TERWORTB
EINEMANN
A Game Theoretic Model of Gun Control
ROBERT
Department
TAYLOR
of Economics and Finance, College of Business,
University, Johnson City, Tennessee
East Tennessee
State
I. Introduction
With the passage of the Brady Bill in 1993 and the approval of the Feinstein Amendment the following year, gun control has once again become a subject of heated
public debate in the United States. Procontrol
forces point to high levels of gun
ownership as the cause of much, if not most, violence in America, and argue that gun
control will inevitably reduce violence by reducing the number of guns in circulation.
Anticontrol
forces respond that high levels of gun ownership are a response to, not
a cause of, violent crime, and that gun control disarms law-abiding
citizens, not
criminals.
Certain facts are clear. Approximately
half of all American households are armed,
giving the U.S. the highest rate of gun ownership in the world. Switzerland is probably second, with one-third of its households armed (Kleck 1991, 19). The relationship between levels of gun ownership and violence is harder to discern, however, as
is the connection
between gun control and violence.
A large number of empirical assessments of gun control legislation have appeared
over the past two decades.’ A recent article by Kleck and Patterson (1993) gives a
useful overview of this literature.
Kleck and Patterson note that the overwhelming
majority of the studies either were inconclusive or found that gun control laws had no
significant impact on violent crime. They conclude their literature
review with the
following statement:
“taking prior research as a whole, it would be fair to say at this
point that a consistent, credible case for gun control efficacy in reducing violence has
not yet been made.”
Given that the empirical literature
has largely failed to resolve the gun control
controversy,
researchers
might be well advised to take a more theoretical approach
to the problem. The construction
of relatively simple models of victim-criminal
interaction
might provide interesting
insights about the desirability
of gun control
legislation and the mechanisms of its influence. Rarely in the gun control debate has
either side stated its assumptions
clearly. Formal modeling would require these as-
I thank Phil Cook, Don Kates, Kip Viscusi, and two anonymous referees for their very helpful comments on this
paper. 1 am also grateful to Bill Gentry, Jay Hamilton, Mike Meurer. and the participants at the Duke University
Public Finance Workshop for their suggestions.
‘Some recent contributions include Kellerman (1988), Lester (1988). Jung and Jason (1988), Mundt (1990). Rich
et al. (1990), Loftin et al., (199 I). Mauser and Holmes (1992), and Lester and Leenaars (1993).
International Review of Law and Economics 15:26%288,
8 1995 by Elsevier Science Inc.
655 Avenue of the Americas, New York, NY 10010
1995
0144-8188/95/$10.00
SD1 0144-8188(95)00018-4
A game theoretic m&l
270
of gun control
sumptions
to be explicitly stated. We could then test the robustness
of any conclusions reached to changes in these assumptions.
This paper will undertake just such a project. The first model presented will be an
attempt to formalize conclusions reached in a recent article by Polsby ( 1993).2 Polsby
asserts that even if universal disarmament
is optimal, gun control measures that
merely move us in that direction may reduce social welfare. He is, in effect, claiming
that the relationship
between gun control and social welfare is nonmonotonic:
social
welfare may first decrease, then later increase, with increasingly
stringent gun control laws. Moreover, he suggests that gun control laws stringent enough to attain
universal
disarmament
may be politically
infeasible
or otherwise
unattainable.
Clearly, such conclusions
put gun control in a very bad light.3
The second model presented relaxes two key assumptions
of the first model. In the
resulting model, modest gun control measures may be capable of improving
social
welfare, even though the relationship
between gun control and social welfare remains fundamentally
nonmonotonic.
However, the success of gun control in this
second model is highly dependent
upon both the character of the control measures
adopted and the motivation of criminals to obtain a gun.
The paper finishes with a discussion of the second model’s policy implications,
with
a look at empirical work that lends support to the second model’s structure, and with
some suggestions for future research. As we will see, the second model helps to make
sense of recent policy changes at the state level that appear to move in opposite
directions (e.g., background
checks and the liberalization
of concealed-carry
laws).
II. The Basic Model: Gun Control May Backfire
Most arguments
1.
2.
for gun control
make two implicit
Universal disarmament
is an optimal state.
Gun control, by moving us toward this optimal
welfare.
assumptions:
state, will necessarily
increase
Even if we accept the validity of the first assumption
(which is not itself uncontroversial),4 the second assumption
does not necessarily hold-social
welfare may first
decrease, then much later, increase, with increasingly
stringent gun control laws. The
following simple model of victim-criminal
interaction
will generate this unusual result.
Setting Up the Model
Assume that there are two equal-sized populations
of potential victims and potential
criminals. The word “potential” is used here because, as we shall see, “victims” may
not be attacked and “criminals” may refrain from attacking. Further assume that all
‘For a more complete rendering of P&by’s views, see Polsby (1994).
sPublic finance economists may notice a resemblance between Polsby’s arguments and those made in the tax
reform literature. Specifically, a system of lump-sum taxation may be perfectly nondistortionary
(efficient), but
merely wuwing t0ward.s
a system of lump-sum taxation by way of incremental tax reforms may be efficiency-reducing
(Feldstein 1976 and Dixit 1975).
%ee, for instance, Keeley (1983). For a work that describes the difficulties of nuclear disarmament using the
tools of game theory, see Brams and Kilgour (1988), especially chapter 4.
TAYLOR
271
victims are identical and all criminals are identica1.s Finally, assume that all players,
both victims and criminals, are risk-neutral,
i.e., they value a lottery at its expected
value. This assumption
will make the calculation of payoffs straightforward.
In this model, each victim is randomly paired up with a criminal. The two then
play the “game” that is outlined in Figure 1. In this simultaneous-move
game, the
victim and criminal each have two options. The criminal (C) must decide whether to
attack (A) or not attack (NA). If the attack (A) option is chosen, the criminal will purchase
a gun and attack with it. This key assumption-that
the criminal will always attack with
a gun-will
be relaxed in the next section. For the time being, think of this game as
being a representation
of some type of crime in which guns are used a majority of the
time, such as homicide (Kleck 1991,43). As for the other player, the victim (V) must
decide whether to purchase a gun (G) or not to purchase a gun (NC) for self-defense.
Note again that the game is one of simultaneous
moves-victims
and criminals cannot observe each other’s choices before choosing themselves.
This assumption
is
quite plausible: criminals usually do not know if they will be facing an armed victim,
and victims usually do not know if they are going to be unlucky enough to be
victimized. Nevertheless,
this assumption will also be relaxed later in the section, with
interesting
results.
