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<!DOCTYPE html>
<meta charset="utf-8">
<html>
<head>
<link rel="stylesheet" type="text/css" href="css/bootstrap.css">
<link rel="stylesheet" type="text/css" href="css/style.css">
<script src="js/jquery-1.10.1.js"></script>
<script src="js/d3.v3.js"></script>
<script src="js/lodash.js"></script>
<script src="js/sidenote.js"></script>
</head>
<body>
<div class="btn-group addRemoveColumn">
<button class="btn"><a href="javascript:Sidenote.more()">more columns</a></button>
<button class="btn"><a href="javascript:Sidenote.less()">fewer columns</a></button>
</div>
<div class="headerbox">
<h1>Ultra Rationality and the Prisoner’s Dilemma</h1>
</div>
<ul id="breadcrumbs" class="breadcrumb"></ul>
<div id='column-container'>
<a href="javascript:Sidenote.goBack()"><img id="left-arrow" class="hide-arrow" src="img/left_arrow.png"></a>
<a href="javascript:Sidenote.goForward()"><img id="right-arrow" class="hide-arrow" src="img/right_arrow.png"></a>
<div class='column' id="column0"></div>
</div>
<div class="content-storage">
<div id='proof3'>
<h3>Proof of Theorem 3</h3>
<ul>
<li>Assume:<ul>
<li>You are player <em>A</em>, and</li>
<li>You are ultra-rational</li>
</ul>
</li>
<li>Being content, you optimize your strategy for the other player’s penalty</li>
<li>If player <em>B</em> defects, your best strategy is to remain silent. In which case<ul>
<li>Player <em>B</em> experiences zero years in prison</li>
<li>You experience three years in prison, but do not interpret it as a penalty</li>
<li>Thus the total global penalty is zero</li>
</ul>
</li>
<li>If player <em>B</em> remains silent, your best strategy is to remain silent. In which
case:<ul>
<li>Player <em>B</em> experiences one year in prison (which has a maximum penalty of
one year in prison)</li>
<li>You experience one year in prison, but do not interpret it as a penalty</li>
</ul>
</li>
<li>Thus the total maximum global penalty is one year in prison</li>
</ul>
</div>
<div id='dilemma'>
<h2>The Prisoner’s Dilemma</h2>
<p>The prisoner’s dilemma is a fascinating and well-known paradox in economics.
It illuminates the heart of fundamental problems with our current economic models.</p>
<p>In the dilemma, two prisoners, <em>A</em> and <em>B</em>, are charged with the same crime and
offered the same bargain.</p>
<p>From the <a href="https://en.wikipedia.org/wiki/Prisoner's_dilemma">Wikipedia article</a>:</p>
<ul>
<li>If <em>A</em> and <em>B</em> both betray the other, each of them serves 2 years in prison</li>
<li>If <em>A</em> betrays <em>B</em> but <em>B</em> remains silent, <em>A</em> will be set free and <em>B</em> will serve 3 years in prison (and vice versa)</li>
<li>
<p>If <em>A</em> and <em>B</em> both remain silent, both of them will only serve 1 year in prison (on the lesser charge)</p>
</li>
<li>
<p>The <a href="javascript:Sidenote.openColumnLoud('#dilemma','#rational','rational%20solution')">rational solution</a> is to always defect</p>
</li>
<li>The <a href="javascript:Sidenote.openColumnLoud('#dilemma','#superrational','superrational%20solution')">superrational solution</a> is to remain silent when playing
with another super-rational player</li>
<li>The <a href="javascript:Sidenote.openColumnLoud('#dilemma','#ultrarational','ultra-rational%20solution')">ultra-rational solution</a> is to always remain silent</li>
</ul>
</div>
<div id='futurework'>
<h2>Future work</h2>
<p>We intend to pursue future work along a number of dimensions. We intend to:</p>
<ol>
<li>Investigate <a href="javascript:Sidenote.openColumnLoud('#futurework','#partial','partial%20ultra-rationality')">partial ultra-rationality</a></li>
<li>Investigate ultra-rationality in <a href="javascript:Sidenote.openColumnLoud('#futurework','#other-contexts','other%20contexts')">other contexts</a></li>
