## Pure game theory — an irrelevant tautology

from **Lars Syll**

*Applied game theory* is a theory of real-world facts, where we use game theoretical definitions, axioms, theorems and (try to) test if real-world phenomena ‘satisfy’ the axioms and the inferences made from them. When confronted with the real world we can (hopefully) judge if game theory really tells us if things are as postulated by theory.

But there is also an influential group of game theoreticians that think that game theory is nothing but pure theory, an axiomatic-mathematical scientific theory that presents a set of axioms that people have to ‘satisfy’ by definition to count as ‘rational.’ Instead of confronting the theory with real-world phenomena it becomes a simple matter of definition if real-world phenomena are to count as signs of ‘rationality.’

This makes for ‘rigorous’ and ‘precise’ conclusions — but never about the real world. *Pure game theory* does not give us any information at all about the real world. It gives us absolutely irrefutable knowledge — but only since the knowledge is purely definitional.

Mathematical theorems are tautologies. They cannot be false because they do not say anything substantive. They merely spell out the implications of how things have been defined. The basic propositions of game theory have precisely the same character.

Pure game theorists, like Ken Binmore, give us analytical truths — truths by definition. That is great — from a mathematical and formal logical point of view. In science, however, it is rather uninteresting and totally uninformative! Even if pure game theory gives us ‘logical’ truths, that is not what we are looking for as scientists. We are interested in finding truths that give us new information and knowledge of the world in which we live.

Scientific theories are theories that ‘refer’ to the real-world, where axioms and definitions do not take us very far. To be of interest for an economist or social scientist that wants to understand, explain, or predict real-world phenomena, the pure theory has to be ‘interpreted’ — it has to be ‘applied’ theory. A ‘pure’ game theory that does not go beyond proving theorems and conditional ‘if-then’ statements — and do not make assertions and put forward hypotheses about real-world individuals and institutions — is of little consequence for anyone wanting to use theories to better understand, explain or predict real-world phenomena.

Pure game theory has no empirical content whatsoever. And it certainly has no relevance whatsoever to a scientific endeavour of expanding real-world knowledge.

Sen Wyden (D.Or) called the 9-page Republican tax bill a “right-wing fantasy document that paves the way for trillions in handouts to corporations and the very wealthy,” a con-job. Sen Gillibrand (D.Ny) called it a scam.

This is the result when sociopathy, game theory, and religious absolutism are mixed and added to racism and ignorance. One more in a long history of upheavals Americans have had to deal with.

First, let me say, I agree with what you say. Just want to make a couple of clarifying comments. Science as one form of human knowledge is distinguished by its formality. That doesn’t mean scientists don’t research and learn tacitly and informally. They do. In fact, science could not exist or produce knowledge without such informal understanding and action. All scientific work, explicit or implicit focuses on the expression of facts. Facts are defined as being what is revealed in experience. Scientists can organize such facts into theories, many theories. Theories that can conflict with one another, and even conflict with earlier versions of the same theory. But nowhere in this work does science or scientists intersect with “truth.” Truth is here defined as the property (as of a statement) of being in accord with undisputable reality. Truth is things, events, facts that are immutable. Any scientist may believe, even believe strongly s/he is in contact with truth. But since science is empirically based such beliefs are outside the scope of science. They cannot be science, per se. Game theorists have the truth defined by their theories. This is not scientific. These theories may however, provide a starting point for scientific work. Or, they may fail, sometimes totally to provide such starting points. Just a matter of chance. Examining whether theories provide such help by applying them in random empirical situations is one way scientists use theories in a worthwhile way. But in whatever ways theories are used they remain just educated guesses. Aligning in part with your conclusion: “Pure game theory has no empirical content whatsoever. And it certainly has no relevance whatsoever to a scientific endeavor of expanding real-world knowledge.”

Frankly, I think the whole of this discussion so far is rubbish. Mathematics enables one to express relations, including the dynamic relations we call “know-how”. The fact that its theorems may be tautological does not mean that they are true. There may very well be false theorems, which if one acts on them will not produce the expected results. (Contra Ken, if there is logical doubt about an addition one can always prove the result by decomposing the terms into units and empirically counting these). Truth, so far as I have been able to discover, began as a judgement of the verticality of a building, without which it is at risk of falling down. There are also theorems which express the limits of what is possible, as in Shannon’s theorems, built on his definition of the binary way of expressing information capacity, which show that the rate of sampling a message must be at least twice the rate of change of the message or we will miss systemic variations between samples. The point, anyway, is that maths reveals and expresses relationships which we can’t simply see as “following from definitions” and which we need to be aware of when we are trying to do things.

