tag:theconversation.com,2011:/africa/topics/prisoners-dilemma-17294/articlesPrisoners dilemma – The Conversation2018-11-26T11:49:37Ztag:theconversation.com,2011:article/1075322018-11-26T11:49:37Z2018-11-26T11:49:37ZWill Theresa May’s Brexit deal survive? Game theory has an answer<p>The European Union may have approved Theresa May’s withdrawal agreement, but getting her own MPs to do so seems impossible. And her approach is confusing. On the one hand, she has been telling pro-Remain MPs that they need to vote for her deal or they’ll be left with <a href="https://edition.cnn.com/2018/09/02/europe/uk-brexit-theresa-may-referendum/index.html">no deal at all</a>. On the other, she’s been warning eurosceptics that they could face another referendum or election unless they back her deal. As has been <a href="https://www.independent.co.uk/voices/theresa-may-jeremy-corbyn-pmqs-brexit-a8644891.html">pointed out</a>, both of these threats cannot be true simultaneously; either the prime minister has gone crazy or there is a reason. </p>
<p>Assuming the latter, why on Earth would this be a viable strategy? Mathematics offers one way to understand the situation. In game theory, there is something called the <a href="https://en.wikipedia.org/wiki/Prisoner%27s_dilemma">the prisoner’s dilemma</a> that can help explain what May is up to. This particular game demonstrates that two rational players – in this case, the prisoners – will choose to snitch on each other even though it would appear to be in their best interests to cooperate.</p>
<p>Here is the set up: two thieves are arrested for a jewellery theft. They are guilty, but the police can’t find the jewels and there is no hard evidence to link them with the crime. The police therefore need a confession. Each prisoner has two strategies – blame their partner or stay loyal. If they both point the finger at the other one, they both get convicted with four years of prison. If both stay silent, circumstantial evidence will put them in prison for one year. If one cooperates and the other doesn’t, the one who stays silent gets five years of prison and the one who points the finger is free. </p>
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<p>These rules are known to the thieves, and the police keep them in separate rooms. And now the paradox reveals itself. Cooperating and staying loyal would be the best option for them as a pair, as it would lead to the least amount of prison time (two years in total). But for the individual, it is more complicated. If you trust your partner to cooperate by staying silent, you may as well snitch and get off free. And if you don’t trust your partner, you should definitely snitch. Ultimately the cost of staying silent is the highest, with potentially five years in prison. Rational players will therefore always defect rather than cooperate.</p>
<p>While counter intuitive, this works because the numbers are rigged. The prisoner’s dilemma wouldn’t necessarily lead prisoners to snitch if when both confessed, for example, they’d end up in prison for ten years (rather than four) and get one to five years in prison for staying silent.</p>
<h2>The Tory MPs dilemma</h2>
<p>The prisoner’s dilemma can be used to model <a href="https://www.sciencenews.org/article/game-theory-suggests-current-climate-negotiations-won%E2%80%99t-avert-catastrophe">many social and political situations</a>. With the Brexit strategy, the role of the prisoners is played by the two Tory factions, the Remainers and Brexiteers. Each group has the option of voting for or against the deal – cooperating or not. The cost of each decision is a political one. </p>
<p>If both factions vote against the deal, there will be some political cost since the deal probably won’t pass. But it is completely unclear what will happen afterwards. This might trigger a leadership election, a general election, a new referendum or crashing out with no deal. All MPs like at least one of these options – so if everybody thinks there is a good chance they will get what they actually want (which is impossible), the political cost will be perceived as low.</p>
<p>If both factions vote in favour of this deal, it will have a high probability of passing (there are always willing Labour MPs to vote against their party line). Each of the Tory groups can blame the other for the “bad deal” that they were “forced” to sign, but it could arguably be a higher cost strategy as everybody fundamentally dislikes something about it. So it may seem the best bet would be for the two factions to cooperate to vote against it.</p>
<p>But the PM could place her MPs in a prisoner’s dilemma situation to make sure they don’t cooperate with each other to scupper her deal. She could do this by convincing both factions that voting in favour of her deal is good for them but bad for the others. To do that, she would need to rig the value of the perceived cost so that it will be higher if they vote against it than if they vote in favour – independent of what the political opponents in the other group choose to do. In this way, if both factions behaved in the most rational way possible, they would end up voting in favour, not only to get the other group in trouble, but to save grace themselves.</p>
<p>May is already telling MPs that if they vote for her then they get what they want (or rather they don’t get what they don’t want). The other group can be cast as fanatical or enemies of the people (depending which faction she is talking to). The cost modulation involves removing each team’s favourite option from the table – getting Remainers to think a second referendum is impossible and getting Brexiteers to think EU will never improve on the deal – which is <a href="https://uk.reuters.com/article/uk-britain-eu-may/may-eu-wont-give-us-better-brexit-deal-if-parliament-rejects-current-one-idUKKCN1NS1G1">exactly what the PM has been doing</a>.</p>
