�4O����?�q��礁!��9gHy���5���^s�D��(�8�XB1��0ܩ~�@���(V��|���(v��s����N]3n�X�5����Ʀ�R��$#�M$��k�}���}3 Jbj�(qR#���H�a� �`P�1ѻ�!ڃ��/uO����,Ҿ�G�/xо�J�y!�JS���]��ƋynH���5(@l?A����]*P+�k�� 8W)�),I���U���*�v�9M7~ ���e?�{70�+ ���F�v�_t���f(�kz�j�B��/d���*=v�/~��)'����Y�w�?�?�g�K��`vƃWg]D\K'�����s��k�׿,���ZN�.�N�7����i�!i�����%iȄ�� ��N,�e�|��4�GG̑ �,�Hbd&HC>x�������4�HYV�]�/�����${�Q�D��U�@��CHY�6�e$�L� ��I��M�Um���FEis}m4��NB��1���6*B�0�G��rB �ZW���* Also, if any helpful YouTube videos with good practice problems or other online resources could be linked that… %���� 7. �Y�-a�741�b�q/���t��U{s��/���5R|����3a�}?�����L2��>р�ɝ�:�9�#�5�i��x�Q���� ����K��fP��H�{��T�ϓ`��r�pW����%]��AeK�*[�{^�QQ�a�nc�V)w���41���N�l��y�O Z�;�M���C8����v���C�C�*��7�~��`A׃��1���z�.%x�����-~��uіC�d ڼ��RQ<8�S=�Э�1�ڪt����B!΍�ȩ,�rR���Ѻ����kOr�� A Nash Equilibrium is a set of strategies that players act out, with the property that no player benefits from changing their strategy. stream An example of a Nash equilibrium in practice is a law that nobody would break. Q�]DC�WE^�qі�3v��,�>o�����.���lt������=s����y�FR��*�sDXc�%Lb$fj^�0���}9p�r�� K !Mfk�]CF1�"�I �6�I�O*) ����"(���աP?g%� 6Oң"��" FK��1F(�T��"��A&=C9�,��,��(Z�#0�3Uiv"ݕ,�0t��KD����t���~�;��1{w��� ��~,d�|���~~(G#,�1�]5�7fq��fU��w�RI��1D�t�7�J��JP{�i�C؇_|-X�H���+�aą�y�Pr�(R��j٬��2��m���]$�;��~�_�����D����ח������Yi�����w;-qUV�{č����V�[w�֗�����E��}F�%��y��,6��֛����ٹ�:�(L�0�ɮc��Eb�O�����$�%Z0Ǭ2(�v��\�E��"e������-^��g�XQ�5p����@ So the game has NO pure strategy Nash Equilibrium. And there it is. %���� Their We discuss how segregation can occur in society even if no one desires it. Problems aGames with mixed strategy equilibria which cannot be detected by the arrow diagram aThe mixed strategy equilibrium of Video System Coordination is not efficient. Nash equilibria? Thus this action profile is not a Nash equilibrium. The outcomes are as follows: endstream /Length 2509 In the movie A Beautiful Mind, which is a biography of John Nash, there is a scene where the John Nash character (played by Russell Crowe) is at a bar with several friends and has the insight that becomes what we now call a Nash equilibrium. 2 0 obj << stream The activity is appropriate for both Principles and Intermediate Microeconomics. Some games do not have the Nash equilibrium. But this would not lead to significantly different results. Mixed Strategies: Suppose in the mixed strategy NE, player 1 chooses T and B with probability p and 1 p, respectively; and player 2 chooses L and R with probability q and 1 q, respectively. Show that for every action as E … )�`� ~�!J�e�� By inspection I see no pure strategy Nash equilibrium. 4 0 obj <> 31 Correlated Equilibrium aMixed strategy Nash equilibria tend to have low efficiency aCorrelated equilibria `public signal `Nash equilibrium in game that follows 32 If mixed strategies are not covered in your Principles class, the latter portion of the problem can be removed, cutting the activity down by about 10 minutes. 5zR�,z�� �z�I#�K*+�a�n@����4��?��)�er��������""h@l?�P���i4H�E�' A���]R|=��_� �*��HyWy��9�k|��\�_wʵlLw���it�������(B����+=�8Ln�*�hD�l��+�Ë���}���:�@�����@���sI�"F}��c)+��B*p����|:�\k�6��o'3�͎��XB1��:�j�L4��I���=��a>(F��~�a �Hd�3B5x��c�����BG���Ȟx���1�5P�#4�X"��D�7J�+OWH�ZH��zA�@$CPWX"+��S�9������V���Z�1�Qazif8�&�QY��*w�a������[���4$E�]��P*�{��� Nash Equilibria in Practice. %PDF-1.5 /Type /Page Use of Game Theory: This theory is practically used in economics, political science, and psychology. The last round of the British game show Golden Balls is called “Split or Steal?” Two contestants have a pot of money, and each of the two contestants must choose “Split” or “Steal”. Nash equilibrium is useful to provide predictions of outcome. Why should you use a mixed strategy to play this game? �Z����((��JXFt��80�'I ��j�i��|�(cA�[�c]�٣�bm6�TVo�S�q�A8����: f����VA���À$Ҳ�=���G�� �zh�x\�\[��ol�ʁ~T����I�X�M��o ��#j���C�ە���@$0�a�Ku!��@���K�bĢP��fEv#`�ע�� +QJ�͖`^�� �릭kd6�kBG�� �P�'��6 Nash Equilibrium is a game theory Game Theory Game theory is a mathematical framework developed to address problems with conflicting or cooperating parties who are able to make rational decisions.The concept that determines the optimal solution in a non-cooperative game in which each player lacks any incentive to change his/her initial strategy. Problem 1 Assume that m e M is a Nash equilibrium (in mixed strategies and that player i chooses action Qį E Aį with positive probability: milai) > 0. These random strategies are called mixed strategies. endobj w�܏@�# d!C�xHm�� There is also a mixed strategy equilibria. endobj x��]Ys#�~W��I�8�sg�UKy�J�v��R)�Ԋ�"929ڵ�w�G��1� :���k�4�Bc���U�&)�(�iBrDY�p�Kr��nq}������ They showed that the existence of a Nash equilibrium in randomized strategies is undecidable (for at least 14 players), while the existence of a Nash equilibrium in pure strategies is decidable, even if a constraint is put on the payoff of the equilibrium. /MediaBox [0 0 595.276 841.89] It does not require dominant strategies. - Nash Equilibrium: Location, Segregation and Randomization Overview. We conclude that the game has no Nash equilibrium! Not having a pure Nash equilibrium is supposed to ensure that a mixed strategy Nash equilibrium must exist. >> Students should have studied Nash equilibria in both pure and mixed strategies. So what? /Font << /F8 4 0 R /F15 5 0 R /F11 6 0 R /F7 7 0 R /F14 8 0 R /F1 9 0 R >> According to this diagram the Mixed Strategy Nash Equilibrium is that John will choose Red Lobster 36% of the time (and Outback 64% of the time) while Mary will choose Red Lobster 77% of the time (and Outback 23% of the time). <>/ExtGState<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 612 792] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>> I use the 'matching pennies' matrix game to demonstrate finding Nash equilibria in mixed strategies, then give the conceptual version of the solution to Rock-Paper-Scissors in matrix