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← 16. Backward induction: reputation and duels

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Showing Revision 1 created 07/17/2012 by Amara Bot.

  1. Professor Ben Polak: So
    this is what we did last time:
  2. we looked at a game involving
    an entrant and an incumbent in a
  3. market;
    and the entrant had to decide
  4. whether to enter that market or
    not;
  5. and if they stayed out the
    incumbent remained a monopolist;
  6. and the monopolist made 3
    million in profit.
  7. If the entrant goes in,
    then the incumbent can decide
  8. whether to accommodate the
    entrant and just settle for
  9. duopoly profits,
    making a million each;
  10. or the incumbent can fight,
    in which case the incumbent
  11. makes no money at all and the
    entrant loses a million dollars.
  12. We pointed out a number of
    things about this game.
  13. One was that when we analyzed
    it in a matrix form we quickly
  14. found that there were two Nash
    Equilibria, that Nash
  15. Equilibrium were:
    in and not fight;
  16. and out and fight.
    But we argued that backward
  17. induction tells us that the
    sensible answer is in and not
  18. fight.
    Once the incumbent knows the
  19. entrant is in they're not going
    to fight because 1 is bigger
  20. than 0, and the entrant
    anticipating this will enter.
  21. When we talked a little bit
    more we said this other
  22. equilibrium, this out fight
    equilibrium--it is an
  23. equilibrium because if the
    entrant believes the incumbent's
  24. going to fight then the entrant
    is going to stay out,
  25. and it's costless for the
    incumbent to "fight" if in fact
  26. the entrant does stay out
    because they never get called
  27. upon to fight anyway.
    So the idea of this was that
  28. for the incumbent to say they're
    going to fight is an "incredible
  29. threat."
    That's terrible English.
  30. It's the way it is always
    taught in the textbooks.
  31. It needs to be called a "not
    credible" threat.
  32. And that "not credible" threat
    is: he's not really going to
  33. fight if the entrant comes in,
    and therefore,
  34. the entrant should come in and
    in fact the incumbent will
  35. accommodate it.
    So what we've shown here is
  36. that, if we believe this
    argument, then the entrant will
  37. come in and the incumbent is
    going to let him in.
  38. At the end, we started talking
    about this in a slightly more
  39. elaborate setting,
    so let's just remind you of
  40. what that more elaborate setting
    is.
  41. The more elaborate setting is
    suppose that there is one firm,
  42. one monopolist,
    and that monopolist holds a
  43. monopoly in ten different
    markets.
  44. So we'll have our monopolist be
    Ale.
  45. So here's Ale.
    He's our monopolist,
  46. and he owns pizzeria monopolies
    in ten different markets.
  47. And each of these ten different
    markets are separate,
  48. they are different towns.
    And in each of those ten
  49. markets he thinks--he knows he's
    going to face an entrant and
  50. those entrants are going to come
    in order.
  51. So let's just talk about who
    those entrants are going to be.
  52. The entrants are going to be
    this person, this person and so
  53. on.
    Let's find out who they are,
  54. so your name is?
    Student: Isabella
  55. Professor Ben Polak: Where
    are you from?
  56. Student: Miami.
    Professor Ben Polak:
  57. Miami, so Miami is one of the
    markets.
  58. Your name is?
    Student: Scott.
  59. Professor Ben Polak:
    From where?
  60. Student: Wisconsin.
    Professor Ben Polak:
  61. Where in Wisconsin?
    Student: Madison.
  62. Professor Ben Polak:
    Madison: we've got two towns.
  63. We're just going to do towns
    now.
  64. Student: My name is
    Lang.
  65. I'm from Bridgeport,
    Connecticut.
  66. Professor Ben Polak:
    Okay, we've got three towns.
  67. Student: I'm from Miami
    too.
  68. Professor Ben Polak:
    Talk about Yale diversity.
  69. Well we'll pretend you're from
    somewhere else.
  70. Put him in New Orleans or
    something.
  71. Student: Chris from
    Boston.
  72. Professor Ben Polak:
    From Boston, all right.
  73. Student: From Orange,
    Connecticut.
  74. Professor Ben Polak:
    From Orange, Connecticut so just
  75. down the road.
    Student: St.
  76. Louis, Missouri.
    Professor Ben Polak: All
  77. right, have we got ten yet?
    I'm not quite at ten.
