gamet




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help for gamet
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Game Theory

gamet , payoff(#U111,#U211, ... ,#U11_c,#U21_c ... ,#U11_C,#U21_C \ ... \ #U1_r1,#U2_r1, ... ,#U1_r_c,#U2_r_c ... ,#U1_r_C,#U2_r_C \ ... \ #U1_R1,#U2_R1, ... ,#U1_R_c,#U2_R_c ... ,#U1_R_C,#U2_R_C)

[ ls1(lab_s1) ls2(lab_s2) player1(rlab1 ... rlab_r ... rlab_R) player2(clab1 ... clab_c ... clab_C)

domist elids neps nefms maximin gtree

npath aspect(#) mlabpls(clockpos) mlabppm(clockpos) mlabpp1(clockpos) mlabpp2(clockpos) savingpf(filename) textpp(textsizestyle) texts(textsizestyle) msizepp(relativesize) msizes(relativesize)

scatter_options ]

Description

gamet represents the extensive form (game tree) and the strategic form (payoff matrix) of a non-cooperative game and identifies the solution of a non-zero and zero-sum game through: dominant and dominated strategies, iterated elimination of strongly dominated strategies, Nash equilibrium in pure and fully mixed strategies. Further, gamet is able to identify the solution of a zero-sum game through maximin criterion and the solution of extensive form through backward induction.

Payoff matrix

-------------------------------------------------------------------------------------- | lab_s2 lab_s1 | clab1 ... clab_c ... clab_C ----------+--------------------------------------------------------------------------- rlab1 | (#U111; #U211) ... (#U11_c; #U21_c) ... (#U11_C; #U21_C) ... | ... ... ... ... ... rlab_r | (#U1_r1; #U2_r1) ... (#U1_r_c; #U2_r_c) ... (#U1_r_C; #U2_r_C) ... | ... ... ... ... ... rlab_R | (#U1_R1; #U2_R1) ... (#U1_R_C; #U2_R_C) ... (#U1_R_C; #U2_R_C) --------------------------------------------------------------------------------------

payoff(...) is not optional and provides a way to input, row after row, a general R by C payoff matrix (help matrix input), where

#U1_r_c is the utility for lab_s1 if lab_s1 chooses strategy r and lab_s2 chooses strategy c #U2_r_c is the utility for lab_s2 if lab_s1 chooses strategy r and lab_s2 chooses strategy c

with r = 1,2, ..., R and c = 1,2, ..., C

Remark

gamet is an immediate command given that obtains data not from the data stored in memory but from numbers typed as arguments (help immed).

Options

ls1(lab_s1) attaches a label to the set of strategies for player 1. The default is S1.

ls2(lab_s2) attaches a label to the set of strategies for player 2. The default is S2.

player1(rlab1 rlab2 ... rlab_r ... rlab_R) attaches a label for each strategy of player 1. The default is A1, B2, C3 and so on.

player2(clab1 clab2 ... clab_c ... clab_C) attaches a label for each strategy of player 2. The default is a1, b2, c3 and so on.

domist seeks strongly dominated and dominant strategies for each player. elids eliminates iteratively all strongly dominated strategies for each player.

neps seeks Nash equilibrium in pure strategies.

nefms seeks Nash equilibrium in fully mixed strategies (0<p<1 and 0<q<1). It works only if R and C are equal to 2.

maximin seeks the saddle-point through the minimal column maximum for player 1 and maximal row minimum for player 2. It works for zero-sum games. That is, #U1_r_c + #U2_r_c == 0.

gtree seeks the equilibrium path through backward induction (player 1 moves first). It produces a graphical representation of a sequential game, called game tree.

savingpf(filename) saves the variables obtained by the conversion of the payoff matrix in a file. If the option elids is specified savingpf() saves one file(filename#) for each iteration.

npath specifies no equilibrium path on the game tree.

aspect(#) modifies the aspect ratio (height/widht) of the plot region. By default is set to 1 (equal height and width) so the plot region is a square. See graph_display.

mlabpls(clockpos) specifies the position for label lab_s1 and lab_s2 on the game tree. Use clockpos to make changes from the default (9).

mlabppm(clockpos) specifies the position for #U1_r_c, #U2_r_c on the game tree. Use clockpos to make changes from the default (3).

mlabpp1(clockpos) specifies the position for strategies' labels on the game tree for player 1. Use clockpos to make changes from the default (12).

mlabpp2(clockpos) specifies the position for strategies' labels on the game tree for player 2. Use clockpos to make changes from the default (9).

textpp(textsizestyle) specifies the text size style for lab_s1, lab_s2 and (#U1_r_c; #U2_r_c). Use textsizestyle to make changes from the default (medium).

texts(textsizestyle) specifies the text size style for strategies' labels. Use textsizestyle to make changes from the default (small).

msizepp(relativesize) choices for sizes for objects lab_s1, lab_s2 and (#U1_r_c; #U2_r_c). Use relativesize to make changes from the default (2).

msizes(relativesize) choices for sizes for strategies' labels. Use relativesize to make changes from the default (2).

scatter_options are options of scatter.

Examples

. gamet, payoff(2, 2, 0, 1 \ 3, 0 , 1, 1) player1(High Low) player2(Buy Not_buy) /// ls1(Provider) ls2(Customer) domist

. gamet, pay(3, 0, 0 , 2 , 0, 3\2, 0 , 1, 1 , 2, 0 \ 0, 3 , 0 , 2, 3, 0 ) /// ls1(C1) ls2(C2) player1(x1 y1 z1) player2(x2 y2 z2) elids

. gamet, payoff(0,0,12,8,18,9,36,0\ 8,12,16,16,20,15,32,0\9,18,15,20,18,18,27,0\0,36,0,32,0,27,0,0)/// player1(H M L N) player2(h m l n) ls1(Firm_I) ls2(Firm_II) elids

. gamet, payoff(3, 1, 0, 0\0, 0, 1, 3) player1(Football Ballet) player2(Football Ballet) /// ls1(Boy) ls2(Girl) neps

. gamet, pay(0, 0, -10, 10 \ -1, 0, -6, -90) player1(Not_inspect Inspect) /// player2(Comply Cheat) ls1(I) ls2(II) nems

. gamet, payoff(2, 2, 0, 1 \ 3, 0 , 1, 1) player1(High Low) /// player2(Buy Not_buy) ls1(I) ls2(II) gtree

. gamet, payoff(0,0,12,8,18,9,36,0\ 8,12,16,16,20,15,32,0\9,18,15,20,18,18,27,0\0,36,0,32,0,27,0,0)/// player1(H M L N) player2(h m l n) ls1(Firm_I) ls2(Firm_II) gtree

. gamet, payoff(-5,5,3,-3,1,-1,20,-20\5,-5,5,-5,4,-4,6,-6\-4,4,6,-6,0,0,-5,5) /// player1(1 2 3) player2(1 2) maximin

Authors

Nicola Orsini, Institute of Environmental Medicine, Karolinska Institutet, Stockholm, Sweden. Debora Rizzuto, Department of Public Health, University of Siena, Italy. Nicola Nante, Department of Public Health, University of Siena, Italy.

