<p>This introduces a general cooperative game with preferences and a maximal element model for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\alpha\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-cores. For games with infinitely many players, some necessary and sufficient conditions are given for the existence of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\alpha\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-cores of general cooperative games with preferences. It is revealed the underlying causes that some infinite-player-games have no <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\alpha\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-core in references. It is also shown that the acyclicity of a preference implies the coalitionally transfer quasi-concavity of some general cooperative games. For the cases with finite players, a sufficient condition is given for the existence of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\alpha\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-cores of general cooperative games with preferences. As an application, the existence of TU <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\alpha\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-core of a normal form game is deduced.</p>