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</head><body><h2 id="home"><a href="./index.html">home</a></h2>
<p><em>author: niplav, created: 2024-09-20, modified: 2024-09-20, language: english, status: draft, importance: 5, confidence: certain</em></p>
<blockquote>
<p><strong>In small <a href="https://en.wikipedia.org/wiki/Normal-form_game">normal-form
games</a>, taking away
an option from a player usually improves the payoff of that player
<⅓ of the time, but improves the payoffs for the other player ½
of the time. The numbers depend on the size of the game; plotted
<a href="#Plots">here</a>.</strong></p>
</blockquote><div class="toc"><div class="toc-title">Contents</div><ul><li><a href="#Frequency">Frequency</a><ul></ul></li><li><a href="#Plots">Plots</a><ul><li><a href="#Improved_Payoffs">Improved Payoffs</a><ul></ul></li><li><a href="#Unchanged_Payoffs">Unchanged Payoffs</a><ul></ul></li><li><a href="#Worsened_Payoffs">Worsened Payoffs</a><ul></ul></li></ul></li><li><a href="#Interpretation">Interpretation</a><ul></ul></li><li><a href="#Postscript">Postscript</a><ul></ul></li><li><a href="#See_Also">See Also</a><ul></ul></li></ul></div>
<h1 id="How_Often_Does_Taking_Away_Options_Help"><a class="hanchor" href="#How_Often_Does_Taking_Away_Options_Help">How Often Does Taking Away Options Help?</a></h1>
<p>There's
<a href="https://www.lesswrong.com/posts/6dSitnwYPg8i8NHn3/burch-s-law">been</a>
<a href="https://www.overcomingbias.com/p/rational_agent_html">some</a>
<a href="http://www.overcomingbias.com/2006/11/asymmetric_pate.html">discussion</a>
<a href="https://musingsandroughdrafts.wordpress.com/2022/03/29/paternalism-is-about-outrage/">about</a>
<a href="https://www.overcomingbias.com/p/paternalism_is_html">the origins</a>
<a href="https://www.lesswrong.com/posts/wg5YpAofgbEg587d2/don-t-take-bad-options-away-from-people">of</a>
<a href="https://www.overcomingbias.com/p/heres_my_openinhtml">paternalism</a>.</p>
<p>I believe that there's another possible justification for paternalism:
Intervening in situations between different actors to bring about
Pareto-improved games.</p>
<p>Let's take the <a href="https://en.wikipedia.org/wiki/Chicken_(game)">game of
chicken</a> between Abdullah
and Benjamin. If a paternalist Petra <em>favors</em> Abdullah, and Petra has
access to Abdullah's car before the game, Petra can remove the steering
wheel to make Abdullah's commitment for them — taking an option
away. This improves Abdullah's situation by forcing Benjamin to swerve
first, and guaranteeing Abdullah's victory (after all, it's a <a href="https://en.wikipedia.org/wiki/Dominant_Strategy">strictly
dominant strategy</a>
for Benjamin to swerve).</p>
<p>In a less artificial context, one could see <a href="https://en.wikipedia.org/wiki/Minimum_wage_law">minimum wage
laws</a> as an
example of this. Disregarding potential effects from <a href="https://en.wikipedia.org/wiki/Minimum_Wage#Supply_and_demand_model">increased
unemployment</a>,
having higher minimum wage removes the
temptation of workers to accept lower wages. <a href="https://en.wikipedia.org/wiki/Braess_Paradox">Braess'
paradox</a> is another case
where taking options away from people helps.</p>
<h2 id="Frequency"><a class="hanchor" href="#Frequency">Frequency</a></h2>
<p>We can determine a low-information prior on how often taking away
options from a player improves their game by running a <a href="https://en.wikipedia.org/wiki/Monte-Carlo_simulation">Monte-Carlo
simulation</a>.</p>
<p>First, start by generating random normal form games
with payoffs in <code>$[0,1]$</code>. Then, compute the <a href="https://en.wikipedia.org/wiki/Nash_Equilibrium">Nash
equilibria</a>
for both players via <a href="https://en.wikipedia.org/wiki/Vertex_enumeration_problem">vertex
enumeration</a>
of the <a href="https://en.wikipedia.org/wiki/Best_response">best response</a>
<a href="https://en.wikipedia.org/wiki/Polytope">polytope</a> (with
<a href="https://www.theoj.org/joss-papers/joss.00904/10.21105.joss.00904.pdf">nashpy</a>)—the
<a href="https://en.wikipedia.org/wiki/Lemke-Howson_algorithm">Lemke-Howson
