Game Theory in Practice, on the (Non) Discovery of Social Equilibria

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What follows is an attempt to view (elementary) game theory through the prism of recent developments in philosophy of mathematics. In particular, if a model (in the sense of model theory)on an appropriate domain can be constructed from a formal theory and it can be established that homomorphic mappings between model structures are possible then such a maneuver undermines the Realistic case for solution concepts since it makes the denotation of such concepts problematic since the denotation ceases to be unique. This creates philosophical problems within the Philosophy of mathematics which affects communities of mathematicians working in particular mathematical areas. (Groups, Rings, Vector Spaces, Graphs and so on) However when it comes to game theory, problematic denotation affects not only Professionals (mathematicians, economists, game theorists) but also ‘ordinary’ people in the real world who interact with each other in social contexts amenable to game theoretic analysis I note briefly that Game theory has difficulties, still, with equilibrium selection and simple 2X2 games fail to predict and these problems have triggered extensive debates. I do not specify any specific causal connection between referential failure and any given issue within game theory I do however suggest that there is clearly a need to supplement the semantics of equilibrium concepts (here the minimax theorem) and I note that even simple 2X2 games typically come with labels, prisoner, husband-wife, hawk and so on. These perhaps provide hooks into more comprehensive semantics. I also briefly discuss conventions as an additional mechanism. Finally I suggest that if this analysis is on the right lines it might be extended to include more complex social equilibrium mechanisms like Nash equilibrium or general economic equilibrium in micro economics. The style is deliberately informal.