Coming Soon:

The following books by Robert Paul Wolff are available on Amazon.com as e-books: KANT'S THEORY OF MENTAL ACTIVITY, THE AUTONOMY OF REASON, UNDERSTANDING MARX, UNDERSTANDING RAWLS, THE POVERTY OF LIBERALISM, A LIFE IN THE ACADEMY, MONEYBAGS MUST BE SO LUCKY, AN INTRODUCTION TO THE USE OF FORMAL METHODS IN POLITICAL PHILOSOPHY.
Now Available: Volumes I, II, III, and IV of the Collected Published and Unpublished Papers.

NOW AVAILABLE ON YOUTUBE: LECTURES ON KANT'S CRITIQUE OF PURE REASON. To view the lectures, go to YouTube and search for "Robert Paul Wolff Kant." There they will be.

To contact me about organizing, email me at rpwolff750@gmail.com




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Saturday, April 29, 2017

BALM FOR THE SOUL

Someone, I think it was Chris, described this as a “Marxist/anarchist blog,” which I suppose is fair enough, inasmuch as I identify myself here as a Marxist and an anarchist [also an atheist, a husband, a father, a grandfather, and a violist – this last something of a reach – but whatever, as young people say.]  However, that is not how I think of myself.  I am a philosopher, a teacher, and, more than anything else, a life-long lover of powerful, simple ideas, so lucidly and elegantly expressed that their beauty can be seen by all.  The exigencies of the present political situation have compelled me to venture very far from my true calling, but my mental health requires that I return from time to time to the realm of ideas to remind myself what I most love.

Which brings me to the subject of my musings during this morning’s walk.  Can it be, I found myself wondering, that a term in a language should always be misused?  I am not here merely expressing my inner pedant.  Like many, I cringe when some television talking head says that this or that “begs the question,” meaning that it compels us to ask the question, not that it assumes what is to be proved.  Or when another deep thinker says that it is impossible to underestimate the importance of something, meaning of course that its importance is so great that it is impossible to overestimate that importance.  My favorite example of this linguistic pickiness is Harry Levin, the great Harvard Shakespeare scholar of half a century ago.  When my first wife, Cynthia Griffin Wolff, mailed a complete draft of her doctoral dissertation on Samuel Richardson to Levin, who was her Director, he sent it back without comment, but he had clearly read it, because throughout the text, he had countlessly times changed “shall” to “will” and “will” to “shall.”  Once those essential alterations were made, she was good to go.

No, I am talking about a made-up phrase, coined more than seventy years ago by John von Neumann – zero-sum game.  [Strictly speaking, the term should be credited both to von Neumann and to his co-author, economist Oskar Morgenstern.  Morgenstern was a very interesting thinker, the author, among other things, of a delightful book titled On the Accuracy of Economic Observations, which I recommend to you all, but von Neumann was one of the authentic geniuses of the twentieth century, so I shall imitate my fellow Marxists, who tend to attribute all the ideas of Marx and Engels to Marx, and speak as though Game Theory was von Neumann’s creation alone.]  Everybody uses the phrase “zero-sum game,” and everybody, without exception, misuses it.  Is that even linguistically possible?  Here are just two examples.  The first is from Barack Obama’s farewell address this past January.  The second is an older misuse by Paul Krugman who, think of him what you will, is a Nobel Laureate in Economics and should know better.

Obama:   “Our economy doesn't have to be a zero-sum game.”

Krugman:  “Unlike war, trade is not a zero-sum game.”

I shall now explain to you exactly what “zero-sum game” means in several thousand well-chosen words.  I am well aware that at this point I shall be losing virtually all of my readers, but in a desperate effort to hold a few of you before you surf away to your favorite revolutionary blog, I will simply observe that the term has its roots in the successful attempt by neo-classical economists to purge their “scientific” discipline of its radical redistributionist roots.

In 1944, von Neumann and Morgenstern published a brilliant book, Theory of Games and Economic Behavior, which created the new mathematical/economic sub-field of Game Theory.  The centerpiece of the book is a powerful theorem concerning a certain sub-set of two-person games.  A game is defined as a series of moves leading, by means of a termination rule, to a determinate outcome.  [The positing of a termination rule eliminates the possibility of a game with an infinite number of repetitive moves.  In Chess, for example the rules stipulate that if a position occurs three times, or if fifty moves are made without a piece being taken or a pawn being promoted to the eighth rank, the game is declared a draw.]

von Neumann posits that each player has a complete, transitive utility function that assigns a utility index, invariant up to an affine transformation, to each possible outcome of the game.  [Invariance up to an affine transformation makes it possible to assign cardinal indices, not merely ordinal indices, to the outcomes.  A familiar example of an affine transformation is the rule that allows us to figure out what the Fahrenheit equivalent is of a temperature give in degrees Celsius.  The rule is Degrees F = 9/5 Degrees C + 32.   It tells us that if the TV in the Paris airport, as we deplane, says it is going to be 20 degrees Celsius today, that means it will be 68 degrees F, so no jacket needed before catching a cab to the hotel.]

