Econ 223 Midterm Examination: September 6, 2004
(questions are in blue, suggested answers in black...after i grade the exam I will explian answers and grading in more detail...this is just in case you would like to know how I would have answered the exam.....)
Grade distribution
Q1 pure types of games (18 Marks)
Dixit and Skeath (DS), the author's of the textbook for the course, classify games into a number of various "pure types" by asking and answering some interesting questions. Identify these types and briefly explain the key concepts/distinctions used to interpret and understand each type. (By "briefly" I mean in no more than 1-2 sentences per key concept; you do not have to provide example games in your explanations.)
Q2 Strategies in sequential games (3 marks each part, 12 in total)
2.a The game tree below (Figure 2.1) represents a stylized 3 player (A,B,C) sequential game (no payoffs are shown at the crosshatched terminal nodes). How many strategies does each player, A, B and C, have in this game? List B's strategies (you do not have to list A's or C's).
A has 2 strategies, B has 4 strategies, C has 24 strategies
B's 4 strategies are : uu,ud,du,dd : eg the strategy labeled du means at node b.1 she will choose "d", and at node b.2 she will choos "u"e
JFs comments - about 1/3 of the class still has difficulty figuring out the number of strategies and how to describe them - I'll develop a video clip to explain this answer to help you out.
Figure 2.1
Q2 Information sets and strategies:
The game tree below (Figure 2.2) represents another stylized 3 player (A,B,C) sequential game (no payoffs are shown at the crosshatched terminal nodes). The grey regions enclosed in dashed lines are information sets.
2.b: What does player C know when he makes his move?
Player C can't tell whether B has chosen up or down, but can tell whether A has chosen up or has chosen down. Note also : B doesn't know what C will do (C moves after B) nor does B know what A will do - the information set around B's nodes b.1 and b.2 means that B can't tell which node she's at- ie can't tell/observe what A has done.
2.c: How many strategies does each player, A, B and C, have in this game? List C's strategies (you do not have to list A's or B's).
A has 2 strategies
B has 2 strategies (only 2 choices at her ONE information set)
C has two information sets and at each information set she has two things she can do so 2x2=4 strategies in total: uu,ud,du,dd : eg the strategy labeled du means at her top information set she will choose d, and at her bottom information set she will choose u
2.d: Sketch and label the game tree for this game with the following, changed, information structure: B knows what A does when she moves and C knows what B does but not what A does when it is his turn to move. Note: There are no information sets for B; when B chooses "u", C can observe that, but she can't see whether A has chosen "u" or "d", so we link nodes c.1 and c.3 into one information set. Similar reasoning for nodes c.2 and c.4
Q3 Alternating offer bargaining. ( 2 parts ; 20 marks total; 7/13 for a/b)
Spring is on us and the grass needs cutting. The neighborhood Kid (K) and an Old Lady (OL) are bargaining today, Saturday, about how much the OL is going to pay the Kid to cut the grass. The OL wants the grass cut, and indeed is prepared to pay up to $40 to get the job done today. Being self-interested she would prefer to pay the Kid less than more, and she calculates her net personal payoff as $40, her top-dollar value, minus whatever she has to offer the Kid to get the job doneÉas long as it's done today. For example if the Kid takes the job for $18 the Old Lady's payoff is $40-$18=$22. If the Kid rejects her offer, the game ends, the grass doesn't get cut and the Old Lady's payoff is 0, as is the Kid's payoff. The Kid wants the money but doesn't like to work. If he gets offered $x to do the job and accepts it, he calculates his net personal payoff as $x minus $15, $x-$15. For example, if the OL offers him $18 for cutting the grass his net personal payoff, if he accepts the work, is $18-$15=$3. The Kid generally won't accept any work unless he gets a strictly positive net personal payoffÉbut he is desperate enough, and self interested enough, to accept the job if his net personal payoff is something positive rather than zero.
Q3.a If the OL moves first, making a wage offer (in dollar amounts from 1,2,3Éto 40) and the Kid moves second, either accepting or rejecting, use rollback reasoning to predict what strategies will be chosen and what the resulting payoffs will be.
