基于等级依赖期望效用理论的斗鸡博弈问题新解

m00 �m d iH�E)�?�n Vol.00 , No. 0000\ I Systems Engineering — Theory & Practice ., 0000 Lb�": 1000-0000(0000)00-0000-00 M=iX%&eV�5+BfvQ/ dA� �=t �4&�yX?Xf, �4RH,411201 a W h5LTb+2Æo%.ntre<8V_E}�N'�O, �ntwh { 6 , �J } Æ { �J, 6 } ,�uk1℄^O} Nash $�ÆD��\ 0.4 }�0q9�6 �k1}� �k1$��;�,{%.re<8V_E�,gb~�[(5LTb+2Y!�P:M2ntre <}8V_E>��whuk1℄^O} Nash $���zntre<}8V_E��nt} ��k1 Nash $��:whrx}aS�>�ntD��B) U�Z�B) U�-Z�B l U�jntwh��k1 Nash $��, z>�,re<�Bl }, j{7�{G3}8V pC}|CnÆxd 1 �, ntpwh��k1$��vK��doI*U}re<�Fsre< 8VpC}m��nt��k1$��" Q`} Cm��/�mZ+2��h +A�q� Wint}#sl" rx}r%�o� �(� RDEU +2; �nt; 8V C gH�0! C934 KO B6 A New Solution for Chicken Game under Rank-Dependent Expected Utility Theory Gong Ri-zhao School of Business, Hunan Science and Technology University, Hunan, Xiangtan 411201 Abstract Under the conditions of the expected utility theory and without considering the game players’ emotional factors, there are two pure strategy Nash equilibrium and a mixed strategy equilibrium, that is, {forward , backward}, {back, forward} and all sides adopt�forward�strategy based on 0.4 probability in chicken game. However, when considering players’ emotional factors and according to the rank-dependent utility theory to establish a new chicken game model, it found that there exists two Nash equilibrium solutions in the sense of pure strategy whether the participants have emotional and preferences or not. But the player’s sentiment index play on a great impact on the existence of equilibrium of chicken game if considering the players’ emotion and preference factors on the chicken game. If the game both sides are optimistic or one is an optimist and the other is pessimistic, then all exist the mixed strategy Nash equilibrium solution but if the two participants are pessimistic, then there exists mixed strategy equilibrium C�?;: 2010-00-00 ihP::,f��L�yT (09BTJ012), wC4~`��*>L�yT (07JA790084) ;V��[^>npyT (2008ZK 2002), ;V���L�yT (08YBB278) 4;V���[P�N{_H^>zsh� jd'+:��B (1966-), W, 1�, ;V��~, *�.