John Quiggin gave a splendid presentation at the 25 Australasian Economic Theory workshop at ANU yesterday. His title was Can Game Theory Be Saved?Â (a joint paper with Flavio Menezes). The basic argument is succinctly put on the first page of their paper:

Experience has shown that in most situations, it is possible to tell a game-theoretic story to fit almost any possible outcome. Although this point is only occasionally acknowledged in the formal literature, it is much more widely accepted in informal discussion.

It’s not my area of specialty, so I didn’t follow all the twists and turns, but I came away thinking that the answer to the question was ‘probably not’. One of Menezes and Quiggin’s arguments was that game theorists should pay more attention to the strategy space. This seemed to get short shrift from the audience, who thought that this wouldn’t get far with the current crop of editors and referees.

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I haven’t had time to read the paper in detail. As such, I have only had a quick skim of parts of the paper. However, my preliminary assessment is that the Menezes-Quiggin critique is not particularly convincing.

In analysing games, we do assume that the strategy space is fixed. The choice of strategy space is an important part of modelling any situation. If you choose a different strategy space you may get diferent results. You are, after all, considering a different game. So what? All this says is that we need to give careful thought to the choice of the rules of the game. This problem occurs in other areas of economics as well. In both partial-equilibrium and general equilibrium competitive models, the agents are restricted to choosing only quantities, taking prices as given. One of the reasons for the development of game theory is that we don’t believe that this restriction is reasonable in some cases. In decision theory, the set of outcomes over which simple lotteries are defined is also set by the modeller. (Indeed, I seem to recall reading a comment along these lines on the earlier work of Menezeas and Quiggin on this topic. I suspect that the comment was probably made on one of John Quiggins posts at his own blog, although I’m not sure that that was the source.) As such, if the critique of Menezes and Quiggin has any validity whatsoever, then it is a critique of economics in general. Indeed, it is probably a critique of much of social science in general.

As an economist who specialises in microeconomic theory and game theory, I doubt very much that I will loosing too much sleep over this critique.

“As an economist who specialises in microeconomic theory and game theory, I doubt very much that I will loosing too much sleep over this critique. ”

I think this will be the general response, but that doesn’t mean it’s justified.

As you say “The choice of strategy space is an important part of modelling any situation. If you choose a different strategy space you may get diferent results. You are, after all, considering a different game. So what? All this says is that we need to give careful thought to the choice of the rules of the game.”

So, can you point me to examples of papers where you think this job has been done well? I’m asking seriously, since we’d like to include examples of good practice. I’ll post our paper which includes fields where we don’t think much serious thought has been given to the problem on my blog when I get a moment.

Andrew since reading your blog I have started wondering about the perception I have that economists seem to concentrate on mathematical models to explain how complex systems work.

I may be wrong but many other areas of study have moved away from complex mathematical representations and more towards simulations and computer models and then to let these simulations and models run with variations on different parameters and to treat the exercise more like an experiment.

I doubt if there are many meteorologists or engineers who are working on general mathematical models to explain the workings of complex systems. They are using mathematics to model small parts of the problem where there is a good understanding of what happens. Trying to represent non linear complex systems with mathematical formula and reasoning based on those representations is unlikely to be productive in understanding the systems.

Quiggin’s paper has reinforced that belief – or perhaps I have misunderstood both what economists seem to spend their time on and what Quiggin’s paper was really saying.

John, first let me apologise if the tone of my previous comment was too strident. It was not meant to be a criticism of the authors of the paper. It was just meant to express my immediate reactions to the theme of the paper based on a quick skim of only some parts of the paper.

Some examples from IO that come to mind are:

1. The choice of strategy space in asymmetric cost Bertrand pricing games. The common assumption of assuming no minimimum unit of account results in a pure strategy space is a continuum. If the market is split evenly when both firms charge the same price, then there is no Nash equilibrium in this game because of a trivial openness problem. (This is possibe because the game is not finite, so that Nash’s existence theorem does not apply.) However, if there is a minimum unit of account, say one cent, then the pure strategy set can be modelled as finite (if you place an upper bound somewhere way above the monopoly price). This ensures that an equilibrium exists. The low cost firm simply charges the largest allowable price that is less than the other firms cost. Another way to avoid the trivial openness problem is to assume that the low cost firm captures the entire market when both firms charge the same price. This seems to me to be less reasonable than the minimum unit of account approach, however, although one might justify it as the limiting outcome in the sequence of games in which the minimum unit of account gets smaller and smaller.

2. The justification for believing that long-run profits will, in general, be a function of output (or, if you prefer, capacities). The two key papers in this literature are the Kreps-Scheinkman paper that you cite and a follow-up paper by Davidson and Deneckere. Both of these papers look at a sequential-game in which firms simultaneously choose capacities in the first period and then choose prices in the second period. The Kreps-Scheinkman paper shows that Cournot-like outcomes are a subgame perfect equilibrium to this game if demand is rationed efficently, while the Davidson-Deneckere paper shows that if an alternative rationing rule (or rules?) is used, then the outcome is (potentially?) much more competitive, but long-run profits are still a function of capacities. My view on this is that, while imperfect competition may not be exactly Cournot in nature, it is almost certainly not Bertrand. As such, two firms is probably not going to give you competitive outcomes and the Bertrand paradox is avoided.

