Probability Model of Ties in Multistage Decision Process

A C T A U N I V E R S I T A T I S L O D Z I E N S I S

F O L IA O E C O N O M IC A 175, 2004

W i e s ł a w P a s e w i c z * , Wi e s ł aw W a g n e r * *

P R O B A B IL IT Y M O D E L O F T IE S IN M U L T IS T A G E D E C IS IO N P R O C E S S

Abstract. Probability m odel on m ultistage decision process is discussed with particular emphasis on special case using the rule R(4, 2). A n idea o f im portance graph ties is presented. Possibility recording probability o f success in multistage decision process as linear com bination others probabilities o f the decision process is presented as well.

Key words: M ultistage decision process, importance o f ties, rule R (k, Я).

1. IN T R O D U C T IO N

T h eo ry o f gam es developed essentially in tw o directio ns, co n tain in g so called extended gam es and m atrix games. T h e first case consists o f grap hs with ties an d edges. W alking from one tie o f the g ra p h to the next one is a decision process because the graph has a hierarchical stru c tu re o f m ultistage gam e. It is im p o rta n t to know probability o f en din g the decision process for each tic o f the graph. Such p erm anent m o n ito rin g perm its us to u n d ertak e ra tio n a l decisions.

In the p ap e r we present a probab ility m odel for a p a rtic u la r schem e o f m ultistage decision process. Besides, we give an a m easure o f im p o rtan ce m easure for ties o f the graph.

2. A R U L E O F R EA C H IN G A SU C C E S S O N T H E G R A PH

Let us assum e th a t the decision process is leaded o n g ra p h G ( T , E ) . T h e grap h consists o f ties T and edges E. F ro m each o f th e tie th ere are going o u t tw o edges nam ed strategies L (left) an d R (right). E ach o f these

* P hD , Agricultural U niversity, Szczecin.

tw o edges is going to the one o f tw o different tics with p rob abilities p and q = 1 - p , respectively.

T h e player tak in g decision (strategies L o r R) is ran d o m w alking on the graph. Ih e gam e is finished when the player reached a success according to the follow ing rule R (к , X):

The player ought to obtain at least к strategies L. The number o f these strategies has to exceed the number m strategies R with advantage at least X, that is k - m ^ X, X = 1, 2, 3, ...

We note th a t the g ra p h G ( T , E) depends o f param eters к an d X. T ak in g к = 4 and X = 2 we have the rule R (4, 2), w hich is used in tennis. T h e rule R (4 , 2) will be considered in the next point o f the paper.

3. P R O B A B IL IT Y O F S U C C E S S IN M U L T IS T A G E D E C ISIO N P R O C E S S U SIN G T H E RULE /1(4, 2)

Let us consider the rule R (k, A), w here к = 4 and X = 2. F ig u re 1 show s grap h G ( T , E ) in the case.

С И )

T hese ties d en o te states a : b , where a(b) is the n u m b er o f strategies L(R) reached by the player on the graph. T here are th ree states finishing the gam e w ith victory the player, th a t is 4 : 0 , 4:1 and 4 : 2 . A fter state 3 : 3 the gam e is contin u ed until (a + 2): a, for a = 3 ,4 , ... w ith th e player as a w inner.

T h e m ain idea considered problem is calculate p ro bab ilities P ( a : b) th a t the player reach a success in decision process depend ing on states, illustrated in Fig. 1. In recent p ap ers e.g. B. P. I i s i and D. M. B u r y c h (1971), L. M. R i d d l e (1988), W. P a s e w i c z a nd W. W a g n e r (2000) we can see the follow ing form ulas connected with tennis:

P(0 : 0 ) = /?4(1 + 4q + 10q 2) + ^ ~ 2pq ~ Pr ° k a bility o f success the player in state 0 :0 ,

20p 2q 3 - p ro b a b ility th a t the process starts in state 0 : 0 and ends in state 3 :3 ,

2

v = - p ro b a b ility o f success the player afte r state 3 :3 . 1 - 2 pq

A n aly sis o f sta te s fo r every stage decision p ro c ess fro m g ra p h G(T, E) gives th e general ru le fo r th e p ro b a b ility success en d in g the gam e o f the form

F o r in sta n c e if a = 0 an d b = 2, we h av e P ( 0 :2 ) = p 4 + 4 p 3qv, because there is only one p ath 0 : 2| 1 : 2 |2 : 2 |3 : 2 |4 : 2 from state to 0 : 2 state 4 : 2 and fo u r path s: 0 : 2 | 0 : 3 |1 : 3 |2 : 3 |3 : 3; 0 : 2| 1 : 2| 1 : 3 | 2 : 3 |3 :3; 0 : 2 | 1 : 2 | 2 : 2 | 2 : 3 | 3 : 3 a nd 0 : 2 | 1 : 2 | 2 : 2 | 3 : 2 | 3 : 3 from state 0 : 2 to state 3: 3.

4. D E F IN IT IO N O F T H E IM P O R T A N C E T IE S IN M U L T IS T A G E D E C ISIO N PR O C E SS

1 he player beings in one o f ties can choose the follow ing ties ch o osing strategies L o r R. H e can be interested in so called im p o rtan ce o f th e ties, as well. C. M o r r i s (1977) has introduced the follow ing d efin ition o f im portance tie I as a difference betw een tw o co n d itio n al p ro b ab ilities th a t is (see C r o u c h e r 1998):

1 = P(S(GD) I the player chooses strategy L ) - P ( S ( G D ) \ the player chooses strategy R), w here S(GD) denotes “ the player reaches a success o n the g rap h decision” . Im p o rta n c e m easure o f tie a : h we calculate using the follow ing fo rm u la

where a, b = 0, 1 ,2, 3 or a = 4, 5, an d h = a - 1, a, a + 1 and P ( a : h) is given by (1).