In a rollback equilibrium OL anticipates what response K will make. OL can offer K $16, and he will accept. If she offers him any less he will refuse, giving her a payoff of $0, while if she offers any more he will accept, but her net personal payoff will be lower than $24. K's choice is quite simple, a binary yes or no - with no giving him a net payoff of $0 and yes to any offer exceeding $15 a positive net payoff. No need to draw the game tree.
Q3.b Let's complicate the game a little by adding two more possible bargaining rounds and a strict order of bargaining - as we did in our classroom games. The Kid knows the OL is anxious to get the job done this Saturday rather than next Saturday, or even later. So he might consider rejecting her offer to cut the grass this week and suggest waiting till next week, and making a counter-offer himself in terms of payment and seeing whether the OL will accept or reject. The OL can then accept or reject this. If she accepts the game ends, the grass is cut and payments are made; but if she rejects there is one last opportunity - she can make a counter-counter offer for the job to be done 2 weeks away. (All wage payments, if any, take place todayÉ.even if the work doesn't get done until later). They both know that the OL is not willing to pay as much as $40 for a job done next week, or the following week. She is only willing to pay at most $30 top-dollar today for the job to be done next week, and at most $20 top-dollar today for the job to be done the week after that. As above she calculates her net personal payoffs by subtracting what she has to pay the Kid from her "top-dollar" value. So for example, if she and the Kid made an agreement for him to cut the grass two weeks away for $18 say, her net personal payoff is$20-$18=$2. If the Kid rejects her counter offer to do the job on the last Saturday the game ends and they both receive $0 in net personal payoffs. Assume offers and counter offers are made in units of $1 (ie no fractions) and that in the case of indifference , eg between working or not, or between getting the lawn cut or not, either party has to be offered a positive net personal payoff to motivate them to change from one transaction to another. The question here is: Use your knowledge of game theory to predict the rollback equilibrium strategies, outcomes, and payoffs. Explain your reasoning.
As with all rollback reasoning we start at the end. Go to the last round of the game two Saturdays away É..and figure out what should happen. If we get to the last round of the game we're basically in the 3.a situation with $20 as a top-dollar value for OL (this is an "ultimatum" game). She will offer the Kid $16 for cutting the lawn two Saturdays away, and he will accept. But her net payoff is, at that stage, only $4. Anticipating that, the Kid at round 2 can ask her to pay him $25 and he'll do the job one Saturday away. If the OL rejects this she can carry the game to the final round and expect receive a payoff of $4, whereas if she accepts she can receive a payoff of $30-$25=$5. So she would accept if we did get to that stage, since a net gain of $5 is better than a net gain of $4. The Kid, if the game is at this stage, receives a payoff of $25-$15=$10, far better than the $1 he obtains in the final round. If the Kid tries for any higher payment, eg $26 or above, the OL will reject and he'll (expect to) end up doing the job 2 Saturdays away for$16, a net personal payoff of $1. . If he considers for any lower payment, eg $24 or below, the OL will accept but the Kid will be getting less net personal payoff. Anticipating that the Kid can guarantee himself a net personal payoff of $10 at round 2, the OL will offer him a wage $26 at round 1. This net gain of $26-$15=$11 is higher than the $10 he could otherwise expect, so the Kid accepts at round 1: ie he does the work for $26 today. For her part, the OL gets the job done today for a net payoff to herself of $14=$40-$26. We never actually get to rounds 2 and 3 in this prediction of the outcome.
This was a part I was going to ask but cut out:I include it here for "fun": Q3.c What difference, if any, would it make to your answer to Q3.b if the Kid moved first (and last) and the old Lady moved second ?
If the Kid moves first, he also moves last in this 3 round game. As with all rollback reasoning we start at the end. At round 3 the most he can expect to get paid is $19 (the OL will reject the idea of paying him $20), for a net gain to himself of $19-$15=$4. At round 2 the OL will anticipate that she needs to make him an offer that gives him a slightly higher net personal payoff :a wage of $20 will do , giving the Kid $20-$15=$5 in net personal payoff. He will accept rather than take the game to the last round (where he can only expect to get $4 in net personal payoff), and the OL's net personal payoff is $30-$20=$10. The Kid, at round 1, anticipating that the OL can get $10 net personal payoff for herself if she carries the game to round 2, can make sure she gets $11 at round 1 by suggesting a wage of $29. ($40-$29=$11). Of course the kid benefits $29-$15 equal to $14. So if the Kid moves first we'd expect him to suggest a wage of $29 for doing the job right away and for the OL to accept, each player receiving payoffs $14 and $10 respectively: the Kid benefits and the OL is worse off, compared to 3b, when the Kid moves first.
