This paper focuses on a new model to reach the existence of equilibrium in a pure exchange economy with fuzzy preferences (PXE-FP). The proposed model integrates exchange, consumption and the agent’s fuzzy preference in the consumption set. We set up a new fuzzy binary relation on the consumption set to evaluate the fuzzy preferences. Also, we prove that there exists a continuous fuzzy order-preserving function in the consumption set under certain conditions. The existence of a fuzzy competitive equilibrium for the PXE-FP is confirmed through a new result on the existence of fuzzy Nash equilibrium for fuzzy non-cooperative games. The payoffs of all strategy profiles for any agent are fuzzy numbers in fuzzy non-cooperative games. Finally, we show that the fuzzy competitive equilibrium could be characterized as a solution to an associated quasi-variational inequality, giving rise to an equilibrium solution.
We address the question: What may be the conceptual sources of Chinese elementary students’ correct and incorrect solutions to multiplicative problem situations? We focus on situations for eliciting evidence about their ability to add and subtract sets of composite units—mental structures that underlie conceiving of whole numbers as a single entity composed of smaller units (e.g., “3” is composed of three “1s”). A conception postulated to underlie this ability is termed Same-Unit Coordination (SUC). We attribute student errors to reasonable-to-them spontaneous use of a previously established, repetitively practiced way of operating on 1s (“Ones”) contained within composite units—a conception we term Totaling. We analyze qualitative data to illuminate this phenomenon and quantitative data to depict its scope. These analyses support our claim that student solutions, seen by an observer as correct or erroneous, can be explained as (a) reasonable from the students’ frame of reference and (b) possibly arising from instructional focus on mastering multiplication facts to find totals of 1s in equal-size sets.
This paper focuses on a new model called fuzzy exchange economy (FXE), which integrates fuzzy consumption, fuzzy initial endowment and the agent's fuzzy preference (vague attitude) in the fuzzy consumption set. Also, the existence of the fuzzy competitive equilibrium for the FXE is verified through a related pure exchange economy. We define a core-like concept (called weak fuzzy core) of the FXE and prove that any fuzzy competitive allocation belongs to the weak fuzzy core. The fuzzy replica economy, which is the r-fold repetition of the FXE, is considered. Finally, we show that the weak fuzzy core of the r-fold fuzzy replica economy, i. e., the set of all fuzzy allocations which cannot be blocked by any coalition of agents, converges to the set of fuzzy competitive allocations of the FXE as r becomes large.
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