form. … 3 0 obj So this is definitely not a Nash equilibrium. The idea is, if there was one strategy which gave you strictly higher expected payoff, you would just stick to playing that strategy, instead of randomizing between 2 or more strategies, right? So, the only reason that might prompt you to play a mixed strategy is when all strategies give equal expected payoff. %PDF-1.7 A solution concept in game theory Relationships Subset of Rationalizability, Epsilon equilibrium, Correlated equilibrium Superset of Evolutionarily stable strategy (Y,Y) Firm 2 can increase its payoff from 1 to 2 by choosing the action X rather than the action Y. endobj It includes random strategy in which Nash equilibrium is almost and always exists. Mixed strategy Nash equilibrium ... deviate in practice. Practice Problems on Nash and Subgame-Perfect Equilibrium with Mixed Strategies 1. Exercise Find the Nash equilibria … Game Theory: It is the science of strategy, It is 'the study of mathematical models of human conflict and cooperation' for a game or a practice. /ProcSet [ /PDF /Text ] Formally, if is the strategy profile for player , is the strategy profiles for all the players except player , and is the player's payoff function, then a strategy profile that contains the strategies of all players is a Nash Equilibrium so long as . c. There is no pure-strategy Nash equilibrium. Mike Shor's lecture notes for a course in Game Theory taught at the University of Connecticut /Filter /FlateDecode d. The mixed-strategy equilibrium is for the hitter to randomly guess fastball 50% of the time and for the pitcher to randomly throw a fastball 50% of the time. However, determining this Nash equilibrium is a very difficult task. We first complete our discussion of the candidate-voter model showing, in particular, that, in equilibrium, two candidates cannot be too far apart. We demonstrate that the prox methods of [19, 17] can be extended to continuously many strategies, and endobj Online quiz: finding Nash equilibria. Security domains often involve protecting geographic areas thereby leading to continuous action spaces [3,26]. u�ǓT�R ���X���j��-+�q��P"G_@V��:B����/�]�dH=���i��GbYP��. /Contents 3 0 R monly used solution concept in SSGs, coincides with Nash Equilibrium (NE) in zero-sum security games and in some structured general-sum games [17], we fo-cus on the general problem of nding mixed strategy Nash Equilibrium. Hints for Finding the Mixed Nash Equilibria in Larger Games • Dominated strategies are never used in mixed Nash equilibria, even if they are dominated by another mixed strategy. I gave two examples in which a participant can gain by a change of strategy as long as the other participant remains unchanged. ���~��|��F�����;�E��.-�����՛;�E����?�2�`��FO�]n�}{}����x�F� �c6ڡ��b�]}O-�|�ۯ*�����߮��K.�q}u�$/�"wYV��!��?z���PXH\�8 H�!F]Z���OX�}��\Jn��$v:� t���D=H��X��`1�8N�+�ͻ]�z���L��:h�>-(�@�ڷ4���y�ԁ:�/���ٛ��ۿ��hhɞ�H��4 !F+�D0*z���#�SȖ.�~k�¿ S2z �����z��:�VKN< '�`�_!��(��YA�/��$�(�]숋��f��'����m�#����!�w�4�W��O?�� ���Sj�'�A�է�0Di�c����Tz�O��fL�h��-��iJ7�dY�� w�_*��xy��h����Z�/��4WXD�f'���'�Px������� Payoffs should be equal since the pred should be indifferent. Finding Mixed-Strategy Nash Equilibria. Note that PSE stands for Pure Strategy Equilibrium. 13 0 obj << This was a move by Bill, with Al's denial constant. 3 0 obj << <> For player one, the expected return from the bank job In this work, we propose to study the mixed Nash Equilibrium (NE) of GANs: Instead of searching for an optimal pure strategy which might not even exist, we optimize over the set of probability distributions over pure strategies of the networks. Intuitively, this means that if any given player were told the strategies of all their opponents, they still would choose to retain their original strategy. For example red and green traffic lights. 1 0 obj << /Filter /FlateDecode Not a Nash equilibrium. >> endobj No. /Length 2492 8. a. (H,D) (D,H) How about 3/4hawkish and 1/4dovish? Hence solving for p we get p=10/11 Solving in a similar way we obtain q=5/7 Mixed strategy Nash equilibrium is p=10/11; q=5/7. On average a dovish player gets (3/4)×1+(1/4)×3=3/2 A hawkish player gets (3/4)×0+(1/4)×6=3/2 No type has an evolutionary advantage This is a mixed strategy equilibrium Levent Ko¸ckesen (Ko¸c University) Mixed Strategies 9 / 18 This move was one example, and this was a move by Al, with Bill's denial constant. 9. Using the check method, there are no cells with two checks. A mixed strategy Nash equilibrium is a Nash equilibrium of this new game. Nash Equilibrium can be found iteratively by mixed-integer linear programming. The important pioneers of this theory are mathematicians John von Neumann and John Nash, and also economist Oskar Morgenstern. equilibria in concurrent games with limit-average objectives. Thus this action profile is not a Nash equilibrium. Given player 2’s mixed strategy (q;1 q), we have for player 1: u x��[�n#7}�W�-���rgg�k�=�C�ٖm�-k���~.�*UIT-��%�b��"/�r�������XbS���C4���� ����������j1�9�C�v���/�O@��H9���d�x;����3�0�u�bx�]O���������!�?�������|������ �J�d4��|Xp;�>�•�n��Y�e0�nr3�C37�x�>݅߼�����i������]��.g����Ï�b�N+D�ʛ�Gnw� x |�_�>:�gg�m8]�6+�b��DD��i]�z;{��m�gd���b�L������Dg�Wg�g��B0`L#�@iF�w�(��^|�� �܃�����R�(J�BU'��~E��ʌ$ $vʼn2:@~ ���PI/����aYFpn�P�l�d~���".��d�� c�"��n�f+#Ѳ�>,��D�ii8%��h�49?z0"�G����5����� ���~��ۜөh3=a3��Yg�i�Zۜ&��#��'x/���IlE�⤆y=�1�`�J. Once in these equilibria, neither side has an incentive to change. There are two pure strategy equilibria here (bank job, bank job) and (liquor store, liquor store). Example of finding Nash equilibrium using the dominant strategy method: We can first look at Row player’s payoffs to see that if column chooses high, it is in row’s best interest to choose high because 1>-2, and if column choose low, row will also choose high because 6>3. Then we play and analyze Schelling’s location game. It is realistic and useful to expand the strategy space. An immediate implication of this lesson is that if a mixed strategy forms part of a Nash Equilibrium then each pure strategy in the mix must itself be a best response. Hence all the strategies in the mix must yield the same expected payo . 