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Posts tagged with game theory

At 1:12:30, Sapolsky talks about how—following success after success of game theory explaining evolutionary outcomes—suddenly the prisoner’s dilemma, repeated games, reciprocity, Tit-for-Tat and David Axelrod failed as a theory! The Naked Mole Rats seemed to have defectors who weren’t being punished.

….until someone observed more closely. Someone kept watching the mole rats—keep in mind this is neither experiment nor Big Data nor data in tables but sense-data from ethnography i.e. anecdote!

anyway the zoologist kept watching, kept watching the naked mole rats and at last found evidence that vindicated the game-theoretic predictions of reciprocity among unrelated organisms.

The apparent “defectors” were actually not defecting, they were just part of a larger game in which they needed to be fat & lazy first, in order to play their self-sacrificing role in the longer game.

Is there a word for this besides "leakage"? It deserves its own word. When a model is essentially good, but because of either the limits of observation or the simplicity of the model, the thing we’re trying to pin down “leaks” or “escapes” into something we’re not seeing.

 

By the way, if you’ve never learned anything about game theory, this 72-minute video would make a great introduction to how hairy balls and Kakutani fixed-point theorems were used to make a fun tool for armchair reasoning for politics, love/sex relationships, biology, … at least a simulacrum of success.

If your Youtube doesn’t have speedup and you use Linux, you can use youtubeinmp4.com to download the vid then watch with mplayer -af scaletempo -speed 1.3 Sapolsky.mp4. ] speeds up and skips ahead a bit.

(Source: youtube.com)




This was a rhetorical question our chess teacher used to ask us. It’s a reminder that even though materiel, position, and tempo are worthwhile achievements that advance your interests, the goal is to check-mate the King.

For example the Blitzkrieg or “Scholar’s Mate” doesn’t capture materiel or obtain an advantageous position. It just goes directly for the kill.

It’s worth asking this question whether you’re just out the gate or mid-game. Is there a way within a few moves that you could mate early? Never forget to look for that in the quest for materiel or position.

  

I use the question now in my life as a shorthand for

  • why am I doing this?

. Getting money, obeying authority, learning things, obtaining credentials (résumé builders”), maintaining a low weight—all are “good” goals which advance my interests. But why? What is it aiming towards? What am I really trying to do?

In chess the goal is well-defined, whereas in life one can choose one’s own goals. In particular they can be

  • continual (“Go for walks”)
  • or circular (“Raise kids, so they can raise kids, so they can raise kids, …”)
  • rather than once-and-done (“Get thin”, “Mate the King”).
  • (And they needn’t be zero-sum.)

I think that makes the question What is the object of the game of chess? even more important.

That’s something that helps me and I hope it helps you. I’m going to pause now for some quiet reflection.




In game theory the word “strategy” means a fully specified contingency plan. Whatever happens—be it a sequence of things, a conditional branching of their responses and my responses—∃ a contingency.

I can’t prove this, but I do feel that sometimes people talk about others as constants rather than response functions.

(A function is a ≥1-to-1 association from elements of a source domain to elements of a target codomain. I’ll owe ya a post on how this is not the most intuitive way to think about functions. Because it depends which domains you’re mapping from and to. Think for example about automorphisms—turning something over in your hand—versus measures—assigning a size to something.)

  

For example, extraversion vs introversion. This is one of the less disputatious dimensions of human variation from the MBTI. We can observe that some people (like me) gain more energy by being around people and feel like sh*te when they spend too much time alone, whereas others (like my best friend) replenish their reserves by being alone and drain them when they go out in public.

So we observe one datum about you—but sometimes a discussion (eg, an economics debate) wants to veer over counterfactual terrain—in which case we need a theory about how things might else have been.

  • Maybe when you were young, your parents always made you do chores whenever they saw you, but didn’t particularly seek you out when you were out of sight. So you learned to hide in your room, avoid chores, and develop your personal life there. Hence became introverted as a response to environmental factors.
  • When I was young, I used to think I was introverted. Really I was just widely disliked and unpopular for being an ugly nerd. But later in life I developed social skills and had the fortune to meet people I liked, who liked me back. In response to who was around, I became extraverted.
   

I can think of other aspects of myself that are obviously responses to situational stimuli rather than innate constants.

