A Matrix Representation of an n-person 0-1 Game and Its 0-1 Tail Algorithm to Find (Strictly) Pure Nash Equilibria

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1 Mathematcs ad Compter Scece 26; (): do:.648/j.mcs.26.2 A Matrx Represetato of a -Perso - Game ad Its - Tal Algorthm to Fd (Strctly) Pre Nash Eqlbra Day Jag Isttto of Game Theory ad Its Applcato, Haha Isttte of Techology, Laygag, Cha Emal address: jagday425@26.com To cte ths artcle: Day Jag. A Matrx Represetato of a -Perso - Game ad Its - Tal Algorthm to Fd (Strctly) Pre Nash Eqlbra. Mathematcs ad Compter Scece. Vol., No., 26, pp do:.648/j.mcs.26.2 Receved: Aprl 2, 26; Accepted: Aprl, 26; Pblshed: May 9, 26 Abstract: A -perso doble acto game,.e., a -perso strategy game,.e., every player has ad oly has two actos, s a typcal ad sefl game. It has bee proved that a -perso doble game, every player s two actos ca be deoted by ad. A -perso doble acto game,.e., every player s acto set s deoted as {,}, s sad to be a -perso - game. I ths paper, we frst gve a matrx represetato of a -perso - game ad the gve a ew ad smpler algorthm to fd all the (strctly) pre Nash eqlbra for a -perso - game, called - tal algorthm. Specally, ths algorthm ca be smplfed f the game s symmetrcal. Some examples are gve to show the algorthm. Keywords: -perso Doble Acto Game, -perso - Game, Symmetry, Matrx Represetato, - Tal Algorthm, Symmetrcal 3-perso PD, Symmetrcal 3-perso Game of Ratoal Pgs. Itrodcto Some classcal Examples sch as prsoers dlemma, battle of the sexes, hawk-dove game, game of warrors, game of ratoal pgs, ad game of patrollg by poor ad rch me occr may lteratres. These games are all 2 2 b-matrx games. They are the smplest ad the most typcal. There are oly 2 players ad each of them has oly 2 pre actos these games. If we exted mber of players a 2 2 bmatrx game to ay atral mber to be greater tha ad every player has exactly 2 actos, the we obta other smple ad typcal games, -perso doble acto games. These games have may backgrods. For example, the most mportat ad the most typcal oe of them s famos - perso prsoer's dlemma (See [-4]). It s obvos that - perso doble acto games have very strog character ad so there are specal solvg methods ot to be cotaed that of geeral -perso strategy games. Jag (25) [5] proved that every doble acto game ca be deoted as a - game whch s smpler ad ca be easly dealt wth by bary mbers. That s, every player s two actos ca be deoted as ad, respectvely. It s a mportat topc to fd all (strctly) pre Nash eqlbra for a -perso doble acto game. Jag (25) [5] gves a seve method, bt the method s complex. I ths paper, we have two tasks. Frst oe of them s to gve a matrx represetato of a -perso - game ad the secod oe s to gve a ew algorthm, called - tal algorthm, to fd all (strctly) pre Nash eqlbra for a - perso - game. Ths paper s dvded to for sectos. Secto 2 gves a matrx represetato of -perso - games. Secto 3 gves a - tal algorthm to fd all (strctly) pre Nash eqlbra. Fally, secto 4 defes symmetrcal - games ad smplfes - tal algorthm. 2. A Matrx Represetato of -perso - Games A -perso - game ca be wrtte as the formal strctre [{,2,, },{,}, b b b Γ ], where s the -th player s seral mber, =, 2,,, ad are the player 's two actos, b b b s the player 's tlty der the stato bb b, b {,}, =,2,,. More precsely, whe every player ses hs or her acto b, =,2,,, the bb b player s proft s. As everyoe kows, Schellg (98) [6] wrote a 2-perso b2b - game Γ [{,2},{,}, ] as

