GSOE9210 vicj@cse.unsw.edu.au www.cse.unsw.edu.au/~gs9210 Maximin and minimax regret 1 2 Indifference; equal preference 3 Graphing decisin prblems 4 Dminance
The Maximin principle Maximin and minimax Regret are similar principles: ne cnsiders riginal values and the ther regrets The Maximin criteria is the main decisin methd used under cmplete uncertainty We ve seen Maximin and minimax Regret n decisin tables, but what abut mre cmplex decisin prblems (e.g., multiple decisin pints)? Multi-stage decisins Example (Prduct develpment) Yu head the R&D department f a small manufacturing cmpany which is cnsidering develping a new prduct. The cmpany must decide whether t prceed with prttype develpment and, if develpment is successful, subsequently determine the prductin scale (i.e., the size f the factry) based n prjected demand fr the prduct. Questins What des Maximin r minimax Regret mean in this prblem? Is there a decisin-table representatin?
Multi-stage decisins Eliminating uncertainty L S h l h 10 4 5 L S 4 l 4 5 h 5 l 8 L 4 What is Maximin mean in a tree? Maximin eliminates branches in chance ndes (i.e., prunes the tree) Reduces prblem t that f certainty S 5 5
Outline 1 2 Indifference; equal preference 3 Graphing decisin prblems 4 Dminance Prblem representatin: decisin tables u F D v A ($0) B ($30) C ( $10) Observatin: Each actin and state uniquely determine an utcme Mdel as a binary functin: ω : A S Ω Represented as a table: A ω F D Decisin tables: S A A B C rw = actin clumn = state Interpretatin: B = ω(d, ) means B is the utcme f actin D in state ;
Trees and tables u F D A ($0) v B ($30) C ( $10) F $0 $0 D $30 $10 u F D w v A A B C Multiple trees may crrespnd t the same table Ging frm tables (nrmal frm) t trees (extensive frm) is straight frward, but the cnverse can be tricky Which representatin is better: trees r tables? Which representatin facilitates decisin analysis mst? Multi-stage decisins
Multi-stage decisins Example (Prduct develpment) Yu head the R&D department f a small manufacturing cmpany which is cnsidering develping a new prduct. The cmpany must decide whether t prceed with prttype develpment and, if develpment is successful, subsequently determine the prductin scale (i.e., the size f the factry) based n prjected demand fr the prduct. Questins What des Maximin r minimax Regret mean in this prblem? Is there a decisin-table representatin? Actins t strategies In a decisin tree: Recall that a decisin table is a representatin f the utcme mapping ω : A S Ω Observatin: fllwing a path frm the rt t a leaf leads t a unique utcme Therefre: A state cnsists f all the cnditins alng this path An actin cnsists f all the chices alng the path Definitin (Strategy) A strategy (r plicy r plan) is a prcedure that specifies the selectin f an actin at every reachable decisin pint.
States: s 1 s 2 s 3 s, h s, l f A strategy must specify an actin at each reachable decisin pint; e.g., Authrise develpment (Au), if develpment succeeds (s), then build large factry (L) encded Au;s/L Encding: α/a says: Example: Au;s/S: At the decisin nde reached via path α chse actin A. After authrising develpment (Au), in the event that develpment succeeds (s), chse t build a small factry (S). Strategies fr this prblem: A 1 Au;s/L Au;s/S A 2 A 3 Ab
Cde fc pc lp mp be ldc sat dis sq Descriptin full capacity partial capacity large prfits mderate prfits break even lse dev. csts demand satisfied dissatisfactin status qu s, h s, l f Au;s/L fc,lp,sat pc,be,sat ldc Au;s/S fc,mp,dis fc,mp,sat ldc Ab sq sq sq Value functin: ω V fc,lp,sat 10 pc,be,sat 4 ldc 1 fc,mp,dis 5 fc,mp,sat 8 sq 0 Decisin table: s, h s, l f Au;s/L 10 4 1 Au;s/S 5 8 1 Ab 0 0 0 Exercises What are the Maximin and minimax Regret strategies fr this prblem?
Outline Indifference; equal preference 1 2 Indifference; equal preference 3 Graphing decisin prblems 4 Dminance Indifference; equal preference Indifference: equal preference Which actin belw is preferred abve under Maximin? s 1 s 2 A 1 0 B 0 1 Definitin (Indifference) If tw actins A and B are equally preferred then the agent is said t be indifferent between A and B. Definitin (Weak preference) Actin A is weakly preferred t B iff it A preferred t B r the tw are indifferent; i.e., the agent prefers A at least as much as B.
