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120 CHAPTER 4 Bayesian Games ·························································································· · · · · · · · If the set of players, the sets of players' actions, and the sets of players' types are finite, · · then there necessarily exists at least one mixed Bayesian equilibrium. · · Theorem 139: Existence Theorem This theorem is due to Harsanyi (1967) and the proof directly follows from the Nash theorem (Nash, 1950). In fact, this theorem also holds for the interim model (which is not consistent). Example 140 Assume a game with two players (e.g., mobile phone manufacturers): K = {1, 2}. Each player has two possible actions lying in the set S 1 = S 2 = { WiMax , LTE }. Additionally, each player has two possible types lying in T 1 = T 2 = { cooperative , non-cooperative }. The type cooperative (C) means that the player in question wants to reach a consensus on the standard used whereas non-cooperative (NC) corresponds to the oppositive behavior. Denote by q i ( -i ) the belief of player i about the type of player -i, that is, the probability player -i chooses type -i T -i . Assuming the beliefs to be independent, the game can be described by the following matrix form. q 2 (C) = 1 2 LTE 2 q 1 (C) = 3 q 2 (NC) = 1 2 WiMax LTE LTE WiMax WiMax LTE WiMax (2, 1) (0, 0) LTE (0, 0) (1, 2) WiMax (2, 0) (0, 1) LTE (0, 2) (1, 0) WiMax q 1 (NC) = 1 3 LTE WiMax (0, 1) (1, 0) (2, 0) (0, 2) LTE WiMax (0, 0) (1, 1) (2, 2) (0, 0) Depending on the four possible pairs of types, different games are played, as represented by four utility matrices. In the chosen example, player 1 believes player 2 is going to cooperate 2 with probability 3 , player 1 believes player 2 is going to not cooperate with probability 1 , 3 etc. As the beliefs are independent, one can express the players' utilities in the expected game. For instance, the expected utility player 1 gets when the action profile is ( LTE , LTE ) is given by: u B (( LTE , LTE )) = 1 2 1 2 1 1 1 1 1 4 × × 2 + × × 2 + × × 0 + × × 0 = . 3 2 3 2 3 2 3 2 3 (4.5) By computing the expected utilities for all game outcomes, we obtain the following matrix form for the expected game. LTE LTE WiMax 4 , 1 3 2 1 , 1 3 2 WiMax 2 3 ,1 2 3 ,1