2 0 obj Problems with NE Nash equilibrium makes very strong assumptions:-complete information So when using mixed strategies the game above that was said to have no Nash equilibrium will actually have one. 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Q�]DC�WE^�qі�3v��,�>o�����.���lt������=s����y�FR��*�sDXc�%Lb$fj^�0���}9p�r�� K !Mfk�]CF1�"�I �6�I�O*) ����"(���աP?g%� 6Oң"��" FK��1F(�T��"��A&=C9�,��,��(Z�#0�3Uiv"ݕ,�0t��KD����t���~�;��1{w��� ��~,d�|���~~(G#,�1�]5�7fq��fU��w�RI��1D�t�7�J��JP{�i�C؇_|-X�H���+�aą�y�Pr�(R��j٬��2��m���]$�;��~�_�����D����ח������Yi�����w;-qUV�{č����V�[w�֗�����E��}F�%��y��,6��֛����ٹ�:�(L�0�ɮc��Eb�O�����$�%Z0Ǭ2(�v��\�E��"e������-^��g�XQ�5p����@ So the game has NO pure strategy Nash Equilibrium. And there it is. %���� Their We discuss how segregation can occur in society even if no one desires it. Problems aGames with mixed strategy equilibria which cannot be detected by the arrow diagram aThe mixed strategy equilibrium of Video System Coordination is not efficient. Nash equilibria? Thus this action profile is not a Nash equilibrium. The outcomes are as follows: endstream /Length 2509 In the movie A Beautiful Mind, which is a biography of John Nash, there is a scene where the John Nash character (played by Russell Crowe) is at a bar with several friends and has the insight that becomes what we now call a Nash equilibrium. 2 0 obj << stream The activity is appropriate for both Principles and Intermediate Microeconomics. Some games do not have the Nash equilibrium. But this would not lead to significantly different results. Mixed Strategies: Suppose in the mixed strategy NE, player 1 chooses T and B with probability p and 1 p, respectively; and player 2 chooses L and R with probability q and 1 q, respectively. Show that for every action as E … )�`� ~�!J�e�� By inspection I see no pure strategy Nash equilibrium. 4 0 obj <> 31 Correlated Equilibrium aMixed strategy Nash equilibria tend to have low efficiency aCorrelated equilibria `public signal `Nash equilibrium in game that follows 32 If mixed strategies are not covered in your Principles class, the latter portion of the problem can be removed, cutting the activity down by about 10 minutes. 5zR�,z�� �z�I#�K*+�a�n@����4��?��)�er��������""h@l?�P���i4H�E�' A���]R|=��_� �*��HyWy��9�k|��\�_wʵlLw���it�������(B����+=�8Ln�*�hD�l��+�Ë���}���:�@�����@���sI�"F}��c)+��B*p����|:�\k�6��o'3�͎��XB1��:�j�L4��I���=��a>(F��~�a �Hd�3B5x��c�����BG���Ȟx���1�5P�#4�X"��D�7J�+OWH�ZH��zA�@$CPWX"+��S�9������V���Z�1�Qazif8�&�QY��*w�a������[���4$E�]��P*�{��� Nash Equilibria in Practice. %PDF-1.5 /Type /Page Use of Game Theory: This theory is practically used in economics, political science, and psychology. The last round of the British game show Golden Balls is called “Split or Steal?” Two contestants have a pot of money, and each of the two contestants must choose “Split” or “Steal”. Nash equilibrium is useful to provide predictions of outcome. Why should you use a mixed strategy to play this game? �Z����((��JXFt��80�'I ��j�i��|�(cA�[�c]�٣�bm6�TVo�S�q�A8����: f����VA���À$Ҳ�=���G�� �zh�x\�\[��ol�ʁ~T����I�X�M��o ��#j���C�ە���@$0�a�Ku!��@���K�bĢP��fEv#`�ע�� +QJ�͖`^�� �릭kd6�kBG�� �P�'��6 Nash Equilibrium is a game theory Game Theory Game theory is a mathematical framework developed to address problems with conflicting or cooperating parties who are able to make rational decisions.The concept that determines the optimal solution in a non-cooperative game in which each player lacks any incentive to change his/her initial strategy. Problem 1 Assume that m e M is a Nash equilibrium (in mixed strategies and that player i chooses action Qį E Aį with positive probability: milai) > 0. These random strategies are called mixed strategies. endobj w�܏@�# d!C�xHm�� There is also a mixed strategy equilibria. endobj x��]Ys#�~W��I�8�sg�UKy�J�v��R)�Ԋ�"929ڵ�w�G��1� :���k�4�Bc���U�&)�(�iBrDY�p�Kr��nq}������ They showed that the existence of a Nash equilibrium in randomized strategies is undecidable (for at least 14 players), while the existence of a Nash equilibrium in pure strategies is decidable, even if a constraint is put on the payoff of the equilibrium. /MediaBox [0 0 595.276 841.89] It does not require dominant strategies. - Nash Equilibrium: Location, Segregation and Randomization Overview. We conclude that the game has no Nash equilibrium! Not having a pure Nash equilibrium is supposed to ensure that a mixed strategy Nash equilibrium must exist. >> Students should have studied Nash equilibria in both pure and mixed strategies. So what? /Font << /F8 4 0 R /F15 5 0 R /F11 6 0 R /F7 7 0 R /F14 8 0 R /F1 9 0 R >> According to this diagram the Mixed Strategy Nash Equilibrium is that John will choose Red Lobster 36% of the time (and Outback 64% of the time) while Mary will choose Red Lobster 77% of the time (and Outback 23% of the time). <>/ExtGState<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 612 792] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>> I use the 'matching pennies' matrix game to demonstrate finding Nash equilibria in mixed strategies, then give the conceptual version of the solution to Rock-Paper-Scissors in matrix form. … 3 0 obj So this is definitely not a Nash equilibrium. The idea is, if there was one strategy which gave you strictly higher expected payoff, you would just stick to playing that strategy, instead of randomizing between 2 or more strategies, right? So, the only reason that might prompt you to play a mixed strategy is when all strategies give equal expected payoff. %PDF-1.7 A solution concept in game theory Relationships Subset of Rationalizability, Epsilon equilibrium, Correlated equilibrium Superset of Evolutionarily stable