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356 lines
17 KiB
HTML
356 lines
17 KiB
HTML
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<!-- Mirrored from www.mathsisfun.com/sets/game-theory.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:45:39 GMT -->
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<title>Game Theory</title>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Game Theory Introduction</h1>
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<p class="center"><img src="images/control-center.jpg" alt="control center" height="159" width="600"></p>
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<p>Game Theory can help us find the ...</p>
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<ul>
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<li><b>best</b> decision in a competitive situation, or</li>
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<li><b>fairest</b> decision in a cooperative situation</li></ul>
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<p>... where the outcome for each player depends on <b>their</b> decision and the decisions of <b>other</b> players.</p>
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<p>It is useful in business, military, sports, finance, personal life, games and more.</p>
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<p>Let's have a look at an example to see how Game Theory can help us find the best decision.</p>
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<h2>Prisoner's Dilemma</h2>
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<p>Casey and Dana are arrested after a burglary. They are in separate rooms and cannot cooperate.</p>
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<p>Casey has been told:</p>
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<ul>
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<li>if you both stay quiet you both get <b>1 month</b> in prison for trespass</li>
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<li>if you accuse Dana then you <b>go free</b>, and Dana gets 10 months for theft</li>
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<li>if Dana accuses you, then you get <b>10 months</b> (and Dana goes free)</li>
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<li>if you both blame each other you both get <b>6 months</b></li></ul>
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<p>What do you advise Casey to do?</p>
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<p class="center"><i>... think about it for a bit ...</i></p>
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<p>So maybe both Casey and Dana should keep quiet, right? They get only 1 month each that way..</p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/unstable-rock.jpg" alt="unstable rock" height="224" width="300"></p>
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<p><br></p>
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<p>But that outcome is called <b>unstable.</b></p>
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<p>Because either side can <b>do better</b> by making the "I go free, you get 10 months" decision.</p>
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<p>So what to do?</p>
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<div style="clear:both">
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</div>
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<p>Well, sadly, Casey is better off blaming Dana.</p>
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<p>We can see it in a table like this:</p>
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<div class="simple">
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<table align="center" border="1">
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<tbody>
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<tr>
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<td style="background: transparent;"><br>
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</td>
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<td style="background: transparent;"><br>
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</td>
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<td rowspan="1" colspan="2" style="text-align:center; font-weight: bold; color:blue;">Dana</td></tr>
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<tr>
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<td style="background: transparent;"><br>
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</td>
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<td style="background: transparent;"><br>
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</td>
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<td style="width: 130px; text-align:center; font-weight: bold; color:blue;">Stay Quiet</td>
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<td style="width: 130px; text-align:center; font-weight: bold; color:blue;">Blame Casey</td></tr>
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<tr>
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<td style="text-align:right; font-weight: bold; color:orange;">Casey</td>
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<td style="text-align:right; font-weight: bold; color:orange;">Stay Quiet</td>
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<td style="width: 130px; text-align:center; font-size:140%;"><span style="color:orange;">-1</span>, <span style="color:blue;">-1</span></td>
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<td style="width: 130px; text-align:center; font-size:140%;"><span style="color:orange;">-10</span>, <span style="color:blue;">0</span></td></tr>
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<tr>
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<td><br>
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</td>
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<td style="text-align:right; font-weight: bold; color:orange;">Blame Dana</td>
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<td style="width: 130px; text-align:center; font-size:140%;"><span style="color:orange;">0</span>, <span style="color:blue;">-10</span></td>
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<td style="width: 130px; text-align:center; font-size:140%;"><span style="color:orange;">-6</span>, <span style="color:blue;">-6</span></td></tr></tbody></table></div>
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<p>Casey risks getting 10 months by staying quiet!</p>
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<p>They will most likely get 6 months each though.</p>
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<div class="words">
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<p>Each players choice is called a <b>strategy</b>.</p>
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<p>A table entry such as <b>-1,-1</b> is called a <b>set of strategies</b></p>
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</div>
