D iscrete vs. C ontinuous V ariables and CK

1. Discrete vs. Continuous Variables and CK

2. Recall, Norm against Chemical Weapons

4. Why were chemical weapons the red line? Why not 100,000 deaths? Why not wanton murder of civilians?

5. This norm reared its head many times before…

6. “These flamethrowers were used to kill Japanese holed into pillboxes, buildings and caves. “ -Wiki entry for Battle of Iwo Jima

7. Why flame throwers? “A strong military case was made for the use of gas before America’s attack on the island of Iwo Jima; Japanese defenders in caves and tunnels would have been particularly vulnerable. Franklin Roosevelt rejected the idea.” -The Economist, The History of Chemical Weapons

8. >26,000 American casualties, Iwo Jima, WWII

9. To save civilians?

10. “Because all the civilians had been evacuated, there were no civilian casualties at Iwo Jima”

11. More humane than chemical weapon?

12. Video Clip: Japanese Soldier Burnt Alive by Flame Thrower

13. We ask… Why does the norm not attend to the thing that actually matters (number of innocent deaths? degree of suffering?) Why does the norm pay undue attention to a detail (the type of weapon used) even when the detail isn’t always that consequential?

14. We will use our common knowledge result to show… • We can’t coordinate based on continuous variables (like how much suffering has occurred) • We can coordinate based on discrete variables (like whether a chemical weapon was used) Then we will discuss a bunch of other applications and insights…

15. Intuition: Even if, say, 100,000 suffered isn’t common knowledge that more than, say, 10 suffered. Why? Perhaps other has slightly lower estimate. (Still above 10. So second order believe more than 10 suffered, but what about third order?) Then he thinks, perhaps you have slightly lower estimate, in which case, you have slightly lower estimate…he has estimate below 10! (so have, say, 1000th order knowledge more than 10 suffered, but not 1001th order knowledge Not common knowledge By our theorem, can’t act differently than if 10 had suffered

16. Let’s formalize this…

17. A B 5,5 0,0 Model: 1) Consider a random variable x x~U[0,1] e.g. x = # civilian casualties 2) Players gets a signal of x , si(x) si(x)~U[x-.01,x+.01] i.e. si(x)= body count by i’s surveyors, i=U.S.,France 3) Then they play a coordination game A 0,0 1,1 B • i.e. choose whether or not to attack Syria, • (assumes worth attacking iff other country attacks)

18. There does not exist a Nash Equilibrium where play A if get low signal and B if get high signal I.e. if don’t attack when body count is low, cannot attack when body count is high

19. Why? Because 1 believes s2(x)~U[3.05-.01,3.05+.01] So s2 < 3 with probability ¼ Why? Because we assumed 2 would play A when he got a low signal Sketch of proof: -Assume σi(si)=A, if si<.3 i.e. play A when get low signal -Suppose s1 < .305 i.e. suppose get slightly higher signal μ(s2<.3 |s1)>1/4 i.e. still likely that other got low signal μ(σ2(s2)=A |s1)>1/4 i.e. still likely that other will play A σ1(s1)=A i.e. so 1 still want to play A -Suppose .305<σ1< .31 i.e. now suppose get slightly higher signal μ(s2<.305 |s1)>1/4 i.e. still likely that other got signal not lower than original threshold, but lower than threshold from previous step μ(σ2(s2)=A |s1)>1/4 (by above logic!) σ1(s1)=A Let’s check if the proof is sensitive to… -What if s1=.7? -What is relation to common p-belief? For which value of p? -What if signals more noisy? -What if signals less noisy? -What if U(A,A)=(2,2)? -What if U(A,A)=(.5,.5)? -What if noise is not uniform? -What if x isn’t uniform? -What if U(B,B) increases with x? Why? If play A, get ¼ × 5 = 1.25 If play B, get ¾ ×1 = .75

20. Why? Because 1 believes s2(x)~U[3.025-.005,3.025+.005] So s2 < 3 with probability ¼ Suppose signals half as noisy si(x)~U[x-.005,x+.005] -Assume σi(si)=A, if si<.3 i.e. play A when get low signal -Suppose s1 < .3025 i.e. suppose get slightly higher signal μ(s2<.3 |s1)>1/4 i.e. still likely that other got low signal μ(σ2(s2)=A |s1)>1/4 i.e. still likely that other will play A σ1(s1)=A i.e. so 1 still want to play A -Suppose .3025<σ1< .305 i.e. now suppose get slightly higher signal μ(s2<.3025|s1)>1/4 i.e. still likely that other got signal not lower than original threshold, but lower than threshold from previous step μ(σ2(s2)=A |s1)>1/4 (by above logic!) σ1(s1)=A … Only difference… proof has twice the number of iterations -What if U(B,B) increases with x?

