Lecture 3: Graphical analysis- Arms race
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1 Lecture 3: Graphical analysis- Arms race Daniel Chan UNSW Semester Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
2 Scenario Event 1 Saruman has taught the orcs the secrets of making inter-continental ballistic firebombs (ICBF). Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
3 Scenario Event 1 Saruman has taught the orcs the secrets of making inter-continental ballistic firebombs (ICBF). Event 2 In response, Gandalf has taught the hobbits how to make ICBFs. Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
4 Scenario Event 1 Saruman has taught the orcs the secrets of making inter-continental ballistic firebombs (ICBF). Event 2 In response, Gandalf has taught the hobbits how to make ICBFs. Problem Discuss the effects of these changes. Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
5 Scenario Event 1 Saruman has taught the orcs the secrets of making inter-continental ballistic firebombs (ICBF). Event 2 In response, Gandalf has taught the hobbits how to make ICBFs. Problem Discuss the effects of these changes. We wish to see how maths can be used to give an enlightened answer to this problem. We first need to build a mathematical model. Step 1 is Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
6 Scenario Event 1 Saruman has taught the orcs the secrets of making inter-continental ballistic firebombs (ICBF). Event 2 In response, Gandalf has taught the hobbits how to make ICBFs. Problem Discuss the effects of these changes. We wish to see how maths can be used to give an enlightened answer to this problem. We first need to build a mathematical model. Step 1 is Identify variables of interest Let h = the number of hobbit ICBFs and o = the number of orc ICBFs Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
7 Hypotheses To study these variables, we will need to assert some hypotheses. This part may or may not be mathematical in nature. Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
8 Hypotheses To study these variables, we will need to assert some hypotheses. This part may or may not be mathematical in nature. Let s take the hobbit perspective and for sake of argument posit Assumption (hippy) 1 We hobbits wish to minimise costs. 2 Orcs are peace loving hippies who would never launch an ICBF. Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
9 Hypotheses To study these variables, we will need to assert some hypotheses. This part may or may not be mathematical in nature. Let s take the hobbit perspective and for sake of argument posit Assumption (hippy) 1 We hobbits wish to minimise costs. 2 Orcs are peace loving hippies who would never launch an ICBF. Q What should we hobbits do? Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
10 Hypotheses To study these variables, we will need to assert some hypotheses. This part may or may not be mathematical in nature. Let s take the hobbit perspective and for sake of argument posit Assumption (hippy) 1 We hobbits wish to minimise costs. 2 Orcs are peace loving hippies who would never launch an ICBF. Q What should we hobbits do? Some points about modelling We will often need to impose assumptions so that we can perform some mathematical analysis. The more realistic the assumptions, the better the model should be. However, if the assumptions are too complicated, then the model may be too hard to analyse mathematically. Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
11 Hypotheses To study these variables, we will need to assert some hypotheses. This part may or may not be mathematical in nature. Let s take the hobbit perspective and for sake of argument posit Assumption (hippy) 1 We hobbits wish to minimise costs. 2 Orcs are peace loving hippies who would never launch an ICBF. Q What should we hobbits do? Some points about modelling We will often need to impose assumptions so that we can perform some mathematical analysis. The more realistic the assumptions, the better the model should be. However, if the assumptions are too complicated, then the model may be too hard to analyse mathematically. Historical note: China has a no nuclear first strike policy. Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
12 Deterrent hypothesis Let s posit a more reasonable hypothesis, that the hobbits will use the ICBFs as a deterrent. Assumption (deterrent) 1 Hobbits wish to minimise costs. 2 Hobbits wish to retain enough ICBFs that they can do enough damage to orc society, that the orcs will be discouraged from launching a strike. 3 Orcs will adopt the optimal strategy to frustrate the aims in 2) above. Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
13 Deterrent hypothesis Let s posit a more reasonable hypothesis, that the hobbits will use the ICBFs as a deterrent. Assumption (deterrent) 1 Hobbits wish to minimise costs. 2 Hobbits wish to retain enough ICBFs that they can do enough damage to orc society, that the orcs will be discouraged from launching a strike. 3 Orcs will adopt the optimal strategy to frustrate the aims in 2) above. Q What is the orc strategy? DISCUSS Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
14 Deterrent hypothesis Let s posit a more reasonable hypothesis, that the hobbits will use the ICBFs as a deterrent. Assumption (deterrent) 1 Hobbits wish to minimise costs. 2 Hobbits wish to retain enough ICBFs that they can do enough damage to orc society, that the orcs will be discouraged from launching a strike. 3 Orcs will adopt the optimal strategy to frustrate the aims in 2) above. Q What is the orc strategy? DISCUSS Q What does enough damage to orc society mean? Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