3. It is worth noting that Davidson and Deneckere have another paper which employs the Kreps-Scheinkman assumption of the efficient rationing rule. However, after firms choose capacities, they play anb infinitely repeated pricing game. This paper looks at the role of excess capacity in attempts to sustain collusion.

On a different point, I am not sure that the comment on equilibrium refinements is entirely correct. Most of the mnajor equilibrium refinements are nested within broader equilibrium concepts. For example, for any given game:

(i) the set of Nash-equilibria is a subset of the set of correlated equilibria;

(ii) the set of subgame perfec t equilibria is a subset of the set of Nash equilibria;

(iii) The set of PBE’s is a subset of the set of of SPNE;

(iv) The set of sequential equilibria is a subset of the set of PBE’s;

(v) The set of extensive form trewmbling hand perfect equilibria is a subset of the set of SE.

At this point, it is common to move from the extensive form to the normal form. This introduces a bit of a problem for the nesting of further refinements within the set of EFTHPE unless the game is a simulataneous move game (or unless you are willing to work with the agent-normal form, rather than the standard normal form). Nonetheless, fudging from here, the set of normal form refinements include proper equilibria, the Cho-Kreps intuitive criterion, the Banks-Sobel (I think they were the authors) divine equilibrium and the various concepts of strategic stability (Kohlberg and Mertens, Hillas, Mertens Part 1 and Mertens Part 2). I think all of these require that acceptable equilibria be normal-form trembling hand perfect (or something very similar). The strongest of these concepts is strategic stability. If we are willing to fudge the issues that arise when we move from the extensive form to the normal form, we can think of the set of strategically stable equilibria as being approximately a subset of the set of EFTHPE. There is a nice chapter on equilibrium refinements by Hillas and Kohlberg in the Hanbook of Game Theory with Economic Applications Volume 3. I think they say somethink like all strategically stable equilibria are “quasi-perfect”.

Anyway, the poiunt of this ratyher long detour is that, while the equilibrium refinement literature did not generate a single-valued equilibrium concept, it did result in a number of restrictions on the set of admissable equilibria. Furthermore, it is not the caee that an equilibrium refinement exists that can justify just about any outcome of a game. If the outcome is not a Nash equilibrium in the first place, it will not satisfy any of the higher order refinements mentioned above. Do refinements alway reduce the size of the set of acceptable equilibria? Probably not. Do they sometimes reduce the size of the set of acceptable equilibria? Definitely.

There is also a literature on equilibrium selection, although admittedly this literature is not as well developed as the equilibrium refinements literature, as far as I am aware.

Game theory has generated some very interesting non-existence results in the area of Bayesian learning. Kalai and Lehrer showed that under cetain conditions, Bayesian learning would lead to a Nash equilibrium. However, Nachnar showed that this result was non-generic. In essence, the best that can be hoped for is that Bayesian learning will lead to correlated equilibria. (I think someone may have established that Bayesian learning leads to correlated equilibria, but I am not sure who.)

The previous comment on this topic that Imention in my first comment on this thread is the following one by Joseph Clarke on a post over at John Quiggin’s blog:

http://johnquiggin.com/index.php/archives/2005/10/13/whats-wrong-with-game-theory/#comment-35133 .

Game theory provides very good answers to those practitioners who take the time to study their own strategy space and formalize their utilities. I believe that the actual practitioners usually have a very good idea of what the “rules of the game” are, whether formally or informally. Game theory then can and is used to optimize own behavior, and perhaps more importantly to anticipate the actions of the competitors. In many cases the actions of the competitors are non-intuitive, and this is the case when the benefits of game-theoretic analyses are greatest.

Now for the critique of Menezes and Quiggin. What they are essentially saying is that when the rules of the game are not observable, but only the actions of the practitioners it is always possible to find such rules of the game as would fit the observed actions ex-post. This renders useless not game theory but the armchair economist: at most he can expect to infer what the rules of the game must have been to explain the observations.

Menezes and Quiggin seem to agree that it is not the non-existance of well-defined rules of the game, but of the researchers ignorance of them that is at fault:

> In general, it is necessary to bring to bear extrinsic information about the

>â€˜rules of the gameâ€™if useful predictions about outcomes are to be obtained.

> Such information may be either institutional or behavioral. Institutional

information may be related to knowledge about the political and economic

> environment (for example, some â€˜ actionsâ€™might be ruled out by law or social

> norms).

…

> With more attention to

the determination of the strategy space, incorporating a mixture of institu-

> tional analysis and choice theory, the range of outcomes consistent with a

> reasonable Nash equilibrium may be limited to an extent that allows useful predictions.

There was a further distinct point raised in the paper and on this blog about the multiple equilibria, even in a well defined game. I would like to hear more on this, but here is my 2c. The real world is not deterministic, and the field of finance and some subfields of economics have come to respect that but not all. Something like the nuclear catastrophe in the Cuban crisis genuinely may or may not have happened. we can complain not so much that the theory does not predict whether cooperation or annihilation will happen but that it does not predict the probability. Game theory, even when extended to account for games with imperfect information, and randomness driving the outcomes (as in poker) still cannot predict which equilibrium will be reached. Perhaps this is ok, though it seems to me unique equilibria are much more common in real life situations.

This thread reminds me of the aphorism, “game theorists have made a lot of great contributions, but they’ve all been about game theory”.

“Kohlberg and Mertens, Hillas”

FWIW, Hillas is a University of Queensland graduate.

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