tag:theconversation.com,2011:/id/topics/applied-mathematics-36638/articlesApplied mathematics – The Conversation2024-02-05T13:34:48Ztag:theconversation.com,2011:article/2213522024-02-05T13:34:48Z2024-02-05T13:34:48ZHow bats ‘leapfrog’ their way home at night – new research<figure><img src="https://images.theconversation.com/files/572171/original/file-20240130-27-o1vlrb.jpg?ixlib=rb-1.1.0&rect=0%2C10%2C3413%2C2539&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">The greater horseshoe bat is one of the UK's 18 bat species. </span> <span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/flying-bat-hunting-forest-greater-horseshoe-1494098204">Rudmer Zwerver/Shutterstock</a></span></figcaption></figure><p>A silent ballet takes place above our heads at night as Britain’s bat populations leave their roosts to forage for food. Although their initial movement away from roosts is fairly well understood, until recently little was known about how they returned home. </p>
<p>But our <a href="https://link.springer.com/article/10.1007/s11538-023-01233-5">new research</a> shows how bats may use a “leap-frogging” motion to make their way home, something which could help conservationists in future.</p>
<p>As they flit through the darkness, bats play a crucial role in the health of our ecosystems. From keeping insect populations in check to dispersing seeds and pollinating plants, they provide a multitude of benefits. </p>
<p>In the UK alone, the 18 bat species devour agricultural pests such as cockchafers with impressive efficiency. So, it is imperative that we not only understand and appreciate bats, but also actively support and safeguard their populations for the wellbeing of our planet.</p>
<p>But bat populations are vulnerable to pollution, climate change and loss of roosting locations. Habitat fragmentation and light pollution can also interrupt how bats feed. This is particularly important during the maternity season <strong>in early summer</strong>, when bats gather together to have and raise their young.</p>
<p>An integral aspect of effective bat conservation lies in unravelling the mysteries of how bats move. This not only helps us understand how bats navigate and use their environment, but also helps in identifying and protecting their roosts. </p>
<h2>Radio-tracking</h2>
<p>Conventional methods for pinpointing bat roosts primarily hinge on radio-tracking surveys. This arduous process involves capturing bats, attaching small radio transmitters to them before releasing them and following the signals throughout the night. </p>
<p>Our team conducted a radio-tracking survey in Devon which monitored 12 greater horseshoe bats over 24 nights. The trajectories of seven of those bats over 14 nights were extracted from the data for analysis, ensuring that in each case, a bat’s beginning and ending roost were the same.</p>
<p>Using this data, we measured the population’s average distance from the roost. We found two distinctive patterns in the data we analysed: an initial spread of bats within the first one to two hours after sunset and a gradual return to the roost afterwards.</p>
<p>The initial spread reflects the expected random dispersal of bats leaving their roosts to forage after sunset. The return to the roost, occurring two to eight hours after sunset, is more complicated. </p>
<p>This prompted us to explore two potential mechanisms influencing the bats’ return. First, a “pull mechanism”, where the roost attracts the bats home, and second, a mechanism pushing the bats who range furthest away back to the roost.</p>
<p>We modelled the pushing mechanism as a leapfrog process. Imagine this as a cascade effect, where the outermost bats begin their return. Once the “outer” bats have passed or “leapt over” bats that are closer to the roost, the “inner” bats become the furthest out causing them to return too.</p>
<p>This motion unfolds systematically, like a synchronised dance, as each bat from the periphery of the foraging range follows suit in returning to the roost after being “leapfrogged”.</p>
<p>But what causes the bats to return in this manner? One plausible explanation underscores how bats rely on each other for effective navigation, like tiny radar signals. If a bat experiences prolonged silence or predominantly hears calls from one direction, it might decide to move closer to the roost, anticipating the presence of other colony members. </p>
<p>But a bat might return more slowly, prolonging foraging, if it perceives the presence of bats beyond its current location. So, it is the outer bats that would drive the return as they would not be surrounded by calls.</p>
<h2>How does this research help bats?</h2>
<p>The significance of these findings extends beyond just describing the movements of bats. They have laid the foundation for work that promises easier discovery of new bat roosts, potentially reducing the need for labour-intensive bat tracking surveys in the future. </p>
<p>One of the immediate effects of our research includes informing a measurement of the “core sustenance zone” for greater horseshoe bats. This is where most of their foraging occurs, so it’s important in bat ecology, conservation and construction planning.</p>
<p>The leapfrogging mechanism also allows us to ascribe intention to bat movements. Namely, through using surrounding bat calls they can identify where the population is relative to their position, suggesting whether or not they are on the periphery of the group, which is an indicator of their vulnerability. Should they be furthest from the roost they move back towards the bulk of the population and closer to the roost.</p>
<p>While these interpretations hold promise, further rigorous testing is essential. And we need to think about the safety and wellbeing of the bat population.</p>
<p>Our observations are also specific to greater horseshoe bats during the summer months. Different bat species have distinct flight patterns and habitat preferences, with the same species displaying diverse behaviours at different times of the year. </p>
<p>So, while we have taken some crucial first steps, we still have a lot of work to do in unravelling the characteristics of bat motions in general.</p><img src="https://counter.theconversation.com/content/221352/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Fiona Mathews receives funding from Devon Area of Outstanding Natural Beauty, Devon County Council and the Natural Environment Research Council. She is affiliated with the UK Mammal Society, Mammal Conservation Europe, Ecotype Genetics and Ecology Search Services Ltd. </span></em></p><p class="fine-print"><em><span>Thomas Woolley does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Maths plays a crucial role in new research which finds that bats “leapfrog” their way home at night.Thomas Woolley, Senior Lecturer in Applied Mathematics, Cardiff UniversityFiona Mathews, Professor of Environmental Biology, University of SussexLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/2042232023-04-25T14:06:58Z2023-04-25T14:06:58ZDobble: what is the psychology behind the game?<figure><img src="https://images.theconversation.com/files/522365/original/file-20230421-16-7hgsp2.jpg?ixlib=rb-1.1.0&rect=8%2C0%2C5982%2C3997&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Dobble is a card game with rules that makes it sound easier than it actually is.</span> <span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/dobble-card-game-kids-billereaquitainefrance-08232021-2029552409">Ana Belen Garcia Sanchez/Shutterstock</a></span></figcaption></figure><p>Following birthday and Christmas presents, families often have a glut of new games to learn and play. Many of these games involve computers or games consoles, but with concerns about children’s <a href="https://www.forbes.com/health/family/how-much-screen-time-kids/">screen time</a> there has been a recent <a href="https://www.washingtonpost.com/business/2022/12/24/board-game-popularity/">increase</a> in the popularity of traditional board and card games.</p>
<p>One non-electronic card game that has made its way into our homes is Dobble. It’s a game of observation, articulation and speed that was first released in France in 2009. </p>
<p>While the <a href="https://www.petercollingridge.co.uk/blog/mathematics-toys-and-games/dobble/">mathematics</a> behind the workings of this game is interesting, as cognitive psychologists we were also fascinated by the underlying cognitive processes that make this simple game so absorbing and challenging to play.</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/VTDKqW_GLkw?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">How does Dobble work mathematically?</span></figcaption>
</figure>
<p>The aim of the game is to be the first player to get rid of all their cards by discarding them one at a time into a central pile. Players do that as soon as they can identify, and announce, the single common symbol between the card in their hand and that on top of the pile. </p>
