Exam Rules

The rules for the exam are as follows: You can bring whatever notes you like to class. You are not allowed to bring books nor technology that substitutes for books including web surfing technology. Unfortunately, I cannot allow calculators except for "old-school" versions that only do basic operations. What this means is that you are free to use a calculator if you like, but only calculators that have no internet/kindle etc capability are allowed. So if you use your smartphone as a calculator, I'm afraid you'll have to spring the $3 or so for a basic one if you want to use it on this exam.

Extra Study Materials

1) Reminder: if you are planning to come to office hours, please try to email me the past exam questions you want to go over beforehand so that I can read through them.

2) Here is a list of previous exam questions that I would not spend too much time on because they are moral hazard questions (or include a moral hazard red herring).
2011 #1,2
2010 #2
2007 #1
2006 #1
2004 #2


3) Below is a problem set and solution key from last year.  The last two problems are cheap talk problems.

problem set
solutions

4) Professor Morgan has provided a few extra problems to practice with:

Question 1: The Modigliani-Miller theorem provides conditions where a firm’s capital structure is irrelevant to its investment decisions. Yet, in practice, a firm’s capital structure does seem to profoundly affect investment decision making as well as the valuation placed on the company in capital markets. Using tools and concepts from the field of contract theory, offer some explanations for the following questions.

1. What determines the optimal stake in a company for outside equity to take in an owner-operated firm? How does the structure of the stake (debt versus equity) affect this decision?

2. What determines when a firm should make a seasoned equity offering (i.e. sell additional equity in the capital market after previously issuing equity at some point in the past)? How should markets respond to seasoned equity offerings in determining the valuation of a firm?

3. A firm is competing with several other firms in trying to acquire some other company that is on the market. What determines the optimal bidding strategy? How should the market respond to a successful acquisition in terms of the share price for the acquirer and the target?

4. Suppose that a firm is selling some assets and faces bidders offering equity stakes. When would it prefer to sell for equity versus selling in exchange for cash or debt? 

In formulating your answers, please describe one or more of the key economic tradeoffs associated with each of the questions and describe how these tradeoffs are affected by some of the details of the situation. In particular, your answers should focus on how the organization will react to changes in its balance sheet in terms of effort undertaken in the organization, projects pursued or not pursued, and acquisitions pursued or not pursued.

In doing so, you may, at your discretion, provide a “toy model” to illustrate your answer. You may also, at your discretion, highlight key existing work illuminating the tradeoffs you have identified.

And

Question 2: (All work and no play makes Jack an investment banker) There are two types of investment banking analysts, high or low with probabilities .8 and .2 respectively. Analysts can choose two levels of work intensity, normal and psychotic. Partners in investment banks observe levels of intensity (but not types) and then decide between the actions fire or promote. The analysts payoffs are the sum of two elements: she obtains 2 units if she is promoted and 2 units if she works at her preferred level of intensity. Low types prefer to work at a normal intensity level; whereas high types prefer to work at a psychotic intensity level. The partner receives a payoff one for firing low types and a payoff of one for promoting high types. Otherwise, the partner’s payoff is zero.

Use the notion of weak perfect Bayesian equilibrium to answer the following questions.

            a. Characterize the set of separating equilibria (if any) in this game. If there are no such equilibria, then prove that this is the case.

            b. Characterize the set of pooling equilibria (if any) in this game. If there are no such equilibria, then prove that this is the case.

            c. Use belief-based refinements to argue that at least one of the above equilibria is unreasonable. (You can be somewhat informal in these arguments should you wish.)

This week

By popular request, the review session that was scheduled to take place during regular section time on Tuesday has been replaced by extended office hours.  Office hours for Megan this week will be from 3:30-5:00 on Tuesday and 3:10-4:00 on Wednesday in 204 Giannini.    


Final Exam Solution Sets:

2004

2006

2007

2008

2010

2011

Point-by-point maximization

For those who were in section today, here is why point-by-point maximization works.  As discussed in class, maximizing the integral of f(s(x))dx by choosing s(x) is the same as maximizing f(s(x)) by choosing s(x).  The concern that was brought up in class was that you might choose an s*( ) that would maximize f for some values of x but not for others.  However this would contradict s*( ) being the argmax of f(s(x))!  We set no limitations on what s( ) can be - it can be any sort of function at all, piecewise, discontinuous, whatever - and s*( ) is defined as the function s( ) which maximizes f at every point.