Now, we all remember that Jeremy Bentham brought into Political Economy the notion of a social calculation of the pleasure [or utility] and pain [or disutility] promised by a proposed law, along with the principle that we should always seek in our legislating to produce the Greatest Happiness for the Greatest Number, a principle that rapidly acquired the label Utilitarianism.  What we may not so readily recall is that when Bentham proposed this now familiar principle, it was intended by him and understood by others to be a shockingly radical, not to say revolutionary, idea.  Bentham stipulated that each was to count for one, which meant that the pleasures and pains of the peasants would weigh as heavily as those of the aristocrats.  This was utterly unacceptable to the toffs, who protested that since their sensibilities were ever so much more refined than those of the rude masses, their delights and discomforts should carry greater weight in the social calculus [the Princess and the Pea Principle].  But there were so many peasants and so few aristocrats that no such weighting could overcome the tendency of the misery of the masses to outweigh the pleasures of the classes.   It was a proposal that had the power to overturn the established order, and Bentham knew it.

Bentham’s godson, John Stuart Mill, did his best to contain the damage, arguing in Utilitarianism for a distinction between higher and lower pleasures, but that was a desperation rearguard action, akin to laying a few landmines during the retreat to Dunkirk.  The real solution was advanced by the Economists, who latched onto an arcane doctrine in English Philosophy about the impossibility of interpersonal comparisons of utility.  This gave rise to Indifference Maps, Pareto Preference, and all manner of highly successful defenses against the totally unacceptable suggestion that one person’s utility should be added to another’s.  The firewall against the demands of the lower classes was given its theoretical imprimatur in Lionel Robbins’ classic 1932 book, Essay on the Nature and Significance of Economic Science.

Enter von Neumann.  In Theory of Games and Economic Behavior, von Neumann assumed that each of the players in the two-person game had a cardinal utility function, but that in general nothing could be said about the relationship between one player’s evaluation of the outcome of a game and that of the other player.  However, he proved that in one very special set of circumstances, it was possible to make such a comparison, and in fact actually to add one player’s assignment of utility to another’s.  In short, one could give meaning to the notion of the sum of their evaluations, despite not making any assumptions about interpersonal comparisons of utility.

The key was the assumption that the two players had strictly opposed preference orders not merely for the finite set of outcomes of the game, but even for probability mixes of those outcomes, which can be called “lotteries.”  Let me explain.  There are, by the stipulation of a termination rule, a finite set of outcomes, over which each player is assumed to be able to define a utility function.  Von Neumann added to this the further assumption that each player could express a complete and transitive preference over the infinitely varied ways in which one could assign probabilities to those outcomes [each assignment to sum to 1, of course].  Think of these assignments as lottery wheels, with the size of each slice of the lottery wheel corresponding to the weight being assigned to the outcome that the slice represents.  Spin the wheel, and one prize will win, with the probability of that win a function of the size of the slice.  Von Neumann now made a second assumption, that each player could express consistent preferences not only over prizes and lotteries of prizes, but even over lotteries of lotteries of prizes – what are called compound lotteries [The most famous example of a compound lottery is the Irish Sweepstakes, in which the prizes were not amounts of money, but betting tickets on horse races.  A ticket in the Sweepstakes was a bet on a bet, as it were.]

Now, by mathematical rules quite familiar to probability theorists, compound lotteries can be reduce to simple lotteries in which the two prizes are the least and most favored outcomes.  [This reduction calculation is equivalent to the assumption that the players have no pure preference for or aversion to risk itself, independent of the probabilities.  That is a powerful assumption, by the way.  I, for example, have an aversion to risk.  Offer me the certainty of one dollar, or a fifty-fifty chance of getting nothing or two dollars, and I will take the sure dollar every time.]