<p>Players must be quick as the top card will change every time your opponent(s) are able to match and discard one of their cards before you. There are 55 cards, each containing eight symbols out of a possible 57. And in any pair of cards, only one symbol matches. </p>
<p>The first task in the game is to visually search the symbols on both the card in your hand and that on the top of the central pile to find the single match. Colour, size and location are typical <a href="https://psycnet.apa.org/record/1996-04250-015">cues</a> we use when searching. But this task is more difficult than it seems due to the number and variety of symbols. Their shared features sometimes give rise to false alarms when scanning quickly. For example, the lips, heart, maple leaf and fire symbols are all red in colour. </p>
<p>The fact the target items will likely be of a different size and orientation on each card also means that we <a href="https://books.google.co.uk/books?hl=en&lr=&id=HktnDAAAQBAJ&oi=fnd&pg=PA26&dq=perception+object+match+different+orientation&ots=wBLoJvKx-H&sig=IqWZ-6H9R4HpaOscqGXlbkjA3W4#v=onepage&q&f=false">perceive</a> the same symbol slightly differently. So a match is more difficult to identify. </p>
<p>Unlike, for example, <a href="https://www.independent.co.uk/arts-entertainment/books/news/where-s-the-brains-behind-wally-6261459.html">Where’s Wally?</a>, where the object of the search is clearly defined, with Dobble we do not know on any round which item we are searching for. Indeed, this will be different for each player. </p>
<p>The task requires dividing attention by searching two visual scenes in parallel. And also holding in memory the symbols that you have viewed on one card for comparison with those on the other. </p>
<p>We may <a href="http://matt.colorado.edu/teaching/highcog/fall8/m3.pdf">switch</a> between different strategies such as scanning the symbols on both cards in the hope that the match will just “pop out”. Or we may adopt a more structured approach where we peruse each symbol in turn. </p>
<p>When demands on attention are high, we are more likely to suffer <a href="https://books.google.co.uk/books?hl=en&lr=&id=Z2Sz7YgWIpQC&oi=fnd&pg=PA55&dq=inattentional+blindness+divided+attention&ots=2rrM836Idb&sig=IPiM1lTPKa-JlXAJ9QUHbZHmPvw#v=onepage&q=inattentional%20blindness%20divided%20attention&f=false">inattentional blindness</a>. That’s the phenomenon of “looking but not seeing”, whereby the item we are fixating on does not receive enough attention for us to actually notice it.</p>
<h2>Say the name</h2>
<p>Once you have found the matching symbol you must quickly announce what it is before placing your card down on the pile. This again sounds simple, but, just like producing the correct word in everyday speech, it requires the <a href="https://mybrainware.com/blog/brainware-safari-cognitive-skills-development-and-learning-to-read/">processes</a> of linking the desired concept – the symbol on the cards – with the name that represents it. </p>
<p>Also, you have to ensure that you select the appropriate word, for example saying “tortoise” rather than “turtle”. Plus you must select the correct sounds to utter that word, before finally saying it out loud. In the urgency of the game, you may find these processes don’t happen as quickly as you want them to.</p>
<figure class="align-center ">
<img alt="A pair of hands holds a collection of round cards. There is another pile of round cards on the table beneath the hands." src="https://images.theconversation.com/files/522130/original/file-20230420-18-n8asch.jpg?ixlib=rb-1.1.0&rect=0%2C0%2C6016%2C4016&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/522130/original/file-20230420-18-n8asch.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=401&fit=crop&dpr=1 600w, https://images.theconversation.com/files/522130/original/file-20230420-18-n8asch.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=401&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/522130/original/file-20230420-18-n8asch.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=401&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/522130/original/file-20230420-18-n8asch.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/522130/original/file-20230420-18-n8asch.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/522130/original/file-20230420-18-n8asch.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Dobble - it’s not as easy as saying what you see.</span>
<span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/verona-italy-february-2nd-2021-detail-1921867253">Claire Adams/Shutterstock</a></span>
</figcaption>
</figure>
<p>Once you have correctly articulated the matching symbol and played your card, the whole process starts again. Given the low chance of that previous symbol being the next correct match, you must inhibit (stop yourself thinking about) this recent item – its name, its location, even its colour – so that you can be open to a new search. However, you must not inhibit it completely as there is still a chance it could appear next. </p>
<p>Inhibition is also required if your opponent calls out a symbol on their card first. Even if you were about to articulate a match, you must now inhibit this vocalisation and instead restart the search for a new pairing since the reference card in the centre has now changed. This ability to switch between searches and inhibit unwanted information is one of a number of <a href="https://www.tandfonline.com/doi/abs/10.1080/13803395.2010.533157">“executive”</a> organisational cognitive processes that help us in the planning and coordination of activities.</p>
<h2>Under stress</h2>
<p>And of course, all of this occurs under time pressure. Stress can increase when it seems your opponent is discarding their cards quicker. We know that increased stress levels impair our <a href="https://www.researchgate.net/profile/Tony-Buchanan/publication/320312261_Tip_of_the_Tongue_States_Increase_Under_Evaluative_Observation/links/59e0d8b1aca2724cbfd5e271/Tip-of-the-Tongue-States-Increase-Under-Evaluative-Observation.pdf">word-finding ability</a>, attention to information, inhibition of responses and <a href="https://pubmed.ncbi.nlm.nih.gov/28690203/">ability to adapt</a> to changing circumstances. All of those are vital to performing well in Dobble. </p>
<p>The bad news for parents is that many of the processes we have described <a href="https://link.springer.com/content/pdf/10.1038/s41598-020-80866-1.pdf">decline</a> as we get older, meaning that children may have the competitive edge at Dobble.</p><img src="https://counter.theconversation.com/content/204223/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>The authors do not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and have disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Dobble is a card game that originated in France in 2009. It involves observation, articulation and speed.Nick Perham, Reader in Applied Cognitive Psychology, Cardiff Metropolitan UniversityHelen Hodgetts, Reader in Applied Cognitive Psychology, Cardiff Metropolitan UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1992312023-02-20T16:32:43Z2023-02-20T16:32:43ZHow queuing leads to city centre violence and what our research says about preventing night-time brawls<p>People go out at night because they want to socialise, drink and be entertained. Unfortunately, all too often that leads to violent behaviour in our towns and city centres. But the events that lead to such violence are poorly understood. </p>
<p>We set out to explore some of the possible explanations of night-time violence using data on Cardiff city centre footfall (the number of people in the city centre) and assault-related attendances at the nearby University Hospital of Wales. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9779416/">We found</a> that a break down in the unwritten etiquette of queuing may be one of the reasons behind increases in violence at night.</p>
<p>When revellers gather to enjoy themselves at night, they often drink alcohol and possibly take drugs. This typically sets the activity apart from other places where people gather, such as transport centres or places for other commercial activity such as shopping. </p>
<p>People are attracted to night-time environments based on the <a href="https://journals.sagepub.com/doi/abs/10.1177/004728759303200204">total number of social opportunities they provide</a>, whether it’s going clubbing or visiting pubs. So, while entertainment venues compete against one another for trade, they also collectively market to attract patrons from near and far.</p>
<p><strong>The relationship between footfall and assault related injuries</strong></p>
<figure class="align-center ">