  78. One, two, three,
    four, five, six,
  79. seven.
    Student: Saffron,
  80. New York.
    Professor Ben Polak: All
  81. right.
    Student: Hong Kong.
  82. Professor Ben Polak:
    Hong Kong, that's way away.
  83. Student: Long Island.
    Professor Ben Polak:
  84. Long Island.
    I think I've probably got ten
  85. markets here.
    So Ale owns a pizza shop.
  86. He's the monopoly pizza shop
    owner in each of these ten
  87. markets.
    And what we're going to see is
  88. we're going to see what happens
    as, sequentially,
  89. these entrants try to enter.
    The way that this game's going
  90. to work is that they're lined
    up--we know the order in which
  91. the entrants are going to come.
    They're going to start off,
  92. the first person who is going
    to have to make a decision is-
  93. Student: Enter.
    Isabella.
  94. Professor Ben Polak: Is
    Isabella, right?
  95. We're going to see how our
    monopolist responds.
  96. So let's have a look at this.
    So Isabella who is in which
  97. market again?
    Student: Miami.
  98. Professor Ben Polak: In
    Miami, okay what are you going
  99. to do?
    Student: Enter.
  100. Professor Ben Polak:
    What are you going to do?
  101. Student: I will fight.
    Professor Ben Polak: Oh
  102. dear, so you owe me a million
    dollars.
  103. So one person's down a million
    dollars, let's see what happens
  104. next.
    Student: I'm going to
  105. stay out.
    Professor Ben Polak:
  106. Which market was this?
    Student: Scott. Madison.
  107. Professor Ben Polak:
    Madison stayed out.
  108. Student: I'm going to
    stay out.
  109. Professor Ben Polak:
    Staying out.
  110. So Bridgeport stayed out.
    Student: I guess I'll
  111. stay out.
    Professor Ben Polak:
  112. Stayed out again.
    Student: Stay out.
  113. Professor Ben Polak:
    Which market are we up to now,
  114. somewhere in Orange County
    wasn't it?
  115. Where were we?
    Student Orange,
  116. Connecticut.
    Stay out.
  117. Professor Ben Polak:
    Stay out.
  118. Student: I think I'll
    stay in.
  119. Professor Ben Polak:
    You'll come in okay,
  120. and which market is this?
    Student: St.
  121. Louis, Missouri.
    Professor Ben Polak: St.
  122. Louis, Missouri.
    So you owe me a million dollars
  123. as well, okay.
    A couple of million dollars is
  124. a good class.
    We're going to have plenty of
  125. money for lunch.
    Student: I'm also going
  126. to fight.
    Professor Ben Polak:
  127. You're going to fight,
    which market is that?
  128. Student: Saffron,
    New York.
  129. Professor Ben Polak:
    Where abouts?
  130. Student: Saffron.
    Professor Ben Polak:
  131. Saffron, New York.
    Where are we at?
  132. One, two, three,
    four, five, six,
  133. seven, eight,
    Ale?
  134. TA: Fight.
    Professor Ben Polak: You
  135. fight?
    So you owe me a million dollars
  136. too.
    That was eight, nine?
  137. Student: Out.
    Professor Ben Polak: Out
  138. and ten?
    Student: Stay out.
  139. Professor Ben Polak:
    Stays out, okay.
  140. Now let's just notice something
    here, which was the tenth
  141. market?
    What town were you?
  142. Student: Long Island.
    Professor Ben Polak:
  143. Whereabouts in Long Island?
    Student: Huntington.
  144. Professor Ben Polak: So
    if Huntington,
  145. Long Island our last market had
    come in, suppose you'd said in,
  146. what would Ale have said?
    TA: I would not have
  147. fought.
    Professor Ben Polak:
  148. Would not have fought,
    aha!
  149. Okay, so what happened here?
    When we analyzed this last time
  150. as an individual market,
    we argued that each entrant
  151. should come in just as our first
    entrant came in,
  152. and our monopolist should not
    fight:.
  153. That's what we have up on the
    board.
  154. That's what backward induction
    suggests.
  155. But in fact, Ale fought.
    A whole bunch of people came
  156. in, and a whole bunch of them
    stayed out, is that right?
  157. Now why?
    Why was Ale fighting these guys
  158. and why were they staying out?
    Let's talk about why,
  159. and what market were you again?