Reference

Myerson, R. B. 1991. Game Theory: Analysis of Conflict, Harvard University Press, Cambridge (MA).

Support

http://nicolaorsini.altervista.org nicola.orsini@imm.ki.se

Also see

On-line: help for matrix, _variables, tabdisp, macrolists

Worked examples for gamet

Click here to run or save the do-file

. capture net install http://nicolaorsini.altervista.org/stata/gamet

. . which gamet .\gamet.ado *! N.Orsini, D.Rizzuto, N.Nante *! Version 1.0 - November 4, 2004 *! Version 1.1 - November 8, 2004 *! Version 1.2 - February 19, 2005

. . help gamet

. . * DOMINANT AND DOMINATED STRATEGIES . . gamet , payoff(2, 2, 0, 1 \ 3, 0 , 1, 1) player1(High Low) player2(Buy Not_bu > y) ls1(Provider) ls2(Customer) domist

---------------------------- | Customer Provider | Buy Not_buy ----------+----------------- High | (2; 2) (0; 1) Low | (3; 0) (1; 1) ----------------------------

DOMINATED AND DOMINANT STRATEGIES Dominated strategy for Provider is High No dominated strategy for Customer Dominant strategy for Provider is Low No dominant strategy for Customer

. . * ITERATED ELIMINATION OF STRONGLY DOMINATED STRATEGIES . . gamet, payoff(0, 0, 12, 8, 18, 9, 36, 0\ 8, 12, 16, 16, 20, 15, 32, 0 \ 9, 18 > , 15, 20, 18, 18, 27, 0\0, 36, 0, 32, 0, 27, 0, 0) /// > player1(H M L N) player2(h m l n) ls1(Firm_I) ls2(Firm_II) elids

-------------------------------------------------- | Firm_II Firm_I | h m l n ----------+--------------------------------------- H | (0; 0) (12; 8) (18; 9) (36; 0) M | (8; 12) (16; 16) (20; 15) (32; 0) L | (9; 18) (15; 20) (18; 18) (27; 0) N | (0; 36) (0; 32) (0; 27) (0; 0) --------------------------------------------------

Iteration 1

DOMINATED AND DOMINANT STRATEGIES Dominated strategy for Firm_I is N Dominated strategy for Firm_II is n No dominant strategy for Firm_I No dominant strategy for Firm_II

---------------------------------------- | Firm_II Firm_I | h m l ----------+----------------------------- H | (0; 0) (12; 8) (18; 9) M | (8; 12) (16; 16) (20; 15) L | (9; 18) (15; 20) (18; 18) ----------------------------------------

Iteration 2

DOMINATED AND DOMINANT STRATEGIES Dominated strategy for Firm_I is H Dominated strategy for Firm_II is h No dominant strategy for Firm_I No dominant strategy for Firm_II

------------------------------ | Firm_II Firm_I | m l ----------+------------------- M | (16; 16) (20; 15) L | (15; 20) (18; 18) ------------------------------

Iteration 3

DOMINATED AND DOMINANT STRATEGIES Dominated strategy for Firm_I is L Dominated strategy for Firm_II is l Dominant strategy for Firm_I is M Dominant strategy for Firm_II is m

-------------------- | Firm_II Firm_I | m ----------+--------- M | (16; 16) --------------------

Iteration 4

DOMINATED AND DOMINANT STRATEGIES No dominated strategy for Firm_I No dominated strategy for Firm_II Dominant strategy for Firm_I is M Dominant strategy for Firm_II is m

. . * NASH EQUILIBRIM IN PURE STRATEGIES . . gamet , payoff(2, 2, 0, 1 \ 3, 0 , 1, 1) player1(High Low) player2(Buy Not_bu > y) ls1(Provider) ls2(Customer) domist neps

---------------------------- | Customer Provider | Buy Not_buy ----------+----------------- High | (2; 2) (0; 1) Low | (3; 0) (1; 1) ----------------------------

DOMINATED AND DOMINANT STRATEGIES Dominated strategy for Provider is High No dominated strategy for Customer Dominant strategy for Provider is Low No dominant strategy for Customer

NASH EQUILIBRIUM IN PURE STRATEGIES 1. Low Not_buy (1; 1)

. . * NASH EQUILIBRIM IN MIXED STRATEGIES . . gamet , payoff(2, 2, 0, 1 \ 3, 0 , 1, 1) player1(High Low) player2(Buy Not_bu > y) ls1(Provider) ls2(Customer) neps

---------------------------- | Customer Provider | Buy Not_buy ----------+----------------- High | (2; 2) (0; 1) Low | (3; 0) (1; 1) ----------------------------

NASH EQUILIBRIUM IN PURE STRATEGIES 1. Low Not_buy (1; 1)

. . gamet, pay(0, 0, -10, 10 \ -1, 0, -6, -90) player1(Not_inspect Inspect) playe > r2(Comply Cheat) ls1(I) ls2(II) nefms

---------------------------------- | II I | Comply Cheat ------------+--------------------- Not_inspect | (0; 0) (-10; 10) Inspect | (-1; 0) (-6; -90) ----------------------------------

NASH EQUILIBRIUM IN FULLY MIXED STRATEGIES p = 0.90 q = 0.80

(0.90*[Not_inspect]+0.10*[Inspect], 0.80*[Comply]+0.20*[Cheat])

Expected equilibrium payoff for I 0.80*0+(1-0.80)*-10 = -2 0.80*-1+(1-0.80)*-6 = -2

Expected equilibrium payoff for II 0.90*0+(1-0.90)*0 = 0 0.90*10+(1-0.90)*-90 = 0

