# On the core of normal form games with a continuum of players : a correction

## TL;DR

This paper proves the nonemptiness of the weak-core in continuum games without side payments, showing it can be larger than the Aumann's alpha-core and linking it to finite game approximations.

## Contribution

It establishes the existence of the weak-core in continuum games with strategy-dependent payoffs and clarifies its relation to regularity conditions and finite game limits.

## Key findings

- Weak-core is nonempty for payoff-dependent strategies.
- Weak-core can be larger than Aumann's alpha-core.
- Regularity conditions for pure Nash equilibria are irrelevant for weak-core non-vacuity.

## Abstract

We study the core of normal form games with a continuum of players and without side payments. We consider the weak-core concept, which is an approximation of the core, introduced by Weber, Shapley and Shubik. For payoffs depending on the players' strategy profile, we prove that the weak-core is nonempty. The existence result establishes a weak-core element as a limit of elements in weak-cores of appropriate finite games. We establish by examples that our regularity hypotheses are relevant in the continuum case and the weak-core can be strictly larger than the Aumann's $\alpha$-core. For games where payoffs depend on the distribution of players' strategy profile, we prove that analogous regularity conditions ensuring the existence of pure strategy Nash equilibria are irrelevant for the non-vacuity of the weak-core.