An arms buildup is thought to have been a contributing factor to World War I. An arms buildup is thought to have been a contributing factor to World War I.
An arms buildup is thought to have been a contributing factor to World War I. The naval arms race between Germany and Great Britain is par- ticularly noteworthy. In 1889, the British adopted a policy for maintain- ing naval superiority whereby they required their navy to be at least two- and-a-half times as large as the next-largest navy. This aggressive stance induced Germany to increase the size of its navy, which, according to Britain’s policy, led to a yet bigger British navy, and so forth. In spite of attempts at disarmament in 1899 and 1907, this arms race fed on itself. By the start of World War I in 1914, the tonnage of Britain’s navy was 2,205,000 pounds, not quite 2.5 times that of Germany’s navy, which, as the second largest, weighed in at 1,019,000 pounds.10 With this scenario in mind, let us model the arms race between two countries, denoted 1 and 2. The arms expenditure of country i is denoted and is restrictedxi
V2(x1, x2) � x2 � 20x1x2.
V1(x1, x2) � x1 � 10x1x2
qi(pi � 10 � 5xi). 1,000 � xi � pi;
qi(10 � 5xi). qi
10 � 5xi.
xipi
e0 if p1 � p2 (p2 � 15)(100 � p2) if p2 6 p1
.
Exercises 177
to the interval [1,25]. The benefit to a country from investing in arms comes from security or war-making capability, both of which depend on relative arms expenditure. Thus, assume that the benefit to country 1 is
so it increases with country 1’s expenditure relative to total expenditure. The cost is simply so country 1’s payoff function is
and there is an analogous payoff function for country 2:
These payoff functions are hill shaped. a. Derive each country’s best-reply function. b. Derive a symmetric Nash equilibrium.
9. Players 1 and 2 are playing a game in which the strategy of player i is de- noted and can be any nonnegative real number. The payoff function for player 1 is
and for player 2 is
These payoff functions are hill shaped. Find all Nash equilibria.
10. The wedding anniversary of a husband and wife is fast approaching, and each is deciding how much to spend. Let denote the amount that the husband spends on his wife and the amount the wife spends on her husband. Assume that they have agreed that the most each can spend is 500. A players’ strategy set is then the interval [0,500]. A spouse enjoys giving a bigger gift, but doesn’t like spending money. With that in mind, the husband’s payoff function is specified to be
The payoff function can be understood as follows: The benefit from ex- changing gifts is captured by the term Since “men are boys with bigger toys,” this benefit increases with the size of the wife’s gift:
The “warm glow” the husband gets from giving his wife a gift is reflected in the term which increases with the size of his gift:
0(50gH � (14)gHgW) 0gH
� 50 � a1 4 b gW 7 0.
50gH � ( 1 4)gHgW,
0(50gH � (14)gHgW) 0gW
� a1 4 b gH 7 0.
50gH � ( 1 4)gHgW.
VH(gH, gW) � 50gH � a14b gHgW � a12b(gH)2.
gW gH
V2(z1, z2) � (80 � z1 � z2)z2.
V1(z1, z2) � (100 � z1 � z2)z1
zi
V2(x1, x2) � 36 a x2x1 � x2b � x2.
V1(x1, x2) � 36 a x1x1 � x2b � x1, x1,
36( x1
x1 � x2),
178 CHAPTER 6: STABLE PLAY: NASH EQUILIBRIA IN CONTINUOUS GAMES
Alas, where there are benefits, there are costs. The personal cost to the husband from buying a gift of size is represented by the term
or in his payoff function. Thus, we subtract this cost from the benefit, and we have the husband’s payoff function as de- scribed. The wife’s payoff function has the same general form, though with slightly different numbers:
These payoff functions are hill shaped. a. Derive each spouse’s best-reply function and plot it. b. Derive a Nash equilibrium. c. Now suppose the husband’s payoff function is of the same form as the
wife’s payoff function:
Find a Nash equilibrium. (Hint: Don’t forget about the strategy sets.)
11. Players 1, 2, and 3 are playing a game in which the strategy of player i is denoted and can be any nonnegative real number. The payoff function for player 1 is
for player 2 is
and for player 3 is
These payoff functions are hill shaped. Find a Nash equilibrium.
12. Players 1, 2, and 3 are playing a game in which the strategy of player i is denoted and can be any nonnegative real number. The payoff function for player 1 is
for player 2 is
V2(y1, y2, y3) � y2 � y1y2 � (y2) 2,
V1(y1, y2, y3) � y1 � y1y2 � (y1) 2,
yi
V3(x1, x2, x3) � x1x2x3 � 1 2
(x3) 2.
V2(x1, x2, x3) � x1x2x3 � 1 2
(x2) 2,
V1(x1, x2, x3) � x1x2x3 � 1 2
(x1) 2,
xi
VH(gH, gW) � 50gH � 2gHgW � a12b(gH)2.
VW (gH, gW) � 50gW � 2gHgW � a12b (gW)2.
�(gH) 2,�gH � gH,
gH
References 179
and for player 3 is
These payoff functions are hill shaped. Find a Nash equilibrium. (Hint: The payoff functions are symmetric for players 1 and 2.)
