On Human Utility and Social Welfare: part 1

Being a rather unemotional person in general, and rather unmoved by social welfare issues and personal stories of pain and suffering, I figured the best way to compensate for this weakness is to create an analytical method through which I can analyze public good.

To develop a welfare function intrinsically linked to the psychology of human beings is a great challenge, since the dimensionality of a human being’s utility function is very high. I’m sure that there are many papers on this topic already, but here’s to a quick brainstorm session from a systems modeling perspective.

Let’s begin with a simplified model: Maslow’s hierarchy of needs.

At the bottom of the pyramid are physiological needs: food, water, breathing, sleep, sex, homeostasis, excretion. However, mapping these onto a utility function over 7 variables is not simple. For instance, one aspect of these variables are that they operate at different timescales. For example, a human being faints within 4 minutes without air, suffers permanent brain damage after 6-7 minutes, and dies within approximately 8 minutes. On the other hand, a human being can go without water or sleep for a few days, and without food for weeks. Excretion can also go for several hours to a couple days. Homeostasis–I suppose this has long term effects on decaying health. Sex is probably at the top of the physiological ladder. It is only until you are able to meet all the other needs that you reproduce.

One can view this as a workload scheduling problem with many deadlines. For example, X moles of oxygen needs to be delivered into your bloodstream every minute on average, or your blood oxygen level will start depleting until you die. Water and sleep function on a slower time scale. Requirements come in by hours or days, and their deadlines are much more lenient. And so on.

Although an imperfect model, one could consider human reaction to needs as a rate monotonic (RM) scheduling for periodic tasks. In RM scheduling, tasks are assigned priorities according to the frequency of occurrence, and the highest frequency tasks preempt lower frequency ones. That is, if you receive a task from your brain periodically, every 5 seconds or so, to take a breath, you will always schedule breathing over eating, sleeping, or sex first. Let’s just do an arbitrary prioritization of needs: breathing, water, sleep, excretion, food, homeostasis, and sex.

On the other hand, someone who is highly concerned with ensuring that he is top notch healthy may employ a deadline monotonic (DM) schedules tasks with earliest deadline first. Suppose you get hungry frequently, but don’t get thirsty very often. You are likely to eat first because you are hungry. However, because eating has a much longer deadline, you decide to delay your afternoon snack to ensure that you are properly hydrated first.

The complication occurs when there are shared resources. For example, suppose you attend a concert, and during intermission, many people are squeezing in line for the restroom. A foresighted person would, of course, make sure that he claims the resource to relieve himself when there are plenty of free stalls, not when there are plenty of people.

So, I didn’t have time to get to the higher levels of the pyramid, which include much fuzzier concepts. Safety can still be defined in terms of resource rates and uncertainty. Love? Not so easily. There is also a need to determine the actual utility function that determines a person’s level of satisfaction or happiness. For instance, suppose a person has his physiological needs met, and has portions of the other areas met. Suppose that there is no change, no spontaneity in his life. He may grow increasingly miserable over time. Spontaneity is therefore a dynamical function over time, where each triggering event generates a positive utility spike followed with monotonically decreasing values over time.

All of this is so fascinating and yet, so inexplicably complex. And I haven’t even gotten to the plethora of social welfare functions out there! (weighted sum, Nash, max-min?) But to figure out what are the right policies to enact, what to support and what to protest, one would need a rigorous, detailed model of what is good and what is bad for the people. Here’s to the beginning of such an attempt. =p

Appendix to “Creation vs Evolution”

This post is an appendix to the previous “Evolution vs Creation” debate post, which takes a step back from the original post. Based on the comments received, perhaps the original post had poor language usage and lost its intended wider meaning, and sadly may have offended both researchers in the field of biology and also believers in literal creationism. To better “position” myself on this issue, here is the original intent of the post:

My post is primarily a philosophical one–not one that’s focused on biology or even statistical theory, although I do mention them quite alot in the argument. I did not claim to be a 7 day creationist, nor an evolutionist. I also do not claim to know any more about biological theories than a high school graduate who took an AP course. That being said, this is the main point of my post:

I don’t believe that all creationists are anti-intellectual. Sure, many creationists we know of will accept the bible and the literal use of the word “day” with absolute faith, and as a result there is no room for discussion about scientific evidence that suggests the contrary (you can’t discuss theories about the big bang or the origins of the universe, which I find absolutely fascinating): hence a lack of intellectual discussion due to failure to find common ground. However, two creationists may be able to engage in a mutually interesting, though non-scientific, discussion about the Garden of Eden and the creation account in the bible. They may also be arguing about interpretation. This is all an intellectual exercise, no less than scientists arguing about the theories of evolution. Just because the majority believe in one thing doesn’t make the minority less intellectual. They simply use different axioms.