Von Neumann now asks the following question:  What, if anything, can we say about the utility assignments to outcomes of a game between two players who have strictly, exactly opposite preferences not only for the outcomes but also for all compound lotteries of the outcomes?  His answer was simply gorgeous.  First, he said, perform an affine transformation on each player’s utility function so that the least preferred outcome for that player is assigned a utility of 0 and the most preferred outcome is assigned a utility of 1.  Under these very restrictive and special conditions – a two person game with a finite set of outcomes in which the players have strictly opposed preferences for compound lotteries of the outcomes – it is possible fairly easily to show that the sum of the utilities assigned by the two players to any outcome or lottery of outcomes will sum to 1.  [Check my other blog for the proof.]  If one then performs one final affine transformation, this time transforming player 2’s utility function so that it runs from -1 to 0 rather than from 0 to 1, then the sum of the utilities assigned by the two players to outcomes or compound lotteries of outcomes will always sum to zero.

THIS, AND ONLY THIS, IS WHAT IS MEANT BY A “ZERO-SUM GAME.

In particular, no game or game-like situation with three or more players can be a zero-sum game.  Furthermore, it is a mistake to conclude that all other games are positive sum, or negative sum, or variable sum games.  The concept of the sum of a game, assuming the impossibility of interpersonal comparisons of utility, is simply undefined for all such games.  And this is true, regardless of what Barack Obama, Paul Krugman, and everyone else says.

So everyone always misuses the phrase “zero-sum game.”




13 comments:

howie berman said...

So much for rectitude, how about flexible pragmatism?
An idea is only as good as how we the living can put it to good use.
In the period of the Enlightenment, Aristotelian realism did not always imply adherence to no matter without form; people knew what you meant.
The idea of markets in early economic theory if I remember Economics 101 correctly referred to a market place not as today an aggregation of markets, and models extended supply and demand to larger and larger markets.
If an idea as in Piaget does not assimilate and accommodate to the real world, what's the point?
So Von Neumman's idea has been applied to novel yet similar situations in which games are played.
What's wrong with that? Krugman could probably whip out a pocket full of equations for you.
Sense and reference may evolve. Still they have a comfortable nook in our grab bag.
I wouldn't expect you to ply a hardline with game theory any more than with the bill of rights

I. M. Flaud said...

"In particular, no game or game-like situation with three or more players can be a zero-sum game. "

I agree that the popular use of the term 'zero-sum game' is a source of fremdscham.

However, I'm going to disagree that the term necessarily refers to two-player games. For evidence, I'm going to quote my Sixtieth Anniversary Edition of "The Theory of Games and Economic Behavior." In Chapter XI, General Non-Zero-Sum Games, von Neumann and Morgenstern discuss the zero-sum extension of an n-player game. In section 56.2.1, page 505, they write, "We have formulated the program of linking the theory of the general game in some way to the theory of the zero-sum games. It will actually be possible to do more: any given general game can be re-interpreted as a zero-sum game.
This may seem paradoxical since the general games form a much more extensive family than the zero-sum games. However, our procedure will be to interpret an n-person game as an n+1-person zero-sum game. Thus the restriction caused by the passage from general games to zero-sum games will be compensated for--indeed made possible--by the extension due to the increase in the number of participants."

This is from pages 505-506 of my edition of TGEB. Chapter and verse would be evidence enough against the charge of that zero-sum games with more than two players is meaningless, but possibly not against the charge of triviality. I would hesitate to call any mathematical procedure of von Neumann's trivial, but this is an argument from imagined intimidation. It turns out the extension is not so trivial--it even has an interesting interpretation.

In the next section, 56.2.2, the extension of a general n-person game to an (n+1)-person zero-sum game "...consists of introducing a--fictitious--(n+1)-st player who is assumed to lose the amount which the totality of the other--real--players wins, and vice versa." We can leave out the symbolism--it's enough to say that the payoff function of the (n+1)-st player is the negative of the sum of the payoffs to the preceding n players.

Why isn't this trivial? Giants once walked the earth--they did indeed. In TGEB section 56.3.2, von Neumann and Morgenstern wrote, "The fictitious player was introduced as a mathematical device to make the sum of the amounts obtained by the players equal to zero. It is therefore absolutely essential that he should have no influence whatever on the course of the game. ...We must nevertheless put to ourselves the question whether the fictitious player is absolutely excluded from all transactions connected with the game."

Now I wish to suggest that von Neumann and Morgenstern characterized the role of the taxpayer in the bailout of the financial crisis of 2008, and more generally the role of the taxpayer in socializing the losses of the important players in our plutocracy--a claim that would not have been associated with the right-wing von-Neumann and that would have ben unthinkable during the period in which TGEB was written and for some time after, but I am getting ahead of myself.

