We provide an epistemic foundation for the core of a cooperative game by proof theory . Given a cooperative game, we first transform each payoff vector into a decision problem ( accept or reject ) for each player . Then we use a modified KD-system in epistemic logic to describe a player's belief/ knowledge, decision-making criterion, and reasoning process . Especially , we define  C-acceptability to capture the criterion for  a core payoff vector. Within this syntactical framework, we characterize the core of a cooperative game in terms of each player's knowledge. Based on this result, we discuss  epistemic inconsistency behind Debreu-Scarf Theorem  .
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 .
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, 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