Spring 06 Assignment 3: Solution
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1 Spring 06 Assignment 3: Slutin Questins t Rng Yan(yanrng@cs.cmu.edu) Out: 2/21/06 Due: 3/7/06 Name: Andrew ID: Please turn in yur answers n this assignment (etra cpies can be btained frm the class web page). This assignment must be turned in at the beginning f class at 1:30pm n March 7. Please write yur name and Andrew ID in the space prvided n the first page, and write yur Andrew ID in the space prvided n each subsequent page. This is wrth 5 pints: if yu d nt write yur name/andrew ID in every space prvided, yu will lse 5 pints. This assignment has n cding prblems Late plicy. Yur assignment is due at 1:30pm n 3/7. Submitting yur wrk late will affect its scre as fllws: If yu submit it after 1:30pm n 3/7 but befre 1:30pm n 3/8, it will receive 90% f its scre. If yu submit it after 1:30pm n 3/8 but befre 1:30pm n 3/9, it will receive 50% f its scre. If yu submit it after 1:30pm n 3/9, it will receive n scre. Cllabratin plicy. Yu are t cmplete this assignment individually. Hwever, yu are encuraged t discuss the general algrithms and ideas in the class in rder t help each ther answer hmewrk questins. Yu are als welcme t give each ther eamples that are nt n the assignment in rder t demnstrate hw t slve prblems. But we require yu t: nt eplicitly tell each ther the answers nt t cpy answers nt t allw yur answers t be cpied In thse cases where yu wrk with ne r mre ther peple n the general discussin f the assignment and surrunding tpics, we ask that yu specifically recrd n the assignment the names f the peple yu were in discussin with (r nne if yu did nt talk with anyne else). This is wrth five pints: fr each prblem, space has been prvided fr yu t either write peple s names r nne. If yu leave any f these spaces blank, yu will lse five pints. This will help reslve the situatin where a mistake in general discussin led t a replicated weird errr amng multiple slutins. This plicy has been established in rder t be fair t the rest f the students in the class. We will have a grading plicy f watching fr cheating and we will fllw up if it is detected. 1 The Rbt Mtin Planning Prblem (15 pints) References (names f peple I talked with regarding this prblem r nne ): Cnsider the rbt planning prblem illustrated in the fllwing figure. 1
2 Andrew ID: (5 pints) Draw (apprimately) the path fund by using ptential fields. Slutin: Since we d nt define the eact frm f the ptential field functin, we accept multiple crrect answers fr this prblem. Abve are tw pssible paths. 1.2 (5 pints) Draw (apprimately) the path fund by using apprimate cell decmpsitin (there are several pssibilities). Briefly discuss what are the advantages and disadvantages f the apprimate cell decmpsitin technique. Slutin: Advantage: 1. limited assumptin n bstacle; 2. used in practice; 3. find bvius slutins quickly. Disadvantage: 1. N clear ntin f ptimality; 2. tradeff f cmpleteness/cmputatin; 3. difficult t be used in high dimensin. 1.3 (5 pints) Draw (eactly) the path fund by using the visibility graph technique. Draw the initial visibility graph. Briefly discuss what are the advantages and disadvantages f the visibility graph technique.