<img alt="A graph showing days of the week on the horizontal axis and numbers on both vertical axes. A line loops up and down." src="https://images.theconversation.com/files/508108/original/file-20230203-2880-nenoi7.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/508108/original/file-20230203-2880-nenoi7.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=338&fit=crop&dpr=1 600w, https://images.theconversation.com/files/508108/original/file-20230203-2880-nenoi7.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=338&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/508108/original/file-20230203-2880-nenoi7.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=338&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/508108/original/file-20230203-2880-nenoi7.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=424&fit=crop&dpr=1 754w, https://images.theconversation.com/files/508108/original/file-20230203-2880-nenoi7.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=424&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/508108/original/file-20230203-2880-nenoi7.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=424&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">A graph showing the relationship between footfall and assault related injuries.</span>
<span class="attribution"><span class="source">author provided</span></span>
</figcaption>
</figure>
<p>There has been plenty of research into what reduces or promotes night-time violence in city centres. One of the clear signals of danger is that the larger the footfall in the area, <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9779416/">the larger the chance for assaults to occur</a>. </p>
<p><a href="https://www.sciencedirect.com/science/article/pii/S1359178908000451?via%3Dihub">Crowding</a> and <a href="https://link.springer.com/article/10.1057/cpcs.2012.12">noise</a> are associated with increases in violence in city centres at night. And, in Australia, it has been shown that when <a href="https://www.tandfonline.com/doi/full/10.1080/10826084.2021.2019772">trading hours are restricted</a> there is a <a href="https://onlinelibrary.wiley.com/doi/10.1111/dar.12123">decrease in violence</a>.</p>
<p>But our research shows the correlation between footfall and assault is not linear. In other words, if we double the footfall, we do not simply double the number of assaults. The relationship between these two factors is more complicated, so we decided to investigate what could account for that.</p>
<h2>Queue etiquette</h2>
<p>One particular aspect we considered was the role drunkenness has to play because it <a href="https://royalsocietypublishing.org/doi/full/10.1098/rstb.2008.0028?casa_token=m93tA6ou_hwAAAAA:BprjJMUf5LNgIp6ftJlEBAnpBA-rAIQMvv6gvvCh4-wtDDkZFtYAN8LRNdVRDs2CCS7U8EkiV4PzGQ">affects how people cooperate</a>, for example when queuing. Queues are a social response to resource competition, whether that resource is nightclub entry, a pint of beer or a taxi. </p>
<p>However, since queuing is a social phenomenon, the people waiting in line have <a href="https://pubsonline.informs.org/doi/abs/10.1287/opre.35.6.895">expectations about how others should behave</a>, such as not skipping to the front. </p>
<p>When a violation of those unwritten rules occurs, people queuing in an orderly fashion will seek to defend the queue’s order, with the most vocal complaints stemming from those who are <a href="https://psycnet.apa.org/buy/1987-04011-001">closest to where the person jumps into the line</a>. Although even those ahead of the intrusion <a href="https://onlinelibrary.wiley.com/doi/full/10.1111/j.1559-1816.2008.00396.x?casa_token=xrR27tjG5jcAAAAA%3A-ub-j1Onmr9Senw4N2aGVljAgfifCQvqv_qWTGqPbTm437LilyojU30Mq1vyB2X_lOv3A_KMxLiSPo8">may also react to the injustice</a>. </p>
<p>However, whether there’s a queue violation or not, waiting in line makes people stressed. This increases <a href="https://link.springer.com/article/10.1177/0092070396244005">the longer they believe they have been waiting</a>. In turn, such stress can lead to aggression.</p>
<p>To understand the role queues play in the relationship between footfall and assault, we used a mathematical model to help predict what would happen in a variety of night-time scenarios. </p>
<p>We assumed the average arrival time of people into a queue is constant. We also assumed the rate at which they are served and leave the queue is constant, but also rises and falls in line with the number of servers, such as bar staff, taxi drivers or similar. </p>
<p>We also adjusted the models to take account of various other factors, such as weather, bank holidays and whether there were Six Nations or other international rugby matches being played at Cardiff’s Principality Stadium.</p>
<p>We found there was a significant relationship between the number of people in the city centre and the number of assaults recorded in the hospital’s accident and emergency department. The relationship relating footfall with assaults we saw from our queuing models performed better than the simple linear relationship. This is why doubling footfall does not double assaults.</p>
<p>Our study also found events such as bank holidays and rugby matches led to an increase in violence, beyond what might be expected from the impact of footfall alone. Additionally, warmer weather also increased the likelihood of assaults but more rain did not have a significant effect.</p>
<h2>Cutting city centre violence</h2>
<p>Our mathematical models show that by reducing queuing time, stress and related violence drops too. The average waiting time drops dramatically as the number of servers increases. So when pubs, taxi services or similar are understaffed, that increases the competition between people queuing.</p>
<p>The <a href="https://www.legislation.gov.uk/ukpga/2003/17/contents">UK Licensing Act 2003</a> places a duty on licensed premises to prevent crime and disorder and to maintain public safety. But there are no provisions on how licensed activity should increase as the number of patrons increases. </p>
<p>If further research confirms our observations, then there is a need to address the design and operation of night-time services, not only of bars, but of other areas where queues of revellers might form, such as taxi ranks and fast food outlets.</p><img src="https://counter.theconversation.com/content/199231/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>James White receives funding from the NIHR, MRC, Health and Care Research Wales, Scottish Chief Scientist Office, and Department for Education. </span></em></p><p class="fine-print"><em><span>Simon C Moore receives funding from NIHR, Youth Endowment Fund, ESRC, MRC. </span></em></p><p class="fine-print"><em><span>Thomas Woolley does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Recent research shows how the relationship between alcohol consumption, queuing and crowds can lead to violent behaviour in city centres at night.Thomas Woolley, Senior Lecturer in Applied Mathematics, Cardiff UniversityJames White, Chair professor, Cardiff UniversitySimon C Moore, Professor of Public Health Research, Co-Director of Crime and Security Research Institute and Director of the Violence Research Group, Cardiff UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1277302019-12-23T20:18:26Z2019-12-23T20:18:26ZMerry Christmaths: the statistics of Secret Santa<figure><img src="https://images.theconversation.com/files/305327/original/file-20191205-38984-5lgzaz.jpg?ixlib=rb-1.1.0&rect=21%2C7%2C4764%2C3298&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><span class="source">Stokkete/Shutterstock</span></span></figcaption></figure><p>Last Christmas, my family gathered to organise our Kris Kringle. My sister drew her husband, but they were already buying presents for each other, so we decided to draw again. No one in my family (except me) is particularly interested in mathematics or statistics, but my brother fatefully asked: </p>
<blockquote>
<p>What are the chances that all the partners draw each other?</p>
</blockquote>
<p>At the time, my family consisted of eight people: my mother and her partner, my older sister and her husband, my younger brother and his new girlfriend, my grandpa (widowed and never remarried), and me (I was single). So there were three sets of partners and two singles.</p>
<p>I started thinking out loud about how to answer my brother’s question. One way to find a probability is to calculate the fraction</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/303924/original/file-20191127-112484-3i8krn.GIF?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/303924/original/file-20191127-112484-3i8krn.GIF?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=59&fit=crop&dpr=1 600w, https://images.theconversation.com/files/303924/original/file-20191127-112484-3i8krn.GIF?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=59&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/303924/original/file-20191127-112484-3i8krn.GIF?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=59&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/303924/original/file-20191127-112484-3i8krn.GIF?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=74&fit=crop&dpr=1 754w, https://images.theconversation.com/files/303924/original/file-20191127-112484-3i8krn.GIF?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=74&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/303924/original/file-20191127-112484-3i8krn.GIF?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=74&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption"></span>
