HW 2 Hints

For chapter 1 number 10, consider some specific examples. For example, if $s=15$ and my previous bid is 10, the other player has just made a higher bid, does it make sense for me to respond with a bid of 26 or 27 or 28 or ...? If I pass I lose $10$. If I bid 26 or more, my final bid is also at least 26. Then if I am the final bidder (and get the stakes) I end up losing my final bid minus the stakes (15), so I lose at least 11. If I am not the final bidder and don't get the stakes I lose my final bid which is at least 26. In either case I am better off passing then bidding 26 or higher. Give this argument in the setting of general $s$ for answering this problem. (Use a slight variation for the second part when the conservative convention is assumed.)

More specifically for part (a), assume that the stakes are 10 player 1 has bid 7, player 2 has responded and now it is player 1's bid again. We want to show that it is not rational for player 1 to bid 18 or more on the next bid. If player 1 passes their loss is 7. Assume player 1's next bid is n > 17 = 10 + 7 (i.e., 18 or more). Then player 1's final bid f in the game is at least n so f >= n. If player 1 `wins' (makes the final bid) then they pay f and get back 10, for a loss of f - 10. But f - 10 >= n - 10 > 10 + 7 - 10 = 7. So the loss in this case is greater than 7. If player 1 `losses' (does not make the final bid) then they pay f. But f >= n = 10 + 7 > 7. In each case the loss is greater than that obtained by passing. So it is not rational for player 1 to bid 18 or more.

For the homework, a general argument with stakes s (instead of 10) and previous bid p (instead of 7) needs to be made.

For number 12, you are also asked to give the answer to the closest number of millions. For example, to give the answer to the closest number of thousands, if the exact figure is 53,439 you say 53 thousand.