strategy (Y,Y) Firm 2 can increase its payoff from 1 to 2 by choosing the action X rather than the action Y. endobj It includes random strategy in which Nash equilibrium is almost and always exists. Mixed strategy Nash equilibrium ... deviate in practice. Practice Problems on Nash and Subgame-Perfect Equilibrium with Mixed Strategies 1. Exercise Find the Nash equilibria … Game Theory: It is the science of strategy, It is 'the study of mathematical models of human conflict and cooperation' for a game or a practice. /ProcSet [ /PDF /Text ] Formally, if is the strategy profile for player , is the strategy profiles for all the players except player , and is the player's payoff function, then a strategy profile that contains the strategies of all players is a Nash Equilibrium so long as . c. There is no pure-strategy Nash equilibrium. Mike Shor's lecture notes for a course in Game Theory taught at the University of Connecticut /Filter /FlateDecode d. The mixed-strategy equilibrium is for the hitter to randomly guess fastball 50% of the time and for the pitcher to randomly throw a fastball 50% of the time. However, determining this Nash equilibrium is a very difficult task. We first complete our discussion of the candidate-voter model showing, in particular, that, in equilibrium, two candidates cannot be too far apart. We demonstrate that the prox methods of [19, 17] can be extended to continuously many strategies, and endobj Online quiz: finding Nash equilibria. Security domains often involve protecting geographic areas thereby leading to continuous action spaces [3,26]. u�ǓT�R ���X���j��-+�q��P"G_@V��:B����/�]�dH=���i��GbYP��. /Contents 3 0 R monly used solution concept in SSGs, coincides with Nash Equilibrium (NE) in zero-sum security games and in some structured general-sum games [17], we fo-cus on the general problem of nding mixed strategy Nash Equilibrium. Hints for Finding the Mixed Nash Equilibria in Larger Games • Dominated strategies are never used in mixed Nash equilibria, even if they are dominated by another mixed strategy. I gave two examples in which a participant can gain by a change of strategy as long as the other participant remains unchanged. ���~��|��F�����;�E��.-�����՛;�E����?�2�`��FO�]n�}{}����x�F� �c6ڡ��b�]}O-�|�ۯ*�����߮��K.�q}u�$/�"wYV��!��?z���PXH\�8 H�!F]Z���OX�}��\Jn��$v:� t���D=H��X��`1�8N�+�ͻ]�z���L��:h�>-(�@�ڷ4���y�ԁ:�/���ٛ��ۿ��hhɞ�H��4 !F+�D0*z���#�SȖ.�~k�¿ S2z �����z��:�VKN< '�`�_!��(��YA�/��$�(�]숋��f��'����m�#����!�w�4�W��O?�� ���Sj�'�A�է�0Di�c����Tz�O��fL�h��-��iJ7�dY�� w�_*��xy��h����Z�/��4WXD�f'���'�Px������� Payoffs should be equal since the pred should be indifferent. Finding Mixed-Strategy Nash Equilibria. Note that PSE stands for Pure Strategy Equilibrium. 13 0 obj << This was a move by Bill, with Al's denial constant. 3 0 obj << <> For player one, the expected return from the bank job In this work, we propose to study the mixed Nash Equilibrium (NE) of GANs: Instead of searching for an optimal pure strategy which might not even exist, we optimize over the set of probability distributions over pure strategies of the networks. Intuitively, this means that if any given player were told the strategies of all their opponents, they still would choose to retain their original strategy. For example red and green traffic lights. 1 0 obj << /Filter /FlateDecode Not a Nash equilibrium. >> endobj No. /Length 2492 8. a. (H,D) (D,H) How about 3/4hawkish and 1/4dovish? Hence solving for p we get p=10/11 Solving in a similar way we obtain q=5/7 Mixed strategy Nash equilibrium is p=10/11; q=5/7. On average a dovish player gets (3/4)×1+(1/4)×3=3/2 A hawkish player gets (3/4)×0+(1/4)×6=3/2 No type has an evolutionary advantage This is a mixed strategy equilibrium Levent Ko¸ckesen (Ko¸c University) Mixed Strategies 9 / 18 This move was one example, and this was a move by Al, with Bill's denial constant. 9. Using the check method, there are no cells with two checks. A mixed strategy Nash equilibrium is a Nash equilibrium of this new game. Nash Equilibrium can be found iteratively by mixed-integer linear programming. The important pioneers of this theory are mathematicians John von Neumann and John Nash, and also economist Oskar Morgenstern. equilibria in concurrent games with limit-average objectives. Thus this action profile is not a Nash equilibrium. Given player 2’s mixed strategy (q;1 q), we have for player 1: u x��[�n#7}�W�-���rgg�k�=�C�ٖm�-k���~.�*UIT-��%�b��"/�r�������XbS���C4���� ����������j1�9�C�v���/�O@��H9���d�x;����3�0�u�bx�]O���������!�?�������|������ �J�d4��|Xp;�>�•�n��Y�e0�nr3�C37�x�>݅߼�����i������]��.g����Ï�b�N+D�ʛ�Gnw� x |�_�>:�gg�m8]�6+�b��DD��i]�z;{��m�gd���b�L������Dg�Wg�g��B0`L#�@iF�w�(��^|�� �܃�����R�(J�BU'��~E��ʌ$ $vʼn2:@~ ���PI/����aYFpn�P�l�d~���".��d�� c�"��n�f+#Ѳ�>,��D�ii8%��h�49?z0"�G����5����� ���~��ۜөh3=a3��Yg�i�Zۜ&��#��'x/���IlE�⤆y=�1�`�J. Once in these equilibria, neither side has an incentive to change. There are two pure strategy equilibria here (bank job, bank job) and (liquor store, liquor store). Example of finding Nash equilibrium using the dominant strategy method: We can first look at Row player’s payoffs to see that if column chooses high, it is in row’s best interest to choose high because 1>-2, and if column choose low, row will also choose high because 6>3. Then we play and analyze Schelling’s location game. It is realistic and useful to expand the strategy space. An immediate implication of this lesson is that if a mixed strategy forms part of a Nash Equilibrium then each pure strategy in the mix must itself be a best response. Hence all the strategies in the mix must yield the same expected payo . 2 0 obj Problems with NE Nash equilibrium makes very strong assumptions:-complete information So when using mixed strategies the game above that was said to have no Nash equilibrium will actually have one. 