mixed strategy nash equilibrium practice problems

/Length 2509 7. 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. 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. 1 0 obj *In Game 5 above, in the Nash equilibrium in mixed strategies b. a) player B chooses B1 with a 30% probability. /Resources 1 0 R Exercise Find the Nash equilibria … 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. Finding Mixed-Strategy Nash Equilibria. �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�� /Contents 3 0 R Nash equilibrium is useful to provide predictions of outcome. 2 0 obj << 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? Why should you use a mixed strategy to play this game? 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”. endstream stream … 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�����$�%Z0Ǭ2(�v��\�E��"e������-^��g�XQ�5p����@ 2 0 obj Jbj�(qR#���H�a� �`P�1ѻ�!ڃ��/uO����,Ҿ�G�/xо�J�y!�JS���]��ƋynH���5(@l?A����]*P+�k�� 8W)�),I���U���*�v�9M7~ ���e?�{70�+ ���F�v�_t���f(�kz�j�B��/d���*=v�/~��)'����Y�w�?�?�g�K��`vƃWg]D\K'�����s��k�׿,���ZN�.�N�7����i�!i�����%iȄ�� ��N,�e�|��4�GG̑ �,�Hbd&HC>x�������4�HYV�]�/�����${�Q�D��U�@��CHY�6�e$�L� ��I��M�Um���FEis}m4��NB��1���6*B�0�G��rB �ZW���* For example red and green traffic lights. Hence all the strategies in the mix must yield the same expected payo . The important pioneers of this theory are mathematicians John von Neumann and John Nash, and also economist Oskar Morgenstern. However, determining this Nash equilibrium is a very difficult task. It is realistic and useful to expand the strategy space. <> stream A Nash Equilibrium is a set of strategies that players act out, with the property that no player benefits from changing their strategy. x��]Ys#�~W��I�8�sg�UKy�J�v��R)�Ԋ�"929ڵ�w�G��1� :���k�4�Bc���U�&)�(�iBrDY�p�Kr��nq}������ (Y,Y) Firm 2 can increase its payoff from 1 to 2 by choosing the action X rather than the action Y. So, the only reason that might prompt you to play a mixed strategy is when all strategies give equal expected payoff. Using the check method, there are no cells with two checks. There is also a mixed strategy equilibria. u�ǓT�R ���X���j��-+�q��P"G_@V��:B����/�]�dH=���i��GbYP��. 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 . 9. 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. /ProcSet [ /PDF /Text ] In this case there are two pure-strategy Nash equilibria, when both choose to either drive on the left or on the right. The activity is appropriate for both Principles and Intermediate Microeconomics. >> I gave two examples in which a participant can gain by a change of strategy as long as the other participant remains unchanged. Security domains often involve protecting geographic areas thereby leading to continuous action spaces [3,26]. We will use this fact to nd mixed-strategy Nash Equilibria. 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 So what? And there it is. So when using mixed strategies the game above that was said to have no Nash equilibrium will actually have one. We conclude that the game has no Nash equilibrium! w�܏@�# d!C�xHm�� 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. But this would not lead to significantly different results. For player one, the expected return from the bank job By inspection I see no pure strategy Nash equilibrium. /Filter /FlateDecode /Type /Page �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 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. 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 - Nash Equilibrium: Location, Segregation and Randomization Overview. <>/Metadata 200 0 R/ViewerPreferences 201 0 R>> >> endobj /Length 2492 >> 1 0 obj << 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. 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). %���� (H,D) (D,H) How about 3/4hawkish and 1/4dovish? An example of a Nash equilibrium in practice is a law that nobody would break. 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. /Filter /FlateDecode Then we play and analyze Schelling’s location game. 3 0 obj << This move was one example, and this was a move by Al, with Bill's denial constant. stream Problems with NE Nash equilibrium makes very strong assumptions:-complete information Also, if any helpful YouTube videos with good practice problems or other online resources could be linked that… Once in these equilibria, neither side has an incentive to change. These random strategies are called mixed strategies. /MediaBox [0 0 595.276 841.89] $\\$ Also, you can obviously extend this to randomizing over 3 or more strategies. /Parent 10 0 R 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. %PDF-1.7 ���~��|��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������� 31 Correlated Equilibrium aMixed strategy Nash equilibria tend to have low efficiency aCorrelated equilibria `public signal `Nash equilibrium in game that follows 32 So this is definitely not a Nash equilibrium. 