<p>For the prisoners dilemma, the assumptions are simple: both players act rationally to their best interest and there are only two possible strategies involved. If each group believes what May has been telling them – so it is her deal or no other deal – then she has set up the prisoner’s dilemma perfectly. Remainers will support her deal as the lowest cost option and the one that will stop the Brexiteers getting a no deal outcome. The Brexiteers will back her too, partly to stop the Remainers getting a second referendum.</p>
<p>So if May is persuasive, game theory suggests the deal will pass. Of course, this is just one of several mathematical models. And there are certain assumptions involved that may not completely capture the complexity of reality. The players are not individuals but groups. Each MP has different levels of ambition and dedication to the country. And whether she can guarantee that her factions will act rationally is the biggest question of all. After all, Brexit is a highly emotive issue. </p>
<p> </p><img src="https://counter.theconversation.com/content/107532/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Nicos Georgiou receives funding from EPSRC. </span></em></p>Theresa May could game Conservative MPs by placing them into a ‘prisoner’s dilemma’.Nicos Georgiou, Senior Lecturer in Mathematics, Probability and Statistics, University of SussexLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/619362016-07-05T09:16:48Z2016-07-05T09:16:48ZHow primitive emotions may explain Brexit<p>Evolution occurs in the long term, politics in the short. For this reason, evolutionary perspectives are usually little use for analysing day-to-day political developments. Still, some events ignite tendencies that are wired very deep in human psychology. Brexit is one of these. </p>
<p>The decision by the UK electorate to leave the European Union has had an electrifying effect on the media, on financial markets and even on <a href="http://www.theguardian.com/lifeandstyle/2016/jun/27/brexit-family-rifts-parents-referendum-conflict-betrayal">interpersonal relationships</a>. Can evolutionary psychology do anything to explain the extraordinary impact of Brexit? </p>
<p>The answer is: yes, within limits. Research by political scientists suggests that many standard political divisions – such as those between conservatives and liberals – are driven be <a href="https://books.google.co.uk/books?id=y9-GG5gPzgwC&printsec=frontcover&dq=jonathan+haidt+the+righteous+mind&hl=en&sa=X&ved=0ahUKEwiv-8yc5MzNAhWMDcAKHbusAOIQ6AEIJTAA#v=onepage&q=jonathan%20haidt%20the%20righteous%20mind&f=false">pre-existing personality differences</a>, rather than the outcome of conscious deliberation. Given this, it is reasonable to think that an event as momentous as Brexit is more than just a difference of opinion. In fact, it can be argued that it embodies the most fundamental problem of cooperation: the <a href="http://science.sciencemag.org/content/162/3859/1243">tragedy of the commons</a>.</p>
<p>The tragedy of the commons is a multi-player example of the <a href="http://plato.stanford.edu/entries/prisoner-dilemma/">prisoner’s dilemma</a>, a scenario described by mathematicians in the 1950s to account for the apocalyptic logic of the Cold War. The idea is that it can be rational to pursue self-interest, even when everyone would benefit more from cooperation. Though framed using mathematical game theory, the tragedy of the commons is easy to understand. </p>
<p>Imagine that a collective action – say, cleaning a beach – is proposed to a group of people. Though everyone benefits from clean beaches, it requires hard work that will need to be done regularly. The result is that some people – free riders – will decide to let others clean the beach and enjoy the results while making no effort themselves. After all, the beach is public property, and no one can be forced to work against their will. Still, if too many people do this, beach-cleaning activities break down and everyone gets stuck with a dirty beach. The result is that a perfectly justifiable decision not to cooperate leads to bad results for everyone, even the non-cooperator. </p>
<h2>Punishing free riders</h2>
<p>What does this have to do with Brexit? The core problem posed by Brexit is that of the free rider. There is experimental evidence that human beings have an <a href="http://www.ncbi.nlm.nih.gov/pubmed/22268815">innate disposition</a> to <a href="http://www.sciencedirect.com/science/article/pii/S1090513804000054">identify and punish free riders</a>. This is backed up by long-standing historical prejudices <a href="https://books.google.co.uk/books?id=4zksAwAAQBAJ&pg=PA54&lpg=PA54&dq=prejudice+against+nomads&source=bl&ots=rfCOj8vHO4&sig=PzTgsnfVKcVknSMfjD1VKD_47RE&hl=en&sa=X&ved=0ahUKEwjciqvYjM3NAhUsDcAKHbK2BawQ6AEIHDAA#v=onepage&q=prejudice%20against%20nomad">against mobile populations</a> such as <a href="http://romistika.pspace.cz/docs/Kovats_The-politics-of-Roma-identity-between-nationalism-and-destitution.pdf">the Roma</a>, <a href="https://books.google.co.uk/books?id=nSO07i4wXTcC&printsec=frontcover&dq=historical+anti+semitism&hl=en&sa=X&ved=0ahUKEwjp--GnjM3NAhWLJcAKHWHBBI0Q6AEIOTAF#v=onepage&q=historical%20anti%20semitism&f=false">Jews</a> and <a href="http://sf.oxfordjournals.org/content/81/3/909.short">immigrants</a>, which can be explained as fear of people moving on before they pay their fair share. What this suggests is that human beings may be wired by evolution to be suspicious of anyone who they think might take from the collective welfare without giving in return. </p>