A t present, we will show three exam ples ap plication o f fo rm u la (2). Let I ( a : b) I ( c : d) d enotes th a t tie a : h is “m ore im p o rta n t” th a n tie c : d.

T h e follow ing inequalities: (a) / ( 2 : 3) > / ( 1 : 2), (b) / (3 : 2) > / ( 2 : 1) and (c') 1(2: 3) > 1 ( 2 :2 ), if p ^ 1/2, (c") / ( 2 : 3 ) < 7 (2 :2 ), if p < l / 2 are tru e, because of: 1 ( 2 : 3) = P ( 3: 3) - P ( 2 :4 ) = v - 0 = v, /(1 :2 ) = P ( 2 :2) - P(1 : 3) = p 2 + 2pqv - p 2v = (1 - 2pq)v = (1 - p 2)v, I ab = P(a + l - . b ) - P ( a - . b + l )

(2)

N a tu ra l generalization o f the rule R (4, 2) is R(k, 2), w here к = 2, 3, 4, T hen th e equality (1) will be form

Pk( a : b ) = p* 0

(3)

Let us w rite d ow n a few p artic u la r cases o f eq uality (3) fo r a, b = 0, 1 and к = 2, 3, 4, 5. W e have

P 2(0 : 0) = p 2 + 2pqv,

P з(° : 0) = p 3( l + 3q) + 6p 2q 2v,

P 4( 0 : 0 ) = p 4( 1 + 4 9 + 10 q2) + 20p3,?3 v, P 5(0 : 0) = p s(l + 5 9 + 15q2 + 35q3) + 70p * q \ .

O f course the pro b ab ilities P 4( 0 : 0 ) , P 4(l :0 ) and P 4( 0 : 1) we could o b tain using te eq uality (1).

N ow , we prove the follow ing relation: Similarly, P 2( 1 :0 ) = p + qv, P 3( l : 0 ) = p 2(l + 2q) + l p q 2v, P 4(l : 0) = p 2(\ + 3q + 6q 2) + 10p 2q 3v, P 5( 1 :0 ) = p \ \ + 4 q + 10 q2 + 20<?3) + 35 p 3q \ and

v,

₍₄₎

+ 4 1 + 2 V V 2v = pP*(l :0) + q Pk(0: 1).

In th e sam e m a n n e r we can show that:

Pk(0 : 0) = p 2Pk( 2: 0) + 2pqPk(\ : 1) + q 2Pk(0 : 2), Pk(0 : 0 ) = p 3P*(3 :0 ) + ) p 2qPk( 2 :1 ) + 3M 2P t (l : 2) + q 3P * (0 : 3), and generally Pjł(0 : 0) = £ ( П) р я- тятР к( ( п - т ) :т) (5) m = 0 \ m/ for n = 1, 2, ... and n > m .

T h u s P t ( 0 :0 ) is a linear com b in atio n o f p rob ab ilities reach ing a success in the decision process w hen the player is in the tie with state ( n - m ) : m and P k((n — m ) : m ) we calculate according to fo rm u la (3).

6. C O N C L U SIO N

In this p ap e r was considered p robability m odel o f m u ltistage decision process using the rule R(k, A) w ith special case for к = 4 an d A = 2. T he player w alking on g ra p h G( T , E) (Fig. 1) is tak in g decisions (strategies L or R) w hich are d ep e n d en t on A, к and ties o f the g rap h . H reach a success (for A = 2) if n u m b er к o f strategies L will be a t least by tw o m ore th a n n u m b er m o f strategies R.

C ase for Я = 1 is a sim ple case and form ula (3) reduces to the form

E specially interestin g case is for X = 3. T h en the p ro b a b ility m odel o f m ultistage decision process is m ore com plicated. P rob lem R(k , 3) a u th o rs will be present in the next paper.

C r o u c h e r J. S. (1998), D eveloping S trategies in Tennis, [in:] J. B e n n e t t (ed.), S ta tistics in S p o rt, A rnold, N ew Y ork, 157-171.

H s i B. P., B u r y c h D . M. (1971), Games o f Two Players, J. R. Statist. Soc., Series C, 20, 86-92. M o r r i s C. (1977), The M o s t Im p o rta n t P oint in Tennis, [in:] S. P. L a d a n y , R. E. M a c h o l

(eds.), O ptim a l Stra teg ies in Sports, N o rth -H o lla n d , N ew Y ork, 131-140.

P a s e w i c z W. , W a g n e r W. (2000), C h a ra kterystyka m odeli probabilistycznych tv opisie gem a i seta tv tenisie ziem n ym (english: D escription o f the Probability M o d e li o f G am e and S et in Tennis), „W yzw ania i D ylem aty Statystyki XX I wieku” , A kadem ia E konom iczna, W rocław, 140-147.

R i d d l e L. H. (1988), P robability m odels f o r tennis scoring system s, Appl. Statist., 37, 1, 63-75.

W artykule rozw ażany jest model probabilistyczny w ielostopniow ego procesu decyzyjnego ze specjalnym uw zględnieniem przypadku użycia reguły R(4, 2). Z aprezentow ano ideę wiązań w grafach oraz m ożliw ość przedstawienia praw dopodobieństw a sukcesu w w ielostopniow ym procesie decyzyjnym jako liniow ą kombinację innych prawdopodobieństw w procesie decyzyjnym.

for k = 1 , 2 , 3, ...

R EFEREN CES

Wiesław Pasewicz, Wiesław Wagner PR O B A B IL IST Y C Z N Y M O D E L W IĄ ZA Ń W W IE L O E T A PO W Y M PR O C E SIE D E C Y Z Y JN Y M