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A study of Correlated Equilibrium Polytope By: Jason Sorensen

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The set of Correlated Equilibrium is a polytope Polytope has two equivalent definitions: 1.)The Convex hull of a finite set of points in R^n 2.)The bounded intersection of a finite set of closed half-spaces The correlated equilibrium constraints are Nash Equilibrium are Correlated -> non-empty

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Linear programming Can find maximal correlated equilibrium by linear programming Use constraints given, player i maximizing hyperplane is Easily solved by any LP program for reasonable sized games

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The Shapley game 12 (Non-trivial) LP constraints 6 of them reduce to x1 = x2 = x5 = x6 = x7 = x9 Reduces original 8 dimensional polytope to 9 dimensions Unique Nash Equilibrium at xi = 1/9 This corresponds to LP minimum for all utility hyperplanes 123 11,00,10,0 2 1,00,1 3 0,01,0

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The Shapley polytope

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Chicken or Dare? Game matrix is: 3 Nash Equilibria at x3 = 1, x2 = 1, and one mixed Each Nash Equilibirium is an LP minimum for a utility hyperplane Can reduce polytope dimension to 3 by utilizing equality constraint DC D0,07,2 C2,76,6

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The Chicken Polytope

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Theorem time! All Nash Equilibrium lie on the boundary of the Correlated Equilibrium Proof on the board! … But may not by too great if polytope is not full dimensional (boundary != relative boundary)

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Are CE always better than NE? In the two cases we studied, all utility hyperplanes were minimized at an NE Is this generally the case? NO! In fact, there is no way to find NE by linear systems in general

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Correlated Equilibrium get Bested Game of “Poker” (?!) Unique Nash Equilibrium - with irrational coordinates NE does not occur at a vertex (all vertices are rational) Value of NE for player one:.890 Value of worst case CE for player one:.5833 Value of best case CE for player one: 1.467 The CE Polytope is full 7 dimensional (not graphable) 1LR T(3,0,2)(0,2,0) B(0,1,0)(1,0,0) 2LR T (0,1,0) B(0,3,0)(2,0,3)

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Putting it all together We can guarantee convergence to CE by natural (no-regret) learning processes But CE may not always be better than Nash Equilibrium Which CE do these “natural learning processes” converge to? How long does convergence take? Will investigate further in next 2 weeks

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A Conundrum Using learning processes to find optimal responses may result in being “bullied” by opponent (it’s always the smart ones who get picked on) In chicken or dare, if the opponent is rational and knows you are learning, there is no reason not to always play dare How do we decide whether to learn or bully for optimal payoffs? Each “playing strategy” (learning or not) combated against each other in a game results in a certain payoff for each player after convergence Model this situation as a new game, where each “move” is a learning strategy and learn the optimal strategy Have we really learned anything?

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Open Problems Compare learning strategies to figure out optimal method Figure out properties of general game polytopes (the number of faces) In which situations is the polytope full dimensional? In which situations are the NE the LP minimizing vertices for all utility hyperplanes?

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