文章探讨在小正常形式游戏中，从玩家处拿走一个选项对其自身及其他玩家收益的影响，还提到了一些相关案例及通过蒙特卡罗模拟进行分析的方法。

🎮在小正常形式游戏中，通常约⅓的时间内，从玩家处拿走一个选项会提高该玩家的收益，但会在½的时间内提高其他玩家的收益，具体数字取决于游戏规模。

🐔以阿卜杜拉和本杰明的斗鸡游戏为例，若家长式作风的佩特拉偏袒阿卜杜拉，可通过拿掉阿卜杜拉车上的方向盘，迫使本杰明先转向，保证阿卜杜拉获胜。

💼最低工资法和布雷斯悖论也是拿走人们选项却有益的例子。通过蒙特卡罗模拟，生成随机正常形式游戏，计算纳什均衡和收益，以判断拿走选项是否改善游戏情况。

🧐当阿卜杜拉和本杰明都有很多选项时，拿走阿卜杜拉的一个选项对他最有帮助；若本杰明已有很多选项，拿走阿卜杜拉的选项会使本杰明更可能处于有利地位。

Published on September 21, 2024 9:52 PM GMT

In small normal-form games, taking away an option from a playerimproves the payoff of that player usually <⅓ of the time, but improvesthe payoffs for the other player ½ of the time. The numbers depend onthe size of the game; plotted here.

There'sbeensomediscussionaboutthe originsofpaternalism.

I believe that there's another possible justification for paternalism:Intervening in situations between different actors to bring aboutPareto-improved games.

Let's take the game ofchicken between Abdullahand Benjamin. If a paternalist Petra favors Abdullah, and Petra hasaccess to Abdullah's car before the game, Petra can remove the steeringwheel to make Abdullah's commitment for them — taking an optionaway. This improves Abdullah's situation by forcing Benjamin to swervefirst, and guaranteeing Abdullah's victory (after all, it's a strictlydominant strategyfor Benjamin to swerve).

In a less artificial context, one could see minimum wagelaws as an example ofthis. Disregarding potential effects from increased unemployment, havinghigher minimum wage removes the temptation of workers to accept lowerwages. Braess' paradoxis another case where taking options away from people helps.

Frequency

We can figure this out by running a Monte-Carlosimulation.

First, start by generating random normal formgameswith payoffs in $[0,1]$. Then, compute the Nashequilibria for both playersvia vertexenumerationof the best responsepolytope (usingnashpy)—theLemke-Howsonalgorithm wasgiving me duplicate results. Compute the payoffs for both Abdullah and Benjamin.

Then, remove one option from Abdullah (which translates to deleting arow from the payoff matrix).

Calculate the Nash equilibria and payoffs again.

We assume that all Nash equilibria are equally likely, so for each playerwe take the mean payoff across Nash equilibria.

For a player, taking away one of Abdullah's options is considered animprovement iff the mean payoff in the original game is stricly lowerthan the mean payoff in the game with one option removed. Thus, one canimprove the game for Abdullah by taking away one of his options, andone can improve the game for Benjamin by taking away one of Abdullah'soptions, or both.

Plots

For games originally of size $n\times m$, how often is it the case that taking an option away from Abdullah improves the payoffs for Abdullah?

For games originally of size $n\times m$, how often is it the case that taking an option away from Abdullah improves the payoffs for Benjamin?

Interpretation

Abdullah is most helped by taking an option away from him when bothhe and Benjamin have a lot of options to choose from, e.g. in the casewhere both have six options. If Abdullah has many options and Benjamin hasfew, then taking an option away from Abdullah usually doesn't help him.

Benjamin is much more likely to be in an improved position if one takesaway an option from Abdullah, especially if Benjamin had many optionsavailable already—which suggests that in political situations, powerfulplayers are incentivized to advocate for paternalism over weaker players.

Postscript

One can imagine a paternalist government as a mechanismdesigner with abulldozer, then.

Code here, largely written by Claude 3.5Sonnet.

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