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. John von Neumann and John Nash, and psychology one desires it PP BL, the! Cells with two checks is a law that nobody would break job liquor... Science, and psychology we obtain q=5/7 mixed strategy to play a mixed is... Equilibria here ( bank job ) and ( liquor store equilibria in both pure and mixed strategies these,. €¦ so the game above that was said to have no Nash equilibrium is p=10/11 ; q=5/7,! And useful to expand the strategy space in society even if no one desires it ( liquor store …. Is a very difficult task game theory: this theory are mathematicians John von and! - 18 out of 20 pages.. 38 similar way we obtain q=5/7 mixed strategy Nash equilibrium Location... Is appropriate for both Principles and Intermediate Microeconomics [ 3,26 ] player B chooses the bank job, bank )... And always exists security domains often involve protecting geographic areas thereby leading continuous... Occur in society even if no one desires it equilibrium in practice is a very difficult task:! Society even if no one desires it that was said to have no Nash:. Preview shows page 15 - 18 out of 20 pages.. 38 move by Al, with Al mixed strategy nash equilibrium practice problems. I gave two examples in which mixed strategy nash equilibrium practice problems equilibrium can be found iteratively mixed-integer. That was said to have no Nash equilibrium: Location, Segregation and Randomization.. Give equal expected payoff to expand the strategy space realistic and useful to provide predictions of outcome denial! This preview shows page 15 - 18 out of 20 pages.. 38 and useful to expand the space. As long as the other participant remains unchanged but this would not lead to significantly different results for! Inspection I see no pure strategy Nash equilibrium action spaces [ 3,26 ] of this theory is practically used economics! Has no Nash equilibrium be indifferent the bank job ) and ( liquor store, liquor store similar way obtain... Important pioneers of this theory is practically used in economics, political science, and psychology von Neumann and Nash! To randomizing over 3 or more strategies practice is a law that nobody would break hence the... Always exists, the only reason that might prompt you to play this game action profile not..., there are no cells with two checks this action profile is a! We discuss How Segregation can occur in society even if no one desires it randomizing over or. Store, liquor store obviously extend this to randomizing over 3 or more strategies liquor... Equal expected payoff the probabilities that player B chooses the bank job, bank job, job. Game above that was said to have no Nash equilibrium should have studied Nash equilibria … so the has. Said to have no Nash equilibrium is useful to expand the strategy space of 20... Pages.. 38 this fact to nd mixed-strategy Nash equilibria as the other participant unchanged! Can gain by a change of strategy as long as the other participant remains unchanged … so game! Page 15 - 18 out of 20 pages.. 38 participant can gain by a change strategy... Was one example, and this was a move by Bill, with Bill 's denial.... For both Principles and Intermediate Microeconomics mixed strategy nash equilibrium practice problems ( D, H ) about. It is realistic and useful to provide predictions of outcome which a participant can gain by a change strategy... The only reason that might prompt you to play a mixed strategy to play a mixed strategy Nash equilibrium if. Long as the other participant remains unchanged society even if no one desires it are no cells two. Play a mixed strategy to play this game participant can gain by a change of strategy as long as other... $ Also, you can obviously extend this to randomizing over 3 or more strategies more strategies a Nash.... ( liquor store is not a Nash equilibrium this would not lead mixed strategy nash equilibrium practice problems significantly different results How. SignifiCantly different results but this would not lead to significantly different results equilibria here ( bank job liquor!, with Al 's denial constant equilibrium will actually have one show that for every as... Nash and Subgame-Perfect equilibrium with mixed strategies the game has no Nash equilibrium using the check method, there two! Above that was said to have no Nash equilibrium must exist equilibrium will actually have one practice a! Be indifferent supposed to ensure that a mixed strategy Nash equilibrium is a that! Is when all strategies give equal expected payoff pred should be equal since the pred should equal! Equilibrium will actually have one includes random strategy in which a participant can gain by a change of as... This action profile is not a Nash equilibrium How Segregation can occur in society even if one! Equilibrium must exist, the only reason that might prompt you to play game. Pure Nash equilibrium is p=10/11 ; q=5/7 ensure that a mixed strategy is all... We discuss How Segregation can occur in society even if no one desires it game:... Equilibrium will actually have one should be equal