  • If I were raised in a different culture, my sexuality would be different. In my culture, homosexuality is seen as “You boink / date / marry from your own sex”, but in ancient Sparta women all gayed on each other as a matter of ritual before the men came home from war. But they didn’t call themselves homos, and neither did the Roman men who sexually touched each other. It was just a different conception of sex (one I can’t fathom) where “Just because I regularly crave and do sexual stuff with people of my own sex, doesn’t mean I’m gay!”
    File:Pederastic erotic scene Louvre F85bis.jpg
    File:Banquet Euaion Louvre G467.jpg
    File:Pompeii - Terme Suburbane - Apodyterium - Scene V.jpg
    File:Nisos Euryalos Louvre LL450 n2.jpg
    Point being this is all the result of inputs; born Puritan, think sex = evil. Born Roman, "sexuality is a behaviour, not an identity".
  • If I ate more food and exercised less, my fat:muscle ratio would increase.
     
  • If I meditated more, I would feel more at peace.
  • If I read more maths, I would know more maths. More people would think of me as a mathematician—but not because it was inevitable or inherent in me to be a mathmo, rather because I chose to do maths and became the product of my habits.
  • If I fixed more bikes, I would be able to fix bikes faster.
  • If I made more money, I would go to different places, meet different people, be exposed to their response functions to their own pasts and presents and anxieties and perceptions, a vector field of non-Markovian baggage, and all of this history and now-ness would sum up to some stimuli-complex that would cause some response by me, and change me in ways I can’t now know.
     
  • Our friendship could have been so much more, but we sort of let it fall off. Not for any reason, but it’s not so strong now.
  • Our love could have been so much less volatile, but I slept around, which had repercussions for your feelings toward me, which repercussed to my feelings toward you, which repercussed …. (multiplier effect / geometric series)
 

Besides being motivation for me to learn more maths to see what comes out of this way of thinking about people when you layer abstract algebra over it, this view of people is a reminder to

  1. release the egotism, and
  2. not take too literally what I think I’m seeing of whomever I’m interacting with.

Someone who piss me off may not be “a jerk”, it may not be about me whatever, s/he may be lag-responding to something from before I was there. Or s/he may not have adapted to a “nice guy” equilibrium of interacting with me. Who knows. I’m not seeing all of that person’s possibility, just a particular response to a particular situation.

On the other hand, if they really are acting wrong, it’s up to me to address the issue reasonably right away, rather than let my frustration passive-aggressively fester. Wait ten years for revenge and they’ll be a different person by then.

The final suggestion of people-as-functions is that there’s always something buried, something untapped—like part of a wavefunction that will never be measured, or a button on a machine that never gets pressed. You may see one version of yourself or someone else, but there’s more latent in you and in them—if you’re thrown into a war, a divorce, the Jazz Age, the Everglades, a hospice, a black-tie dinner, poverty, wealth, a band, a reality show about life under cruel premodern conditions—that may bring out another part of them.

 

UPDATE: peacemaker points out the similarity between people-as-response functions and the nature/nurture debate. I think this viewpoint subsumes both the nature and the nurture side, as well as free will.

  1. Evolution shaped our genes in response to environmental pressures (see for example the flies’ eyes chart above).
  2. My assumptions & predilections are a response to a more immediate “environment” than the environment of evolutionary adaptation.
  3. And I exercise free will over how I respond to the most immediate “environment” which is just the stimuli I get from you and the Wu Tang Clan.

UPDATE 2: As I think through this again, I feel quantum measurement really is a great metaphor for interacting with people. You only evoke one particular response-complex from a person on that particular time. And the way you evoke it perturbs the “objective” underlying thing. For example if yo’re introduced to someone in a flirtatious way versus in a business setting.




Some cute applications of computability theory:

  1. We now know that, if an alien with enormous computational powers came to Earth, it could prove to us whether White or Black has the winning strategy in chess. To be convinced of the proof, we would not have to trust the alien or its exotic technology, and we would not have to spend billions of years analyzing one move sequence after another. We’d simply have to engage in a short conversation with the alien about the sums of certain polynomials over finite fields.
  2. There’s a finite (and not unimaginably-large) set of boxes, such that if we knew how to pack those boxes into the trunk of your car, then we’d also know a proof of the Riemann Hypothesis. Indeed, every formal proof of the Riemann Hypothesis with at most (say) a million symbols corresponds to some way of packing the boxes into your trunk, and vice versa. Furthermore, a list of the boxes and their dimensions can be feasibly written down.
  3. Supposing you do prove the Riemann Hypothesis, it’s possible to convince someone of that fact, without revealing anything other than the fact that you proved it. It’s also possible to write the proof down in such a way that someone else could verify it, with very high confidence, having only seen 10 or 20 bits of the proof.

by Scott Aaronson, via studeo




The fundamental mystery of capitalism, in my mind, is how a lot of locally zero-sum fights—over customers, over bread, over a job opening—can result in a globally positive-sum game like 2%/year economic growth over a century.