I. M. Flaud said...

"This caveat is not at all superfluous. As soon as \overline{\Gamma} [\Gamma is the notation for the n-player game, \overline{\Gamma} that of its zero-sum extension] involves three or more persons, the game is ruled by coalitions, as we observed at an early analysis. A participation of the fictitious player in any coalition--which is likely to involve the payment of compensations between the participants--would be completely contrary to the spirit in which he is introduced. ..."

In section 56.3.3, the objection that there might be side deals among the real and fictitious players looks like a non-starter. After all, the payoffs to the real players don't depend on any amounts that the fictitious player controls. [I'm paraphrasing.]
von Neumann and Morgenstern then write, "It may appear at first that this argument has some merit. The conditions described make it seem that any coalition of real players is just as well off without the fictitious player as with him. Is he anything but a dummy? If this were so, the theory of \Gamma could be applied without any further qualifications to \overline{\Gamma}. However this is not the case."

I'll jump to the statement at 56.4.3 "Hence we must conclude that the zero-sum game \overline{\Gamma} cannot be considered am unqualified equivalent to the general game \Gamma.
What are we then to do? In order to answer this question, it is best to return to the analysis of the specific example of 56.4.1 [omitted here], where the difficulty was expressed fully."

Well, we're dealing at this point with (n+1)-player zero sum games. It turns out that the extension of a general n-player game, defined in The Theory of Games and Economic Behavior, to a zero-sum (n+1)-player game doesn't permit the application of the theory of zero-sum games. So one cannot rule out the extension as a trivial exception to the assertion that no game with greater than two players can be a zero-sum game.

I. M. Flaud said...

Correction, "doesn't permit the application of the theory of zero-sum games without restriction." It's important to get at least that right. Meaning that if you look at the new structure [the zero-sum (n+1)-player extension of an n-player game] and analyze it according to von Neumann and Morgenstern's zero-sum theory, then without including into the description of the game some mathematical notion of "fictitious" versus "real", the player we designated outside the formalism as fictitious will nevertheless get into the game "...in spite of his inability to influence the its course directly by moves of his own. Indeed, it is just this impotence which determines his policy of offering compensations to others, and thus sets the above mechanism into motion."

Daniel Langlois said...

I could just pithily drop the quote here, but I'll expand about what we might call the Humpty-Dumpty theory of language. No, actually, I like the idea of just pithily dropping the quote here..

--
'"I don't know what you mean by 'glory'," Alice said.

Humpty Dumpty smiled contemptuously. "Of course you don't- till I tell you. I meant 'there's a nice knock-down argument for you!'"

"But 'glory' doesn't mean 'a nice knock-down argument'," Alice objected.

"When I use a word," Humpty Dumpty said, in rather a scornful tone, "it means just what I choose it to mean- neither more nor less."

"The question is," said Alice, "whether you can make words mean so many different things."

"The question is," said Humpty Dumpty, "which is to be master-that's all."

Alice was too much puzzled to say anything; so after a minute Humpty Dumpty began again. "They've a temper some of them- particularly verbs: they're the proudest- adjectives you can do anything with, but not verbs- however, I can manage the whole lot of them! Impenetrability! That's what I say!"'
--

Daniel Langlois said...

btw, 'the Princess and the Pea Principle].':

I get it -- Princess Pea is a girl of principles too. How did the queen know the princess was a real princess? She hid peas at the bottom of the 20 mattresses and blankets and the princess could still feel it.

Look in your pillows. Look under your pillows. What did you find?

--
-“In the morning she was asked how she had slept. ‘Oh, very badly!’ said she. ‘I have scarcely closed my eyes all night. Heaven only knows what was in the bed, but I was lying on something hard, so that I am black and blue all over my body. It’s horrible!’ Now they knew that she was a real princess because she had felt the pea right through the twenty mattresses and the twenty eider-down beds. Nobody but a real princess could be as sensitive as that. So the prince took her for his wife, for now he knew that he had a real princess.”
--

I say that I get it, but when I was a kid, I’m pretty sure I came away from the story thinking that the princess was a spoiled brat. But today, at 47, I find myself sympathizing with the princess. She’s a victim. Privilege isn’t always a privilege. She's the kind of woman whose morning is ruined if the new guy at Starbucks messes up her soy-latte; the kind of delicate flower whose entire day is ruined if her favorite yoga instructor calls in sick; the kind of therapy-junkie whose entire week is ruined if her therapist cancels her weekly appointment; the kind of absentee-parent who has a panic attack when the nanny quits because she really doesn’t know how to take care of her own kids. Look at her: she’s pathetic.