Figure 1 also enumerates
the “payoffs” that the victims and criminals will receive
for different combinations
of attack and armament
decisions. For example, if we
ignore the letter “C” for a moment, a criminal attack on an armed victim (denoted
[A,GJ) gives a payoff of X for the criminal and W for the victim. Before discussing the
ordering
of these payoffs, first note that the payoffs themselves are expressed as
changes from the payoffs involved in the “default” combination
[NA,NG], in which
criminals find another line of work and victims do not bother to arm themselves.
Using this utopian combination
as a reference point is simply a matter of preference.
The choice of zero as the reference
payoff for both victim and criminal is also
somewhat arbitrary.
Neither of these modeling choices should affect the results in
any significant way.
The gross payoffs to victim and criminal are ordered as follows: Z > 0 > X > W > Y,
where
Z is the criminal’s payoff from an attack on an unarmed
victim,
X is the criminal’s payoff from an attack on an armed victim,
W is the armed victim’s payoff from an attack, and
Y is the unarmed
victim’s payoff from an attack.
The criminal receives a positive gross payoff of Z from an attack on an unarmed
victim; all other payoffs are negative. This feature emphasizes two important
points:
namely, that the only person who can possibly improve on the default combination
[NA,NG] payoff is the criminal and that “crime pays” only if the victim is unarmed.
An armed confrontation
between criminal and victim (combination
[A,G]) yields
payoffs of X and W, respectively. X is assumed to be greater than W, given that the
victim’s payoff if
criminal has the element of surprise.6 Finally, Y is the unarmed
attacked. Y is assumed to be less than W, because crime statistics reveal that resistance
‘This assumption will be relaxed in a later footnote. Surprisingly, the relaxation of this assumption may not
radically alter the conclusions reached.
6A reader has pointed ow that X may in fact be less fhan W for a number of reasons, including that criminals
at-e more likely than victims to be impaired in a gunfight by the consumption of alcohol or other psychoactive
272
A game theoretic moaW of gun control
c
V
G A
NG
G
--____-V
A
NA
x-c
0
-Ic
W-C
Z-C
NG
0
Y
0
I
1
Normal Form
Extensive Form
FIG. 1. The
basic model.
with a gun generally results in fewer injuries for the victim than either nonresistance
or resistance with other weapons (Kleck 1991, 149, Table 4.4).’
One element of the payoffs remains to be discussed: the cost of obtaining a gun, C.
This cost is equal to the sum of two components,
C* and R. The constant C* may be
thought of as the market price of a generic firearm in the absence of any form of gun
control. The variable R, therefore, is the additional cost, monetary or otherwise, of
obtaining a firearm due to gun control. It would therefore include not only taxes and
the higher prices artificially induced by gun control restrictions,
but also the timecost involved in finding a gun on the black market and the risk of punishment
if
caught violating a gun control ordinance.
Including
R as a component
of a gun’s cost allows us to incorporate
gun control
explicitly into the structure
of our game. This method of incorporation
is very
realistic. After all, as the U.S. war on drugs has amply demonstrated,
government
is
incapable of eliminating
a product that many desire. What government
is capable of
doing is increasing
the cost of obtaining that product. Treating R as a cost component merely emphasizes that no matter how draconian
gun control becomes, guns
will be available at some price, though that price might be quite high.
Returning
to Figure 1, the cost of obtaining
a gun C is subtracted
from gross
payoffs wherever a decision entails the purchase of a gun. Notice that the same C is
subtracted from gross payoffs whether a victim or a criminal is involved. That is, gun
control laws impose identical cos.!.son both criminals and ~victims.This key assumption,
like
substances (Kates 1991, 163, footnote 165). The only defense offered against this charge is that the ordering of X
and W turns out to be inessential to the conclusions reached.
‘This assumption will turn out to be a vital one. Some researchers (such as Yeager et al. 1976 and Cook 1986)
have come to a different conclusion. As Kleck (1991) notes, however, these researchers made the mistake of
“lumping gun resistance in with other forms of forceful resistance”; this practice is very misleading because “other
forms of forceful self-protection are far more risky than resisting with a gun.”
Furthermore, note that if Y > W (i.e.. if the victim is better off being unarmed when attacked), then the 50%
of American households that remain armed are in serious error, at least if they are keef&g guns for self-defense.
This radical implication would seem to cast at least a little doubt on the original assumption.
TAYLOR
273
the assumption
of the always-armed
criminal, will be relaxed in the next section. Still,
this assumption
may hold for certain types of gun control (e.g., a waiting period
without a background
check).
Equilibrium Concepts
What outcome should we expect from the game just described?
The first step in
identifying
an outcome is to examine the game and see if any strategies are “dominated,” i.e., see if there are any strategies that, if pursued, would be unambiguously
inferior to another strategy, regardless of what the other player does. Return to the
normal form depiction of the game in Figure 1. One can easily see that if Z < C (i.e.,
the payoff from an attack on an unarmed victim is less than the full cost of the gun),
then attacking is a dominated
strategy for the criminal. Attacking will always yield a
negative payoff, regardless
of the armament
decision of the victim, whereas not
attacking would always yield a payoff of zero. Thus, a rational criminal will decide
never to attack. In response, the victim will never go armed (choosing G is a dominated strategy for the victim once it is known that no criminals are attacking), and the
default combination
[NA,NG] will be realized. Given that we want to talk about a
world where criminals do sometimes attack, we will assume that Z > C when C is low,
i.e., when the costs of obtaining a gun are low.
Similarly, if Y > W - C (i.e., if the unarmed
victim’s payoff from an attack is
greater than the armed victim’s payoff, inclusive of the cost of obtaining a gun), then
arming oneself is a dominated
strategy for the victim. Victims will therefore choose
to go unarmed.
Criminals, recognizing
this, will always attack (choosing NA will be
a dominated
strategy for them), and the combination
[A,NG] will be realized. Again,
given that we want to talk about a world where victims sometimes arm themselves, we
will assume that Y < W - C when C is low.
Because in the real world only some potential victims arm themselves
in selfdefense and only some potential criminals decide to attack, we need to find an
equilibrium
concept that allows for different strategies within the populations
of
players. One such concept is the mixed-strategy
Nash equilibrium
(or MSNE).8 Before describing
an MSNE, let us first define some terms. Let OLbe the fraction of
victims who always arm themselves (i.e., always choose G); thus, 1 - (Y of victims
never arm themselves (i.e., always choose NC). Similarly, let B be the fraction of
criminals who always attack (i.e., always choose A); thus, 1 - p of criminals never
attack (i.e., always choose NA). So, the victim and criminal populations
are divided
into subpopulations
whose memberships
are based on strategies chosen.