More importantly however, note the key words that I used in the previous paragraph: “evidence” and “suggests”. The scientific paradigm is one that rests on the burden of “proof”. However, unless you are in a purely mathematical field (i.e. playing with abstract ideas), scientific “proof” rests on inductive axioms that can never really be “proven”. It can only be supported by statistical evidence and models constructed based on such statistics, and much of it rests on the assumption that what was observed in the past still holds today. At least, this is how it’s done in the engineering discipline, in theory and in practice. You can only engage in an intellectual discussion if both of you believe in the theories/models/techniques behind it. And personally, I wouldn’t be employing models at work if I did not believe in them (with high probability  ). As I said before, I am not a biologist and am therefore unfamiliar with how much evidence exists to support different evolutionary statements about the origins of different species that exist today or in the past. That analysis is up to the experts in the community. However, my claim is that the scientific philosophy rests on these fundamental assumptions about observability and regularity.

Along those lines however, I am also stating that “miracles” by definition cannot be scientifically proven (except with a set of probability zero or near zero) because they are things that are observed perhaps only once in a lifetime, if at all. If miracles could be reproduced in practice, they would cease to be miracles and at best would be scientific “phenomenons” (i.e. things that can be made to occur, but the cause is not well understood). 7-day creationism (if true), Noah’s ark (if true), Jesus’s resurrection from the dead (if true), would fall under the category of “miracles”. They must be accepted by faith because they are inherently non-reproducible.

What is my conclusion? Basically, open-mindedness. I am not asserting that you have to doubt what you currently believe, whether by religious faith or by scientific faith. I am not advocating some type of relativism. However, it would probably benefit you to put on two different thinking caps when discussing issues of religion versus issues of science, because they rest on two fundamentally different axioms that seem to be contradictory, but at their core are not. A scientist should be able to resolve the idea of “miracles” , since not everything is reproducible (at the moment at least, e.g. the Planck epoch of the Big Bang). Likewise, I do not see a reason for a Christian to take offense when entertaining ideas based on observation and statistics, including evolution. After all, you probably use similar assumptions all the time in your card games or dice games. All else being equal, if God exists, so do unreproducible miracles. If He does not, or He simply chooses not to intervene with the universe that He created, then our best guesses can only be constructed by what is more or less, regularly observed.

Penalty and morality

At the risk of possibly contradicting a number of my prior posts, and also overanalyzing something silly, here’s something that recently crept up in my mind when reading random blogs. I came across someone’s entry about how she recently found out that she was almost aborted as a baby, but she felt indifferent and maintained her stance as being pro-choice. Then there were a ton of comments from random people saying that having an abortion and using a condom is no different, or people accusing her of being indifferent, etc.

So here’s when the thought occured: why is it that people seem to always draw a binary (black or white) line between when it is right to terminate a potential life, and when it is wrong? As I see it, pro-lifers are very rigid with regard to a “moral line” and believe that upon conception, a new life is born. Hence, knowingly removing this life, whether by a morning after pill or by late term abortion, is equivalent to murder. However, some pro-choicers similarly draw a rigid “lawful” line, who believe that abortion at any stage should be tolerated, prior to childbirth. And then we have people in between, the fuzzies: For example, 3rd trimester abortions are wrong, but stem cell research and early stage abortions are ok. Or abortion only in cases of rape, incest, or health hazard should be legal. Or whatever else is out there.

Regardless, this is a very complicated issue, but at the risk of sounding offensive, here are a few things that I’ve been hearing underneath the arguments:

Pro-choice on the surface:
-Nobody’s pro-abortion, but a women should be given the right to make decisions regarding her body and her future, especially in poor economic situations. It’s not something to be taken lightly, but the option should be open.