. . * GAME TREE - BACKWARD INDUCTION . . gamet , payoff(2, 2, 0, 1 \ 3, 0 , 1, 1) player1(High Low) player2(Buy Not_bu > y) ls1(Provider) ls2(Customer) gtree

---------------------------- | Customer Provider | Buy Not_buy ----------+----------------- High | (2; 2) (0; 1) Low | (3; 0) (1; 1) ----------------------------

BACKWARD INDUCTION Equilibrium path: High Buy Payoffs pair: (2; 2)

Game tree #1 . . gamet, payoff(2, 2, 0, 1 \ 3, 0 , 1, 1) player1(High Low) player2(Buy Not_bu > y) ls1(I) ls2(II) gtree /// > plotregion(color(yellow)) graphregion(color(yellow)) note(Change layout as m > uch as you like) texts(large) /// > msizes(*12) mlabpp1(9) title(Quality choice example) subtitle(I moves first)

---------------------------- | II I | Buy Not_buy ----------+----------------- High | (2; 2) (0; 1) Low | (3; 0) (1; 1) ----------------------------

BACKWARD INDUCTION Equilibrium path: High Buy Payoffs pair: (2; 2)

Game tree #3 . . gamet, payoff(0, 0, 12, 8, 18, 9, 36, 0\ 8, 12, 16, 16, 20, 15, 32, 0 \ 9, 18 > , 15, 20, 18, 18, 27, 0\0, 36, 0, 32, 0, 27, 0, 0) /// > player1(H M L N) player2(h m l n) ls1(Firm_I) ls2(Firm_II) gtree title(Exten > sive form)

-------------------------------------------------- | Firm_II Firm_I | h m l n ----------+--------------------------------------- H | (0; 0) (12; 8) (18; 9) (36; 0) M | (8; 12) (16; 16) (20; 15) (32; 0) L | (9; 18) (15; 20) (18; 18) (27; 0) N | (0; 36) (0; 32) (0; 27) (0; 0) --------------------------------------------------

BACKWARD INDUCTION Equilibrium path: H l Payoffs pair: (18; 9)

Game tree #2 . . * MAXIMIN FOR ZERO-SUM GAME . . gamet, payoff(-5, 5, 3, -3, 1, -1, 20, -20\5, -5, 5, -5, 4, -4, 6, -6\ -4, 4, > 6, -6, 0, 0, -5, 5) /// > player1(1 2 3) player2(1 2 3 4) maximin

------------------------------------------------------ | S2 S1 | 1 2 3 4 ----------+------------------------------------------- 1 | (-5; 5) (3; -3) (1; -1) (20; -20) 2 | (5; -5) (5; -5) (4; -4) (6; -6) 3 | (-4; 4) (6; -6) (0; 0) (-5; 5) ------------------------------------------------------

ZERO-SUM GAME - MAXIMIN CRITERION Minimal Column Maximum of S1 = 4 Maximal Row Minimum of -{S2} = -4 Saddle-point = 2 3

. . * EXAMPLES . . * example from http://www.cdam.lse.ac.uk/Reports/Files/cdam-2001-09.pdf . . gamet, payoff(2, 2, 0, 1 \ 3, 0 , 1, 1) player1(High Low) player2(Buy Not_bu > y) ls1(Provider) ls2(Customer) elids neps

---------------------------- | Customer Provider | Buy Not_buy ----------+----------------- High | (2; 2) (0; 1) Low | (3; 0) (1; 1) ----------------------------

Iteration 1

DOMINATED AND DOMINANT STRATEGIES Dominated strategy for Provider is High No dominated strategy for Customer Dominant strategy for Provider is Low No dominant strategy for Customer

-------------------------- | Customer Provider | Buy Not_bu ----------+--------------- Low | (3; 0) (1; 1) --------------------------

Iteration 2

DOMINATED AND DOMINANT STRATEGIES No dominated strategy for Provider Dominated strategy for Customer is Buy Dominant strategy for Provider is Low Dominant strategy for Customer is Not_buy

------------------ |Custome | r Provider | Not_bu ----------+------- Low | (1; 1) ------------------

Iteration 3

DOMINATED AND DOMINANT STRATEGIES No dominated strategy for Provider No dominated strategy for Customer Dominant strategy for Provider is Low Dominant strategy for Customer is Not_buy

NASH EQUILIBRIUM IN PURE STRATEGIES 1. Low Not_buy (1; 1)

. . gamet, payoff(2, 2, 0, 1 \ 1, 0 , 1, 1) player1(High Low) player2(Buy Not_bu > y) ls1(Provider) ls2(Customer) elids neps

---------------------------- | Customer Provider | Buy Not_buy ----------+----------------- High | (2; 2) (0; 1) Low | (1; 0) (1; 1) ----------------------------

Iteration 1

DOMINATED AND DOMINANT STRATEGIES No dominated strategy for Provider No dominated strategy for Customer No dominant strategy for Provider No dominant strategy for Customer

NASH EQUILIBRIUM IN PURE STRATEGIES 1. High Buy (2; 2) 2. Low Not_buy (1; 1)

. . gamet, pay(0, 0, -10, 10 \ -1, 0, -6, -90) player1(Not_inspect Inspect) playe > r2(Comply Cheat) ls1(I) ls2(II) nefms

---------------------------------- | II I | Comply Cheat ------------+--------------------- Not_inspect | (0; 0) (-10; 10) Inspect | (-1; 0) (-6; -90) ----------------------------------

NASH EQUILIBRIUM IN FULLY MIXED STRATEGIES p = 0.90 q = 0.80

(0.90*[Not_inspect]+0.10*[Inspect], 0.80*[Comply]+0.20*[Cheat])

Expected equilibrium payoff for I 0.80*0+(1-0.80)*-10 = -2 0.80*-1+(1-0.80)*-6 = -2

Expected equilibrium payoff for II 0.90*0+(1-0.90)*0 = 0 0.90*10+(1-0.90)*-90 = 0

. . gamet, payoff(-10,5, 8,-10\3, -1, 0,0) ls1(Iran) ls2(IAEA) player1(Make_nukes > No_nukes) player2(Inspect Not_inspect) nefms

------------------------------------- | IAEA Iran | Inspect Not_inspect -----------+------------------------- Make_nukes | (-10; 5) (8; -10) No_nukes | (3; -1) (0; 0) -------------------------------------