Another angle here is to wonder what really makes us great -- such as, maybe, feeling the holy ghost or something. And not that we are so used to luxury that we can't sleep on a lumpy mattress. Nobody took the princess for a philosopher king. In other words, there are princesses enough..there, that is a true story..

There occurs to me an interesting thought about short-run phenomena, versus medium-run or long-run phenomena. Short-run being measured in minutes or hours. There are processes which proceed simultaneously, but how to model this? That some processes are infinitely faster. It's a relevant thought, if you're going to try to model 'social equilibrium', maybe. You may require some heroic simplifying assumptions. And the princess is being identified as a marriage prospect -- accurately!

One could prefer to discuss untilitarianism and, maybe, social contract theory etc., but I wouldn't consider it irrelevant that we seem totally incapable of understanding this princess and pea fairy tale, eh? I would have thought somebody was talking a little dumb to appear to be smart. Do you not get this fairy tale?

Daniel Langlois said...

'In particular, no game or game-like situation with three or more players can be a zero-sum game. Furthermore, it is a mistake to conclude that all other games are positive sum, or negative sum, or variable sum games. The concept of the sum of a game, assuming the impossibility of interpersonal comparisons of utility, is simply undefined for all such games. And this is true, regardless of what Barack Obama, Paul Krugman, and everyone else says.'

Okay, so John von Neumann proved an important theorem, i.e. every two-person zero-sum game has a value. And one may present a proof using linear programming dual theory. I think I'm funny, because actually, that's true, but his proof uses the Brouwer fixed-point theorem. But then, the question become whether there are classes of well-behaved two-player games for
which we could hope for positive results in the networked setting. That is, we ask whether there are properties, like of 2-player zerosum games, maybe, that extend to the network setting despite the combinatorial complexity that the networked interactions introduce.

For example, a small variation of zero-sum games are strictly competitive games. I would want us to have a little moment here, and touch base, about whether we agree that finding a Nash equilibrium in polymatrix games with strictly competitive games on their edges
is PPAD-complete. That's lots of jargon, but I'm not trying to be pretentious -- I have an M.S. in computer science. This doesn't make me an accomplished man, I'm just saying -- I disagree with the post and it's not like I'm bored. Anyways, it is often stated that zero-sum games represent “perfect competition”, and such.

But I'll wrap this up by making a remark about what are the 'properties' that I mentioned? --the important properties of two-player zero-sum games? Well, in short, haha, the idea is going to be to try to provide strong support on the plausibility of the Nash equilibrium predictions in this setting. This is maybe one of the most important properties for this:

'if the nodes of the network run any no-regret learning algorithm, the global behavior converges to a Nash equilibrium.'

..which gets us into discussing the simplicity, universality and distributed nature of the no-regret learning algorithms. Anyways, the notion of a no-regret learning algorithm, and the type of convergence used here is, I suggest, to be thought of as being quite standard in the learning literature. Now, I say that I'm not trying to be pretentious and I'm sorry that this stuff is abstruse, but I didn't bring it up. I will add, that I don't think it's right to try to be pretentious. Calling out all the Nobel-prize winning clowns who ought to know better, for example, rather forces upon me the question who is being pretentious?

Daniel Langlois said...

I will add that I do not mean to simply defend Krugman -- especially, though, I have this problem with him, that Krugman can be shrill, laying out personal attacks and portraying anyone who might disagree with him as being motivated by pure evil. Furthermore I am not interested in giving the last word on whether the infamous Housing Bubble was nothing more than a zero-sum game. I don't look to Krugman on the matter. He has an idea that the bubble really was nothing more than a huge wealth transfer, and now the people who got the wealth are selfishly squirreling it away. My hope would be to catch onto the idea that there are debates, here, -- that it's interesting. That maybe it's more interesting when you learn more about it. There is a whole subject here. People work with notions about whether all of the assets created in the boom have exactly the same value that they had before the crash occurred, etc. I'm not able to defend the notion that all of this really is a zero-sum game, as Krugman is claiming, or whatever. I don't know, for example, whether more spending will cure everything by giving the economy more "traction". I see such ideas, and I even think I see them in Krugman, as I recall and I can't distinguish it from the application of circular logic.

