## 1. Introduction

## 2. Preliminaries

**Definition**

**1**

**.**A probability distribution p on S is a correlated equilibrium for the game Γ if for every player $i\in N$,

**Definition**

**2**

**.**A heuristic for a player $i\in N$ is a sequence of mixed strategies:

## 3. Main Results

#### 3.1. Explicit Scheme

**Definition**

**3**

**.**An explicit scheme $\mathcal{E}$ is a single sequence of probability distributions,

**Example**

**1.**

**Remark**

**1.**

**Remark**

**2.**

**Proposition**

**1.**

**Proof.**

**Example**

**2.**

#### 3.2. Convergence

**Definition**

**4.**

**Proposition**

**2.**

**Definition**

**5**

**.**An explicit scheme $\mathcal{E}=\left({p}_{1},{p}_{2},\dots \right)$ converges to the set of correlated equilibria C if for any ${C}^{\prime}\supseteq C$, there exists a $T\in \tau $ such that:

#### 3.3. Characterizing Heuristic Schemes

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

## 4. Discussion

**Proposition**

**3.**

**Proof.**

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Proof**

**of**

**Proposition**

**2.**

**Lemma**

**A1.**

## References

- Fudenberg, D.; Levine, D.K. The Theory of Learning in Games; MIT Press: Cambridge, MA, USA, 1998. [Google Scholar]
- Hart, S.; Mas-Colell, A. Simple Adaptive Strategies; World Scientific Publishing: Singapore, 2013. [Google Scholar]
- Shoham, Y.; Leyton-Brown, K. Mutiagent Systems; Cambridge University Press: New York, NY, USA, 2010. [Google Scholar]
- Foster, D.; Vohra, R. Calibrated learning and correlated equilibrium. Games Econ. Behav.
**1997**, 21, 40–55. [Google Scholar] [CrossRef] - Hart, S.; Mas-Colell, A. A Simple Adaptive Procedure leading to Correlated Equilibrium. Econometrica
**2000**, 68, 1127–1150. [Google Scholar] [CrossRef] - Hart, S.; Mas-Colell, A. A General Class of Adaptive Strategies. J. Econ. Theory
**2001**, 98, 26–54. [Google Scholar] [CrossRef][Green Version] - Blum, A.; Mansour, Y. From external to internal regret. J. Mach. Learn. Res.
**2007**, 8, 1307–1324. [Google Scholar] - Hart, S. Adaptive Heuristics. Econometrica
**2005**, 73, 1401–1430. [Google Scholar] [CrossRef] - Hart, S.; Nisan, N. The query complexity of correlated equilibria. Games Econ. Behav.
**2018**, 108, 401–410. [Google Scholar] [CrossRef][Green Version] - Aumann, R.J. Subjectivity and correlation in randomized strategies. J. Math. Econ.
**1974**, 1, 67–96. [Google Scholar] [CrossRef] - Aumann, R.J. Correlated Equilibrium as an Expression of Bayesian Rationality. Econometrica
**1987**, 55, 1–18. [Google Scholar] [CrossRef] - Rudin, W.A. Principles of Mathematical Analysis, 3rd ed.; McGraw-Hill Publishing: New York, NY, USA, 1976. [Google Scholar]
- Steen, L.A.; Seebach, J.A. Counterexamples in Topology; Dover Publications: New York, NY, USA, 1995. [Google Scholar]
- Baum, L.E.; Katz, M. Convergence rates in the law of large numbers. Bull. Am. Math. Soc.
**1963**, 69, 771–772. [Google Scholar] [CrossRef] - Puri, M.L.; Ralescu, D.A. Strong Law of Large Numbers with Respect to a Set-Valued Probability Measure. Ann. Probab.
**1983**, 11, 1051–1054. [Google Scholar] [CrossRef] - Molchanov, I. Theory of Random Sets, 2nd ed.; Springer: London, UK, 2017. [Google Scholar]

1 | Refer to the book Hart and Mas-Colell [2] for a more elaborate treatment. |

2 | Hart and Nisan [9], in fact, show that randomization is necessary for such algorithms. |

3 | The notation $\left|A\right|$ stands for the number of elements of a set A. |

4 | Further, the space of all probability distributions over any finite set of strategies is a complete metric space. |

5 | For instance, in the case of the usual one-dimensional convergence to a point, m would have the cardinality of $\mathbb{N}$. |

6 | There is work on the rate of convergence for laws of large numbers, for instance Baum and Katz [14], but we do not get into that in this paper. |

7 |

© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).