NASH EQUILIBRIUM IN FULLY MIXED STRATEGIES p = 0.06 q = 0.38

(0.06*[Make_nukes]+0.94*[No_nukes], 0.38*[Inspect]+0.62*[Not_inspect])

Expected equilibrium payoff for Iran 0.38*-10+(1-0.38)*8 = 1.1428571 0.38*3+(1-0.38)*0 = 1.1428571

Expected equilibrium payoff for IAEA 0.06*5+(1-0.06)*-1 = -.625 0.06*-10+(1-0.06)*0 = -.625

. . gamet, pay(50, 50, 80, 20 \ 90, 10, 20 , 80) ls1(Serena) ls2(Venus) player1( > DL CC) player2(DL CC) nefms

------------------------------ | Venus Serena | DL CC ----------+------------------- DL | (50; 50) (80; 20) CC | (90; 10) (20; 80) ------------------------------

NASH EQUILIBRIUM IN FULLY MIXED STRATEGIES p = 0.70 q = 0.60

(0.70*[DL]+0.30*[CC], 0.60*[DL]+0.40*[CC])

Expected equilibrium payoff for Serena 0.60*50+(1-0.60)*80 = 62 0.60*90+(1-0.60)*20 = 62

Expected equilibrium payoff for Venus 0.70*50+(1-0.70)*10 = 38 0.70*20+(1-0.70)*80 = 38

. . gamet , pay(-1, -1, 3, 0 \ 0, 3, 0, 0) neps nefms elids

------------------------------ | S2 S1 | a1 b2 ----------+------------------- A1 | (-1; -1) (3; 0) B2 | (0; 3) (0; 0) ------------------------------

Iteration 1

DOMINATED AND DOMINANT STRATEGIES No dominated strategy for S1 No dominated strategy for S2 No dominant strategy for S1 No dominant strategy for S2

NASH EQUILIBRIUM IN PURE STRATEGIES 1. A1 b2 (3; 0) 2. B2 a1 (0; 3)

NASH EQUILIBRIUM IN FULLY MIXED STRATEGIES p = 0.75 q = 0.75

(0.75*[A1]+0.25*[B2], 0.75*[a1]+0.25*[b2])

Expected equilibrium payoff for S1 0.75*-1+(1-0.75)*3 = 0 0.75*0+(1-0.75)*0 = 0

Expected equilibrium payoff for S2 0.75*-1+(1-0.75)*3 = 0 0.75*0+(1-0.75)*0 = 0

. . gamet , pay(-1, -1, 4, 0 \ 0, 3, 0, 0) neps nefms ls1(MD) ls2(DK) player1(E D > ) player2(E D) elids

------------------------------ | DK MD | E D ----------+------------------- E | (-1; -1) (4; 0) D | (0; 3) (0; 0) ------------------------------

Iteration 1

DOMINATED AND DOMINANT STRATEGIES No dominated strategy for MD No dominated strategy for DK No dominant strategy for MD No dominant strategy for DK

NASH EQUILIBRIUM IN PURE STRATEGIES 1. E D (4; 0) 2. D E (0; 3)

NASH EQUILIBRIUM IN FULLY MIXED STRATEGIES p = 0.75 q = 0.80

(0.75*[E]+0.25*[D], 0.80*[E]+0.20*[D])

Expected equilibrium payoff for MD 0.80*-1+(1-0.80)*4 = 0 0.80*0+(1-0.80)*0 = 0

Expected equilibrium payoff for DK 0.75*-1+(1-0.75)*3 = 0 0.75*0+(1-0.75)*0 = 0

. . * examples from http://www.courses.fas.harvard.edu/~ec1052/lecture/lecture5.p > df . . gamet, pay(1, 1, 0, 4 \ 0, 2, 2 , 1) neps nefms ls1(player1) ls2(player2) pla > yer1(U D) player2(L R) elids

-------------------------- | player2 player1 | L R ----------+--------------- U | (1; 1) (0; 4) D | (0; 2) (2; 1) --------------------------

Iteration 1

DOMINATED AND DOMINANT STRATEGIES No dominated strategy for player1 No dominated strategy for player2 No dominant strategy for player1 No dominant strategy for player2

NASH EQUILIBRIUM IN PURE STRATEGIES None

NASH EQUILIBRIUM IN FULLY MIXED STRATEGIES p = 0.25 q = 0.67

(0.25*[U]+0.75*[D], 0.67*[L]+0.33*[R])

Expected equilibrium payoff for player1 0.67*1+(1-0.67)*0 = .66666667 0.67*0+(1-0.67)*2 = .66666667

Expected equilibrium payoff for player2 0.25*1+(1-0.25)*2 = 1.75 0.25*4+(1-0.25)*1 = 1.75

. . gamet , pay(1, 1, 0, 2,0,4 \0,2,5,0,1,6 \0,2,1,1,2,1) player1(U M D) player2( > L C R) elids

---------------------------------- | S2 S1 | L C R ----------+----------------------- U | (1; 1) (0; 2) (0; 4) M | (0; 2) (5; 0) (1; 6) D | (0; 2) (1; 1) (2; 1) ----------------------------------

Iteration 1

DOMINATED AND DOMINANT STRATEGIES No dominated strategy for S1 Dominated strategy for S2 is C No dominant strategy for S1 No dominant strategy for S2

-------------------------- | S2 S1 | L R ----------+--------------- U | (1; 1) (0; 4) M | (0; 2) (1; 6) D | (0; 2) (2; 1) --------------------------

Iteration 2

DOMINATED AND DOMINANT STRATEGIES Dominated strategy for S1 is M No dominated strategy for S2 No dominant strategy for S1 No dominant strategy for S2

-------------------------- | S2 S1 | L R ----------+--------------- U | (1; 1) (0; 4) D | (0; 2) (2; 1) --------------------------

Iteration 3

DOMINATED AND DOMINANT STRATEGIES No dominated strategy for S1 No dominated strategy for S2 No dominant strategy for S1 No dominant strategy for S2

. . gamet , pay(1, 1, 0, 2,0,4 \0,2,5,0,1,6 \0,2,1,1,2,1) player1(U M D) player2( > L C R) elids

---------------------------------- | S2 S1 | L C R ----------+----------------------- U | (1; 1) (0; 2) (0; 4) M | (0; 2) (5; 0) (1; 6) D | (0; 2) (1; 1) (2; 1) ----------------------------------