15 Deterrent hypothesis Let s posit a more reasonable hypothesis, that the hobbits will use the ICBFs as a deterrent. Assumption (deterrent) 1 Hobbits wish to minimise costs. 2 Hobbits wish to retain enough ICBFs that they can do enough damage to orc society, that the orcs will be discouraged from launching a strike. 3 Orcs will adopt the optimal strategy to frustrate the aims in 2) above. Q What is the orc strategy? DISCUSS Q What does enough damage to orc society mean? Our A There may or may not be mathematical considerations in determining some answer to this question. We will ignore it by letting h 0 be the number of ICBFs which achieves sufficient damage. Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
16 Try to plot the (h, o) graph To begin our analysis we shall attempt to plot the graph of h versus o, by plotting some points. Can we work out any points? Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
17 Try to plot the (h, o) graph To begin our analysis we shall attempt to plot the graph of h versus o, by plotting some points. Can we work out any points? o = 0 intercept Then h = h 0. DISCUSS Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
18 Try to plot the (h, o) graph To begin our analysis we shall attempt to plot the graph of h versus o, by plotting some points. Can we work out any points? o = 0 intercept Then h = h 0. DISCUSS Q Suppose the orcs launch a first strike. How should they aim their ICBFs? Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
19 Try to plot the (h, o) graph To begin our analysis we shall attempt to plot the graph of h versus o, by plotting some points. Can we work out any points? o = 0 intercept Then h = h 0. DISCUSS Q Suppose the orcs launch a first strike. How should they aim their ICBFs? This is easy to answer in Case o = h: Then each orc ICBF should be aimed at a different hobbit ICBF. Q How many hobbit ICBFs remain after the first strike? Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
20 Try to plot the (h, o) graph To begin our analysis we shall attempt to plot the graph of h versus o, by plotting some points. Can we work out any points? o = 0 intercept Then h = h 0. DISCUSS Q Suppose the orcs launch a first strike. How should they aim their ICBFs? This is easy to answer in Case o = h: Then each orc ICBF should be aimed at a different hobbit ICBF. Q How many hobbit ICBFs remain after the first strike? That depends on the probability p of each hobbit ICBF withstanding an attack from one orc ICBF. Then ph ICBFs remain and we need h 0 = ph, that is h = h0 p and we can plot the point ( h0 p, h0 p ). Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
21 The relationship between h and o Let s plot more points Case o = 2h: Orcs aim 2 ICBFs at each hobbit ICBF. Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
22 The relationship between h and o Let s plot more points Case o = 2h: Orcs aim 2 ICBFs at each hobbit ICBF. Now only p 2 h ICBFs remain so h 0 = p 2 h and h = h0 p 2 Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
23 The relationship between h and o Let s plot more points Case o = 2h: Orcs aim 2 ICBFs at each hobbit ICBF. Now only p 2 h ICBFs remain so h 0 = p 2 h and h = h0 p 2 Case o = nh, n Z: Orcs aim n ICBFs at each hobbit ICBF. Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
24 The relationship between h and o Let s plot more points Case o = 2h: Orcs aim 2 ICBFs at each hobbit ICBF. Now only p 2 h ICBFs remain so h 0 = p 2 h and h = h0 p 2 Case o = nh, n Z: Orcs aim n ICBFs at each hobbit ICBF. Now only p n h ICBFs remain so h 0 = p n h and h = h0 p n. Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
25 The relationship between h and o Let s plot more points Case o = 2h: Orcs aim 2 ICBFs at each hobbit ICBF. Now only p 2 h ICBFs remain so h 0 = p 2 h and h = h0 p 2 Case o = nh, n Z: Orcs aim n ICBFs at each hobbit ICBF. Now only p n h ICBFs remain so h 0 = p n h and h = h0 p n. Since n = o h, our above analysis suggests the following Relationship h = h 0 p o/h Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
26 The relationship between h and o Let s plot more points Case o = 2h: Orcs aim 2 ICBFs at each hobbit ICBF. Now only p 2 h ICBFs remain so h 0 = p 2 h and h = h0 p 2 Case o = nh, n Z: Orcs aim n ICBFs at each hobbit ICBF. Now only p n h ICBFs remain so h 0 = p n h and h = h0 p n. Since n = o h, our above analysis suggests the following Relationship h = h 0 p o/h N.B. Here h is not an explicit function of o, but we can plot points of this graph easily enough by looking at various values of n. Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
27 Plot Let s pick h 0 = 100, p =.4. Then we can use the computer to plot Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
28 Properties of the graph We can use calculus to study the shape of the graph. It is easier to study o as a function of h. From h = we see h0 p o/h Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
29 Properties of the graph We can use calculus to study the shape of the graph. It is easier to study o as a function of h. From h = we see h0 p o/h p o/h = h 0 h o h ln p = ln h 0 ln h o = h ln h 0 h ln h ln p Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
