On Generalized Tian Ji s Horse Racing Strategy

Transcription

1 Source: Iterdiscipliary Sciece Reviews, Vol. 37, o., pp , ; DOI:.79/3888Z.4 O Geeralized Tia Ji s Horse Racig Strategy Jia-Ju SHU School of echaical & Aerospace gieerig, ayag Techological Uiversity, 5 ayag Aveue, Sigapore ABSTRACT: Tia Ji s horse racig strategy, a faous Chiese leged, costitutes a proisig cocept to be applied to iportat issues i today s copetitive eviroet; this strategy is elaborated o ad aalyzed by exaiig the geeral case. The atheatical forulatio cocerig the calculatio of wiig, drawig or losig cobiatios ad probabilities is preseted to illustrate the iterestig isights o how aciet philosophies could proote thikig i busiess copetitiveess, i particular, the wisdo behid sacrificig the part for the beefit of the whole or sacrificig the short-ter objectives i order to gai the log-ter goal. Keywords: Tia Ji s horse racig strategy; uleria uber; Chiese leged. Itroductio I the oder world today, the survival of the fittest holds for geeral people, but the survival of the cuig is possible for itellectuals. owadays, with the developet of society, ost copaies face ore itese copetitio, especially for the weaker ad saller oes. I actuality, the stroger ad larger copaies possess ay ore advatages ad better talets tha the weaker ad saller oes. Despite the fact, the strogest of warriors has his Achilles heel. I such a copetitive world whereby alost everythig ca be rivaled i, it is ecessary for the weaker ad saller copaies to pla the order of egageet i order to copete with stroger ad larger rivals. This is just like playig a tea sport. The coach of the weaker tea ust carefully orgaize his players i suitable order of appearace. The idea is to sacrifice the part for the beefit of the whole or to sacrifice the short-ter objectives i order to gai the log-ter goal. This procedure is very siilar to Tia Ji s horse racig strategy, a faous Chiese leged. I this study, the geeralized Tia Ji s horse racig strategy is aalyzed atheatically to provide a iterestig isight ito decisio akig i dyaic ad highly copetitive global eviroets. The correct applicatio of the priciple of gaiig overall victory with partial loss ay eable the iferior side to gai superiority ad wi victory with a surprise ove. Tia Ji s horse racig strategy I aciet Chia, there was a era kow as Warrig States Period (43 BC BC) durig which Chia was ot a uified epire but divided by idepedet Seve Warrig States with coflictig iterests, oe of which was Qi State located i easter Chia. Fro 356 BC to 3 BC, the ruler of Qi State was Tia Yi-Qi (378 BC 3 BC), Kig Wei of Qi. The story of Tia Ji s horse racig

2 strategy, which is well-kow ad popular i Chia today, was origially recorded [] i the biography of Su Bi (? 36 BC), as a ilitary strategist i Qi State ruled by Kig Wei of Qi: Geeral Tia Ji, a high-rakig ary coader i Qi State, frequetly bet heavily o horse races with Kig Wei of Qi. Observig that their horses, divided ito three differet speed classes, were well-atched, Su Bi the advised Tia Ji, Go ahead ad stake heavily! I shall see that you wi. Takig Su Bi at his word, Tia Ji bet a thousad gold pieces with the Kig. Just as the race was to start, Su Bi couseled Tia Ji, Pit your slow horse agaist the Kig s fast horse, your fast horse agaist the Kig s ediu horse, ad your ediu horse agaist the Kig s slow horse. Whe all three horse races were fiished, although Tia Ji lost the first race, his horses prevailed i the ext two, i the ed gettig a thousad gold pieces fro the Kig. Aazedly, the victorious strategy (as did Tia Ji after followig Su Bi s advice) was rearkable to be achieved 3 years log before operatios research ad gae theory were iveted []. This was oly oe way that Tia Ji could clai a victory over the Kig, as illustrated i Figure. All the other optios would preset Tia Ji with loss. Su Bi s victorious advice, sice called Tia Ji s horse racig strategy, ca be exteded to a sceario where Tia Ji ad the Kig would race horses withi a disjoit stratificatio of a arbitrary uber differet speed classes. I order to facilitate the aalysis of the geeralized Tia Ji s -horse racig strategy, the horses owed by two players: Tia Ji (T ) ad the Kig ( K ) are deoted respectively by T ad K, where the subscript,,, is defied as player T s or K s horse i the th speed class. I this sceario of -horse racig, T s horse i the faster class is able to beat K s horse i the slower class, but T s horse is uable to beat K s horse i the sae or faster class. Without losig geerality, the relative racig capabilities of horses are T K T K for ay,,,, where the sybol eas uable to beat ad the larger subscript correspods to the slower class.