Iteration 1

DOMINATED AND DOMINANT STRATEGIES No dominated strategy for S1 Dominated strategy for S2 is C No dominant strategy for S1 No dominant strategy for S2

-------------------------- | S2 S1 | L R ----------+--------------- U | (1; 1) (0; 4) M | (0; 2) (1; 6) D | (0; 2) (2; 1) --------------------------

Iteration 2

DOMINATED AND DOMINANT STRATEGIES Dominated strategy for S1 is M No dominated strategy for S2 No dominant strategy for S1 No dominant strategy for S2

-------------------------- | S2 S1 | L R ----------+--------------- U | (1; 1) (0; 4) D | (0; 2) (2; 1) --------------------------

Iteration 3

DOMINATED AND DOMINANT STRATEGIES No dominated strategy for S1 No dominated strategy for S2 No dominant strategy for S1 No dominant strategy for S2

. . * examples from http://web.mit.edu/14.12/www/02F_lecture102.pdf . . gamet, pay(-1, -1, 1, -10\ -10, 1, 2, 2) player1(C D) player2(C D) elids neps > nefms

------------------------------ | S2 S1 | C D ----------+------------------- C | (-1; -1) (1; -10) D | (-10; 1) (2; 2) ------------------------------

Iteration 1

DOMINATED AND DOMINANT STRATEGIES No dominated strategy for S1 No dominated strategy for S2 No dominant strategy for S1 No dominant strategy for S2

NASH EQUILIBRIUM IN PURE STRATEGIES 1. C C (-1; -1) 2. D D (2; 2)

NASH EQUILIBRIUM IN FULLY MIXED STRATEGIES p = 0.10 q = 0.10

(0.10*[C]+0.90*[D], 0.10*[C]+0.90*[D])

Expected equilibrium payoff for S1 0.10*-1+(1-0.10)*1 = .8 0.10*-10+(1-0.10)*2 = .8

Expected equilibrium payoff for S2 0.10*-1+(1-0.10)*1 = .8 0.10*-10+(1-0.10)*2 = .8

. . * examples from http://web.mit.edu/14.12/www/02F_slides5_602.pdf (there is a > range) . . * gamet, pay(2, 2, 4, 0 \0, 4, 5, 0) player1(Rabbit Stang) player2(Rabbit Sta > ng) elids neps nefms . . * examples from http://www.virtualperfection.com/gametheory/3.1.Nash%20Equili > brium.1.0.pdf . . gamet, payoff(9,9,0,8 \ 8,0,7,7) player1(U D) player2(l r) elids neps nefms

-------------------------- | S2 S1 | l r ----------+--------------- U | (9; 9) (0; 8) D | (8; 0) (7; 7) --------------------------

Iteration 1

DOMINATED AND DOMINANT STRATEGIES No dominated strategy for S1 No dominated strategy for S2 No dominant strategy for S1 No dominant strategy for S2

NASH EQUILIBRIUM IN PURE STRATEGIES 1. U l (9; 9) 2. D r (7; 7)

NASH EQUILIBRIUM IN FULLY MIXED STRATEGIES p = 0.88 q = 0.88

(0.88*[U]+0.13*[D], 0.88*[l]+0.13*[r])

Expected equilibrium payoff for S1 0.88*9+(1-0.88)*0 = 7.875 0.88*8+(1-0.88)*7 = 7.875

Expected equilibrium payoff for S2 0.88*9+(1-0.88)*0 = 7.875 0.88*8+(1-0.88)*7 = 7.875

. . gamet, payoff(2,1,0,0 \ 0,0,1,2) player1(U D) player2(l r) elids neps nefms

-------------------------- | S2 S1 | l r ----------+--------------- U | (2; 1) (0; 0) D | (0; 0) (1; 2) --------------------------

Iteration 1

DOMINATED AND DOMINANT STRATEGIES No dominated strategy for S1 No dominated strategy for S2 No dominant strategy for S1 No dominant strategy for S2

NASH EQUILIBRIUM IN PURE STRATEGIES 1. U l (2; 1) 2. D r (1; 2)

NASH EQUILIBRIUM IN FULLY MIXED STRATEGIES p = 0.67 q = 0.33

(0.67*[U]+0.33*[D], 0.33*[l]+0.67*[r])

Expected equilibrium payoff for S1 0.33*2+(1-0.33)*0 = .66666667 0.33*0+(1-0.33)*1 = .66666667

Expected equilibrium payoff for S2 0.67*1+(1-0.67)*0 = .66666667 0.67*0+(1-0.67)*2 = .66666667

. . * examples from http://www.virtualperfection.com/gametheory/3.2.NashEq2x2Norm > alFormGames.1.0.pdf . . gamet, payoff(1,-1,3,0 \ 4,2 ,0,-1) player1(U D) player2(l r) ls1(A) ls2(B) e > lids neps nefms

---------------------------- | B A | l r ----------+----------------- U | (1; -1) (3; 0) D | (4; 2) (0; -1) ----------------------------

Iteration 1

DOMINATED AND DOMINANT STRATEGIES No dominated strategy for A No dominated strategy for B No dominant strategy for A No dominant strategy for B

NASH EQUILIBRIUM IN PURE STRATEGIES 1. U r (3; 0) 2. D l (4; 2)

NASH EQUILIBRIUM IN FULLY MIXED STRATEGIES p = 0.75 q = 0.50

(0.75*[U]+0.25*[D], 0.50*[l]+0.50*[r])

Expected equilibrium payoff for A 0.50*1+(1-0.50)*3 = 2 0.50*4+(1-0.50)*0 = 2

Expected equilibrium payoff for B 0.75*-1+(1-0.75)*2 = -.25 0.75*0+(1-0.75)*-1 = -.25

. . gamet, payoff(1,1.5,3,1 \ 4,2 ,3,3) player1(U D) player2(l r) ls1(A) ls2(B) e > lids neps nefms

------------------------------ | B A | l r ----------+------------------- U | (1; 1.5) (3; 1) D | (4; 2) (3; 3) ------------------------------

Iteration 1

DOMINATED AND DOMINANT STRATEGIES No dominated strategy for A No dominated strategy for B No dominant strategy for A No dominant strategy for B

NASH EQUILIBRIUM IN PURE STRATEGIES 1. D r (3; 3)

NASH EQUILIBRIUM IN FULLY MIXED STRATEGIES p = 0.67 q = 0.00

(0.67*[U]+0.33*[D], 0.00*[l]+1.00*[r])