<p>If this is so, the acrimony that followed the Brexit vote can be explained in terms of free rider detection. Young people see the leave vote by older people as <a href="http://www.theguardian.com/politics/2016/jun/24/meet-the-75-young-people-who-voted-to-remain-in-eu">inter-generational free riding</a>, whereby the opportunities the elder generation received when younger are not paid forward. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/129167/original/image-20160704-19124-cvkbmj.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/129167/original/image-20160704-19124-cvkbmj.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/129167/original/image-20160704-19124-cvkbmj.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/129167/original/image-20160704-19124-cvkbmj.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/129167/original/image-20160704-19124-cvkbmj.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/129167/original/image-20160704-19124-cvkbmj.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/129167/original/image-20160704-19124-cvkbmj.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">
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<span class="caption">Free rider?</span>
<span class="attribution"><a class="source" href="http://www.shutterstock.com/dl2_lim.mhtml?src=JwzJawF1h2k1XgKxaj5o4A-1-4&clicksrc=download_btn_inline&id=282635600&size=medium_jpg&submit_jpg=">Tinxi/Shutterstock.com</a></span>
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<p>Ethnic nationalists, meanwhile, argue that the EU allows free riding by immigrants who use public services they have not contributed to. Metropolitan and cultural elites feel shamed by the leave vote, which they feel identifies them as the <a href="http://www.bbc.co.uk/news/uk-scotland-scotland-politics-36649450">free rider to international partners</a>. Free market proponents see the EU as a fake collective action project, in which Brussels free riders exploit the language of reciprocity for their own advantage while hobbling the UK with red tape. Educated voters with a <a href="http://www2.warwick.ac.uk/fac/soc/economics/staff/eproto/workingpapers/rpd_iq.pdf">high tendency to cooperate</a> interpret the leave vote by the less well educated as the satisfaction of short-term emotional preferences at the expense of long-term collective advantages. </p>
<p>In other words, the Brexit vote goes far beyond the tribalism of everyday politics. By channelling the tragedy of the commons, it taps into many of the Stone Age emotions that evolved to make cooperation possible. And because every participant in the debate can be framed as a free rider by another participant, these emotions are amplified to the maximum degree. </p>
<p>Will Brexit leave a permanent mark on UK politics? Apart from Scottish independence, probably not. Powerful feelings are hard to sustain, and unpleasant emotions <a href="https://www.researchgate.net/profile/W_Walker3/publication/230233641_Autobiographical_Memory_Unpleasantness_Fades_Faster_Than_Pleasantness_over_Time/links/54d3b28b0cf25013d026372a.pdf">fade more quickly than pleasant ones</a>. Nevertheless, politicians would do well to remember that stirring up Palaeolithic passions for short-term strategic gain can have very unpredictable consequences.</p><img src="https://counter.theconversation.com/content/61936/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>James Carney 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>An evolutionary psychologist’s perspective on Brexit.James Carney, Senior Research Associate (Psychology), Lancaster UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/612262016-06-22T12:31:00Z2016-06-22T12:31:00ZGame theory offers a better way forward in Britain’s EU drama<p>The way that the EU referendum campaigns – both for and against British membership of the bloc – have been handled has been redolent of game playing. As an academic who studies game theory, a number of parallels are evident. And, from the displays of nastiness on both sides of the campaign, it is clear that Britain needs to forge a more productive path forward in its relationship with the EU – whether it remains or leaves. My work on a new type of game theory may offer some insights.</p>
<p>From the moment David Cameron went to Brussels in February 2016 to secure better terms for Britain’s EU membership, the games began. Having already promised a referendum on Britain’s EU membership, he was no doubt hoping to use the shadow of a Brexit vote as a bargaining chip in his negotiations. </p>
<p>Essentially, he argued that if the other leaders agreed to the UK’s demands for concessions, he would be able to convince the British public to vote to remain in the EU. If the UK didn’t get what it wanted, the implication was that Britain would exit and weaken the EU for the remaining nations. The looming referendum was designed to increase the UK’s bargaining power, but it fell flat and the other leaders called his bluff, <a href="https://theconversation.com/the-uk-deal-with-the-eu-explained-what-it-says-and-what-it-means-55052">making limited concessions</a>. </p>
<h2>A credible threat?</h2>
<p>An important concept in game theory when it comes to winning negotiations is the idea of making “<a href="http://chicagounbound.uchicago.edu/cgi/viewcontent.cgi?article=2378&context=law_and_economics">credible threats</a>”. For Cameron’s threat to be credible (and therefore effective) it required the other EU leaders to believe he had such influence over the referendum outcome that he could determine a vote to remain or leave. With the referendum now looming and victory for Remain far from certain, it would seem as though his threat was not credible. </p>
<p>Meanwhile, members of the Leave camp appear to be using the referendum in similar ways – to secure better terms of EU membership for the UK. Former leader of the Conservative Party, Michael Howard, <a href="http://www.theguardian.com/politics/2016/feb/26/former-tory-leader-michael-howard-call-uk-leave-eu">has argued</a> that the February negotiations failed and that, if the UK votes to leave the EU, this will “shake Europe’s leaders out of their complacency”. He said: </p>
<blockquote>