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<h2>Nash Equilibrium</h2>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/john-nash.jpg" height="279" width="248"></p>
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<p>The set of strategies <b>(-6,-6)</b> where Casey and Dana both blame each other is a <b>Nash Equilibrium</b>, named after John Nash (the subject of the movie "A Beautiful Mind").</p>
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<p>It is when <b>no player is better off</b> when they change only their strategy.</p>
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<p><br></p>
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<p>In the above example: at <b>(-6,-6)</b> Casey is <b>not better off</b> by changing to "quiet", and Dana is also <b>not better off</b> by changing to "quiet", so this is a Nash Equilibrium.</p>
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<p>To put it another way: if <b>any</b> player <b>is</b> better off changing then it is <b>not</b> a Nash Equilibrium.</p>
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<div class="example">
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<h3>Example: Jade and Page travel by train to new places to earn money</h3>
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<ul>
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<li>if Jade takes a camera and Page a printer they can take people's portraits and earn $300 each.</li>
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<li>or they can take their own cleaning gear and clean windows for $200 total.</li>
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<li>but they can't carry two lots of things.</li></ul>
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<p>The strategies look like this:</p>
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<div class="simple">
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<table align="center" border="1">
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<tbody>
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<tr>
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<td style="background: transparent;"><br>
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</td>
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<td style="background: transparent;"><br>
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</td>
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<td rowspan="1" colspan="2" style="text-align:center; font-weight: bold; color:blue;">Page</td></tr>
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<tr>
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<td style="background: transparent;"><br>
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</td>
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<td style="background: transparent;"><br>
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</td>
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<td style="width: 130px; text-align:center; font-weight: bold; color:blue;">Printer</td>
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<td style="width: 130px; text-align:center; font-weight: bold; color:blue;">Cleaning</td></tr>
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<tr>
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<td style="text-align:right; font-weight: bold; color:orange;">Jade</td>
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<td style="text-align:right; font-weight: bold; color:orange;">Camera</td>
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<td style="width: 130px; text-align:center; font-size:140%;"><span style="color:orange;">300</span>, <span style="color:blue;">300</span></td>
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<td style="width: 130px; text-align:center; font-size:140%;"><span style="color:orange;">0</span>, <span style="color:blue;">200</span></td></tr>
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<tr>
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<td><br>
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</td>
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<td style="text-align:right; font-weight: bold; color:orange;">Cleaning</td>
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<td style="width: 130px; text-align:center; font-size:140%;"><span style="color:orange;">200</span>, <span style="color:blue;">0</span></td>
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<td style="width: 130px; text-align:center; font-size:140%;"><span style="color:orange;">100</span>, <span style="color:blue;">100</span></td></tr></tbody></table></div>
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<p>At (300,300) Jade is <b>not</b> better off changing to (200,0). And Page is <b>not</b> better of changing to (0,200). So this <b>is</b> a Nash Equilibrium.</p>
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<p>At (0,200) Jade <b>is</b> better off changing to (100,100). So <b>not</b> a Nash Equilibrium, and we don't need to check any more.</p>
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<p>At (200,0) Page <b>is</b> better off changing to (100,100). So <b>not</b> a Nash Equilibrium.</p>
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<p>At (100,100) Jade is <b>not</b> better off changing to (0,200). And Page is <b>not</b> better of changing to (200,0) So this <b>is</b> a Nash Equilibrium.</p>
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<p>So in this example there are <b>two</b> Nash Equilibria!</p>
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</div>
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<p>The previous example shows that players can end up stuck in a less effective strategy (100,100) vs (300,300) that can be more about habit than anything else.</p>
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<h2>No Police Needed</h2>
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<p>One way of thinking about Nash Equilibria is that (for rational players!) no police are needed to keep the rules. The players will naturally "self-police".</p>
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<div class="example">
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<h3>Example: Intersection</h3>
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<p>Imagine two people arrive at an intersection from different sides.</p>
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<ul>
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<li>If they both drive they <i>crash</i>, with $9,000 worth of damage each</li>
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<li>If one stops, the other drives with a benefit of $1</li>
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<li>But if they both stop they will be sitting there a long time and cost them $10</li></ul>
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<div class="simple">
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<table align="center" border="1">
