SITUATION: TRAGEDY OF THE COMMONS—THE EXTINCTION OF THE WOOLLY MAMMOTH
At the end of the Pleistocene era, there was a mass extinction of more than half of the large-mammal species in the Americas, including the woolly mammoth. One prominent hypothesis is that the extinction was caused by hunting. Evidence of a large human population in the Americas dates to around 13,400 years ago, and it was roughly only 1,200 years later that the wave of extinctions occurred.
P̂HP � 50.67.
P̂HP � 40 � .25 � 42.67,
P̂Dell � 42.67.
P̂Dell(1 � .0625) � 40,
P̂Dell � 30 � 10 � .0625P̂Dell,
Now assume that both Dell and HP have a marginal cost of 20. Dell’s payoff function is
while HP’s payoff function is
Find all Nash equilibria.
VHP(PDell, PHP) � (PHP � 20)(100 � 2PHP � PDell).
VDell(PDell, PHP) � (PDell � 20)(100 � 2PDell � PHP),
6.2 CHECK YOUR UNDERSTANDING
Hunting a Woolly Mammoth
TH E
G R A
N G
ER C
O LL
EC TI
O N
, N EW
Y O
R K
6.3 Solving for Nash Equilibria with Calculus (Optional) 165
A recent computer simulation modeling the interaction between primitive humans and their environment supports this hypothesis.7 One of those simu- lations is shown in FIGURE 6.14; the thick black line represents the size of the human population, and each of the other lines represents a species hunted by humans. Most of those species saw their population size go to zero—which means extinction—and the median time between human beings’ arrival and extinction was 1,229 years, strikingly close to the evidence.
FIGURE 6.14 The Size of the Human Population (Thick Black Line) and Prey Species (Thin Gray Lines)
To explore how humans may have hunted species into extinction, let’s go back in time to the ice age and see what mischief primitive hunters can get into. Suppose there are n hunters and each hunter decides how much effort to exert in hunting woolly mammoths. Let denote the effort of hunter i, and as- sume that so that the strategy set comprises all nonnegative (real) num- bers. (One could imagine that effort is the number of hours exerted, in which case putting some upper bound on the strategy set would be appropriate.)
The total number of mammoths that are killed depends on how much ef- fort is exerted by all hunters. Letting denote the com- bined effort of all hunters, we find that the total number of mammoths killed (measured, say, in pounds) is which is plotted in FIGURE 6.15. Note that E increases and then decreases with total effort. There are two forces at work here. For a given population size of mammoths, more effort by hunters means more dead mammoths. However, because there are then fewer mammoths to reproduce, more effort also results in a smaller population of mammoths to kill. When total effort is sufficiently low slightly(E 6 500),
E(1,000 � E),
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166 CHAPTER 6: STABLE PLAY: NASH EQUILIBRIA IN CONTINUOUS GAMES
more effort means more mammoth meat; the first force is then bigger than the second one. When total effort is sufficiently high slightly more ef- fort means less mammoth meat, so the second force is the dominant one. A total effort exceeding 500 is a point of overexploitation: humans are killing mammoths faster than they can replenish themselves.
What we’ve described thus far is how total effort determines the total num- ber of kills. There is still the matter of how the meat is allocated among the hunters. An egalitarian approach would be for each hunter to get the same share, Some might say that a more equitable approach would be for a hunter who exerts more effort to get more meat. This approach could be carried out by the judgment of the tribal leader, assuming that he observes all the hunters’ efforts. Alternatively, suppose that hunters are acting individ- ually (or in small groups) and consume the meat of the mammoths they actu- ally kill, in which case a hunter who exerts more effort kills more and thus eats more.
Going with the equitable approach, we find that the fraction of meat re- ceived by hunter i is assumed to equal his fraction of the total effort:
Total output is and i’s share of it is Finally, we need to net out the personal cost of effort to hunter i, which is assumed to equal A hunter’s payoff function is then
Fixing the effort of the other hunters, we portray the payoff function for hunter i in FIGURE 6.16. Note that it is hill shaped.*
n � 1
Vi(e1, . . . , en) � ei 31,000 � (e1 � e2 � p � en) 4 � 100ei. 100ei.
ei/E.E(1,000 � E)
aei E b E(1,000 � E) � ei(1,000 � E).
E(1,000 � E)/n.
(E 7 500),
Po un
ds
0 500 1,000 E
E(1,000 � E )
FIGURE 6.15 The Relationship Between Effort in Hunting Woolly Mammoths and the Amount of Mammoth Meat
*Note also that 02Vi(e1, . . . en)/0e2i � �2 6 0.