The function s*( ) is a rule chosen for every single value of x such that for every single value of x, we have maximized f (and therefore maximized the integral of f with respect to x.)

If you are still confused come grab me.

Problem Set 2 Solutions

PS 2 solutions can be found here.

Next week in section I will be covering moral hazard.

Class #9 Highlights

Whew. At long last we finally get to signaling. Here, we studied an education game with two types where education is a pure costly signal--it has no productivity value. In this class, we looked for a pooling equilibrium, this is an equilibrium in which all types choose the same signal and therefore the posteriors of the receiver are the same as the priors.

To find such an equilibrium, we needed to specify strategies as well as beliefs on the part of the receiver. To determine the full set of such equilibria, we relied on pessimistic beliefs. In our setting, these beliefs are a bit like a grim trigger strategy in repeated games--anything not supportable by these beliefs cannot be supported by any other beliefs.

With this specification, we determined that the upper bound occurred where the low type was indifferent between the equilibrium education level, e*, and choosing e = 0 and being branded as a low type hence making r_L.

These beliefs are a WPBE and, with an appropriate specification of perturbations, a sequential equilibrium as well. To support these as an SE, suppose that L types choose a random e with probability epsilon. H types follow the same strategy with probability epsilon^2. Taking limits as epsilon -> 0, it then follows that the posterior beliefs of the receiver put infinitely more weight on low types for all off equilibrium choices of e. But these are precisely the pessimistic beliefs we specified.

In a sense, these beliefs are goofy. Why would low types be infinitely more likely to choose higher, and hence more costly, education levels than high types?

To deal with this, one refinement that is commonly used is the intuitive criterion. It asks the following question: Suppose a given type chose an education level (i.e. a signal) e and got to pick what it would like the receiver to believe. Given these beliefs, the receiver would then choose an optimal action. If the result is that this type still earns less than her equilibrium payoffs, then the receiver should never believe that such an e came from that type.

In terms of our game, choose an education level e' such that, if the receiver R believed that it came from a high type for sure, an L type would be indifferent between and the equilibrium e*. Formally, e' solves:

r_H - e' theta_L = r_bar - e* theta_L.

Rewriting:

r_H - r_bar = theta_L (e' - e*)

Clearly, all education levels above e' could not have come from low types. But notice that, for a high type,

r_H - r_bar > theta_H (e' - e*)

since theta_H < theta_. But this means H types could profitably deviate from e* if they are recognized as H types.

The intuitive criteria requires that beliefs following choices e > e' must place all weight on H types. As a consequence, none of the pooling equilibria satisfy the intuitive criterion.


Key things should be able to do after this class:

1. Know how to construct a pooling equilibrium.
2. Make an argument using the intuitive criterion to rule out any pooling equilibrium.

Class #8 Highlights

In this class, we studied equilibrium concepts in sequential games with private information. The logical extension of Nash equilibrium to private information settings is Bayes-Nash equilibrium. This is simply a best response to the expectation of a rival's strategy, where the expectation is formed using Bayes' Rule where possible. The two key difficulties with this solution concept are:

1. It places no restriction on out of equilibrium actions.
2. It places no restriction on out of equilibrium beliefs.

Problem 1 is a familiar one from sequential games of complete information. The usual issue is that non-credible out of equilibrium actions can be used to affect on equilibrium actions. We saw a private information version of this idea in the entry game used to motivate the study of solution concepts. The point here was that the incumbent could threaten to fight out of equilibrium even though this was a dominated strategy.

Weak Perfect Bayesian Equilibrium takes care of the first objection. It requires that all strategies be sequentially rational; that is, on and off equilibrium actions must be best responses to beliefs, which are again formed using Bayes' rule where possible.

But this does not deal with the second objection. One could still use "goofy" beliefs to support sequentially rational out of equilibrium actions.

Sequential Equilibrium attempts to address this concern. This is a bit like the private information analog of trembling hand equilibrium. The idea here is to consider perturbations of the equilibrium strategies that are completely mixed. This forces the use of Bayes' rule on and off the equilibrium path. If there exists a sequence of such perturbations that converges to a WPBE, then we have found a sequential equilibrium.

