This input file gives examples of procedure PNE and PNEv regarding PURE Nash\ equilibria ---------------------------------------- For Barry's Problem 1(a), the pure equlibria are given by {[1, 1], [2, 2]} a verbose version is Recall that the matrix of our game is [[4, 4] [0, 3]] [ ] [[3, 0] [2, 2]] Player Column has, 2, possible strategies. Let's find Row's best response for each of them Since we are only interested right now in Row's pay-offs, let's list them he\ re [4 0] [ ] [3 2] For Column's strategy Number, 1, the respective pay-offs for each of Row's, 2, strategies are [4, 3] among the numbers in the list, [4, 3], the following places have the largest value, {1} Hence, Row's best response(s) is(are), {1} For Column's strategy Number, 2, the respective pay-offs for each of Row's, 2, strategies are [0, 2] among the numbers in the list, [0, 2], the following places have the largest value, {2} Hence, Row's best response(s) is(are), {2} To sum up, the table of Row's best responses for each of Column's, 2, strategy are, respectively [{1}, {2}] Recall that the matrix of our game is [[4, 4] [0, 3]] [ ] [[3, 0] [2, 2]] Player Row has, 2, possible strategies. Let's find Column's best response for each of them Since we are only interested right now in Column's pay-offs, let's list them\ here [4 3] [ ] [0 2] For Row's strategy Number, 1, the respective pay-offs for each of Column's, 2, strategies are [4, 3] among the numbers in the list, [4, 3], the following places have the largest value, {1} Hence, Column's best response(s) is(are), {1} For Row's strategy Number, 2, the respective pay-offs for each of Column's, 2, strategies are [0, 2] among the numbers in the list, [0, 2], the following places have the largest value, {2} Hence, Column's best response(s) is(are), {2} To sum up, the table of Column's best responses for each of Row's, 2, strategy are, respectively [{1}, {2}] We are considering Strategy Number , 1, of player Row we saw above that the set of Column's best reponses to it , {1} We are considering Strategy Number , 1, of player Column Since strategy, 1, is in the above set there is hope, but now we have to chec\ k the second condition Recall that the set of best responses of Row to Column's strategy, 1, is , {1} Yea!, Row's Strategy, 1, belongs to the set of Best Reponses of Column's Strategy, 1 and vice-versa Yea!, Column's Strategy, 1, belongs to the set of Best Reponses of Row's Strategy, 1 Hence , [1, 1], is a Pure Nash Equilibrium We are considering Strategy Number , 2, of player Column Since strategy, 2, is NOT in the above set, we alredy know that, [1, 2], is NOT a Pure Nash Equilibrium We are considering Strategy Number , 2, of player Row we saw above that the set of Column's best reponses to it , {2} We are considering Strategy Number , 1, of player Column Since strategy, 1, is NOT in the above set, we alredy know that, [2, 1], is NOT a Pure Nash Equilibrium We are considering Strategy Number , 2, of player Column Since strategy, 2, is in the above set there is hope, but now we have to chec\ k the second condition Recall that the set of best responses of Row to Column's strategy, 2, is , {2} Yea!, Row's Strategy, 2, belongs to the set of Best Reponses of Column's Strategy, 2 and vice-versa Yea!, Column's Strategy, 2, belongs to the set of Best Reponses of Row's Strategy, 2 Hence , [2, 2], is a Pure Nash Equilibrium To sum up we found that the set of pure Nash Equilibria for the above game is, {[1, 1], [2, 2]} ---------------------------------------- ---------------------------------------- For Barry's Problem 1(b), the pure equlibria are given by {[1, 2]} a verbose version is Recall that the matrix of our game is [[1, 8] [7, 9]] [ ] [[3, 4] [5, 5]] Player Column has, 2, possible strategies. Let's find Row's best response for each of them Since we are only interested right now in Row's pay-offs, let's list them he\ re [1 7] [ ] [3 5] For Column's strategy Number, 1, the respective pay-offs for each of Row's, 2, strategies are [1, 3] among the numbers in the list, [1, 3], the following places have the largest value, {2} Hence, Row's best response(s) is(are), {2} For Column's strategy Number, 2, the respective pay-offs for each of Row's, 2, strategies are [7, 5] among the numbers in the list, [7, 5], the following places have the largest value, {1} Hence, Row's best response(s) is(are), {1} To sum up, the table of Row's best responses for each of Column's, 2, strategy are, respectively [{2}, {1}] Recall that the matrix of our game is [[1, 8] [7, 9]] [ ] [[3, 4] [5, 5]] Player Row has, 2, possible strategies. Let's find Column's best response for each of them Since we are only interested right now in Column's pay-offs, let's list them\ here [8 9] [ ] [4 5] For Row's strategy Number, 1, the respective pay-offs for each of Column's, 2, strategies are [8, 9] among the numbers in the list, [8, 9], the following places have the largest value, {2} Hence, Column's best