The multi-player nonzero-sum Dynkin game in discrete time
We study the infinite horizon discrete time N-player nonzero-sum Dynkin game (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$N \ge 2$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>N</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math> </EquationSource> </InlineEquation>) with stopping times as strategies (or pure strategies). The payoff depends on the set of players that stop at the termination stage (where the termination stage is the minimal stage in which at least one player stops). We prove existence of a Nash equilibrium point for the game provided that, for each player <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$\pi _i$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi mathvariant="italic">π</mi> <mi>i</mi> </msub> </math> </EquationSource> </InlineEquation> and each nonempty subset <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$S$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>S</mi> </math> </EquationSource> </InlineEquation> of players that does not contain <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$\pi _i$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi mathvariant="italic">π</mi> <mi>i</mi> </msub> </math> </EquationSource> </InlineEquation>, the payoff if <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$S$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>S</mi> </math> </EquationSource> </InlineEquation> stops at a given time is at least the payoff if <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$S$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>S</mi> </math> </EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$\pi _i$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi mathvariant="italic">π</mi> <mi>i</mi> </msub> </math> </EquationSource> </InlineEquation> stop at that time. Copyright Springer-Verlag Berlin Heidelberg 2014
