also strictly less than n k . But this means that RAV should have favored c over w in the k-th round, a contradiction.

Perfect Representation
A key application of multi-winner voting is parliamentary elections, where an important goal is to select a committee that reflects as fairly as possible the different opinions or preferences that are present in a society. Fairness in this context means that each committee member should represent approximately the same number of voters and as many voters as possible should be represented by a committee member that they approve. From this perspective, the best-case scenario is when each voter is represented by a candidate that she approves and each winning candidate represents exactly the same number of voters. Thus, we may want our voting rules to output committees with this property whenever they exist. This motivates the following definition.
Definition 2. Perfect representation (PR) Consider a ballot profile A = (A 1 , . . . , A n ) over a candidate set C, and a target committee size k, k ≤ |C|, such that k divides n. We say that a set of candidates W , |W | = k, provides perfect representation (PR) for (A, k) if it is possible to partition N into k pairwise disjoint subsets N 1 , . . . , N k of size n k each and assign a distinct candidate from W to each of these subsets in such a way that for each ∈ [k] all voters in N approve their assigned member of W . An approval-based voting rule satisfies PR if for every profile A and every target committee size k the rule outputs a committee that provides PR for (A, k) whenever such a committee exists.
An example of a voting rule that satisfies PR is the Monroe rule: a committee that provides perfect representation for an n-voter ballot profile has the Monroe score of n, i.e., the maximum possible score, whereas the Monroe score of any committee that does not provide perfect representation is at most n − 1.
We note that the PR axiom is quite demanding from a computational perspective: the problem of deciding whether there exists a committee that provides PR for a given pair Theorem 2. Given a ballot profile A and a target committee size k, it is NP-complete to decide whether there exists a committee that provides PR for (A, k).
Proof sketch. To show containment in NP, we reduce the problem of checking whether a given committee W provides perfect representation for (A, k) to the problem of finding a b-matching in a bipartite graph that can be associated with A and W ; the latter problem admits a polynomial-time algorithm (Anstee 1987). To prove NP-hardness, we adapt a reduction of Procaccia et al. (2008), which shows that finding a winning committee under the Monroe rule is NP-hard. The details can be found in the appendix.
Remark 1. Theorem 2 immediately implies that, unless P=NP, RAV and Greedy Monroe fail PR; it is also not hard to construct specific examples on which these rules fail PR. As for PAV, Theorem 3 below implies that it also fails PR.
Viewed from a different perspective, PR is a rather weak axiom: it only constrains the behavior of a voting rule on inputs that admit a committee that provides PR. In particular, this axiom has no bite if k does not divide n. Also, unlike EJR, PR does not engage with the idea that a voter may benefit from being represented by more than one candidate. Thus, we may want a voting rule to satisfy both PR and another representation axiom, such as, e.g., EJR. However, this turns out to be impossible: PR and EJR are incompatible.
Theorem 3. There exists a ballot profile A and a target committee size k such that the set of committees that provide PR for (A, k) is non-empty, but none of the committees in this set provides EJR.
Proof. Let C = {c 1 , . . . , c 6 }, and consider a ballot profile Observe that W = {c 1 , c 2 , c 3 , c 4 } is the unique committee of size 4 that provides PR for (A, 4). However, W fails to provide EJR: {A 5 , A 6 , A 7 , A 8 } is a 2cohesive set of voters, but each of these voters only approves one candidate in W .
This motivates the following question: can we find a weakening of the EJR axiom that still provides meaningful guarantees to large cohesive groups of voters, yet is compatible with PR? We address this question in the next section.

Proportional Justified Representation
The EJR axiom provides the following guarantee: at least one member of an -cohesive group has at least representatives in the committee. This focus on a single group member does not quite reflect our intuition of what it means for a group to be well-represented. A weaker and perhaps more natural condition is to require that collectively the members of an -cohesive group are allocated at least representatives. This idea is captured by the following definition. Definition 3. Proportional justified representation (PJR) Given a ballot profile A = (A 1 , . . . , A n ) over a candidate set C and a target committee size k, k ≤ |C|, we say that a set of candidates W , |W | = k, provides proportional justified representation (PJR) for (A, k) if for every ∈ [k] and every -cohesive set of voters N * ⊆ N it holds that |W ∩ ( i∈N * A i )| ≥ . We say that an approval-based voting rule satisfies proportional justified representation (PJR) if for every ballot profile A and every target committee size k it outputs a committee that provides PJR for (A, k).