The hidden assumption:
-A fetus is most likely unaware of its own existence, and thus there is nothing logically different from aborting a fetus (at any stage, morning after or late-stage), versus using a condom. Therefore, what is most important is the impact it has on the lives of cognizant beings, primarily the mother.
-Marginally pro-choice people who also believe in the “sanctity of life” simply believe that the mother’s well-being trumps the life card in most cases, after factoring in subsequent utility losses from childbirth due to anticipated parental neglect or abuse, or financial hardships.

Criticism:
-In a poorer country where safe abortion techniques do not exist, is killing a baby 1 minute after childbirth (due to severe perceived hardships) any different from abortion? A related question is whether the “morality” of abortion and “baby-killing” is contingent upon the technological advances of that community. If so, what is the “guiding principle” behind moral laws?
-Why not reform foster care and adoption agencies instead?

Pro-life on the surface:
Upon conception, the fetus is a life with greater value than any type of pain in and after child-rearing and should therefore be protected.

The hidden assumption:
-There is some intrinsic value placed on human life, given by some greater power or principle such as a moral God. Moreover, this intrinsic value begins at the point of conception.

Criticism:
-Those who don’t believe in the same “God-given right to life” at conception are likely not to reason in a similar manner, and thus convincing the agnostic, and even some of the religious, community in such a manner is meaningless.
-Most arguments that are aimed at making people realize the “faultiness” of the alternative rely on slippery-slope arguments, ironically, similar to the “criticism” I described above for pro-choice (baby-killing).

Solution?

This type of “drawing-the-line” problem occurs not only in abortion, but occurs on many different moral issues as well. Why is this the case?

Well, let’s consider a type of  “penalty function” for abortion, a continuous, monotonically increasing function over fetal age, where using a condom and avoiding conception is perfectly moral, and killing a baby after it is born is perfectly immoral (equivalent to a full-scale murder). This reflects a type of “grey scale” morality, where you can be only 40% wrong, or 60% wrong, with respect to abortion.

Somewhere along the line, every person sets a threshold, where above that threshold, the value of morality is quantized to “right”, and below the threshold, as “wrong”, i.e. you are either 0% wrong and 100% wrong. I’m not saying that there aren’t people who are go-in-betweens, who might conceptualize multiple levels of “wrongness” for key stages of development (as they have been so  defined by our society), but why is a fluid mechanism design not well accepted in the community? Well, viewing morality as a semi-fluid model is too similar to paying taxes using “tax brackets”. It counts the effect, not the reason, as the basis for moral law.

Fluidity obscures morality simply because morality transcends quantitative analysis. It is at its very heart qualitative: a principle of matters, a spirit of the law, a condition of the heart. It is not so much quantitative, although the effects of immorality can be measured quantitatively. Had there been two Hitlers, one who came into power in Germany, and another who had power only to kill 10 innocent people, would this be a proper metric for evaluating the “badness” of Hitler? Sure the effects of the first Hitler was far more devastating, but at the heart of the matter the two men were the same. One just had a power amplifier attached to him.

Tune in next time as I try to quantify other unquantifiable axioms.

Obama-rama and lifting bans on abortion

Probably my most straightforward opinion on this is that I hope he’s thinking practically for the short-term only, and not for any other reasons. A president who focuses on giving people hope should not be encouraging abortions as a permanent solution for welfare. Other than the obvious pitfall of a government deciding what human beings are actually human beings, saying that abortion, 95% of which in the US is NOT a result of incest or rape, is part of the solution to bettering the lives of citizens, is giving up hope on the reformation of institutions such as adoption, foster care, and orphanages, many of which could give the promise of a good life to abandoned children. Sure, birth control should be heavily emphasized in various parts of the world. Sure, abortion may be a short-term solution for third-world countries where a family is too poor to have 10 children anyway, and half of them die before reaching age 7. But if the whole world should converge to the living conditions of a first world country, well, I would hope that the practice of abortion (though not necessary the legal aspect of it) would diminish significantly. The future unborns of this country need better options.

