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2.2 Equilibrium Existence > 2.2.3 Potential Games - Pg. 46

46 CHAPTER 2 Playing with Equilibria in Wireless Non-Cooperative Games sub-modular games (Topkis, 1979) and super-modular games (Milgrom and Roberts, 1990; Topkis, 1998). The following discussion will give the definition, an existence theorem, and an example of a game for these types of games. The notation a b will mean that each entry of the vector a is greater than or equal to the entry having the same index in the vector b. 2.2.3 Potential Games Potential games have been introduced by Monderer and Shapley in Monderer (1996). They proposed four types: weighted potential games, exact potential games, ordinal potential games, and generalized potential games. Since then, other types of poten- tial games have been introduced. For instance, Voorneveld introduced best-response potential games in Voorneveld (2000). As far as the existence of a Nash equilibrium is concerned, distinguishing between these types of potential games is not relevant. The reasons for their inclusion are twofold: depending on the game under consideration, proving that the game is potential can be easier for a given type; and, although exis- tence is guaranteed for all the mentioned types, they do not have the same properties