It is immediate that every committee that provides PJR also provides JR: the PJR condition for = 1 is exactly JR. Also, it is easy to see that every committee that provides EJR also provides PJR: the condition "|A j ∩ W | ≥ for some i ∈ N * " in the definition of EJR implies the condition "|W ∩ ( i∈N * A i )| ≥ " in the definition of PJR. To summarize, we obtain the following proposition. Proposition 1. EJR implies PJR, and PJR implies JR.
Moreover, unlike EJR, PJR is compatible with PR. Theorem 4. For every profile A = (A 1 , . . . , A n ) and every target committee size k, if a set of candidates W , |W | = k, provides PR, then W also provides PJR.
Proof. Observe that because W provides PR, k divides n. Let W = {w 1 , . . . , w k }. As W provides PR, there exist k pairwise disjoint subsets N 1 , . . . , N k of size n k each such that all voters in N i approve w i for each i ∈ [k]. Consider a set of agents N * ⊆ N and a positive integer such that |N * | ≥ n k . By the pigeonhole principle, N * has a nonempty intersection with at least of the sets N 1 , . . . , N k . As each voter in N * ∩ N i approves w i , it follows that the the number of candidates in W approved by some voter in N * must be greater than or equal to .
Another advantage of PJR is that a committee that provides PJR can be computed in polynomial time as long as the target committee size k divides the number of voters n; indeed, under this condition both the Monroe rule and Greedy Monroe (the latter of which is polynomial-time computable) provide PJR. We note that PAV satisfies EJR and hence PJR even if k does not divide n; however, computing the output of PAV is NP-hard. Theorem 5. Consider a ballot profile A = (A 1 , . . . , A n ). If the target committee size k divides n then the outputs of Monroe and Greedy Monroe on (A, k) satisfy PJR.
Proof. We provide a proof for Greedy Monroe; the proof for Monroe can be found in the appendix.
Let s = n k ; note that s ∈ N. Suppose for the sake of contradiction that the set W output by Greedy Monroe fails PJR for some ∈ [k] and some -cohesive set of voters N * ; we can assume that |N * | = · s. Consider a candidate c ∈ C \ W that is approved by all voters in N * .
By the pigeonhole principle N * has a non-empty intersection with at least of the sets N 1 , . . . , N k constructed by Greedy Monroe (the integrality of s is crucial here); let the first of these sets be N i1 , . . . , N i with i 1 < · · · < i . For each t = 1, . . . , , pick a voter in N it ∩ N * ; note that all these voters are assigned to different candidates in W . Now, if each of these voters approves the candidate she is assigned to, we are done, as we have identified distinct candidates in W each of which is approved by some voter in N * . Otherwise, let j = min{i t : the voter we chose in N it ∩ N * does not approve of c it }. By our choice of j, not all voters in N j approve c j , yet the pair (N j , c j ) was chosen at step j. Among the sets N 1 , . . . , N j−1 there are at most − 1 sets of size s each that have a non-empty intersection with N * , so at step j at least s voters in N * are present in N (N is the set of unsatisfied voters; see the definition of Greedy Monroe in section 2). Now, candidate c, together with s voters from N * ∩ N , would be a better choice for Greedy Monroe than (N j , c j ), a contradiction.
