```-------------------------------------------------------------------------------
help for gamet
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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
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) ///

. 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

```