On risk attitude and optimal yacht racing tactics

Ocean Engineering 90 (2014) 149-154

E L S E V I E R

C o n t e n t s lists available at ScienceDirect

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Ocean Engineering

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / o c e a n e n g

On risk attitude and optimal yacht racing tactics

F. Tagliaferri A.B. Philpott^ I.M. Viola ^ R.G.J. Flay^

' Institute for Energy Systems, School of Engineering The University of Edinburgh, United Kingdom ^ Yacht Research Unit, Department of Engineering Science, The University of Auckland, New Zealand " Yacht Research Unit, Department of Mechanical Engineering, The University of Auckland, New Zealand

(D

A R T I C L E I N F O

Article history:

Received 14 November 2013 Accepted 30 July 2014

Available online 10 September 2014

Keywords: Yacht race Tactics Risk aversion

A B S T R A C T

W h e n the future w i n d direction is uncertain, the tactical decisions o f a yacht skipper involve a stochastic routing problem. The objective of this problem is to maximise the probability of reaching the next mark ahead of all the other competitors. This paper describes some numerical experiments that explore the effect of the skipper's risk attitude on their policy w h e n match racing another boat. The tidal current at any location is assumed to be negligible, w h i l e the w i n d direction is modelled by a Markov chain. Boat performance in different w i n d conditions is deflned by the output of a velocity prediction program, and we assume a Icnown speed loss for tacking and gybing. We compare strategies that minimise the average time to sail the leg w i t h those that seek to maximise the probability of winning, and show that by adopting different attitudes to risk when leading or trailing the competitor, a skipper can improve their chances of w i n n i n g .

© 2014 Elsevier Ltd. A l l rights reserved.

1. I n t r o d u c t i o n I n t h i s p a p e r w e m o d e l a n d analyse t h e p r o b l e m f a c e d b y a s k i p p e r w h o w a n t s t o sail a n u p w i n d l e g o f a y a c h t race, r o u n d i n g t h e m a r k b e f o r e h i s o p p o n e n t . This p r o b l e m f a l l s i n t o t h e c a t e g o r y o f s t o c h a s t i c s h o r t e s t - p a t h p r o b l e m s , w h e r e t h e cost f u n c t i o n t o be m i n i m i s e d is t h e t i m e n e e d e d t o reach t h e m a r k , a n d i t d e p e n d s o n s t o c h a s t i c q u a n t i t i e s such as w i n d d i r e c t i o n . M a n y p r o b l e m s f a l l i n t o t h i s c a t e g o r y a n d i n v o l v e r o u t i n g f o r e m e r g e n c y response, b o t h c i v i l (Yamada, 1 9 9 6 ) a n d m i l i t a r y (Resch e t al., 2 0 0 3 ) , a n d a p p l i c a t i o n s i n l o g i s t i c s ( F l e i s c h m a n n et al., 2 0 0 4 ) a n d t r a n s p o r t ( S h u x i a , 2 0 1 2 ) . The a i m is t o f i n d a p a t h b e t w e e n t w o v e r t i c e s o f a g r a p h s u c h t h a t t h e s u m o f its c o n s t i t u e n t edges, o f t e n r e p r e s e n t -i n g a cost, -is m -i n -i m -i s e d . W h e n cost d e p e n d s o n r a n d o m q u a n t -i t -i e s t h i s b e c o m e s a stochastic p r o b l e m , a n d t h e s t a n d a r d o b j e c t i v e is t o m i n i m i s e e x p e c t e d costs ( w h e r e costs i n c l u d e t i m e ) (Bertsekas a n d Tsitsildis, 1991). For y a c h t races, m o d e l s w h i c h m i n i m i s e t h e e x p e c t e d t i m e t o f i n i s h , o r t o reach t h e n e x t m a r k , have b e e n s t u d i e d i n a n u m b e r o f p a p e r s ( P h i l p o t t a n d M a s o n , 2 0 0 1 ; P h i l p o t t , 2 0 0 5 ) . T h i s m i g h t be a p p r o p r i a t e i n f l e e t races w h e r e c o r r e c t e d t i m e o v e r a n u m b e r o f races f o r m s a basis f o r s c o r i n g p o i n t s . E v e n so, s u c h s c o r i n g systems assign r a n k i n g s i n each race a n d i t is w e l l k n o w n t h a t r a n k - b a s e d s c o r i n g leads t o d i f f e r e n t i n c e n t i v e s t h a n t h o s e f r o m p e r f o r m a n c e o n average ( A n d e r s o n , 2 0 1 2 ) .

* Corresponding author.

E-mail addresses: f.tagliaferrl@ed.ac.uk (F. TagliaferrI),

a.philpott@auckland.ac.nz (A.B. Philpott), l.m.vlola@ed.ac.uk (I.M. Viola), r.nay@auckland.ac.nz (R.G.J. Flay).

http://dx.doi.Org/10.1016/j.oceaneng.2014.07.020 0 0 2 9 - 8 0 1 8 / © 2014 Elsevier Ltd. All rights reserved.

As o b s e r v e d i n P h i l p o t t ( 2 0 0 5 ) r a n k - b a s e d s c o r i n g takes i t s m o s t e x t r e m e f o r m i n m a t c h r a c i n g , w h e r e t h e o b j e c t i v e is t o m a x i m i s e t h e p r o b a b i l i t y o f a r r i v i n g b e f o r e t h e c o m p e t i n g y a c h t . I n d e e d t h e t i m e d i f f e r e n c e b e t w e e n t h e t w o boats is n o t o f i n t e r e s t , as o p p o s e d t o i t s s i g n . I n t h i s c o n t e x t , t h e a t t i t u d e t o w a r d s r i s k o f t h e s k i p p e r assumes a g r e a t e r i m p o r t a n c e . The a i m o f t h i s w o r k is to s h o w t h a t b y c h a n g i n g t h e s k i p p e r ' s a t t i t u d e t o r i s k , i t is possible t o d e f i n e a s t r a t e g y t h a t p e r f o r m s b e t t e r i n m a t c h races t h a n strategies a i m e d a t m i n i m i s i n g t h e e x p e c t e d t i m e t o f i n i s h . Of course, i n m o s t f o r m s o f m a t c h r a c i n g , t h e i n t e r a c t i o n b e t w e e n t h e boats is i m p o r t a n t . A l e a d i n g y a c h t w i l l a t t e m p t t o cover a t r a i l i n g y a c h t , n o t o n l y f o r t a c t i c a l reasons, b u t also t o s p i l l t u r b u l e n t a i r o n t h e t r a i l i n g y a c h t ' s sails t o r e d u c e t h e i r d r i v e . F o r c i n g a n o t h e r b o a t t o t a c k t o a v o i d a c o l l i s i o n is also a t a c t i c a l p l o y t o increase a yacht's advantage. I n t h i s p a p e r w e choose t o i g n o r e these e f f e c t s , as w e l l as a s s u m i n g i d e n t i c a l yachts a n d c r e w e x p e r t i s e . T h i s is d o n e f o r m o d e l l i n g c o n v e n i e n c e as w e l l as s i m p l i c i t y . By f o c u s i n g solely o n l y o n r i s k a t t i t u d e w e can see t o w h a t e x t e n t t h i s is i m p o r t a n t , o t h e r e f f e c t s b e i n g e q u a l .

