Mixed Strategies Keep em guessing Mixed Strategy Nash Equilibrium A mixed strategy is one in which a player plays his available pure strategies with certain probabilities probabilities Mixed strategies are best understood in the context of repeated games where each player s aim is to keep the other player s guessing for example Rock Scissors Paper If each player in an n player game has a finite number of pure strategies then there exists at least one equilibrium in strategies possibly mixed strategies Nash proved this pure strategy gy equilibria q there must be a If there are no p unique mixed strategy equilibrium However it is possible for pure strategy and mixed strategy Nash equilibria to coexist as in the Chicken game Example 1 Tennis Let p be the probability that Serena chooses DL so 1 p is the probability that she chooses CC Let q be the probability that Venus positions herself for DL DL so 1 q 1 q is the probability that she positions herself for CC To find mixed strategies we compute the p mix and q mix options Venus Williams Serena Williams DL CC q mix DL 50 50 80 20 50q 80 1 q CC 90 10 20 80 90q 20 1 q 50 9 50p 10 1 p 8 20p 80 1 p p mix i Row Player s Optimal Choice of p Chose p so as to equalize the payoff your opponent receives from playing either pure strategy This requires understanding how your opponent s payoff varies with your choice of p Graphically p y in the Tennis example p For Serena s choice of p Venus s expected payoff from playing DL is 50p 10 1 p and from playing CC is 20p 80 1 p Venus i made is d indifferent if Serena chooses p 70 Algebraically g y Serena solves for the value of p that equates Venus s payoff from positioningg herself for DL or CC p 50p 10 1 p 20p 80 1 p or 50p 10 10p 20p 80 80p or 40p 10 80 60p or 100p 70 so p 70 100 0 100 70 0 If Serena plays DL with probability p 70 and CC with probability 1 p 30 1 p 30 then Venus s Venus s success rate from DL 50 70 10 30 38 Venus s success rate from CC 20 70 80 30 38 Since this is a constant sum game Serena s success rate is 100 Venus s success rate 100 38 62 Column Player Player ss Optimal Choice of q Choose q so as to equalize the payoff your opponent receives from playing either pure strategy This h requires understanding d d h how your opponent s payoff ff varies with h your choice of q Graphically in our example For Venus s choice of q Serena s Serena s expected payoff from playing DL is 50q 80 1 q and from playing CC is 90q 20 1 q Serena is made indifferent if Venus chooses q 60 60 Algebraically Venus solves for the value of q that equates Serena s payoff p y from playing p y g DL or CC 50q 80 1 q 90q 20 1 q or 50q 80 80q 90q 20 20q or 80 30 70 80 30q 70q 20 20 or 60 100q so q 60 100 60 60 If Venus positions herself for DL with probability q 60 and CC with probability 1 q 40 then Serena s success rate from DL 50 60 80 40 62 Serena s DL 50 60 80 40 62 Serena s success rate from CC 90 60 20 40 62 Since this is a constant sum game Venus s success rate is 100 Serena s success rate 100 62 38 The Mixed Strategy gy Equilibrium q A strictly mixed strategy Nash equilibrium in a 2 player 2 choice 2x2 game is a p 0 and a q 0 such that p is a best response by the row player to column player s choices and q is a best response by the column player to the row player player ss choices choices In our example p 70 q 60 The row player s payoff Serena was 62 and the column player s payoff Venus was 38 38 Serena wins 62 62 Venus 38 38 Pure strategies can now be understood as a special case of mixed strategies where p is chosen from the set 0 1 and d q is chosen h ffrom the h set 0 1 For example if p 0 and q 1 then row player always plays CC and column player always plays DL Keeping the Opponent Indifferent Why is this a useful objective for determining the mixing probability In constant sum games such as the tennis example making your opponent indifferent in expected payoff terms is equivalent l to minimizing your opponents ability bl to recognize and exploit systematic patterns of behavior in your own choice In constant sum games keeping your opponent indifferent is equivalent to keeping yourself indifferent The same objective works for finding mixed strategy equilibria in non constant sum games as well where players interests are not totally opposed to one another Necessarily suggests that the game is played repeatedly repeatedly Best Response p Functions Another way to depict each player s choice of the mixing probability Recall p Pr DL by Serena q Pr DL by Venus Shows strategic best response of q f p and p g q p q 0 is always play CC p q 1 is always play DL Venus s Best Response Function Serena s Best Response Function 1 1 p q 0 0 0 p 70 1 0 q 60 1 Construction of Best Response Functions Use the h graphs h off the h optimal i l choices h i off q f p f and d p g q Venus s Success Rate From Positioning for DL or CC Against Serena s p mix Serena s Success Rate From Positioning for DL or CC Against Venus s q mix 80 60 40 DL 20 CC 0 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 Serena s choice of p 1 Sern na s Success Rate Venu us s Success Rate 100 100 80 60 40 DL 20 CC 0 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 Venus s choice of q Combining the Best Response Functions R Reveals l the h Mixed Mi d Strategy S Nash N h Equilibrium E ilib i Serena s Best Response Function Equilibria pure or mixed obtain wherever the best response functions intersect Combined Best Response Functions Nash equilibria are mutual best responses Mixed Strategy Nash h Equilibrium ilib i Occurs at p 70 q p q 60 1 q 60 0 0 p 70 1 Venus s Best Response p Function Note that there is no equilibrium in pure strategies in this game Intersections in the middle of the box are mixed strategy equilibria intersections at the …