</figcaption>
</figure>
<p>The “event of interest” here is that all three partners draw each other’s names and, because there are no other possibilities, my grandpa and I draw each other. This can only happen in one way, so the top line of the fraction above must be 1. </p>
<figure class="align-right ">
<img alt="" src="https://images.theconversation.com/files/303934/original/file-20191127-112517-1e8145p.jpg?ixlib=rb-1.1.0&rect=0%2C47%2C482%2C629&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/303934/original/file-20191127-112517-1e8145p.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=1066&fit=crop&dpr=1 600w, https://images.theconversation.com/files/303934/original/file-20191127-112517-1e8145p.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=1066&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/303934/original/file-20191127-112517-1e8145p.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=1066&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/303934/original/file-20191127-112517-1e8145p.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1339&fit=crop&dpr=1 754w, https://images.theconversation.com/files/303934/original/file-20191127-112517-1e8145p.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1339&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/303934/original/file-20191127-112517-1e8145p.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1339&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">My grandpa unwrapping his new weed sprayer on Christmas day.</span>
<span class="attribution"><span class="license">Author provided</span></span>
</figcaption>
</figure>
<h2>What’s the bottom line?</h2>
<p>Unfortunately, the bottom line is much trickier to calculate. You want to count the total number of ways that eight people can draw names from a hat, without drawing themselves. For example, one possible event outcome is: I draw my mum, who draws my brother, who draws my sister, who draws her husband, who draws my brother’s new girlfriend, who draws mum’s partner, who draws my grandpa, who draws me.</p>
<p>My brother’s new girlfriend interrupted my thinking by asking whether the answer isn’t simply eight factorial (which is written in mathematical notation as “8!”). </p>
<p>For those who aren’t familiar,</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/303925/original/file-20191127-112522-70gzyp.GIF?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/303925/original/file-20191127-112522-70gzyp.GIF?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=39&fit=crop&dpr=1 600w, https://images.theconversation.com/files/303925/original/file-20191127-112522-70gzyp.GIF?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=39&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/303925/original/file-20191127-112522-70gzyp.GIF?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=39&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/303925/original/file-20191127-112522-70gzyp.GIF?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=49&fit=crop&dpr=1 754w, https://images.theconversation.com/files/303925/original/file-20191127-112522-70gzyp.GIF?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=49&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/303925/original/file-20191127-112522-70gzyp.GIF?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=49&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption"></span>
</figcaption>
</figure>
<p>My family let out a collective “ooooooh!” at what they perceived as a challenge to my mathematical prowess from my brother’s new girlfriend. </p>
<p>Her guess was sensible but not quite right, because it included the outcomes in which someone draws their own name. </p>
<p>Without pen or paper handy I decided to file the problem away in the back of my mind and rejoin the conversation, which had swiftly moved on.</p>
<h2>When in doubt, bring in the experts</h2>
<p>I found the perfect place to resurrect this problem, at a recent <a href="http://www.mathscraft.org/">MathsCraft</a> event, which brought mathematicians, teachers and students together to explore problems like a research mathematician. </p>
<p>My University of Melbourne colleagues <a href="https://ms.unimelb.edu.au/study/mslc/people">TriThang Tran</a>, <a href="https://ms.unimelb.edu.au/people/profile?id=2657">Sam Povall</a> and <a href="https://acems.org.au/our-people/rhys-bowden">Rhys Bowden</a> are all experts in different mathematics research fields. I couldn’t think of anyone more qualified to count things than these three – surely they must be able to answer my Christmas conundrum. </p>
<p>However, a short brainstorm session revealed that the Kris Kringle question is harder than initially expected.</p>
<p>But lo, a possible saviour was delivered unto us in the form of <a href="https://acems.org.au/our-people/nigel-bean">Professor Nigel Bean</a>, an expert in applied probability. Like the star guiding the Three Wise Men, hopefully Nigel would be able to guide us puzzled mathematicians to the solution. </p>
<p>“That’s easy!” Nigel confidently announced. But seconds later, his face fell as he too came to the realisation that this problem is deceptively difficult. </p>
<p>We threw ideas around the table, our voices rising in excitement. This captured the attention of two secondary school maths teachers, Amy Xue and Callum Johnson, who came over to investigate. While Sam brought Amy and Callum up to speed, Nigel and Rhys worked on the whiteboard, drawing tables and defining notation to help them. TriThang took a different approach, working at the table and using pictures to illustrate different outcomes.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/303930/original/file-20191127-112531-tkd6b4.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/303930/original/file-20191127-112531-tkd6b4.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=344&fit=crop&dpr=1 600w, https://images.theconversation.com/files/303930/original/file-20191127-112531-tkd6b4.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=344&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/303930/original/file-20191127-112531-tkd6b4.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=344&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/303930/original/file-20191127-112531-tkd6b4.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=432&fit=crop&dpr=1 754w, https://images.theconversation.com/files/303930/original/file-20191127-112531-tkd6b4.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=432&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/303930/original/file-20191127-112531-tkd6b4.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=432&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Mathematicians working on the Kris Kringle problem together. From left to right: Nigel, Rhys, Sam, Amy, Callum and TriThang.</span>
</figcaption>
</figure>
<h2>The solution</h2>
<p>It took an hour and a half – and a coffee break – but Nigel, Rhys and (independently) TriThang finally arrived at the same solution. The chance that each of the couples in my family draw each other’s names in our annual Secret Santa is 1 in 14,833, or about 0.007%. </p>
<p>However, it was Callum who came up with the answer the fastest. Recalling the topic of <a href="http://mathworld.wolfram.com/Derangement.html">derangements</a> that he learned nine years ago at university and encountered again in the final question of the 2016 year 12 NSW Extension 2 maths paper, he produced his solution:</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/304161/original/file-20191127-112489-l00gye.GIF?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/304161/original/file-20191127-112489-l00gye.GIF?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=69&fit=crop&dpr=1 600w, https://images.theconversation.com/files/304161/original/file-20191127-112489-l00gye.GIF?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=69&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/304161/original/file-20191127-112489-l00gye.GIF?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=69&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/304161/original/file-20191127-112489-l00gye.GIF?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=87&fit=crop&dpr=1 754w, https://images.theconversation.com/files/304161/original/file-20191127-112489-l00gye.GIF?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=87&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/304161/original/file-20191127-112489-l00gye.GIF?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=87&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption"></span>
</figcaption>
</figure>
<p>where the bottom line of this equation is just a particularly unusual way of writing 14,833 (the <em>e</em> refers to <a href="https://www.mathscareers.org.uk/article/calculating-eulers-constant-e/">Euler’s number</a>).</p>
