What’s wrong with game theory

The latest Nobel Prize award to Aumann and Schelling has generated a bit of discussion about the value or otherwise of game theory. Generally speaking, economists are enthusiastic about game theory and other social scientists less so. Although I admire the work of Aumann and (even more) Schelling, as economists go, I’m a game-theory sceptic, for a fundamental reason I’ll try to explain.

My main problem has to do with the idea of a strategy and role in equilibrium concepts such as the famous Nash equilibrium. A game outcome is a Nash equilibrium if no player can gain by varying their own strategy, assuming that other players stick to their equilibrium strategy.

The problem here is to say what a “strategy” is. In a game like chess or poker, this is easy: the rules say what each player can do and when they can do it. The same is true in some special kinds of economic situations, such as auctions. But most of the time, there is no book of rules, so the set of strategies has to be described as part of the model.

If we look back at the Nash equilibrium idea, and put ourselves in the position of one of the players, it can be seen that there’s really no difficulty with the definition of our own strategy. We can look at the outcomes that are available, given the other player’s strategy, and pick whichever one is best for us. The way in which we label our choices doesn’t matter.

The critical problem is in the phrase “given the other player’s strategy”. In the absence of a rulebook, we can only know this if we know how the other player is going to react to this move (and the same holds in reverse for the other player). There was a large literature on this issue of “conjectural variations” before the rise of game theory, but it was generally felt to have ended in failure[1].

I’ve been trying to make this point for a decade or more, along with Simon Grant and Flavio Menezes, in a string of papers, some published and some not, most notably here (PDF files).

Although it’s easy enough to make the point in specific instances that if we choose a different assumption about strategies we get a different equilibrium, economic theorists strongly resist the argument that this is a general problem and that economic interactions with a well-defined strategy space are the exceptions rather than the rule.

At least, that’s the case in relation to finite games. For infinitely-repeated games, the strategy space gets very large, and the (in)famous Folk Theorem says, roughly, that anything can happen. This is really just the same problem in a different form.

This doesn’t mean game theory can’t be a useful source of insight, something Schelling in particular has shown. But it’s unlikely, in most cases, to yield definite and reliable predictions.

fn1. Digging around, I see the claim that we have witnessed in recent years a revival of Conjectural Variations in Game Theory

24 thoughts on “What’s wrong with game theory

  1. Its interesting the way fashions come and go in economics. When I was an undergrad in the late 1960s, game theory was considered a dead end. Then it re-emerged as a central part of microeconomics. Most hard-headed business schools currently teach game theory for example.

    Now once again questions are being raised about its value.

    I think its role in understanding basic issues in business (coordination, commitment, cooperation) is indeniable. And if pressed I have problems challenging intellectually the idea of a Nash equilibrium — it does describe ‘best responses’ so why assume anything else. Also work in experimental economics supports the idea that players play Nash strategies at least in many simple games. The fact that subgame perfection doesn’t describe actual behavior in simple sequential games such as the ultimatum game suggests something about how individuals do form their judgements in such situations. And even if the theory does not describe behaviour it has the saving grace of having prescriptive value.

    I always thought of the idea of conjectural variations as a synthetic way of describing a variety of game theoretic reactions — Cournot, Bertrand, Stackelberg etc. And I thought the sort of response you followed reflected your environment. You would be a dominant player if you were big, a Bertrand competitor if you declared your price bid before you produced and a Cournot competitor if you decided output before price because of a rigid production technology or because of high costs of changing inventories. In most business situations it is evident which CV is applicable.

    I have taught business students doing MBAs game theory and microeconomics for more than a decade. The bit that sticks with them (and which they enjoy most) is the game theory. It helps them understand the business world.

  2. The problem of defining the strategy sets is not unique to game theory. Any model with agents making choices suffers exactly as much. Same for arbitrariness of equilibrium. Criticisms of game theoretic equilibria do not invalidate game theory any more than criticisms of economic equilibrium concepts invalidate economics.

    I find it difficult to accept criticisms of game theory because I see game theory as an extension of current models rather than a replacement. The useful part of game theory is not the equilibrium concepts, but the rich and flexible language for describing interactive situations.

  3. what do you mean “rich and flexible” language? Name me one thing that game theory can describe which cannot be described in a optimization framework. equilbrium theory is just as easily applicable to diverse situations and flexible as game theory.

  4. Mr_Hayuk —

    One thing game theory gives which is not provided in an optimization framework is the notion that an interaction is multi-threaded, ie, that there are multiple threads of control, one for each participant in the interaction. The outcome of an interaction is decided by all these threads, interacting with one another, which might (might!) be equivalent to a God’s-eye-view of the interaction. But who and where is this controlling entity? From the point of view of the participants, the control of the interaction is by the participants interacting with one another, not anywhere else. If you’re a participant, engaged in an interaction and trying to decide what to do, you don’t have access to God’s viewpoint.

    This multi-threaded idea may seem a small idea, but it has revolutionized computer science, and it is still an idea lots of people just don’t get.