Now consider the decision that a typical criminal must make. This representative
criminal must decide whether to join the subpopulation
that always attacks (chooses
A) or never attacks (chooses NA). How will this decision be made? A welfaremaximizing criminal will choose the subpopulation
(and, therefore, the strategy) that
offers the higher expected payoff. Suppose initially that this strategy is always attack
(choose A). The criminal will therefore join this subpopulation.
Of course, other
potential criminals will eventually notice this difference in expected payoffs, and they
‘We know that this equilibrium concept is relevant here because every finite strategic-form
(Fudenberg and Tirole 1991, 29-30).
game has an MSKE
A game theoretic model of gun control
274
will move into the subpopulation
that always attacks+
will increase, and 1 - B will
necessarily fall. This process will continue until there is no longer an advantage
to
moving; that is, the process will continue until expected payoffs change in such a way
as to make criminals indifferent
about their choice of subpopulations
(strategies).g A
similar process occurs with victims. Readers may recognize a kinship between the
above process and the elimination
of economic profits in a long-run
competitive
equilibrium
through the entry of firms.
The dynamic process described above is what generates an MSNE. In essence;an
MSNE is defined as the set of population
fractions a* and B* that result when the
movement of individuals
across groups finally grinds to a halt. When a* of victims
arm themselves and p* of criminals attack, both victims and criminals are indifferent
about their choice of subpopulations
(strategies), so that there is no tendency for
individuals
to switch groups.
How can we formally define a* and B*? The following two equations implicitly
define these fractions for criminals and victims, respectively:
a(X- C) + (1 - a)(Z - C) =
a(0)
+ (1 -
a)(O)
= 0,
(1)
and
p(w - C) + (1 - B)(-C)
= BY + (1 - B)(O) = BY.
(2)
Notice that these equations are mathematical
representations
of the indifference
of
players regarding
their choice of strategies. Take, for example, equation 2. The
left-hand side of the equation is the expected payoff for a victim from joining the
subpopulation
that always arms itself (chooses G). Because each victim is randomly
paired up with a criminal, the victim will get a criminal that always attacks with
probability B and one that never attacks with probability
1 - B. The victim’s payoff
from an armed confrontation
with a criminal is W - C, while the payoff from
arming oneself needlessly is - C. Therefore,
the expected payoff is B(W - C) + (1
- B)( -C). The same logic applies to the right-hand
side, which is the expected
payoff for a victim from joining the subpopulation
that never arms itself (chooses
NC). Thus, setting the left-hand side equal to the right-hand
side is equivalent
to
saying that victims are indifferent
about their choice of subpopulations
(strategies),
which is what we would expect in an MSNE.
By solving equations
1 and 2 for a and B, we obtain the MSNE population
fractions: a* = (C - Z)/(X - Z) and B* = C/(W - Y). Thus, in the MSNE, a* of victims
will always arm themselves, etc. Now that we know how criminals and victims will
divide themselves among the various subpopulations
(strategies), we can calculate the
expected payoff for a representative
criminal and victim in the MSNE:
Criminal:
a*P*(X- C) + (1 - a*)P*(Z
- C) +
a*(1
-
p*)(O)
+ (1 -
a*)( 1 -
P*)(o) = 0
Victim:
a*B*(W
-
C) + a*(1 - f3*)( -C) + (1 - a*)P*Y
p*)(o) = p*y = CY/(W - Y).
+ (1 -
a*)(1
-
The expected payoff for a representative
player is calculated by summing the payoffs
from each combination
of strategies multiplied by the probability of that combina‘For elaboration, see Fudenberg and Tirole (1991). 27.
275
TAYLOR
tion occurring.
These payoffs are (by the definition of an MSNE) equivalent
to the
payoffs obtained in equations
1 and 2. Thus, victims expect to receive a payoff of
p*Y, regardless
of whether they are armed, because in an MSNE they must be
indifferent
between their choice of strategies (subpopulations).
Notice that the representative
criminal’s expected payoff in the MSNE is zero-the
same as it is in the default combination
[NA,NG]. This feature is not a coincidence:
the criminal, who has the initiative, can always decide not to attack; therefore,
the
payoff in the MSNE cannot be any lower than it is in the default combination.
Drawing Conclusions
We will define social welfare as the welfare of the victim-the
welfare of the criminal
will be ignored. lo Specifically, social welfare will be defined as the expected payoff of
the representative
victim. Armed with this definition and the above model, we can
draw the following conclusions:
COI\‘CLUSION1: A movement from the MSNE
to [NG,NA]
would increase social welfare.
The representative
victim would clearly be made better off by a move from the
MSNE to [NG,NA] because the MSNE payoff, which is CY/(W - Y), is less than the
payoff from the [NG,NA] combination
of strategies, which is 0 (given our assumption
that 0 > W > Y). Thus, if the government
could somehow eliminate all weapons, so
that criminals would be unable to attack and victims would be unable to arm themselves, we could achieve the social welfare optimum of universal disarmament.
This
conclusion
therefore
represents
a potentially
powerful argument
in favor of gun
is incapable of eliminating
control. ’’ As we noted earlier, however, government
weapons-it
can only make them more expensive. As the next conclusion
demonstrates, however, attempts by government
to approach the social welfare optimum by
increasing the costs of obtaining a gun may, paradoxically,
reduce social welfare.
COKCLUSION 2: In the MSNE,
an increase in gun costs reduces social welfare.
As noted earlier in this section, C, which is the cost of obtaining a gun, is equal to C*
plus R, where C* is the cost of a generic firearm in the absence of any gun control
regulations
and R is any additional
cost, monetary or otherwise, imposed by gun
control. If we take the derivative of the representative
victim’s expected payoff,
CY/(W - Y), with respect to R, we obtain Y/(W - Y), which is less than 0. In other
words, stricter gun control reduces the representative
victim’s expected payoff, and,
therefore,
social welfare. This outcome occurs because higher gun costs reduce the
fraction of victims who arm themselves (a*) and increase the fraction of criminals
who attack (p*). That is, in the MSNE, gun control serves to disarm victims and
“‘See Lenin and Trumbull (1990). This definition would seem to be particularly appropriate when one is
dealing with violent crimes.
