------------------------------------------------------------------------------- help forgamet-------------------------------------------------------------------------------

Game-theoretic calculations

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)

domistelidsnepsnefmsmaximingtree

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

scatter_options]

Description

gametrepresents 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 strictly dominated strategies, Nash equilibrium in pure and fully mixed strategies. Further,gametis 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_s2lab_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 generalRbyCpayoff matrix (help matrix input), where

#U1_r_cis the payoff forlab_s1iflab_s1chooses strategyrandlab_s2chooses strategyc#U2_r_cis the payoff forlab_s2iflab_s1chooses strategyrandlab_s2chooses strategycwith

r= 1,2, ...,Randc= 1,2, ...,C

Remark

gametis 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.

domistseeks strictly dominated and dominant strategies for each player.elidseliminates iteratively all strictly dominated strategies for each player.

nepsseeks Nash equilibrium in pure strategies.

nefmsseeks Nash equilibrium in fully mixed strategies (0<p<1 and 0<q<1). It works only ifRandCare equal to 2.

maximinseeks 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.

gtreeseeks 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 optionelidsis specifiedsavingpf()saves one file(filename#) for each iteration.

npathspecifies 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 labellab_s1andlab_s2on the game tree. Use clockpos to make changes from the default (9).

mlabppm(clockpos)specifies the position for#U1_r_c,#U2_r_con 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 forlab_s1,lab_s2and (#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 objectslab_s1,lab_s2and (#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_optionsare options of scatter.

Examples

. gamet, payoff(2, 2, 0, 1 \ 3, 0 , 1, 1) player1(High Low) player2(Buy No> t_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(Fo> otball 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) nefms

. 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

AuthorsNicola 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.

ReferenceMyerson, R. B. 1991.

Game Theory: Analysis of Conflict, Harvard University Press, Cambridge (MA).

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

Also seeOn-line: help for matrix, _variables, tabdisp, macrolists