Q4 2 Tough Love (20 marks, 10 for each part, 2 parts)
JF 2004 I marked a and b out of 10. About 1/3 of the class didn't develop the right payoff table for the sequential game in part b and got zero for that part. so 10/0 was a common grade. All I can say is that almost every midterm in the past incuded one of these types of questions, and there are model answers too.
Teenager/parent relationships are complex. But lets examine a simplified version of one small piece of interaction we might call the "tough love" game . A young teenager has to make a choice between staying out late on a school night (LATE) or being home early (EARLY). Her parent has to decide between being TOUGH, aggressively challenging and interrogating their daughter, or being SOFT, avoiding any confrontation and not mentioning the incident. IF the parent and teenager were playing a simultaneous game the payoffs for the various strategy combinations would look like those in the following table, with a poor dominance solvable Nash Equilibrium with the parent being tough and the teen coming home late (those mutually preferred payoffs with the teen coming home early and the parent being soft 3,4 just aren't an equilibrium). Being tough doesn't work here because the teen "doesn't care" ie has a dominant strategy to stay out late, in this game.
|
PARENT |
||
TEENAGER |
SOFT |
TOUGH |
|
EARLY |
3 , 4 |
1 , 2 |
|
LATE |
4 , 1 |
2 , 3 |
But perhaps the real game being played here is sequential not simultaneous, with the Teenager moving first, and the Parent moving second. Using the same payoffs as in the Table:
Q4a Draw and clearly label the game tree for this sequential game and analyse the game pruning relevant branches to show the rollback path of play and outcomes.
Q4b Take the sequential game in 4a and analyse it as a simultaneous game: ie, identify the strategies for all players, draw an appropriate payoff matrix for the game, and analyse this simultaneous game. Find all of the pure strategy Nash equilibria for this game and briefly explain the similarities and differences from your answer to part 4a.
The graph below helps "explain" the answers . For 4a the Teen will come home early and the parent will be soft, with payoffs of (3,4) (see left hand side top game tree, labeled Subgame perfect NE). Path of play is shaded as Early then Soft. For 4b you need to construct the 2x4 payoff table just below the game tree . There are two NE when we analyze the game as a simultaneous game using the appropriately defined player's players strategies (I have replicated the game trees and payoff table on left and right hand sides to separate out the 2 Nash equilbria). The parent has 4 strategies, shown as lists, the teen has 2. X's and O's mark the best responses, and the NE occur at the mutual best responses. One NE, E=early for the teen and (S,T) for the parent , Soft if early and Tough if Late , with payoffs (3,4) corresponds to the Sub Game perfect NE we find via Rollback reasoniong. The other NE, on the right, L=Late for the teen and T,T , always tough for the parent, with payoffs 2,3 just as in the simultaneous game, corresponds to the sequential game on the right where the parent plays tough if the teen comes home earlyÉbut why would a rational parent do this,and a rational teen knowing her parents preferences expect this. It's not a "sub game perfect " thing to do, or expect. So what looks like a reasonable dominance argument for always being tough breaks down when we look at a different information structure for the game and letting teens move first.
Q5 Simultaneous games Multiple Choice.
The 10 payoff matrices below describe various types of 2 player simultaneous games. Payoffs to players are the numbers 1,2,3,4 indicating preference rank. Higher is better, so for example 4=best & 1=worst. In the answer sheet provided write down the option, or set of options, from the following list (A through H) that most accurately describes the game AND identify all pure strategy equilibrium strategies for each game (in some cases more than one option may be appropriate). You do NOT have to rewrite the payoff matrix in your answer booklet.
a) A prisoner's dilemma game
b) A game with a dominant strategy equilibrium
c) A game with no Nash Equilibrium in pure strategies
d) A dominance solvable game
e) An assurance game
f) A pure coordination game
g) A game of chicken
h) A game of battle of the sexes
i) A constant sum game