<p>There would be a significant chance that they [the EU] would ask us to think again. When Ireland and Denmark voted to reject EU proposals, the EU offered them more concessions and, second time round, got the result they wanted.</p>
</blockquote>
<p>So, in a sense, this Leave campaigner and the UK prime minister have been using the referendum in similar ways – to secure better terms of EU membership. Where they differ, however, is that Michael Howard is keeping his threat alive. Vote Leave on June 23, he says and use this as a bargaining chip against an EU <a href="http://www.huffingtonpost.co.uk/2016/06/21/the-waugh-zone-june-21-20_n_10584034.html">desperate for Britain to stay</a>.</p>
<p>With the polls neck and neck on the cusp of the EU referendum and no sign of further concessions being made by the EU, we appear to be witnessing another key game theory concept at play: <a href="http://www.academia.edu/17950372/Brexit_A_game_of_Chicken">the game of chicken</a>. </p>
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<img alt="" src="https://images.theconversation.com/files/127716/original/image-20160622-7158-15v9jqg.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/127716/original/image-20160622-7158-15v9jqg.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=370&fit=crop&dpr=1 600w, https://images.theconversation.com/files/127716/original/image-20160622-7158-15v9jqg.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=370&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/127716/original/image-20160622-7158-15v9jqg.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=370&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/127716/original/image-20160622-7158-15v9jqg.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=465&fit=crop&dpr=1 754w, https://images.theconversation.com/files/127716/original/image-20160622-7158-15v9jqg.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=465&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/127716/original/image-20160622-7158-15v9jqg.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=465&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
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<span class="caption">Game playing.</span>
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<p>This is the well-known game where two players drive at speed towards each other on a single-track road. Whoever swerves first loses. Of course, if neither player swerves, then they crash into each other and both lose. Some have asked whether it is sensible for the UK to play this game <a href="http://www.libdemvoice.org/playing-chicken-with-a-27-headed-opponent-49570.html">with a 27-headed opponent</a>.</p>
<h2>Tapping into our emotions</h2>
<p>Where traditional game theory falls down in analysing Britain’s relationship with the EU is the way that it is based on the idea that humans are all fully rational, unemotional, self-interested people who get the most value out of everything we do, with no regard for others. </p>
<p>Step in behavioural economics and behavioural game theory, which incorporates psychology, emotions and social preferences – such as our sense of fairness, trust and empathy – into the standard economics models. It recognises that people may care about others – and particularly working with others for a common goal. </p>
<p>Take the classic <a href="http://www.econlib.org/library/Enc/PrisonersDilemma.html">prisoner’s dilemma</a> – where the obvious, rational strategy for each individual in the game results in the worst outcome for both. Research <a href="http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2021113">shows</a>, however, that if we add empathy into models of the prisoner’s dilemma – where each player cares sufficiently about the other – it can transform the game into one of mutual cooperation, enabling a win-win situation for both players.</p>
<p>This can be true for the EU and Britain. If both sides were to develop greater empathy for the other side – this would entail closer relationships between all of the nations, more awareness of each other’s needs, greater kindness towards each other, softer negotiating approaches, and a better view of humanity – both will be better placed to reach a mutually beneficial outcome.</p><img src="https://counter.theconversation.com/content/61226/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Richard Fairchild 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>Britain will continue negotiating the terms of its relationship with Europe, whatever the outcome of its referendum.Richard Fairchild, Senior Lecturer in Corporate Finance, University of BathLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/461662015-08-26T09:50:50Z2015-08-26T09:50:50ZHow understanding the prisoner’s dilemma can help bridge liberal and conservative differences<figure><img src="https://images.theconversation.com/files/92944/original/image-20150825-15907-fqxzfj.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">How do people make social choices?</span> <span class="attribution"><a class="source" href="https://www.flickr.com/photos/gregloby/3515990945/in/photolist-6mGnBx-ohnH9-7KjjUa-47DMxQ-pZTtRL-iugy48-kApF4K-7t6x1w-bwZGq4-4mQ92F-gi2Y8-aepoHj-4dWDy1-bBkSEv-7YLToc-7zmDvJ-p8AeCU-82iUui-4YEcB5-rv5BBz-johxfJ-a8eatm-4yHvxq-hE2BNj-4p26pe-wKPZuX-oaswe8-7ZhZny-dpdfS4-oGTUn-9cszRw-3sZr4-7YTS1d-wykvLA-82j8tP-6kNWxw-9xNg55-zhW2-5LTZ3x-7vALtD-jh3o27-a7nCTX-ajCMFJ-7fSgv3-aT8Fgz-bAcXyf-msAE56-5YNZLo-76XUCA-oojdXx">Greg Lobinski</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span></figcaption></figure><p>In my social psychology class, <a href="https://www.washingtonpost.com/posteverything/wp/2015/07/20/why-i-give-my-students-a-tragedy-of-the-commons-extra-credit-challenge/">I pose an extra credit question</a> where students choose between having two points or six points added onto their final term paper grade, with the stipulation that if more than 10% of the class chooses six points, no one gets any points. </p>