3 Andrew ID: 3 Slutin: Advantage: 1. always find the shrtest path; 2. easy t implement. Disadvantage: 1. try t stay clse t the bstacle; 2. easily lead t cllisin; 3. cmplicate in high dimensin. 2 The Game Play Prblem(40 pints) References (names f peple I talked with regarding this prblem r nne ): Gmku, g-mku, r gbang is an abstract strategy bard game with tw players. It is traditinally played with black and white stnes n a 19 * 19 g bard. One player has black stnes and the ther has white stnes. T simplify, we assume the game are played n a 10*10 bard with crsses() representing black stnes and circles() representing white stnes. In this game, black plays first and players alternate in placing a stne f their clr n an empty space. The winner is the first player t get an unbrken rw f five stnes hrizntally, vertically, r diagnally. Sme eamples f the winning situatins fr black stnes are shwn as fllws. Please see fr mre details abut this game. This game is an eample f 2-player zer-sum deterministic game f perfect infrmatin. 2.1 (5 pints) Any tw-player zer-sum deterministic game can be represented by the fllwing quintuple: (S, I, Succ, T, V), where S is the entire space f game states, I is the initial state, Succ is the successr functin, T are the
4 Andrew ID: 4 terminal states, V maps frm terminal states t its payff/utility. Please describe what is each element f the quintuple in the Gmku game. Slutin: S: all pssible stne cnfiguratins n the bard, where the difference between the numbers f black stnes and white stnes is n mre than 1. I: an empty bard. Succ: place an additinal stne n the previus bard state. T: either player has five stnes cnnected hrizntally, vertically r diagnally. V: maps black(white) wins t 1, white(black) wins t -1, a draw t (10 pints) Design tw different heuristic evaluatin functins fr this game given the current state f the bard. In this prblem, the evaluatin functin shuld be able t handle the intermediate states f the game. Fr eample, yu cannt define +1 fr black wins, -1 fr white wins, 0 therwise because it cannt prvide useful infrmatin fr the intermediate states. (Hints: yu can cnsider the number f cnnected stnes n the bard.) Slutin: There are many ways t define the heuristic evaluatin functins. The fllwing functins are three eamples f evaluatin functins fr the player using black stne SB SB SB SB SB SB SW 5 6 SW 4 2 SW P B P B P W 4 6 P W 3 where SB i (SW i ) is the number f i-cnnected black(white) stnes n the bard, P B i (P W i ) is the number f live i-cnnected black(white) stnes. Live cnnected stnes means n ther stnes are placed n bth end f the cnnected stnes. Therefre, a live 4-cnnected stne will guarantee t win. In the fllwing discussins, let us assume we have arrived t the fllwing state f bard. Player is ging t mve net. 2.3 (10 pints) Slve the whle game and find the net mve fr with the mini-ma algrithm. Rather than using heuristic evaluatin functins, we will evaluate the bard state based n the simple win/lss between and (i.e., +1 fr wins, -1 fr lses and 0 therwise). Fr the purpse f grading, yu nly need t cnsider the ndes labeled in the fllwing figure (frm 1 t 4) t epand and must use the ascending rder fr epansin, i.e., n each level f epansin, yu must epand the lwest remaining numbered nde befre epanding the higher numbered ndes. Any nde where a player have 5 stnes cnnected is the leave nde and has n children. Yu d *nt* need t draw the entire game tree in this prblem, but yu need t draw the best path fund by the mini-ma algrithm (if there are several best paths, yu shuld shw the first ne epanded). Fr each nde in the path, indicate the inde f the nde epanded and the evaluated value cmputed/prpagated by