It is then natural to ask if there is a polynomial-time computable voting rule that satisfies PJR for all values of n and k. Interestingly, it turns out that the answer to this question is 'yes': in a very recent paper, Brill et al. (2017) describe an approval-based multi-winner rule developed by the Swedish mathematician Lars Edvard Phragmén more than 100 years ago, and show that a sequential variant of this rule, which they refer to as seq-Phragmen, is polynomialtime computable and provides PJR. Another voting rule with this combination of properties is the ODH rule, which has been proposed by Sánchez-Fernández et al. in a recent arXiv preprint (Sánchez-Fernández et al. 2016). Interestingly, both rules are extensions of the D'Hondt seat allocation method (Farrell 2011) to approval-based multi-winner elections.
Moreover, PJR inherits a useful feature of EJR: it characterizes PAV within the class of w-PAV rules.

Average Satisfaction
A useful measure in the context of justified representation is that of average satisfaction: given a ballot profile (A 1 , . . . , A n ), a committee W , and a group of voters N * ⊆ N , we define the average satisfaction of the voters in N * as 1 While it is maybe impossible to ensure that every group of voters has high average satisfaction, it is natural to ask if we can provide some guarantees with respect to this measure to groups of voters that are large and cohesive.
Our first observation is that if a committee W provides JR, we can derive a lower bound on the average satisfaction of such groups. Proposition 4. Consider a ballot profile (A 1 , . . . , A n ), and suppose that the target committee size k divides n. Let W be a committee of size k that provides JR, and let be a positive integer. For every -cohesive group of voters N * we have Proof. Let s = n k . Since W provides JR, we can find a voter i 1 ∈ N * with |A i ∩ W | ≥ 1. If |N * | > s then the set N * \{i 1 } satisfies |N * \{i 1 }| ≥ s, | i∈N * \{i1} A i | ≥ , so, applying the JR condition to this set, we can conclude that there is another voter i 2 = i 1 that approves some candidate in W . By repeating this argument, we conclude that at most s − 1 voters in N * approve no candidate in W . Hence, However, for voting rules that satisfy EJR we can obtain a much stronger guarantee. Theorem 6. Consider a ballot profile (A 1 , . . . , A n ), and suppose that the target committee size k divides n. Let W be a committee of size k that provides EJR, and let be a positive integer. Then for every -cohesive group of voters N * ⊆ N it holds that Proof. Let s = n k and let |N * | = n * . EJR implies that every subset of N * of size · s contains a voter who approves candidates in W . Let i 1 be some such voter. If n * > · s then the set N * \ {i 1 } satisfies |N * \ {i 1 }| ≥ · s, | i∈N * \{i1} A i | ≥ , so, applying the EJR condition to this set, we can conclude that there is another voter i 2 = i 1 that approves candidates in W . By repeating this argument, we can construct a subset N ⊆ N * of size n * − · s + 1 such that each voter in N approves at least candidates in W . Now, consider the set N * \ N . We have |N * \ N | = · s − 1 = ( − 1) · s + (s − 1), | i∈N * \N A i | ≥ . Thus, by the same argument, the set N * \ N contains at least s voters each of which approves at least − 1 candidates in W ; let N −1 be a set of exactly s such voters.
The average satisfaction of voters in N * \ N is at least whereas the average satisfaction of the voters in N is at least . As the average satisfaction of voters in N * is a convex combination of these two quantities, it is at least −1 2 . In contrast, the worst-case guarantee provided by PJR is not any stronger than the one provided by JR alone. Example 2. Consider a ballot profile (A 1 , . . . , A n ) over a candidate set C = {c 1 , . . . , c n , d 1 , . . . , d n } where A i = {d 1 , . . . , d n , c i } for i ∈ [n]. For k = n, the committee {c 1 , . . . , c n } provides PJR (and PR), but the average satisfaction of the voters in N (which form an n-cohesive group) is only 1.

Discussion
We consider JR to be an important axiom in a variety of applications of multi-winner voting. Thus, the result of Section 3 shows that RAV should not be ruled out on these grounds if the target committee size is small. In some applications, such as, e.g., shortlisting candidates for a job, k ≤ 5 may be a reasonable assumption (Barberà and Coelho 2008;Elkind et al. 2017). We find it surprising that the threshold value of k turns out to be 5 rather than 2 or 3; it would be interesting to see a purely combinatorial proof of this fact.
