Game Theory Basics - Department of Mathematics

1.13. EXERCISES FOR CHAPTER 1 23ending condition, but that nevertheless is close to nim and ends in finite time whenplayed well.]Exercise 1.7 Consider the following network (in technical terms, a directed graph or “digraph”).Each circle, here marked with one of the letters A–P, represents a node of thenetwork. Some of these nodes (here A, F, G, H, and K) have counters on them, whichare allowed to share a node, like the two counters on H. In a move, one of the countersis moved to a neighbouring node in the direction of the arrow as indicated, for examplefrom F to I (but not from F to C, nor directly from F to D, say). Players alternate, and thelast player no longer able to move loses.DCHBGLAFKPEJOINM(a) Explain why this game fulfils the ending condition.(b) Who is winning in the above position? If it is player I (the first player to move),describe all possible winning moves. Justify your answer.(c) How does the answer to (b) change when the arrow from J to K is reversed so that itpoints from K to J instead?Exercise 1.8 Consider the game chomp from Exercise 1.3 of size 2 × 4, added to a nimheap of size 4.What are the winning moves of the starting player I, if any?[Hint: Represent chomp as a game in the normal play convention (see exercise 1.3(b),by changing the dot pattern), so that the losing player is not the player who takes the“poisoned cookie”, but the player who can no longer move. This will simplify finding thevarious nim values.]