response(s) is(are), {2} For Row's strategy Number, 2, the respective pay-offs for each of Column's, 2, strategies are [4, 5] among the numbers in the list, [4, 5], the following places have the largest value, {2} Hence, Column's best response(s) is(are), {2} To sum up, the table of Column's best responses for each of Row's, 2, strategy are, respectively [{2}, {2}] We are considering Strategy Number , 1, of player Row we saw above that the set of Column's best reponses to it , {2} We are considering Strategy Number , 1, of player Column Since strategy, 1, is NOT in the above set, we alredy know that, [1, 1], is NOT a Pure Nash Equilibrium We are considering Strategy Number , 2, of player Column Since strategy, 2, is in the above set there is hope, but now we have to chec\ k the second condition Recall that the set of best responses of Row to Column's strategy, 2, is , {1} Yea!, Row's Strategy, 1, belongs to the set of Best Reponses of Column's Strategy, 2 and vice-versa Yea!, Column's Strategy, 2, belongs to the set of Best Reponses of Row's Strategy, 1 Hence , [1, 2], is a Pure Nash Equilibrium We are considering Strategy Number , 2, of player Row we saw above that the set of Column's best reponses to it , {2} We are considering Strategy Number , 1, of player Column Since strategy, 1, is NOT in the above set, we alredy know that, [2, 1], is NOT a Pure Nash Equilibrium We are considering Strategy Number , 2, of player Column Since strategy, 2, is in the above set there is hope, but now we have to chec\ k the second condition Recall that the set of best responses of Row to Column's strategy, 2, is , {1} Since strategy, 2, is NOT in the above set, we alredy know that, [2, 2], is NOT a Pure Nash Equilibrium To sum up we found that the set of pure Nash Equilibria for the above game is, {[1, 2]} ---------------------------------------- ---------------------------------------- For Barry's Problem 1(c), the pure equlibria are given by {} a verbose version is Recall that the matrix of our game is [[2, 8] [3, 5]] [ ] [[8, 2] [1, 7]] Player Column has, 2, possible strategies. Let's find Row's best response for each of them Since we are only interested right now in Row's pay-offs, let's list them he\ re [2 3] [ ] [8 1] For Column's strategy Number, 1, the respective pay-offs for each of Row's, 2, strategies are [2, 8] among the numbers in the list, [2, 8], the following places have the largest value, {2} Hence, Row's best response(s) is(are), {2} For Column's strategy Number, 2, the respective pay-offs for each of Row's, 2, strategies are [3, 1] among the numbers in the list, [3, 1], the following places have the largest value, {1} Hence, Row's best response(s) is(are), {1} To sum up, the table of Row's best responses for each of Column's, 2, strategy are, respectively [{2}, {1}] Recall that the matrix of our game is [[2, 8] [3, 5]] [ ] [[8, 2] [1, 7]] Player Row has, 2, possible strategies. Let's find Column's best response for each of them Since we are only interested right now in Column's pay-offs, let's list them\ here [8 5] [ ] [2 7] For Row's strategy Number, 1, the respective pay-offs for each of Column's, 2, strategies are [8, 5] among the numbers in the list, [8, 5], the following places have the largest value, {1} Hence, Column's best response(s) is(are), {1} For Row's strategy Number, 2, the respective pay-offs for each of Column's, 2, strategies are [2, 7] among the numbers in the list, [2, 7], the following places have the largest value, {2} Hence, Column's best response(s) is(are), {2} To sum up, the table of Column's best responses for each of Row's, 2, strategy are, respectively [{1}, {2}] We are considering Strategy Number , 1, of player Row we saw above that the set of Column's best reponses to it , {1} We are considering Strategy Number , 1, of player Column Since strategy, 1, is in the above set there is hope, but now we have to chec\ k the second condition Recall that the set of best responses of Row to Column's strategy, 1, is , {2} Since strategy, 1, is NOT in the above set, we alredy know that, [1, 1], is NOT a Pure Nash Equilibrium We are considering Strategy Number , 2, of player Column Since strategy, 2, is NOT in the above set, we alredy know that, [1, 2], is NOT a Pure Nash Equilibrium We are considering Strategy Number , 2, of player Row we saw above that the set of Column's best reponses to it , {2} We are considering Strategy Number , 1, of player Column Since strategy, 1, is NOT in the above set, we alredy know that, [2, 1], is NOT a Pure Nash Equilibrium We are considering Strategy Number , 2, of player Column Since strategy, 2, is in the above set there is hope, but now we have to chec\ k the second condition Recall that the set of best responses of Row to Column's strategy, 2, is , {1} Since strategy, 2, is NOT in the above set, we alredy know that, [2, 2], is NOT a Pure Nash Equilibrium To sum up we found that the set of pure Nash Equilibria for the above game is, {}