<p>I was proud to finally reveal the solution to my brother: it was incredibly unlikely that all three partners would have drawn each other that night at dinner - about as unlikely as tossing a coin 14 times and coming up with 14 heads. However, he wasn’t quite as excited about knowing the answer as I was. (I told you my family aren’t particularly interested in maths.)</p>
<h2>A Christmas twist</h2>
<p>Here’s the thing, though: the Kris Kringle question is a known problem, related to the <a href="https://standardwisdom.com/softwarejournal/2010/01/10-people-and-10-hats-an-old-problem/">old hats problem</a>. Why didn’t we just Google it? </p>
<p>Well, we all really enjoyed working on the problem. And Christmas is supposed to be fun. So I have to agree with Nigel when he asked: “Where is the fun in that?”</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/303933/original/file-20191127-112489-atz4rs.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/303933/original/file-20191127-112489-atz4rs.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=338&fit=crop&dpr=1 600w, https://images.theconversation.com/files/303933/original/file-20191127-112489-atz4rs.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=338&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/303933/original/file-20191127-112489-atz4rs.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=338&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/303933/original/file-20191127-112489-atz4rs.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=424&fit=crop&dpr=1 754w, https://images.theconversation.com/files/303933/original/file-20191127-112489-atz4rs.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=424&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/303933/original/file-20191127-112489-atz4rs.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=424&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Nigel (left) and Rhys solving the Kris Kringle problem on the whiteboard.</span>
</figcaption>
</figure>
<hr>
<p><em>Thanks to all the mathematicians who worked on this problem, <a href="https://acems.org.au/">ACEMS</a> for hosting the event that brought us together, and my family for coming up with the question.</em></p><img src="https://counter.theconversation.com/content/127730/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Rheanna Mainzer does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>How likely is it that everyone in a family Kris Kringle will draw their own partner? It took a roomful of mathematicians to find out.Rheanna Mainzer, Melbourne Early Career Academic Fellow (Statistics Tutor), The University of MelbourneLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/749092017-04-19T22:35:25Z2017-04-19T22:35:25ZThe extraordinary return of sea otters to Glacier Bay<figure><img src="https://images.theconversation.com/files/165688/original/image-20170418-32716-dgt62a.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">A sea otter floats in Kachemak Bay, Alaska.</span> <span class="attribution"><a class="source" href="http://www.apimages.com/metadata/Index/Sea-Otter-Bounty/d17ce8d14b9f4427a9a917fe952b0233/13/0">AP Photo/Laura Rauch</a></span></figcaption></figure><p>Human beings have a long history of persecuting apex predators such as wolves, tigers and leopards. The loss of these predators – animals at the top of the food chain – has resulted in ecological, economic and social <a href="http://science.sciencemag.org/content/333/6040/301">impacts around the globe</a>. Rarely do the predators fully recover from human oppression, and, when they do, we often lack data or tools to assess their recovery.</p>
<p>The sea otters in Glacier Bay, Alaska, are an exception. In <a href="http://dx.doi.org/10.1002/ecy.1643">a recent study</a>, our team chronicled the incredible return of sea otters to an area where they’ve been absent for at least 250 years. </p>
<p>Our approach – which fuses mathematics, statistics and ecology – can help us better understand the role of sea otters in marine ecosystems and the ability of apex predators to return to an ecosystem after they’ve been absent. It may even help us learn what a changing climate means for many other species.</p>
<h2>Return to Glacier Bay</h2>
<p>Although not typically viewed in the same vein as wolves, tigers and leopards, sea otters are an apex predator of the nearshore marine ecosystem - the narrow band between terrestrial and oceanic habitat.</p>
<p>During the commercial maritime fur trade in the 18th and 19th centuries, sea otters were nearly hunted to extinction across their range in the North Pacific Ocean. By 1911, only a handful of small isolated populations remained.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/165689/original/image-20170418-32726-fwtx0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/165689/original/image-20170418-32726-fwtx0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/165689/original/image-20170418-32726-fwtx0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=458&fit=crop&dpr=1 600w, https://images.theconversation.com/files/165689/original/image-20170418-32726-fwtx0.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=458&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/165689/original/image-20170418-32726-fwtx0.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=458&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/165689/original/image-20170418-32726-fwtx0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=576&fit=crop&dpr=1 754w, https://images.theconversation.com/files/165689/original/image-20170418-32726-fwtx0.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=576&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/165689/original/image-20170418-32726-fwtx0.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=576&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Historic range (gray shading) and 1911 remnant populations (red icons) of sea otters. The populations at Queen Charlotte Islands and San Benito Islands went extinct by 1920.</span>
<span class="attribution"><a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>But sea otter populations have recovered in many areas, thanks to a few changes. The <a href="http://celebrating200years.noaa.gov/events/fursealtreaty/welcome.html">International Fur Seal Treaty</a> in 1911 protected sea otters from most human harvest. Wildlife agencies also made an effort to aid sea otter recolonization.</p>
<p>Eventually, sea otters began to increase in abundance and distribution, and they made their way to Glacier Bay, a tidewater glacier fjord and national park in southeastern Alaska. Glacier Bay is functionally one of the largest marine <a href="https://www.nps.gov/glba/index.htm">protected areas</a> in the northern hemisphere. </p>
<figure class="align-right ">
<img alt="" src="https://images.theconversation.com/files/164959/original/image-20170411-26751-7y67vg.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/164959/original/image-20170411-26751-7y67vg.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=1246&fit=crop&dpr=1 600w, https://images.theconversation.com/files/164959/original/image-20170411-26751-7y67vg.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=1246&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/164959/original/image-20170411-26751-7y67vg.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=1246&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/164959/original/image-20170411-26751-7y67vg.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1566&fit=crop&dpr=1 754w, https://images.theconversation.com/files/164959/original/image-20170411-26751-7y67vg.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1566&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/164959/original/image-20170411-26751-7y67vg.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1566&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Glacier Bay National Park, southeastern Alaska.</span>
<span class="attribution"><span class="source">Map used with permission from the National Park Service</span></span>
</figcaption>
</figure>
<p>Glacier Bay was completely covered by glacier ice until approximately 1750 – about the same time sea otters vanished from the surrounding area due to over-harvest. It then endured the most rapid and extensive tidewater glacier retreat in recorded history. After glacier retreat, a rich environment emerged. This new environment supported high concentrations of wildlife, including sea otter prey species – such as crabs, mollusks and sea urchins – that were able to increase in size and abundance in the absence of sea otters. </p>
<p>Sea otters first reappeared at the mouth of Glacier Bay in 1988. Here they encountered vast habitat, abundant prey populations and protection from all human harvest.</p>
<h2>Our approach</h2>