In terms of our entry game, if the entrant deviated and entered the game, we saw that it would rationally choose a fight strategy and hence one could not justify the beliefs supporting the fight strategy by the incumbent following a sequence of such perturbed strategies.

As we will see in class #9, even sequential equilibrium does not rule out all goofy beliefs. We might still be clever in constructing sequences of deviations satisfying this equilibrium concept yet not satisfying common sense (or empirical testing in the lab or the field).

Technical footnote: Sequential equilibrium is only formally defined for games with finite actions, types, etc. For games with continuous types, the right extension s a matter of some debate. The notion of perfect Bayesian equilibrium, which is a strengthening of WPBE is one way to deal with this.


Key things should be able to do after this class:


1. Know what a BNE, WPBE, and SE are and use these concepts to analyze sequential games of private information.
2. Recognize that the "correct" solution concept depends on the game being studied. For many games, WPBE works just fine. Use your judgment rather than being dogmatic.

Problem Set 3

You can download problem set 3 here.
It's due on April 19th.
Youssef

Section 4

The slides for section 4 on cheap talk can be found here.

Key points:

* There is no guarantee that you will wind up in a Nash equilibrium.
* Cheap talk and other forms of signaling can help solve coordination problems and facilitate arrival at an equilibrium.
* When theory predicts multiple equilibria, laboratory evidence can be a useful way to gain intuition about what kind of equilibria are more robust in action.
* The papers we looked at showed some evidence that equilibria which are cognitively simpler - pure strategies as opposed to mixed, or strategies which aren't too dependent on future punishment or precise communication - may be more common in practice.
* Furthermore, the papers we looked at provided evidence that the details of the message space are important and have impact on the equilibrium outcome.
* The dynamics of how an equilibrium is reached is studied by evolutionary game theorists.  If you are interested in learning more, here is an article called "Do people play Nash equilibrium? Lessons from evolutionary game theory" by George J. Mailath.

http://www.jstor.org/discover/10.2307/2564802?uid=3739560&uid=2129&uid=2&uid=70&uid=4&uid=3739256&sid=47698855016007

Class #7 Highlights

In this class, we studied dominant strategy implementation. These mechanisms are of interest since truth-telling is robust to irrationality. An outcome is an equilibrium in weakly dominant strategies if there exists a strategy s(theta) for a type theta such that, regardless of the strategies (types) of the other individuals, s(theta) is better than any alternative s'. The revelation principle of dominant strategies says that, given any indirect mechanism and any allocation that arises as a dominant strategy equilibrium, the same allocation is implementable via a direct mechanism under truth-telling. The proof is identical to that for BNE.

With that definition in hand, we studied ex post efficient allocations under the following class of preferences:

U_i = v_i(x_i, theta_i) - T_i

Under these preferences, each individual only cares about her portion of the vector x, her own type, and her own transfer. 

We said that an allocation x* is ex post efficient if, for every realization of theta,

Sum[v_i(x*_i, theta_i),{i,1,n}] >= Sum[v_i(x'_i, theta_i),{i,1,n}]

for all  x' ~= x*. 

An example of an ex post efficient allocation is to give the object to the highest valued bidder in a single under auction. 

A theorem by Groves showed that any ex post efficient allocation is DS implementable using the transfer rule

T_i = -Sum[v_j(x*_j, theta_j),{j~=i}] + h_i[theta-i]

Here, h[.] is an arbitrary function. The point is that such a transfer rule aligns private and social preferences of each individual. 

Of particular interest was the Vickrey-Clarke version of h. The idea of these mechanisms is that everyone should pay the amount of their externality on the others. Define x-i* to be an ex post efficient allocation when individual i is absent. Then the externality that i imposes on the others is simply

Sum[v_j(x-i*_j, theta_j) - v_j(x*_j, theta_j), {j~=i}]

In words, this is the difference between the surplus (excluding transfers) that the other individuals enjoy when  i doesn't exist less the surplus they enjoy when i does exist. 

An example makes this clear. Suppose there were 3 individuals numbered {1,2,3}. Suppose each individual values the object by their index, so individual 1 places value 1 for the object, individual 2 values it at 2, and so on. Then, when 3 is not present, the item goes to individual 2, who earns 2 utils. When 3 is present, ex post efficiency allocates the object to her and none of the others earn any utils. Therefore, 3's externality on the others is 2. Similarly, if 2 is absent, the item goes to 3 and the combination of 1 and 3 earn 3 utils. When 2 is present, the item still goes to 3 and individuals 1 and 3 still earn 3 utils in total. Thus, 2 exerts no externality from her presence. 