Another reason why I would hope for more future restrictions on abortion is that: many people who are normally against abortion, upon facing the humiliation brought before them when their daughter is about to have a baby, will immediately go to the doctor and have that child aborted. Faced with a terrible conundrum and an option for escape, an option that is becoming more and more socially acceptable, the notion of responsibility on the part of each citizen is diminished. We’re allowed to do whatever we want without any real repercussions. We no longer need to live within the confines of morality. In the extreme case, we are playing God.

Which leads me to a theory about 90% of the crap that happens in our country, and 90% of the unfortunate or misdirected policies made in response to this crap: it all comes from human greed. We thirst for money. We want lower taxes. We want better benefits. We want to be spoiled. We want freedom to do whatever we like. We want popularity. We want power. So what is the real ailment behind our quality of life? Is it Republicans cutting taxes, starting wars, and overextending our sphere of influence? Is it Democrats spending too much money on unnecessary programs and promising too much to the American people? Is it whatever extravagant promises each party will give to appease their respective half of the US population? And that turned us into a culture of borrowers, heavily in debt, on the verge of an economic crisis.

Practically speaking, the problem is too hard to manage at a macroeconomic scale, but it’s never too late to start changing as individuals. To start giving to your local charity or church. To start volunteering time and money whereever there is need. I myself am guilty of not doing much, if anything at all, but I would like to start. If anything, the Bible tells that us that people do not need riches. They need love. They need to know that they are God’s children, and have inherent dignity. What better way to mitigate the practice of abortion than to provide other ways out of it? Although most of us are not in the position to influence these matters directly, I believe any impact we have on the general well-being of others will also have an impact on people far and wide. We just have to get off of our lazy, self-absorbed asses, and start moving.

The ladder paradox

My friend and I, neither of whom are physicists, got into a discussion about special relativity recently. Essentially, special relativity leads to many paradoxes that result from time dilation, Lorentz contraction, etc.

The famous example is that of a garage that is 9 feet long, a ladder that is 10 feet long when “stationary” (but remember, it is only “stationary” relative to some frame of reference), and a mover that is holding the ladder and moving at a relativistic speed. If you are in anyway familiar with relativity, Lorentz contraction says that if an object is moving at a speed on the order of the speed of light (relative to your frame of reference), it will begin to look shorter to you. So let’s say that this ladder is moving at a speed of 3/5*c relative to the observer, such that its length shrinks to 8 feet. The garage has doors located in the front and back that are initially open, but when the ladder enters the garage, the doors immediately close. Then they open before the ladder exits the garage. From the observer’s point of view, there is no contradiction, because the ladder 8 feet long can fit within the garage.

But what if you were viewing it from the mover’s point of view? In his world, the garage is moving at 3/5*c, so its length shrinks to 7.2 feet. However, the ladder is 10 feet long! Can he actually fit inside the garage then? Well, thanks to the relativity of simultaneity, there is no contradiction based on what he sees moving at such high speeds. The mover will actually see the front door open, then the back door open. After he slides through, the front door closes, and then the back door closes. Pretty weird.

Now my friend went ahead and proposed another example, to which I might have an answer but will need to verify. Suppose that from the observer’s point of view, the back door is closed. The front garage door open, and the ladder goes in. Then the doors close, and the mover immediately stops moving after the doors close. What will happen inside the garage when the ladder suddenly expands?

I will leave this to the reader as an exercise.

God’s big decomposition problem

I was going back through some old journal entries when I found this one. It was definitely entertaining enough to be blogworthy, so I made a few small edits to it.