<li>Investigate <a href="javascript:Sidenote.openColumnLoud('#futurework','#alternatives','alternative%20formalizations')">alternative formalizations</a></li>
<li>Investigate ultra-rationality <a href="javascript:Sidenote.openColumnLoud('#futurework','#experimentally','experimentally')">experimentally</a></li>
</ol>
</div>
<div id='other-contexts'>
<h2>Other contexts</h2>
<p>In this work, we have only explored the effect of ultra-rationality on
the prisoner’s dilemma.</p>
<p>We intend to explore the effects of ultra-rationality in other economic problems.</p>
</div>
<div id='discussion'>
<h2>Discussion</h2>
<p>We believe that ultra-rationality represents a realistic and desirable
economics model. It seems obvious and self-evident that real humans have the
ability to choose their own personal interpretations of punsihment and have the
capacity to be compassionate. Compassionate and content prisoners exist.</p>
<p>Ultra-rational behavior depends on <em>contentedness</em>. If a player can experience
life joyfully and peacefully regardless of circumstance, then it can behave
selfishly while at the same time seeking globally optimal solutions.</p>
<p>Unfortunately, anecdotal evidence suggests that it takes great effort to
cultivate a subjective state of mind that is free from the experience of penalty.</p>
</div>
<div id='alternatives'>
<h2>Alternative formalizations</h2>
<p>We defined ultra-rationality in terms of two Axioms of Subjective Interpreation
and a definition of ultra rationality. As we explore ultra rationality in
other contexts we may find it is beneficial to define ultra rationality
differently.</p>
</div>
<div id='ultrarational'>
<h2>Ultra-rational Solution</h2>
<p>Here we introduce the concept of ultra-rationality and analyze its strategy.</p>
<h3>Definition of ultra rationality</h3>
<ul>
<li>An ultra-rational player is “compassionate;” it seeks globally optimal solutions</li>
<li>An ultra-rational player is “content;” it does not experience prison time as a penalty</li>
</ul>
<h3>Theorem 3:</h3>
<ul>
<li>When an ultra-rational player plays, the total maximum global penalty is one year in prison</li>
<li>See <a href="javascript:Sidenote.openColumnLoud('#ultrarational','#proof3','Proof%20of%20Theorem%203')">Proof of Theorem 3</a></li>
</ul>
</div>
<div id='superrational'>
<h2>Superrational Solution</h2>
<h3>Objective:</h3>
<p>The <em>superrational</em> objective is identical to the <a href="javascript:Sidenote.openColumnLoud('#superrational','#rational','rational')">rational</a>
objective. That is, A <em>superrational</em> player’s objective is to minimize their
time spent in prison.</p>
<h3>Defintion of superrationality</h3>
<p>A superrational player is a player who has a perfect understanding of the game,
and uses a deterministic strategy.</p>
<h3>Theorem 2:</h3>
<ul>
<li>When two superrational players play together, they both remain silent.</li>
<li>See <a href="javascript:Sidenote.openColumnLoud('#superrational','#proof2','Proof%20of%20Theorem%202')">Proof of Theorem 2</a></li>
</ul>
<h3>See also:</h3>
<ul>
<li>The <a href="javascript:Sidenote.openColumnLoud('#superrational','#ultrarational','ultra-rational%20solution')">ultra-rational solution</a></li>
<li>Wikipedia article on <a href="https://en.wikipedia.org/wiki/Superrationality">Superrationality</a></li>
</ul>
</div>
<div id='rational'>
<h2>Rational Solution</h2>
<h3>Objective:</h3>
<p>A <em>rational</em> player’s objective is to minimize their time spent in prison.</p>
<h3>Theorem 1:</h3>
<ul>
<li>The rational strategy is to always betray the other player.</li>
<li>See <a href="javascript:Sidenote.openColumnLoud('#rational','#proof1','Proof%20of%20Theorem%201')">Proof of Theorem 1</a></li>
</ul>