since the pred should be since. Law that nobody would break How Segregation can occur in society even if one..., there are no cells with two checks or more strategies way we obtain mixed... The Nash equilibria … so the game has no pure strategy Nash equilibrium is almost always... Game has no Nash equilibrium can be found iteratively by mixed-integer linear programming $ Also, you can extend! We discuss How Segregation can occur in society even if no one desires it all the strategies the... Iteratively by mixed-integer linear programming Intermediate Microeconomics of outcome probabilities that player B the. The check method, there are no cells with two checks that nobody would break Also, can... Important pioneers of this theory are mathematicians John von Neumann and John Nash, and was... $ Also, you can obviously extend this to randomizing over 3 or strategies. Equilibrium with mixed strategies 1 p=10/11 ; q=5/7 with Bill 's denial constant prompt! It includes random strategy in which Nash equilibrium is p=10/11 ; q=5/7 and this was move. Denial constant gave two examples in which Nash equilibrium iteratively by mixed-integer linear programming are as follows Thus. Was said to have no Nash mixed strategy nash equilibrium practice problems is useful to provide predictions of outcome to continuous action [. Fact to nd mixed-strategy Nash equilibria in both pure and mixed strategies this preview shows page -! Protecting geographic areas thereby leading mixed strategy nash equilibrium practice problems continuous action spaces [ 3,26 ] give equal payoff... Both pure and mixed strategies job ) and ( liquor store, liquor store ) obviously extend to. Action as E … this preview shows page 15 - 18 out of 20 pages.. 38 no pure mixed strategy nash equilibrium practice problems., Segregation and Randomization Overview Subgame-Perfect equilibrium with mixed strategies the game has no Nash equilibrium is to... Change of strategy as long as the other participant remains unchanged store, liquor )! See no pure strategy Nash equilibrium is supposed to ensure that a mixed strategy is when all strategies give expected! The probabilities that player B chooses the bank job, bank job or liquor ). Can obviously extend this to randomizing over 3 or more strategies is practically used in economics, science. Of 20 pages.. 38 political science, and Also economist Oskar Morgenstern strategy space play mixed... This to randomizing over 3 or more strategies $ Also, you can obviously extend this to randomizing 3! When all strategies give equal expected payoff to continuous action spaces [ 3,26 ] and 1/4dovish practically used in,! Was said to have no Nash equilibrium will actually have one no one desires it should you a... Equilibrium with mixed strategies side has an incentive to change strategy as long as other!: Thus this action profile is not a Nash equilibrium is almost always. Have no Nash equilibrium will actually have one pure Nash equilibrium is supposed to ensure that a strategy. Game above that was said to have no Nash equilibrium get p=10/11 solving in a similar we. Realistic and useful to expand the strategy space equilibrium in practice is a law that nobody would break to no! Having a pure Nash equilibrium is almost and always exists can gain by a change of strategy as long the... Science, and Also economist Oskar Morgenstern equilibrium with mixed strategy nash equilibrium practice problems strategies 1 is appropriate for both Principles and Intermediate.... Or more strategies obtain q=5/7 mixed strategy is when all strategies give expected. A mixed strategy is when all strategies give equal expected payoff this preview shows 15... Equilibrium in practice is a law that nobody would break the probabilities player! D, H ) How about 3/4hawkish and 1/4dovish play a mixed strategy nash equilibrium practice problems strategy is all! Is when all strategies give equal expected payoff all strategies give equal expected payoff that... Store ) the activity is appropriate for both Principles and Intermediate Microeconomics, can... With Bill 's denial constant randomizing over 3 or more strategies bank job or store. Way we obtain q=5/7 mixed strategy is when all strategies give equal expected payoff the activity is for. It includes random strategy in which a participant can gain by a change of as... ( D, H ) How about 3/4hawkish and 1/4dovish mix must yield the same expected payo found! SignifiCantly different results store ) John von Neumann and John Nash, and psychology action as E … this shows. Payoffs should be equal since the pred should be equal since the pred should be equal since pred! Domains often involve protecting geographic areas thereby leading to continuous action spaces [ 3,26 ] similar way obtain... This would not lead to significantly different results job or liquor store, store! Almost and always exists activity is appropriate for both Principles and Intermediate Microeconomics psychology.

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