Expected equilibrium payoff for A 0.00*1+(1-0.00)*3 = 3 0.00*4+(1-0.00)*3 = 3

Expected equilibrium payoff for B 0.67*1.5+(1-0.67)*2 = 1.6666667 0.67*1+(1-0.67)*3 = 1.6666667

. . * examples from http://levine.sscnet.ucla.edu/econ201/game.pdf . . gamet, payoff(1,1,0,0 \ 0,0,1,1) player1(U D) player2(l r) ls1(A) ls2(B) elid > s neps nefms

-------------------------- | B A | l r ----------+--------------- U | (1; 1) (0; 0) D | (0; 0) (1; 1) --------------------------

Iteration 1

DOMINATED AND DOMINANT STRATEGIES No dominated strategy for A No dominated strategy for B No dominant strategy for A No dominant strategy for B

NASH EQUILIBRIUM IN PURE STRATEGIES 1. U l (1; 1) 2. D r (1; 1)

NASH EQUILIBRIUM IN FULLY MIXED STRATEGIES p = 0.50 q = 0.50

(0.50*[U]+0.50*[D], 0.50*[l]+0.50*[r])

Expected equilibrium payoff for A 0.50*1+(1-0.50)*0 = .5 0.50*0+(1-0.50)*1 = .5

Expected equilibrium payoff for B 0.50*1+(1-0.50)*0 = .5 0.50*0+(1-0.50)*1 = .5

. . gamet, payoff(2,2,-10,0 \ 0,-10,1,1) player1(U D) player2(l r) ls1(A) ls2(B) > elids neps nefms

------------------------------ | B A | l r ----------+------------------- U | (2; 2) (-10; 0) D | (0; -10) (1; 1) ------------------------------

Iteration 1

DOMINATED AND DOMINANT STRATEGIES No dominated strategy for A No dominated strategy for B No dominant strategy for A No dominant strategy for B

NASH EQUILIBRIUM IN PURE STRATEGIES 1. U l (2; 2) 2. D r (1; 1)

NASH EQUILIBRIUM IN FULLY MIXED STRATEGIES p = 0.85 q = 0.85

(0.85*[U]+0.15*[D], 0.85*[l]+0.15*[r])

Expected equilibrium payoff for A 0.85*2+(1-0.85)*-10 = .15384615 0.85*0+(1-0.85)*1 = .15384615

Expected equilibrium payoff for B 0.85*2+(1-0.85)*-10 = .15384615 0.85*0+(1-0.85)*1 = .15384615

. . gamet, payoff(6,6,2,7 \ 7,2,0,0) player1(U D) player2(l r) ls1(A) ls2(B) elid > s neps nefms

-------------------------- | B A | l r ----------+--------------- U | (6; 6) (2; 7) D | (7; 2) (0; 0) --------------------------

Iteration 1

DOMINATED AND DOMINANT STRATEGIES No dominated strategy for A No dominated strategy for B No dominant strategy for A No dominant strategy for B

NASH EQUILIBRIUM IN PURE STRATEGIES 1. U r (2; 7) 2. D l (7; 2)

NASH EQUILIBRIUM IN FULLY MIXED STRATEGIES p = 0.67 q = 0.67

(0.67*[U]+0.33*[D], 0.67*[l]+0.33*[r])

Expected equilibrium payoff for A 0.67*6+(1-0.67)*2 = 4.6666667 0.67*7+(1-0.67)*0 = 4.6666667

Expected equilibrium payoff for B 0.67*6+(1-0.67)*2 = 4.6666667 0.67*7+(1-0.67)*0 = 4.6666667

. . * examples from http://www.polisci.ucsd.edu/~bslantch/courses/gt/problem-sets > /w0403s.pdf . . gamet, payoff(2,2,1,3\ 3,1,0,0) ls1(Player1) ls2(Player2) player1(D H) player > 2(D H) elids neps nefms

-------------------------- | Player2 Player1 | D H ----------+--------------- D | (2; 2) (1; 3) H | (3; 1) (0; 0) --------------------------

Iteration 1

DOMINATED AND DOMINANT STRATEGIES No dominated strategy for Player1 No dominated strategy for Player2 No dominant strategy for Player1 No dominant strategy for Player2

NASH EQUILIBRIUM IN PURE STRATEGIES 1. D H (1; 3) 2. H D (3; 1)

NASH EQUILIBRIUM IN FULLY MIXED STRATEGIES p = 0.50 q = 0.50

(0.50*[D]+0.50*[H], 0.50*[D]+0.50*[H])

Expected equilibrium payoff for Player1 0.50*2+(1-0.50)*1 = 1.5 0.50*3+(1-0.50)*0 = 1.5

Expected equilibrium payoff for Player2 0.50*2+(1-0.50)*1 = 1.5 0.50*3+(1-0.50)*0 = 1.5

. . gamet, payoff(1,1,0,0\ 0,0,1,1) ls1(Player1) ls2(Player2) player1(Bridge Barn > ) player2(Bridge Barn) elids neps nefms

-------------------------- | Player2 Player1 | Bridge Barn ----------+--------------- Bridge | (1; 1) (0; 0) Barn | (0; 0) (1; 1) --------------------------

Iteration 1

DOMINATED AND DOMINANT STRATEGIES No dominated strategy for Player1 No dominated strategy for Player2 No dominant strategy for Player1 No dominant strategy for Player2

NASH EQUILIBRIUM IN PURE STRATEGIES 1. Bridge Bridge (1; 1) 2. Barn Barn (1; 1)

NASH EQUILIBRIUM IN FULLY MIXED STRATEGIES p = 0.50 q = 0.50

(0.50*[Bridge]+0.50*[Barn], 0.50*[Bridge]+0.50*[Barn])

Expected equilibrium payoff for Player1 0.50*1+(1-0.50)*0 = .5 0.50*0+(1-0.50)*1 = .5

Expected equilibrium payoff for Player2 0.50*1+(1-0.50)*0 = .5 0.50*0+(1-0.50)*1 = .5

. . gamet, payoff(1,1,0,0\ 0,0,2,2) ls1(Player1) ls2(Player2) player1(Bridge Barn > ) player2(Bridge Barn) elids neps nefms

-------------------------- | Player2 Player1 | Bridge Barn ----------+--------------- Bridge | (1; 1) (0; 0) Barn | (0; 0) (2; 2) --------------------------

