I. M. Flaud quotes extended passages from a later edition of von Neumann and Morgenstern dealing with the extension of the notion of a zero-sum game to n-person games, n greater than 2. This is totally new to me and I need to hunt that up and look at it before I try to respond.

One question to I. M. Flaud: Whom are you quoting? von Neumann died in '57, I think, and a 60th anniversary edition would have appeared in 2004. The language quoted about gains and losses makes it sound as though we are talking not about zero sums of utility but zero sums of money, which are of course interpersonally comparable. So that is an entirely different question.

Can you throw any light on this?

## Saturday, April 29, 2017

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## 12 comments:

I'm quoting von-Neumann and Morgenstern from The Theory of Games and Economic Behavior, pages 505-511. The Sixtieth Anniversary Edition includes an intro by Harold W. Kuhn and an afterward by Ariel Rubinstein, and some critical reviews, but from that text I quote only von Neumann and Morgenstern.

The talk of payoffs and side deals is von Neumann and Morgenstern's. They introduced the theory of characteristic functions from the beginning, in the original TGEB. The business about payoffs to join coalitions is in section 56.3.4, among other places. The next section, 56.4.1, on the limitations of the use of the (n+1)-player extension, is worth going through. Maybe I'll get to it in further comments.

I hope I'm not being a pain in the ass with this. Your blog is an oasis.

Not at all. I am very grateful. I have ordered the edition, in paper, from Amazon [$54!]. It should be here in two days. I will read the relevant passages and try to formulate a coherent response. I should tell you that I studied von Neumann's theorem in Luce and Raiffa's book, GAMES AND DECISIONS, rather than in the original. This is very interesting.

I'll answer my question: I am being a pain in the ass. If by "zero sum game" one means a game in which the players preferences are equal and opposite, then there are only two players, and up to a positive affine transformation the game is zero sum. (The scaling coefficient is positive. The translation can be any real number.)

But in the text of The Theory of Games of Economic Behavior--and elsewhere--there is the notion that a game is zero sum if the sum of the payoffs to the players is zero. (non-cooperative games have payoff functions, cooperative games have characteristic functions...)

This allows the possibility of more than two players, and I believe von-Neumann and Morgenstern wanted to allow this possibility in the hope of extending their zero-sum theory to the general n-player game. So there is talk about the extension of an n-player game to (n+1) zero-sum game. Then they run up against the difficulty that unless their is some mathematical restriction is imposed on the game, the new game introduces new plays for the new player unless some restrictions are imposed.

Now there is the question about usage. I'm not so sure about what people "really" mean when they say "zero-sum game" but it's a safe bet to say they don't have your precise sense in mind. I know I didn't, and I've invented a few games of my own (sure enough, even in those situations, there is a strong tendency for coalitions to form, with side deals). The general usage seems to follow "the sum of the payoffs is zero"--a perfectly sound interpretation, present in the original TGEB. It's probably safe say that the idea of side deals and coalitions probably isn't in what I'll call the journalistic understanding.

These elementary grammatical errors and misspellings of mine are, well, unforgivable.

I once pointed out to some conservative academics that with the zero-sum extension of the n-player game, von Neumann and Morgenstern had captured the role of the taxpayer in socialization the losses of the plutocrats. This is obviously glib, but on account of von Neumann's hawkishness and the association of mathematical economics with right-wing politics, the suggestion elicited outrage and contempt. Very gratifying, in retrospect. At the time I was quaking in my boots. There's the first-order suggestion that the wealth of the plutocrats hasn't been an entirely meritocratic acquisition. Then there is the second-order suggestion that even if the taxpayer is playing the role of the fictitious player, it's non-trivial to show that the real players won't deny themselves some advantage and make some sort of side deal with the fictitious player. This is irritating--why can't the little people be ignored outright? And this is present in some nascent form in von Neumann's foundational text.

I'm interested in seeing this discussion, but a skim of the blog posts hasn't allowed me to find it. Where was the initial discussion, if I may ask?

'The language quoted about gains and losses makes it sound as though we are talking not about zero sums of utility but zero sums of money, which are of course interpersonally comparable. So that is an entirely different question.'

The socratic question is 'how is utility' measured?' But I'll answer that -- utility is measured in terms of money.

Maybe I ought to slow my roll with the assertion that 'utility is measured in terms of money', and just ask: how is utility measured? Although I can add that several economists including Marshall, suggested the measurement of utility in monetary terms.

Utils cannot be taken as a standard unit for measurement as it will vary from individual to individual. And then, maybe the advantage of using monetary values instead of utils is that it allows easy comparison between utility and price paid, since both are in the same units. Of course it is still impossible to measure satisfaction of a person.

Marinus Ferreira, the initial discussion is here and in the comments following.

The discussion on zero-sum games was picked up on Hacker News. The reception is unfriendly. Some commenters there have no taste for intellectual history.

I cherry pick this remark from there:

'What he's really saying, in a roundabout fashion, is that the economy and trade are zero sum and that Obama and Krugman are (respectively) wrong..I'd say Wolff is the one who's wrong. But he does agree with Trump.'

yes and yes, though perhaps this could have been phrased as a query?

Prof Wolff is not wrong about two-player zero-sum games in the sense that they have strong theoretical properties that n-player zero-sum games (n>2) don't have, such as max-min = min max (a robust solution concept), no tendency for players to form coalitions, etc. It's understandable that von Neumann and Morgenstern would attempt some sort of reduction of the n-player game to zero-sum games. Such a reduction would undoubtably have occurred to readers: add a new player whose payoff is the negative sum of the payoffs to the other players. An extended discussion of the non-trivial difficulties is warranted in a foundational work--the extension leads to non-trivial phenomena not present in the original theory.

Still, the tendency to want to generalize "zero-sum" to players who stand for entire classes is not immediately nonsensical. If you believe in a strict sex dichotomy (or believe that the distribution of the sexes is bimodal) you might speak of the two sexes as players in a two-player zero-sum game. The tendency to generalize by analogy in mathematics is often productive. Where entire classes of individuals are involved, one would expect side deals and various accommodations to occur that would have been ruled out under the idealized zero-sum two player setup--a setup that ought to be characterized as a caricature.

von Neumann and Morgenstern were aware of this. And for me, at least, the mechanism of side deals (even among entire classes of relatively powerless players whose payoffs might, for example total the negative sum of the payoffs to the "real" and "important" players) is a source of interest and even hope. It also introduces complications into discussions of relative power dynamics, some of which amount to the two-player caricature.

Daniel Langlois: the libertarian cheerleaders of Hacker News are wrong. Prof Wolff never said that the economy and trade are zero sum.

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

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

Under Prof Wolff's reading of zero-sum (in which all zero sum games have two players):

Obama's assertion is trivially true, because the economy isn't a zero-sum game, so it trivially doesn't have to be one.

Krugman's assertion is a conjunction which may be true or false. If the war is a nuclear war, the outcome, up to a positive affine transformation, is most likely zero for both sides, so trivially zero sum. Under the strict reading, no war involves only two individual players, the conjunct is false and the conjunction false. The U.S. hasn't been winning wars with adversaries that cannot be said to have won either, so those wars have not been zero sum. Since trade isn't zero sum (a asymmetric zero sum exchange is not ordinarily considered a trade), that conjunct is trivially true. One might model trade in cooperative game theory as Shapley market model--not a zero-sum game.

It's too bad about Krugman. My opinion is that he's too boring to be right.

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