I was talking to someone about matching mechanisms and the fact that strategy-proof incentives are often incompatible with efficiency. The question came up as to why we insist upon strategy-proofness, i.e. dominant strategy incentives as a constraint. If there is a trade-off between incentives and efficiency shouldn’t that tradeoff be in the objective function? We could then talk about how much we are willing to compromise on incentives in order to get some marginal improvement in efficiency.
For example, we might think that agents are willing to tell the truth about their preferences as long as manipulating the mechanism doesn’t improve their utility by a large amount. Then we should formalize a tradeoff between the epsilon slack in incentives and the welfare of the mechanism. The usual method of maximizing welfare subject to an incentive constraint is flawed because it prevents us from thinking about the problem in this way.
That sounded sensible until I thought about it just a little bit longer. If you are a social planner you have some welfare function, let’s say V. You want to choose a mechanism so that the resulting outcome maximizes V. And you have a theory about how agents will play any mechanism you choose. Let’s say that for any mechanism M, O(M) describes the outcome or possible outcomes according to your theory. This can be very general: O(M) could be the set of outcomes that will occur when agents are epsilon-truth-tellers, it could be some probability distribution over outcomes reflecting that you acknowledge that your theory is not very precise. And if you have the idea that incentives are flexible, O can capture that: for mechanisms M that have very strong incentive properties, O(M) will be a small set, or a degenerate probability distribution, whereas for mechanisms M that compromise a bit on incentives O(M) will be a larger set or a more diffuse probability distribution. And if you believe in a tradeoff between welfare and incentives, your V applied to O(M) can encode that by quantifying the loss associated with larger sets O(M) compared to smaller sets O(M).
But whatever your theory is you can represent it by some O(.) function. Then the simplest formulation of your problem is: choose M to maximize V(O(M)). And then we can equivalently express that problem in our standard way: choose an outcome (or set of outcomes, or probability distribution over outcomes ) O to maximize V(O) subject to the constraint that there exists some mechanism M for which O = O(M). That constraint is called the incentive constraint.
Incentives appear as a constraint, not in the objective. Once you have decided on your theory O, it makes no sense to talk about compromising on incentives and there is no meaningful tradeoff between incentives and welfare. While we might, as a purely theoretical exercise, comment on the necessity of such a tradeoff, no social planner would ever care to plot a “frontier” of mechanisms whose slope quantifies a rate of substitution between incentives and welfare.
Cheap Talk 
6 comments
Comments feed for this article
November 30, 2011 at 4:43 am
Brittany
Doesn’t the revelation principle give the conclusion (maybe this is a version of it?)? We can characterize all equilibrium outcomes by just looking at those that are strategyproof, so a social planner simply picks the best strategyproof equilibrium.
That is:
if
Z=max V(O) s.t. O=O(M), M is strategyproof
Z’=max V(O) st. O=O(M’), M’ may not be strategyproof
then Z=Z’ because O(M’)=O(M) by revelation principle.
November 30, 2011 at 8:59 am
jeff
In essence yes, however the revelation principle does not apply to all solution concepts. Your theory O may fail the revelation principle. That’s why I phrased it the way I did to make it clear that the point is orthogonal to the revelation principle.
November 30, 2011 at 11:07 am
Tayfun Sonmez
Hi Jeff,
Thanks for this wonderful post. We (Parag Pathak and myself) were just writing about it! There are a number of important applications where strategy-proofness is an important design objective. The point that I will try to make might not be valid for all applications, but it is the certainly the case for some.
We (me and Parag Pathak) are indeed heavily discussing some of these issues in our revision of “School Admissions Reform in Chicago and England: Comparing Mechanisms by their Vulnerability to Manipulation.”
School choice is an application where strategy-proofness is considered by policy makers as a design objective. And indeed precisely because of the “gaming” possibilities, the highly manipulable Boston mechanism as well as a variant called (first preference first system) is banned throughout the England with 2007 code. The points made by British policy makers are mostly about the “complexity” of non-strategy-proof mechanisms and the harm it does to least sophisticated families because they cannot optimize. One of the biggest benefits of adopting a strategy-proof mechanism is not only you can teach most people how to optimize (just by being truthful), but also for most people (unlike what most game theorists think), the default behavior is truth-telling. Policy makers simply do not want to reach outcomes through complicated games. In the case of England, the Boston mechanism is banned on the grounds that it is “unfair” due to the complex game it induces. A similar episode took place in Chicago, and unlike the school choice reform in Boston, economists were not involved in these.
There are other reasons why strategy-proofness is important for some applications (such as school choice and cadet branching). Whenever a mechanism asks preferences over schools, there is the risk that the data that is generated will be used with its face value. Several empirical papers use school choice data with the highly manipulable Boston mechanism for example. This data is often used to evaluate schools, to replicate, or even to close them. In a matter of speaking, these mechanisms create an externality, and unless you are a game theorist, you are very likely to assume the preference data represents actual preferences. There are numerous examples of such misinterpretation in the education literature.
You eliminate this unintended harm by adopting strategy-proof mechanisms.
Let me give another example where the preference data becomes needed but turns out to be useless because of the lack of strategy-proofness. I give a more detailed description of this issue in my paper “Bidding for Army Career Specialties: Improving the ROTC Branching Mechanism.”
U.S. Army has a big problem of racial diversity in its senior ranks. While about 30% of enlisted ranks are African Americans, less than 5% of generals are African Americans. It turns out this is very big deal, and U.S. military is trying to change this unbalance. When they try to understand why that is happening, they conclude that one major reason is African Americans’ lack of ranking key career branches as their top choices when cadets compete for Army specialties in their senior year. In a Rand study Lim et al (2009) they showed that while 30% of African Americans target career fields, that rate is 58% for their white peers. The data they have is ROTC 2007 data. But they realize that ROTC mechanism is manipulable, and African Americans might not target these fields entirely based on strategic reasons. Hence they are unable to offer a solution since the policy implications will be entirely different depending on to what extend the difference of preferences represent a lack of interest of African Americans on career branches. So when one uses highly manipulable mechanisms, there are implications beyond the intended purpose of the mechanism.
I do not know whether this will be surprising but I have never seen policy makers making dramatic changes based on efficiency considerations but on several occasions they changed dramatic changes because the underlying mechanism was highly manipulable. I understand that not all applications are of that nature, but there are important ones where this is big deal.
Tayfun
November 30, 2011 at 11:42 pm
jeff
thanks for the comment tayfun
November 30, 2011 at 4:44 pm
Gabriel
Thanks for this thoughtful post. I feel obliged to suggest a couple reasons why someone might be interested in the incentive-efficiency frontier:
1. A planner might believe that agents are epsilon-truth-tellers, but be uncertain about the value of epsilon. She might think, say, that epsilon is some small value with probability 90%, or some larger value with probability 10%. Then the optimal mechanism will either be a small-epsilon-IC mechanism that guarantees a less-efficient outcome, or a large-epsilon-IC one, that gives a 90% assurance of a more-efficient outcome. So the planner’s choice between these ends up formally looking like a tradeoff between incentives and efficiency, even though at the level of primitives, incentives are indeed a constraint rather than an objective.
2. Even if the planner is certain about epsilon, the theorist might not be. Plotting out the frontier means solving a whole family of potential planners’ problems at once.
November 30, 2011 at 11:40 pm
jeff
Hi Gabriel. Your #1 is basically what I meant when I wrote “possibly wrong” in the title. I actually think the argument is mostly wrong. But still I think it is worth framing it it this way because it clarifies how this tradeoff would be properly modeled and i think it also imposes some discipline.
And I was saying the same thing as your #2 at the very end.