5 Andrew ID: 5 the algrithm. Fr each nde, als indicate if it is frm a ma level r a min level. Mrever, please answer: what shuld be s first mve? hw many ndes des the minima algrithm need t epand in this case? Slutin: The best path is (mve 1, MAX level, value 0) (mve 3, MIN level, value 0) (mve 2, MAX level, value 0) (mve 4, MAX level, value 0). The first mve fr shuld be 1. There are sme ambiguities in the last questin. If yu interpret epand as creating children ndes, the number shuld be 25 r 26 (depends whether yu cunt the initial nde r nt). If yu interpret epand as visiting a nde, the number shuld be 46 r 47. We accept all these answers. 2.4 (10 pints) Slve the whle game with the alpha-beta pruning. The prblem settings are the same as the last prblem. But yu *need* t draw the entire game tree, ecept fr thse parts that yu can prune (yu must leave them ut). Indicate which level f the trees are ma levels r min levels. Fr each nde in the game tree, indicate the inde f the nde epanded and the alpha-beta values discvered. In this case, hw many ndes des the algrithm need t epand? Slutin: The game tree is shwn as fllws. Each cell cntains the mvement and the update histry (frm left t right) f alpha/beta, where A means alpha and B means beta. Fr each nde, alpha is initialized t -1 and beta is initialized t 1. Nte that we nly list alpha fr ndes in the MAX levels because beta will nt be updated in these ndes. Similarly, we nly list beta fr the ndes in the MIN levels. Due t the ambiguity f epand, we accept the answers f 17/18 and 10/11 as the crrect answers fr the last questin. Rt, A: (-1,0) MAX Mve 1, B: (1,0) Mve 2, B: (1,-1) Mve 3, B: (1,-1) B:(1,-1) MIN Mve 2, A: (-1,1) Mve 3, A: (-1,0) A: (-1,0) Mve 1, A: (-1) Mve 1, A: (-1) Mve 1, A: (-1) MAX Mve 3, B: (1) Mve 2, B: (1,0) B: (1,0) Mve 2, B: (1,0) MIN A: (0) A: (0) A: (0) MAX 2.5 (5 pints) Nw let us cnsider a different situatin. The state f the bard is the same as befre. But player need t leave twn fr his net mve (right after the first s mve). S that mve will be handled by his
6 Andrew ID: 6 representative r. But it is well knwn that r will just make a randm mve n the bard (frm 1 t 4)! Gd news is that will return fr his last mve. Therefre, the entire mving sequence is, r,,. Briefly eplain hw yu can mdify the mini-ma algrithm accrdingly t deal with this change. Under this cnditin, des player s first mve change? Please eplain why. Slutin: T handle the randm mve f player, yu shuld mdify the secnd level f yur game tree by cmputing the epectatin f yur heuristic functins under a unifrm prbability distributin fr s mvements, instead f cmputing the minimal value. Hwever, under this cnditin, player s first mve des nt change, because the game value fr t pick mvement 1 is still higher than the ther mvements. 3 Optimal Mied Strategy (10 pints) Cnsider the game with the fllwing matri: (if yu lk carefully, this is a generalizatin f the children s game scissrs, rck, paper ) (5 pints) Shw that the value f that game (i.e., the epected payff that will be btained by either player using the ptimal mied strategy) is 0. In fact, this is true fr any game fr which m ij = m ji, but yu dn t need t prve the general result. Slutin: Suppse player 1 plays a mied strategy that uses strategy i with a prbability p i. Then we shuld prve the maimin payff f player 1 is equal t 0, i.e., ma min(e p1, E p2, E p3 ) = 0, p where E pj is the epected payff f player 1, i.e., m 1j p 1 + m 2j p 2 + m 3j p 3, given the player 1 uses the mied strategy p and player 2 uses the pure strategy j. Substituting the numbers int the frmula, we have t prve, ma min( p 2 + 2p 3, p 1 2p 3, 2p 1 + 3p 2 ) = 0. p Several ways t prve this prpsitin: 1. Rigrusly, yu shuld first shw min( p 2 + 2p 3, p 1 3p 3, 2p 1 + 3p 2 ) 0. If this is nt true, it means E p1 = p 2 + 2p 3 > 0, E p2 = p 1 3p 3 > 0 and E p3 = 2p 1 + 3p 2 > 0, which will lead t a cntradictin by shwing 0 = 3E p1 + 2E p2 + E p3 > 0. Then yu can find (3/6, 2/6, 1/6) is a triplet t satisfy min( p 2 + 2p 3, p 1 3p 3, 2p 1 + 3p 2 ) = 0. This will cmplete the prf. 2. Alternatively, yu can cnsider all the intersectin pints between the pairs f E p1, E p2, E p3, i.e., the pint f (3/6, 2/6, 1/6), and als all the bundary pints f the prbability distributins, i.e., (1, 0, 0), (0, 1, 0), (0, 0, 1). Prve min( p 2 + 2p 3, p 1 3p 3, 2p 1 + 3p 2 ) 0 n these pints and thus it is true fr all the prbability distributins. 3. As anther way t prve, yu can first shw that the ptimal mied strategies f player 1 and player 2 are the same given the game is symmetry (strictly speaking, yu shuld prve this is crrect, but we will still give yu credits if yu just hand wave here). Thus, the epected payff f player 1 in this case is always 0.
