Internal Code: 1HAEF
Problem 1 continues from Problem 1 of Assignment 1. The numbering of sub-parts is kept throughout the problem, so the rst sub-part of this problem is (d).
Consider the following auction that most of you must have encountered before.
There is a prize (e.g. a chocolate rabbit). Each player buys tickets, worth 1 cent each, write his or her name on it, and put them in a box. Then a ticket is drawn at random and the person whose name is on
the ticket receives the chocolate rabbit. If no player bought any tickets (that is, the box is empty), then all players get nothing.
Assume, for all parts of this problem, that there are two players and that the rabbit is worth $1 for both. The auction I am describing is usually used for charity; ignore all aspects of charity and assume that the only thing players care about is getting the rabbit.
Note: In this problem, you may assume that (i) tickets are perfectly divisible (you can buy 0.71486 tickets) or (ii) tickets must be whole non-negative numbers (0 is included in both cases). If you assume (i),the solution would be much nicer, but you will need to deal with a small technical problem. If you assume (ii), a technical problem disappears, but solution is ugly. It’s your choice.
Suppose that the timing of the problem has changed: Player 1 buys the lottery tickets and put them in a transparent box. Player 2 observes the decision of player 1 and decides how many lottery tickets to buy.
(d) [8pt] Model this new situation as a formal game.
(e) [12pt] Find the subgame perfect Nash equilibrium of this game. (Hint: although this game may look slightly dierent to what we have seen so far, approach it in the same way as other games: nd that Player 2 will do given Player 1’s choice; then nd what Player 1 will do knowing the choice of Player 2 that will follow. You will need to write maximisation problems; pay attention what is the decision variable in these problems and what is not.)
Let us now consider a hybrid of (a) and (d). Let us suppose that Player 1 buys tickets and put them in the box, and Player 2 can observe whether the box is empty (Player 1 bought nothing) or has tickets, but cannot observe how many tickets there are inside.
(f) [10pt] Model this situation as a formal game.
(g) [7pt] Give an example of the strategy prole in this game.
(h) [4pt] Explain why the strategy prole in (g) is / is not a Nash equilibrium.
(i) [5pt] Does pure strategy Nash equilibrium exist in this game? Give an example if it does or explain if it does not.
(j) [4pt] Suppose we are looking for a subgame perfect Nash equilibrium. How does you answer to (i) changes?
Suppose there are three players (1, 2 and 3) who need to pick one of the three alternatives: a; b or c. The decision is reached as follows:
Suppose players’ preferences are: a 1 b 1 c; b 2 c 2 a; c 3 a 3 b. All players know the preferences of other players.
(a) [5pt] Suppose that each player votes for alternatives according to their preferences (that is, if i prefers a to b, then i votes for a). Find the chosen alternative w2.
(b) [15pt] Suppose now that each player anticipates the outcome of the second round and votes strategically. That is, players’ strategies form a subgame perfect Nash equilibrium of the overall game. Suppose also that, when player’s vote does not matter, she votes for an alternative she personally prefers. For example, when choosing between a and c, if players 2 and 3 vote for c, then the vote of player 1 does not change the outcome. In that case, we assume that player 1 votes for a.1 When player’s vote matters, the player votes so that she will get the best possible outcome (hint: it involves voting contrary to own preferences in the rst round).
Find the equilibrium strategies and the chosen alternative.
(c) [10pt] Suppose that player 1 sets the agenda; that is, player 1 picks the order in which alternatives are voted for. Can player 1 pick the order so that her favourite outcome (a) gets selected?
Consider the following problem. Fisher (who is a female) has an object that is worth $1 to her and $5 to Henry. Fisher can invest into the object, at the cost of $2; the investment increases the value of the object to $2 for Fisher and to $10 for Henry. After Fisher decides whether to invest or not, Henry observes Fisher’s decision and makes her an oer to buy the object. That is, Henry oers a payment p to Fisher and Fisher can only reject (and consume the object herself) or accept (and consume the payment p for the object).
(a) [7pt] Model it as a game.
(b) [7pt] Find a subgame perfect Nash equilibrium (SPNE).
(c) [6pt] Find a Nash equilibrium (but not necessarily SPNE) where Fisher invests.
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