3 Figure : Tia Ji s horse racig strategy T ad K would choose the sae class of -horse racig, that is, the pairwise racig is T T T. Because T s horse is slower tha K s oe i the sae class K K K T K,,,,, T s horses would lose all races. The essece of Tia Ji s horse racig strategy is that the origially-classified racig appearace of T s horses should be shifted oe place i order to achieve the T s best result. The best strategy for the geeralized Tia Ji s -horse racig, T T T suggested by Su Bi, should be the pairwise racig. T would clai a victory K K K of the -horse racig with oe loss ad wis. If K always chooses the racig appearace K, K,, K agaist T s best respose T, T,, T, T would wi every bet. aturally, K would soo realize that the racig appearace K, K,, is resultig i recurret losses. K would becoe a active player ad cosider a alterative racig appearace to tur the racig aroud. A copetitive situatio is ecoutered for each player copetig T T T with a total of! cobiatorial pairwise racig (where is the K K K perutatio) available to T ad K. The above explaatio idicates that T would lose all races for the uit perutatio (as T s worst strategy) ad T would clai a victory with oe loss ad wis for the shift perutatio (as T s best strategy). The the atural K 3

4 questio is what is T s wiig probability for radoly-pairwise racig betwee T s ad K s horses. The equivalet questio is how ay perutatios are available to T as T s victorious strategies. Theore : The uber of T havig exactly wis i -horse racig is the uleria uber [3],!,, the uber of perutatios o,,, with!! exactly excedaces. Proof A excedace of the perutatio o,, ad the uleria uber,, wi is K T, that is,, is defied as ay idex such that,, is defied as the uber of the perutatio o, with exactly excedaces [4]. It is obvious that the existece coditio of T havig oe for ay idex. To deterie the uber of T s wis is equivalet to coutig the uber of, which is a excedace of the perutatio i the parlace of cobiatorics. So the uber of T havig exactly wis i -horse racig is the,. The detailed proof for the suatio forula of the uleria uber, uleria uber!,, is give as follows.!! There are two ways of gettig a -perutatio with excedaces fro a - perutatio by isertig the etry. ither the -perutatio has excedaces, ad the isertio of does ot for a ew excedace, or excedaces, ad the isertio of does for a ew excedace. I the first case, the etry is placed at the ed, or at the positio of oe of the excedaces ad the replaced oe is oved ito the ed. I the secod case, the etry is placed at the positio of oe of the o-excedaces ad the replaced oe is oved ito the ed. The desired recurrece is obtaied as,,, for all. (T)! ote that for bioial coefficiet,,!!. Therefore, we have 4

5 5,,,,,, where the last step uses (T) ad,,. By atheatical iductio o, the above expressio ca be used to prove the Worpitzky s idetity., all for (T) Usig (T) with,, ad, we get.,,,,,!!! The theore is proved. I view of the syetry property of the uleria uber, that is,,,, Theore ca be expressed equivaletly as follows i Theore. Theore : The uber of T havig exactly o-wis i -horse racig is the uleria uber.

6 I both theores, it is iterestig to ote that the geeralized Tia Ji s horse racig strategy, as the extesio of the faous Chiese leged, ca be viewed as a practical deostratio of applyig the uleria uber. There are oly three outcoes for T, aely the wiig cobiatio (, )! (, ) odd eve with probability! (, ) (,! ) odd, eve the drawig cobiatio (, ) odd eve with probability! (, ) odd, eve or the losig cobiatio (, )! odd eve with probability! (, ) odd, eve which are show i Table. horses Table : Cobiatio ad probability with variable uber of horses Total! Wiig Drawig Losig Cobiatio (with probability) (%) (%) (%) (%) (5%) (5%) 3 6 (7%) (%) 5 (83%) 4 4 (4%) (46%) (5%) 5 7 (3%) (%) 93 (78%) (8%) 3 (4%) 36 (5%) (6%) (%) 378 (74%) (%) 569 (39%) 6 (5%) (9%) (%) (7%) (4%) 3354 (36%) 844 (5%) 6