If there ever was a God who is simulaneously good, omniscient, and sovereign, then it’s obvious that He does not operate on this world as a centralized controller. Human beings are not like the ocean’s waves that can be calmed by a single command, or the mountains that can be moved by a simple act of faith, or the universe that can be spoken into existence. Rather, we are autonomous agents with a significant amount of free will–so much in fact that our very existence has led many to doubt at least one of 3 characteristics of God described above. But what if instead, God worked in a decentralized manner? Then it is then possible for God to be good, omniscient, and sovereign, without having to control the outcome of every situation. I will use a model that stems from decomposition theory in operational research. Hopefully, tough questions like these that are hard to answer using conventional apologetics can be answered by modeling God’s purpose as a decentralized optimization problem. Suppose that God is trying to maximize some function, which we will denote “God’s glory”. “God’s glory” depends largely on human actions, but does not necessarily correlate with what makes each individual happy.* However, provided that the distributed mechanism is sound, the world will converge to “God’s maximum glory”. There are many popular distributed optimization approaches that exist, but for the sake of argument, I will just stick to the two most popular decomposition methods in engineering: 1) Dual decomposition: This is the mechanism that can be used to reveal God’s righteousness through judgment/punishment. Herein lies the idea of karma: if you break the rules, you get what you deserve. In dual decomposition, there are no constraints to free will. God does not say that you are incapable of sinning, only that if you sin, you will be “taxed” or “priced” for sinning. This provides both awareness and incentive for righteousness. A prime example of God’s dual decomposition involves the old testament nation of Israel: Seek the Lord, and be blessed; turn away from the Lord, and be punished. “Then you will know that I am the Lord your God, who brought you out of Egypt, out of the land of slavery.” 2) Primal decomposition: If God decides instead not to punish people, but to directly influence/change people’s nature, He can accomplish this through primal decomposition. In primal decomposition, each person’s utility is not only a function of his own actions, but also an external coupling variable introduced by a higher (possibly centralized) agent. Primal decomposition can also have numerous levels of hierarchy. Then what is God’s primal decomposition? This would be His kingdom coming through His spirit and love, a direct allocation of resources to anyone who believes. First, consider how His kingdom is built: it started with Jesus Christ, who dispersed knowledge of God by teaching 12 disciples and many others during his ministry. Through his act of love on the cross, he demonstrated God’s love to sinners. His believers went out and shared this message with others, some of whom became believers as well. Likewise, God’s spirit is what marks us as saved. This comforter and helper is the variable that couples us with God. After all, Christianity is all about the relationship. =) What are important lessons to take away from decomposition: Scale: Different things require different amounts of time to come to fruition, depending on how knowledge, righteousness, love, and spirit of God are distributed among agents. Different hierarchical levels may also communicate at different frequencies, thereby dispersing the knowledge/blessing at different times–possibly even years into the future. In other words, God has his own timing, and a believer in the distributed nature of God’s workmanship will know to wait on Him. Temporary setbacks: Distributed optimization does not always require the state to improve during each iteration, only that in the end, “God’s glory” will be maximized. In other words, there is a place for suffering in a distributed environment, and suffering can often be on a path that leads to good. Therefore, one should not give up hope just because he is going through a tough trial. Miscellaneous distributed structures in Christianity: -The church (or gathering of believers) is and has always been run in a distributed fashion, usually with a shepherd periodically overlooking them. Just think about the apostle Paul and letters he writes to various churches. -Prayer: this is the channel given to distributed agents to communicate directly with the centralized agent, or God. If you believe in prayer, meaning that God listens and answers, then this is an extremely powerful means of communication. Some thoughts left out of this rambling that may be considered in the future: -I did not consider the dynamic nature of human beings. In particular, there are other game theoretical results for multi-agent systems where future reward is considered. In general, temporally-minded people have beta < 1, meaning reward that can be modeled in their minds as R(0)+beta*R(1)+beta^2*R(2)+…., where R(0) is the immediate reward, and R(t) is the reward at time t in the future (a continuous model can be used). Eternally-minded people have beta = 1. In general, people have beta < 1. God’s timelessness probably corresponds to time-averaging (beta = 1), so it is often difficult for humans with myopic vision to discern the will of God.

Clarification on my neutrality towards prop 8

Most of you might think that I would be a “yes” on prop 8 due to my Christian beliefs, but actually I am neither for nor against. As I see it, extending the *legal* definition of marriage to include couples of the same sex is very much a cultural issue that depends very much on how people view marriage. And as far as I see, on this issue, people do not vote based on pure reason, but based on their desires for equality of rights, for recognition, for protecting their children, free speech, whatever. Here’s how I see the so-called “justifications” for and against prop 8.

1) Yes on prop 8: This can include a wide range of people, from conservative parents to various religious groups. I will use Christianity as I am most familiar with what the bible says about marriage.