21. Hence, can’t “coordinate on continuous variables” Why? Because continuous variables can never create common p-beliefs

22. So what kinds of signals can we coordinate on?

23. Discrete Here’s the intuition…

24. If get signal chemical weapon was used pretty sure other got signal chemical weapon was used in which case, other would be pretty sure you got signal chemical weapon was used • in which case… You p-believe that other p-believes that you p-believe… Is common p-believed By our theorem, can act differently depending on whether or not get signal chemical weapons were used

25. Formally…

26. B A 5,5 0,0 1) Consider a random variable x x~U{-1,1} e.g. x = -1 iff Assad didn’t use chemical weapons 2) Players get a signal of x si(x)=x w/ prob .9 and -x w/ prob .1 i.e. 10% chance think use chemical weapon when don’t and vice versa 3) Play a coordination game A 0,0 1,1 B • i.e. choose whether or not to attack Syria, • worth attacking iff other country attacks

27. There is a NE where play A if get one signal and B if get other signal! I.e. can attack iff surveyors conclude chemical weapons were used!

28. Sketch of Proof: Suppose both Play A iff get signal -1. We will show neither can deviate. Hence, this is a NE If get signal of -1 large chance other got same signal large chance other will play A best response to play A If get signal 1… -What is relation to common p-beliefs? -What if large chance of error? -What if U(A,A)=(10,10)? -What if x can take on 3 values? -What if x can take on 1000 values?

29. Hence, Can “coordinate on discrete variables” Why? Discrete variables can create common p-beliefs

30. That’s all fine and good in theory… But who is calculating common p-beliefs? Well…even if no one is calculating, a continuous norm would “unravel” over time Let’s see why…

31. Suppose that we had a norm that said “attack if you estimate 100 civilians had been killed.” One day you will get a signal that 100 civilians had been killed. You attack…And no one else does, since they got signal only 99 had been killed. So you quickly learn not to attack at 100. (Or perhaps you intuit this immediately, and don’t need to learn from experience.) Shortly thereafter, you get a signal that 101 civilians had been killed…since no one attacks even when they get signal 100 had been killed, you again are likely to find yourself being the only only attacker…so you learn not to attack then either… Pretty soon (OK it might take a while)… we learn not to attack even if 100,000 civilians had been killed

32. In contrast, a discrete norm can be sustainable… If we all attack when we get signal chemical weapon was used…then it is very rare that we are the lone attackers, so we don’t “learn” to change our strategy…even though this norm is “second best”

33. Let’s discuss some insights…

34. What will happen over time if we try and enforce norm against wantonly killing civilians? What is the cost/benefit of enforcing a norm against chemical weapons in Syria? Which international treaties should we expect to last? What would be the benefit of creating a military arm to the U.N.?

35. Here are some of the many examples where we condition on discrete variables, even when seems like unimportant detail And ignore continuous variables when seem to be the thing that actually matters

36. Utilitarianism vs rule based ethics! Try this: cooperate with your friends whenever the benefits to them exceeds the costs to you (What about with family?) Why do we call it murder when you shoot a healthy 20 year old or a 80 year old who has been in a coma for 5 years? (Do we feel equally as sad? Why not?) Why do we start wars over a single assassination or boat being sunk? Why do we start revolutions after a tea tax? (What should a fascist gov do to prevent a revolution? What should the CIA do to start one?) Why does the NRA care so much about a ban on assault weapons? Or background checks? Why do we care so much about gay marriage? (Are we wrong to care?)

37. One last insight: Recall from our H-D-B lecture

38. Note the following pattern for uncorrelated asymmetries (we’ll come back to this later in the class) Yes: Who got there first No: Who is taller Yes: “One drop” or “one grandparent” rule No: Darker skin Yes: Reserved for handicapped or pregnant No: Reserved for people older than you discrete continuous discrete continuous discrete continuous Can condition whether play Hawk on discrete variable, but not on continuous

39. The power of this model: -We have a deeper sense of which slopes are slippery -We have a deeper sense of when we will slide -We have an ultimate explanation -We can see the similarities between all these examples -We can see the similarities between continuous variables and omission and innuendos and…

40. Next class:

41. Recall from H-D-B Thus, we expect animals to pay attention to who arrived first! Even if arriving first has NO impact on value of resource or likelihood of winning combat

42. Note: We could have written model where play Hawk if arrive second Play Hawk if second would be unique symmetric equilibrium in that game as well But we don’t ever observeHawk if arrive second. Why not? We’ll find out later in the semester!

43. We will present one last model related to CK that explains this (Will ALSO help us resolve a few other loose ends!)