    Student: Madison,
  160. Wisconsin.
    Professor Ben Polak: So
  161. why did Madison,
    Wisconsin stay out?
  162. Student: Well we talked
    about it last time how he has an
  163. incentive to fight now because
    there's more that just our
  164. analysis up there in terms of
    establishing that he'll fight to
  165. keep people out.
    Professor Ben Polak: All
  166. right, so it looks like there
    might be some reason for
  167. fighting to keep you out.
    So let's just talk about it a
  168. little bit more,
    let's go to the third guy.
  169. Which market are you again?
    Student: Bridgeport.
  170. Professor Ben Polak:
    Bridgeport, so why did you stay
  171. out?
    Student: Because I knew
  172. he was going to fight.
    Professor Ben Polak: You
  173. knew he was going to fight.
    Now how did you know he was
  174. going to fight?
    Student: Because he has
  175. an incentive to establish,
    he established that he was
  176. going to fight for every single
    market and so I was going to
  177. lose out.
    Professor Ben Polak: All
  178. right, so we know--we think we
    know that Ale is--you know he's
  179. this tough Italian pizzeria
    owner and we think he's going to
  180. try and establish what:
    a reputation as being a tough
  181. pizzeria owner by fighting these
    guys,
  182. perhaps fighting a few guys
    early on in order to keep these
  183. guys out.
    In fact, he had to fight the
  184. first person but he kept out
    2,3, 4,5, 6 and this person came
  185. in, so 7 and 8 came in,
    but then 9 and 10 he kept out.
  186. So he kept a lot of people out
    of the market by fighting early
  187. on.
    Now this argument sounds right:
  188. it seems to ring true.
    It's about establishing
  189. reputation, but now I want to
    show you that there's a worry
  190. with this argument.
    The worry is this is a
  191. sequential game and like all
    sequential games of perfect
  192. information we've seen in the
    class, we should analyze this
  193. game how?
    Now that wasn't loud enough,
  194. how?
    Backward induction.
  195. So where's the back?
    Where's the back of this game?
  196. Way back here--sorry for the
    guys in the balcony.
  197. Way back here we have the last
    market in town--which was the
  198. last market?
    And if we look at this last
  199. market, we in fact saw that if
    the last market came in,
  200. Ale in fact gave in.
    Ale gave in.
  201. Now why did Ale give in,
    in the last market?
  202. Let's have a look back on the
    board.
  203. So on the board we can see what
    that last market looks like.
  204. With ten markets this is a very
    complicated game.
  205. This would be the first market,
    and then there's three versions
  206. of the second market depending
    on what Ale did in the first
  207. market,
    and so there's nine versions of
  208. the third one.
    The tree for this game is
  209. horrendous.
    But nevertheless,
  210. once we get to the end of the
    game, the tenth market--which
  211. was what?
    Bridgeport or something,
  212. I've forgotten where it was at
    now--wherever it was.
  213. Once we get to that last market
    this tree pretty well describes
  214. that last market--is that
    correct?
  215. There isn't another market
    afterwards.
  216. There's only ten markets.
    So, in this last market,
  217. what do we know Ale's going to
    do?
  218. In this last market if the
    entrant enters Ale is going to
  219. not fight, which is what exactly
    what Ale did do.
  220. So Ale is that right?
    So when in fact we discussed
  221. the tenth guy coming in,
    you chose to?
  222. TA: I would have chosen
    not to fight.
  223. Professor Ben Polak:
    Would have chosen not to fight.
  224. That's exactly what the model
    predicts.
  225. He has no incentive to
    establish a reputation for the
  226. eleventh market because there
    isn't an eleventh market.
  227. He's done at ten.
    Is that right?
  228. So we know that in the last
    market, the tenth market,
  229. Ale actually is not going to
    fight.
  230. Therefore, the person who's the
    entrant in the tenth market
  231. should know that they can safely
    enter and Ale won't fight them.
  232. But now we're in trouble.
    Why are we in trouble?
  233. Well let's go back to the ninth
    market, the second to last
  234. market.
    So I've forgotten where it was.
  235. Raise your hand,
    the second to the last market.
  236. Okay, the second to last market
    is this guy?
  237. You're the tenth market?
    So this guy who is in the Hong
  238. Kong market, he should know he's
    the second to last market.
  239. He knows that no matter what he
    does the tenth market's going to
  240. enter and Ale's going to give in
    to the tenth market.