<h3>See also:</h3>
<ul>
<li>The <a href="javascript:Sidenote.openColumnLoud('#rational','#superrational','superrational%20solution')">superrational solution</a></li>
<li>The <a href="javascript:Sidenote.openColumnLoud('#rational','#ultrarational','ultra-rational%20solution')">ultra-rational solution</a></li>
</ul>
</div>
<div id='experimentally'>
<h2>Experimental reserarch</h2>
<p>We are interested in finding test subjects who embody the characteristics of
ultra rationality and evaluate their actual performance in realistic games.</p>
</div>
<div id='proof1'>
<h2>Proof of Theorem 1</h2>
<ul>
<li>Assume you are player <em>A</em> and you want to minimize your time in prison</li>
<li>If player <em>B</em> defects, your best strategy is to also defect (you therefore
serve two years in prison instead of three)</li>
<li>If player <em>B</em> remains silent, your best strategy is to defect (you therefore
serve zero years in prison instead of one)</li>
</ul>
</div>
<div id='proof2'>
<h3>Proof of Theorem 2</h3>
<ul>
<li>Assume:<ul>
<li>You are player <em>A</em>,</li>
<li>You are superrational, and</li>
<li>You want to minimize your time in prison</li>
</ul>
</li>
<li>Because you have a perfect understanding of the game, you will know if you are
playing with another superrational player.</li>
<li>You are therefore assured that you will both use the same strategy (
because you both have the same understanding of the game and and you both
use a deterministic strategy)</li>
<li>Thus, you will either <em>both defect</em>, or you will <em>both remain silent</em></li>
<li>Since, the silent strategy yields a superior payoff for youself you remain
silent</li>
<li>The other player follows the same strategy, you both remain silent, and you
each receive one year in prison</li>
</ul>
</div>
<div id='main'>
<p>We present a tentative new economics model, <em>ultra-rationality</em>, and evaluate its
relationship with the Prisoner’s Dilemma problem.</p>
<ul>
<li><a href="javascript:Sidenote.openColumnLoud('#main','#dilemma','The%20Prisoner&#8217;s%20Dilemma')">The Prisoner’s Dilemma</a></li>
<li><a href="javascript:Sidenote.openColumnLoud('#main','#rational','Rational%20solution')">Rational solution</a></li>
<li><a href="javascript:Sidenote.openColumnLoud('#main','#superrational','Superrational%20solution')">Superrational solution</a></li>
<li><a href="javascript:Sidenote.openColumnLoud('#main','#ultrarational','Ultra-rational%20solution')">Ultra-rational solution</a></li>
<li><a href="javascript:Sidenote.openColumnLoud('#main','#discussion','Discussion')">Discussion</a></li>
<li><a href="javascript:Sidenote.openColumnLoud('#main','#futurework','Future%20work')">Future work</a></li>
<li>This work is dedicated to the public domain.</li>
<li>Hosted on <a href="https://github.com/mikegagnon/ultra-rationality/">Github</a>.</li>
</ul>
</div>
<div id='partial'>
<h2>Partial ultra-rationality</h2>
<p>Strict ultra-rationality seems to be both uncommon and difficult to achieve.</p>
<p>To address these problems we intend to investigate notions of <em>partial</em>
ultra-rationality. The basic idea is that we belive it is possible for
players to achieve incremental benefits from following weaker models of ultra
rationality. We hope to describe formal models of partial
ultra-rationality that are realistic and representative of real-world economics.</p>
<p>Towards this end, we intend to espouse the goal of partial ultra-rationality
in our work. We intend to study and practice partial ultra-rationality and we
invite all other scholars to collaborate on our research by studying and
practicing partial ultra-rationality with us.</p>
<p>We hypothesize that such an approach to research will yield superior research
products at lower cost.</p>
</div>
</div>
</body>
</html>