Is an economy is a zero-sum game? The question is so abstract and informal that it seems to me to just be playing with words. I ask the question, not knowing what I mean. Sometimes, in any case, one catches the impression from Krugman, superficially, that he really is claiming that his intellectual opponents reach their conclusions because they are moralistic bastards. It's hard to know what to make of this stuff -- people seem to argue that all you have to do is pay people to dig ditches, and eventually the economy will turn around and the ditch diggers can get a real job. It's hard to follow such stuff. This isn't for the casual dilettante -- I mean, understanding Lord Keynes and such. I don't claim to understand any of it particularly well. I'm belaboring the point, perhaps, that these matters are complicated. But they are. I myself might forget how little I know what I am talking about, even though I seem to be the flipping genius who knows anything about computer science and John Nash. A little knowledge makes me humble, and ought to make me humbler.

The thing to focus on, maybe, is the familiar claim that the extraordinary and marvelous thing about free trade in a global economy - it's not a zero-sum game. Nations party to free-trade agreements win, and the generated economic gains accrue across all sectors of the participating economies. As President Kennedy aptly observed about expanding economies, "A rising tide lifts all boats."

This is a platitude, by now, that in what has been the most dynamic era of economic development in human history, trade has become the basis for a prosperous world economy. Is this really quite transparent, though? The talk about whether some free trade agreement will generate significant gains is rather an esoteric point, -- it looks different if you 'believe' in Adam Smith and such...by the way, I think Adam Smith is brilliant and I am honestly ashamed that I haven't spent more time reading and learning that stuff. I think he's brilliant. We ought to get into Adam Smith sooner or later, if we are pretending to be interested in economics -- let's be doing what we take ourselves to be doing..want to talk about economics? I have much to learn, about economics...

Daniel Langlois said...

'myself might forget how little I know what I am talking about, even though I seem to be the flipping genius who knows anything about computer science and John Nash. A little knowledge makes me humble, and ought to make me humbler. '

meant with respect for those whose little knowledge makes them humbler too. I see that I am not the only one who arrives to the thread with some interest in the topic. Respectfully!

A few points. What is 'game theory'? Game theory is a way to mathematically describe strategic reasoning — of competitors in a market, or drivers on a highway or predators in a habitat. Note that the Nobel Prize in economics has occasionally been awarded to game theorists for their analyses of multilateral treaty negotiations, price wars, public auctions and taxation strategies, among other topics. Now, this is probably rather strange, but in game theory, a “game” is any mathematical model that correlates different player strategies with different outcomes.

One of the simplest examples is the penalty-kick game: In soccer, a penalty kick gives the offensive player a shot on goal with only the goalie defending. --In the game-theory version, the goalie always wins if both players pick the same half of the goal, and the shooter wins if they pick different halves. So each player has two strategies — go left or go right — and there are two outcomes — kicker wins or goalie wins.

The best strategy for both players is to randomly go left or right with equal probability; that way, both will win about half the time.

That is the “Nash equilibrium” for the game -- the point in a game where the players have found strategies that none has the incentive to change unilaterally. In this case, for instance, neither player can improve her outcome by going one direction more often than the other.

Of course, most games are more complicated than the penalty-kick game, and their Nash equilibria are more difficult to calculate.

However, Nash was the first to prove that every game must have a Nash equilibrium. Of course, in the real world, competitors in a market or drivers on a highway don’t (usually) calculate the Nash equilibria for their particular games and then adopt the resulting strategies. Actually, for some games, the Nash equilibrium is so hard to calculate that all the computers in the world couldn’t find it in the lifetime of the universe.

Nevertheless, consider that we are talking about competitors in a market or drivers on a highway, in the real world. And okay, hypothetically, consider this: they will tend to calculate the strategies that will maximize their own outcomes given the current state of play. But if one player shifts strategies, the other players will shift strategies in response, which will drive the first player to shift strategies again, and so on.

And, furthermore, this kind of feedback will eventually converge toward equilibrium: in the penalty-kick game, for example, if the goalie tries going in one direction more than half the time, the kicker can punish her by always going the opposite direction.

Note that approximations of the Nash equilibrium for two-player poker have been calculated, and professional poker players tend to adhere to them — but, exhaustively characterizing a given player’s set of strategies is complicated enough in itself, but to the extent that professional poker players’ strategies in three-player games can be characterized, they don’t appear to be in equilibrium.

There's lots to get into here, and maybe that in itself can be my point. For example, there is a 30-year-old problem in economics, a generalization of work that helped earn the University of Chicago’s Roger Myerson the Nobel Prize in economics. That problem was how to structure auctions for multiple items so that, even if all the bidders adopt strategies that maximize their own returns, the auctioneer can still extract the greatest profit.