“Many game theoretic models of violence seem to stop at this point. For example, Moulin (1986) uses a
Prisoner’s Dilemma to model aggression. In such a model, Moulin notes, “war is the likely outcome. Decentralization of strategic choices [i.e., the decision to aggress or be peaceful] has a high collective cost.” The implication,
of course, is that a centralized imposition of peaceful strategies would be welfare-improving-an
argument similar
to those used by gun control proponents.
276
A game thoretic
model of gun control
encourage
criminal predation.‘*
Hence the old age adage, “if guns are outlawed,
only outlaws will have guns.”
The implications of this conclusion for gun control policy are best seen with the aid
of an example. Suppose that in the absence of gun control a generic firearm would
cost $200 (C*). Also, assume that the type of crime in question is homicide, so that the
gross payoffs for criminal and victim are fairly large in absolute value. Specifically,
assume that the criminal’s gross payoffs are $1,200 for attacking an unarmed victim
(Z) and - $500 for attacking an armed victim (X), and assume that the victim’s gross
payoffs if attacked are - $600 if armed (W) and - $1,500 if unarmed
(Y).t3
Given these assumptions,
how would increasingly
stringent gun control laws affect
social welfare, i.e., the representative
victim’s expected payoff? Figure 2 plots the
relationship
between R (the additional
cost that gun control imposes on gun purchasers) and social welfare. At R = $0 (no gun control), a mixed-strategy
Nash
equilibrium
exists. In this MSNE, the representative
victim’s expected payoff is approximately
- $333 (= CY/[W - Y]). As R increases, social welfare falls: fewer
victims arm themselves and more criminals attack.
Social welfare continues to fall until R reaches $700. At this point, gun control has
increased
the price of a generic firearm by 350%. In fact, the cost of a generic
firearm has become so high that any further increases will make W - C < Y; that is,
further
increases will make arming oneself a dominated
strategy for victims, as
discussed above. Victims will therefore stop arming themselves; criminals, realizing
this, will always choose to attack. Thus, [A,NG] will be the outcome of all confrontations, and the representative
victim’s payoff will be Y = - $1,500.
Social welfare remains at - $1,500 (the curve in Figure 2 flattens out) until R
finally reaches $1,000. At this point, gun control has increased the cost of a generic
firearm by 500%. The cost has become so high that any further increases will make
C > Z; that is, further increases will make attacking a dominated
strategy for criminals, as discussed above. Criminals will therefore choose not to attack, and [NA,NG]
will be the outcome of all confrontations.
The representative
agent’s payoff will jump
discontinuously
to $0 and remain there for all further increases in R.14 Of course,
further increases in R have no impact anyway, because at this point universal disarmament has been achieved.15
This example has several important
implications
for gun control policy. First and
“Victims may also respond to higher gun costs by substituting cheaper guns for mm-e expensive guns (e.g.,
substituting revolvers for automatic pistols). The cheaper guns will, in general, be less effective self-defense weap
OllS.
IsThese payoffs represent the noney value of the different outcomes to criminals and victims. These values
incorporate such things as fear, physical pain, etc. Readers are encouraged to try other values if they find these to
be extremely unrealistic.
“What sort of effect would relaxing the assumption of identical criminals have in this example? Suppose that
criminal i receives an identical gross payoff of Q (rather than 2) for attacking an unarmed victim. Let fl be a
normally distributed random variable with mean 2 and standard deviation o. In this case, the graph displayed in
Figure 2 would become “smoothed out” around R = $1,000 (see the light-dotted curve in Figure 2). There is
currently a discontinuity in the graph at this point because all criminals, being identical, switch to the NA strategy
at the same time. With criminal heterogeneity, however, some criminals would switch to the NA strategy at levels
of R below $1,000, while some would switch above it. Note that this light-dotted curve would become more
“stretched out” horizontally with increases in the dispersion parameter o.
IsNote that if the absolute value of Y is especially high (as may be the case with homicide), the curve in Figure
2 may not have time to flatten out before reaching R = 2 - C*, which is the point where criminals all switch to
the NA strategy.
TAYLOR
Representative
Expected
PZ@f
277
Victim’s
(See F-
14)
41500 -R=S700
$0
Parameter Settings:
w = -$6oo;x
= -$500;
FIG.2. An example
Y = -$IJoo;Z
of “Backfiring”
R=SlOOO
= $1,200; c*
R
= $200
gun control.
foremost is the implication
that increasingly
stringent gun control laws may make
things worse for victims long before they make things better; that is, the relationship
between gun control stringency and victim welfare may be fundamentally
nonmonotonic. In short, gun control can easily backfire by putting victims at a strategic disadvantage in confrontations
with criminals.
A second implication
is that although increasingly
stringent gun control laws can
theoretically
achieve universal disarmament,
the stringency necessary to do so may
be politically infeasible. In the above example, the cost of obtaining a gun must be
increased
by 500% to disarm the entire population.
One can easily imagine gun
control laws strict enough to do this (warrantless
house-to-house
searches for firearms, the summary execution of those found with guns, etc.), but the political feasibility of such measures is questionable,
especially in the United States and other
Western democracies.
Note also that if gun control laws cannot be made strict
enough to bring about universal disarmament,
then the best gun control policy is
none at all (R = $0). In short, our obsession with the unachievable
first-best solution
of universal disarmament
may be blinding
us to achievable second-best
solutions
currently at hand.
COKCLUSION 3. A movement from the MSNE
to [G,NA]
would increase social welfare.
A movement
from the MSNE to [G,NA] would unambiguously
increase the representative victim’s expected payoff, because - C > CY/(W - Y). Under what circumstances would such a movement take place? Return to the extensive form depiction
of the game in Figure 1. Suppose that criminals could observe the armament
decisions of victims. If the victim chose to go armed (G), the rational criminal would
refrain from attacking (NA) because 0 > X - C. Similarly, if the victim chose not to
278
A gametheoretic model of gun control
go armed (NG), the rational criminal would most certainly attack (A) because Z - C
> 0 (for low values of C, at least). Of course, the victim, being rational, is perfectly
aware of the criminal’s incentives here. The victim will therefore always choose to go
armed, knowing that a failure to do so will provoke a criminal attack. Thus, the
criminal will never attack, and the combination
[G,NA] will be the outcome of every
confrontation.
lf3
If criminals had some relatively low-cost means of ascertaining
whether a potential
victim was armed, the outcome described above might very well emerge. This conclusion suggests that laws requiring the concealment
of firearms may be misguided.
After all, a firearm is a more effective deterrent
to attack if the criminal knows the
victim is carrying one. Thus, in the absence of governmental
restrictions,
most individuals who carried firearms would probably carry them openly, and the armaments decision would be public, producing
the outcome outlined above.”