T h e p a p e r is l a i d o u t as f o l l o w s . I n t h e n e x t s e c t i o n w e describe t h e m o d e l o f t h e y a c h t a n d basic s a i l i n g s t r a t e g y f o r t h e u p w i n d leg o f a m a t c h race. V^e t h e n r e v i e w d y n a m i c p r o g r a m m i n g as a n a p p r o a c h t o f i n d i n g t h e s t r a t e g y t h a t m i n i m i s e s t h e e x p e c t e d t i m e t o reach t h e n e x t m a r k . T h e f o l l o w i n g s e c t i o n s h o w s h o w t h i s is i m p l e m e n t e d i n a r o u t i n g m o d e l t h a t a c c o u n t s f o r d i f f e -r e n t -risk a t t i t u d e s o f t h e s k i p p e n W e t h e n p -r e s e n t t h e -results o f s o m e s i m u l a t i o n s o f t h e strategies t h a t e m e r g e f r o m t h e r o u t i n g m o d e l .

150 F. Tagliaferri et al. / Ocean Engineering 90 (2014) 149-154

J.1. Sailing Strategy

T h e speed o f a s a i l i n g y a c h t depends o n t h e w i n d speed a n d o n t h e angle b e t w e e n b o a t h e a d i n g a n d w i n d d i r e c t i o n . I t is u s u a l l y expressed as a p o l a r d i a g r a m l i k e t h e one s h o w n i n Fig. 1. T h e n u m b e r s a r o u n d t h e s e m i c i r c l e r e p r e s e n t d i f f e r e n t t r u e w i n d angles, w h i l e t h e r a d i a l ones r e p r e s e n t t h e b o a t speed. T h e r e d l i n e c o r r e s p o n d s t o t h e p l o t o f boat speed f o r a p a r t i c u l a r t r u e w i n d speed. W h i l e n o d i r e c t course is possible s t r a i g h t i n t o t h e w i n d , i t is possible t o sail u p w i n d w i t h a n angle b e t w e e n w i n d d i r e c t i o n a n d sailed course w h i c h is u s u a l l y b e t w e e n 3 0 ° a n d 5 0 ° . S a i l i n g closer t o t h e w i n d d i r e c t i o n ( l o w e r angle) m a k e s t h e course shorter, b u t w h e n s a i l i n g at h i g h e r angles a b o a t is faster. V e l o c i t y m a d e g o o d ( V M G ) is t h e c o m p o n e n t o f y a c h t v e l o c i t y i n t h e w i n d d i r e c t i o n . W i t h a c o n s t a n t w i n d d i r e c t i o n f r o m t h e t o p m a r k , a n o p t i m a l p o l i c y m a x i m i s e s V M G . This is t y p i c a l l y a t t a i n e d at a t r u e w i n d angle o f a r o u n d 4 0 - 4 5 ° (as i n t h i s e x a m p l e ) . I n a p o l a r d i a g r a m l i k e t h e one i n Fig. 1, i t is possible t o f i n d t h e m a x i m u m V M G f o r a g i v e n w i n d speed b y f i n d i n g t h e i n t e r s e c t i o n b e t w e e n t h e p o l a r c o r r e s p o n d i n g to t h e w i n d speed a n d t h e l i n e p e r p e n d i -c u l a r t o t h e u p w i n d d i r e -c t i o n . For t h i s reason t h e -c o m m o n r o u t e t o w a r d s an u p w i n d m a r k , o r i n general t o w a r d s t h e d i r e c t i o n f r o m w h i c h t h e w i n d b l o w s , is a zigzag r o u t e . Such a r o u t e r e q u i r e s changes o f d i r e c t i o n w h i c h are called tacl<s. W h e n m a n o e u v r i n g f o r a tack, a b o a t p o i n t s f o r a f e w seconds d i r e c t l y i n t o t h e w i n d , t h e r e f o r e c a u s i n g a t e m p o r a r y decrease i n b o a t speed. I f t h e w i n d is c o n s t a n t d u r i n g t h e race a n d a l l over t h e r a c i n g area, t r y i n g

to do t h e m i n i m u m n u m b e r o f tacks is t h e b e s t choice. Fig. 2(a) shows t w o possible routes. I n a c o n s t a n t w i n d , t h e r o u t e o n the l e f t is faster because i t i n v o l v e s j u s t o n e t a c k . Fig. 2 ( b ) s h o w s a s i t u a t i o n i n w h i c h t h e w i n d s h i f t s t o w a r d s t h e l e f t o v e r the d u r a t i o n o f t h e leg. The best p o l i c y i n t h i s case is t o go t o t h e l e f t o f t h e course ( r e f e r r e d t o as b e i n g o n starboard tack), a n d t h e n tack a n d p o i n t t o w a r d s t h e m a r k , w h i l e a m y o p i c p o l i c y t h a t begins t h e race g o i n g to t h e r i g h t ( r e f e r r e d t o as b e i n g o n port tack) t u r n s o u t t o be s u b o p t i m a l .