30 Properties of the graph We can use calculus to study the shape of the graph. It is easier to study o as a function of h. From h = we see h0 p o/h Differentiating and noting h 0 h gives p o/h = h 0 h o h ln p = ln h 0 ln h o = h ln h 0 h ln h ln p do dh = ln h 0 ln h 1 = ln(h 0/h) 1 > 0 ln p ln p Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
31 Properties of the graph We can use calculus to study the shape of the graph. It is easier to study o as a function of h. From h = we see h0 p o/h Differentiating and noting h 0 h gives p o/h = h 0 h o h ln p = ln h 0 ln h o = h ln h 0 h ln h ln p do dh = ln h 0 ln h 1 = ln(h 0/h) 1 > 0 ln p ln p Hence dh do > 0 and h is an increasing function of o. Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
32 Properties of the graph We can use calculus to study the shape of the graph. It is easier to study o as a function of h. From h = we see h0 p o/h Differentiating and noting h 0 h gives p o/h = h 0 h o h ln p = ln h 0 ln h o = h ln h 0 h ln h ln p do dh = ln h 0 ln h 1 = ln(h 0/h) 1 > 0 ln p ln p Hence dh do > 0 and h is an increasing function of o. Furthermore, do dh increases with h, so the o versus h graph is concave up. Thus the h versus o graph is concave down. Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
33 The orc story To understand the complete situation, we need to consider the orc story. Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
34 The orc story To understand the complete situation, we need to consider the orc story. Let s suppose that the orcs also adopt the deterrent strategy. Then we may swap the roles of o and h. Let o 0 be the number of ICBFs required to do unacceptable damage to the hobbits and q be the probability an orc ICBF withstands an attack from a hobbit ICBF. Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
35 The orc story To understand the complete situation, we need to consider the orc story. Let s suppose that the orcs also adopt the deterrent strategy. Then we may swap the roles of o and h. Let o 0 be the number of ICBFs required to do unacceptable damage to the hobbits and q be the probability an orc ICBF withstands an attack from a hobbit ICBF. From the orc perspective, we have Relationship o = o 0 q h/o Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
36 Combined plot Picking o 0 = 200, q =.2, we have the following combined plot Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
37 Combined plot Picking o 0 = 200, q =.2, we have the following combined plot The intersection point is (o, h) (315, 280). We predict that the number of ICBFs will be 315 for the orcs and 280 for the hobbits. Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
38 Combined plot Picking o 0 = 200, q =.2, we have the following combined plot The intersection point is (o, h) (315, 280). We predict that the number of ICBFs will be 315 for the orcs and 280 for the hobbits. DISCUSS Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
39 Qualitative analysis After the model is built, quantitative analysis may be impossible because of limitations in ascertaining accurate values for parameters. Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
40 Qualitative analysis After the model is built, quantitative analysis may be impossible because of limitations in ascertaining accurate values for parameters. Nevertheless, leaving the parameters arbitrary, it may be possible to perform a qualitative analysis by studying the graph. For example Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
41 Qualitative analysis After the model is built, quantitative analysis may be impossible because of limitations in ascertaining accurate values for parameters. Nevertheless, leaving the parameters arbitrary, it may be possible to perform a qualitative analysis by studying the graph. For example Question What will happen if hobbits decide to house their ICBFs in a more protected environment? Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
42 Qualitative analysis After the model is built, quantitative analysis may be impossible because of limitations in ascertaining accurate values for parameters. Nevertheless, leaving the parameters arbitrary, it may be possible to perform a qualitative analysis by studying the graph. For example Question What will happen if hobbits decide to house their ICBFs in a more protected environment? A This increases the parameter p. Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
43 Qualitative analysis After the model is built, quantitative analysis may be impossible because of limitations in ascertaining accurate values for parameters. Nevertheless, leaving the parameters arbitrary, it may be possible to perform a qualitative analysis by studying the graph. For example Question What will happen if hobbits decide to house their ICBFs in a more protected environment? A This increases the parameter p. The h-intercept remains the same but from do dh = ln(h0/h) 1 ln p increases so dh do decreases i.e. the curve flattens out. we see that do dh Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
44 New graph Assuming p increases from.4 to.5, we obtain Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
45 New graph Assuming p increases from.4 to.5, we obtain Conclusion The hobbit strategy graph flattens from the blue curve to the red. We see ICBFs reduce for both the hobbits and orcs. Daniel Chan (UNSW) Lecture 3: Graphical analysis Semester / 12
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