III. A More Complex Model: Gun Control, Gaod and Bad
The simple model described in the previous section succeeded in generating
Polsby’s
results: namely, that the relationship
between gun control stringency
and victim
welfare is nonmonotonic,
and that gun control measures harsh enough to attain
universal disarmament
may very well be politically infeasible. The above analysis
suggests that the elimination
of gun control may be the best attainable policy option.
This section will continue to follow the research agenda outlined in the introduction. The conclusions of the last section will be tested for robustness to changes in the
model’s assumptions.
As we will see, some elements of Polsby’s analysis will remain
intact in a more complex model, while others will have to be modified.
Setting Up the Game
The two key assumptions
of the model presented in the previous section-that
gun
control laws impose equal costs on criminals and victims, and that criminals always
highly unrealistic and should be modified. Let us look at the
attack with a gun -are
assumption
of equal costs first. While this assumption
may be true for some gun
checks), it is certainly not
control policies (e.g., waiting periods without background
true for others. Some gun control policies may raise costs for victims more than costs
for criminals. For example, a prohibition
on the sale of firearms would force all
individuals,
victims and criminals alike, to purchase weapons on the black market.
Presumably, criminals would have better access to the black market and better knowl-
‘6Technically. making the armaments decisions of victims public converts the game depicted in Figure 1 into
one of perfect information. Applying backwards induction to this new game will yield [G,NA]; this outcome is a
subgame-perfect
Nash equilibrium (Fudenberg and Tirole 1991, 74; Tirole 1988, 429).
“One possible justification for concealed carry laws is that a concealed gun provides a more general deterrent
effect than an unconcealed gun. Guns carried openly are, in effect, a private good: they deter criminals from
attacking the person who is carrying. Guns carried concealed are, however, a public good: they deter criminals
from attacking people in general, because the criminal cannot tell who is armed and who is not. However, economic
theory suggests that public goods will tend to be underprovided
in the absence of corrective subsidies. So, people
will have a tendency to “free ride” on the concealed carrying of others; this free riding may lead to low levels of
firearms carrying and therefore low levels of general deterrence. Therefore,
public subsidies for concealed carrying might be required.
279
TAYLOR
edge about prices on it. Thus, prohibition
would increase costs for both groups, but
presumably
less so for criminals.
On the other hand, some gun control policies may raise costs for criminulr more
check before a firearms
than costs for victims. For example, requiring a background
purchase
is allowed will almost certainly impose heavier costs on criminals than
given to criminals who
victims. Also, add-on penalties (i.e., additional punishment
use firearms in the commission of a crime) would impose potentially heavy costs on
criminals without imposing any costs on victims.
In short, any realistic model of gun control should allow for the possibility of
policies that have disproportionate
impacts on criminals and victims. The model in
this section will allow for this possibility by setting the criminal’s cost of obtaining a
gun, C,, equal to the sum of C* and R and by setting the victim’s cost of obtaining a
gun, C,, equal to the sum of C* and yR, where y is some constant greater than or
equal to zero. For gun control measures that affect the two groups equally, y = 1. For
those that affect criminals more than victims, y z (0,l). Finally, for measures that
affect victims more than criminals, y E (1 ,~a).
Now let us look at the assumption
that criminals always attack with a gun. Clearly,
all crimes, even homicides and commercial robberies, are often carried out by criminals not armed with a gun. Again, any realistic model of gun control should allow
for the possibility of attacks without a gun.
This section’s revised model is presented in Figure 3. This model is identical to the
one described in the last section except for two major changes. First, the criminal
now has three options rather than two: refrain from attacking (NA), attack with a gun
(G), or attack without a gun (NC). Second, the two groups now face differential costs,
C, and C,, which are defined above.
The gross payoffs S, T, U, V, W, X, Y, and Z are ordered as follows: X > W, V >
U,Z>V>O>X>T,andO>S>U>W>Y
where
Z is the armed criminal’s payoff for attacking an unarmed
victim,
V is the unarmed
criminal’s payoff for attacking an unarmed
victim,
X is the armed criminal’s payoff for attacking an armed victim,
T is the unarmed
criminal’s payoff for attacking an armed victim,
S is the armed victim’s payoff from an unarmed criminal’s attack,
v
G
A
NG
G
G
NG
NA
X-G
T
0
w-cv
s-cv
V
V
Z-CC
NG
Extensive
Y
-cv
U
0
0
Normal Form
Form
FIG. 3.
A more complex model.
280
A game theoretic model of gun control
U is the unarmed victim’s payoff from an unarmed criminal’s attack,
W is the armed victim’s payoff from an armed criminal’s attack, and
Y is the unarmed victimTs payoff from an armed criminal’s attack.
Again, the criminal is assumed to have the advantage in any encounter where the two
players are equally armed (X > W) or equally unarmed (V > U). The criminal gets
a higher gross payoff for an armed attack than for an unarmed attack, regardless of
whether the victim is armed (X > T) or unarmed (Z > V). Finally, the victim gets a
higher gross payoff for going armed than for not going armed, regardless of whether the criminal is armed (W > Y) or unarmed (S > U).”
Equilibrium
Concepts
This model has two distinct mixed-strategy Nash equilibria (MSNEs). When R is low
(i.e., gun control laws are liberal), the criminal population chooses between the G
(attack with a gun) and NA (no attack) strategies. When R is high, however, criminals
choose between the NG (attack without a gun) and NA strategies. One can immediately see how including an NG option in the revised model makes gun control
relatively more attractive: gun control may promote “harm reduction” by encouraging unarmed rather than armed attacks.lg
At what level of R will there be a “switch” between the two MSNEs? Returning to
the normal form depiction of the model in Figure 3, the switch will occur when R
becomes so large that Z - C, < V, i.e., when gun control becomes so stringent that
the additional gross payoff a criminal gets for attacking with rather than without a
gun no longer compensates for the cost of obtaining a gun. Solving this equation for
R, we get R > Z - V - C*. So, when R < Z - V - C*, criminals will choose between
G and NA, whereas when R > Z - V - C*, criminals will choose between NG and
NA.
The two MSNEs are justified and derived in precisely the same way that they were
in the last section. For each MSNE, we want to find a pair (cr,p) of population
fractions that make criminals and victims indifferent about their choice of strategies.
This indifference, as we noted above, results from the movement of criminals and
victims across subpopulations until the advantages of any given strategy have been
dissipated away. Once we obtain this pair of population fractions we can proceed to
calculate the expected payoffs to a representative victim and criminal in that MSNE.