algorithm</a> was
giving me duplicate results. Compute the payoffs for both Abdullah
and Benjamin.</p>
<p>Then, <em>remove</em> one option from Abdullah (which translates to deleting a
row from the <a href="https://en.wikipedia.org/wiki/Payoff_Matrix">payoff matrix</a>).</p>
<p>Calculate the Nash equilibria and payoffs again.</p>
<p>We assume that all Nash equilibria are equally likely, so for each player
we take the unweighted mean payoff across Nash equilibria.</p>
<p>For a player, taking away one of Abdullah's options is considered an
<em>improvement</em> iff the mean payoff across equilibria in the original
game is stricly lower than the mean payoff in the game with one option
removed. Thus, one can improve the game for Abdullah by taking away one
of his options, and one can improve the game for Benjamin by taking away
one of Abdullah's options, or both.</p>
<h2 id="Plots"><a class="hanchor" href="#Plots">Plots</a></h2>
<p>I run this analysis on games with up to nine actions, leaving out games
with two actions/one option in the reduced game (since nashpy requires
at least two options for each player, and a player with only one option
can't be really said to play the game).</p>
<h3 id="Improved_Payoffs"><a class="hanchor" href="#Improved_Payoffs">Improved Payoffs</a></h3>
<p><img alt="" src="./img/options/A_i.png"/></p>
<p><em>For games originally of size <code>$n \times m$</code>, how often is it the case that taking an option away from Abdullah improves the payoffs for Abdullah?</em></p>
<p><img alt="" src="./img/options/B_i.png"/></p>
<p><em>For games originally of size <code>$n \times m$</code>, how often is it the case that taking an option away from Abdullah improves the payoffs for Benjamin?</em></p>
<h3 id="Unchanged_Payoffs"><a class="hanchor" href="#Unchanged_Payoffs">Unchanged Payoffs</a></h3>
<p><img alt="" src="./img/options/A_n.png"/></p>
<p>Removing an option from Abdullah usually changes their payoffs, unless
they had many options already.</p>
<p><img alt="" src="./img/options/B_n.png"/></p>
<p>Similarly, removing an option from Abdullah usually changes the payoffs
for Benjamin, unless they had very few options.</p>
<h3 id="Worsened_Payoffs"><a class="hanchor" href="#Worsened_Payoffs">Worsened Payoffs</a></h3>
<p><img alt="" src="./img/options/A_w.png"/></p>
<p>Abdullahs payoffs are usually worsened, most often when Benjamin has
many options.</p>
<p><img alt="" src="./img/options/B_w.png"/></p>
<p>Benjamins payoffs are rarely worsened by taking an option away from
Abdullah.</p>
<h2 id="Interpretation"><a class="hanchor" href="#Interpretation">Interpretation</a></h2>
<p>Abdullah is <em>most</em> helped by taking an option away from him when both
he and Benjamin have a lot of options to choose from, e.g. in the case
where both have six options. If Abdullah has many options and Benjamin has
few, then taking an option away from Abdullah usually doesn't help him.</p>
<p>Benjamin is much more likely to be in an improved position if one takes
away an option from Abdullah, especially if Benjamin had many options
available already—which suggests that in political situations, powerful
players are incentivized to advocate for paternalism over weaker players.</p>
<h2 id="Postscript"><a class="hanchor" href="#Postscript">Postscript</a></h2>
<p>One can imagine a paternalist government as a <a href="https://en.wikipedia.org/wiki/Mechanism_Design">mechanism
designer</a> with a
bulldozer, then—it can only remove options from people, but that
hurts those people more than it helps them, unless the intervention is
carefully chosen.</p>
<p>Code <a href="./code/options/options.py">here</a>, largely written by <a href="https://www.anthropic.com/news/claude-3-5-sonnet">Claude 3.5
Sonnet</a>.</p>
<h2 id="See_Also"><a class="hanchor" href="#See_Also">See Also</a></h2>
<ul>
<li>Discussions
<ul>
<li><a href="https://www.lesswrong.com/posts/2AdfJygXGCEaaTS63/how-often-does-taking-away-options-help">LessWrong</a></li>
</ul></li>
</ul>
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