# ECON 308: Midterm Examination – Fall 2010

## Section I: True/False/Uncertain and Explain.

1. When randomizing or mixing strategies, players should never use rules or patterns to pick their strategies. Players should instead rely solely on pure randomization devices, like rolling a die to determine which strategy to play.
False. It is important that you use a pseudorandom device to mix your strategies. It is often infeasible to carry or use a randomization device when making strategic decisions. Moreover, even computers rely on pseudorandom techniques to generate random numbers. It is important that your opponent not be able to figure out the pseudorandom process used to mix your strategies, so using rules and patterns to mix is acceptable, so long as your opponent cannot determine how you are mixing. Dixit and Nalebuff use the example of a pitcher choosing whether to throw a fastball or curveball: if the second hand on the pitcher's watch points to an even number, he throws a fastball, otherwise he throws a curve. So long as the pitcher's watch and the batter's watch are not synchronized, even if the batter picks up on the fact that the pitcher is using his watch, he will be unable to exploit the pattern to his own advantage.
Answers of "false" which show an understanding that hiding any pattern of mixing is crucial to success receive full credit. Answers of "true" that show an understanding of what it means to mix optimally receive partial credit.
2. Being laid off is significantly worse for an employee than losing his job due to his company going bankrupt.
True. Being laid off (being fired) signals to future prospective employers that you were a low productivity worker at your last firm. When picking which employees to let go, the firm will keep its most productive workers on board. If the company fails, everyone loses their job, so being unemployed carries no informative signal for future employers.
An answer of "true" which mentions the Akerloff "lemon problem," asymmetric information, or specifically adverse selection, received full credit. Some substituted "division" for "company" or "firm" in my answer, and that is OK. One mentioned that government layoffs occur by seniority, so being laid off carries a weaker signal (if any) about an employee's productivity. An answer of "false" that mentioned unemployment insurance as the main reason bankruptcy can be worse than being fired received partial credit.
3. The following game has a unique Pure Strategy Nash Equilibrium.
Cathy
XY
RichardX4,43,1
Y1,37,7
False. This is a coordination game, and thus the game has two Nash Equilibria. The set of Nash Equilibria $$\mathrm{N.E}. = \{(\mathrm{X},\mathrm{X}),(\mathrm{Y},\mathrm{Y})\}$$.
The answer above received full credit. It was difficult to give partial credit for this problem, since it is a simple application of the definition of Nash Equilibrium. If the answer was not as above, it is likely because: 1) you did not understand the concept of Nash Equilibrium or 2) you misread the question. If you answered "true" and named one of the Nash Equilibria, you received partial credit.
4. Tit-for-tat is an evolutionarily stable strategy because it beats each strategy it competes against.
True. An evolutionarily stable strategy, if adopted by a population, will defeat any invading strategy. TFT is evolutionarily stable, since any invader will quickly be converted to using TFT, or will exit the population. This is because TFT is provocable, nice, and forgiving.
This question was trickier than I intended. Therefore, answers like the above received full credit. Answers of "false" which mentioned that TFT lost to individual strategies in Axelrod's tournament also received full credit. Answers of true or false with an explanation which indicated understanding of TFT and repeated Prisoner's Dilemma received partial or full credit, depending on their quality.
5. Competitive markets are the only way to achieve an optimal allocation of scarce resources.
False. Competitive markets are not the only way to achieve an optimal allocation of scarce resources. There are other coordination methods, including mutual adjustment, direct supervision, and standardization (of process, of skills, of output, and of norms). These other, non-price coordination methods are often mixed with price coordination, but they also represent alternatives to the price system. They are used when the price system fails, as when markets for the desired product do not exist, or when significant frictions prevent the market itself from achieving an optimal allocation.
An answer of "false" with an explanation that revealed an understanding that different coordination mechanisms exist received full credit. Answers of "false" which explained that sometimes monopolies naturally occur, such that monopolies are the best market we have available, received partial credit. Answers of "true" which simply list the four assumptions of perfect competition received the least credit, since they do not begin to address the question.
6. In sealed-bid first-price auctions (or Dutch auctions), a bidder's optimal bid is his true value for the good at auction.
False. Bidding your true value is an optimal bid in a sealed-bid second-price auction (Vickrey auction), since because you pay the second highest bid, bidding less than your true value is never advantageous (the strategy of bidding less than your true value is dominated by bidding your true value). In a sealed-bid first-price (or Dutch auction), you should shade (or reduce) your bid. The optimal bid is one that shades to just above the second-highest bidder's value, since this implies you are bidding as if you'd won (you can't change your bid without regret).
Answers of "false" which mentioned shading received partial credit. Answers of "false" which mentioned the optimal bid being just above the second highest bid received full credit. Of this kind of answer, one mentioned the Revenue Equivalence Theorem (RET) which is a shortcut to explaining why the optimal bids in both the Vickrey and Dutch auctions lead to bidders acquiring the item for just above the second highest bidder's value.

