Unformatted text preview: ORIE 4350 1/27/2012 Two Exercises You will not be asked to submit these (at this time, maybe later), but you should work on them, write out your answers, and bring it to class on 1/31. 1. In class on 1/26 we worked out the decision tree for Nim12 (2 rows, one with one token and one with two tokens). This is equivalent to a version of the Rooks Game that we looked at on 1/24: the board has 2 (columns) and 4 (rows) with initial positions given by white rooks in cells (1,2) and (2,1) and black rooks in cells (1,4) and (2,4). On 1/26 we drew and analyzed a decision tree for Nim12. Also on 1/24 we looked at the 3‐by‐4 version of the Rooks Game with initial positions given by: white rooks in cells (1,2), (2,1) and (3,1) and black rooks in cells (1,4), (2,4) and (3,3), which is equivalent to Nim121. The figure below gives the basic form of the decision tree for this game. Each non‐terminal node corresponds to a decision point (some for White, some for Black) and each branch corresponds to an action in the form of an ordered pair (i,j) where i is the column number and j is the number of rows moved. Indicate which nodes are decision nodes for White, which nodes are decision nodes for Black, and which nodes are terminal nodes. You may find it convenient to use two very distinctive colors other than black and white to mark the decision nodes. You may also find it convenient to refer to the 1/26 analysis of the N12 tree. Indicate on each edge the corresponding ordered pair (i,j). For each terminal node indicate whether the outcome is a win for White (label with a V) or a loss for White (label with an L). Now starting with the win/loss designations at the terminal nodes, determine at the decision nodes that immediately precede the terminal nodes whether rational play (by both players) leads to a win for White (label with a V) or a loss for White (label with an L) at these nodes. Repeat this process inductively to determine for all decision nodes (including the root) whether rational play from that node leads to a V or an L. 2. At the end of class on 1/26 we looked briefly at a 2‐player, win‐lose game in which there are six playing cards, 2 each of value 1,2 and 3. Players I and II alternate taking one of the remaining cards until the sum of the values of the cards taken is 10 or greater, at which time the game ends and the player who took the last card wins. Informally analyze this game (without drawing a decision tree). Assuming player II is rational, can player I win with an initial choice of a card of value 2? Assuming I initially chooses a card of value 1 and plays rationally in her later choices, can player II win? Assuming I initially chooses a card of value 3 and plays rationally in her later choices, can player II win? If you are feeling frisky and you have a large piece of paper, draw and analyze the decision tree for this game in the spirit of exercise 1. Keep in mind that the actions available at a decision node depend on more than the sum of the values of the cards taken in prior moves. (For example the actions available to I after I opens with a 2 and II follows with a 2 are not the same as the actions available to I after I opens with a 1 and II follows with a 3, even though the total value of the cards taken is 4 in both cases.) ...
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