"Nash equilibrium is the strategy of choosing their best response by a player given the otherplayers strategy. In other words it is showing each player best response considering otherplayer’s strategy fixed, so there is no incentive to deviate from that point or equilibrium byany player.The following payoff matrix given as below:For calculating mixed strategy nash equilibrium, first we calculate expected payoff of firm2:Let the probability of choosing low price by firm 2 is P, so probability of not choosing lowprice or choosing high price is (1 – P) and probability of choosing low price by firm 1 is Qso probability of choosing high value is (1 – Q). U Low,Low and U Low,Low? ? ? ? ? ? ? ? 1 2 showing utility of firm 1 and utility of firm 2 respectively, where firm 1 choosing low priceand firm 2 also chooses low price, the others showing the same.Expected payoff of firm 2 is:E F = P×Q× U Low,Low +P× 1-Q × U Low,High ? ? ? ? ? ? ? ? ? ? ? ? 2 2 2 + 1-P ×Q× U High,Low + 1-P × 1-Q × U High,High ? ? ? ? ? ? ? ? ? ? ? ? ? ? 22 =P×Q×2+P× 1-Q ×1+ 1-P ×Q×0+ 1-P × 1-Q ×6? ? ? ? ? ? ? ? =2PQ+ P-PQ +0+6 1-P-Q+PQ ? ? ? ? =7PQ-5P-6Q+6 So, expected payoff of firm 2 is .7PQ – 5P – 6Q ? 6 Expected payoff of firm 1 is:E F = P×Q× U Low,Low +Q× 1-P × U Low,High ? ? ? ? ? ? ? ? ? ? ? ? 1 1 1 + 1-Q ×P× U High,Low + 1-P × 1-Q × U High,High ? ? ? ? ? ? ? ? ? ? ? ? ? ? 11 =P×Q×0+P× ?1-Q ?×2+ ?1-P ?×Q×7+ ?1-P ?× ?1-Q ?×6 =0+ 2P-2PQ +7Q-7PQ+6 1-P-Q+PQ ? ? ? ? = -3PQ - 4P + Q + 6 So, expected payoff of firm 1 is -3PQ - 4P + Q + 6 .From these expected payoffs we calculate both firms’ best response as we did below:E F =7PQ-5P-6Q+6 ? ? 2 dE ?F ? 2 =7Q-5 dP 7Q-5=0 5 Q= 7Similarly we calculate value of P:E F = -3PQ-4P+Q+6 ? ? 1 dE F ? ? 1 = -3P+1 dQ -3P+1=0 1 P= 3 1 5 2 2 So, value of P is , value of Q is , value of (1-P) is and value of (1-Q) is .3 7 3 7 For getting mixed strategy nash equilibrium we put all values in expected value, as weknow the above calculated values are probabilities so they ranges from 0 to 1 and somecases make to solve this for firm 2:5 1) Q ? 7 In this case value of Q is less than5/7, so we put Q=0 in expected value of firm 2 then:E F = 7PQ-5P-6Q+6 ? ? 2 = -5P+6 Here if P is 0 then expected value is 6 and when P is 1 then expected value is 0. And bothare provide pure strategy nash equilibrium.5 2)Q > 7 In this case Q is greater than 5/7, so we put Q=1 in expected value of firm 2 then:E F = 7PQ-5P-6Q+6 ? ? 2 = 7P-5P-6+6 = 2P Here if P is 0 then expected value is 0 and when P is 1 then expected value is 2. And bothare provide pure strategy nash equilibrium.5 3) Q = 7 In this case Q is equal to 5/7, so we put Q=5/7 in expected value of firm 2 then:E F = 7PQ-5P-6Q+6 ? ? 2 30 = 5P-5P- +67 12 =7 12 Here expected value isirrespective the value of P and it is mixed strategy nash7 equilibrium. Similarly we did the same with firm 1 as we do for firm 2:1 1) P ? 3 In this case value of P is less than1/3, so we put P=0 in expected value of firm 2 then:E F = -3PQ-4P+Q+6 ? ? 1 = Q+6 Here if Q is 0 then expected value is 6 and when Q is 1 then expected value is 7. And bothare provide pure strategy nash equilibrium.1 2)P > 3 In this case P is greater than 1/3, so we put P=1 in expected value of firm 2 then:E ?F ?= -3PQ-4P+Q+6 1 = -3Q-4+Q+6= -2Q+2 Here if Q is 0 then expected value is 2 and when Q is 1 then expected value is 0. And bothare provide pure strategy nash equilibrium.1 3) P = 3 In this case P is equal to 1/3, so we put P=1/3 in expected value of firm 2 then:E F = -3PQ-4P+Q+6 ? ? 1 4 = -Q- +Q+63 14 =3 14 Here expected value isirrespective the value of Q. 3 1 5 So, we conclude that there are two mixed strategy nash equilibrium P ? and Q ? .3 7 For getting mixed strategy nash equilibrium, the expected values should come incombination of both players probability like we get in our question -3PQ - 4P + Q + 6 and7PQ-5P-6Q+6 for both firms. "