I n real races t h e e v o l u t i o n o f t h e w i n d c a n be m u c h m o r e c o m p l i c a t e d t h a n these examples, w i t h t e m p o r a r y s h i f t s o r gusts t h a t a sailor seeks t o take advantage of. M o r e o v e r w i n d has a r a n d o m c o m p o n e n t . W h i l e r a c i n g , i t is d i f f i c u l t t o k n o w h o w t h e w i n d is b e h a v i n g a t a n o t h e r l o c a t i o n , o r t o foresee h o w i t w i l l behave once t h a t p o i n t is reached. I n t h e presence o f r a n d o m n e s s t h e o p t i m a l course i n Fig. 2 ( b ) m i g h t t u r n o u t t o be w o r s e t h a n a m y o p i c p o l i c y t h a t tacks o n every w i n d s h i f t . For t h i s reason sailors t e n d t o t r y a n d stay i n t h e c e n t r e o f t h e course t o enable s h i f t s i n w i n d d i r e c t i o n t o be e x p l o i t e d b y t a c k i n g , w h i l e a v o i d i n g t h e r i s k o f o v e r l a y i n g t h e m a r k .

I n t h e presence o f a c o m p e t i t o r , a p o l i c y t h a t avoids t h e course b o u n d a r i e s w h i l e s t a y i n g close t o t h e c o m p e t i t o r reduces t h e r i s k o f b e i n g beaten, at least w h e n t h e c o m p e t i t o r is t h e t r a i l i n g boat. O n t h e o t h e r h a n d , w h e n t h e c o m p e t i t o r is l e a d i n g , i t c a n m a k e sense f o r a s k i p p e r t o t a k e a r i s k a n d e x p l o r e t h e corners o f t h e course h o p i n g f o r a f a v o u r a b l e w i n d s h i f t . This is t h e p h e n o m e n o n t h a t w e seek t o m o d e l i n t h i s p a p e n

Fig. 1. Example of a polar diagram (velocities in m/s and angles in degrees). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

Fig. 2. Example of upwind routes, (a) Constant wind and (b) left wind shift.

1.2. Dynamic programming

F i n d i n g an o p t i m a l set o f tacks w h e n t h e w i n d varies r a n d o m l y requires a stocliastic dynamic o p t i m i s a t i o n m o d e l . I n c o n t r a s t t o t h e d e t e r m i n i s t i c case, a s o l u t i o n does n o t consist o f a single o p t i m a l p a t h f o r a specific w i n d r e a l i s a t i o n , b u t a policy t h a t is o p t i m a l o v e r a range o f w i n d realisations. Policies can be c o m -p u t e d a -p r i o r i a n d res-pect t h e -p r i n c i -p l e o f o -p t i m a l i t y : an o -p t i m a l p o l i c y has t h e p r o p e r t y t h a t w h a t e v e r t h e i n i t i a l state a n d i n i t i a l d e c i s i o n are, t h e r e m a i n i n g decisions m u s t c o n s t i t u t e a n o p t i m a l p o l i c y w i t h regard t o t h e state r e s u l t i n g f r o m t h e first d e c i s i o n ( B e l l m a n , 1957). A p o l i c y t h a t respects t h i s p r i n c i p l e can b e f o u n d w i t h dynamic programming (Bertsekas, 1995). D y n a m i c p r o g r a m -m i n g has been successfully a p p l i e d i n s a i l i n g i n b o t h ocean races a n d s h o r t course r a c i n g (see P h i l p o t t a n d M a s o n , 2 0 0 1 ; P h i l p o t t , 2 0 0 5 ) . I n t h i s w o r k w e a d a p t t h e s h o r t - c o u r s e m o d e l d e s c r i b e d i n P h i l p o t t a n d M a s o n ( 2 0 0 1 ) a n d P h i l p o t t ( 2 0 0 5 ) w i t h t h e a i m o f i n c o r p o r a t i n g t h e skipper's a t t i t u d e t o w a r d s risk i n t h e i r actions. The r i s k t h a t a s k i p p e r is w i l l i n g t o t a k e is u s u a l l y i n f l u e n c e d b y his p o s i t i o n w i t h respect t o t h e o p p o n e n t . A c o m m o n b e h a v i o u r a l p a t t e r n is t o be c o n s e r v a t i v e , o r risk averse, w h e n i n a l e a d i n g p o s i t i o n , w h i l e b e i n g risk seeking w h e n l o s i n g . H e r e w e i n t e r p r e t r i s k a v e r s i o n as b e i n g p e s s i m i s t i c a b o u t w i n d s h i f t s , b e l i e v i n g t h a t any s h i f t s w e o b s e r v e w i l l n o t be t o o u r advantage. I n c o n t r a s t , a r i s k - s e e k i n g s k i p p e r w i l l be o p t i m i s t i c a b o u t w i n d s h i f t s a n d act as i f these are m o r e l i k e l y t o be t o his advantage. Such a t t i t u d e s can be m o d e l l e d b y a l t e r i n g t h e t r a n s i t i o n p r o b a b i l i t i e s o f t h e process t h a t d e f i n e s w i n d s h i f t s .

To u n d e r s t a n d t h e e f f e c t o f risk-averse o r risk-seeking s k i p p e r s , w e d e v e l o p a race m o d e l l i n g p r o g r a m ( R M P ) f o r s i m u l a t i n g races b e t w e e n t w o boats. The flrst RMP w a s d e v e l o p e d i n 1987 f o r t h e A m e r i c a ' s Cup syndicate Stars a n d Stripes a n d is d e s c r i b e d i n Letcher e t al. ( 1 9 8 7 ) . Since t h e n , RMPs have b e e n used m a i n l y i n A m e r i c a ' s Cup a p p l i c a t i o n s t o c o m p a r e d i f f e r e n t designs (see e.g. P h i l p o t t e t al., 2 0 0 4 ) . I n o u r case, since w e are i n t e r e s t e d i n c o m p a r i n g tactical choices, w e m o d e l t w o i d e n t i c a l boats (i.e. t h e y have t h e same p o l a r d i a g r a m ) .