  5. Was it Schelling who set an exam question along the lines of “You are told to meet someone in a city (New York?) on a particular day, but not told a time or a place. They are told the same information. Where and when do you go?”

  6. Mr Hayuk, sure everything Game Theory shows can be achieved with optimisation framework, but dynamic programming and optimal control methods can get extremely complicated very quickly. Extensive Bayesian games are a far easier way to get to the same result.

  7. Wasn’t game theory useful in avoiding WWIII? I have an inkling that von Neuman designed the nuclear weapons security and launching procedures utilising game-theoretic insights. Certainly the US system was robust and designed with sufficient redundancy to provide for both “idiot-proof” launch and “fail safe” accident prevention.

    On the other hand, game theory almost started WWIII. Von Neumann was notorious for advocating the US launch a preventive nuclear war against the USSR, which he did on the basis of zero-sum calculations.

    Von Neumann…gained a certain notoriety as an analyst at the RAND Corporation in the 1950s when he used game theory to “prove” that the United States should engage in “preventative war” against the Soviet Union; assuming that the Soviets were thinking along the same lines, he reasoned, one of the two countries was almost certainly going to get toasted, and given that, better that it should be them than us. Thankfully, his view did not prevail, but only with the benefit of hindsight can we safely call the notion madness.

    Nixon had a game-theoretic strategy, which was the “madman theory” of international relations.Act like you are unpredictable and capable of great violence and careless of consequences. Opponents will give you a wide berth. Apparently it helped in arms control and cease fire negotiations.

    Also I believe game theory has been helpful in modelling the evolutionarily stable strategy (ESS) for selecting breeding partners.

  8. dave Says: October 14th, 2005 at 6:38 am

    Was it Schelling who set an exam question along the lines of “You are told to meet someone in a city (New York?) on a particular day, but not told a time or a place. They are told the same information. Where and when do you go?�

    00:00 hrs on the front steps of Grand Central railway station? You want to get there at the earliest time because that ~halves the possibility that you will be late. You want to choose GC railways station because that is the most visible arrival point.

    I guess you would have to be carrying some sort of sign saying that “I am looking for someone who does not know what I look like.”

  9. Pick the top likely places to meet, get there early and then go to each place and stick a note somewhere it will be seen giving your planned movements for the day. Eg, “I’ll be waiting at Grand Central from 10-11, then from 11-12 I’ll be checking all my notes in counter-clockwise order along the following route, then from 12-1 I’ll wait at grand central again, then from 1-2 I’ll check notes again, etc.

    Increase your chances- give every bum you see near the meeting points a picture of your friend and a copy of your schedule and $5 with a promise of another $5 if he sees your friend and gives him the note.

  10. That might be best if one is unconstrained. But I think that the problem may be constrained to a single place at a specified time range. Also, it is a little messy and complicated.

    Going for the middle is the most normal thing to do. The most logical place is a prominent transport juncture – mid-city. And the most logical time is 12:00 hrs – midday.

  11. I’d suggest that the problem is not particularly interesting if you’re constrained to a single location at a specified time.

    Sure, pick the “most likely” place/time given what you know about the person you are going to meet – but the probability of actually meeting will be pretty low. Too little information. Not very entertaining.

    I have a standing protocol with my wife and kids that relates to this problem. If anyone gets lost, they go to the place you were last together and waits there. Has paid off on more than one occasion.

  12. Where I find game theory useful is in modelling evolutionary processes. We are not assuming that players adopt a strategy consciously, or even that they are capable of doing so. We are assuming that everything works so that each does what maximizes inclusive fitness, where what maximizes inclusive fitness is dependent upon the behavior of the other players. In ths context, game theory does yield definite predictions, and many of these predictions are borne out by observation. I suspect evolutionary theory just is in large part game theoretic.
    Game theory is also useful in understanding certain collective action problems and the limits of rationality. Why do peole persist in behaviors that are second-best? Because of their predictions about how others will behave.

  13. “Nixon had a game-theoretic strategy, which was the “madman theoryâ€? of international relations.Act like you are unpredictable and capable of great violence and careless of consequences. Opponents will give you a wide berth.”

    Many believe this is precisely the strategy that North Korea is consciously following.

  14. I guess it’s easier for some cities than others. In Adelaide, it would be the Mall’s Balls at noon. Not all cities have a readily identifiable modal meeting point.

    Why am I not surprised that JQ and Simon Grant have written a paper on the topic?

  15. The madman theory is interesting, because it requires meta-reasoning about your opponent, something like this:

    “If my opponent assumes I am rational then he will arrive at the following optimal strategy using the latest and greatest game theory. Hmm, that’s not so good for me, so let’s not be rational”

  16. Dave, Jack — Not even New York City has a generally-agreed meeting point. Grand Central Station is the place people from the areas north and east of NYC naturally think of (ie, people from places such as Harvard and Yale), because this is where their trains terminate, but this is not the obvious place to someone coming from New Jersey.