<p>This exercise is a classroom demonstration of the <a href="http://perspicuity.net/sd/sd.html">commons dilemma</a>, and similar to the prisoner’s dilemma. Essentially, people are forced to choose between what would maximize their personal outcomes (more points) and what would be best for the group as a whole (fewer points). </p>
<p>It’s worth noting that this <a href="http://www.tandfonline.com/doi/abs/10.1207/s15328023top1704_5#.VdyNZ9NViko">exercise</a> was developed 25 years ago. I first learned it from my college psychology professor <a href="http://drigotas.socialpsychology.org/">Steve Drigotas</a> over a decade ago. I have been using it since 2008. </p>
<p>But recently, after a student of mine tweeted the dilemma of the extra credit question, it went viral in a way that I had never expected. So why is it only now starting to resonate with so many people worldwide? And why are people <a href="http://www.buzzfeed.com/rachelzarrell/dont-be-greedy#.knMAzn4B5">connecting this exercise</a> to concerns about greed or selfishness? </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/167094/original/file-20170427-15121-1t458qe.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/167094/original/file-20170427-15121-1t458qe.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=530&fit=crop&dpr=1 600w, https://images.theconversation.com/files/167094/original/file-20170427-15121-1t458qe.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=530&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/167094/original/file-20170427-15121-1t458qe.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=530&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/167094/original/file-20170427-15121-1t458qe.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=666&fit=crop&dpr=1 754w, https://images.theconversation.com/files/167094/original/file-20170427-15121-1t458qe.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=666&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/167094/original/file-20170427-15121-1t458qe.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=666&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">A student’s reaction.</span>
<span class="attribution"><span class="source">Twitter/Seanhin</span></span>
</figcaption>
</figure>
<h2>The prisoner’s dilemma</h2>
<p>Let’s analyze this class exercise. At first glance, it would seem that the obvious choice would be to pick two points – for then, everyone is sure to get the points. </p>
<p>But this requires a great deal of social trust. And that is not always apparent between strangers. </p>
<p>Thus, some students choose six points (greater than 10% of students in all of my classes have done so, except one class which hit 10% exactly). In fact, I would argue, picking six is a “rational” choice, because the likelihood of your own choice directly affecting the group is very <a href="http://mindyourdecisions.com/blog/2015/07/14/an-awesomely-evil-test-question-and-its-mathematical-answer-game-theory-tuesdays/">small</a>.</p>
<p>Now let’s look at the big picture. </p>
<p>Imagine if everyone in a group uses this line of “rational” reasoning. Then everyone would proceed to behave in a way that maximizes their own lot. The point here is that choosing six is “rational,” but only when we consider how <em>individual</em> actions impact the group.</p>
<p>In the aggregate, when thousands (or millions) of people behave this way, the consequences are disastrous. </p>
<p>This exercise is <a href="https://www.washingtonpost.com/posteverything/wp/2015/07/20/why-i-give-my-students-a-tragedy-of-the-commons-extra-credit-challenge/">analogous</a> to real-world behavior involving consumption of public resources (water, food, oil, electricity, etc). The “rational” mindset is how we end up with overharvesting, water or food shortages, pollution, climate change, etc.</p>
<h2>What the exercise revealed</h2>
<p>It’s important to note that most students in my class (around 80% each semester) end up choosing two points. While many students choose the “rational” six-point option, they are still in the minority. </p>
<p>I believe this is because most people do understand the importance of being communal. In other words, most people are happy to behave in a way that benefits others around them. </p>
<p>Here’s a real-world example at work: <a href="https://www.youtube.com/watch?v=V3PIXy9lBgo">Honest Tea gave people the opportunity to pay for tea using the “honor system.”</a> People can choose to take a bottle of tea without paying (the selfish option), or voluntarily pay for their tea by putting money into a jar (the communal option). </p>
<p>Again, the “rational” choice is to take the tea without paying. But <a href="http://www.usatoday.com/story/money/business/2014/08/18/honest-tea-national-honesty-index-marketing-public-relations/14235077/">a majority of people pay for their tea</a>, even if they don’t have to. </p>
<p>Why is it so? Humans are prosocial creatures – which means they like to help each other. </p>
<p>That most students chose the prosocial option in my class is notable. It inspires me and gives me hope for the future. However, the fight is not over, and we still need to reduce excessive consumption. </p>
<h2>People crave reciprocity</h2>
<p>So, learning from this exercise, how can we increase cooperation on a mass scale? </p>
<p>Psychological science may provide some potential solutions.</p>