  241. Ale's going to let the tenth
    entrant in.
  242. Is that right?
    So the ninth market actually
  243. knows that nothing Ale's going
    to do here is going to establish
  244. a reputation to keep the tenth
    guy out.
  245. So therefore in fact,
    he should what?
  246. He should come in, right?
    He should come in,
  247. and in fact if he had come in,
    Ale would have had to give in
  248. because there's no way that Ale
    can keep the tenth guy out.
  249. We just argued the tenth guy's
    coming in by backward induction.
  250. So since we know that the tenth
    guy's coming in anyway,
  251. and in fact,
    Ale's going to concede to them,
  252. there's no point Ale trying to
    scare off the tenth guy.
  253. So in fact, Ale's going to say
    no fight to the ninth guy.
  254. But now we go to the eighth guy.
    We've just argued that the
  255. tenth guy's coming in anyway and
    Ale's going to give in to him.
  256. We've argued the ninth guy's
    coming in, so Ale's going to
  257. give in to this guy as well
    because you can't put off the
  258. tenth guy.
    And therefore we know that once
  259. we get to the eighth guy,
    once again, he can safely come
  260. in because Ale knows by backward
    induction he can't keep the
  261. ninth and the tenth guy out
    anyway,
  262. and so this guy should come in
    as well, and if we do this
  263. argument all the way back,
    what do we get?
  264. He lets everybody in.
    Everybody should come in and he
  265. should let everybody in.
    So we have a problem here.
  266. We have a problem.
    Backward induction says,
  267. even with these ten markets,
    Ale in fact should let
  268. everybody in.
    Everyone should know that,
  269. so they should come in.
    So there's a disconnect here.
  270. There's a disconnect between
    what the theory is telling
  271. us--backward induction is
    telling us Ale can't keep people
  272. out by threatening to fight,
    by establishing a reputation
  273. --, and what we actually just
    saw, what happened,
  274. which was Ale did fight and did
    keep people out,
  275. and we know that other
    monopolist's do that as well.
  276. So how can we make rigorous
    this idea of reputation?
  277. It's not captured by what we've
    done so far in the class.
  278. So how can we bring back what
    must be true in some sense--it's
  279. intuition that,
    by fighting,
  280. Ale could keep people out and
    therefore will keep people out.
  281. So to make that idea work I
    want to introduce a new idea.
  282. And the new idea is that,
    with very small probability,
  283. let's say 1% chance,
    Ale is crazy.
  284. So stand up a second,
    so he looks like a normal kind
  285. of guy but there's just 1%
    chance that he's really bonkers.
  286. There's a 1% chance that he's
    actually Rahul.
  287. So now let's redo the
    analysis--And what do I mean by
  288. bonkers?
    By bonkers, I mean,
  289. with 1%, Ale is the kind of guy
    who likes to fight.
  290. So with 1% chance,
    he's actually not got these
  291. payoffs at all;
    he's actually got some
  292. different payoffs,
    which are payoffs of somebody
  293. who--okay he'll lose money--but
    he so much enjoys a fight he
  294. gets +10 here.
    That's the bonkers guy's payoff.
  295. But there's only 1% chance he's
    this bonkers guy.
  296. So now what happens?
    Let's just walk it through.
  297. With 1% chance,
    if there was only one market,
  298. not the ten markets.
    So there's only one market and
  299. this one market was--I've
    forgotten your name?
  300. Student: Isabella.
    Professor Ben Polak:
  301. Isabella, who was in which
    market, I've forgotten.
  302. Student: Miami.
    Professor Ben Polak: In
  303. Miami, then she doesn't really
    much care about the 1% chance
  304. that Ale is actually Rahul.
    That doesn't really bother her
  305. very much, why?
    Because with 99% chance Ale's
  306. going to give way and that's
    good enough odds.
  307. With 99% chance she's happy to
    come in.
  308. So if there was only one market
    here, we'd be done.
  309. But with ten markets things are
    a little different.
  310. Why? Let's see why.
    So suppose in fact that
  311. Isabella in Miami thinks that
    Ale--and everybody else thinks
  312. Ale is a pretty sane guy.
    With 99% probability he's a
  313. sane guy, and Isabella enters
    and everyone sees this.