Let us now calculate the population fractions and expected payoffs for each of the
MSNEs:
‘slf one ignores the criminal’s NA option and the costs of obtaining a gun, the model above looks very much like
a Prisoner’s Dilemma. For this statement to hold true, however, it must be the case that U > W, i.e., the victim must
prefer a confrontation
without guns to a confrontation
with guns. One can easily imagine situations where this
assumption
would not hold. For instance, a woman might prefer a gunfight to an unarmed confrontation
with a
rapist who is physically stronger than she is. As we shall see, however, this assumption
is not essential to the results
that follow.
19An anonymous
referee has pointed out that this statement
may be an oversimplification
for the following
reason: gun control may lead criminals to substitute knives for guns. Victims of gun crimes are far less likely to be
injured than victims of knife crimes (probably because the simple display of a gun is an effective inducement
to
cooperation),
though a gun injury is obviously much more likely to be fatal than a knife injury (Saltzman 1992). For
the purpose of this model, we will simply assume that the expected payoff to a victim from a knife crime (higher
probability of [less serious] injury) is greater than that from a gun crime (lower probability of [more serious] injury).
If this assumption
were false, then gun control would look particularly
unattractive
(at least within the context of
this model).
Low -
TAYLOR
281
a(X - C,) + (1 - a)(Z - C,) = a(0) + (1 - a)(O) = 0
(3)
p(w - C,.) + (1 - P)(-C,)
(4)
R MSNE
Indifference
Population
(R c Z -
V -
C*)
Equations
= pY + (1 - P)(O) = pY
Fractions
a* = (C, - Z)l(X - Z) and p* = C,I(W - I’)
Expected
Payoffs
Victim:
a*P*(W
- C,) + a*( 1 - p*)(-C,)
= C,Y/(W
Criminal:
High -
(5)
a*P*(X
R MSNE
Indifference
(R 2 Z -
+ (1 - a*) p*Y = p*Y
- Y)
- C,) + (1 - a*)P*(Z
V -
- C,) = 0
C*)
Equations
(f-3
aT+(l-a)V=a(O)+(l-a)(O)=0
p(s - C,.) + (1 - P)(-C,,)
Population
= pu + (1 - P)(O) = pu
(7)
Fractions
a* = - V/(T - V) and p* = C,/(S - U)
Expected
(8)
Payoffs
Victim:
a*P*(S - Cv) + a*(1 - p*)(-Cv)
+ (1 - a*)P*U
= p*U
= CvU/(S -U)
Criminal:
a*P*T
+ (1 - a*)P*V
= 0
Note that the Low-R MSNE is identical to the MSNE in the previous
that this one has differential
costs for obtaining a gun.
section,
except
Drawing Conclusions
The two new features of this model (differential
costs and an NG option
criminal) change the simple model’s results in the following ways:
1.
A lower y reduces victim welfare losses from
for the
increases in R. The ability to target
criminals with gun control laws reduces the welfare losses associated with increasingly stringent gun control. Return to the curve plotted in Figure 2. The
downward-sloping
part of this curve has a slope of Y/(W - Y), which is the
derivative of the representative
agent’s expected payoff with respect to R. Suppose now that we introduce differential costs. In this case, the slope of the curve
would be the derivative of C,Y/(W - Y) with respect to R, which is merely yYI(W
282
A game theoretic model of gun control
- Y). Thus, as we reduce y (i.e., change the composition
of gun control laws in
such a way as to target criminals more), the slope of the curve will rise: the curve
will become more shallowly sloped.
This result means that increases in R will have less of a tendency to disarm victims.
In fact, if y = 0 (i.e., gun control laws target criminals exclusively), then increases in
R will impose no welfare losses on victims-the
curve in Figure 2 would be completely flat in this case (slope = 0). For example, if the only gun control laws in
existence were the add-on penalties discussed earlier, stiffening such penalties would
not reduce the expected payoff of the representative
victim (because, by definition,
the law acts only to disarm criminals).
2.
The existence of an NG option means that the “switch point” occurs at a much lower R. In
Figure 2, guns are not eliminated
until R reaches a very high level (specifically,
R = Z - C*). In the revised model, however, the possibility of attack without a
gun means that much more modest gun control laws (specifically, R = Z - V C*) can lead to the disarming of criminals, as long as criminals do not place a
high premium on gun use in the attack. This premium is just equal to Z - V, the
difference in gross payoffs between attacks with and without a gun (when the
victim is disarmed).
The importance
of these changes is best illustrated with a specific example. Suppose again that in the absence of gun control a generic firearm would cost $200 (C*).
Also, assume that the type of crime in question is rape, so that the loss to the victim
is large, but the premium on gun use (Z - V) is relatively small. Specifically, assume
that the criminal’s gross payoffs are $500 for an armed attack on an unarmed victim
(Z) and $100 for an unarmed attack on an unarmed victim (V), and assume that the
unarmed
victim’s gross “payoff’ for an armed attack is -$l,OOO (Y). Furthermore,
assume that y = 0.2, so that every $5 increase in R increases the criminal’s cost of
obtaining a gun by $5 but increases the victim’s cost by only $1. Thus, the mix of gun
control laws used here targets criminals very well.*’
Given these assumptions,
how would increasingly stringent gun control laws affect
the representative
victim’s expected payoff? Figure 4 plots the relationship
between
R (the additional
cost that gun control imposes upon gun purchasers)
and social
welfare. At R = $0 (no gun control), we are at a low-R MSNE. The representative
victim’s expected payoff is - $363 here. As R increases (i.e., as gun control becomes
increasingly
stringent),
social welfare falls: fewer victims arm themselves and more
criminals attack.
This continues
until R = $200 and the victim’s expected payoff has fallen to
- $436. At this point, gun control has doubled the cost of firearms for criminals but
increased costs only 20% for victims. Notice that when R = $200, Z - C, = V; i.e.,
the criminal is indifferent between an armed and an unarmed attack (on an unarmed
victim). Any further increase in R will disarm criminals completely
and lead to a
“switch” to the high-R MSNE, where criminals choose between the unarmed
attack
(NG) and no attack (NA) strategies. The representative
victim’s expected payoff will
the expected payoff with no gun
jump to -$320
(which is greater than -$363,
2”The values chosen for the other gross payoffs are listed under Figure 4. They meet all of the restrictions
enumerated earlier in the section.