<p>One of the biggest theoretical developments in moral psychology in recent years has been <a href="http://www.moralfoundations.org/">Moral Foundations Theory</a>, which suggests that there are <a href="http://www-bcf.usc.edu/%7Ejessegra/papers/GHKMIWD.inpress.MFT.AESP.pdf">several intuitive systems</a> that feed into our judgments of right and wrong. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/92946/original/image-20150825-15912-3s8yts.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/92946/original/image-20150825-15912-3s8yts.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/92946/original/image-20150825-15912-3s8yts.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/92946/original/image-20150825-15912-3s8yts.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/92946/original/image-20150825-15912-3s8yts.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/92946/original/image-20150825-15912-3s8yts.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/92946/original/image-20150825-15912-3s8yts.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">Mostly humans behave in ways that helps others around them.</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/thearches/5907952019/in/photolist-a14NiH-eiyJ19-e31Xe5-5yFrKB-9sg3u8-5Lak4P-7FeDmK-4L8eyq-5UQWT-7yRaQT-dqTT7E-auVHF5-ei7smM-diQrxf-D7LMt-qVmCa1-42CNjv-nPQkq-3W1Vv4-rHxoRL-6QRf9D-3kGgNd-aqAKNw-7zDRUU-8qHyZM-qxkirC-azVume-9ePsf7-57H7r4-51m3oq-jwXdpD-nTo9KF-nNThRF-oxKgD3-ozKiXW-k9nL2t-pE16Z7-7Dr9Zr-i6v4DU-9GEQsk-2kwhmV-nypipm-5QNavA-8Woeay-eibJrt-5WFXE9-81PyAd-bJC5qz-fucYfP-qcb2AU">TheArches</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>One of these is a concern about fairness/cheating. </p>
<p>People crave reciprocity with others. If someone does us a favor, we feel compelled to repay the kindness: or if they hurt us, we crave revenge. </p>
<p>Fairness manifests in justice and equality (eg, right to a fair trial), and in principles like the “Golden Rule” (treating people the same way you want them to treat you). </p>
<p>Another moral virtue is in-group loyalty. </p>
<p>Every community and nation has important symbols of unity (eg, the national flag), songs, pledges of allegiance, legends and monuments to its founders, sacred documents (eg, the Constitution), and institutions designed for the group’s protection (eg, the military). </p>
<p>In recent history, <a href="http://www-bcf.usc.edu/%7Ejessegra/papers/GrahamHaidtNosek.2009.Moral%20foundations%20of%20liberals%20and%20conservatives.JPSP.pdf">liberals</a> have tended to strongly emphasize the importance of fairness and justice in building a strong society.</p>
<p>Consider the equal rights movements for African Americans, women, and LGBTQ folks, especially in light of the <a href="http://www.nytimes.com/2015/06/27/us/supreme-court-same-sex-marriage.html?_r=0">recent Supreme Court ruling</a>. Liberals are fighting to close wage gaps, end discrimination, and promote equal rights. </p>
<p>In contrast, conservatives have tended to emphasize the importance of <a href="http://www-bcf.usc.edu/%7Ejessegra/papers/GrahamHaidtNosek.2009.Moral%20foundations%20of%20liberals%20and%20conservatives.JPSP.pdf">group loyalty</a>. Consider the emphasis on keeping America strong, protecting the homeland from foreign threats, bolstering the military and respecting national symbols like the flag.</p>
<h2>Challenges outside the classroom</h2>
<p>So how do these moral virtues apply to the commons dilemma game? </p>
<p>Well, if you want the extra points, you’re relying on other people to cooperate. So, think about the ethic of fairness. </p>
<p>Pick the same choice that you would want others to take. Let your own desires for others’ behavior guide your own personal decisions – if you want others to choose two points, you should do the same. </p>
<p>Additionally, if you want your group (eg, your school, your community) to thrive, you must personally contribute. If you care about the health and the spirit of your culture, that sentiment must be reflected in your own actions. </p>
<p>If we consider the ethic of group loyalty, then choosing two points is not only cooperative, it’s patriotic. Making a conscious effort to limit one’s consumption of resources (by using less water, for example) is a duty to the flags, symbols and pledges of allegiance that unite us.</p>
<p>Outside of the classroom setting, there are environmental problems that must be solved, and there are moral virtues that can help bridge across the ideological aisle. </p>
<p>While liberals and conservatives may differ in their perspectives on political issues, there are ways to build common ground by drawing on their respective moral concerns. </p>
<h2>Solving the world’s problems</h2>
<p>Fairness and loyalty are two different paths toward reining in selfishness and making cooperation more possible. </p>
<p>If we can harness the power of these moral virtues together, we just might have a shot at solving some of the world’s toughest ecological problems. </p>
<p>A case in point is the Pentagon. Usually in charge of military matter, it <a href="http://www.scientificamerican.com/article/immediate-risk-to-national-security-posed-by-global-warming/">now considers climate change a national security threat</a>. </p>
<p>In order to combat climate change, we need to use all the tools available in our moral toolkit. Everyone must sacrifice for the common good of preserving our great nation, and it is essential that we view our neighbors as equal partners in this endeavor.</p><img src="https://counter.theconversation.com/content/46166/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Dylan Selterman 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>A professor’s extra credit question goes to show how, as humans, we do care for each other. The challenge is: how do we apply it to more pressing problems of the world?Dylan Selterman, Lecturer, University of MarylandLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/423332015-05-26T10:06:30Z2015-05-26T10:06:30ZJohn Nash: a beautiful mind and its exquisite mathematics<p>John Nash, mathematician and <a href="http://www.nobelprize.org/nobel_prizes/economic-sciences/laureates/1994/">Nobel laureate in economics</a>, died in a taxi accident on May 23; he was 86. His wife, Alicia, was with him and also did not survive the crash. The Nashes were on their way home to Princeton from Norway, where John was honored as a recipient (along with Louis Nirenberg) of this year’s <a href="http://www.abelprize.no/">Abel Prize</a> in mathematics.</p>