Thanks for your comment, Dave. Particularly the rubbish part. Yes, mathematics is one instrument humans invented to express the relationships they thought they saw around them. It could be used to express some but not every relationship, logical and otherwise. Never forgetting that humans also invented logic. In fact, humans have a long history of inventing self-defining systems to express relationships, including human-human relationships. This includes science and empiricism generally. But each of these has the same limit. They are merely and always human inventions. Using one or more of these systems and within the limits of time, humans express what they currently can accept as a “fact” as well as the facts they believe will never change. And, as you suggest many of these “facts” were not planned, and were sometimes unexpected. But they still derive through the same invented human systems. That this is not apparent results from the systems being incomplete and many aspects of the systems existing tacitly. But that’s just how humans work. Or, that’s at least the current thinking.

In an earlier post, I explained how game theory actually acts as a blindfold, which prevents economists from understanding the realities of human behavior. In the simple Ultimatum Game, game theory makes four predictions about human behavior, all of which are false. In the linked paper (Empirical Evidence against Neoclassical Utility Theory) I also show that factors which make an important difference for human behavior, do not matter in a game theoretical analysis. Thus game theory systematically blinds economists to the real world — it cannot be tautological.

https://weapedagogy.wordpress.com/2017/03/10/behavioral-vs-neoclassical-economics/

Asad, which “reality of human behavior” are you writing about? After all, game theory is a human invention. So, it can be considered real, a fact. And as behaviorists like B. F. Skinner demonstrated over 50 years ago, with the correct conditioning protocol human behavior can for a period be made to conform to just about any model. And, as you suggest each model has the tendency to obscure the view of other possible models, at least for some period, but never permanently. The more extreme models, those that deny basic human needs such as food, shelter, or emotional support tend to either destroy human communities or be subverted before that occurs. In any case, they seldom leave any footprint that could influence the future. So, the relevant question is which other models for human culture does game theory obscure? Since it’s impossible that it obscures all others. The more important concern is that destructive models come back not because of current examples but because humans too often commit the error Edmund Burke warned us against, “Those who don’t know history are doomed to repeat it.” It’s become clear of late that no one in the administration of Donald Trump knows history. Thus, we have fascists rising to power once again in a major western nation.

Ken, when I referred to “rubbish” I was thinking of Lars’ arguments (i.e. not person) as well as yours. Asad’s is, I feel, much nearer the mark. Capitalist economics is a game the bookmakers are persuading us to play, and it behoves us to realise that: to understand what a game is, how those in a position to make or alter the rules of the game can play games with those gaming, and how if the object of the game is to reliably make money, only the bookmakers can be sure of winning.

So what is a game, and why are we willing to play them? (Some people aren’t). From my own experience playing a simple card game like ‘Sevens’ I would say it can be a mental challenge to get the best outcome one can with the cards one has been given. Those who play for money seem to enjoy the excitement of being in danger of losing, which for some is a side-effect of the possibility of winning which can be rationally given up if they win, but for others becomes an irrational addiction to what they will inevitably lose. Those who are interested in the theory of games (not just the rules of a particular game) see the rules creating numerous options, and playing the choice of a strategy or sequence of moves which will lead to the desired outcome. At the highest level one such strategy is a sequence of moves which leads one’s opponents to mistake your own strategy: the classic “red herring”. That, I suggest, is the strategy of bookmakers and kings who (since Charles I lost his head) have kept their heads below the parapet and got politicians playing their games instead.

Telling “The Story of Mathematics”, Richard Mankievicz says “There are games in which there is an optimal strategy, and once this has been found, then the game becomes essentially trivial. For example, noughts and crosses (tic-tac-toe) is a popular children’s game, but one the strategy has been understood and each player plays rationally, then every game is drawn and interest wanes. Nash proved that even chess has an optimum strategy, but the game is so complex that the optimum strategy has not yet been found”.

Discussion of RAND leads on to “In a non-zero sum game like the stock market, winning and losing are relative, and the game is more about maximising one’s own winnings that beating an opponent”. However, it seems a Nash equilibrium is created by not playing games but giving way to those most disadvantaged, cf. my “parable” of a roundabout at a road junction.

The mathematical proof of that would be irrelevant, but the truth of the conclusion isn’t.

Dave, according to the textbooks and journals, game theory is “the study of mathematical models of conflict and cooperation between intelligent rational decision-makers”. Game theory is mainly used in economics, political science, and psychology, as well as logic, computer science and biology. (Myerson, Roger B. (1991). Game Theory: Analysis of Conflict, Harvard University Press, p. 1) Originally, it addressed zero-sum games, in which one person’s gains result in losses for the other participants. Today, game theory applies to a wide range of behavioral relations, and is now an umbrella term for the science of logical decision making in humans, animals, and computers. In other words, game theory is just one more effort by humans to explain themselves and their actions. Interesting is its self-definition, “the science of logical decision making.” Game theory, or the science of logical decision making suffers the same limitations as the other schemes invented by humans to explain humans. First, those who invent and use them must invent or choose what is and is not included in the scheme. Second, the scheme and all it parts are always incomplete and will remains always incomplete. Third, each scheme’s explanations supposedly know things that can’t be known and don’t know these things are unknowable. If we accept these limitations, as we must, each scheme can be useful, in some circumstances and in some periods. With usefulness established situationally and historically. From all this derives the multiple words and claims of a scheme such as game theory – logical, scientific, optimal, mathematical, rational, etc.

