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Taxes, Game Theory, and Python (Part 1 of 2)

Be­fore I gave up on be­com­ing an ed­u­cat­ed man, I stud­ied math. And to this day it piss­es me that noone has fig­ured out how to make math in­ter­est­ing to the math­-a­verse. Here's a small at­temp­t.

Let's con­sid­er the fol­low­ing sce­nar­i­o, based com­plete­ly on things I know, not things I do ;-)

Sup­pose that in a city called San Isidro, there is a house. Hous­es in San Isidro pay a mu­nic­i­pal tax, in ex­change for the ser­vice of garbage col­lec­tion, street sweep­ing, tree trim­ming, and street light­ing.

It's a very small tax, but let's say it's $100 a month be­cause it's a nice, easy to han­dle num­ber.

Al­so, San Isidro is in a coun­try called Ar­genti­na. In that coun­try there are sev­er­al laws that af­fect the home own­er­s:

  1. You can't sell a house if you owe any tax­es.

  2. The own­er has a 1% chance of wan­t­ing to sell the house each month.

  3. Debts ex­pire af­ter 5 years.

  4. If you are sued and you lose, you pay they oth­­er guy's lawyer fees.

  5. Lawyer fees are capped to 25% of the mon­ey be­ing dis­­put­ed.

  6. Lawyers are re­luc­­tant to help you sue some­one if they get very lit­­tle mon­ey (de­fined as less than $2000)

  7. If sued by the city for owed tax­es, the own­er al­ways los­es.

  8. Un­­paid tax­es ac­crue 2% com­­pound in­­ter­est mon­th­­ly. So, if you don't pay your $100, you will owe $102, then $104.04, $106.0128 etc.

With all those el­e­ments in place, let's play a game called "Tax Golf"!

The game is played by an in­de­ter­mined num­ber of play­ers called own­ers and one spe­cial play­er called city.

The game is played to 100 "month­s" or un­til all prop­er­ty has been sol­d.

The goal of the game, for the own­er­s, is to pay as lit­tle mon­ey as they can. The score is cal­cu­lat­ed like this: amount of mon­ey you paid di­vid­ed by the time you owned the house.

The own­er with the low­est score is the win­ner.

The goal of the game for the city, is to get as much mon­ey as he can. He's not com­pet­ing against the play­er­s, for him it's a sort of soli­taire where he com­petes against his past per­for­mance.

This, my friends is math. Math is a tool that helps you (a­mong oth­er things) do the right thing in this sort of com­pli­cat­ed, ar­bi­trary, re­al life sce­nari­o.

So, what's a good strat­e­gy for a own­er, and for the city?

In a sec­ond post next wednes­day, I will give some an­swers to those ques­tion­s, us­ing python.

Juan BC / 2012-09-05 15:01:

yo encontre en un momento el algoritmo para gtheory con zero-value y hice esto

Ran / 2012-09-05 18:41:

You did not give us all thr rules! What are the players doing at each turn? What are the allowed actions and possible rewards/punishment?

Roberto Alsina / 2012-09-05 18:50:

Ok, filling some blanks:

1) The owner can pay any number of the months he owes. So, if he owes 5 months, he can pay one or more. Each one he pays includes the interest accumulated from that month.

2) The city can sue or not sue, as long as the lawyer gets paid enough. If the city sues, it gets paid everything up to 5 years (60 months) back.

Those are the only possible moves.

John Lenton / 2012-09-05 22:12:

so the actual selling of the house doesn't happen?

Roberto Alsina / 2012-09-05 23:51:

Yes, if it happens you stop paying tax, you have to pay all your debt, and your score is calculated.

Joe / 2012-09-05 22:06:

"The goal of the game for the city, is to get as much money as [it] can. [It]'s not
competing against the players, for [the city] it's a sort of solitaire where [it]
competes against [its] past performance."

I don't think so. The city is a fictitious entity and the "city" player(s) are actually the politicians and bureaucrats that "run" the city government. Their goal is not to get as much money as they can, but various sub-goals depending on each individual that has input regarding running the city. Some individuals may crave raw "power" (over other persons), others prestige, others may believe they are doing "the right thing" by coercing their neighbors to collectively pay for the services provided.

You also don't mention whether the monthly taxes are the same for everybody or whether it varies according to the size (or estimated resale value) of the house they live in. Either way, even if those who live in larger houses or more affluent neighborhoods are assessed a higher tax, it's usually them who are the winners because ultimately they can afford the tax much more than those whose monthly earnings are toward the lower end of the scale. In addition, the former individuals are more likely to have influence or even be part of the city governmen, which adds to their "winnings".

Roberto Alsina / 2012-09-05 22:09:

Yes, this is not a simulation of the universe ;-)

For the purposes of this game, each house is worth the same, and pays exactly the same $100 in tax every month.