2 . M e t h o d

2.1. Dynamic programming

W e consider an u p w i n d leg o f 6 0 0 0 m ( c o r r e s p o n d i n g t o 3.24 n a u t i c a l m i l e s , w h i c h a p p r o x i m a t e s t h e l e n g t h o f t h e 2013 A m e r -ica's Cup course), a n d 4 0 0 0 m w i d e . I n t h e c o o r d i n a t e s y s t e m used t h e s t a r t i n g l i n e is l o c a t e d o n t h e x-axis, a n d c e n t r e d a r o u n d t h e o r i g i n , w h i l e t h e u p w i n d m a r k is l o c a t e d o n t h e y - a x i s . T h e r a c i n g area is discretised i n t o a r e c t a n g u l a r g r i d w i t h N = 2 0 i n c r e m e n t s A x across t h e course a n d M = 4 0 0 i n c r e m e n t s A y i n t h e d i r e c t i o n o f t h e course, as s h o w n i n Fig. 3. The J V - 1 lines d e f i n i n g t h e g r i d t h a t are p e r p e n d i c u l a r t o t h e y - a x i s w i l l be r e f e r r e d t o i n t h e f o l l o w i n g as "cross sections". The d y n a m i c p r o g r a m is at stage i w h e n t h e y a c h t crosses t h e i t h cross s e c t i o n .

T h e state variables are t h e yacht's p o s i t i o n X(, t h e w i n d d i r e c -t i o n Wi o b s e r v e d a-t s-tage i , a n d -t h e c u r r e n -t -tack z ( w h e r e z = 0 d e n o t e s s t a r b o a r d t a c k a n d z = l d e n o t e s p o r t t a c k ) . T h e w i n d d i r e c t i o n w,- is r a n d o m a n d satisfies t h e Marl<ov p r o p e r t y , n a m e l y t h a t t h e p r o b a b i l i t y d i s t r i b u t i o n f o r t h e v a r i a b l e w,-, c o n d i t i o n e d o n a l l t h e p r e v i o u s values, is equal t o t h e d i s t r i b u t i o n f o r t h e v a r i a b l e Wi c o n d i t i o n e d j u s t o n t h e last event: w h e r e w ' is t h e w i n d d i r e c t i o n t h a t is o b s e r v e d at stage i + 1. N o w w e can d e f l n e t h e r e c u r s i o n as f o l l o w s : P(Wi = V | W i _ i = V i _ i , W i _ 2 = Vf. = P ( W i = V | W , . _ , = V , _ i ) , W o = V o ) (1) f o r e v e r y i > 0 a n d f o r e v e r y w,- i n t h e state space.

T h e actions at each stage are w h e t h e r t o tack t h e b o a t (i.e. c h a n g e z t o 1 - z ) o r c o n t i n u e o n t h e same tack. As m e n t i o n e d i n t h e I n t r o d u c t i o n , a t a c k i n g m a n o u v r e i m p l i e s a t i m e loss t h a t w i l l be d e n o t e d as t . G i v e n a yacht's p o l a r a n d its l o c a t i o n , w e c a n c o m p u t e t ( i , x , x ' , w , z ) , d e f i n e d t o be t h e t i m e t o sail f r o m l o c a t i o n ( x , i A y ) t o ( x ' , ( i + l ) A y ) i f i t is o n t a c k z a n d t h e o b s e r v e d w i n d d i r e c t i o n is w . W e d e f i n e t h e v a l u e f u n c t i o n Tf(Xi, w.-.z) t o be t h e m i n i m u m e x p e c t e d t i m e t o sail f r o m l o c a t i o n x,- o n cross s e c t i o n i t o t h e t o p m a r k g i v e n w i n d o b s e r v a t i o n w,-, a n d c u r r e n t t a c k z. Clearly TM(x,Wi,z) = 0 w h e n l o c a t i o n x is at t h e t o p m a r k , a n d w e choose TM(X,Wi,z) = oo o t h e r w i s e . W e c o m p u t e r o ( x o , W o , z ) f o r ( X o , W o , z ) c o r r e s p o n d i n g t o t h e boat's p o s i t i o n a n d t a c k o n t h e s t a r t l i n e , u s i n g a d y n a m i c p r o g r a m m i n g r e c u r s i o n . First d e f l n e at stage i t h e f u n c t i o n

F(i, X, X', w , z) = t(i, X, X', w , z)+Ew'[T"i+1 (x', w / , z ) | w ] , (2)

5' s rj(Xi,Wi,z) = m i n ^ m i n F ( i , x , , X i + i , W i , z ) Xl+1 eX r+ m i n FCi.Xj.Xj+i,w,-, 1 • Xi+x eX •Z) (3)

Fig. 3. Schematic representation of the course.

w h e r e X is t h e set o f x c o o r d i n a t e s o f p o s i t i o n s ( x , + ] , ( ! + l ) A y ) t h a t c a n be reached at stage i + l f r o m p o s i t i o n x,- at stage i . M o r e details o n t h e r e c u r s i v e p r o c e d u r e d e f l n e d b y Eqs. ( 2 ) a n d ( 3 ) c a n be f o u n d i n P h i l p o t t a n d M a s o n ( 2 0 0 1 ) .

2.2. Wind modelling

W e assume t h e w i n d speed t o be c o n s t a n t d u r i n g t h e race, f o c u s i n g o n t h e changes i n w i n d d i r e c t i o n . As discussed i n t h e p r e v i o u s section t h e d y n a m i c p r o g r a m m i n g a l g o r i t h m w e use assumes t h a t t h e w i n d d i r e c t i o n satisfles t h e M a r k o v p r o p e r t y . A l t h o u g h m o r e r e f i n e d w i n d m o d e l s are b e i n g d e v e l o p e d (see f o r instance t h e r e c e n t r e v i e w s b y Costa et al., 2 0 0 8 a n d B i t n e r -Gregersen e t al., 2014), M a r k o v m o d e l s are c o m p u t a t i o n a l l y v e r y e f f l c i e n t a n d can s t i l l c a p t u r e m o s t o f t h e s t a t i s t i c a l p r o p e r t i e s t h a t are r e l e v a n t i n c e r t a i n a p p l i c a t i o n s ( S h a m s h a d et al., 2 0 0 5 ; S a h i n a n d Sen, 2 0 0 1 ) .