Note this:

http://news.mit.edu/2012/comp-sci-econ-0625

Daniel Langlois said...

'THIS, AND ONLY THIS, IS WHAT IS MEANT BY A “ZERO-SUM GAME.'

I will meditate on what is the point that I am missing, here. Beyond that maybe a caps-lock key is broken. The dots are gonna have to be connected together more patiently for me, though apparently, patience is running out.

I suppose we all agree, at least, about some 'Wealth of Nations' themes. No, I guess that we don't agree! Maybe we agree that they *are* 'Wealth of Nations' themes..?

One of these themes, is what might be called 'the benefit of free exchange'. And this idea, is that when the market is left to itself and exchanges are free, both sides benefit. Indeed, no one would enter into an exchange that comes at a loss to them. In foreign commerce, this means that imports and exports can both be very valuable to a society. One society's wealth does not have to come at the expense of another society. A society has more to gain if its trading partners are wealthy.

I'm glad we agree?

Daniel Langlois said...

'THIS, AND ONLY THIS, IS WHAT IS MEANT BY A “ZERO-SUM GAME.'


In game theory and economic theory, a zero-sum game is a mathematical representation. We need to have a notion about what is 'utility'. I mean, in economics. The concept is an important underpinning of rational choice theory in economics and game theory. First of all, the idea is that one cannot directly measure benefit, satisfaction or happiness from a good or service. This is how it has become a 'thing', for economists to have have devised ways of representing and measuring utility in terms of measurable economic choices. Economists have attempted to perfect highly abstract methods of comparing utilities by observing and calculating economic choices. We could say that economists consider utility to be revealed in people's willingness to pay different amounts for different goods, but this would maybe be 'in the simplest sense'. I'm talking about some notion of 'revealed preference', and, again, it was recognized that utility could not be measured or observed directly, so instead economists devised a way to infer underlying relative utilities from observed choice.

So let's start over, and see if we agree that 'utility' is a term used by economists to describe the measurement of "useful-ness" that a consumer obtains from any good. The utility of any object or circumstance can be considered. Some examples include the utility from eating an apple, from living in a certain house, from voting for a specific candidate, from having a given wireless phone plan. Utility may measure how much one enjoys a movie, or the sense of security one gets from buying a deadbolt. Utility can be seen as a measure of how much one values a particular good.

And then, in economics, they usually say that an individual is "rational" if that individual maximizes utility in their decisions. Note here, that the rationality assumption may seem trivial, but if we could not assume rationality, it would be impossible to say what, when presented with a set of choices, an individual would select.

With emphasis: there are no real methods of measuring utility outside of a purely theoretical framework. Again, 'utility' is an abstract theoretical concept rather than a concrete, observable quantity.Now, these sorts of thoughts are used to explain how and why individuals and economies aim to gain optimal satisfaction in dealing with scarcity.

I wonder if we all are happy to stipulate things, here, like that maybe something or other can be said to demonstrate the law of diminishing marginal utility. It is said that the first unit of consumption for any product is typically highest, with every unit of consumption to follow holding less and less utility. This is basic stuff, but I am not sure whether we are already seeming to have wandered into contentious territory? Talking about 'diminishing prices' is, I am confident, already going to be *not* basic uncontroversial stuff, right? And that is, the law of diminishing marginal utility directly relates to the concept of diminishing prices. As the utility of a product decreases as its consumption increases, consumers are willing to pay smaller dollar amounts for more of the product. For example, assume an individual pays $100 for a vacuum cleaner. Because he has little value for a second vacuum cleaner, the same individual is willing to pay only $20 for a second vacuum cleaner..

Again, An individual can purchase a slice of pizza for $2; she is quite hungry and decides to buy five slices of pizza. But, the individual is so full from the first four slices that consuming the last slice of pizza results in negative utility. The five slices of pizza demonstrate the decreasing utility that is experienced upon the consumption of any good. Controversial stuff? Or not? I'm willing to be instructed on whether this seems relevant to anything, but only if it is *not* controversial stuff..

Daniel Langlois said...

'THIS, AND ONLY THIS, IS WHAT IS MEANT BY A “ZERO-SUM GAME.'