<p>Thanks to A Beautiful Mind, Sylvia Nasar’s chronicle of Nash’s life, and its film adaptation starring Russell Crowe, Nash was one of the few mathematicians well known outside the halls of academia. The general public may remember the story of Nash’s mental illness and eventual recovery from paranoid schizophrenia. But Nash’s influence goes far beyond the Hollywood version of his biography. His <a href="http://mobile.nytimes.com/2015/05/25/science/john-nash-a-beautiful-mind-subject-and-nobel-winner-dies-at-86.html">colleagues count his mathematical innovations</a>, particularly on noncooperative games (the work that would earn him his Nobel Prize), among the great economic ideas of the 20th century.</p>
<h2>Noncooperative games</h2>
<p>Nash is best known for his work in game theory. In mathematics, a game involves two or more “players” who earn rewards or penalties depending on the actions of all the participants. Some games are called <em>zero-sum</em>, which means that one player’s gain is another player’s loss. Nash’s work applied to <em>noncooperative games</em>. In these situations, players may unilaterally change strategy to improve (or worsen) their own outcome without affecting the other players.</p>
<p>The prototypical example of such a game is the basic Prisoner’s Dilemma. Two criminals have been captured and detained in separate cells, unable to communicate with each other. The prosecutors do not have sufficient evidence to convict them on the primary charge, but they can convict them on a lesser charge which comes with a one-year sentence. The prisoners are offered a deal: Testify against the other defendant (ie, defect) and go free while he serves three years. However, if both defendants betray each other, both will serve two years. If neither betrays the other (ie, they cooperate), then they will both be convicted of the lesser charge and serve the one year. The outcomes may be summarized in a <em>payoff matrix</em>.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/82862/original/image-20150525-32558-yqpwyc.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/82862/original/image-20150525-32558-yqpwyc.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=144&fit=crop&dpr=1 600w, https://images.theconversation.com/files/82862/original/image-20150525-32558-yqpwyc.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=144&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/82862/original/image-20150525-32558-yqpwyc.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=144&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/82862/original/image-20150525-32558-yqpwyc.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=181&fit=crop&dpr=1 754w, https://images.theconversation.com/files/82862/original/image-20150525-32558-yqpwyc.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=181&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/82862/original/image-20150525-32558-yqpwyc.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=181&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">The payoff matrix for the Prisoner’s Dilemma. In parentheses, the sentences are ordered (Player A, Player B).</span>
<span class="attribution"><span class="source">generated by the author</span></span>
</figcaption>
</figure>
<p>What Nash discovered is that any such game has a strategy, now called a <em>Nash equilibrium</em>, where any unilateral change in strategy by a player results in a worse outcome for that player. In the case of the Prisoner’s Dilemma, there are two such equilibria, the upper-left and lower-right squares in the payoff matrix. Indeed, in the situation in the lower-right corner, if either player changes his strategy unilaterally and decides not to defect, he will increase his sentence and thereby end up with a worse outcome. This example is particularly vexing because the upper-left strategy is clearly the best way to go for the prisoners (they should remain silent), but purely rational players will end up at the lower-right position.</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/CemLiSI5ox8?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">A Beautiful Mind famously dramatized Nash’s discovery of what’s now called the Nash equilibrium.</span></figcaption>
</figure>
<p>Game theory has applications in many fields, including economics and political science. Many scenarios in international relations may be modeled as noncooperative games. For example, the development of nuclear programs during World War II can be modeled as a sort of Prisoner’s Dilemma, in which the two sides each decide to pursue a bomb for fear the other side will do so. This, of course, led to the less-desirable outcome of nuclear proliferation, analogous to both prisoners defecting.</p>
<h2>Embedding theorems</h2>
<p>While Nash is best known globally for his work on game theory, most mathematicians think of his results on embeddings of Riemannian manifolds as his most innovative and important. In this subspecialty of geometry, an <em>n-manifold</em> is a space which locally looks like <em>n</em>-dimensional Euclidean space (the typical three spatial dimensions we’re used to form a 3-dimensional Euclidean space). For example, a surface, such as a sphere or hollow donut, is a 2-manifold since any point on the surface has a small disc around it; to a small bug standing at the point, the surface looks like a flat 2-dimensional plane (hence the ancient belief that the Earth is flat).</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/82863/original/image-20150525-32551-19u3u9h.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/82863/original/image-20150525-32551-19u3u9h.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=186&fit=crop&dpr=1 600w, https://images.theconversation.com/files/82863/original/image-20150525-32551-19u3u9h.