1. "With perfect competition… each [person] is a price taker… Only in such circumstances can we say prices acts as sufficient statistics that convey all the necessary information." (DS p.63) Being a price taker means there are so many others in the market that you do not expect your actions in the market will change the price. But in order for price to be a sufficient statistic (to carry all the relevant information needed to make allocative decisions), it is important for the actions of individuals to affect the price. Reconcile these seemingly conflicting claims.
These claims are entirely commensurable, once properly understood. Douma and Schreuder are correct in asserting that in competitive markets, individuals are price takers. Being a price taker implies only that an individual may be "sorted" along a demand curve or supply curve based on willingness-to-pay or willingness-to-produce. Since supply and demand analysis relates prices to quantities, any change in other relevant variables lead to a shift in demand or supply, causing price to adjust. Therefore, individuals acting in concert, responding to some change in their demand (other than price), can affect price. It is the aggregate action of individuals in the market place which trigger information transfer through price.
Answers which reveal an understanding of the difference between aggregate changes and individual decision-making received full credit. Answers which reveal an understanding of the price system and its efficiency properties received partial credit.
2. Suppose there are four lily pads on a pond and four frogs who live at the pond. Each lily pad can hold one frog, and each frog wants to occupy one lily pad. Further suppose frogs like to be next to other frogs. Each frog leaves in the morning to frolic in the forest, doing froggy things. Each frog arrives back at the pond in the evening, one at a time, looking for a lily pad on which to rest his weary bones. (You might want to draw a diagram and label each lily pad to make your answer clearer.)
1. Which lily pad should the first frog choose to maximize his happiness? (5 points)
The first frog should choose the second or third lily pad.
2. Which lily pad should the second frog choose to maximize his happiness? (5 points)
The second frog should choose the second or third lily pad, whichever the first frog did not take.
Any of the answers to the previous parts which were consistent with the explanation in this part received full credit. The explanation received full credit if it was understandable and unambiguous. If it was clear that you did not understand the question from your answer, you received less credit. If you answered a different question than the one asked, you received the least credit.
3. Consider a penalty kick in soccer. We simplify by supposing the goalie only has three options when facing a kicker: diving left, diving right, or staying in the center of the goal (which is where she must start before the ball is kicked). Likewise, a kicker can kick to a goalie's right, down the center, or the goalie's left. The payoffs are listed in the "normal form" or "strategic game" below. Think of the payoffs as representing a percentage chance that each succeeds.
1. List the set of players, the strategies for each player, and the payoff for each outcome. (5 points)
The players $$N=\{\mathrm{Kicker,Goalie}\}$$. The strategies for each player are the same, $$X_g=X_k=\{L,C,R\}$$. The payoff are as follows:
 $$u(L,L)=(60,40)$$ $$u(L,C)=(70,30)$$ $$u(L,R)=(98,2)$$ $$u(C,L)=(100,0)$$ $$u(C,C)=(0,100)$$ $$u(C,R)=(100,0)$$ $$u(R,L)=(97,3)$$ $$u(R,C)=(60,40)$$ $$u(R,R)=(50,50)$$
Answers as above received full credit. Answers which did not have all three parts as above received partial credit. Answers which did not show an understanding of how to read or interpret a strategic (or normal) form game, or those that simply copied the game, received the least credit.
2. Find the optimal mix (mixed strategy equilibrium) for the goalie only. That is, in what proportions should the goalie play each strategy available to her in order to make the kicker indifferent between the strategies available to her? (10 points)
The steps for solving this game are exactly as outlined in class. We will proceed as follows:
Algorithm
1. Set up how the goalie will mix and find the expected payoff to the kicker.
2. Equate the expected payoff from kicker playing strategy $$L$$ to the expected payoff of kicker playing strategy $$C$$. Solve for $$p$$.
3. Equate the expected payoff from the kicker playing strategy $$R$$ to the expected payoff of kicker playing strategy $$C$$. Solve for $$q$$.
4. Insert the expression found for $$q$$ into the expression found for $$p$$ (or vice versa). Solve for the variable that remains, $$p$$ (or $$q$$).
5. Insert the solution for $$p$$ into the expression for $$q$$ (or vice versa). Solve for $$q$$.
6. Insert $$p$$ and $$q$$ into $$1-p-q$$ and solve.
7. Write our answer in the form $$(p,q,1-p-q)$$.
8. Check your answer by inserting $$p$$ and $$q$$ into the expected payoff functions for kicker to show that kicker is indifferent when goalie uses the mix $$(p,q,1-p-q)$$ (the expected payoffs should be equal). Check that $$p<1$$ and $$q<1$$ and that $$(1-p-q)+p+q=1$$.
Calculation