�, ;V�[X���Gw&, q�T �o*A��:*A� �1 jqA:15073230906;E-Mail: grzh661205@163.com 2 j I � F * A � � o m � if and only if the sum of the reciprocals of their sentiment index is greater than 1. In addition, For the different types of participants,the game has a mixed strategy equilibrium solution corresponding function variation with the participation of sentiment index changes. The results of this theory have very great reference value to choice the game participants in the practical management decision. 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Nash �8�U�E� (4.2) d}"�UM �gi;0) (4.4) 3 (4.5) MaKbv�)il"|sqB"�UMuo� Z5 g0) (4.4)3 (4.5)MaKbv�)il"|sqB"�UMuob?�eG�L0) (4.5)d jÆ 8 j I � F * A � � o m � [1 2 |;0) (4.5) �8�k jÆ (1) |(d8) β 6= 0�0)rt�.#so (1�0); (2) �- β > 0, O0)MEo (0,0) "2�d�l limx→+∞ y = +∞ �RQ�BCÆi) ^ 0 < β < 1 �, UM$o x∗ ∈ (0,+∞) � 0)M (0, x∗) XZK0)�M (x∗,+∞) XZ�0)� (3) �- β < 0, O lim x→+∞ y = − 3 2 , lim x→0+ y =   0, β ∈ (−1, 0) 1, β = −1 +∞, β ∈ (−∞,−1) lC�k jÆi) ^ β ≤ −1 �rt"��[rnmdK0), l�- β < −1�O limx→0+ dy dx = −∞�� - β = −1�O limx→0+ dy dx = − 52�ii) ^ −1 < β < 0 �UM$o x∗ ∈ (0,+∞) � 0)M (0, x∗) XZ�0 )�M (x∗,+∞) XZK0), l limx→0+ dy dx = +∞. 7;_ dsg��.)d℄M.E?��1.�L (4.5) �0)�dMz, �kM 4 3M 5Æ M 4. M β < 0 dDqk0)� (4.5) d'?p!M M 5. M β > 0 dDqk0)� (4.5) d'?p!M \"�t(a^ r2 > 0 �'D β ≡ 1 r2−1 d BZ β ∈ (−∞, 0)∪ (0,+∞), -R, M 4 k −1 ≤ β < 0 d m��&1>�<� �Æ.) 2 3J0) (4.4) drt=Y�F.<�0) (4.5) 3 (4.4) d" dRs j�;"�eG� �"3 a�k|�Æ [1 3 MaKbv�)il"|sqBX�|;0) (4.4) 3 (4.5) 8 (1) �- α > 0, β < −1�O3rtUMY"uo (x∗, y∗) ∈ (0, 1)× (0, 1); m d ��B: L 0, β > 0, O^l�^ 0 < α < β ��3rtUMY"uo (x∗, y∗) ∈ (1,+∞) × (1,+∞), �O�3rtM (0,+∞) X1UMuo; (3) �- α < 0, β > 0, O3rtUMY"uo (x∗, y∗) ∈ (1,+∞)× (0, 1); (4) �- α < 0, β < −1,O^l�^ β+1 < α < 0��3rtUMY"uo (x∗, y∗) ∈ (0, 1)× (1,+∞), �O�3rtM (0,+∞) X1UMuo� t(a�kY�gi%iÆ � α > 0, β < −1⇔ 0 < r1 < 1, 0 < r2 < 1� � β > α > 0⇔ r1 > 1, r2 > 1 l 1 r1 + 1 r2 > 1; α < 0, β > 0⇔ 0 < r1 < 1, r2 > 1� β + 1 < α < 0⇔ r1 > 1, 0 < r2 < 1� � x = 1⇔ p = 0.4; 0 < x < 1⇔ p > 0.4 g� ;", �ÆO? 1 3.) 