Please stop [trying to?] muddying the waters, Ken. You can do better than that. “The” textbooks and journals? “Today, game theory applies to a wide range of behavioral relations, and is now an umbrella term …”? Changing the apparent meaning of words is the tactic of rogues who wish to persuade the innocent that evil is good, and of lawyers paid to defend a guilty client. “If we accept these limitations, as we must [?], each scheme can be useful, in some circumstances and in some periods”. Yes, in the circumstances of lawyers, bookmakers and other entrepreneurs treating the rest of us as pawns and muddying the waters in the hope we won’t notice.

Wow, David. I don’t think I was in any way obtuse or “muddying.” Textbooks from which game theory is taught seems, to me at least a good place to begin understanding game theory. Let me try to make clearer the limits game theory, and all such social science or systems theories face. Our species lives by time and place. That is, each human’s life is in a time and place. And can’t be otherwise. But our species also possesses an imagination. As far as we know, the only species on Earth that does. By that imagination humans can consider and explain other places and other times. And invent theories and methods such as game theory. These inventions are assessed in terms of their usefulness in working through actual problems. Whether those problems are in a classroom, laboratory, corporation, or union.

See Asad Zaman’s brilliant post on The Shifting Battleground: i.e. the matador waving a red flag at a bull. If you want to play games, write a book persuading victims that the game is other than it will be: e.g. about political/religious conflict rather than bankers gambling in their own casinos with marked cards.

When mathematical economic theory fails it is never to blame to Maths. It is always to blame that the premises made in modeling are wrong. Maths is pure logic, it never fails. But any conclusions are valid only, and only if, the chosen premises fit the real world system under investigation. The main problem with most models at all today is that the banking system and its interdependenices with real GDP is not referred to correctly. If at all (e.g. SFC models) only commercial and retail banking is in its premises, but the (today even much larger) investment banking and its interdependencies with real GDP is excluded. Thus such models will never fit real world economics well.

Yes; but perhaps there are two ways of looking at that. Either the model is inadequate, or the investment banking is not part of real economics. Which it isn’t. It’s a scam.

Heribert, your faith in mathematics is misplaced. Per C.S. Peirce we can know “only in an uncertain and inexact way,” and, “there are three things to which we can never hope to attain by reasoning, namely, absolute certainty, absolute exactitude, absolute universality.” Neither “pure logic” nor mathematics can overcome these limitations. Mathematics and logic are fallible just like any other cultural creation of humans. We need to take them and use them for what they are, rather than what they can never be.

Of course, investment banking is just as much in the economic cycle of saving and investing as commercial banking. Those who decide on investment products no longer have the money for real investments. Finally, returns on investment products must always be drawn from GDP. At the latest on the due bail-out of non-performing financial products.

Nov. 22,2017,

M. Zimmerman,

I think your rather self-righteous & dismissive “wisdom” of Nov. 02, 03, and Nov. 04, could stand a few corrections.

{1} I’m not sure you understand what TACIT KNOWLEDGE is: Although you’re not doing too badly about the intent, apps and limitations of Game Theory. If you really are on the level about learning what “tacit knowledge” is, I suggest that you find out who {1} Michael Polanyi {not his rather shallow unreal brother,.Karl } is. Try the TACIT DIMENSION, 1967 based on the Terry Lectures of 1962. {2} I also suggest Friedrich Hayek’s thought on the same subject. If you’re not familiar with Hayek, try THE ROAD to SERFDOM {1944} for starters. It was published the same year as the Neumann -Morgenstern book on Game theory. Keynes loved it. .

{2} It was not Edmund Burke who said that. It was George Santayana.. Burke is famous for: “All that is necessary for the triumph of evil, is for good men to do nothing”. Or as Martin Luther King put it: “He who passively accepts evil without protest, is as much involved in it as the perpetrators of it”..

Please accept my suggestions in the didactic spirit intended.

With respect: Norman L. Roth

Norman, if my comments came off as self-righteous and dismissive, I apologize. That was not my intention. But your surmise that I dislike game theory is correct. It’s practitioners in economics are too often pretentious and pushy. Mathematics, of any sort is not the only or even a preferred pathway into understanding economic actions. Although it can help in counting the beans, after it’s decided what’s a bean and what is not. As to Michael Polanyi, know his works well. They were a central part of my Anthropology dissertation. I’ve read Hayek, including The Road to Serfdom. Rather limited and stilted view of things, in my view. One of those “had his mind made up before the research” books, also in my view. As to the quote, you’re correct that Santayana’s version of it is probably the most famous. But similar quotes have many authors. One is Edmund Burke. His meaning was that history is the greatest teacher, if studied wisely and completely. It shows human successes and failures. We need to study it. Can’t avoid another Hitler if we’re ignorant about the first one.