2 6 Day Jag: A Matrx Represetato of a -Perso - Game ad Its - Tal Algorthm to Fd (Strctly) Pre Nash Eqlbra (, 2 ) (, 2 ). (, 2 ) (, 2 Based o Schellg (98) [6], McCa (24) [7] wrote a bb 3 2b 3-perso - game Γ [{,2,3},{,}, ] as two Schellg matrces as follows. (, 2, 3 ) (, 2, 3 ), ad (, 2, 3 ) (, 2, 3 (, 2, 3 ) (, 2, 3 ). (, 2, 3 ) (, 2, 3 The frst matrx deotes the case that whe the player 3 ses hs or her acto ad the secod does that whe the player 3 ses hs or her acto. Jag (25) [5] ses the symbols bb b (,,, ), =,2,,. b b b b b b bb b to wrte a -perso - game, where ca be ay atral mber to be greater tha. Now let s gve two matrx represetatos of a -perso - game as follows. The frst oe s 2,,2,,,2,,,2, ad secod oe s trasposto of the frst oe,.e.,,,,,,,,,, As a example, we oly expla the frst oe becase the secod s smlar. Row mbers of the matrx are deoted by bary mbers wth the legth accordg to ther sze order. Ths the matrx has 2 rows ad ther decmal mbers are from to 2. The j -th colm deotes the player j, j =,,,. A row mber wth bary mber deotes a stato. For example, for a 3-perso - game, deotes that the player ad 3 se ther actos ad that the 2 does. The elemet b b b bb 2, of the,. matrx deotes the player s tlty der the stato bb b bb, where b =,, =,2,, A - Tal Algorthm to Fd (Strctly) Pre Nash Eqlbra For a mber b {,}, we wll wrte b = b,.e.,, f b =, b =, f b =. For a -perso - game [{,2,, },{,}, b b b Γ ], f there exsts a stato sch that bb b, * b {,}, =,2,,, * * bb b b bb b b, =,2,,, (3-) the bb b s called a pre Nash eqlbrm. A set of all the pre Nash eqlbra for the -perso - game Γ s wrtte as PNE( Γ ). Whe every eqaltes (3-) s strct,.e., * * bb b b bb b b >, =,2,,, (3-2) the bb b s called a strctly pre Nash eqlbrm. A set of all the strctly pre Nash eqlbra for the -perso - game Γs wrtte as SPNE( Γ ). The above cocepts ad marks ca be fod from Jag (25) [5]. As everyoe kow, Schellg (98) [6] gave a streak method to fd pre Nash eqlbra for a b-matrx game, or 2-perso doble acto game. Jag (25) [5] gave a seve method to fd pre strctly Nash eqlbra for a -perso - game. However, the seve method s complex. Based o or matrx represetato, we ca obta a ew algorthm to fd (strctly) pre Nash eqlbra, called - tal algorthm. Theorem 3.(- Tal Algorthm to Fd all Pre Nash Eqlbra) All pre Nash eqlbra for a -perso - game ca be fod by the followg algorhm. Frst, wrte ot the matrx of the gve -perso - game. The execte the followg algorthm. () Set bb b. (2) Set. > (3) If bb b b 2b, bb b b2b,, the go to (4); else go to(7). (4) Wrte (called a tal) after the matrx elemets wth the row mber bb b bb 2 (called a head), ad the let +. (5) If, the go to (3); else go to (6).

3 Mathematcs ad Compter Scece 26; (): (6) Let tal of bb b bb 2 be. (7) Let tal of bb b bb 2 be. (8) Takg the top bb b bb 2 of bary mbers wthot tal ad the go to (2) tl every bary row mber (.e., head) gets ts tal or. A stato bb b bb 2 s a pre Nash eqlbrm f ad oly f ts tal s marked. If every strctly eqalty b b b b2b, > bb b b2b, the step (3) s rewrtte as the correspodg o-strct oe bb b bb 2, bb b bb 2,, the a set of strctly pre Nash eqlbra ca be obtaed by the algorthm as well. Proof: For every statob b b, t eeds be checked that b b b > from =. They ca be exected, b b b, by the steps (2) ad (3). If the eqalty holds, the the stato b b b s ot a pre Nash eqlbrm ad so ts tal shold be marked by. If does ot, amely, b b b,, the the b b b, stato b b b s ot a pre Nash eqlbrm ad hece ts tal shold be marked by as well as. Ths ca be exected by the steps (3), frst part of (4) ad (7). Whe the eqalty > s satsfed, we b b b, b b b, shold check the ext player +,.e., we eed let +, whch s the secod part of the step (4). If ths ew meets the codto, the we eed recheck the eqalty b b b >. Whe the eqalty does ot, b b b, hold,.e., >, all the eqaltes >, =,2,, bb b b2b, bb b b2b, have bee checked. Ths the stato b b b s a pre Nash eqlbrm ad so ts tal shold be marked by. Those