Joe / 2012-09-05 22:21:

Granted, but then it would have been better to propose the game in a more abstract way, not with reference to a specific city, but instead more like a game of Monopoly, with the "banker" player having special privileges (collect $100 every time the other players pass GO) and some players ("agents") only assisting others to avoid paying the dues or assisting the banker to collect them.

Roberto Alsina / 2012-09-05 23:52:

Feel free to start your own variant :-)

Joe / 2012-09-06 01:08:

My point is that you should not posit this "hypothetical" game which just "coincidentally" happens to criticize the greedy (goal == "get as much money as it can") government of a particular city. If you introduce into your "game" some aspects of reality (San Isidro, Argentina, municipal taxes to pay for services, cap on lawyers' fees), then I think criticism of your model not being entirely realistic is fair. If it was purely abstract, nobody could argue with your game rules.

Roberto Alsina / 2012-09-06 01:18:

It's not a criticism. I fully expect and encourage efficiency in tax collection.

This is really just setting up a scenario to investigate some amateur-level game theory ideas, so don't get hung up on the details too much. For example:

If selling the property were voluntary you can "win" by selling immediately, which is why I use a random chance for that.

It may be in the city's interest to sue random owners all the time even if it lost money on lawyers to "encourage" the others to pay cheaper.

It would *definitely* be on the city's interest to make some lawyers into employees so their fees are not capped.

There are dozens of similar wrinkles but I have tried to set the parameters so that they fit "reality" as much as I could without this getting incredibly convoluted: the city hardly ever sues anyone over this, and selling the property is not completely something you do voluntarily.

On the other hand, Monopoly uses Atlantic City locations, and noone complains about it being unrealistic :-)

Joe / 2012-09-06 01:39:

Here's where we part sides: as a voluntaryist/libertarian, I'm at the very opposite of your wishes. Without government's forced "services", those less fortunate would have an option of not paying the $100 tax, and dealing with garbage collection, etc. in some other way, maybe setting up a cooperative for pickup, paying someone (or bartering with them) to trim their trees or doing it themselves, etc.

Roberto Alsina / 2012-09-06 01:42:

Don't want to get into a boring debate-with-a-libertarian, sorry.

Joe / 2012-09-06 01:58:

No need to be sorry. I'm only disappointed to find that you believe that some people are "more equal" than others ( ).

Roberto Alsina / 2012-09-06 02:00:

Fuck off.

Joe / 2012-09-06 02:02:


Roberto Alsina / 2012-09-06 02:04:

Hey, you compare me to the pigs in Animal Farm, that's the only valid answer.

You know what? Congratulations, you are the first person I ever blacklisted in the comments in over 10 years.

Eivind Drivdal / 2012-09-06 07:16:

What does rule 6 actually mean ? "Are reluctant to", does this mean you *cannot* sue someone if the expected payoff to lawyers are less than $2000 ?

Roberto Alsina / 2012-09-06 12:23:

In practical terms it means you may have to pay the lawyer under the table in order to be able to sue someone for small amounts of money. In game terms, it means you can't sue.

Eivind Drivdal / 2012-09-06 07:32:

For a home-owner, it seems rational to not pay any taxes until their debt is precisely such that even with the accrued interest, the debt won't top $8000 before the oldest of the debt becomes 5 years old. Thereafter they should pay taxes exactly often enough to keep the debt at all times *slightly* under $8000 (including the accrued interest).

Aslong as the odds of wanting to sell are 1%/month, the game will certainly run to 100 months. (well, close enough to certain to make no difference), unless there's just a tiny amount of houses in the city. (on the average, about 35% of the houses are unsold after 100 months)

For the city, collect whenever they can, obviously ? They always win, so why not ?

Roberto Alsina / 2012-09-06 12:23:

You are on the right path :-)

Eivind Drivdal / 2012-09-07 06:12:

If the lawyers reluctance is due to taking home less than $2000, it would be beneficial for the city to pay the rest out-of-pocket in some situations. If they sue for $6000, then the lawyers part comes to $1500, so the home-owner must pay $7500 in total, the city could hand over $500 from their winnings to the lawyer, and still come out ahead. (I guess that's against the rules of this game, though)

2% interest monthly is 27% a year. This is an order of magnitude higher interest than the city can get elsewhere, and an *excellent* investment. As such, I retract my earlier statement that the city should always sue when it can. The city should always sue when it can -- *but* it should wait as long as possible, until waiting longer would mean that the oldest debts expire.

So if a person owes $8000, but the oldest of that debt is only 3 years, the city should wait until the oldest of the debt is 4.99 years old before suing. (or until the game is about to end, if that happens sooner)

Roberto Alsina / 2012-09-07 16:28:

That's exactly the kind of thinking I want to encourage!

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