For t a c t i c a l p u r p o s e s w e are i n t e r e s t e d i n changes i n w i n d d i r e c t i o n t h a t s i g n i f l c a n t l y a f f e c t t h e r a c i n g t i m e . W e t h e r e f o r e d e f l n e a flnite n u m b e r o f w i n d d i r e c t i o n states: n a m e l y - 4 5 ° , - 4 0 ° , . . . , 0 ° , - h 5 ° , . . . , 4 - 4 5 ° , w h e r e 0 ° r e p r e s e n t s t h e w i n d d i r e c t i o n at w h i c h t h e u p w i n d m a r k is set, a n d t h e o t h e r states r e p r e s e n t s h i f t s o f + 5 ° f r o m t h a t d i r e c t i o n .

For a s y s t e m w i t h a finite n u m b e r o f states t h e stochastic process is u n i q u e l y d e f l n e d w i t h a n i n i t i a l d i s t r i b u t i o n f o r W q a n d a t r a n s i t i o n m a t r i x P . T h e m a t r i x e l e m e n t s Pjk r e p r e s e n t t h e p r o b -a b i l i t y t h -a t t h e s y s t e m -at time step i is i n st-ate k c o n d i t i o n e d o n t h e f a c t t h a t i t w a s i n state j at t h e p r e v i o u s time step i - l :

Pjt = P(Wi = k | W i _ , = j ) I n o r d e r t o o b t a i n a r e a l i s t i c t r a n s i t i o n m a t r i x w e c o n s i d e r e d a t i m e series o f w i n d m e a s u r e m e n t s f r o m a w e a t h e r s t a t i o n i n s t a l l e d o n t h e N e w c a s t l e U n i v e r s i t y research vessel, a n d t h e n b u i l t t h e m a t r i x P u s i n g a m a x i m u m l i k e l i h o o d e s t i m a t o r As w e use f o r t h e m o d e l a g r i d w i t h 15 m r e s o l u t i o n i n t h e u p w i n d d i r e c t i o n a n d t h e decisions are t a k e n e v e r y t i m e t h e y a c h t reaches a cross section, t h e w i n d is m o d e l l e d u s i n g a time step o f 3 s, w h i c h is t h e time s p e n t o n average t o m o v e b e t w e e n t w o c o n s e c u t i v e cross sections. T h e r e c o r d e d w i n d d i r e c t i o n s i g n a l was s a m p l e d e v e r y t h r e e seconds, a n d t h e c o r r e s p o n d i n g w i n d d i r e c t i o n s w e r e p l a c e d i n K b i n s o f a m p l i t u d e 5 ° . T h e n u m b e r o f j u m p s f r o m b i n j t o b i n k d i v i d e d b y t h e t o t a l n u m b e r o f j u m p s o u t o f b i n J d e f l n e s t h e v a l u e Pjk,j,k=l,2,...,K, i n t h e t r a n s i t i o n m a t r i x . G i v e n a t r a n s i t i o n m a t r i x P , Eq. ( 2 ) b e c o m e s k = K

F(i,x,x',Wj,z) = t(i,x,x:,Wj,z)+ 2 P,fcr,+i(x',Wk,z). ( 4 )

k = l 2.3. Risk modelling W e n o w t u r n o u r a t t e n t i o n t o t h e r i s k a t t i t u d e o f t h e y a c h t skipper. T h e r e is a n e n o r m o u s l i t e r a t u r e o n m o d e l l i n g r i s k ( f o r a r e c e n t i n t r o d u c t i o n see A n d e r s o n , 2013). To m o d e l risk a v e r s i o n , w e a d o p t a n a p p r o a c h based o n t h e t h e o r y o f coherent r i s k m e a s u r e s ( A r t z n e r e t al., 1 9 9 9 ) . As s h o w n i n A r t z n e r e t a l . ( 1 9 9 9 ) c o h e r e n t risk m e a s u r e s c a n be expressed as t h e w o r s t - c a s e e x p e c t a t i o n o v e r a c o n v e x set o f p r o b a b i l i t y d i s t r i b u t i o n s t o g i v e a risk-adjusted e x p e c t a t i o n . G i v e n t h e c u r r e n t w i n d d i r e c t i o n state,

152 F. Tagliaferri et al. / Ocean Engineering 90 (2014) 149-154 t h e p r o b a b i l i t y d i s t r i b u t i o n t h a t w e w o r k w i t h is t h e c o r r e s p o n d -i n g r o w o f t h e t r a n s -i t -i o n m a t r -i x . To m o d e l r -i s k a v e r s -i o n w e choose t h e w o r s t p a s s i b l e t r a n s i t i o n p r o b a b i l i d e s f r o m a c o n v e x set V o f t r a n s i t i o n m a t r i c e s . I n o t h e r w o r d s , ( 4 ) b e c o m e s k = K F ( ) , x , x ^ W j , z ) = t ( ! , x , x ' , W j , z ) - h m a x 2 PjkTi+\{x',Wk,z). (5) A n I n t e r p r e t a t i o n o f ( 5 ) is i l l u m i n a t i n g . A b o a t s k i p p e r w h o is w i n n i n g w i l l be r i s k averse. She w i l l t r y t o b e h a v e safely, t r y i n g t o stay ahead a n d t o m i n i m i s e h e r losses i n b a d w i n d o u t c o m e s . U s i n g (5) i n a r e c u r s i o n is p e s s i m i s t i c a b o u t t h e n e x t w i n d s h i f t a n d assigns a h i g h e r p r o b a b i l i t y to t h e w o r s t o u t c o m e s (i.e. h e a d -i n g s h -i f t s ) . B e -i n g p e s s -i m -i s t -i c a b o u t r a n d o m o u t c o m e s reduces r-isk, at s o m e loss i n e x p e c t e d p e r f o r m a n c e .

