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)
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 strictly 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 payoff for lab_s1 if lab_s1 chooses strategy r and lab_s2 chooses strategy c #U2_r_c is the payoff 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 strictly dominated and dominant strategies for each player. elids eliminates iteratively all strictly 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 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
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