Game Theory:-

Game Theory is an interesting as mathematics. Its applicability is, however, controversial.

Foundations

The classic work is by von Neumann and Morgenstern. Unfortunately, it goes beyond mathematical game theory as such to consider applications to economics.

In order to apply any mathematics to any empirical subject, one needs a ‘mathematization’ of the subject, which is usual based on the views of the domain experts. Such mathematizations are only as reliable as the experts’ views on which it is based. As views change, mathematizations and hence applications of the mathematics need to change. But even though the particular mathematization of this seminal work may seem hopelessly inadequate, the methodology still provides a good example. Further, by taking a logical approach it identifies the key assumptions (‘axioms’) behind the old theory, which still provide a good starting point if one wants to understand the weaknesses of the old theory.

More generally, there has been a tendency to apply game theory by mathematizing some view of a situation, analysing it (using game theory) and then making appropriate recommendations. The problem here is that this is often a part of an attempt to make a significant change to outcomes, which will often destabilise the game, for example by changing coalitions in multi-player games. Thus the application of the analysis, at least in the long-run, undermines its own assumptions. But this is not to say that game theory cannot be applied, only that one should not take too much for granted and should consider taking a higher-level view.

Stability

Fixed games often lead to equilibrium solutions, typically with probabilistic behaviours. In the short-run, as in economics, there can be an incentive to rely on such equilibrium behaviour, but if too many players do this it can destabilise play: a kind of ‘tragedy of the commons‘. There is also a kind of uncertainty principle: if players act on their short-term incentives then as behaviours tend to equilibrium the behaviours of the players get more predictable but the overall behaviour becomes more unpredictable.

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