Iteration 1

DOMINATED AND DOMINANT STRATEGIES No dominated strategy for Player1 No dominated strategy for Player2 No dominant strategy for Player1 No dominant strategy for Player2

NASH EQUILIBRIUM IN PURE STRATEGIES 1. Bridge Bridge (1; 1) 2. Barn Barn (2; 2)

NASH EQUILIBRIUM IN FULLY MIXED STRATEGIES p = 0.67 q = 0.67

(0.67*[Bridge]+0.33*[Barn], 0.67*[Bridge]+0.33*[Barn])

Expected equilibrium payoff for Player1 0.67*1+(1-0.67)*0 = .66666667 0.67*0+(1-0.67)*2 = .66666667

Expected equilibrium payoff for Player2 0.67*1+(1-0.67)*0 = .66666667 0.67*0+(1-0.67)*2 = .66666667

. . gamet, payoff(0,1,1,0\ 1,0,0,1) ls1(Player1) ls2(Player2) player1(A B) player > 2(A B) elids neps nefms

-------------------------- | Player2 Player1 | A B ----------+--------------- A | (0; 1) (1; 0) B | (1; 0) (0; 1) --------------------------

Iteration 1

DOMINATED AND DOMINANT STRATEGIES No dominated strategy for Player1 No dominated strategy for Player2 No dominant strategy for Player1 No dominant strategy for Player2

NASH EQUILIBRIUM IN PURE STRATEGIES None

NASH EQUILIBRIUM IN FULLY MIXED STRATEGIES p = 0.50 q = 0.50

(0.50*[A]+0.50*[B], 0.50*[A]+0.50*[B])

Expected equilibrium payoff for Player1 0.50*0+(1-0.50)*1 = .5 0.50*1+(1-0.50)*0 = .5

Expected equilibrium payoff for Player2 0.50*1+(1-0.50)*0 = .5 0.50*0+(1-0.50)*1 = .5

. . gamet, payoff(1,-1,3,0\ 4,2,0,-1) ls1(Player1) ls2(Player2) player1(D H) play > er2(D H) elids neps nefms

---------------------------- | Player2 Player1 | D H ----------+----------------- D | (1; -1) (3; 0) H | (4; 2) (0; -1) ----------------------------

Iteration 1

DOMINATED AND DOMINANT STRATEGIES No dominated strategy for Player1 No dominated strategy for Player2 No dominant strategy for Player1 No dominant strategy for Player2

NASH EQUILIBRIUM IN PURE STRATEGIES 1. D H (3; 0) 2. H D (4; 2)

NASH EQUILIBRIUM IN FULLY MIXED STRATEGIES p = 0.75 q = 0.50

(0.75*[D]+0.25*[H], 0.50*[D]+0.50*[H])

Expected equilibrium payoff for Player1 0.50*1+(1-0.50)*3 = 2 0.50*4+(1-0.50)*0 = 2

Expected equilibrium payoff for Player2 0.75*-1+(1-0.75)*2 = -.25 0.75*0+(1-0.75)*-1 = -.25

. . gamet, payoff(0,0,1,0\ 0,1,3,3) ls1(Player1) ls2(Player2) player1(D H) player > 2(D H) elids neps nefms

-------------------------- | Player2 Player1 | D H ----------+--------------- D | (0; 0) (1; 0) H | (0; 1) (3; 3) --------------------------

Iteration 1

DOMINATED AND DOMINANT STRATEGIES No dominated strategy for Player1 No dominated strategy for Player2 No dominant strategy for Player1 No dominant strategy for Player2

NASH EQUILIBRIUM IN PURE STRATEGIES 1. D D (0; 0) 2. H H (3; 3)

NASH EQUILIBRIUM IN FULLY MIXED STRATEGIES p = 1.00 q = 1.00

(1.00*[D]+0.00*[H], 1.00*[D]+0.00*[H])

Expected equilibrium payoff for Player1 1.00*0+(1-1.00)*1 = 0 1.00*0+(1-1.00)*3 = 0

Expected equilibrium payoff for Player2 1.00*0+(1-1.00)*1 = 0 1.00*0+(1-1.00)*3 = 0

. . * examples from http://www.andrew.cmu.edu/user/xinming/gametheory/ . . * gamet, pay(-1,-1, -9,0 \ 0, -9, -6,-6) ls1(Prisoner1) ls2(Prisoner2) player > 1(mum confess) player2(mum confess) elids neps nefms . . gamet, pay(-1,1,1,-1\1,-1,-1,1) player1(Head Tail) player2(Head Tail) elids n > eps nefms

---------------------------- | S2 S1 | Head Tail ----------+----------------- Head | (-1; 1) (1; -1) Tail | (1; -1) (-1; 1) ----------------------------

Iteration 1

DOMINATED AND DOMINANT STRATEGIES No dominated strategy for S1 No dominated strategy for S2 No dominant strategy for S1 No dominant strategy for S2

NASH EQUILIBRIUM IN PURE STRATEGIES None

NASH EQUILIBRIUM IN FULLY MIXED STRATEGIES p = 0.50 q = 0.50

(0.50*[Head]+0.50*[Tail], 0.50*[Head]+0.50*[Tail])

Expected equilibrium payoff for S1 0.50*-1+(1-0.50)*1 = 0 0.50*1+(1-0.50)*-1 = 0

Expected equilibrium payoff for S2 0.50*1+(1-0.50)*-1 = 0 0.50*-1+(1-0.50)*1 = 0

. . gamet, pay(2, 1, 0, 0 \ 0,0, 1, 2) ls1(Chris) ls2(Pat) player1(Opera Prize_Fi > ght) player2(Opera Prize_Fight) elids neps nefms

-------------------------------------- | Pat Chris | Opera Prize_Fight ------------+------------------------- Opera | (2; 1) (0; 0) Prize_Fight | (0; 0) (1; 2) --------------------------------------

Iteration 1

DOMINATED AND DOMINANT STRATEGIES No dominated strategy for Chris No dominated strategy for Pat No dominant strategy for Chris No dominant strategy for Pat

NASH EQUILIBRIUM IN PURE STRATEGIES 1. Opera Opera (2; 1) 2. Prize_Fight Prize_Fight (1; 2)

