Compatible beliefs and equilibrium

In this paper I investigate the nature of the beliefs which agents must hold (at least implicitly) in order to justify their considering various alternatives, in two distinct settings: the Walrasian model without production (with competitive equilibrium), and the sell-all version of the Shapley-Shubik market game (with Nash equilibrium). For this purpose I introduce a weak consistency requirement on behavior, one which I refer to as (having) compatible beliefs. My main conclusion is that, in this respect, these two versions of market allocation are essentially identical. For both, contemplating different choices requires varying the associated set of values of the variables defining compatible beliefs. And - though prima facie very different - it turns out that both equilibrium concepts can be recast entirely in terms of having compatible beliefs. My analysis also leads unequivocally to the interesting conclusion that, in the Walrasian model (even elaborated to encompass production, financial markets, and so on), budget constraints must hold, ab initio, with equality. This has one very important consequence: the First Basic Welfare Theorem, as usually stated, is false, as I demonstrate with two distinct counterexamples, the second of which is (in classical terms) unexceptional.