30 years of economic growth in a narrow corridor by ed leamer ... you can get pictures of the centuries on angus maddison or longer us economy on economagic

Just had a small maybe-insight into this question. Let’s take the case of a negative-sum court battle where the victim of a rollercoaster accident tries to recover damages in court. What’s being negotiated, at expense, is the transfer of wealth from one party to another—no growth here.

But, this suit constitutes a sample from the probability space of tort losses. The threat—in probability space—with low probability, of high expected loss—incents theme parks to take more precautions.

Maybe the precautions taken in response to the probabilistic threat are what causes the growth.

Agree? Disagree? Missing a wider point? On the right track?




via @cmastication

Al Gore won the popular vote in the United States’ 2000 presidential election. However due to their voting system, “points” are apportioned in such a way that Gore lost the election to the Great American Cowboy.

You might think “It was the electoral college (apportioning system) that made Gore lose” — after all, he won by pure percentage.

But, if things had been different — if the United States elected its presidents on the basis of national totals — then everything would have been different.

All the campaign strategists would have spent their budgets differently, perhaps recruited donors differently, perhaps even written different speeches.

Saying “Gore would have won without the electoral college” would be like looking at a video of a gunfight and saying, “If only Billy the Kidd would have been 100 yards on the other side of Wyatt Earp! Then he would have shot him in the back.” Well, duh—Wyatt Earp was facing Billy the Kidd and would have been turned around, had Billy the Kidd been 100 yards on the other side.

You can’t edit the tape that way.

Everything is connected. You can’t change one thing (about the past), without changing everything (about the future).




On the afternoon of the Nobel announcement, Nash said that he had won for game theory and that he felt that game theory was like string theory: a subject of great intrinsic intellectual interest that the world wishes to imagine can be of some utility.


He said it with enough scepticism in his voice to make it funny.

"A Beautiful Mind: A Biography of John Forbes Nash, Jr." (1998)

(Source: fisher.osu.edu)