Notice that an individual only exerts an externality when she is pivotal, i.e. when the allocation is different when she is present than when she is absent. The Vickrey-Clarke (pivotal) mechanism has the property that only pivotal individuals are obliged to make payments. 

We showed that the same idea can be applied to public goods settings, such as the go/no-go decision to build a discrete public good like a bridge. Vickrey invented the auction version of the mechanism while Clarke invented the public goods version. Groves noticed the deeper connection. He showed:

Theorem: The pivot mechanism is the cheapest mechanism to DS implement the ex post efficient allocation. 

This is useful for two reasons. First, if money taken out of the system represents waste, then this tells us the least wasteful way to achieve ex post efficiency. Second, it tells us that we cannot have DS implementation of ex post efficient allocations and budget balance at the same time. We have to choose one or the other. 

Mechanisms where individuals pay their externality are commonly called Vickrey-Clarke-Groves or VCG mechanisms in honor of the three individuals making key discoveries about them. 

Rebates

Now, the alert student of information economics may wonder why we need to take the money out of the system. After all, there are many schemes whereby rebating money back in lump-sum fashion can get out of problems like these. But this will not work here. If we amend the mechanism and say that we will give back (say) equal shares of all the money collected to the individuals in a society, then it will wreck the mechanism since an individual who is pivotal will now pay less than her externality. This is the case since a portion of her payments, when she is pivotal, will be returned. Anticipating this, truth-telling will no longer be incentive compatible.

To see the problem, consider an auction setting with 2 bidders where the auctioneer will give back all the money collected equally to the two individuals. Suppose that individual 1 values the item at 1.9 while individual values it at 2. Suppose 2 tells the truth, then 1 will, optimally, want to report a value in excess of 2 since she will then receive the object, pay 2 for it, but get 1 back via the rebate. This yields a net surplus of 0.9 (1.9 - 2 + 1) versus a payoff of zero from truth-telling. Other rebating schemes run into similar problems.

Key things should be able to do after this class:

1.  Explain how the revelation principle works for weakly dominant strategies.
2. Know what an ex post efficient allocation is
3. Apply the VCG mechanism to implement an ex post efficient allocation. Compute the payments made by each player in the game. 

Class #6 Highlights

In this class, we saw the RET in action. The key is to note that the RET implies even stronger conditions about individuals participating in the mechanism. Notably, any mechanism that induces the same allocation x and gives the same utility to the lowest type produces the same utility to each type. This arises from the utility formula offered in the previous class highlights which shows that a type's equilibrium utility only depends on x and on the utility of the lowest type.

Moreover, under the same conditions, a given type makes the same expected payment under any mechanism. In an auction setting, this means that the expected payment under any auction that allocates the good to the same type and has the same utility for the lowest type will be the same as that under the Vickrey auction. Since the Vickrey is quite easy to analyze, this equivalence is very handy to compute things in a given mechanism without having to go through all the calculations to obtain equilibrium.

An example of this is a winner-take-all labor tournament. There are several equivalent interpretations for this mechanism. (1) Suppose that effort is directly measured but that higher types can produce a given level of effort at lower cost than lower types. (2) Output, which is a combination of effort and ability is measured. Effort costs the same for all types, but higher types produce more output per unit of effort than lower types. (3) Effort is measured and all individuals produce effort at the same cost but higher types value the prize more than lower types. Case (3) is obviously the least realistic, but the easiest to map directly into our RET framework. From this, we can deduce the expected payment for each type. Since types "pay" their effort, then the expected payment is the same as the equilibrium effort of each type. Furthermore, since all three settings are isomorphic, this is also the equilibrium effort under either of the other interpretations as well.

The point of the example is twofold: First, it illustrates that settings that are not, recognizably auctions, can be fruitfully modeled in a way that permits us to use the RET to analyze. Second, it illustrates that equilibrium calculations that are otherwise annoying to compute may be calculated quite directly using other implications of the RET.


Key things should be able to do after this class:


1. Explain how and why the RET works.
2. Use the RET to compute equilibria in auction-like games.

Problem set 1 solutions

You can find the solutions to problem set 1 here.  

Youssef