3, � a�ks)Æ �1 4 M RDEU )�kdtO-�P k�-�UMF6S? Nash �8, ^l�^�9U A 3 B dm�d)C�kDqa"Æ (1) �- 0 < r1 < 1, 0 < r2 < 1�U�9Uw"'& d�O-�UMUMY"dF6 Nash �8 {p∗, q∗}, C 0.4 < p∗, q∗ < 1; (2) �- r1 > 1, r2 > 1, U�9Uw"�& d�lm�d)BC 1 r1 + 1 r2 > 1 , O-�UMUMY" dF6 Nash �8 {p∗, q∗}, C 0 < p∗, q∗ < 0.4; (3) �- 0 < r1 < 1, r2 > 1, U�9U A "'& d�B "�& d�O-�UMUMY"dF6 Nash �8 {p∗, q∗}, C 0 < p∗ < 0.4 < q∗ < 1; (4) �- r1 > 1, 0 < r2 < 1 �U�9U A "�& d��9U B "'& d�O-�UMY"dF6 Nash �8 {p∗, q∗}, C 0 < q∗ < 0.4 < p∗ < 1� �Æs) 1-4�� L�kO�Æ I4 1 M RDEU )�kdtO-�P k, �-�9Uw"�& d (r1 > 1, r2 > 1) lm�d)B C 1 r1 + 1 r2 ≤ 1, O-�1UMF6S?�8 {p∗, q∗}, C 0 < p∗, q∗ < 1�eUMQS?�8 {p∗, q∗} = {1, 0} 3 {0, 1}� kK�L"��Bd�-Æ h��9Uw"�& d�8 r1 = 5 2 , r2 = 2, m{C 1 r1 + 1 r2 = 9 10 ≤ 1, �ÆO� 1�-�1UMF6S?�8 {p∗, q∗}, C 0 < p∗, q∗ < 1�Z5�℄W"|��eG�TO? 1 d-:���B g℄1� ���, R�|/ α = 1.2, β = 1, 0) (4.4) Z y = x1.2, 0) (4.5) Z y = 2x 3x+ 2 (x− 1), x > 0. �Æ.) 2, �`W3�0)drt�-UMuo�Ouoe�YMqk x ∈ (1,+∞) X�\����� &℄MMqk x ∈ (1,+∞) X x1.2 > 2x 3x+ 2 (x − 1) 9D., f*�J0) y = x1.2 drt}M0) y = 2x3x+2 (x− 1), x > 0 drta���|R��&s�1g �gi'Z 3x1.2 + 2x0.2 − 2x+ 2 > 0 (4.7) 10 j I � F * A � � o m � 8 F (x) = 3x1.2 + 2x0.2 − 2x+ 2, x ∈ (1,+∞), O F (1) = 5 lm{8 dF (x) dx = 3.6x0.2 + 0.4x−0.8 − 2 > 0 |( x ∈ (1,+∞) D.�WI ,M5 (4.7) D.� =+k��ÆO� 1�B� L�k|�Æ M M RDEU )�kdtO-�P k, �-�9Uw"�& d (r1 > 1, r2 > 1) lm�d)Rw uG (r1 = r2 = r > 1), O^ r ≥ 2 �-�1UMF6S?�8eUMQS?�8 {p ∗, q∗} = {1, 0} 3 {0, 1}� �Æs) 2 3s) 3 dO`℄M.E, B�& Lm�d)|�8}d0w%iÆ I4 2 M RDEU )�kdtO-�P k, 8Æ (1) �-�9U A Z'& l'&m�d)"s �O7"�9U B �N�i��S?d�8�>"| �m�d)d[rnP0)�u/k��9U A �N�i��S?d�8�>"|�m�d)d[rnm0 )� (2) �-�9U A Z�& l�&m�d)"s �O7"�9U B �N�i��S?d�8�>"% ;|�m�d)d[rnm0)�G���9U A �N” i�” S?d�8�> "%;|�m�d)d[ rnm0)� (3) �-3�9Uk8"}"fm�( �O7"�9U�N�i��S?d�8�>Zs 0.4�| ��N�i��S?d�8�>"%;|�m�d)d[rnm0)� (4) �-3�9U�8RwuGdm�d)�O7G�N�i��S?d�8�>ug�l"%;m �d)d[rnm0)� �Æ�( g�eG�&�L�-��k} (�9U) dm�1�0wxF6S?�8dUM �l |�8S?dW~ws>�7Wd0w� 5 A3&F �Æ�{<��o}F6S?�8�e��%l�Æu/dm�d)℄1L α, β |/d �{:�Æ0 ) (4.4) 3 (4.5) oLrtuo�) (x∗, y∗), LF.u/d'DU� LF6S?�8} {p∗, q∗}� =SX: �-3�9U�8G( dm��Uw"'& dIw"�& d�\m�d)1G�tk* 1 kZ* dm�d) � aF6S?�8} {p∗, q∗}�p* 1Æ * 1. uGm�( -�F6S?�8�-|- Table 1. The results of mixed strategy equilibrium under the same emotional type �9Uw"'& (r1, r2 < 1) �9Uw"�& (r1, r2 > 1, 1 r1 + 1 r2 > 1) r1 r2 p ∗ q∗ r1 r2 p ∗ q∗ 0.2 0.4 0.5024 0.6684 1.1 1.3 0.3134 0.3648 0.3 0.5 0.4968 0.6249 1.2 1.4 0.2799 0.3273 0.4 0.6 0.4840 0.5897 1.5 2.0 0.0867 0.1538 0.6 0.8 0.4470 0.5271 1.9 1.8 0.0449 0.0380 0.8 0.9 0.4269 0.4212 2.0 1.5 0.1538 0.0867 S* 1 �&�LÆ�-3�9Uw"'& d�\^3}dm�d)G�PW�U3}wH#H1' &�O3}M�8w:k�N�i��d�>w'~�p�; 0.4. =S�: �-3�9U�81G( dm��tk* 2 kZ* dm�d) � aF6S?