    Jack — On the question of game theory and WW III: Philip Mirowski in his great book “Machine Dreams: Economics Becomes a Cyborg Science” noted that around 1960 lots of senior US military officials, from the Preident on down, made public speeches saying GT was of no use for military strategy. They didn’t stop funding it, though.

    What were they doing? Was GT really of no use? Or were they trying to send a signal to the Soviets? If so, what signal? That it was of no use, so don’t bother using it? Or, that it really is of use, so we are trying to misinform you? Or, that it really is of use, and we know you won’t believe what we say, so when we say it’s of no use, you will use it, and then we can be sure that we are both playing the same game in this cold war?

  17. John

    Interesting comments about the utility of game theory – there really is very little evidence that game theory actually works in dealing with conflict situations.

    Kesten Green from Monash Uni and Scott Armstrong from Wharton have done some good work in this area – simulated interactions and structured analogies work best in forecasting decisions. Game theory is about the same as chance.

    see http://www.kestencgreen.com/ for some papers


  18. Economists borrowed (stole?) the word from philosophers, who for 2300 years have been using “rational” to mean “reason-based” (argument, decisions, interactions, etc). It is nice to see that some economists (eg, Sen in his 2002 book) have realized recently that there is value in the philosophers’ definition, after all.

    I agree with you that many uses of the word are ideological — to brand those whom the speaker targets as “irrational”, or “emotional”, etc. Even logical consistency can only ever be with respect to some logic (eg, axioms and inference rules), the choice of which is ultimately value-dependent.

  19. Re “North Korea exemplifies the madman theory of international relations”:
    Supposedly the Bush doctrine of permanent nuclear superiority and asymmetric deterrence codified a strategy sufficient to guarantee peace for America, if not the world as a whole. The Iraqis then learned that refraining from actual attempts to overturn the Bush doctrine´s premises was rewarded with military attack and an occupation regime that doubled unemployment in the country and could not create that minimum of law and order conventionally characterized as a functioning night-watchman state.
    So the U.S. followed a rule of “acting like you are unpredictable and capable of great violence and careless of consequences”. North Korea took notice and responded with a challenge to the Bush doctrine (not with an attack on the U.S.) The question then arises whether that qualifies as rational. We´d probably insist it doesn´t, yet we´d still have to realize that it represents a logical response. This type of interaction should properly be labeled “reciprocated lunacy” rather than being viewed as purely simulated irrationality.

    There is a big difference between Nixon and Bush. Nixon specialized in clandestine operations and didn´t really emulate Krushchev (to whom provocation seems to have come rather naturally and who may thus be considered the dean of the “putting-on-the-airs-of-a-madman” school of international relations), whereas Bush has yet to put his boots back on – something an actor would presumably not forget about if he wanted to signal a switch from hysterical performance back to everyday mode.

  20. Some years ago I had the pleasure of listening to James Dolan presenting his work on modelling dynamic games using category theory. There were a sufficient number of category theorists in the room to scrutinise the application of this body of knowledge such that visitors could focus on the game theory part only.

    What is an appropriate solution concept for a dynamic game?

    Dolan defined ‘success for player p’ as the state (date-event pair) where player p has a move left.

    I like it. I like it because of its generality, its simplicity and, possibly unintended, its humanity. For example, the survivor in an earth quake is a successful player in the game of life against nature. The person who does not commit suicide but battles on is a successful player (against himself). (I wonder whether knowledge of Dolan’s solution concept might influence a person’s decision.) Old ‘wisdom’, such as ‘while there is life there is hope’ is preserved in Dolan’s definition of success in a dynamic game. A person who does not have a move left because he is killed is ‘unsuccessful’. The notion of ‘success’ preserves respect for life. The solution concept is meaningful to economics which is concerned with the material welfare of humans, as distinct from ‘economic rationalism’ which seems to confuse ‘everything with everything’. I don’t know whether it is meaningful to religious people but the correspondence between ‘success’ and ‘life preserving’ does not seem to contradict any of the tenants of those religions I have heard about. All other notions of ‘success’ (monetary, status, power, …., meeting with someone who has sufficiently similar probability assessments of a ‘focal point’ in New York…..) are at most special cases of a ‘successful player’ defined as one that has a move left. It is universal.

    Dolan’s special interest at this seminar was amnesia games. (eg p’s boss changes the rules of the game retrospectively to affect the pay-off matrix.)

    Dolan claimed: An amnesia strategy is successful if and only if the original strategy is successful. The assumption on the analytical abilities (‘rationality’) of the players used to establish the claim may not hold in many situations. However, it seems to me prior knowledge of the notion of a sub-game perfect equilibrium is helpful in being a successful player in an amnesia game.

    (Applied: p can’t be successful ‘in reality’ because p’s boss has the power to ignore rationality. True. In ‘real life’ a law which penalises dishonesty and which is accessible at a ‘reasonable price’(ie one that is affordable by p) would seem to be helpful. )

    I don’t think it is helpful to ‘value’ (prejudge) the usefulness of any knowledge which is true in the logic of mathematics.

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