7 Probability Probability Figure : Tred of probability for odd-ubered horses Wiig Drawig Losig odd-ubered horses Figure 3: Tred of probability for eve-ubered horses Wiig Drawig Losig eve-ubered horses 7

8 Probabilities for odd- or eve- ubered horses are plotted respectively i Figures ad 3. Fro the results illustrated above, the probabilities follow the uleria distributio ad there are two detectable characteristics. First, the case of odd-ubered horse racig has o drawig, which drawig happes oly i the case of eve-ubered horse racig. Secod, the losig probabilities of ay eve-ubered horse racig are always at the costat 5% regardless of horse uber ivolved. I the odd-ubered horse racig, the wiig ad losig probabilities coverge to the costat 5% as horse uber icreases due to o drawig; whereas i the eve-ubered horse racig, the wiig ad drawig probabilities coverge to a costat 5% as horse uber icreases due to the costat 5% losig. Overall, the wiig probability of the odd-ubered horse racig is uch higher tha that of the adjacet eve-ubered cases, for exaple, 3% of 5 is uch higher tha 4% of 4 ad 8% of 6. This shows that odd-ubered horse racig gives a opportuity of wiig better tha a eveubered case does. This iplies that the best cobat uits should be odd-ubered. Of course, as the cobat efficiecy would be gettig better as icreases, the difficulty of cotrollig uch large cobat uits would be ecoutered. o drawig occurs i the odd-ubered horse racig, which eas that a decisive outcoe ust be reached istead of a staleate. ore iportatly, Figures ad 3 suggest that the ore horses ivolved, the larger, the higher is the wiig probability. Philosophically, it is typically the epitoe of wiig i ubers. 3 Cocludig Rearks This paper is a geeralizatio of the atheatical versio of Tia Ji s horse racig strategy ivolvig a oe-to-oe cotest betwee two sets of racig horses withi a disjoit stratificatio of speed classes. The forulatio of deteriig the wiig, drawig ad losig probabilities of the geeralized Tia Ji s horse racig strategy for ay give uber of racig horses is discussed. Based o the uleria uber, the way of calculatig the uber of havig exactly wis i -horse racig is straightforward, thereby eablig us to fid the probability of wiig a etire gae by havig ore wis tha losses. The wisdo behid Tia Ji s horse racig strategy is to sacrifice the part for the beefit of the whole or to sacrifice the short-ter objectives i order to gai the log-ter goal. As a exaple of the geeralized Tia Ji s horse racig strategy, the atheatical treatet of how to sacrifice could proote philosophical thikig i dealig with coplicated situatios. I busiess, product diversificatio requires very differet fiacial, hua ad techological resources. Trade-offs are ievitable. A copay has to decide which part should be sacrificed for the beefit of the whole or which short-ter objectives should be sacrificed i order to gai the log-ter goal. Studyig the theory of Tia Ji s horse racig strategy provides iterestig isight ito decisio akig i dyaic ad highly copetitive global eviroets. Refereces [] Si-a, Qia (9 BC) Su Zi Wu Qi Lie Zhua (Biographies of Su Zi ad Wu Qi) i Shi Ji (Records of History), 65(5), (i Chiese). glish traslatio by Yag, Xia Yi ad Yag, Gladys (8) Selectios fro Records of the Historia, I, Foreig Laguages Press. 8

9 [] Shu, Jia-Ju, Wag, Qi-We ad Yog, Kia-Ya () DA-based coputig of strategic assiget probles, Physical Review Letters, 6(8), 887. [3] uler, Leohard (755) Istitutioes Calculi Differetialis cu eius vsu i Aalysi Fiitoru ac Doctria Serieru, Ipesis Acadeiae Iperialis Scietiaru, Petropolitaae, (i Lati). glish traslatio by Blato, J.D. () Foudatios of Differetial Calculus, Spriger. [4] Rose, Keeth H. () Hadbook of Discrete ad Cobiatorial atheatics, CRC Press. 9