On the space of players in idealized limit games

RETRACTED ARTICLE: Poisson-type inequalities for growth properties of positive superharmonic functions

“…For similar results with respect to Schrödinger operator, we refer the reader to the literature (see [5, 9]).…”

Modeling infinitely many agents

“…The proofs of the claims in these examples are given in Appendix A.2. 8 A rather involved example provided by Qiao and Yu (2014) shows the existence of a convergent sequence of exact Nash equilibria in a sequence of finite-agent games such that the idealized limit game of the sequence does not have any Nash equilibrium. 9 Khan et al (2013) and Qiao and Yu (2014) considered the existence issue of pure-strategy Nash equilibria and the closed graph property in large games with traits via a saturated agent space; for details about saturated agent spaces, see Section 4.5.…”

“…8 A rather involved example provided by Qiao and Yu (2014) shows the existence of a convergent sequence of exact Nash equilibria in a sequence of finite-agent games such that the idealized limit game of the sequence does not have any Nash equilibrium. 9 Khan et al (2013) and Qiao and Yu (2014) considered the existence issue of pure-strategy Nash equilibria and the closed graph property in large games with traits via a saturated agent space; for details about saturated agent spaces, see Section 4.5. More generally, the nowhere equivalence condition allows the possibility to work with both saturated and nonsaturated agent spaces and unifies various earlier approaches (including the distributional approach, the standard representation, and the saturated probability spaces).…”

“…In the setting of large games (with or without traits), Keisler and Sun (2009) and Khan et al (2013) considered the necessity result for the existence of Nash equilibria, and Khan et al (2013) and Qiao and Yu (2014) studied the necessity result for the closed graph property. All these papers worked with saturated agent spaces, which exclude the Lebesgue unit interval.…”

“…26 The determinateness and existence problems in large games were resolved in Khan and Sun (1999) by working with a hyperfinite agent space; see Section 4.4 for more discussions. 27 The closed graph property for large games with traits was shown in Khan et al (2013) and Qiao and Yu (2014) by working with a saturated agent space, while Green (1984) considered a similar property based on the distributional approach without specifying the finite-agent spaces. Similar to Remark 1, our closed graph property for large games still holds when one removes the restriction that k is finite for each k ∈ Z + , which also implies that D F (G) is closed.…”