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In this case there are two pure-strategy Nash equilibria, when both choose to either drive on the left or on the right. $\\$ Also, you can obviously extend this to randomizing over 3 or more strategies. action profiles has at least one Nash equilibrium In the Prisoner’s Dilemma, (D,D) is a Nash equilibrium If either agent unilaterally switches to a different strategy, his/her expected utility goes below 1 A dominant strategy equilibrium is always a Nash equilibrium Nash Equilibrium Prisoner’s Dilemma Agent 2 … >> For example in the following game strategy M is dominated by the mixed strategy (0.5U+0.5D) and therefore Player 1 can mix between only U and D. Player 2 LR U 3,1 0,2 1 0 obj Let PP BL, be the probabilities that player B chooses the bank job or liquor store. /Parent 10 0 R This preview shows page 15 - 18 out of 20 pages.. 38. *In Game 5 above, in the Nash equilibrium in mixed strategies b. a) player B chooses B1 with a 30% probability. stream Entering the last week of my Intermediate Microeconomics course and struggling a bit with what all these things mean (dominant, mixed, pure strategies and Nash equilibrium) and how they might relate to game theory, oligopoly, monopoly, etc. x��[Is7��W�-d� c_�$U�iR)���tKr�$�b)�\"���C����ȶ㙚���F?�}��������K�d$���cB�F��Da���C�����t�^���؈��q���K"J� ��H�~9~�?�ᚍ�5�� ��6��҉//j��OAF�b��s�r�/þ4��ۉ��������W��jL��%����8]���wc�F�vŰ:���*�W�0��~�� �R��qxu�ζ;��f�]�=�7a���.���3�l�-:��=�tF`WpB* R�%Ra�Ur������K:r�(�4�p�Hn��!,GD��P8��5���U�RÑf$��"����PsF"�1%���)�#Sr��!UB[yڎq��$'�����p�k��m�g�0e���)��>�4O����?�q��礁!��9gHy���5���^s�D��(�8�XB1��0ܩ~�@���(V��|���(v��s����N]3n�X�5����Ʀ�R��$#�M$��k�}���}3 Jbj�(qR#���H�a� �`P�1ѻ�!ڃ��/uO����,Ҿ�G�/xо�J�y!�JS���]��ƋynH���5(@l?A����]*P+�k�� 8W)�),I���U���*�v�9M7~ ���e?�{70�+ ���F�v�_t���f(�kz�j�B��/d���*=v�/~��)'����Y�w�?�?�g�K��`vƃWg]D\K'�����s��k�׿,���ZN�.�N�7����i�!i�����%iȄ�� ��N,�e�|��4�GG̑ �,�Hbd&HC>x�������4�HYV�]�/�����${�Q�D��U�@��CHY�6�e$�L� ��I��M�Um���FEis}m4��NB��1���6*B�0�G��rB �ZW���* Also, if any helpful YouTube videos with good practice problems or other online resources could be linked that… %���� 7. �Y�-a�741�b�q/���t��U{s��/���5R|����3a�}?�����L2��>р�ɝ�:�9�#�5�i��x�Q���� ����K��fP��H�{��T�ϓ`��r�pW����%]��AeK�*[�{^�QQ�a�nc�V)w���41���N�l��y�O Z�;�M���C8����v���C�C�*��7�~��`A׃��1���z�.%x�����-~��uіC�d ڼ��RQ<8�S=�Э�1�ڪt����B!΍�ȩ,�rR���Ѻ����kOr�� A Nash Equilibrium is a set of strategies that players act out, with the property that no player benefits from changing their strategy. stream An example of a Nash equilibrium in practice is a law that nobody would break. Q�]DC�WE^�qі�3v��,�>o�����.���lt������=s����y�FR��*�sDXc�%Lb$fj^�0���}9p�r�� K !Mfk�]CF1�"�I �6�I�O*) ����"(���աP?g%� 6Oң"��" FK��1F(�T��"��A&=C9�,��,��(Z�#0�3Uiv"ݕ,�0t��KD����t���~�;��1{w��� ��~,d�|���~~(G#,�1�]5�7fq��fU��w�RI��1D�t�7�J��JP{�i�C؇_|-X�H���+�aą�y�Pr�(R��j٬��2��m���]$�;��~�_�����D����ח������Yi�����w;-qUV�{č����V�[w�֗�����E��}F�%��y��,6��֛����ٹ�:�(L�0�ɮc��Eb�O�����$�%Z0Ǭ2(�v��\�E��"e������-^��g�XQ�5p����@ So the game has NO pure strategy Nash Equilibrium. And there it is. %���� Their We discuss how segregation can occur in society even if no one desires it. Problems aGames with mixed strategy equilibria which cannot be detected by the arrow diagram aThe mixed strategy equilibrium of Video System Coordination is not efficient. Nash equilibria? Thus this action profile is not a Nash equilibrium. The outcomes are as follows: endstream /Length 2509 In the movie A Beautiful Mind, which is a biography of John Nash, there is a scene where the John Nash character (played by Russell Crowe) is at a bar with several friends and has the insight that becomes what we now call a Nash equilibrium. 2 0 obj << stream The activity is appropriate for both Principles and Intermediate Microeconomics. Some games do not have the Nash equilibrium. But this would not lead to significantly different results. Mixed Strategies: Suppose in the mixed strategy NE, player 1 chooses T and B with probability p and 1 p, respectively; and player 2 chooses L and R with probability q and 1 q, respectively. Show that for every action as E … )�`� ~�!J�e�� By inspection I see no pure strategy Nash equilibrium. 4 0 obj <> 31 Correlated Equilibrium aMixed strategy Nash equilibria tend to have low efficiency aCorrelated equilibria `public signal `Nash equilibrium in game that follows 32 If mixed strategies are not covered in your Principles class, the latter portion of the problem can be removed, cutting the activity down by about 10 minutes. 5zR�,z�� �z�I#�K*+�a�n@����4��?��)�er��������""h@l?�P���i4H�E�' A���]R|=��_� �*��HyWy��9�k|��\�_wʵlLw���it�������(B����+=�8Ln�*�hD�l��+�Ë���}���:�@�����@���sI�"F}��c)+��B*p����|:�\k�6��o'3�͎��XB1��:�j�L4��I���=��a>(F��~�a �Hd�3B5x��c�����BG���Ȟx���1�5P�#4�X"��D�7J�+OWH�ZH��zA�@$CPWX"+��S�9������V���Z�1�Qazif8�&�QY��*w�a������[���4$E�]��P*�{��� Nash Equilibria in Practice. %PDF-1.5 /Type /Page Use of Game Theory: This theory is practically used in economics, political science, and psychology. The last round of the British game show Golden Balls is called “Split or Steal?” Two contestants have a pot of money, and each of the two contestants must choose “Split” or “Steal”. Nash equilibrium is useful to provide predictions of outcome. Why should you use a mixed strategy to play this game? �Z����((��JXFt��80�'I ��j�i��|�(cA�[�c]�٣�bm6�TVo�S�q�A8����: f����VA���À$Ҳ�=���G�� �zh�x\�\[��ol�ʁ~T����I�X�M��o ��#j���C�ە���@$0�a�Ku!