Correlated equilibrium

An $N$-player strategic game $\backslash displaystyle\; (N,\backslash \{A\_i\backslash \},\backslash \{u\_i\backslash \})$ is characterized by an action set $A\_i$ and utility function $u\_i$ for each player {{nowrap|$i$.}} When player $i$ chooses strategy $a\_i\; \backslash in\; A\_i$ and the remaining players choose a strategy profile described by the {{nowrap|$(N-1)$-tuple}} $a\_\{-i\}$, then player $i$'s utility is $\backslash displaystyle\; u\_i(a\_i,a\_\{-i\})$.

A strategy modification for player $i$ is a function $\backslash phi\_i\backslash colon\; A\_i\; \backslash to\; A\_i$. That is, $\backslash phi\_i$ tells player $i$ to modify his behavior by playing action $\backslash phi\_i(a\_i)$ when instructed to play $a\_i$.

Let $(\backslash Omega,\; \backslash pi)$ be a countable probability space. For each player $i$, let $P\_i$ be his information partition, $q\_i$ be $i$'s posterior and let $s\_i\backslash colon\backslash Omega\backslash rightarrow\; A\_i$, assigning the same value to states in the same cell of $i$'s information partition. Then $((\backslash Omega,\; \backslash pi),P\_i,s\_i)$ is a correlated equilibrium of the strategic game $(N,A\_i,u\_i)$ if for every player $i$ and for every strategy modification $\backslash phi\_i$:

:$\backslash sum\_\{\backslash omega\; \backslash in\; \backslash Omega\}\; q\_i(\backslash omega)u\_i(s\_i(\backslash omega),\; s\_\{-i\}(\backslash omega))\; \backslash geq\; \backslash sum\_\{\backslash omega\; \backslash in\; \backslash Omega\}\; q\_i(\backslash omega)u\_i(\backslash phi\_i(s\_i(\backslash omega)),\; s\_\{-i\}(\backslash omega))$

In other words, $((\backslash Omega,\; \backslash pi),P\_i)$ is a correlated equilibrium if no player can improve his or her expected utility via a strategy modification.

{{Payoff matrix | Name = A game of Chicken

| 2L = Dare | 2R = Chicken out |

1U = Dare | UL = 0, 0 | UR = 7, 2 |

1D = Chicken out | DL = 2, 7 | DR = 6, 6 |

Width = 33% }}

Consider the game of chicken pictured. In this game two individuals are challenging each other to a contest where each can either dare or chicken out. If one is going to dare, it is better for the other to chicken out. But if one is going to chicken out, it is better for the other to dare. This leads to an interesting situation where each wants to dare, but only if the other might chicken out.

In this game, there are three Nash equilibria. The two pure strategy Nash equilibria are (D, C) and (C, D). There is also a mixed strategy equilibrium where both players chicken out with probability 2/3.

Now consider a third party (or some natural event) that draws one of three cards labeled: (C, C), (D, C), and (C, D), with the same probability, i.e. probability 1/3 for each card. After drawing the card the third party informs the players of the strategies assigned to them and to their opponent on the card. Suppose a player is assigned D, they would not want to deviate supposing the other player played their assigned strategy since they will get 7 (the highest payoff possible). Suppose a player is assigned C. Then the other player will play C with probability 1/2 and D with probability 1/2. The expected utility of Daring is 7(1/2) + 0(1/2) = 3.5 and the expected utility of chickening out is 2(1/2) + 6(1/2) = 4. So, the player would prefer chickening out.

Since neither player has an incentive to deviate, this is a correlated equilibrium. The expected payoff for this equilibrium is 7(1/3) + 2(1/3) + 6(1/3) = 5 which is higher than the expected payoff of the mixed strategy Nash equilibrium.

The following correlated equilibrium has an even higher payoff to both players: Recommend (C, C) with probability 1/2, and (D, C) and (C, D) with probability 1/4 each.

Then when a player is recommended to play C, they know that the other player will play D with (conditional) probability 1/3 and C with probability 2/3, and gets expected payoff 14/3, which is equal to (not less than) the expected payoff when they play D. In this correlated equilibrium, both players get 5.25 in expectation. It can be shown that this is the correlated equilibrium with maximal sum of expected payoffs to the two players.

One of the advantages of correlated equilibria is that they are computationally less expensive than Nash equilibria. This can be captured by the fact that computing a correlated equilibrium only requires solving a linear program whereas solving a Nash equilibrium requires finding its fixed point completely.{{cite journal |first1=Christos H. |last1=Papadimitriou |first2=Tim |last2=Roughgarden |title=Computing correlated equilibria in multi-player games |journal=J. ACM |volume=55 |issue=3 |pages=14:1–14:29 |year=2008 |doi=10.1145/1379759.1379762|citeseerx=10.1.1.335.2634 |s2cid=53224027 }} Another way of seeing this is that it is possible for two players to respond to each other's historical plays of a game and end up converging to a correlated equilibrium.{{cite journal |last1=Foster |first1=Dean P. |first2=Rakesh V. |last2=Vohra |title=Calibrated Learning and Correlated Equilibrium |journal=Games and Economic Behavior |year=1996 }}

{{Reflist}}

{{refbegin}}

• Fudenberg, Drew and Jean Tirole (1991) Game Theory, MIT Press, 1991, {{ISBN|0-262-06141-4}}
• {{Citation | last2=Shoham | first2=Yoav | last1=Leyton-Brown | first1=Kevin | title=Essentials of Game Theory: A Concise, Multidisciplinary Introduction | publisher=Morgan & Claypool Publishers | isbn=978-1-59829-593-1 | url=http://www.gtessentials.org | year=2008 | location=San Rafael, CA }}. An 88-page mathematical introduction; see Section 3.5. [http://www.morganclaypool.com/doi/abs/10.2200/S00108ED1V01Y200802AIM003 Free online] {{Webarchive|url=https://web.archive.org/web/20000815223335/http://www.economics.harvard.edu/~aroth/alroth.html |date=2000-08-15 }} at many universities.
• Osborne, Martin J. and Ariel Rubinstein (1994). A Course in Game Theory, MIT Press. {{ISBN|0-262-65040-1}} (a modern introduction at the graduate level)
• {{Citation | last1=Shoham | first1=Yoav | last2=Leyton-Brown | first2=Kevin | title=Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations | publisher=Cambridge University Press | isbn=978-0-521-89943-7 | url=http://www.masfoundations.org | year=2009 | location=New York }}. A comprehensive reference from a computational perspective; see Sections 3.4.5 and 4.6. [http://www.masfoundations.org/download.html Downloadable free online].
• Éva Tardos (2004) Class notes from Algorithmic game theory (note an important typo) [http://www.cs.cornell.edu/courses/cs684/2004sp/feb20.pdf]
• Iskander Karibzhanov. [http://www.mathworks.com/matlabcentral/fileexchange/25281 MATLAB code] to plot the set of correlated equilibria in a two player normal form game
• Noam Nisan (2005) Lecture notes from the course Topics on the border of Economics and Computation (lowercase u should be replaced by u_i) [http://www.cs.huji.ac.il/~noam/econcs/course/lec8.pdf]

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{{Game theory}}

Category:Game theory equilibrium concepts