We provide an epistemic foundation for the cooperative game by proof theory by studying the knowledge needed for players to unanimously accept only core payoffs. We first transform each cooperative game into a decision problem where each player accept or reject each payoff vector . Then we use a modified KD-system in epistemic logic  to describe a player's knowledge, decision-making criterion, and reasoning process; especially, we define a formula called C-acceptability to describe the criterion for accepting a core payoff vector. Within this syn- tactical framework, we characterize the core of a cooperative game in terms of each player' s knowledge. Based on that result, we discuss an epistemic inconsistency behind Debreu-Scarf Theorem, that is, the increase of the number of replicas , on one hand, has constant require- ment on each participator 's knowledge in the competitive market, while , on the other hand, requires unbounded epistemic ability for at least one player in the cooperative situation .
We provide an epistemic foundation for cooperative games by proof theory via studying the knowledge for players unanimously accepting only core payoffs. We first transform each cooperative game into a decision problem where a player can accept or reject any payoff vector offered to her based on her knowledge about available cooperation . Then we use a modified KD-system in epistemic logic , which can be regarded as a counterpart of the model for non-cooperative games in Bonanno (2008), (2015), to describe a player's knowledge, decision-making criterion, and reasoning process; especially, a formula called C-acceptability is defined to capture the criterion for accepting a core payoff vector. Within this syntactical framework, we characterize the core of a cooperative game in terms of players' knowledge. Based on that result, we discuss an epistemic inconsistency behind Debreu-Scarf Theorem, that is, the increase of the number of replicas has invariant requirement on each participant 's knowledge from the aspect of competitive market, while requires unbounded epistemic ability players from the aspect of cooperative game .
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