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<tbody>
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<tr>
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<td style="background: transparent;"><br>
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</td>
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<td style="background: transparent;"><br>
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</td>
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<td rowspan="1" colspan="2" style="text-align:center; font-weight: bold; color:blue;">Driver B</td></tr>
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<tr>
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<td style="background: transparent;"><br>
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</td>
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<td style="background: transparent;"><br>
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</td>
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<td style="width: 130px; text-align:center; font-weight: bold; color:blue;">Go</td>
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<td style="width: 130px; text-align:center; font-weight: bold; color:blue;">Stop</td></tr>
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<tr>
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<td style="text-align:right; font-weight: bold; color:orange;">Driver A</td>
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<td style="text-align:right; font-weight: bold; color:orange;">Go</td>
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<td style="width: 130px; text-align:center; font-size:140%;"><span style="color:orange;">-9000</span>, <span style="color:blue;">-9000</span></td>
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<td style="width: 130px; text-align:center; font-size:140%;"><span style="color:orange;">-1</span>, <span style="color:blue;">0</span></td></tr>
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<tr>
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<td><br>
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</td>
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<td style="text-align:right; font-weight: bold; color:orange;">Stop</td>
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<td style="width: 130px; text-align:center; font-size:140%;"><span style="color:orange;">0</span>, <span style="color:blue;">-1</span></td>
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<td style="width: 130px; text-align:center; font-size:140%;"><span style="color:orange;">-10</span>, <span style="color:blue;">-10</span></td></tr></tbody></table></div>
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<p>
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So it is better to stop and wait for the other driver rather than risk a bad day.</p>
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</div>
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<p>But an important point:</p>
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<p class="center larger">This all assumes that players are <b>rational</b>.</p>
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<p>In the real world some people do stupid things and cause accidents, so we need police to help keep us safe.</p>
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<h2>Strictly Dominant</h2>
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<p>When a player is better off switching away from a choice (no matter what the other player chooses) then we can eliminate that choice.</p>
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<div class="example">
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<h3>Example: Dandelion Pty Ltd are planning their ad campaign and have calculated their success vs their main competitor (FruitBasket) depending on each side's ad style:</h3>
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<div class="simple">
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<table align="center" border="1">
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<tbody>
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<tr>
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<td style="background: transparent;"><br>
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</td>
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<td style="background: transparent;"><br>
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</td>
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<td rowspan="1" colspan="3" style="text-align:center; font-weight: bold; color:blue;">Dandelion</td></tr>
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<tr>
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<td style="background: transparent;"><br>
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</td>
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<td style="background: transparent;"><br>
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</td>
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<td style="width: 130px; text-align:center; font-weight: bold; color:blue;">Funny</td>
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<td style="width: 130px; text-align:center; font-weight: bold; color:blue;">Straight</td>
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<td style="width: 130px; text-align:center; font-weight: bold; color:blue;">Aggressive</td></tr>
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<tr>
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<td style="text-align:right; font-weight: bold; color:orange;">FruitBasket</td>
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<td style="text-align:right; font-weight: bold; color:orange;">Funny</td>
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<td style="width: 130px; text-align:center; font-size:140%;"><span style="color:orange;">14</span>, <span style="color:blue;">8</span></td>
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<td style="width: 130px; text-align:center; font-size:140%;"><span style="color:orange;">2</span>, <span style="color:blue;">10</span></td>
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<td style="width: 130px; text-align:center; font-size:140%;"><span style="color:orange;">8</span>, <span style="color:blue;">8</span></td></tr>
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<tr>
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<td><br>
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</td>
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<td style="text-align:right; font-weight: bold; color:orange;">Straight</td>
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<td style="width: 130px; text-align:center; font-size:140%;"><span style="color:orange;">7</span>, <span style="color:blue;">4</span></td>
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<td style="width: 130px; text-align:center; font-size:140%;"><span style="color:orange;">4</span>, <span style="color:blue;">8</span></td>