<p>It’s challenging to estimate how populations grow and spread, due to their dynamic nature. Each year, animals move to new areas, increasing the amount of area and effort required to find them. Airplanes searching for sea otters have to cover more ground, usually with the same amount of time and money. Additionally, individuals may move from one area to the next during any time period for a number of reasons, including sea otter social behavior and their reaction to the environment. Because these challenges can interfere with accurate population estimates, it’s important to understand and address them. </p>
<p>Shortly after sea otters arrived in Glacier Bay, scientists from the U.S. Geological Survey began collecting data to document their return. Although the data clearly indicated that sea otters were increasing, we needed novel statistical methods to unveil the extent of this increase.</p>
<p>First, we developed a mathematical model using partial differential equations to describe the growth and spread of sea otters. Partial differential equations are commonly used to describe phenomena such as fluid dynamics and quantum mechanics. Therefore, they were a natural choice to describe how a mass – in our case, the sea otter population – spreads through space and time. </p>
<p>The new approach allowed us to incorporate our current understanding of sea otter ecology and behavior, including habitat preferences, maximum growth rates and where sea otters were first observed in Glacier Bay. </p>
<p>Second, we incorporated our equations within a hierarchical statistical model. Hierarchical models are used to draw conclusions from data that arise from complex processes. They provide flexibility to describe and distinguish among various sources of uncertainty, such as uncertainty in data collection and ecological processes.</p>
<p>Partial differential equations are not new to the field of ecology, dating back to <a href="http://www.math.wm.edu/%7Eshij/math490-2006/skellam.pdf">at least 1951</a>. However, by fusing these equations with formal statistical models, we can reliably infer dynamic ecological processes, while appropriately quantifying the uncertainty associated with our findings. It provides a data-driven way to analyze surveys of sea otter abundance for the past 25 years. </p>
<p>This gave us rigorous and honest estimates of colonization dynamics that incorporated our understanding of the ecological system. </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/164963/original/image-20170411-26730-1e4kmo5.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/164963/original/image-20170411-26730-1e4kmo5.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/164963/original/image-20170411-26730-1e4kmo5.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/164963/original/image-20170411-26730-1e4kmo5.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/164963/original/image-20170411-26730-1e4kmo5.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/164963/original/image-20170411-26730-1e4kmo5.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/164963/original/image-20170411-26730-1e4kmo5.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/164963/original/image-20170411-26730-1e4kmo5.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Group of sea otters in Glacier Bay National Park, 2016.</span>
<span class="attribution"><span class="source">Photo by Jamie Womble</span></span>
</figcaption>
</figure>
<h2>A record-breaking recovery</h2>
<p>Using our new approach, we discovered that the Glacier Bay sea otter population grew more than 21 percent per year between 1993 and 2012. </p>
<p>By comparison, the estimated growth rates of sea otters in other populations in Alaska, who were also recovering, have been limited to 17 to 20 percent. Furthermore, the maximum biological reproductive rate – the fastest rate sea otters can reproduce – is between 19 to 23 percent per year. That means that the Glacier Bay sea otter growth rate was near or at maximum, and greater than any recorded sea otter population in history. </p>
<p>In the wake of glacier retreat, sea otters went from nonexistent to colonizing nearly all of Glacier Bay in a span of 20 years. Today, they are one of the most abundant marine mammals in Glacier Bay. Recent observations have documented large groups of more than 500 sea otters in some parts of lower Glacier Bay, suggesting that prey resources are abundant. </p>
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<p>The fusion of state-of-the-art statistical and mathematical methods depicted, for the first time, just how extraordinary the growth and spread of this population was. </p>
<p>Sea otters had great success in the wake of tidewater glacier retreat in Glacier Bay. While climate-induced loss of sea ice can negatively affect some wide-ranging apex predators – such as polar bears or walruses – other species may benefit from the emergence of newly available habitat and prey resources. </p>
<p>Humans have caused the global decline of apex predators, and these declines are often difficult to reverse. However, our results suggest that, when there is minimal human interference, apex predators can be widely successful at recolonizing suitable habitat.</p><img src="https://counter.theconversation.com/content/74909/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Mevin Hooten receives funding from the National Park Service, U.S. Geological Survey, National Oceanic and Atmospheric Administration, and the National Science Foundation. </span></em></p><p class="fine-print"><em><span>Perry Williams does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Sea otters had been absent from this Alaskan national park for at least 250 years. By marrying math and statistics, scientists map this animal’s successful comeback.Perry Williams, Postdoctoral Fellow in Statistics and Fish, Wildlife, and Conservation Biology, Colorado State UniversityMevin Hooten, Assistant Unit Leader, U.S. Geological Survey, Colorado Cooperative Fish and Wildlife Research Unit; Associate Professor, Fish, Wildlife, and Conservation Biology and Statistics, Colorado State UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/749732017-04-12T00:38:29Z2017-04-12T00:38:29ZBeyond instant runoff: A better way to conduct multi-candidate elections<figure><img src="https://images.theconversation.com/files/164512/original/image-20170407-27621-1e5q4aj.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">A vote is cast in New Hampshire 2012 primary. </span> <span class="attribution"><a class="source" href="http://www.apimages.com/metadata/Index/APTOPIX-New-Hampshire-Primary-2012/de5da42595844cceb510a83af8150039/15/0">AP Photo/Matt Rourke</a></span></figcaption></figure><p>Last November, <a href="https://theconversation.com/maine-ballot-initiative-would-let-voters-rank-candidates-67694">Maine voters approved</a>, by a slim majority, a ballot initiative to adopt a voting system called “instant runoff.” </p>
<p>This system has been proposed as an alternative to our traditional election method – called “plurality voting” – by several politicians, including 2016 Green Party presidential candidate <a href="http://www.jill2016.com/ranked_choice_voting">Jill Stein</a>. It has also been implemented in <a href="http://www.fairvote.org/rcv#where_is_ranked_choice_voting_used">various municipal elections</a> in the United States. </p>
<p>Many other multi-candidate election methods have been proposed. Most of them have the drawback of being complicated, and therefore are probably not politically viable. I want to suggest a method that I believe is much better than both plurality voting and instant runoff, and just as simple as instant runoff. </p>
<h2>Plurality voting and its problems</h2>
<p>In plurality voting, every voter names their favorite candidate, and the candidate named most often wins. </p>
<p>This is the only reasonable thing to do when there are only two candidates, but it becomes problematic when there are more. The problems are <a href="https://electology.org/blog/top-5-ways-plurality-voting-fails">well-recognized</a>. For example, if you were every voter’s second choice among five candidates, you’d be doing very well, quite possibly better than any other candidate by most reasonable measures – yet you would lose. Plurality voting in fact appears to promote the emergence of two-party systems. Political scientists call this <a href="http://scorevoting.net/Duverger.html">Duverger’s Law</a>.</p>
<p>When there are two major candidates and some much weaker third-party candidates, plurality voting leads to “spoiler” problems. The weak candidates can change the outcome, sometimes in ways that their supporters find highly undesirable. For instance, the presence of Green Party candidate Ralph Nader on the presidential ballot in Florida in 2000 may very well have caused Al Gore to lose Florida, and thereby the presidency, even though it’s likely that a large majority of Nader voters preferred Gore to George Bush. </p>
<p>Attempts to improve plurality voting have a long history, with primaries in the U.S. as well as runoff rounds in presidential elections in France, Brazil and other countries. </p>
<h2>Instant runoff and its problems</h2>