Many claim that gay marriage should not be tolerated because it goes against God’s institution for marriage, which should be between one man and one woman. This might be true for “Christian marriages,” but aside from the argument of separation of church and state (which I actually find a weak argument for various reasons, more on that later maybe), there is another reason why I find this argument weak. If one argues that biblically, God ordained marriage to be between one man and one woman, then he must also be willing to accept all other laws regarding marriage that God has ordained. For example, divorce should be outlawed except in the case of (reported) abuse or extramarital affair. Jesus himself said that a divorce is a sin unless the spouse has cheated or abandoned you. But people get divorces just because they “don’t love each other” anymore, or they don’t think it’s “working out”. The question then is, if you are fighting so hard to make gay marriage unconstitutional in order to “protect” marriage, why aren’t you working half as hard to change other marriage and divorce laws?

2) No on prop 8: (Marriage is a right, not a “right thing to do”.) Most arguments stem from the fact that marriage in America seems to mean nothing more than a lifetime commitment between two individuals. In this case, gay marriage is a simple extension of straight marriage, by allowing a minority the same rights as the majority. It is no different from interracial marriage extending from same race marriage. But according to this logic, it also means that incest between a brother and sister is an extension of straight marriage, and hence should be tolerated. And taking it a step further, you have to consider polygamous cultures, such as Muslims and Mormons. Why can’t a wedding involve 3 or 4 people? Is there anything wrong with that? Arguably no, granted that the meaning of this word “marriage” can be amended to include lifetime commitment between a *group* of individuals.

3) Why I am neutral: This has nothing to do with being indifferent, but rather the fact that I see marriage as a purely cultural ideal. I don’t believe laws are made based on logic so much as they are based on a mix of beliefs, power to enact these beliefs, comfort with these beliefs, and personal desires. In America, where we are inspired by a mix of Christian monogamy, separation of church-and-state, and Disney romances, we believe in marriage primarily as a binding commitment of love between two (and not more) individuals, traditionally between man and woman, though less so today. In Muslim countries, as in many ancient cultures, men are accustomed to have multiple wives. In remote regions of China, some women have multiple husbands.

Given different meanings attached to the word marriage, which may stem from numerous religious, philosophical, or cultural ideals around the world, America will eventually choose its meaning. (It will probably coincide with that of the ancient Romans.) Regardless of what is chosen in legislature, there is no difference in the great calling for Christians to live holy lives within, or in pursuit of, God’s definition of matrimony. The question is how far should we go to change restrictive laws regarding marriage, divorce, abstinence vs protection, restricting pornography, etc. Or how far should we go to grant everyone equal opportunity to pursue their desires for “marriage”, whatever it means. I have no real answer for that, so I will just leave it up to you to ponder the slippery slopes of conservatism and liberalism. In the end, however, I speculate that most of us voted based on our own desires more than based on pure logic. It’s human nature.

The moderate independent,

Foo

Joker’s game: An extended game of hope and paranoia

I am sure that there have already been a number of game theorists who have commented on Joker’s ship game in the Dark Knight. For example, the following link analyzes the game while factoring in human desires to survive and to obey morality. Also, it introduces a probability that batman will intervene and save the ship.

However, one shortcoming of this model is that the ship game is treated as a one-shot game, which does not take into account the changes that may occur in human thought and behavior over time. For example, suppose that Joker started the game at 6pm, and the ships are set to detonate at midnight. You are aboard one ship, and you know that the other ship has no intention to blow you up. However, you feel morally obligated to keep the other ship alive as long as possible, in case batman does intervene successfully. Would you end up blowing the other ship up? Well, if you ultimately want to live, you would probably want to blow the other ship up a few seconds before midnight, granted that batman has not yet succeeded. However, at 8pm, you may want to hold onto your remote and wait to see the outcome.

In this entry, I will use a simple mathematical model to predict the change in human behavior over time as a function of their sense of urgency.

Partition the game into T discrete time periods (not necessarily equal) during which Batman has a fixed, independent probability Z of intervening successfully. Call the value of life X, and the value of obeying ones conscience Y. (In our game matrix, we will use 0 for clear conscience, and -Y for a guilty conscience.) If batman successfully intervenes with Joker’s plan first, then the game ends, and both ships survive with clear consciences. The game also ends if one ship destroys the other ship, or vice versa, but the surviving ship is left with a guilty conscience. Finally, we will also assume that if both ships flip the switch during the same time interval, then both ships are annihilated, but no moral consequences are felt.