Risk s e e k i n g b e h a v i o u r has b e e n less w e l l s t u d i e d , a l t h o u g h i t is o f t e n g i v e n as a n e x p l a n a t i o n f o r p a r t i c i p a t i o n i n l o t t e r i e s a n d n e g a t i v e e x p e c t a t i o n gambles, w h e r e o p t i m i s t i c p a r t i c i p a n t s place g r e a t e r w e i g h t o n w i n n i n g p r o b a b i l i t i e s t h a n t h e i r r e a l values. I n o u r c o n t e x t w e m o d e l risk s e e k i n g b y c h o o s i n g t h e b e s t possible t r a n s i t i o n p r o b a b i l i t i e s f r o m a c o n v e x set V o f t r a n s i t i o n m a t r i c e s . I n o t h e r w o r d s , ( 4 ) b e c o m e s F ( ! , x , x ' , W j , z ) = f ( i , x , x ' , W j , z ) - ^ m i n ^ 2 ^ PjvJi+iCx', w ^ . z ) . (6) T h i s has t h e f o l l o w i n g i n t e r p r e t a t i o n . A b o a t s k i p p e r w h o is l o s i n g w i l l seek risk. I f she a d o p t s a m i n i m u m e x p e c t e d f i n i s h time s t r a t e g y a g a i n s t a n o t h e r s k i p p e r w h o m i n i m i s e s his e x p e c t e d t i m e to f i n i s h , t h e n she w i l l t e n d t o m a k e t h e same decisions (unless t h e boats see v e r y d i f f e r e n t w i n d s ) a n d lose t h e race a l m o s t c e r t a i n l y . She w i l l i n s t e a d seek d i f f e r e n t w i n d c o n d i t i o n s f r o m t h e c o m p e -t i -t o r U s i n g ( 6 ) i n a r e c u r s i o n w i l l be o p -t i m i s -t i c a b o u -t -t h e possible a d v a n t a g e o u s w i n d s h i f t s a n d assign a h i g h e r p r o b a b i l i t y t o t h e s e o u t c o m e s (i.e. l i f t i n g s h i f t s ) . B e i n g o p t i m i s t i c a b o u t r a n d o m o u t -c o m e s in-creases risk, as w e l l as i n -c u r r i n g s o m e loss i n e x p e -c t e d p e r f o r m a n c e . W e i m p l e m e n t ( 5 ) a n d ( 6 ) i n t h e r e c u r s i o n b y a d d i n g a t r a n s f o r m a t i o n i n t h e s o l v e r t h a t p o s t m u l t i p l i e s t h e t r a n s i t i o n m a t r i x b y a n o t h e r m a t r i x w h i c h r e d i s t r i b u t e s t h e p r o b a b i l i t i e s . The r e s u l t i n g m a t r i x has t o be n o r m a l i s e d i n o r d e r t o r e p r e s e n t a g a i n a p r o b a b i l i t y d i s t r i b u t i o n . 3. R e s u l t s Fig. 4 s h o w s a g r a p h i c a l r e p r e s e n t a t i o n o f t h e t r a n s i t i o n m a t r i x f o r t h e M a r k o v m o d e l o b t a i n e d w i t h t h e m a x i m u m l i k e l i h o o d e s t i m a t o r as d e s c r i b e d i n t h e p r e v i o u s s e c t i o n . W i t h a n o t a t i o n t h a t w i l l be used t h r o u g h o u t t h i s paper, w e use a g r e y scale t o r e p r e s e n t v a l u e s i n t h e i n t e r v a l [ 0 , 1] w h e r e w h i t e r e p r e s e n t s 0 a n d b l a c k r e p r e s e n t s 1. I t can be n o t i c e d t h a t t h e d i a g o n a l is d o m i n a n t , m e a n i n g t h a t , i n general, i f t h e w i n d is i n state /, t h e m o s t p r o b a b l e state f o r t h e n e x t step is t o r e m a i n i n state i .

Fig. 4. Representation of tlie transition matrix obtained for the wind model.

M o r e o v e r , w h e n t h e w i n d has d e v i a t e d f r o m t h e m e a n , t h e e v e n t o f a s h i f t back t o w a r d s t h e m e a n v a l u e is m o r e l i k e l y t h a n o n e i n t h e same d i r e c t i o n . T h e w i n d f o r t h e s i m u l a t i o n s w a s g e n e r a t e d as d e s c r i b e d i n t h e p r e v i o u s s e c t i o n . T h e M a r k o v c h a i n d e f i n e s a discrete w i n d d i r e c t i o n . T h i s c a n be m a d e c o n t i n u o u s b y s u p e r i m p o s i n g a m e a n -r e v e -r s i o n noise p-rocess (see P h i l p o t t et al., 2 0 0 4 ) . H o w e v e -r w e d i d n o t d o t h i s as w e f o u n d t h a t t h e b e h a v i o u r o f t h e s i m u l a t e d w i n d signal, a c h i e v e d w i t h n o a d d i t i o n a l noise c o m p o n e n t , w a s s i m i l a r t o t h e e m p i r i c a l one, as c a n be seen i n Fig. 5, w i t h close v a l u e s o f m e a n a n d v a r i a n c e o n d i f f e r e n t s u b - i n t e r v a l s . A w i n d h i s t o r y o f 4 0 0 values w a s g e n e r a t e d f o r each o f t h e 4 0 0 0 s i m u l a t e d races.

Fig. 6 s h o w s a h i s t o g r a m o f t h e t i m e n e e d e d b y a y a c h t f o l l o w i n g t h e p o l i c y g e n e r a t e d t o m i n i m i s e t h e e x p e c t e d t i m e o f a r r i v a l , a c c o r d i n g t o t h e w i n d d i s t r i b u t i o n p r e v i o u s l y m o d e l l e d . T h e d i s t r i b u t i o n is a s y m m e t r i c , a n d t h i s is d u e t o t h e f a c t t h a t e v e n w i t h a v e r y f a v o u r a b l e e v o l u t i o n o f t h e w i n d t h e r e is a m i n i m u m t i m e n e e d e d t o c o m p l e t e t h e course. O n t h e o t h e r h a n d , e v e n w i t h a p o l i c y w h i c h is e f f e c t i v e i n t h e m a j o r i t y o f t h e cases, i t is p o s s i b l e to be v e r y unlucl<y a n d n e e d a m u c h h i g h e r time. This p o l i c y w a s g e n e r a t e d u s i n g a r i s k - n e u t r a l t r a n s i t i o n m a t r i x f o r w i n d d i r e c t i o n as p i c t u r e d i n Fig. 4 . W h e n t h e s k i p p e r is r i s k s e e k i n g o r r i s k averse w e replace t h i s w i t h a m o d i f i e d t r a n s i t i o n m a t r i x . A sailor w h o is l o s i n g w i l l seek r i s k . This c o r r e s p o n d s t o i n c r e a s i n g h e r c o n f i d e n c e o f a l i f t i n g w i n d s h i f t w h i l e d i s c o u n t i n g t h e l i k e l i h o o d o f a h e a d i n g w i n d s h i f t . The t r a n s i t i o n m a t r i c e s w e use t o r e p r e s e n t a risk-seeking s k i p p e r are s h o w n i n Fig. 7 ( a ) a n d ( b ) . As s h o w n i n t h e f i g u r e s , a d v a n t a g e o u s s h i f t s (cells b e l o w t h e d i a g o n a l w h e n t h e s k i p p e r is t o t h e l e f t o f t h e o p p o s i t i o n , a n d cells a b o v e w h e n o n t h e right) h a p p e n w i t h h i g h e r p r o b a b i l i t y t h a n i n t h e risk-neutral case. T h e r e m a i n i n g p r o b a b i l i t i e s i n each r o w are r e d u c e d t o a d d t o o n e . T h e t r a n s i t i o n m a t r i c e s f o r a riskaverse s k i p p e r are c o n -s t r u c t e d -s i m i l a r l y . Here b a d w i n d -s h i f t -s ( a b o v e t h e d i a g o n a l w h e n t h e s k i p p e r is t o t h e l e f t o f t h e o p p o s i t i o n , a n d b e l o w t h e d i a g o n a l Empiiical CtDF X