Our results also highlight a difficulty with the notion of EJR: this axiom is incompatible with perfect representation, which is a very desirable property in parliamentary elections and other settings where fairness is of paramount importance. We therefore propose an alternative to this axiom, Proportional Justified Representation, which is motivated by similar considerations (namely, ensuring that large cohesive groups of voters are allocated several representatives), but does not conflict with PR. PJR also has further attractive properties: it is satisfied by several well-known multi-winner rules (for some of these rules we have to additionally require that k divides n), some of which are efficiently computable, and, just like EJR, it provides a justification for using the harmonic weight vector (1, 1 2 , 1 3 , . . . ) as the default weight vector for PAV.
However, the results of Section 6 can be viewed as an argument in favor of EJR: every committee that provides EJR guarantees high levels of average satisfaction to members of large cohesive groups, whereas the guarantee offered by committees that provide PJR is, in general, much weaker. Thus, one can think of EJR as a more pragmatic requirement: for every ballot profile a committee that provides EJR (and, as shown by Aziz et al. (2015a;, such a committee is guaranteed to exist) ensures that members of large cohesive groups are happy on average, at the cost of possibly ignoring other agents. In some applications of multi-winner voting such a tradeoff may be acceptable. Consider, for instance, an academic department where members of different research groups pool their funding to run a departmental seminar. Faculty members have preferences over potential speakers, with members of each research group agreeing on a few candidates from their field. Choosing speakers so as to please the members of large research groups may be a good strategy in this case, even if this means that some members of the department will not be interested in any of the talks. Indeed, if very few talks are of interest to members of a large group, this group may prefer to withdraw its contribution to the funding pool and run its own event series.
The relationship between PR and PJR is more subtle than it may seem at the first sight: while PR implies PJR at the level of committees (Theorem 4), this is not the case at the level of voting rules. Indeed, a voting rule that satisfies PR may behave arbitrarily when the committee size k does not divide the number of voters n, whereas the PJR axiom remains applicable in such scenarios. Of course, a voting rule can satisfy PJR, but not PR: Greedy Monroe, restricted to instances where k divides n, is a case in point. Thus, these two axioms are only loosely related. Indeed, none of the voting rules we consider always satisfies both PJR and PR: the Monroe rule comes closest, but even this rule only satisfies PJR when k divides n (see Example 1). We can, however, construct an artificial voting rule that satisfies both axioms: given (A, k), this rule checks if there is a committee that provides PR for (A, k) and, if so, outputs some such committee (which, by Theorem 4, also provides PJR), and otherwise it runs PAV. This voting rule can be seen as an analogue of the Black rule, which is a single-winner rule that outputs a Condorcet winner if one exists and a Borda winner otherwise. Very recently, Brill et al. (2017) identified a voting rule that provides both PR and PJR for all values of n and k, namely, a maximization version of the Phragmén's rule, which they refer to as max-Phragmen.
We conclude the paper by mentioning some open questions that are raised by our work. First, we do not know what is the complexity of checking whether a given committee provides PJR; we note that this problem is polynomialtime solvable for PR (the first part of Theorem 2) and coNPcomplete for EJR (Theorem 11 of Aziz et al.). Also, it would be useful to derive bounds on average satisfaction provided by committees that are produced by the voting rules considered in this paper, both theoretically and empirically. In particular, it would be interesting to see whether PAV, which satisfies EJR, performs better in this regard than Monroe or Greedy Monroe, which only satisfy PJR.
A Justified Representation and RAV: Omitted Proofs We will now present an explicit example of a ballot profile on which RAV violates JR for k = 6. This profile, which is described in Table 1, is obtained from a solution to the respective linear program, as described in the proof of Theorem 1. The total number of agents in this example is 5992. Since 1000 agents approve c 7 only, JR requires that c 7 should be in the winning set. But this does not happen: in round 1 c 1 is elected with an approval weight of 2000; in round 2 c 2 is elected with an approval weight of 1499; in round 3 c 3 is elected with an approval weight of 1220.5; in round 4 c 4 is elected with an approval weight of 1060.33; in round 5 c 5 is elected with an approval weight of 1017.67; and finally, in round 6 c 6 is elected with an approval weight of 1017.17. There are no ties.