Indifference classes Indifference; equal preference Definitin (Indifference class) An indifference class is a nn-empty set f all actins/utcmes between which an agent is indifferent. Fr a given actin A A, the indifference class f A is given by I(A) = {a A V (a) = V (A)} Different agents will have different preferences ver utcmes/actins, hence different indifference classes Different decisin rules will prduce different indifference classes Outline Graphing decisin prblems 1 2 Indifference; equal preference 3 Graphing decisin prblems 4 Dminance
Visualisatin Graphing decisin prblems s 1 s 2 A 2 3 B 4 0 C 3 3 D 5 2 E 3 5 Let v i (a) = v(a, s i ) be the value f actin a in state s i. Each actin a crrespnds t a pint (v 1, v 2 ), where v i = v(a, s i ). v 2 5 (3, 5) 4 A C 3 (2, 3) (3, 3) D 2 1 B (4, 0) 0 0 1 2 3 4 5 E (5, 2) v 1 Graphing decisin prblems Indifference curves: Maximin Fr the pure actins belw: s 1 s 2 A 2 3 B 4 0 C 3 3 D 5 2 E 3 5 Cnsider the curves f all pints which represent actins with the same Maximin value; i.e., the Maximin indifference curves. v 2 5 E 4 3 A C 2 I(A) D 1 0 0 1 2 3 4 5 3 2 v 1
Graphing regret Graphing decisin prblems Cnsider three actins: v 2 s 1 s 2 A 2 4 B 4 1 C 5 3 Regret values and indifference curves fr minimax Regret shwn in blue 5 4 3 2 1 0 A B C A C B 0 1 2 3 4 5 6 v 1 Outline Dminance 1 2 Indifference; equal preference 3 Graphing decisin prblems 4 Dminance
River example Dminance X A B C Example (River lgistics) Alice s cmpany has a warehuse situated at X n a river that flws dwn-stream frm C t A. Her cmpany delivers gds t a client at C via mtr bats. On sme days a (free) gds ferry perates, travelling up the river, stpping at A then B and C, but nt at X. The fuel required t reach C frm each starting pint: A X B C T C frm: 4 3 2 0 Alice wants t minimise fuel cnsumptin (in litres). River example Dminance X A B C f f A 4 0 B 3 1 C 1 1 Alice cnsiders three pssible ways t get t C (frm starting pint X): A : via A, by flating dwn the river B : via B, by travelling up-stream t B C : by travelling all the way t C Outcmes are measured in litres left in a fur-litre tank. Exercise Let w : Ω R dente fuel cnsumptin in litres. What transfrmatin f : R R is respnsible fr the values v : Ω R in the decisin table?
River example Dminance The axes crrespnd t the payffs in each f the tw states; i.e., payff v 1 in state s 1 = f and v 2 in s 2 = f The actins are graphed belw: v 2 2 B 1 C A 0 0 1 2 3 4 v 1 Clearly ptin C will nt be a better respnse than either f the ther tw under any circumstances (i.e., in any state) Actin C can be disregarded Generalised dminance Dminance Definitin (Strict dminance) Actin A strictly dminates B iff every utcme f A is strictly preferred ver the crrespnding utcme f B. Definitin (Weak dminance) Actin A weakly dminates B iff every utcme f A is weakly preferred ver the crrespnding utcme f B, and sme utcme is strictly preferred. s 1 s 2 s 3 Exercise A 3 4 2 Which actins in the decisin table B 4 4 3 shwn are dminated? C 5 6 3
Dminance Dminance and best respnse Crllary An actin A strictly dminates B iff A is a better respnse than B in each pssible state. Crllary An actin A weakly dminates B iff A is a better respnse than B in sme pssible state and B is nt a better respnse than A in any state. Dminance principle A ratinal agent shuld never chse a dminated actin. Admissibile actins Dminance s 1 s 2 A 4 A 0 4 C 3 D B 3 1 2 B C 2 3 1 D 1 2 v 2 0 0 1 2 3 4 v 1 Definitin (Admissible) An actin is admissible iff it is nt dminated by any ther actin. The set f all admissible actins is called the admissible frntier. Exercises Which actins abve are admissible?
Dminance Dminance: MaxiMax and Maximin Definitin (Dminance eliminatin) s 1 s 2 M m A 2 2 2 2 B 2 1 2 1 C 1 1 1 1 A decisin rule is said t satisfy (strict/weak) dminance eliminatin if it always eliminates actins that are (strictly/weakly) dminated. Dminated actins can be discarded under any rule that satisfies dminance eliminatin Dminance summary Dminance Rules that satisfy strict/weak dminance eliminatin. Rule Strict Weak MaxiMax Maximin Hurwicz s minimax Regret Laplace s Exercise Verify the prperties abve.