We-thinking and 'double-crossing': frames, reasoning and equilibria

The idea of we-thinking, or we-reasoning, is increasingly drawing the attention of more and more economists. The two main contributors are Bacharach and Sugden, and they approach the topic in two different ways. Sugden's aim is to show that we-reasoning is a consistent and logical way of thinking, but he does not face the problem of how we-reasoning can arise. Bacharach's theory is based on frames and his never reached aim (because of his death) was to explain we-thinking in terms of Variable Frame Theory. But some of his intuitions conflict with the logical analysis he proposes. In the present paper, I take a different approach to the way in which we-thinking works. Based on a not fully developed intuition of Bacharach's, i.e. the `double-crossing' problem in Prisoners' Dilemma (PD) game, I propose a framework in which a person is allowed to have both I-thoughts, when she is we-reasoning, and we-concepts, when she is I-reasoning, and develop my analysis in terms of equilibrium concepts.