TAYLOR
Representative
EXpected
283
Victim’s
4320
$363
900
4436
I
SO
R=S200
R
R=SSOO
Parameter Settings:
S= -$l@I;T=
-$7fM;fJ=
-$4@;V=$l~;W=
Y = -$l,ofxI; 2 = $500; c* = $200; y = 0.2
FIG.4. An example of Welfare-Improving
-$45&X=
-&ioO;
gun control.
control”)
but will immediately
begin falling again as R increases. After all, once the
criminals are disarmed,
further tightening
of gun control is counterproductive
(at
least in this model).
Social welfare continues to fall until R = $500. At this point the cost of obtaining
a gun has become so high that S - C, = U: owning a gun (G) has become a
dominated
strategy for victims. All victims will therefore switch to the NC strategy.
Criminals, who are aware of this switch, will themselves switch to their NC strategy
and always attack (though without a gun). The combination
[NG,NG] will be the
outcome of every confrontation,
and victim welfare will remain at U = -$400 for
all R 2 $500.
In the example just described, modest gun control actually increases social welfare.
Victims are slightly better off at R = $200 than at R = $0 (no gun control). Note,
however, that this result is highly dependent
on the two features of the revised model
discussed above--differential
costs and the NC option for criminals. In Figure 4, y is
very low (0.2), meaning that gun control laws target criminals almost exclusively.
Were y equal to, say, 0.5 instead, then at R = $200 the representative
victim’s
expected payoff would jump to only - $400, which is less than the - $363 payoff the
victim would get with no gun control. Thus, gun control laws, to stand any chance of
being successful, must carefully target criminals.
*‘In general, the representative victim’s expected payoff after the jump will be larger than his expected payoff
with no gun control if the following condition holds:
[c* + y (Z - v - c*,]u
s-u
c*y
‘w-y
Notice that the left-hand side of the equation increases with decreases in y. In other words, gun control stands
a better chance of actually improving the lot of victims the more carefully targeted it is toward criminals.
284
A game theoretic model of gun control
Also, modest gun control was successful in disarming criminals in the above example precisely because criminals did not value highly the use of a gun in the
commission of the crime. That is, Z - V, the premium for gun use, was relatively
low. Thus, a mere doubling of the cost of obtaining a gun was enough to make
attacking with a gun a dominated strategy for criminals.
From these results one can draw the following conclusion:
CONCLUSION4: Ina model with differential costs and an NG option for criminals, modest gun
control may lead to welfare improvements, but only if:
1. the measures are narrowly targeted towards criminals and
2. criminals do not value guns very highly in the commission of crime.
The conclusion of this model and the conclusions of the simple model presented
in the last section may seem diametrically opposed. A more careful reading, however, will reveal an important similarity. In the simple model, the chief effect of
increasingly stringent gun control laws was to disarm victims and encourage criminal
predation. The more complex model may also yield this result, even if the two
conditions at the end of Conclusion 4 are satisfied.
This similarity exists because in both models, the relationship between social welfare and the stringency of gun control is fundamentally nonmonotonic. That is,
stricter gun control laws may sometimes increase welfare, but they will often reduce
welfare by disarming victims, thereby placing them at a strategic disadvantage in
confrontations with criminals. Only by carefully targeting such laws toward criminals
do we have any hope of preventing the latter from happening.
IV. Policy Implications
What implications does the more complex (and hopefully more realistic) model
presented in the previous section have for public policy? First note that policy makers
probably have little direct control over the premium Z - V that criminals attach to
using a gun in the commission of a crime. That leaves y, the measure of how
carefully gun control is targeted toward criminals, as the only parameter that can be
directly controlled by policy makers.
The analysis in the last section clearly implies that having a y as close to zero as
possible is highly desirable. That is, gun control laws are more likely to work when
they are very narrowly targeted toward criminals. What kind of laws are like this?
Add-on penalties for the commission of a crime with a firearm are an ideal example.
As noted earlier, such laws impose no cost on victims but impose potentially heavy
costs on criminals. Background checks on gun purchasers impose a modest cost on
criminals while imposing little to no cost on victims (unless the background checks
are very error prone). Similarly, “gun-a-month” purchase restrictions have the potential to impose higher costs on criminals than on victims by reducing the flow of
guns into the black market, as would private transfer/civil liability restrictions that
require the sale of firearms to be processed by a licensed gun dealer. The registration
of gun owners and/or guns might also qualify as a low-y regulation.
Laws that definitely would not qualify include gun bans of any sort (e.g., the
Feinstein Amendment and the proposed “Brady II” ban on certain small-caliber
handguns), waiting periods without background checks, heavy taxes on guns and
ammunition (e.g., U.S. Senator Moynihan’s proposed ammunition tax), and prohibitions (or harsh restrictions) on the carrying of concealed firearms. These laws are
TAYLOR
285
designed to bring about a general reduction in the availability of firearms. Because
they make no effort to distinguish between potential victims and potential criminals,
they may end up hurting the very group they were designed to help.
Interestingly, there are some signs, especially at the state and local levels, that
legislators are beginning to recognize the distinction between these two classes of gun
control laws. Some states, such as Virginia, have recently adopted or are planning to
adopt instant background check systems. 22 At the same time, many states are starting
to liberalize their procedures for granting concealed-carry permits. During the first
half of 1994, four states (Alaska, Arizona, Tennessee, and Wyoming) passed strong
concealed-carry reform laws (NRA-ILA Report 1994).23 Moreover, Chicago’s 1983
freeze on handgun ownership is currently under attack by several of Chicago’s black
political leaders, including the head of the Harold Washington Party (Ford 1994).
The model presented in the previous section helps reconcile these seemingly dissimilar policy movements.
V. Empirical Support
The model developed in section III clearly suggests that gun control laws that carefully target criminals (e.g., add-on penalties, background checks) are more likely to
be effective than those that attempt to induce a general scarcity of firearms (e.g., gun
bans, heavy taxation). The model also suggests that gun control laws will have a
greater impact on crimes in which the advantage of using a gun is low (e.g., rape). Is
there any empirical evidence available that supports these two propositions?
One of the best empirical studies to date on the efficacy of gun control legislation
in reducing violent crime is Kleck and Patterson (1993). Kleck and Patterson examine the impact of nineteen different types of gun control legislation on several measures of violence (including homicide, robbery, aggravated assaults, and rapes) in
170 major U.S. cities. Their research methods are superior to those of previous
studies in several ways. First, Kleck and Patterson’s unit of analysis is the city rather
than the state.24 Thus, their data are less aggregated, and they are able to examine
the effects of very strict local gun control ordinances. Second, unlike most previous
studies, they include a large number of control variables in their regressions. Finally,
Kleck and Patterson not only explicitly measure the prevalence of gun ownership but
also allow for a simultaneous and reciprocal relationship between this variable and
*“APProximately nine states now have instant background check systems in place (Bureau of Justice Statistics
1993, 142).