Fig. 5. Sixty-minute example of artificially generated wind and sixty-minute example of recorded wind.

1500

2000 3000 4000 50OO 6000 7000 1ime(s)

a b

Fig. 7. Modified transition matrices for a risl<-seel<ing sl<ipper. Advantageous wind sliifts occur witli higher probability than disadvantageous ones, (a) Yacht on the left-hand side of competitor and (b) yacht on the right-hand side of competitor.

1000 p 900 • = 500

i

Fig. 8. Histogram of arrival time of B minus arrival time of A.

w h e n o n t h e r i g h t ) h a p p e n w i t h h i g h e r p r o b a b i l i t y t h a n i n t h e r i s k - n e u t r a l case. I n o u r e x p e r i m e n t s w e have o b t a i n e d t r a n s i t i o n m a t r i c e s f o r a risk-averse p o l i c y b y s i m p l y s w a p p i n g t h e m a t r i c e s i n Fig. 7 ( a ) a n d ( b ) . S i m u l a t i o n s w e r e c a r r i e d o u t i n o r d e r t o v e r i f y t h e d i f f e r e n c e s b e t w e e n a r i s k - n e u t r a l p o l i c y t h a t m i n i m i s e s e x p e c t e d a r r i v a l t i m e a t t h e t o p m a r k , a n d a p o l i c y g e n e r a t e d a s s u m i n g e i t h e r risk s e e k i n g o r risk averse b e h a v i o u r Results s h o w e d t h a t p o l i c i e s t h a t m i n i m i s e e x p e c t e d a r r i v a l t i m e w o n m o r e races t h a n e i t h e r b e i n g c o n s i s t e n t l y risk s e e k i n g o r r i s k averse.

However, c o m b i n i n g the strategies together (to a l l o w b o t h risk-seeldng a n d risk-averse behaviour at d i f f e r e n t t i m e s ) can lead t o a significant i m p r o v e m e n t i n the chances o f w i n n i n g . W e s i m u l a t e d races b e t w e e n t w o boats t h a t are denoted as boat A a n d boat B. B o t h boats start the race at t h e same time, o n t w o d i f f e r e n t ( r a n d o m ) points along the s t a r t i n g line. Boat A experiences the simulated w i n d and always f o l l o w s the risk-neutral policy (to m i n i m i s e expected arrival t i m e ) . Boat B experiences the same w i n d as A i f t h e i r distance apart is less t h a n d^in = 10 m , a n i n d e p e n d e n t w i n d i f t h e i r distance is greater t h a n dmax = 100 m , and a linear c o m b i n a t i o n o f A's w i n d a n d a n i n d e p e n d e n t sample i f t h e i r distance is b e t w e e n dmin and d^ax- A t every step o f the s i m u l a t i o n , i f B is m o r e t h a n 15 s b e h i n d A, she uses t h e risk-seeking p o l i c y depending o n t h e side o f the course; i f B is m o r e t h a n 2 0 s ahead o f A, she uses the risk-averse policy, w l i i l e she uses t h e o p t i m u m risk-neutral p o l i c y otherwise. Results o f those s i m u l a t e d races are s h o w n i n Fig. 8.

The X - a x i s s h o w s t h e a r r i v a l time o f b o a t B m i n u s t h e a r r i v a l t i m e o f b o a t A at t h e t o p m a r k . T h e average t i m e d i f f e r e n c e is p o s i t i v e ( a c t u a l l y 16 s i n t h i s p l o t ) . This m e a n s t h a t B a r r i v e s 16 s l a t e r o n average t h a n A , as one w o u l d expect, since A is u s i n g t h e o p t i m u m p o l i c y t o m i n i m i s e t h e average time. H o w e v e r a b o u t 63%

o f t h e race o u t c o m e s are t o t h e l e f t o f zero, m e a n i n g t h a t B w i n s 63% o f t h e time ( a l w a y s b y a s m a l l m a r g i n ) . O f course s o m e t i m e s B is hopelessly outclassed, l o s i n g b y 4 0 0 s ( j u s t a r o u n d 0.01% o f t h e t i m e s , a n d those are e x t r e m e l y u n f a v o u r a b l e e v e n t s ) b u t t h i s is because B takes h i g h r i s k s w h e n b e h i n d . I f w e c o n s i d e r p = 0 . 5 w i n p r o b a b i l i t y as a n u l l h y p o t h e s i s , t h e n t h e p r o b a b i l i t y o f w i n n i n g m o r e t h a n 63% o f 5 0 0 0 races b y chance is t h e p r o b a b i l i t y t h a t a b i n o m i a l r a n d o m v a r i a b l e w i t h m e a n 5 0 0 0 p a n d v a r i a n c e SOOOp ( 1 - p ) exceeds 3150, w l i i c h is n e g l i g i b l e . The s t a n d a r d e r r o r o f t h e v a l u e 0.63 can be e s t i m a t e d u s i n g t h e c e n t r a l l i m i t t h e o r e m t o be a p p r o x i m a t e l y 0 . 0 0 3 5 . So w e can be 97.5% c o n f i d e n t t h a t t h e h y b r i d p o l i c y w i l l w i n at least 62.3% o f t h e races (i.e. 2 s t a n d a r d e r r o r s less t h a n 0.63).