6.3 Solving for Nash Equilibria with Calculus (Optional) 167
With the game now constructed, the next step is to derive a hunter’s best- reply function. We can do this by taking the derivative of his payoff function with respect to his effort and setting that derivative equal to zero:
[6.6]
Next, we solve equation (6.6) for to get the best-reply function:
A Nash equilibrium is n effort levels that satisfy the n equations which en- sure that each hunter is maximizing his payoff:
[6.7]
Notice that the best-reply function has the same form for all hunters; it is 450 minus half of the total effort of all other hunters. Since the game is symmet- ric, this result is not surprising. Let’s then make our life easier by looking for a symmetric Nash equilibrium.
A symmetric Nash equilibrium is a common effort level, call it whereby if the other hunters choose then it is optimal for hunter i to do so ase*,n � 1
e*,
e*n � 450 � a12b(e*1 � p � e*n�1). o
e*2 � 450 � a12b(e*1 � e*3 � p � e*n) e*1 � 450 � a12b(e*2 � p � e*n)
BRi � 450 � a12b(e1 � e2 � p � ei�1 � ei�1 � p � en). ei
� 900 � (e1 � e2 � p � ei�1 � ei�1 � p � en) � 2ei � 0.
0Vi(e1, . . . , en)
0ei � 1,000 � (e1 � e2 � p � en) � ei � 100 � 0
Pa yo
ff
0
0 ei
Vi
FIGURE 6.16 The Payoff Function for Hunter i
well. Substituting for in Equation (6.7), we have n identical equa- tions, all of the form
Now we have to solve only one equation in one unknown:
A Nash equilibrium, then, has each hunter exert an effort of For example, if there are 9 hunters, then a hunter chooses an effort of 90. Note that a hunter’s effort is less when there are more hunters. If there are 10 hunters rather than 9, then each exerts only about 82 units of effort. Less ef- fort is exerted because more hunters are chasing the same set of mammoths. A hunter might then find it more productive to hunt smaller game, such as rabbits, or gather vegetables.
It is also interesting to consider the combined effort of all hunters, which is and thus equals This combined effort is plotted in FIGURE 6.17, where one can see that it increases with the number of hunters.* Although each hunter hunts less when there are more hunters, the addition
of another hunter swamps that effect, so the total effort put into hunting goes up. Furthermore, there is overexploitation of the resource of mam- moths; that is, collectively, hunters hunt past the point that maximizes mammoth meat. Meat pro- duction is maximized when the total effort is 500. (See Figure 6.15.) However, the total equilibrium effort exceeds that value as long as there are at least two hunters:
n 7 1.2
400n 7 500
900n 7 500n � 500
900n n � 1
7 500
900n/ (n � 1).ne*
900/(n � 1).
e* � 900
n � 1 .
e*a2 � n � 1 2
b � 450 e* c1 � a1
2 b(n � 1) d � 450
e* � 450 � a1 2 b(n � 1)e*
e* � 450 � a1 2 b(n � 1)e*.
e*1, . . . , e * ne
*
168 CHAPTER 6: STABLE PLAY: NASH EQUILIBRIA IN CONTINUOUS GAMES
Po un
ds
900
0 4 5 6 7 8 91 2 3 Number of hunters (n)
900n n � 1
FIGURE 6.17 The Relationship Between the Number of Hunters and the Total Amount of Effort when All Hunters Choose Their Equilibrium Level of Effort
*This can be shown as follows:
0ne*
0n �
900(n � 1) � 900n
(n � 1)2 �
900
(n � 1)2 7 0.
6.3 Solving for Nash Equilibria with Calculus (Optional) 169
The resource of woolly mammoths is overexploited by hunters. The exces- sive hunting of mammoths is an example of what Garrett Hardin dubbed the tragedy of the commons.8 A tragedy of the commons is a situation in which two or more people are using a common resource and exploit it beyond the level that is best for the group as a whole. Overfishing Chilean sea bass, exces- sive deforestation of the Amazon jungle, and extracting oil too fast from a com- mon reservoir are examples of the tragedy of the commons. Interdependence between players (and what economists call an “externality”) is at the heart of this problem. When a hunter kills a woolly mammoth, he doesn’t take into ac- count the negative effect his action will have on the well-being of other hunters (i.e., they’ll have fewer mammoths to kill). As a result, from the perspective of the human population as a whole, each hunter kills too many mammoths.
Surely the most important current example of the tragedy of the commons is global climate change. According to the U.S. Environmental Protection Agency, “Since the beginning of the industrial revolution, atmospheric con- centrations of carbon dioxide have increased by nearly 30%, methane concen- trations have more than doubled, and nitrous oxide concentrations have risen by about 15%.”9 During that same period, the average surface temperature of the planet has increased by to 1 degree Fahrenheit and sea level has risen 4–8 inches. Those are the facts about which there is little disagreement. Where controversy lies is whether the atmospheric changes have caused the rise in temperature. If, indeed, it has, then the only way to solve this tragedy of the commons is through coordinated action that limits behavior, such as was pro- posed with the Kyoto Accord.