On the Geometry of Nash and Correlated Equilibria with Cumulative Prospect Theoretic Preferences

Abstract

It is known that the set of all correlated equilibria of an [Formula: see text]-player non-cooperative game is a convex polytope and includes all of the Nash equilibria. Furthermore, the Nash equilibria all lie on the boundary of this polytope. We study the geometry of both these equilibrium notions when the players have cumulative prospect theoretic (CPT) preferences. The set of CPT correlated equilibria includes all of the CPT Nash equilibria, but it need not be a convex polytope. We show that it can, in fact, be disconnected. However, all of the CPT Nash equilibria continue to lie on its boundary. We also characterize the sets of CPT correlated equilibria and CPT Nash equilibria for all [Formula: see text] games, with the sets of correlated and Nash equilibria in the classical sense being a special case.

On the Impossibility of Convergence of Mixed Strategies with No Regret Learning

“…[0, 1] × [0, 1]). Phade and Anantharam [2019] characterize the NE of 2 × 2 games, and, as a consequence, show that all the NE of a game are completely mixed if and only if it is a competitive game.…”

“…Note that the set of zero-sum games with unique NE is strictly contained in the set of competitive games. The results in Phade and Anantharam [2019] imply that, for any competitive game, there exists a corresponding zero-sum game 11 such that the best-response functions of both the players are exactly the same (where the best-response function of a player is a mapping from her beliefs to the action, or set of actions, with the best expected reward given that belief). It is easy to see that if two games have the same best-response functions, then they have the same NE.…”

“…We answer this question in the negative for the set of the games which have only completely mixed NE. These games are designated as competitive games by Calvó-Armengol [2006], and possess one unique NE, which is also the unique correlated equilibrium of the game [Phade and Anantharam, 2019]. These games have been of special interest in the design of experiments to test the performance of NE as a predictor of behavior in games as they have an unambiguous NE 7 .…”

Prospect-theoretic Q-learning

“…To simulate this formulation, i.e., iteration (16) and o.d.e. (17), we vary parameters as in Section 5. As expected, this scheme also converges to the equilibrium points.…”

“…While there has been an interest in extending prospect theoretic models to finance [15], game theory [16,17,18], etc., the works in prospect theoretic reinforcement learning are rare. In addition to those mentioned above, one interesting effort, albeit not in Markov decision theoretic framework, is [19].…”