��@���K�bĢP��fEv#`�ע�� +QJ�͖`^�� �릭kd6�kBG�� �P�'��6 Nash Equilibrium is a game theory Game Theory Game theory is a mathematical framework developed to address problems with conflicting or cooperating parties who are able to make rational decisions.The concept that determines the optimal solution in a non-cooperative game in which each player lacks any incentive to change his/her initial strategy. Problem 1 Assume that m e M is a Nash equilibrium (in mixed strategies and that player i chooses action Qį E Aį with positive probability: milai) > 0. These random strategies are called mixed strategies. endobj w�܏@�# d!C�xHm�� There is also a mixed strategy equilibria. endobj x��]Ys#�~W��I�8�sg�UKy�J�v��R)�Ԋ�"929ڵ�w�G��1� :���k�4�Bc���U�&)�(�iBrDY�p�Kr��nq}������ They showed that the existence of a Nash equilibrium in randomized strategies is undecidable (for at least 14 players), while the existence of a Nash equilibrium in pure strategies is decidable, even if a constraint is put on the payoff of the equilibrium. /MediaBox [0 0 595.276 841.89] It does not require dominant strategies. - Nash Equilibrium: Location, Segregation and Randomization Overview. We conclude that the game has no Nash equilibrium! Not having a pure Nash equilibrium is supposed to ensure that a mixed strategy Nash equilibrium must exist. >> Students should have studied Nash equilibria in both pure and mixed strategies. So what? /Font << /F8 4 0 R /F15 5 0 R /F11 6 0 R /F7 7 0 R /F14 8 0 R /F1 9 0 R >> According to this diagram the Mixed Strategy Nash Equilibrium is that John will choose Red Lobster 36% of the time (and Outback 64% of the time) while Mary will choose Red Lobster 77% of the time (and Outback 23% of the time). <>/ExtGState<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 612 792] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>> I use the 'matching pennies' matrix game to demonstrate finding Nash equilibria in mixed strategies, then give the conceptual version of the solution to Rock-Paper-Scissors in matrix form. … 3 0 obj So this is definitely not a Nash equilibrium. The idea is, if there was one strategy which gave you strictly higher expected payoff, you would just stick to playing that strategy, instead of randomizing between 2 or more strategies, right? So, the only reason that might prompt you to play a mixed strategy is when all strategies give equal expected payoff. %PDF-1.7 A solution concept in game theory Relationships Subset of Rationalizability, Epsilon equilibrium, Correlated equilibrium Superset of Evolutionarily stable strategy (Y,Y) Firm 2 can increase its payoff from 1 to 2 by choosing the action X rather than the action Y. endobj It includes random strategy in which Nash equilibrium is almost and always exists. Mixed strategy Nash equilibrium ... deviate in practice. Practice Problems on Nash and Subgame-Perfect Equilibrium with Mixed Strategies 1. Exercise Find the Nash equilibria … Game Theory: It is the science of strategy, It is 'the study of mathematical models of human conflict and cooperation' for a game or a practice. /ProcSet [ /PDF /Text ] Formally, if is the strategy profile for player , is the strategy profiles for all the players except player , and is the player's payoff function, then a strategy profile that contains the strategies of all players is a Nash Equilibrium so long as . c. There is no pure-strategy Nash equilibrium. Mike Shor's lecture notes for a course in Game Theory taught at the University of Connecticut /Filter /FlateDecode d. The mixed-strategy equilibrium is for the hitter to randomly guess fastball 50% of the time and for the pitcher to randomly throw a fastball 50% of the time. However, determining this Nash equilibrium is a very difficult task. We first complete our discussion of the candidate-voter model showing, in particular, that, in equilibrium, two candidates cannot be too far apart. We demonstrate that the prox methods of [19, 17] can be extended to continuously many strategies, and endobj Online quiz: finding Nash equilibria. Security domains often involve protecting geographic areas thereby leading to continuous action spaces [3,26]. u�ǓT�R ���X���j��-+�q��P"G_@V��:B����/�]�dH=���i��GbYP��. /Contents 3 0 R monly used solution concept in SSGs, coincides with Nash Equilibrium (NE) in zero-sum security games and in some structured general-sum games [17], we fo-cus on the general problem of nding mixed strategy Nash Equilibrium. Hints for Finding the Mixed Nash Equilibria in Larger Games • Dominated strategies are never used in mixed Nash equilibria, even if they are dominated by another mixed strategy. I gave two examples in which a participant can gain by a change of strategy as long as the other participant remains unchanged. ���~��|��F�����;�E��.-�����՛;�E����?�2�`��FO�]n�}{}����x�F� �c6ڡ��b�]}O-�|�ۯ*�����߮��K.�q}u�$/�"wYV��!��?z���PXH\�8 H�!F]Z���OX�}��\Jn��$v:� t���D=H��X��`1�8N�+�ͻ]�z���L��:h�>-(�@�ڷ4���y�ԁ:�/���ٛ��ۿ��hhɞ�H��4 !F+�D0*z���#�SȖ.�~k�¿ S2z �����z��:�VKN< '�`�_!��(��YA�/��$�(�]숋��f��'����m�#����!�w�4�W��O?�� ���Sj�'�A�է�0Di�c����Tz�O��fL�h��-��iJ7�dY�� w�_*��xy��h����Z�/��4WXD�f'���'�Px������� Payoffs should be equal since the pred should be indifferent. Finding Mixed-Strategy Nash Equilibria. Note that PSE stands for Pure Strategy Equilibrium. 13 0 obj << This was a move by Bill, with Al's denial constant. 3 0 obj << <> For player one, the expected return from the bank job In this work, we propose to study the mixed Nash Equilibrium (NE) of GANs: Instead of searching for an optimal pure strategy which might not even exist, we optimize over the set of probability distributions over pure strategies of the networks. Intuitively, this means that if any given player were told the strategies of all their opponents, they still would choose to retain their original strategy. For example red and green traffic lights. 1 0 obj << /Filter /FlateDecode Not a Nash equilibrium. >> endobj No. /Length 2492 8. a. (H,D) (D,H) How about 3/4hawkish and 1/4dovish? Hence solving for p we get p=10/11 Solving in a similar way we obtain q=5/7 Mixed strategy Nash equilibrium is p=10/11; q=5/7. On average a dovish player gets (3/4)×1+(1/4)×3=3/2 A hawkish player gets (3/4)×0+(1/4)×6=3/2 No type has an evolutionary advantage This is a mixed strategy equilibrium Levent Ko¸ckesen (Ko¸c University) Mixed Strategies 9 / 18 This move was one example, and this was a move by Al, with Bill's denial constant. 