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<td style="width: 130px; text-align:center; font-size:140%;"><span style="color:orange;">7</span>, <span style="color:blue;">6</span></td></tr>
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<tr>
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<td><br>
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</td>
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<td style="text-align:right; font-weight: bold; color:orange;">Aggressive</td>
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<td style="width: 130px; text-align:center; font-size:140%;"><span style="color:orange;">0</span>, <span style="color:blue;">20</span></td>
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<td style="width: 130px; text-align:center; font-size:140%;"><span style="color:orange;">3</span>, <span style="color:blue;">18</span></td>
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<td style="width: 130px; text-align:center; font-size:140%;"><span style="color:orange;">9</span>, <span style="color:blue;">0</span></td></tr></tbody></table></div><br>
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<p>Looking at Dandelion's <b>Straight vs Aggressive</b> strategies:</p>
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<ul>
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<li>10 is better than 8 (Straight is better)</li>
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<li>8 is better than 6 (Straight is better)</li>
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<li>18 is better than 0 (Straight is better)</li></ul>
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<p>So for all cases Dandelion is better off using Straight instead of Aggressive style. So we can get rid of their whole Aggressive column:</p>
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<div class="simple">
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<table align="center" border="1">
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<tbody>
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<tr>
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<td style="background: transparent;"><br>
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</td>
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<td style="background: transparent;"><br>
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</td>
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<td style="text-align:center; font-weight: bold; color:blue;" rowspan="1" colspan="2">Dandelion</td>
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</tr>
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<tr>
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<td style="background: transparent;"><br>
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</td>
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<td style="background: transparent;"><br>
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</td>
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<td style="width: 130px; text-align:center; font-weight: bold; color:blue;">Funny</td>
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<td style="width: 130px; text-align:center; font-weight: bold; color:blue;">Straight</td>
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</tr>
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<tr>
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<td style="text-align:right; font-weight: bold; color:orange;">FruitBasket</td>
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<td style="text-align:right; font-weight: bold; color:orange;">Funny</td>
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<td style="width: 130px; text-align:center; font-size:140%;"><span style="color:orange;">14</span>, <span style="color:blue;">8</span></td>
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<td style="width: 130px; text-align:center; font-size:140%;"><span style="color:orange;">2</span>, <span style="color:blue;">10</span></td>
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</tr>
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<tr>
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<td><br>
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</td>
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<td style="text-align:right; font-weight: bold; color:orange;">Straight</td>
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<td style="width: 130px; text-align:center; font-size:140%;"><span style="color:orange;">7</span>, <span style="color:blue;">4</span></td>
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<td style="width: 130px; text-align:center; font-size:140%;"><span style="color:orange;">4</span>, <span style="color:blue;">8</span></td>
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</tr>
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<tr>
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<td><br>
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</td>
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<td style="text-align:right; font-weight: bold; color:orange;">Aggressive</td>
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<td style="width: 130px; text-align:center; font-size:140%;"><span style="color:orange;">0</span>, <span style="color:blue;">20</span></td>
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<td style="width: 130px; text-align:center; font-size:140%;"><span style="color:orange;">3</span>, <span style="color:blue;">18</span></td>
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</tr></tbody></table></div>
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<p>This can make it easier to decide!</p></div>
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<p><br></p>
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<h2>Thinking Clearly</h2>
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<p>By now you will be getting the idea: we set up a table listing the options for each player, then estimate the benefit (or cost) for each entry, and then use logic to work out our player's best strategy.</p>
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<h2>Much More</h2>
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<p>This has just been an introduction, there is much more to learn about Game Theory.</p>
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<p>For example we can change strategies randomly, this makes things interesting, right?</p>
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<p>If you would like us to cover more of this topic please <a href="../contact.html">let us know</a>.</p>
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<div class="related"> <a href="index.html">Sets Index</a>
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<a href="game-theory-tool.html">Game Theory Tool</a>
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