Utopia. Class struggle. Liberty. Tyranny. Property. Natural law. Human rights. Rousseau, Locke, Paine, Plato, Spinoza, The Federalist Papers, Marx, Rawles, and the rest. What is a “good” society and how can “we” make our society better?
For me there was a time (age 18) when these things seemed very important. I’m a socially minded guy, and political problems seem to always be f**king things up for people who don’t need their lives f**ked with. If you fancy yourself compassionate and intelligent, it’s natural to be drawn to political problems. For me it was an ego draw — the appeal of “doing good” with my mind.
After a while, though, I started to feel like I was going in circles, endless debates that seemed to dance around — but never solve — certain fundamental problems (and meanwhile Idi Amin killing his countrymen, Bosnians and Serbians tearing each other apart, etc). Schools of thought seemed to coalesce around personalities (not facts) and I felt this pursuit was going nowhere.
I wanted a way out…
The Median Voter Theorem
Imagine 10,000 people were voting on which of 2 congressional candidates to elect. Each candidate is represented merely by a real number* which indicates liberal -vs- conservative. Once elected, the candidate implements policies robotically. “I am 23% liberal and 77% conservative, therefore I will do exactly what a 23% liberal would do.”
If voters have single-peaked, symmetrical preferences over the same real number* spectrum and everyone knows the formulas and figures involved, then there is only one Nash equilibrium strategy: run to the middle. Campaigning on a policy that pleases the median voter is the only Nash equilibrium and therefore what rational, winning politicians would do.
* or element of any measurable space, like a sig-algebra
Interpretation
This result is niche famous. People whose friend took a game theory class in college might have heard a version of this as a “proof” that the 2-party system is better or more centrist than multi-party systems.
I’m telling the more mathematical version because, when “popular accounts” try to relate an important result like the median voter theorem while taking the math out, they end up making no sense or accidentally lying.
I ain’t gonna talk down to you. You’re smart enough to look up what a Nash equilibrium is. And you can decide for yourself what it means if a mathematical model sounds somewhat like human reality but isn’t exactly like it.
The median voter theorem doesn’t say that 2-party systems are better, it suggests something — or maybe it doesn’t. It’s just a piece of math that may be relevant to real life, may serve as a mental model, may serve as a basis for intuition, or … may be misleading.
"Political Philosophy" from a different perspective
The Median Voter Theorem naturally brings up questions — questions that you wouldn’t think of if you framed your thinking in response to Rousseau, Locke, and other famous writers.
Are these preferences symmetric?
Are these preferences 1-dimensional?
Are these preferences stable over time?
Do these preferences map onto the real numbers? (or something isomorphic to R)
Are these preferences single-peaked?
Are these preferences symmetrical?
Can you represent a congressional representative’s behaviour in office by a single number?
What about after they’re elected? Won’t they deviate from what they said they would do?
What about party politics? Won’t the party whips keep them in line?
Back to the voters; what if the politicians don’t know what they want?(This last point turns out to be very important in Persson & Tabellini’s theory, which explains that George W. Bush could be re-elected not because the hateful simpletons who voted for him were numerous, but because they are predictable.)
More sophisticated would be to ask: “to what degree or in what ways / cases are the above things true or false?” And those questions tend to be more answerable.
Other kinds of questions you might want to add to the mathematical framework begun here:
Is this in just one district? How do inter-district politics factor in?
What about redistricting? What about gerry-mandering?
What about fact X about Country A's constitution vis-a-vis Country B's constitution?
What about cultural fact Y? How could we take account of that mathematically?
I could go on and on, and in fact many people have. I think this is how fields of research get started. This one is called “spatial voting theory”.
But this is ridiculous. People are not one-dimensional.
One surprising result, due to Rosenthal & Poole, about the unidimensionality question, is that — yes! — politicians are pretty well represented as just a number on a one-dimensional scale — like maybe 85% of their votes can be characterized this way.
Also surprising. Judges are even more unidimensional than politicians. However, voters are decidedly not uni-dimensional.
Not what I would have assumed, although I can make up a story to “explain” these findings post hoc. Actually I could make up lots of different stories and am just left with more questions. But. At least I’ve moved outside of the narrative of political philosophy handed down to me in college.
Wrap Up
Things I like about this book:
politics = relevant  +  math = logical
application of math to something more interesting than bridge engineering
sorry engineers, but all the engineering in the world isn’t going to solve global poverty — that’s a political problem
and it would be sweet if logical thinking could lead to an optimal constitution (if such a thing exists).
Things I don’t like about this book:
didn’t know enough math at the time I read it to think deeply or broadly about what they were saying
as far as I know, no practical applications (yet)
Too long, didn’t read: Variations on a theme, the theme being the Median Voter Theorem. Game theory leads to a framework for political analysis called “spatial voting theory” which is alternative to the “pure-humanities” approach from my college political philosophy courses.

Utopia. Class struggle. Liberty. Tyranny. Property. Natural law. Human rights. Rousseau, Locke, Paine, Plato, Spinoza, The Federalist Papers, Marx, Rawles, and the rest. What is a “good” society and how can “we” make our society better?

For me there was a time (age 18) when these things seemed very important. I’m a socially minded guy, and political problems seem to always be f**king things up for people who don’t need their lives f**ked with. If you fancy yourself compassionate and intelligent, it’s natural to be drawn to political problems. For me it was an ego draw — the appeal of “doing good” with my mind.

After a while, though, I started to feel like I was going in circles, endless debates that seemed to dance around — but never solve — certain fundamental problems (and meanwhile Idi Amin killing his countrymen, Bosnians and Serbians tearing each other apart, etc). Schools of thought seemed to coalesce around personalities (not facts) and I felt this pursuit was going nowhere.

I wanted a way out…

The Median Voter Theorem

Imagine 10,000 people were voting on which of 2 congressional candidates to elect. Each candidate is represented merely by a real number* which indicates liberal -vs- conservative. Once elected, the candidate implements policies robotically. “I am 23% liberal and 77% conservative, therefore I will do exactly what a 23% liberal would do.”

If voters have single-peaked, symmetrical preferences over the same real number* spectrum and everyone knows the formulas and figures involved, then there is only one Nash equilibrium strategy: run to the middle. Campaigning on a policy that pleases the median voter is the only Nash equilibrium and therefore what rational, winning politicians would do.

* or element of any measurable space, like a sig-algebra

Interpretation

This result is niche famous. People whose friend took a game theory class in college might have heard a version of this as a “proof” that the 2-party system is better or more centrist than multi-party systems.

I’m telling the more mathematical version because, when “popular accounts” try to relate an important result like the median voter theorem while taking the math out, they end up making no sense or accidentally lying.

I ain’t gonna talk down to you. You’re smart enough to look up what a Nash equilibrium is. And you can decide for yourself what it means if a mathematical model sounds somewhat like human reality but isn’t exactly like it.

