Now letвЂ™s try a somewhat more challenging example and revisit the battle of the sexes. Row is willing to randomize if these are equal, which solves for p = ВЅ. Similarly, the payoff to Row of playing Tails was 1 вЂ“ 2 p. Thus, for example, we computed the payoff to Row of playing Heads, which was 2 p вЂ“ 1, where p was the probability that Column played Heads. The reason is that it is ColumnвЂ™s probabilities that determine the expected payoffs for Row if Row is going to randomize, then ColumnвЂ™s probabilities must be such that Row is willing to randomize. Instead, that equality in RowвЂ™s payoffs will determine the probabilities with which Column plays the various columns. But that equality in RowвЂ™s payoffs doesnвЂ™t determine the probabilities with which Row plays the various rows. Then RowвЂ™s payoffs must be equal for all strategies that Row plays with positive probability. Otherwise, the player would prefer one of them and wouldnвЂ™t play the other.Д¬omputing a mixed strategy has one element that often appears confusing. If a player is supposed to randomize over strategy A or strategy B, then both of these strategies must produce the same expected payoff. Note that randomization requires equality of expected payoffs. A Nash equilibrium in which no player randomizes is called a pure strategy Nash equilibrium A Nash equilibrium in which no player randomizes.ДЇigure 16.14 Mixed strategy in matching pennies involves at least one player playing a randomized strategy and no player being able to increase his or her expected payoff by playing an alternate strategy. In this case, Row could play Heads, could play Tails, or could flip a coin and randomize RowвЂ™s play.ДЄ mixed strategy Nash equilibrium A Nash equilibrium in which at least one player plays a randomized strategy and no player is able to increase his or her expected payoff by playing an alternate strategy. If, on the other hand, 2 p вЂ“ 1 = 1 вЂ“ 2 p, then Row gets the same payoff no matter what Row does.

Similarly, if 2 p вЂ“ 1 < 1 вЂ“ 2 p, then Row is better off playing Tails than Heads. If 2 p вЂ“ 1 > 1 вЂ“ 2 p, then Row is better off, on average, playing Heads than Tails. This is summarized in Figure 16.14 'Mixed strategy in matching pennies'. Similarly, if Row plays Tails, Row gets вЂ“1 with probability p (when Column plays Heads), and 1 with probability (1 вЂ“ p), for an expected value of 1 вЂ“ 2 p. Then if Row plays Heads, Row gets 1 with probability p and вЂ“1 with probability (1 вЂ“ p), for an expected value of 2 p вЂ“ 1.

Suppose that Row believes Column plays Heads with probability p.

Half life 2 randomizer