Gun Control: The Sicko Game
In the wake of the Florida shooting and the resurgent debate on gun control, to focus our attention on solutions it may be helpful to see the problem through the lens of game theory in order to understand how to proceed from a policy point of view. Game theory is an analytical tool that helps us to analyze and provide predictions in settings where behavior is decidedly strategic. The analysis involves identifying the players, their strategy sets, characterizing the nature of the “game” played i.e. whether choices are made simultaneously or sequentially, whether the game has multiple stages or not, whether the game as a whole is played repeatedly in the future or not, and whether there is any uncertainty involved, either with respect to the history of the game, or the payoffs involved or both. The solution concept involved in these games is in one form or another a Nash equilibrium concept or analogue. Such a concept conceives of players as being rational in the sense that they play strategies that they know or believe are best responses to the best responses played by the other players, who in turn are doing the same and this is common knowledge. While the game theoretic models used extensively in economics are advanced and no walk in the park for the uninitiated I will keep this game as simple as possible to focus on the key issues. It is possible to aim for greater realism but this will only limit accessibility to the main points I intend to highlight. So we will keep it simple: Let this game, which I have tagged the Sicko Game, be such that we have two players, a sicko, who has anti-social objectives but which is privately held information so that he or she cannot be easily identified in the public through any non-invasive means of profiling. The other player is an Administrator of a public (or private) facility like a school, health center or similar with populations that could be vulnerable to activities from a sicko. The sicko has a strategy set that is a binary choice on whether or not to attack the public facility (Attack, Not Attack). The Administrator choice is also a binary one to arm staff or not to arm staff (Arm staff, Not Arm staff). Understandably these are not the only choices that the Administrator would have in the real world, but we will assume that alternative choices including things like gating the entire perimeter of the facility and providing for metal and bomb detectors, or having additional plain clothed or uniformed law enforcement officers, are not feasible beyond the status quo security measures already in place, which may be fairly effective as a deterrent or in ensuring the apprehension of a sicko after the fact. At some analytical level, we can have the binary choice as "maintain status quo, and provide more protection", but we want to focus that 'more protection' on the feasible alternative being currently discussed in the media, so we will keep the binary choice as stated above. We assume the Administrator moves first, and the sicko then observes the choice and moves next. The players then receive payoffs based on the outcomes from their combined strategy choices.
There is an added complication to this game in that we assume that the sicko's payoffs are unknown. Such a game, called a game of incomplete information, can be transformed into a game of imperfect information where instead of payoff uncertainty, the uncertainty is with respect to where we are in a bigger game involving different sicko types and where we do not know which sicko is involved, but we know the payoff associated with each sicko type.
Analytically this is done by introducing probabilities that capture beliefs about which sicko type the Administrator might face and where 'nature' assumedly makes the draw from a known probability distribution.
In this now tractable game, because payoffs are now a function of the sicko type, and the Administrator only has beliefs about which type she might face, a perfect bayesian equilibrium provides the solution to this game. This involves solving the game by starting out with the last play. While we do not know for sure which node (the type of sicko) is involved, we can deduce what any type sicko will play when he or she is called on to play, so we work backward to where the Administrator, knowing this, chooses a strategy based on beliefs about which type of sicko is involved and what her expected payoffs will be in each situation. In the perfect bayesian equilibrium that follows from the setup of the game, with plausible payoffs associated with two possible types of sickos, the Administrator will chose to Arm Staff. To keep this simple I have not included the analysis here, but you can find it here. The basic intuition goes thus. Suppose the sicko could be one of two types - a cautious sicko and an aggressive sicko. The idea here being that the cautious sicko does have anti-social destructive objectives but has a stronger preference to commit such an act only if he or she is able to get away with it. The aggressive sicko has no such concerns and possibly even relishes in the idea of going out in a gun fight provided he or she can also succeed in inflicting maximum damage before this happens. Including probabilities associated with both the cautious sicko’s assessment of existing security measures to ascertain his or her chance of success as defined, and with beliefs by the Administrator about which sicko she may face, the result is that the cautious sicko chooses to Not Attack, while the aggressive sicko chooses to Attack.
By comparing her payoff from these choices along with probabilities associated with these, the analysis suggests that the Administrator should Arm Staff unless she is almost certain that the sicko she may face is the cautious type. While the outcomes predicted might look contrived, it does seem apparent that if there is a fairly likely chance the sicko who shows up has intense pleasure in committing anti social behavior and has little regard to his or her life, he or she will be willing to Attack as a dominant strategy. So the Administrator will choose to Arm Staff to avoid this. Is this an advocation for more guns as a solution to the problem? Not necessarily. The game is premised on certain assumptions... for one, we have ruled out the possibility of positively identifying such sickos and keeping them from access to weapons of massive destruction. Much of the debate seems to be based on this initial identification and access control. We have also not addressed the many sided, bigger picture, aspects of the debate including what exactly the second amendment right means in today’s context or/ and whether certain types of weapons should be accessible to the public, and the issue of what age is old enough for access to purchase such weapons. As with all policy issues, there are distributional impacts from any position, and those adversely affected by the status quo or by any policy move from the status quo, will fight hard and provide what they believe to be justifiable arguments for their position. This post does not intend to take a position that could be misinterpreted as partisan. The point here is it does not seem clear why in the midst of this debate many positions that appear to be in opposition cannot be accommodated. If we must keep guns from sickos, then common sense suggests that we should tighten laws that reduce the probability that they can access one including enforcing any already existing laws. In this context any ensuing inconvenience encountered by those who are legitimately seeking gun ownership is but a small price to pay. However because all such laws cannot work with perfect certainty, and because it is possible for people who were initially sane, to lose it, after the fact, a proactive policy that such facilities may choose to adopt should be to arm their staff. While we do not at present have an epidemic of school shootings, the incidence of any happening takes a huge toll on the families and communities that limiting these to zero incidents may be a preferred outcome which arming staff may realize. The belief that this may create greater confusion and a possibly bigger tragedy in the wake of an incident remains for the most part a counterfactual. Idealized environments are never immune to errors, but the best we can do is to try to minimize the likelihood of such friendly fire incidents through adequate training.