�8} {p∗, q∗}�p* 2Æ * 2. 1Gm�( -�dF6S?�8�-|- Table 2. The results of mixed strategy equilibrium under not the same emotional type m d ��B: L 1) A "�& B "'& (r1 > 1, r2 < 1) r1 r2 p ∗ q∗ r1 r2 p ∗ q∗ 0.2 1.5 0.3633 0.8116 1.5 0.6 0.5791 0.3170 0.4 2.0 0.2992 0.7175 2.0 0.5 0.6613 0.2803 0.6 2.5 0.2228 0.6300 2.5 0.4 0.7420 0.2709 0.8 3.0 0.1397 0.5310 3.0 0.3 0.8166 0.2793 0.9 4.0 0.0635 0.4833 4.0 0.2 0.8924 0.2936 S* 2 �&�LÆ"�'& d�9U3"��& d�9U��-���& d�9U�/��2S bi��U�N�i��d�>�u|xW� =S�: �-3�9U�8RwuGdm�d)�Um�d) r1 = r2 = r�tk* 3 kZ* dm� d) � aF6S?�8} {p∗, q∗}�p* 3Æ * 3. RwuGm�d)-�F6S?�8�-|- Table 3. The results of mixed strategy equilibrium under exactly the same sentiment index r < 1 p∗ q∗ r > 1 p∗ q∗ 0.2 0.5947 0.5947 1.2 0.3359 0.3359 0.4 0.5532 0.5532 1.3 0.3005 0.3005 0.5 0.5310 0.5310 1.5 0.2209 0.2209 0.7 0.4828 0.4828 1.7 0.1256 0.1256 0.9 0.4292 0.4292 1.8 0.0699 0.0699 S*��&�L�|;3�9U�8RwuGdm�d)WlRz( dtO-���-7GH�& (m�d)HW)�O7G�N�i��S?d�>H~�-R�-�E�}kLpWewP|d�8|-�^ {�Mp�k7;�8Wlm�d-�K �"1�Yd� =SE: �-3�9UkeG#sfk"℄dm�( �1�h��9U B dm�( #s��tf m�d) r2 = 0.5, 1, 2, |;�9U A d1Gm�d)�-��8|-�p* 4Æ * 4. #s�k} B m�( -�dF6S?�8�-|- Table 4. The results of mixed strategy equilibrium under fixed emotional type of player B r2 = 0.5 B "'& r2 = 1 B fm� r2 = 2 B "�& r1 p ∗ q∗ p∗ q∗ p∗ q∗ 0.2 0.4750 0.6934 0.4000 0.7702 0.3403 0.8383 0.4 0.5151 0.5728 0.4000 0.6415 0.2992 0.7175 0.6 0.5451 0.4964 0.4000 0.5442 0.2620 0.6069 0.8 0.5694 0.4417 0.4000 0.4655 0.2260 0.5018 1.0 0.5900 0.4000 0.4000 0.4000 0.1895 0.4000 1.2 0.6077 0.3668 0.4000 0.3444 0.1512 0.3004 1.4 0.6234 0.3396 0.4000 0.2969 0.1094 0.2023 1.6 0.6373 0.3167 0.4000 0.2558 0.0625 0.1060 1.8 0.6499 0.2972 0.4000 0.1895 0.0128 0.0197 2.0 0.6613 0.2803 0.4000 0.1628 1UM 1UM 2.2 0.6718 0.2665 0.4000 0.1396 2.4 0.6814 0.2525 0.4000 0.1195 2.6 0.6903 0.2408 0.4000 0.1021 2.8 0.6986 0.2303 0.4000 0.0871 5 )5 12 j I � F * A � � o m � gV#$ T�3)�`��5 T�3)�:I05 T�3)�UMd"�xr�Y"x2k} �"��9UY8m�d-��9b?