<p>With instant runoff, every voter ranks the candidates. The candidate who is ranked first by the fewest voters is then removed from the ballots, and candidates who were ranked underneath the removed candidate move up by one notch. Then the process is repeated until only one candidate remains. That candidate wins.</p>
<p>In practice, one would want to allow voters to rank only some, not all, of the candidates, and one would want to allow ties. These are complications that are important, but also easy to deal with. For simplicity, we’ll assume here that all voters rank all candidates, with no ties.</p>
<p>When there are two strong candidates and some much weaker third-party candidates, instant runoff clearly does away with the spoiler problem. Weak candidates are eliminated early on. </p>
<p>For example, if instant runoff had been used in the 2000 presidential election in Florida, Gore would likely have been president. Nader would have been eliminated early on in the process, and those among his 97,421 voters who preferred Gore over Bush would have been counted as Gore voters. Considering that the final official margin by which Bush won Florida was 537, it seems likely that this would have changed the outcome.</p>
<p>Unfortunately, instant runoff – just like plurality voting – also immediately eliminates the candidate who is everyone’s second choice but nobody’s first.</p>
<p>And, just like plurality voting, instant runoff does not work well when there are more than two strong candidates. It can then produce quite arbitrary outcomes. If there are five strong candidates, should you really be eliminated just because 18 percent of voters put you first, while the other four candidates were placed first by 19 to 22 percent of voters? Shouldn’t we look at how many voters put you second, for instance, before ruling you out as the winner? </p>
<h2>Condorcet and Borda</h2>
<p>Two French noblemen of the 1700s thought about how to organize multi-candidate elections: the Marqis de Condorcet and Jean-Charles de Borda. (Condorcet was friends with Thomas Jefferson, who appears to have paid little attention to Condorcet’s writings about voting.) </p>
<p>Condorcet suggested that, if an absolute majority – more than half the voters – prefers Candidate X to Candidate Y, then Candidate Y should not be the winner. That seems very reasonable. Why not make the majority happier by making X the winner? </p>
<p>Unfortunately, when there are more than two candidates, this principle can easily rule out everyone. There can be a situation where, say, 55 percent of voters prefer Candidate A to Candidate B, 60 percent prefer B to C and 65 percent prefer C to A. </p>
<p>Condorcet didn’t say what should happen in such a case. His proposal refers only to situations in which there is a single candidate, the “Condorcet candidate,” who would beat every other candidate in head-to-head contest. He suggested that a Condorcet candidate, if there is one, should win. </p>
<p>As sensible as this sounds, both plurality voting and instant runoff violate it. Take my earlier example of an election with five candidates. If you are ranked second by every single voter, you might well win head-to-head contests against each of your four competitors. But, under plurality voting or instant runoff, you will lose.</p>
<p>Borda proposed <a href="http://www.electoral-reform.org.uk/borda-count">his own election method</a> that allots each candidate points based on their ranking. For instance, if there are five candidates, then Borda proposes to give a candidate five points for first place on a voter’s ballot, four points for second place, and so on. </p>
<p>Unfortunately, Borda and Condorcet can clash in a rather dramatic way. Even if an absolute majority of voters place you first, Borda may have you lose if most of the other voters strongly dislike you.</p>
<p>Borda’s method tends to handicap polarizing candidates. This seems like a good thing. However, if an absolute majority of voters place me first, then I should win, according to Condorcet, and most people would probably agree. When Borda’s method makes me lose because I am strongly disliked by a substantial minority, one could – and Condorcet would – argue that this goes a bit too far.</p>
<h2>Merging Condorcet’s and Borda’s ideas</h2>
<p>Merging Condorcet’s and Borda’s ideas creates an election method which, in my view, is much better than instant runoff, and just as simple. (I discuss this method at greater length in <a href="http://epubs.siam.org/doi/book/10.1137/1.9780898717624">my textbook</a> on this subject.) </p>
<p>In the system I propose, voters rank candidates, as in instant runoff and many other election methods. The outcome is then evaluated in two stages: a “Condorcet stage” where we pick out the strongest candidates, followed by a “Borda stage” where we identify the winner. </p>
<p>In the Condorcet stage, we determine the “strong” candidates. We define the “strong” candidates to be the smallest group of candidates with the property that everybody inside the group would beat everybody outside the group in two-person races. (This is also often called the <a href="http://wiki.electorama.com/wiki/Smith_set">Smith set</a>, after the mathematician <a href="https://en.wikipedia.org/wiki/John_H._Smith_(mathematician)">John H. Smith</a>.) For instance, if there is a Condorcet candidate X, then X is the only strong candidate. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/164701/original/image-20170410-8840-rpb1l5.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/164701/original/image-20170410-8840-rpb1l5.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=441&fit=crop&dpr=1 600w, https://images.theconversation.com/files/164701/original/image-20170410-8840-rpb1l5.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=441&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/164701/original/image-20170410-8840-rpb1l5.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=441&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/164701/original/image-20170410-8840-rpb1l5.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=554&fit=crop&dpr=1 754w, https://images.theconversation.com/files/164701/original/image-20170410-8840-rpb1l5.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=554&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/164701/original/image-20170410-8840-rpb1l5.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=554&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">The Condorcet stage.</span>
<span class="attribution"><span class="source">Christoph Borgers</span>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>We then remove all candidates who are not strong from the ballots, and move on to the Borda stage. The winner is computed with the reduced ballots based on Borda’s method. </p>
<p>In a presidential election, voters would all rank all the candidates: Republicans, Democrats and others. A computer would then determine the strong candidates. (No cause for alarm: Anybody who knows the election results could quite easily verify the computer’s work by hand.) Borda’s method would then decide from among this group. </p>
<p>I believe that many of the people who now support instant runoff should, and would, like this scheme even more. It eliminates weak candidates right away, removing the possibility of spoiler effects. It allows two candidates from the same party to run without interfering with each other so much that neither can win. It allows more than two strong candidates to emerge. When there are several strong candidates, the results are intuitively sensible. The method retains one of the advantages of Borda’s method – namely that polarizing candidates often lose – but, unlike Borda’s method, it does not allow a Condorcet candidate to lose. Equally importantly, the method is simple and transparent, and therefore might be politically viable. </p>
<p>There is no doubt that the issue is important: We cannot value democracy, yet refuse to think about the question how to conduct elections in a fair way.</p><img src="https://counter.theconversation.com/content/74973/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Christoph Borgers does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Some American voters hope that instant runoff can make our elections better. But a mathematician has an idea for another solution.Christoph Borgers, Professor of Mathematics, Tufts UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/742802017-03-14T00:19:13Z2017-03-14T00:19:13ZWhose votes count the least in the Electoral College?<figure><img src="https://images.theconversation.com/files/159985/original/image-20170308-24182-1tbryin.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Weighing up your votes</span> <span class="attribution"><a class="source" href="https://www.flickr.com/photos/northcharleston/15030066870/in/photolist-oUa2eY-6c8HMM-ezLhrE-4Lemjs-antj8G-a4He9p-ezKPY7-ezGraM-dugcMs-49HGS-ezLYQ3-ezH6gz-anthvd-ezK4Sp-ezGDpc-eiwixk-dPcqJ1-eiC5SU-ezJ4Bp-ezGRzi-bppQfJ-bppQhU-ants3E-cPvNh-SruMLA-bCjKs8-97P7">northcharleston/flickr</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span></figcaption></figure><p>In the days following the 2016 presidential election, many pundits and voters alike were stunned by the disparity between the popular vote, which went for Hillary Clinton, and the Electoral College, which favored Donald Trump.</p>