V3(y1, y2, y3) � (10 � y1 � y2 � y3)y3.
REFERENCES 1. Graham Flegg, Numbers: Their History and Meaning (New York: Schoken
Books, 1983).
2. The answer is that these three sites and many more are all operated by the same company, GSI. For details, go to the website of GSI (www.gsicommerce.com/partners/index.jsp). When searching among these different sites, you may think that you’re comparison shopping— but think again!
3. Website of Best Buy, (www.bestbuy.com) (Mar. 1, 2007).
4. Website of Circuit City, (www.circuitcity.com) (Mar. 1, 2007).
5. James D. Hess and Eitan Gerstner, “Price-Matching Policies: An Empirical Case,” Managerial and Decision Economics, 12 (1991), 305–315.
6. Blog of the Mad Canuck, (www.madcanuck.com) (Oct. 17, 2004).
7. John Alroy, “A Multispecies Overkill Simulation of the End-Pleistocene Megafaunal Mass Extinction,” Science, 292 (2001), 1893–1896.
8. Garrett Hardin, “The Tragedy of the Commons,” Science, 162 (1968), 1243–48.
9. Website of the Environmental Protection Agency, <yosemite.epa.gov/oar/ globalwarming.nsf/content/climate.html> (Mar. 1, 2007)
10. Niall Ferguson, The Pity of War (New York: Basic Books, 1999).
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“Obviously the Intruder had acted at night first, then had come out into the open during the day, when Roberto remained awake in his cabin. Should he now revise his plans, giving the impression of sleeping in the daytime and staying awake at night? Why? The other would simply alter his strategy. No, Roberto should instead be unpredictable, make the other unsure, pretend to be asleep when he was awake while he was asleep. . . . He had to try to imagine what the other thought he thought, or what the other thought he thought the other thought he thought. . . .”1
7.1 Police Patrols and the Drug Trade DETERMINED TO CRACK DOWN on the drug trade, a city mayor puts more officers out on patrol to disrupt the business of drug dealers. A drug dealer in a neigh- borhood can work his trade either on a street corner or in the park. Each day, he decides where to set up shop, knowing that word about his location will travel among users. Because a good snitch is lacking, word does not travel to the police. The police officer on the beat then needs to decide whether she will patrol the park or the street corners, while not knowing where the drug dealer is hanging out that day.
The decisions of the officer and the dealer determine the extent of drug trades that day. Assume that, without disruption by the police, 100 trades will occur. A dealer’s payoff equals the number of trades he consummates. For the officer, her payoff is the number of trades she disrupts (which is simply 100 minus the number of trades that occur). If they both end up in the park—the officer patrolling there and the drug dealer selling there—only 40 trades occur, which means that 60 trades are disrupted. Given the size of the park, there is still a fair amount of activity. As shown in FIGURE 7.1, the officer’s payoff is then 60 and the dealer’s payoff is 40. If the drug dealer is in the park, but the offi- cer is out on the streets, then 100 drug deals go down, so the officer’s payoff is zero and the dealer’s is 100. If the officer is patrolling the streets and the dealer
7
181
Keep ’Em Guessing: Randomized Strategies
FIGURE 7.1 Police Patrol and the Drug Trade
80,20 0,100 Police officer
Drug dealer
Street corner
Street corner Park
10,90 60,40Park
182 CHAPTER 7: KEEP ’EM GUESSING: RANDOMIZED STRATEGIES
is out on a street corner, then only 20 trades occur. Finally, if the dealer is on a street corner, but the officer is patrolling the park, then 90 trades occur. (Some drug traffic is disrupted due to patrolling police cars.) All of these pay- offs are illustrated in Figure 7.1.
There is no Nash equilibrium in this game. For example, if the drug dealer is planning to be in the park, then that is where the officer will go. But if the offi- cer is expected to be in the park, then the drug dealer will take to the streets.
This game is an example of an outguessing game, which was introduced in Chapter 4. Each player wants to choose a location that is unanticipated by the other player. A drug dealer would not want to always go to the park, because such predictability would induce the police officer to patrol the park and dis- rupt the dealer’s business. For the same reason, the dealer would not want to always be on a street corner. What seems natural is for the drug dealer to switch his location around—sometimes be on a corner, sometimes in the park. But can we say exactly how he should switch his locations? And given that the dealer is moving his location around, what should the officer do? Clearly, an officer doesn’t want to be predictable either, for if she is always in the park, then the dealer can conduct a brisk business by staying on a street corner.
The objective of this chapter is to learn how to solve outguessing games and, more generally, to derive solutions in which players randomize their be- havior. To achieve that objective, we’ll need to know how to model decision making under uncertainty, which is analyzed in the next section. With that knowledge, we then return in Section 7.3 to the drug trade situation and show how to solve it. Further examples, along with some tips for arriving at a solu- tion, are provided in Section 7.4, while more challenging games are solved in Section 7.5. In Section 7.6, a special property of solutions with randomized play is derived for games of pure conflict.