To show that RAV fails JR k = 7, 8, 9, we adapt the argument from the work of Aziz et al. (2015a;. Specifically, we add to the election shown in Table 1 k−6 additional candidates and 1000(k − 6) additional voters such that for each new candidate there are 1000 voters who approve that candidate only. Note that we still have 1000 > n/k. RAV will select c 1 , . . . , c 6 , followed by k − 6 additional candidates, and c 7 or one of the new candidates will remain unselected.
We remark that we can convert the linear program from the proof of Theorem 1 into an integer linear program that for each value of k finds an election with the minimum number of voters on which RAV never has to break ties and produces a committee that violates JR. Interestingly, while for k = 6 the smallest such example has 108 voters (a valid solution is shown in Table 2), for k = 7 there exists an example with 35 voters (a valid solution is shown in Table 3).

B Perfect Representation: Omitted Proofs
We will now provide a complete proof of Theorem 2. Theorem 2. Given a ballot profile A and a target committee size k, it is NP-complete to decide whether there exists a committee that provides PR for (A, k).
Proof. To show that this problem is in NP, we will argue that, given a ballot profile A = (A 1 , . . . , A n ) and a committee W of size k, where k divides n, we can reduce the problem of deciding whether W provides PR for (A, k) to finding a b-matching in a bipartite graph.
Recall that an instance of a b-matching problem is given by a bipartite graph G with parts L and R and edge set E, where each node u ∈ L∪R is associated with a positive integer κ(u); it is a 'yes'-instance if there is a collection E ⊆ E of edges of G such that each node u ∈ L ∪ R is contained in exactly κ(u) edges of E . It is possible to check in polynomial time if a given instance of the b-matching problem is a 'yes'-instance (Anstee 1987). Now, given a ballot profile A = (A 1 , . . . , A n ) and a committee W of size k, where k divides n, we construct a bipartite graph with parts N and W where there is an edge from i to c if and only if c ∈ A i . We then ask if this graph admits a b-matching if the capacity of each node in N is 1 and the capacity of each node in W is n k . Clearly, W provides PR if and only if such a b-matching exists.
To prove NP-hardness, we modify a reduction in the paper of Procaccia et al. 2008, which shows that finding a committee whose Monroe score is at least as high as a given bound is NP-hard.
Recall that an instance of X3C is given by a set U = (u 1 , . . . , u ν ) of size ν, where ν is divisible by 3, and a collection F = {F 1 , . . . , F r } of size-3 subsets of U . It is a 'yes'-instance if it is possible to find ν/3 sets in F whose union covers the set U .
Given an instance (U, F) of X3C, we construct a set of candidates C, a ballot profile A and a target committee size k as follows. We set n = ν, C = {c 1 , . . . , c r } and k = n/3. For each i ∈ [n], we let A i = {c j | u i ∈ F j }. By construction, n/k = 3.
We claim that (U, F) is a 'yes'-instance of X3C if and only if there exists a committee W that provides perfect representation for (A, k).
Indeed, suppose that W = {c j1 , . . . , c j k } is a committee that provides perfect representation for (A, k), and let (N 1 , . . . , N k ) be respective partition of N such that for each ∈ [k] each voter in N approves c j . Then {F j1 , . . . , F j k } is a cover for U : for each u i ∈ U there exists some ∈ [k] such that i ∈ N ; as i approves c j , this means that u i is covered by F j .
Conversely, if {F j1 , . . . , F j k } is a cover for U , we can set N = {i | u i ∈ F j } for all ∈ [k]: by construction all voters in N approve c j and |N | = 3 for each ∈ [k], so W = {c j1 , . . . , c j k } provides perfect representation for (A, k).

C Proportional Justified Representation: Omitted Proofs
We will now provide the missing part of the proof of Theorem 5, by showing that the Monroe rule satisfies PJR when k divides n. Theorem 5 . Consider a ballot profile A = (A 1 , . . . , A n ).