ca be exected by the steps (5) ad (6). The other parts of the algorthm are trval ad so they ca be omtted. Q. D. E. Example 3. By - tal algorthm, we ca kow that the 4- perso - game has two pre Nash eqlbra ad tals Symmetrcal -perso - Games ad Smplfed - Tal Algorthm Γ Defto 4. We wll say that a gve -perso - game [{,2,, },{,}, b b b ] s symmetrcal f () for ay stato b bj b b, t ca be obtaed b b b b that b = b =,, j =,,, ad j j (2) for ay permtato τ, the followg relato ca be ( ) ( ) obtaed ( b b,, b b τ ) ( b b τ τ =,, b b ). A - tal algorthm abot a symmetrcal - game ca be smplfed by the followg theorem. Theorem 4. If () b b < b ' b ',(2) b ' b ' ca be obtaed by a permtato o b b, ad (3) tal of b b has bee obtaed, the the tal of b ' b ' s the same as that of b b. Example 4. (Game of collectg oysters, refer to Roger (24)[7]). Three persos are collectg oysters Chesapeake Bay. They all kow that there s a oyster bed o the ortheast coast that has ever bee collected. If the oysters are ot harvested before the ext moth, they wll fetch more moey o the market after growg to matrty. These three collectors are crretly cosderg f they wold go collectg (acto ) or wat for some days (acto ). () Whether s the followg game symmetrcal? (2) Fd all the strctly pre Nash eqlbra. Solto: () Payoff matrx of ths game ca be wrtte as It s trval that every row satsfes the codto () for Theorem 4.. Now let s verfy the codto (2). Let τ () =, ad τ 2 () =. The τ (,, ) = τ (7,7,) = (7,,7) 3 2 = (,, ) = (,, ). τ () τ () τ () τ (,, ) = τ (7,7,) = (,7,7) = (,, ) = (,, ). τ 2() τ 2() τ 2() It s smlar for the rows, ad. Ths ths game s symmetrcal.

4 8 Day Jag: A Matrx Represetato of a -Perso - Game ad Its - Tal Algorthm to Fd (Strctly) Pre Nash Eqlbra (2) We have 3 2 tals tals tals tals tals tals By ths comptato we ca kow that they wold go collectg early. However, the stable stato abot dvdal ratoalty s worse tha that they wat for some days to collect. I fact, the game s a symmetrcal 3-perso prsoer s dlemma. Its geeral form s the follows. Example 4.2 (a symmetrcal 3-perso prsoer's dlemma) Three people, 2 ad 3, arrested wth stole property ther possesso, are beg tervewed separately by the polce. They kow that f they keep qet there s ot eogh evdece for them to be covcted of theft, ad so every oe of them gets a years jal setece for beg possesso of stole property. If they cofess to the theft, every oe of them gets c years prso. If oe cofesses ad both keep qet, oe who cofesses wll go free, whle each of the other two wll have d years jal setece. If oe keeps qet ad the other two cofess, the oe who keeps qet wll have e year jal setece, whle each of the other two wll have b year jal setece. Where < a < b < c < d e. If we se to deote keepg qet ad do cofess. The 3 2 tals c c c b b e b e b d d e b b d d d d a a a Therefore ther stable stato wth dvdal ratoalty s that everyoe cofesses. However, the stable stato s worse tha everyoe chooses keepg qet, o-stable. Example 4.3(A symmetrcal 3-perso game of ratoal pgs) Three pgs wth the same szes are pt a Sker box wth a specal pael at oe ed ad a food dspeser at the other. It eeds payc ts of cost to press the pael. Whe the pael s pressed, 3 ts of food are dspesed ad every pg ca eat some food. Whe the three pgs press the pael at the tme, every pg ca eat t of food. Whe two pgs press the pael, each of them ca eat a ts of food ad so the other oe ca eat 3 2a ts. Whe oe presses the pael, t ca eat 3 2b ts ad so each oe of the other two pgs ca eat b ts. Whe oe presses the pael, every pg caot eat aythg. Let s fd set of strctly pre Nash eqlbra. Solto: We have kow that whe the pael s pressed, 3 ts of food are dspesed ad every pg ca eat some food. The whe oe pg presses or two pgs press the pael, there s at least oe pg frst to eat ad ca get more tha t of food ad so every pg who presses the pael ca eat food whch s smaller tha t. Ths < a < ad < 3 2b < (.e., < b < 3 2 ). Sce 3 2 a ( c) = 2( a) + c > 3 2a > c, we ca obta 3 2 tals b b 3 2b c b 3 2b c b 3 2a a c a c 3 2b c b b a c 3 2a a c a c a c 3 2a c c c Case. Whe c 3 2b, the