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=186&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/82863/original/image-20150525-32551-19u3u9h.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=186&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/82863/original/image-20150525-32551-19u3u9h.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=234&fit=crop&dpr=1 754w, https://images.theconversation.com/files/82863/original/image-20150525-32551-19u3u9h.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=234&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/82863/original/image-20150525-32551-19u3u9h.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=234&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Given two vectors in Euclidean space, the angle (theta) between them may be defined via the formulas on the right.</span>
<span class="attribution"><span class="source">generated by the author</span></span>
</figcaption>
</figure>
<p>A manifold is <em>Riemannian</em> if there is a globally consistent way to define angles between vectors tangent to the manifold at a point. In particular, this allows us to define distances between points on the manifold and to find the lengths of curves embedded in the manifold. Euclidean space with its usual notion of angle and distance is the simplest example.</p>
<p>Now imagine trying to put an abstract Riemannian manifold inside Euclidean space. You might twist it up and do all sorts of strange things that end up distorting the angles between tangent vectors on your manifold. The Nash-Kuiper Embedding Theorem asserts that we can always fix this problem; that is, we can find a realization of a Riemannian manifold of dimension <em>n</em> into a Euclidean space of dimension <em>n+1</em> such that the angles are preserved. Then you can compute distances between points on the manifold more easily using the Riemannian structure inherited from Euclidean space.</p>
<p>This may not sound earth-shattering, but the problem had vexed mathematicians for more than a century. That the dimension of the Euclidean space cannot be made smaller than <em>n+1</em> is familiar to anyone who has studied a map – the surface of a sphere cannot be flattened onto a plane without distorting angles.</p>
<p>There are many counterintuitive implications of Nash’s theorem. For example, it implies that any closed surface may be realized inside an arbitrarily small ball in 3-dimensional space. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/82861/original/image-20150525-32575-5qxcow.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/82861/original/image-20150525-32575-5qxcow.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=426&fit=crop&dpr=1 600w, https://images.theconversation.com/files/82861/original/image-20150525-32575-5qxcow.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=426&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/82861/original/image-20150525-32575-5qxcow.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=426&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/82861/original/image-20150525-32575-5qxcow.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=535&fit=crop&dpr=1 754w, https://images.theconversation.com/files/82861/original/image-20150525-32575-5qxcow.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=535&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/82861/original/image-20150525-32575-5qxcow.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=535&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">The board in Hex; blue wins.</span>
<span class="attribution"><span class="source">http://upload.wikimedia.org/wikipedia/commons/3/38/Hex-board-11x11-%282%29.jpg</span></span>
</figcaption>
</figure>
<h2>An actual game</h2>
<p>Nash is also credited with inventing a game, eventually marketed by Parker Brothers as a board game called <em>Hex</em>. This game, played on a parallelogram-shaped field of hexagonal cells, was discovered independently in Denmark around the same time. In Princeton it was called <em>Nash</em>, after its creator, or <em>John</em>, a double entendre involving the fact that it was played on the tiles in the mathematics department’s men’s room floor. There are two players, each of whom has tokens of a single color (red and blue, say). The object is to form an unbroken path from one side of the board to the other before one’s opponent does the same in the opposite direction.</p>
<p>There are <a href="http://www.lutanho.net/play/hex.html">online versions</a> of the game. The first player always has a winning strategy; that is, the player who makes the first move can always win, provided he executes the proper sequence of moves.</p>
<h2>A life’s work remembered</h2>
<p>In any given century there are a handful of mathematicians who stand out, whose work is so original and groundbreaking that it becomes a part of the language. As journalist Erica Klarreich <a href="http://www.pnas.org/site/classics/classics5.xhtml">pointed out</a>, no one cites Nash’s papers any longer because “Nash equilibrium” is standard vocabulary; every mathematician knows what it means. While he published only a small handful of papers, John Nash will be remembered as one of the most original and influential mathematicians of the 20th century, whose work continues to inspire new results and research directions.</p><img src="https://counter.theconversation.com/content/42333/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Kevin Knudson 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>The Nobel Prize winning mathematician made lasting contributions in the fields of game theory and topology. Famously portrayed by Russell Crowe in the movie A Beautiful Mind, he died May 23 at age 86.Kevin Knudson, Professor of Mathematics, University of FloridaLicensed as Creative Commons – attribution, no derivatives.