If the target committee size k divides n then the output of the Monroe rule on (A, k) satisfies PJR.
Proof. Let s = n k . Note that s ∈ N. Suppose that the Monroe rule outputs a committee W ; assume without loss of generality that W = {c 1 , . . . , c k }. Let π be a valid mapping N → W whose Monroe score is equal to the Monroe score of W . Let N j = π −1 (c j ) for j = 1, . . . , k.
Suppose for the sake of contradiction that there exists an > 0 such that PJR is violated for some -cohesive set of voters N * . Note that this means that at least one of the candidates jointly approved by voters in N * does not appear in W ; let c be some such candidate.
Let us say that a set N j is good if at least one voter in N j ∩ N * approves c j , and bad otherwise. Note that if there are at least good sets among N 1 , . . . , N k then there are distinct candidates in W each of which is approved by at least one voter in N * , a contradiction with our choice of N * .
A {c 1 , c 2 , c 4 , c 5 } {c 1 , c 2 , c 4 , c 6 } {c 1 , c 3 , c 5 } {c 1 , c 3 , c 6 } {c 2 , c 3 , c 4 } {c 2 , c 3 , c 5 } {c 2 , c 5 } {c 2 , c 6 } n A  500  500  500  500  222  333  55   Thus, there are at most − 1 good sets, so at most ( − 1)s voters from N * appear in good sets. Hence, there is bad set N j with N j ∩ N * = ∅. Also, there are at least s voters in N * that are assigned to candidates they do not approve; let N ⊆ N * be a set that consists of s such voters.
Let W = (W \ {c j }) ∪ {c}. We now construct a valid mapping π : N → W as follows. We let π (i) = π(i) for all i ∈ N \ (N ∪ N j ). Then, we set π (i) = c for each i ∈ N . Finally, for each voter i ∈ N j \ N we define π (i) so as to obtain a valid Monroe mapping; this can be done, e.g., by establishing a bijection µ between N j \ N and N \ N j and setting π (i) = π(µ(i)).
To obtain a contradiction, it remains to observe that the score of π is higher than the score of π: by our choice of N j , at most s − 1 voters in N ∪ N j approve their assigned candidate under π, whereas all s voters in N are approve their assigned candidate under π, and all voters in N \ (N ∪ N j ) are indifferent between π and π . Proposition 2. The rule w-PAV satisfies PJR if and only if w = (1, 1 2 , 1 3 , . . . ).
Our proof that w-PAV fails PJR if w = (1, 1 2 , 1 3 , . . . ), is very similar to the respective proof for EJR by Aziz et al. (2015a;. First, Aziz et al. show (Lemma 1 in the journal version) that if w j > 1 j for some j > 1 then w-PAV fails JR. Therefore, in such cases w-PAV fails PJR as well.
Second, if w j < 1 j for some j > 1, Aziz et al. construct (Lemma 2 in the journal version) an election in which there exists a subset of candidates C with |C | = j and a set of j n k voters who approve candidates in C only such that w-PAV only elects j − 1 candidates from C . This establishes that w-PAV fails PJR in this case.
Proposition 3. The rule w-RAV fails PJR for each weight vector w.

D A note on Average Satisfaction and PJR
We have shown that the worst-case average satisfaction over cohesive group of voters that is guaranteed by PJR is the same as the one guaranteed by JR. It has to be noted, however, that when the group of voters is "very cohesive" (that is, when all the voters approve exactly the same set of candidates), the average satisfaction that is guaranteed by PJR and EJR is the same and much higher than the one guaranteed by JR. In particular, for an election in which a group of voters of size n/k exists, such that all the voters in such group approve precisely the same set of candidates (of size at least ), a set of winners for such election that provides PJR (or EJR) has a minimum average satisfaction for the "very cohesive" group of voters of . However, a set of winners for such election that provides JR has a minimum average satisfaction for the "very cohesive" group of voters of only 1.