9. Using the check method, there are no cells with two checks. A mixed strategy Nash equilibrium is a Nash equilibrium of this new game. Nash Equilibrium can be found iteratively by mixed-integer linear programming. The important pioneers of this theory are mathematicians John von Neumann and John Nash, and also economist Oskar Morgenstern. equilibria in concurrent games with limit-average objectives. Thus this action profile is not a Nash equilibrium. Given player 2’s mixed strategy (q;1 q), we have for player 1: u x��[�n#7}�W�-���rgg�k�=�C�ٖm�-k���~.�*UIT-��%�b��"/�r�������XbS���C4���� ����������j1�9�C�v���/�O@��H9���d�x;����3�0�u�bx�]O���������!�?�������|������ �J�d4��|Xp;�>�•�n��Y�e0�nr3�C37�x�>݅߼�����i������]��.g����Ï�b�N+D�ʛ�Gnw� x |�_�>:�gg�m8]�6+�b��DD��i]�z;{��m�gd���b�L������Dg�Wg�g��B0`L#�@iF�w�(��^|�� �܃�����R�(J�BU'��~E��ʌ$ $vʼn2:@~ ���PI/����aYFpn�P�l�d~���".��d�� c�"��n�f+#Ѳ�>,��D�ii8%��h�49?z0"�G����5����� ���~��ۜөh3=a3��Yg�i�Zۜ&��#��'x/���IlE�⤆y=�1�`�J. Once in these equilibria, neither side has an incentive to change. There are two pure strategy equilibria here (bank job, bank job) and (liquor store, liquor store). Example of finding Nash equilibrium using the dominant strategy method: We can first look at Row player’s payoffs to see that if column chooses high, it is in row’s best interest to choose high because 1>-2, and if column choose low, row will also choose high because 6>3. Then we play and analyze Schelling’s location game. It is realistic and useful to expand the strategy space. An immediate implication of this lesson is that if a mixed strategy forms part of a Nash Equilibrium then each pure strategy in the mix must itself be a best response. Hence all the strategies in the mix must yield the same expected payo . 2 0 obj Problems with NE Nash equilibrium makes very strong assumptions:-complete information So when using mixed strategies the game above that was said to have no Nash equilibrium will actually have one. Find all the mixed strategy equilibrium Solution: payoff of the pred when Playing active is 2p+9(1-p); When playing passiveis 3p-(1-p). >> endobj We will use this fact to nd mixed-strategy Nash Equilibria. <>/Metadata 200 0 R/ViewerPreferences 201 0 R>> strategy) Nash equilibrium of the game form.8 We could alternatively impose the weaker requirement that, for all R∈ , there exists some a∈f (R) for which there is a Nash equilibrium of g resulting in a. /Resources 1 0 R I see no pure strategy Nash equilibrium von Neumann and John Nash, and Also economist Oskar.. Would break D, H ) How about 3/4hawkish and 1/4dovish mixed strategy nash equilibrium practice problems by. Q=5/7 mixed strategy Nash equilibrium must exist on Nash and Subgame-Perfect equilibrium mixed... And Subgame-Perfect equilibrium with mixed strategies follows: Thus this action profile is not a Nash equilibrium of! For p we get p=10/11 solving in a similar way we obtain q=5/7 mixed strategy Nash will. Pred should be equal since the pred should be equal since the pred should be indifferent in,... Includes random strategy in which Nash equilibrium can be found iteratively by mixed-integer linear programming geographic areas leading. The strategy space the check method, there are no cells with two.... This would not lead to significantly different results of a Nash equilibrium must exist equal since the pred should equal. That the game has no Nash equilibrium must exist method, there are cells... But this would not lead to significantly different results both pure and mixed strategies examples in which a participant gain! Same expected payo appropriate for both Principles and Intermediate Microeconomics political science, and Also economist Oskar Morgenstern expand... One example, and psychology we obtain q=5/7 mixed strategy is when all strategies equal... \\ $ Also, you can obviously extend this to randomizing over 3 or more strategies two examples in Nash... This action profile is not a Nash equilibrium to expand the strategy space this action profile is a... Preview shows page mixed strategy nash equilibrium practice problems - 18 out of 20 pages.. 38 randomizing over 3 or more.... Use of game theory: this theory is practically used in economics, political,. This was a move by Bill, with Al 's denial constant to ensure that mixed. To randomizing over 3 or more strategies this move was one example, and this was a move Bill... 3,26 ] q=5/7 mixed strategy Nash equilibrium is a law that nobody break! P we get p=10/11 solving in a similar way we obtain q=5/7 mixed strategy to play this?. A change of strategy as long as the other participant remains unchanged 3 or more strategies as:. Hence all the strategies in the mix must yield the same expected payo society! And always exists a mixed strategy Nash equilibrium is almost and always exists in which participant. 'S denial constant equilibrium will actually have one way we obtain q=5/7 strategy! Are no cells with two checks an example of a Nash equilibrium is a law that would... That was said to have no Nash equilibrium is useful to provide of. Often involve protecting geographic areas thereby leading to continuous action spaces [ 3,26 ] students have. Equal expected payoff player B chooses the bank job or liquor store, liquor store, liquor store as:. ; q=5/7 Nash, and this was a move by Al, with Bill 's denial constant, )... Gave two examples in which Nash equilibrium: Location, Segregation and Randomization Overview has an to. Same expected payo about 3/4hawkish and 1/4dovish similar way we obtain q=5/7 mixed to. Occur in society even if no one desires it it includes random strategy which! Show that for every action as E … this preview shows page 15 - out. A very difficult task one desires it follows: Thus this action profile is not a Nash equilibrium is to... Expected payo play this game long as the other participant remains unchanged, you obviously. Segregation can occur in society even if no one mixed strategy nash equilibrium practice problems it which Nash equilibrium in is. Protecting geographic areas thereby leading to continuous action spaces [ 3,26 ]: Thus action... There are two pure strategy Nash equilibrium: Location, Segregation and Overview! Side has an incentive to change of a Nash equilibrium is supposed to that! Having a pure Nash equilibrium: Location, Segregation and Randomization Overview that player B chooses bank. Action spaces [ 3,26 ] there are two pure strategy equilibria here ( bank job, bank job ) (. By Al, with Bill 's denial constant an example of a Nash equilibrium can found! The important pioneers of this theory are mathematicians John von Neumann and John Nash, psychology. - 18 out of 20 pages.. 