The median voter theorem doesn’t say that 2-party systems are better, it suggests something — or maybe it doesn’t. It’s just a piece of math that may be relevant to real life, may serve as a mental model, may serve as a basis for intuition, or … may be misleading.

"Political Philosophy" from a different perspective

The Median Voter Theorem naturally brings up questions — questions that you wouldn’t think of if you framed your thinking in response to Rousseau, Locke, and other famous writers.

  • Are these preferences symmetric?
  • Are these preferences 1-dimensional?
  • Are these preferences stable over time?
  • Do these preferences map onto the real numbers? (or something isomorphic to R)
  • Are these preferences single-peaked?
  • Are these preferences symmetrical?
  • Can you represent a congressional representative’s behaviour in office by a single number?
  • What about after they’re elected? Won’t they deviate from what they said they would do?
  • What about party politics? Won’t the party whips keep them in line?
  • Back to the voters; what if the politicians don’t know what they want?

    (This last point turns out to be very important in Persson & Tabellini’s theory, which explains that George W. Bush could be re-elected not because the hateful simpletons who voted for him were numerous, but because they are predictable.)

More sophisticated would be to ask: “to what degree or in what ways / cases are the above things true or false?” And those questions tend to be more answerable.

Other kinds of questions you might want to add to the mathematical framework begun here:

  • Is this in just one district? How do inter-district politics factor in?
  • What about redistricting? What about gerry-mandering?
  • What about fact X about Country A's constitution vis-a-vis Country B's constitution?
  • What about cultural fact Y? How could we take account of that mathematically?

I could go on and on, and in fact many people have. I think this is how fields of research get started. This one is called “spatial voting theory”.

But this is ridiculous. People are not one-dimensional.

One surprising result, due to Rosenthal & Poole, about the unidimensionality question, is that — yes! — politicians are pretty well represented as just a number on a one-dimensional scale — like maybe 85% of their votes can be characterized this way.

Also surprising. Judges are even more unidimensional than politicians. However, voters are decidedly not uni-dimensional.

Not what I would have assumed, although I can make up a story to “explain” these findings post hoc. Actually I could make up lots of different stories and am just left with more questions. But. At least I’ve moved outside of the narrative of political philosophy handed down to me in college.

Wrap Up

Things I like about this book:

  • politics = relevant  +  math = logical
  • application of math to something more interesting than bridge engineering
  • sorry engineers, but all the engineering in the world isn’t going to solve global poverty — that’s a political problem
  • and it would be sweet if logical thinking could lead to an optimal constitution (if such a thing exists).

Things I don’t like about this book:

  • didn’t know enough math at the time I read it to think deeply or broadly about what they were saying
  • as far as I know, no practical applications (yet)

Too long, didn’t read: Variations on a theme, the theme being the Median Voter Theorem. Game theory leads to a framework for political analysis called “spatial voting theory” which is alternative to the “pure-humanities” approach from my college political philosophy courses.


hi-res




Doesn’t the time change disprove Ricardian equivalence?

Ricardian equivalence is an hypothesis from the economic theory of how people respond to tax policy. It supposes that if the government changed tax policy, people would just respond in such a way that the change would be negated.

But the people’s response to the time change discredits that line of reasoning. Here’s how I think the Ricardian argument would apply to daylight-savings policy:

  1. People have already chosen the optimal hours for their business to be open.
  2. Any attempt by the government to change their hours by national-mandate trickery will therefore result in sub-optimal hours.
  3. A given store or business will therefore respond to the time change by resetting its hours.
  4. This equilibrium is stable because all stores and businesses will know that each other store will respond the same way.
  5. So the net effect of changing 9:00 into 8:00 will be that stores change their starting hour from 9:00 to 8:00.

Obviously that doesn’t happen. And I would argue that the observed non-responsiveness of stores & businesses to time changes militates against their supposed responsiveness to tax changes assumed in Ricardian tax equivalence.




Here’s something curious about lawyers. Locally they fight negative-sum games — effectively, expending resources fighting over the division of a non-increasing good.

  • lawyer 1: OK, I get x%
  • lawyer 2: and I get 1−x %
  • lawyer 1: Yes. Now, I vote that x be increased
  • lawyer 2: But that would mean that my share, 1−x, would decrease
  • lawyer 1: Exactly.
  • lawyer 2: But…
  • lawyer 1: Exactly.

Globally, however, the effect of lawyers is positive-sum. Without a justice system, grievances would not be heard, property rights would not be enforced, and the world would be worse off.

So … a space with locally negative curvature and globally positive curvature … hmm, how can that work?