5 Mathematcs ad Compter Scece 26; (): tals b b 3 2b c b 3 2b c b 3 2a a c a c 3 2b c b b a c 3 2a a c a c a c 3 2a c c c It ca be obtaed that the set of strctly pre Nash eqlbra s SPNE( Γ ) = {,,}. I words, whe oe pg who press the pael s proftable, the stable stato s that exact oe of the three pgs wold go to press the pael. Case 2. Whe c > 3 2b, the 3 2 tals b b 3 2b c b 3 2b c b 3 2a a c a c 3 2b c b b a c 3 2a a c a c a c 3 2a c c c Ths, shold oe pg pressg the pael lose more tha ga, oe wold press the pael. I ay case, t s mpossble for more tha oe pg to press the pael, or eqvaletly, at most oe pg wold go to press the pael. Fabac, Radoševc ad Magdalec (24) [8] created a ratoal pgs game exteded (brefly, RPGE), whch the trodcto of a thrd pg etals sgfcat strctral chages. I fact, ths artcle s the frst 3-ratoal pgs game. Jag (25b) [9] establshed L-system of axomatc theory of boxed pgs ad ts dedctve sb-systems were obtaed from t, sch as smple K-systems, stat K-systems ad tmg K- systems, K=,. Fally, the method to chage a game amog may pgs (ca be ftely may) to a game betwee a bg pg ad a small pg ad applcable degree of the method are gve. However, the game amog may ratoal pgs s a approxmate descrpto that ratoal pgs mst be sffcetly may. Jag(25c) [] defes that each of two -perso strategy games s sad to be a egatve game of the other oe f sm of ther tltes s eqal to zero. Sce games based o tradtoal boxed pgs story, sch as Jag (25b) [9], are called boxed pgs games or ratoal pgs games, t meas that they shold be postve, sch as 3 meas +3. Therefore egatve game of a boxed pgs (or ratoal pgs) game shold be called a egatve boxed pgs (or ratoal pgs) game. L, et al (25) [] gves a specal ratoal pgs game. Jag, et al (26) [2] researches the egatve game of the game gve by L, et al (25) [] ad gves a applcato to webste maagemet. Refereces [] Schellg T. C. (978). Mcro-Motves ad Macro-Behavor, W. W. Norto, New York. [2] Molader P. (992). The prevalece of free rdg, Joral of Coflct Resolto. 36: [3] Taylor M. (995). The Possblty of Cooperato. Cambrdge Uversty Press. New York. [4] Jbor Mags (999). Voltary Coerco, Collectve Acto ad the Socal Cotract, Departmet of Phlosophy, Ld Uversty. [5] Jag D. Y. (25a). Stato Aalyss of Doble Acto Games-Applcatos of Iformato Etropy to Game Theory. Detschlad/Germay:Lap Lambert Academc Pblshg. [6] Schellg T. C. (98). The Strategy of Collct. York: Harvard Uversty Press. [7] Roger A. McCa (24). Game Theory: A No-Techcal Itrodcto to the Aalyss of Strategy, Thomso Soth- Wester. [8] Fabac, R., D. Radoševc & I. Magdalec (24). Atogeerator-based modellg framework for developmet of strategc games smlatos: ratoal pgs game exteded. The Scetfc World Joral. [9] Jag D. Y. (25b). L-system of Boxed Pgs ad ts Dedctve Sb-systems------Based o Amal ad Ecoomc Behavor. Colmba (USA): Colmba Iteratoal Pblshg. [] Jag, D. Y. (25). Strct descrptos of some typcal 2 2 games ad egatve games. Ecoomcs. Specal Isse: Axomatc Theory of Boxed Pgs. 4(3-): 6-3. DOI.648/ j.eco.s [] L Q., D. Y., Jag, T. Matshsa, Y. B. Shao, X. Y. Zh (25). A Game of Boxed Pgs to Allow Robbg Food. Ecoomcs. Specal Isse: Axomatc Theory of Boxed Pgs. Vol. 4, No. 3-, pp do:.648/j.eco.s [2] Jag D. Y., Y. B. Shao, & X. Y. Zh (26). A egatve ratoal pgs game ad ts applcatos to webste maagemet. Game Vew. Vol. 2. o.. pp. -6.