7 Andrew ID: (5 pints) Derive the ptimal mied strategy, which is the same fr bth players. If yu have truble with questin 1, just assume that the value f the game is 0 and prceed with this questin. Slutin: (p 1, p 2, p 3 )=(1/2, 1/3, 1/6) 4 Widget Cmpanies (10 pints) Cnsider tw widget cmpanies, Acme and U.S. Widgets, cmpeting fr the same market and each firm must chse a high price ($2 per widget) r a lw price ($1 per widget). Here are the rules f the game: 1. At a price f $2, 5000 widgets can be sld fr a ttal revenue f $10000; 2. At a price f $1, widgets can be sld fr a ttal revenue f $10000; 3. If bth cmpanies charge the same price, they split the sales evenly between them; 4. If ne cmpany charges a higher price, the cmpany with the lwer price sells the whle amunt and the cmpany with the higher price sells nthing; 5. Payffs are prfits revenue minus the $5000 fied cst. (Nte: This is a very special case f smething called the Curnt s game that mdels price cmpetitin. Unfrtunately, it is an verly simplified and unrealistic eample because we frce it t be a zer-sum game.) 4.1 (5 pints) Write dwn the game matri. Slutin: Suppse Acme s strategies crrespnd t the rws and U.S. Widgets strategies crrespnd t the clumns. The game matri fr Acme is shwn as fllws, (5 pints) Is there a pure strategy slutin? Briefly eplain. Slutin: There is ne pure strategy slutin when bth cmpanies are selling the prducts at the price f $1. This is because bth cmpanies will get lwer prfits if they switch t ther strategies in this situtain. 5 The Nim Game(10 pints) A very simple versin f the game f Nim is played as fllws. There are 2 players and, at the start, tw piles n the table in frnt f them, each cntaining 2 matches. In turn, the players take any psitive number f matches frm ne f the piles. The player taking the last match lses. Nw cnsider a mre elabrate versin f the game f Nim is played as fllws. There are 2 players and, at the start, three piles n the table in frnt f them, each cntaining 2 matches. In turn, the players take any psitive number f matches frm ne f the piles. The player taking the last match lses. Sketch a game tree. Shw that the first player has a sure win. Slutin: The game tree is sketched as fllws. In this tree, we nly cnsider the best strategy fr player 1 and d nt epand any redundant ndes fr player 2. Fr eample, in the secnd level, we d nt epand the ndes f (-, ii, i), since it will have the same utcme as (-, i, ii).
8 Andrew ID: 8 Init: (ii, ii, ii) Player 1 (-, ii, ii) Player 2 (-, i, ii) Player 1 (-, i, -) Player 2 (-, -, -) Player 2 (-, -, ii) Player 1 (-, -, i) Player 2 (-, -, -) 6 The Pin Game(10 pints) A game is played as fllws: The tw players, i.e., A and B, simultaneusly hld up either ne r tw pins. A wins if there is a match (the number f pins are the same fr bth players) and B wins therwise. The amunt wn is the number f pins held up by the winner. It is paid t the winner by the lser. 6.1 (5 pints) Write dwn the matri frm f the game and specify the 2 players pssible pure strategies. Slutin: Suppse A s strategies crrespnd t the rws and B s strategies crrespnd t the clumns. The game matri fr A is shwn as fllws, (2 pints) Eplain why there is n pure strategy slutin in this game. Slutin: Because nne f these 4 game states are equilibria. Fr eample, if we start with the state (1,1), B will switch t the strategy 2 in rder t get a higher payff, and then A will switch t the strategy 2 fr the same reasn. This iterative prcess will nt be cnverged. Therefre, there is nt pure strategy slutin in this game. BTW, it is als k t argue that maimin!= minima in this game matri. 6.3 (3 pints) Cmpute the mied strategy slutin and the value f the game. Slutin: Assume A plays a mied strategy {1:p, 2:(1-p)}, and B plays a mied strategy {1:q, 2:(1-q)}. The mied strategy slutin is, p = arg ma min(p (1 p), 2p + 2(1 p)) = 1/2, p q = arg min q ma(q 2(1 q), q + 2(1 q)) = 2/3, and the game value is 0. Nte: when yu are cmputing the mied strategy f q, yu need t use the minima prcess.
9 Andrew ID: 9 7 Mied strategy(5 pints) Cnsider a tw-player game in which each player has 4 pssible strategies (dented 1 t 4) with the matri frm shwn belw. Cmpute a mied strategy slutin, assuming that each player assigns the same prbability t strategies 1, 2, and Slutin: Assume player 1 plays a mied strategy {1:p, 2:p, 3:p, 4:(1-3p)}, and B plays a mied strategy {1:q, 2:q, 3:q, 4:(1-3q)}. The mied strategy slutin is, p = arg ma min(p (1 3p), 3p + (1 3p)) = 1/5 p q = arg min q ma( q + (1 3q), 3q + (1 3q)) = 1/5
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