I n o r d e r t o q u a n t i f y t h e t a c t i c a l i m p r o v e m e n t o n t h e p o l i c y w e c o m p a r e t h e results o b t a i n e d b y boat A a n d b o a t B w i t h a t h i r d b o a t C t h a t has p e r f e c t k n o w l e d g e o f t h e f u t u r e b e h a v i o u r o f t h e w i n d . I n t h i s case w e s i m u l a t e d 1000 races. O b v i o u s l y t h e b o a t w i t h p e r f e c t k n o w l e d g e o f t h e w i n d scenario a l w a y s w i n s a n d t h e increases i n a r r i v a l t i m e o f A a n d B are a l w a y s p o s i t i v e . The s a m p l e average d i f f e r e n c e i n t i m e o f a r r i v a l is 133 s f o r b o a t A w h i l e f o r b o a t B t h e sample average d i f f e r e n c e is 149 s. T h e d i f f e r e n c e is n o t s i g n i f i c a n t because o f h i g h v a r i a n c e a n d l o w s a m p l e size. H o w e v e r t h i s e x p e r i m e n t c o n f i r m s a t h e o r e t i c a l r e s u l t : t h e e x p e c t e d t i m e d i f f e r e n c e f o r b o a t A r e l a t i v e t o C is n e v e r m o r e t h a n t h e e x p e c t e d t i m e d i f f e r e n c e f o r b o a t B r e l a t i v e t o C (see A p p e n d i x f o r p r o o f ) . 4. C o n c l u s i o n s I n t h i s p a p e r w e have p r e s e n t e d a m e t h o d f o r a p p r o x i m a t i n g a s o l u t i o n o f a stochastic s h o r t e s t p a t h p r o b l e m w i t h a p p l i c a t i o n s t o y a c h t r a c i n g . W e s h o w e d t h a t w i t h a n a d e q u a t e s u b d i v i s i o n o f t h e p r o b l e m i t is possible t o find a s o l u t i o n t h a t m i n i m i s e s t h e e x p e c t e d t i m e n e e d e d t o r e a c h a n u p w i n d m a r k d u r i n g a race.

Moreover, w e introduce f o r t h e first time a m o d e l o f the risk attitude o f the sailor. W e showed t h a t i f a slapper o f a trailing boat has a risk-seeking attitude i t enhances the chance to w i n the race. A n i m p o r t a n t result o f the simulations m n t o simulate races w a s t h a t a i m i n g at m i n i m i s i n g t h e expected time t o finish is n o t always t h e best approach: being o n average s l o w e r m i g h t a l l o w a bigger p r o b -ability o f w i n n i n g against a n o p p o n e n t f o l l o w i n g a flxed policy.

The results p r e s e n t e d i n t h i s p a p e r u n d e r i i n e t h a t w h e n t r y i n g to o p t i m i s e a p o l i c y i n o r d e r t o w i n a c o m p e t i t i o n , l o o k i n g at average values is r a r e l y t h e best a p p r o a c h , a n d a c c o u n t i n g f o r d i f f e r i n g risk a t t i t u d e s m i g h t g i v e p o l i c i e s t h a t p e r f o r m s i g n i f i -c a n t l y b e t t e r F u r t h e r w o r k is b e i n g -c a r r i e d o u t i n o r d e r t o v a l i d a t e t h e m o d e l w i t h data r e g i s t e r e d d u r i n g A m e r i c a ' s Cup races, a n d w e are d e v e l o p i n g m e t h o d o l o g i e s f o r l e a r n i n g r i s k p a r a m e t e r s t h a t y i e l d m a x i m u m w i n p r o b a b i l i t i e s .

A c l m o w l e d g e m e n t s

This research has b e e n p e r f o r m e d w i t h i n t h e SAILING FLUIDS p r o j e c t (PIRSES-GA-2012-318924), w i c h is f u n d e d b y t h e E u r o p e a n C o m m i s s i o n u n d e r t h e 7 t h F r a m e w o r k P r o g r a m m e t h r o u g h t h e M a r i e Curie A c t i o n s , People, I n t e r n a t i o n a l Research Staff Exchange Scheme. T h e a u t h o r s w o u l d l i k e t o t h a n k N e w c a s t i e U n i v e r s i t y f o r p r o v i d i n g w i n d data.

A p p e n d i x

P r o p o s i t i o n 1. Minimising the expected arrival time over all

strate-gies will give a policy that is slower than a perfect skipper by the least amount on average.

154 F. Tagliaferri et ai / Ocean Engineering 90 (2014) 149-154

P r o o f . S u p p o s e a p e r f e c t s k i p p e r sails races i n w i n d t h a t she p r e d i c t s p e r f e c t l y . Each race is a r a n d o m s a m p l e o f w i n d a n d so h e r t i m e t o f i n i s h is a n i n d e p e n d e n t i d e n t i c a l l y d i s t r i b u t e d r a n d o m v a r i a b l e T. S u p p o s e she n o w sails a s t r a t e g y s t h a t is n o t c l a i r v o y a n t i n each o f t h e s e s a m e w i n d c o n d i t i o n s . T h e t i m e t o f i n i s h u n d e r t h i s s t r a t e g y is a n i n d e p e n d e n t i d e n t i c a l l y d i s t r i b u t e d r a n d o m v a r i a b l e S{s). N o w t h e d e l a y i n f i n i s h i n g u n d e r s t r a t e g y s v e r s u s t h e p e r f e c t s t r a t e g y is also a n i n d e p e n d e n t i d e n t i c a l l y d i s t r i b u t e d r a n d o m v a r i a b l e D(s) = S(s)-T. The e x p e c t e d d e l a y f r o m s a i l i n g s is t h e n E[D(s)] = E[S(s)]-Em.

To m i n i m i s e t h i s w e s h o u l d m i n i m i s e E[S(s)] as E[T] is a c o n s t a n t . So t h e s t r a t e g y t h a t m i n i m i s e s e x p e c t e d d e l a y a f t e r a c l a i r v o y a n t s k i p p e r is t h e o n e t h a t m i n i m i s e s e x p e c t e d a r r i v a l t i m e . •

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