I had been offering that 'utility' is a psychological concept, as it were. This is different for different people. Therefore, it cannot be measured directly. Which begs the question: 'can utility be measured?' And that is important, because, in economics, 'production' refers to the creation of utilities in several ways. And it's easy to throw abstractions around informally, and lose track of the fact that this is what we are doing. So consider: is utility the same as satisfaction? Is utility pleasure? I'll answer that one: a commodity may have utility but its consump­tion may not give any pleasure to the consumer, e.g., medicine or an injection. An injection or medicinal tablet gives no pleasure, but it is necessary for the patient.

And yet, what about the idea that a want which is unsatisfied and greatly intense will imply a high utility for the commodity concerned to a person..? And I was discussing just a bit, the notion that the more of a thing we have, the less we want it..

Strangely, utility is the function of intensity of want, although this is related to the point that cannot be utility cannot be measured objectively. Also, with emphasis, it cannot be measured directly in a precise manner. Note then, that there is something to consider here with utility being a subjective phenomenon or feeling of a consumer that maybe cannot be expressed in numerical terms. So utility cannot be measured cardinally or numerically.

One may read that 'utility simply means the ability to satisfy a want', and yet, consider that a cigarette has utility to the smoker but it is injurious to his health. So, 'utility' is not the same as 'usefulness', even! and thus many commodities like opium liquor, cigarettes etc. have demand because of utility, even though, they are harmful to human beings. I want to take some time, here, to be sure we are getting familiar with the notion in its ramifications, that utility is always individual and relative. Utility of a commodity varies in different situations in relation to time and place. Even the same consumer may derive a higher or lower utility for the same commodity at different times and different places. For example—a person may find more utility in woolen clothes during the winter than in summer or at Kashmir than at Mumbai.

Utility of a commodity depends on a consumer’s mental attitude and assessment regarding its power to satisfy his particular want. Thus, utility of a commodity may differ from person to person. Psychologically, every consumer has his likes and dislikes and everyone determines his own level of satisfaction.

One may also read about how supposedly, utility has no ethical or moral significance. Consider a macabre example: A commodity which satisfies any type of want, whether moral or immoral, socially desirable or undesirable, has utility, i.e., a knife has utility as a household appliance to a housewife, but it has also a utility to a killer for stabbing some body.

This has been intended as a meditation on the important characteristic features of utility. I am going over the issue, because cannot, actually, find this issue about people supposedly misusing the jargon 'zero-sum game', in spite of this thunder and blood about how 'THIS, AND ONLY THIS, IS WHAT IS MEANT BY A “ZERO-SUM GAME.'

My guess is that we haven't got the concept of 'utility' in mind, such that it is measured in terms of money and it is relative. If I am fishing in the wrong pool, then all this stuff has been utterly uncontroversial,...right?

Daniel Langlois said...

'I am talking about a made-up phrase, coined more than seventy years ago by John von Neumann – zero-sum game.'

No one could dispute that Von Neumann was brilliant. But I suppose that we can say that game-theoretic insights can be found among commentators going back to ancient times. It had occurred to some actual military leaders and influenced their strategies. Also, if you think of how 'the people' can hire an agent—a government—whose job is to punish anyone who breaks any promise, then of course Hobbes pushes the logic of this argument to a very strong conclusion. And we can add, that details of contractarian political philosophy, as they are in fact pursued in the contemporary debates, all involve sophisticated interpretation of the issues using the resources of modern game theory. Furthermore, Hobbes's most basic point is peoples' own need to protect themselves from what game theorists call ‘social dilemmas’. The structure of his argument is that the logic of strategic interaction leaves only two general political outcomes possible: tyranny and anarchy.

I look back at my soccer example, and I am bored. Let's try again: suppose that you are a fugitive of some sort, and waiting on the other side of the river with a gun is your pursuer. He will catch and shoot you, let us suppose, only if he waits at the bridge you try to cross; otherwise, you will escape. We know from experience that, in situations such as this, people do not usually stand and dither in circles forever. However, you must do what your pursuer least expects; but whatever you most expect him to least expect is automatically what he will most expect. There is a rational solution—that is, a best rational action—available to both players.

The point that wasn't really known until the 1940s, was how to find it mathematically.

Now, I've been meditating on what this has to do with slassical economists, such as Adam Smith and David Ricardo, who were mainly interested in the question of how agents in very large markets—whole nations—could interact so as to bring about maximum monetary wealth for themselves. I will point out, then, that it is my underestanding that Smith's basic insight, that efficiency is best maximized by agents freely seeking mutually advantageous bargains, was mathematically verified in the twentieth century. That is something about the foundations of economics, which begs for more, but I'll leave it there..