*sFlorida’s concealed-carry reform legislation has served as a model for many of these states. Kleck (1991)
describes Florida’s reform: “as of October 1, 1987, the state law was changed to a uniform, state-administered,
largely nondiscretionary, ‘shall issue’ permit system. Unless applicants had disqualifying attributes, the state was
required to issue a license if the applicants submitted the $125 licensing fee (for a J-year license), got themselves
fingerprinted, and properly filled out the required forms.” Between Oct. 1,1987, and May 31,1994, over a quarter
million of these concealed-carry permits were issued (Florida Department of State 1994).
Critics of the law believed that its passage would increase the level of violence. There is little evidence that this
has happened. Since 1987, the homicide rate in Florida hasfallen 21%, whereas the U.S. rate has risen 12% over
the same period (NRA-ILA Report 1994). This is not to suggest that the liberalized permitting system has actually
caused the decline, but merely to show that its critics’ worst fears have not been realized.
Approximately twenty-one states (including the four mentioned above) have liberal permitting systems now;
most have “shall issue” laws similar to Florida’s (Bureau of Justice Statistics 1993, 142).
“0nly two other cross-sectional studies have used city data--&&l
et al. (1969) and Cook (1979).
286
A game theoretic model of gun control
the level of violence. 25 This last feature is important
because high levels of violence
might lead to increased gun ownership, just as high levels of gun ownership might
tend to encourage
violence.
The general conclusion of Kleck and Patterson’s empirical research is that “most
gun restrictions
appear to exert no significant
negative effect on total violence
rates.” As they point out, however:
There do appear to be some gun controls which work, all of them relatively
moderate, popular, and inexpensive.
Thus, there is support for a gun control
policy organized
around gun owner licensing or purchase permits (or some
other form of gun buyer screening), stricter local dealer licensing, bans on possession of guns by criminals and mentally ill people, stronger controls over illegal
carrying, and possibly discretionary
add-on penalties for committing
felonies
with a gun.
In terms of the model presented in section III, most of these laws are low-y laws: they
carefully target the criminal element, and do little to prevent potential victims from
obtaining
guns. Kleck and Patterson’s conclusions
therefore provide some support
for our first proposition:
namely, that gun control laws that carefully target criminals
are more likely to be successful than those that try to induce a general scarcity of
firearms. As for the second proposition
that gun control laws will have more of an
impact on crimes in which criminals do not value highly the use of a gun, Kleck and
Patterson’s research indicates that local dealer licensing has a significant negative
effect on robberies and that discretionary
add-on penalties have a significant negative
effect on rape.
We should probably remember
at this point that the two conditions listed in Conclusion 4 above are necessary conditions, not su.cient conditions, for the effectiveness
of gun control legislation. That is, gun control laws may still be ineffective even if they
are well targeted toward criminals (low y) and criminals have little motivation to arm
themselves (low Z - V). They simply are more likely to be effective if these conditions hold. Thus, the negative results obtained by most empirical gun control studies
are not necessarily inconsistent
with Conclusion 4.26
VI. Conclusions and Future Research
This paper developed two game theoretic models of gun control. The first model was
a formalization
of the work of Polsby (1995). It illustrated how increasingly
stringent
gun control laws might actually reduce social welfare by disarming victims and encouraging criminal predation.
Moreover, it pointed out that even if universal disarmament were an optimal state, gun control laws strict enough to attain it might be
politically infeasible.
The second model extended the first by relaxing its two key assumptions-namely,
that criminals always attacked with a firearm and that gun control imposed equal
z30nly four previous studies have attempted to compute an explicit measure of gun prevalence: Zimring (1979),
Cook (1979). Magaddino and Medoff (1984), and Lester (1988). None of these studies, however, allowed for a
simultaneous and reciprocal relationship between guns and violence.
a6Among the many studies that find little or no connection between the stringency of gun control and crime
rates, the following six were written in the past decade: Mauser and Holmes (1992). Mundt (1996). Jung and Jason
(1988). Lester and Murrell(1986).
Magaddino and Medoff (1984). and Loftin and McDowall(l984).
Earlier studies
that find little or no connection include Ceisel et al. (1969) and Cook (1979).
287
TAYLOR
costs on criminals and victims. In this more realistic model, modest gun control laws
were sometimes capable of improving social welfare, but only if the restrictions were
narrowly targeted toward criminals and criminals did not place a high premium on
gun use in the commission of a crime.
The policy implications
of these models are fairly straightforward.
Gun control
laws that carefully target criminals (e.g., add-on penalties, background
checks) are
more likely to be successful than those that attempt to induce a general scarcity of
firearms (e.g., gun bans, high taxes). The empirical work of Kleck and Patterson
(1993), among others, lends some support to the model’s conclusions.
In what ways might the game theoretic models developed
above be extended?
There are several possibilities for future research. First, criminal and victim heterogeneity might be explicitly introduced
into the models. This issue was touched on
above, but never fully developed.
Second, risk aversion might be introduced
into the models, as well. This paper’s
assumption
of risk neutrality-this
is, the notion that people value a lottery at its
expected value-is
not particularly
realistic. Indeed, differences
in the degree of
risk-aversion
between criminals and victims might have important policy implications
(e.g., criminals will be less concerned
with the prospect of punishment
for the violation of gun control laws than victims if they are less risk-averse than victims, ceteti
paribus).
Third, completely different types of gun control models might be tried. The models above essentially depict confrontations
between “good guys” and “bad guys.” One
might also develop models where “good guys” confront “good guys” (these would be
“bar fight” models, which would probably closely resemble straightforward
Prisoner’s Dilemmas) and where “good guys” confront
persons of an unknown
ethical
persuasion
(these would be “misidentification”
models, which might be relevant for
home defense [mistaking family members for criminals at night] and street defense
[mistaking mysterious strangers for criminals]).
Finally, the models presented might be extended to include more than two players
per confrontation.
After all, one of the most commonly
used arguments
for the
liberalization
of concealed-carry
laws is that it would encourage Good Samaritan acts.
Bringing Good Samaritans
into these models might be very difficult both conceptually and technically, but it would do much to improve their realism.*’
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