<p>If the president were elected by popular vote, every voter’s ballot would have been given equal weight, or influence, over the outcome, and Hillary Clinton would have won. But, as evidenced by Donald Trump’s victory, the Electoral College gives different weights to votes cast in different states. What are these weights, and how can we best compare them? </p>
<p>Most people believe the Electoral College weighs ballots in states with large populations much less than those in small states. For example,
<a href="https://www.washingtonpost.com/news/monkey-cage/wp/2016/11/17/the-electoral-college-badly-distorts-the-vote-and-its-going-to-get-worse/">as the Washington Post</a> noted shortly after the election, Wyoming has three electoral votes and a population of 586,107, while California has 55 electoral votes and 39,144,818 residents. Distributing the electoral vote evenly among each state’s residents suggests that individual votes from Wyoming carry 3.6 times more influence, or weight, than those from California. </p>
<p>The electoral vote total for each state is determined by its population relative to other states, plus two more votes equal to its representation in the Senate. Yet focusing on state population is not the most useful way to determine the relative weight accorded each state’s ballots. It does not help us understand how the weights assigned to voters by the Electoral College differ from the equal weights given to all voters in a popular vote. That’s because the popular vote weighs each vote according to the total turnout, not the total population. </p>
<p>As a professor who studies how mathematics can be used to model weather using computers, I was curious to make an apples-to-apples comparison between the Electoral College and the popular vote. I did this by using the number of ballots cast, rather than population, to compare the weight given to voters in each state by the Electoral College. </p>
<p>Large states such as California, Texas and New York do comparatively well under this analysis; it is the midsized states that fare the worst. These unexpected results help us understand whose votes carry the least weight in U.S. presidential elections. </p>
<h2>Crunching the numbers</h2>
<p><a href="http://www.electproject.org/home/voter-turnout/voter-turnout-data">Roughly 136 million people</a> voted in the 2016 presidential election. If we divide the total number of electoral votes – 538 – by the total number of voters, we can determine how much an individual vote counted toward an Electoral College vote. It turns out that, on a national average, each individual’s vote counted for about four millionths of one full Electoral College vote.</p>
<p>To find the relative weight of a vote in each state, I divided each state’s electoral vote total by the total number of ballots cast in that state, and then divided again by the exact fraction of an Electoral College vote accorded the average American voter (roughly four millionths). Let’s call this the “vote weight,” or simply the weight.</p>
<p>Note that, in a hypothetical system where the total electoral vote for each state equals the precise fraction of the total nationwide ballots cast in that state, votes in all states would be assigned weights of one, the same as in a national popular vote.</p>
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<p>My calculations show voters in Wyoming did indeed receive the most weight, 2.97, for their votes. Voters in Florida came out on the bottom, with a voting weight of just 0.78. The weight given to the votes in Louisiana exactly matched the national average of one.</p>
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<p>Two of the largest states, California and New York, came out only slightly below the national average. Votes from Texas, the second most populous state, actually received an above-average weight of 1.07.</p>
<p>Except for Florida, the states with the smallest weights are midsized, with between seven and 20 electoral votes each. For these states, there is no systematic relationship between vote weight and each state’s electoral vote total. </p>
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<p>The surprisingly low weights carried by votes in many midsized states are partly explained by the difference between the population – which determines the number of electoral votes – and the actual number of eligible voters in each state. Eligible voters do not include those who are too young to vote, noncitizens and, in some states, prisoners or former prisoners. </p>
<p>But it is voter turnout that primarily explains the low vote weights in states with seven or more electoral votes. In fact, the state-to-state difference in voter turnout was the most important factor in determining the variation of vote weights in midsized and large states in the 2016 presidential election.</p>
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<p>In contrast to the weak relationship between a state’s weight and its electoral vote total, the weight attached to each vote clearly tends to decrease as voter turnout within a state increases.</p>
<p>It is hardly surprising that higher turnout within a state decreases the weight accorded to each ballot, because the fixed number of electoral votes for any given state must be shared among the total number of ballots cast. But it does seem remarkable that the link between turnout and weight is so much stronger than the link between the number of electoral votes and weight. </p>
<p>For example, consider the difference between Oklahoma and Oregon. Both states have seven electoral votes, but their weights, 1.22 and 0.89, are quite different. That’s primarily because only 52 percent of eligible voters turned out in Oklahoma – much less than the 66.6 percent turnout in Oregon. That gave Oklahomans a greater weight per vote in the Electoral College than their fellow citizens in Oregon. </p>
<p>In another example, South Carolina saw a relatively low 56.8 percent turnout, versus the much higher 69.8 percent turnout in Colorado. Both states have nine electoral votes. But, as a consequence of voter turnout, ballots in South Carolina received a weight of 1.09, while those in Colorado were given a much lower weight of 0.82.</p>
<p>As a final example, consider the pair of states with 29 electoral votes each: New York and Florida. New York’s weight of 0.97 exceeds Florida’s weight of 0.78, mostly because turnout in New York was 55.7 percent, while that in Florida was 64.5 percent. </p>
<h2>Untangling the Electoral College</h2>
<p>My analysis does not completely capture the many ways in which the Electoral College modifies the influence attached to individual votes. </p>
<p>For example, it does not take into account our winner-take-all system, where all of the electoral votes from each state are awarded to whoever wins the majority of the popular vote. (The congressional district method used in Maine and Nebraska has the same effect, just aggregated into smaller units.) </p>
<p>Consider the election’s “battleground states,” where the election was decided <a href="https://www.nytimes.com/2016/12/19/upshot/why-trump-had-an-edge-in-the-electoral-college.html">by about one percentage point or less</a>. The relative weight of votes in four out of five of these states was less than 0.85. </p>
<p>The low weights for these states are largely due to high turnout, which was likely increased by the residents’ awareness of the significance of their votes. While this increase in turnout did lower the weight of each vote in a battleground state, few would assert that the voters in these states had less opportunity to influence the presidential election than voters living in less competitive states with a high vote weight. </p>
<p>The way the Electoral College rewires American presidential elections in comparison to a simple popular vote is clearly complex. The Electoral College does add extra weight to votes cast in the least populated states. But the way this system treats voters in the remaining states is not well-understood. In states with seven or more electoral votes, it tends to weigh votes based on that state’s voter turnout, rather than its number of electoral votes. </p>
<p>Whatever one’s political affiliation, it is hard to be enthusiastic about a system that penalizes voters in high-turnout states.</p><img src="https://counter.theconversation.com/content/74280/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Dale R. Durran is a registered Democrat.</span></em></p>The 2016 election made clear that the Electoral College does not weigh votes from all states equally. A new analysis suggests the power of your vote is closely linked to voter turnout in your state.Dale R. Durran, Professor of Atmospheric Sciences and Adjunct Professor of Applied Mathematics, University of WashingtonLicensed as Creative Commons – attribution, no derivatives.