Now that we have described our algorithm, let us solve for the optimal mix. Column player will mix with $$(p,q,1-p-q)$$, which is to say column player will play $$L$$ with probability $$p$$, $$C$$ with probability $$q$$, and $$R$$ with probability $$1-p-q$$. The expected payoffs to the kicker are below.

Goalie
$$p$$$$q$$$$1-p-q$$$$E(u_k(x_k))$$
LCR
L60,4070,3098,2$$60p+70q+98(1-p-q)$$
KickerC100,00,100100,0$$100p+100(1-p-q)$$
R97,360,4050,50$$97p+60q+50(1-p-q)$$
We set $$E(u_k(L))=E(u_k(C))$$ and solve for $$p$$. $60p+70q+98-98p-98q = 100-100q$ $60p-98p = 100-100q-70q-98+98q$ $-38p = 2-72q$ $$\label{p} p = {72q-2 \over 38}$$ Now we set $$E(u_k(R))=E(u_k(C))$$ and solve for $$q$$. $97p+60q+50-50p-50q = 100-100q$ $110q = 100-97p-50+50p$ $$\label{q} q = {50-47p \over 110}$$ Now we substitute $$q$$ into our previous expression for $$p$$ and solve again for $$p$$. $p = {72\left({50-47p \over 110}\right)-2 \over 38}$ $38p = 72\left({50-47p \over 110}\right)-2$ $38p = {72(50)-72(47)p \over 110} - 2$ $38(110)p = 72(50)-72(47)p - 220$ $p(38(110)+72(47)) = 72(50) - 220$ $p = {72(50) - 220 \over (38(110)+72(47))} = 0.4469$

Now we solve for $$q$$. $q = {50-47p \over 110} = {50-47(0.4469) \over 110} = 0.2636$

And lastly we solve for $$1-p-q$$. $1-p-q = 1 - 0.4469 - 0.2636 = 0.2895$

Therefore the goalie's optimal mix is $$(0.4469,0.2646,0.2895)$$. Indeed, these sum to one, and each is less than one, so they do not violate the definition of probability.

We can check that this mix makes kicker indifferent by checking whether $$E(u_k(L))=E(u_k(C))=E(u_k(R))$$. $E(u_k(L)) = 60p+70q+98(1-p-q) = 73.64$ $E(u_k(C)) = 100-100q = 73.64$ $E(u_k(R)) = 97p+60q+50(1-p-q) = 73.64$

The kicker is indifferent, so this is goalie's optimal mix.

Answers which obtained the right answer using the right method received full credit. Answers which obtained the right answer for the wrong reason received partial credit. Answers which did not obtain an answer, but which showed a good faith effort in setting up the problem and attempting to solve it, also received partial credit. Answers which simply copied down the first step, or those which only mentioned that an optimal mix makes an opponent indifferent received the least credit. Answers which checked for indifference after finding $$p$$ and $$q$$ received an extra point.