NASH EQUILIBRIUM IN FULLY MIXED STRATEGIES p = 0.67 q = 0.33

(0.67*[Opera]+0.33*[Prize_Fight], 0.33*[Opera]+0.67*[Prize_Fight])

Expected equilibrium payoff for Chris 0.33*2+(1-0.33)*0 = .66666667 0.33*0+(1-0.33)*1 = .66666667

Expected equilibrium payoff for Pat 0.67*1+(1-0.67)*0 = .66666667 0.67*0+(1-0.67)*2 = .66666667

. . gamet, pay(50, 90, 50, 100 \0, -10, 100, -100) ls1(Employee) ls2(Manager) pla > yer1(Work Shirk) player2(Monitor Not_monitor) elids neps nefms

------------------------------------ | Manager Employee | Monitor Not_monitor ----------+------------------------- Work | (50; 90) (50; 100) Shirk | (0; -10) (100; -100) ------------------------------------

Iteration 1

DOMINATED AND DOMINANT STRATEGIES No dominated strategy for Employee No dominated strategy for Manager No dominant strategy for Employee No dominant strategy for Manager

NASH EQUILIBRIUM IN PURE STRATEGIES None

NASH EQUILIBRIUM IN FULLY MIXED STRATEGIES p = 0.90 q = 0.50

(0.90*[Work]+0.10*[Shirk], 0.50*[Monitor]+0.50*[Not_monitor])

Expected equilibrium payoff for Employee 0.50*50+(1-0.50)*50 = 50 0.50*0+(1-0.50)*100 = 50

Expected equilibrium payoff for Manager 0.90*90+(1-0.90)*-10 = 80 0.90*100+(1-0.90)*-100 = 80

. . gamet, pay(6,4,2,6 \3,3,6,1) ls1(Player1) ls2(Player2) player1(T B) player2(L > R) elids neps nefms

-------------------------- | Player2 Player1 | L R ----------+--------------- T | (6; 4) (2; 6) B | (3; 3) (6; 1) --------------------------

Iteration 1

DOMINATED AND DOMINANT STRATEGIES No dominated strategy for Player1 No dominated strategy for Player2 No dominant strategy for Player1 No dominant strategy for Player2

NASH EQUILIBRIUM IN PURE STRATEGIES None

NASH EQUILIBRIUM IN FULLY MIXED STRATEGIES p = 0.50 q = 0.57

(0.50*[T]+0.50*[B], 0.57*[L]+0.43*[R])

Expected equilibrium payoff for Player1 0.57*6+(1-0.57)*2 = 4.2857143 0.57*3+(1-0.57)*6 = 4.2857143

Expected equilibrium payoff for Player2 0.50*4+(1-0.50)*3 = 3.5 0.50*6+(1-0.50)*1 = 3.5

. . * Prisoner's dilemma . . gamet , payoff(-5, -5, 0 , -6 \ -6, 0 , -1, -1) player1(Confess Not_confess) > player2(Confess Not_confess) domist neps

-------------------------------------- | S2 S1 | Confess Not_confess ------------+------------------------- Confess | (-5; -5) (0; -6) Not_confess | (-6; 0) (-1; -1) --------------------------------------

DOMINATED AND DOMINANT STRATEGIES Dominated strategy for S1 is Not_confess Dominated strategy for S2 is Not_confess Dominant strategy for S1 is Confess Dominant strategy for S2 is Confess

NASH EQUILIBRIUM IN PURE STRATEGIES 1. Confess Confess (-5; -5)

. . gamet, payoff(3, 3, 0, 5 \ 5, 0 , 1, 1) player1(C D) player2(C D) ls1(Provid > er) ls2(Customer) domist neps

-------------------------- | Customer Provider | C D ----------+--------------- C | (3; 3) (0; 5) D | (5; 0) (1; 1) --------------------------

DOMINATED AND DOMINANT STRATEGIES Dominated strategy for Provider is C Dominated strategy for Customer is C Dominant strategy for Provider is D Dominant strategy for Customer is D

NASH EQUILIBRIUM IN PURE STRATEGIES 1. D D (1; 1)

. . gamet, payoff(-3, -3, 0, -5 \ -5, 0 , -1, -1) player1(C D) player2(C D) dom > ist neps

------------------------------ | S2 S1 | C D ----------+------------------- C | (-3; -3) (0; -5) D | (-5; 0) (-1; -1) ------------------------------

DOMINATED AND DOMINANT STRATEGIES Dominated strategy for S1 is D Dominated strategy for S2 is D Dominant strategy for S1 is C Dominant strategy for S2 is C

NASH EQUILIBRIUM IN PURE STRATEGIES 1. C C (-3; -3)

. . gamet, payoff(-10,5, 8,-10\3, -1, 0,0) ls1(Player_1) ls2(Player_2) player1(Up > Down) player2(Left Right) elids neps nefms

------------------------------ | Player_2 Player_1 | Left Right ----------+------------------- Up | (-10; 5) (8; -10) Down | (3; -1) (0; 0) ------------------------------

Iteration 1

DOMINATED AND DOMINANT STRATEGIES No dominated strategy for Player_1 No dominated strategy for Player_2 No dominant strategy for Player_1 No dominant strategy for Player_2

NASH EQUILIBRIUM IN PURE STRATEGIES None

NASH EQUILIBRIUM IN FULLY MIXED STRATEGIES p = 0.06 q = 0.38

(0.06*[Up]+0.94*[Down], 0.38*[Left]+0.62*[Right])

Expected equilibrium payoff for Player_1 0.38*-10+(1-0.38)*8 = 1.1428571 0.38*3+(1-0.38)*0 = 1.1428571

Expected equilibrium payoff for Player_2 0.06*5+(1-0.06)*-1 = -.625 0.06*-10+(1-0.06)*0 = -.625

. . gamet , payoff(5, 5, 0 , 6 \ 6, 0 , 1, 1) player1(Confess Not_confess) player > 2(Confess Not_confess) domist neps

-------------------------------------- | S2 S1 | Confess Not_confess ------------+------------------------- Confess | (5; 5) (0; 6) Not_confess | (6; 0) (1; 1) --------------------------------------

DOMINATED AND DOMINANT STRATEGIES Dominated strategy for S1 is Confess Dominated strategy for S2 is Confess Dominant strategy for S1 is Not_confess Dominant strategy for S2 is Not_confess

NASH EQUILIBRIUM IN PURE STRATEGIES 1. Not_confess Not_confess (1; 1)


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