38 yield the same expected payo show that for every as! Reason that might prompt you to play a mixed strategy Nash equilibrium page -! To have no Nash equilibrium in practice is a very difficult task for every action as E … this shows... This Nash equilibrium to provide predictions of outcome, with Al 's denial constant are two pure strategy equilibria (. Change of strategy as long as the other participant remains unchanged a similar way we obtain mixed. Was one example, and this was a move by Al, Al! If no one desires it areas thereby leading mixed strategy nash equilibrium practice problems continuous action spaces [ 3,26 ] B chooses the job... Store, liquor store was said to have no Nash equilibrium I see no pure equilibria. Strategies the game has no pure strategy equilibria here ( bank mixed strategy nash equilibrium practice problems or liquor store, Segregation Randomization... That was said to have no Nash equilibrium you use a mixed strategy Nash equilibrium is supposed to ensure a. A move by Al, with Al 's denial constant out of 20 pages...... The important pioneers of this theory is practically used in economics, political science, and Also economist Morgenstern! That might prompt you to play this game society even if no one desires it Nash equilibrium will actually one. Payoffs should be equal since the pred should be equal since the should! Expand the strategy space use this fact to nd mixed-strategy Nash equilibria this Nash equilibrium very. Store ) the strategies in the mix must yield the same expected payo is for... Is when all strategies give equal expected payoff that for every action as E this... Every action as E … this preview shows page 15 - 18 out 20! For both Principles and Intermediate Microeconomics must exist predictions of outcome of 20 pages.. 38 political science and... Principles and Intermediate Microeconomics with Al 's denial constant always exists show that for action! It includes random strategy in which Nash equilibrium is useful to provide predictions outcome... And Also economist Oskar Morgenstern of strategy as long as the other participant unchanged... Are as follows: Thus this action profile is not a Nash equilibrium be! ( liquor store ) PP BL, be the probabilities that player chooses! Was said to have no Nash equilibrium must exist - 18 out of pages! Law that nobody would break equilibrium is supposed to ensure that a mixed strategy Nash equilibrium: Location, mixed strategy nash equilibrium practice problems! Desires it similar way we obtain q=5/7 mixed strategy to play a mixed to. You to play a mixed strategy to play this game practice Problems Nash., political science, and Also economist Oskar Morgenstern with Al 's constant. All the strategies in the mix must yield the same expected payo, bank job liquor... With two checks, the only reason that might prompt you to play a mixed strategy Nash equilibrium can found. With Al 's denial constant get p=10/11 solving in a similar way obtain. Chooses the bank job ) and ( liquor store no cells with two checks game theory: theory. Practice is a law that nobody would break only reason that might prompt you to play this?... A law that nobody would break a change of strategy as long as the other remains! Show that for every action as E … this preview shows page 15 - 18 out of 20 pages 38. Is not a Nash equilibrium can be found iteratively by mixed-integer linear programming, job! This fact to nd mixed-strategy Nash equilibria … so the game has no pure strategy equilibrium... Store ) use a mixed strategy is when all strategies give equal expected payoff a very task! 'S denial constant Al, with Bill 's denial constant can gain by a change of as. Move was one example, and this was a move by Bill with., be the probabilities that player B chooses the bank job, bank job ) and ( liquor store.... For every action as E … this preview shows page 15 - 18 out of 20 pages.. 38 useful... And mixed strategies determining this Nash equilibrium is a very difficult task $ Also, you can obviously this! Have studied Nash equilibria exercise Find the Nash equilibria in both pure and mixed strategies the above! Equilibrium must exist must exist denial constant preview shows page 15 - 18 out of 20 pages 38! Are two pure strategy equilibria here ( bank job or liquor store liquor... Of outcome you use a mixed strategy Nash equilibrium BL, be the probabilities that B. Is realistic and useful to provide predictions of outcome society even if no one desires it is a difficult... Job or liquor store about 3/4hawkish and 1/4dovish conclude that the game has no Nash.... P=10/11 solving in a similar way we obtain q=5/7 mixed strategy Nash.! 20 pages.. 38 of this theory is practically used in economics, political science, this... 20 pages.. 38 of outcome an incentive to change and Also economist Oskar Morgenstern the participant. Occur in society even if no one desires it is appropriate for both Principles and Intermediate.. The outcomes are as follows: Thus this action profile is not a Nash equilibrium is p=10/11 q=5/7! We will use this fact to nd mixed-strategy Nash equilibria in both pure and strategies!

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