Some Trivial Cases:

Case X > Y, Z = 0. Suppose that batman does not exist, and people do not have a conscience, or a conscience that is worth less than life. Then Joker’s repeated game has a dominant strategy: blow the other ship up as soon as you get the trigger.

Case Y > X, Z = 0. Suppose that batman does not exist, and people have a collective conscience greater than life. If the penalty of violating the conscience exceeds that of dying, then the dominant strategy is to wait until midnight, and let Joker blow them up.

A more interesting case, the game of hope:

Case X > Y, Z > 0. Here batman has a chance Z during each time interval of saving both ships. Hence, at the beginning of the game, the value of the game is a geometric sum from t = 0 to T-1 of Z(1-Z)^t. Provided that batman defeats joker, the resulting utility derived by both ships is X, since both conscience and life are satisfied. Hence, at each time interval t between 0 and T-1, the matrix is given by:

	blow ship	not blow ship
blow ship	0, 0	X -Y, 0
not blow ship	0, X -Y	(1-(1-Z)^(T-t))*X, (1-(1-Z)^(T-t))*X

where the bottom right entry is the expected value derived if both ships decide never to pull the trigger.

When t becomes large enough, the value of not blowing the other ship may become less than X – Y, which leads to a weakly dominant strategy of blowing up the other ship. Note that in the very last interval T-1 (right before midnight), the game reduces to:

	blow ship	not blow ship
blow ship	0, 0	X -Y, 0
not blow ship	0, X -Y	Z*X, Z*X

Hence, if the chance of batman’s intervention is greater than (X-Y)/X, the dominant strategy throughout the entire game is to not blow the other ship. However, if the chance of batman is smaller than (X-Y)/X, somewhere along the way, maybe at 10pm, or 11pm, or 11:30pm, the expected value of not pulling the trigger will become less than pulling the trigger. The best strategy then becomes to blow the other ship up, because it is time to give up hope on Batman.

The game of paranoia

However, this game can be extended once more! Suppose that you also know that the other ship is also reasoning using the same selfish game theoretic principles as you are. You know that at 10pm, both of you are going to give up hope on Batman, and hence blow each other up. In this case, your real deadline is not midnight, but rather 10pm! What happens then? Should you try to blow the other ship up at 9:59pm, knowing that if you do not, then you will die?

However, the other ship also knows that you are thinking this, and hence decides that it would be better to blow you up at 9:58pm. I mean, 1 minute is not enough for Batman to intervene right?

Using this argument recursively, we see that this game becomes one of paranoia, and a dismal game this is! Each ship comes to the conclusion that it is best to press the trigger upon getting it, to minimize the chance of the other ship pressing it first!

The game of mutual trust or cooperation

I do not have a nice rigorous argument for this, but: If there were such a thing as a game of cooperation, it seems likely that the two ships will talk to one another at t=0 and decide to throw out the triggers, even if each ship values life over morality. (Of course, the movie suggests that they did so for moral reasons.) By tossing out the triggers on each ship at time 0, granted that Batman has a sufficiently high probability of intervening, they play a one-shot game based on the expectation that Batman will successfully intervene before midnight. Furthermore, by preventing an extended game, they avoid the terrible equilibrium induced by paranoia.

Of course, because the ships are unable to talk to each other, this decision can only be reached based on some form of “mutual trust” or “faith in humanity”.

—-

Other notes:

In reality, each human has a different weight attached to life and morality, but here I simplified the problem into a symmetric game assuming that the law of large numbers reduces the variance of each ship’s collective value system, such that the utilities are nearly identical.

Also, Joker could have been lying, just for the heck of it. But such a game would not be meaningful to play, would it?

Sounds heard around our lab

I decided to write out my most vivid audio memories of the lab:

sound1

sound2

sound3

Well, the last one is my own. More to come when I remember what the others sound like.

anti-caffeine

I’m guilty of having a little caffeine in my system at times, but I’m adamantly against coffee and espresso.