�K3�)� gtO-�d Nash �8}b?� L5?p81G d|�� "�K�-�k�9Udm�0wxF6S?�8}dUM ��--�+�w"'& d�IU"" �'&7Q"��&�O#{UM3�QS?()kd Nash �83"�F6S?�8��--�+�w" �&d�O^l�^7Gdm�d)d_)a3W; 1�U 1 r1 + 1 r2 > 1 �5UMF6S?�8��O�eU M3�QS?()kd Nash �8�W"o"?F�d|�"Rw1a6d�I ,M5m�|�8}UM 0w� 7"�K��9Udm��sxF6S?�8�>dW~��-"�m�d)zsl"'&m�dD qk�7"�dm�d)HW�Of�N�i��S?d�>$�HW� �"�N�i��d�>$�H W�,MMzs5+}'&di>k�Z$�"+}�'&�l��gf6\�k�2Sbi��5YH xWd$*� _d|�|')�n�87Wdd`�3��h�83h! A3 Bs�Q"x�tO-���9�e G�"3h�h! A F.Ql��d�hsb`}h! B 8?-�d�9U"'& d�l`bf�B dm�d) (h� r2 = 0.5 )�Oh! A Mzs8gby-�d�9U���Mhd�N�"�N��& d�9U�lH�&H2�RQ�p��Gk�N�;-��6`�{^�-�g�W"-�)�w�8 u/dd`()� ^{� _d|-"Mh�-�8?}dm�0)ZJ0)l�30) u(x) = x dDqk ad� �- =m�d0)"f8( dC ω(0) = 0, ω(1) = 1 d[rnP0)��30) u(x) Zd)�30 )�Om{ ad|-E861G�\"� _y>�5R[d� g!j�|f7� ud�U�"3 � b?�8xWd8�i ��8℄u.Ad�3� �.KO [1] Von Neumann, J. and Morgenstern, O., Theory of Games and Economic Behavior[M]. Princeton: Princeton University Press,1944. [2] Quiggin,J., A Theory of Anticipated Utility[J]. Journal of Economic Behavior and Organization, 1982,3:323-343. [3] Schmeidler,D., Subjective Probability and Expected Utility without Additivity[J]. Econometrica, 1989,57: 571- 587. [4] Starmer,C., Developments in Non-Expected Utility Theory: The Hunt for a Descriptive Theory of Choice under Risk[J].Journal of Economic Literature, 2000,8:332-382. [5] Crowley,P.H., Hawks, Doves and Mixed- symmetry Games[J]. Journal of Theoretical Biology, 2000, 204: 543-563. [6] Machina,M.J., Expected Utility Theory without the Independent Axiom[J]. Econometrica, 1982,50:277-323. [7] Quiggin J. Comparative statics for rank-dependent expected utility theory [J]. Journal of Risk and Uncertainty, 1991,4: 339-350. [8] Enrico D.,Peter.P.W.,On the Intuition of Rank-Dependent Utility[J].The journal of Risk and Uncertainty,2001,23: 281-298. [9] Machina,M.J.,Choice under Uncertainty: Problems Solved and Unsolved[J]. Journal of Economic Perspectives,1987, 1:121-154. [10] Matsumura,S.and Kobayashi,T., A Game Model for Dominance Relations among Group- living Animals[J]. Behavioral Ecology Sociobiology,1998,42:77-84. [11] Maynard,Smith and Price,G.R.,The Log