  314. And to everyone's surprise,
    rather than doing the sane
  315. thing, which is letting Isabella
    in and switching to a duopoly in
  316. Miami,
    what happens,
  317. in fact, after Isabella comes
    in is that Ale fights.
  318. Okay, so now it's too late for
    Isabella, she's lost her money,
  319. but our second market is,
    what's your name again?
  320. Student: Scott.
    Professor Ben Polak:
  321. Scott, which market were you?
    Student: Madison.
  322. Professor Ben Polak: So
    Scott in Madison has observed
  323. what happened in Miami and
    initially he thought that Ale
  324. was Ale.
    99% probability Ale was this
  325. sane, nice, calm,
    Italian guy.
  326. But on the other hand,
    he just saw this sane,
  327. calm, Italian guy fight,
    as he shouldn't have fought
  328. because of backward
    induction--fought the entrant in
  329. Miami.
    So now Scott thinks to himself:
  330. hmm, I'm not so sure that Ale
    is this sane guy.
  331. Maybe I should update my
    beliefs in the direction of
  332. thinking that Ale might actually
    be the insane guy.
  333. So maybe--we're up to maybe a
    probability 1/3 that Ale's
  334. actually insane.
    So he thinks:
  335. okay, probability 1/3 that's
    still not very much,
  336. I'll still risk it,
    he comes in,
  337. and Ale fights him again.
    It's a probability 1/3 he's
  338. sane.
    He's going to give in to me.
  339. He comes in--Ale fights him
    again.
  340. So now we're in the third
    market, which was which market?
  341. Student: Bridgeport.
    Professor Ben Polak:
  342. Bridgeport.
    And Bridgeport's seen this
  343. horrible fight going on in Miami
    and this horrible fight going on
  344. in Madison,
    and now he's getting pretty
  345. sure that this nice,
    calm, looking Ale is not nice,
  346. calm, looking Ale.
    He's crazy Rahul.
  347. There's lot of evidence that
    he's crazy Rahul.
  348. He's fought the last two
    markets making huge losses.
  349. It must be that Ale likes to
    fight.
  350. So what does he do?
    He says, I'm going to stay out
  351. of here.
    I'm convinced that this guy may
  352. be crazy, so I'll stay out.
    And all the subsequent markets,
  353. they think: oh well you know he
    fought these first two markets,
  354. that means he must be a crazy
    guy or at least there's a high
  355. probability that he's a crazy
    guy,
  356. so they all stay out which is
    exactly what happened until we
  357. got to here.
    And even here when they came in
  358. Ale acted liked crazy Rahul.
    So what made that argument
  359. possible was what?
    What made that argument
  360. possible was the small
    possibility, the 1% possibility
  361. that Ale is bonkers.
    But you know,
  362. how well do you know Ale?
    There's a 1% chance he's
  363. bonkers.
    How many of you think you're
  364. really sure that he's a sane
    guy?
  365. He supports Italian football
    teams, he's got to be pretty
  366. crazy, right?
    So what happened here?
  367. This small possibility that Ale
    is crazy allowed him to build up
  368. a reputation that kept all these
    guys out,
  369. but actually the argument is
    stronger than that.
  370. Let's try and push this
    argument harder.
  371. Suppose, in fact,
    that Ale is not crazy.
  372. Suppose that Ale is the sane
    Ale, the nice,
  373. calm, Ale we all know and love.
    But we've just argued that if
  374. Ale acts as if he's the crazy
    guy then you're going to be
  375. convinced that he is the crazy
    guy,
  376. so by acting crazy he might be
    able to convince you that he is
  377. crazy and therefore keep you
    out.
  378. So we argued before that,
    if there's some chance that
  379. Ale's crazy, by acting crazy
    early on,
  380. he's going to deter these late
    entrants from entering the
  381. market because they think
    they're fighting Rahul and they
  382. don't want to fight Rahul.
    But we said these early guys,
  383. they had probability .99 that
    he was sane;
  384. and .6 that he was sane;
    and maybe even .5 he was sane
  385. here--so they thought of coming
    in.
  386. But now we're arguing that even
    if Ale is sane,
  387. even if he's that sane guy,
    a rational guy,
  388. he should behave as if he's
    crazy in order to keep these
  389. late guys out.
    And these early players knowing
  390. that even the sane version of
    Ale is going to fight them,
  391. should also stay out.
    Now notice something's happened
  392. here.
    They're not staying out because
  393. they think Ale's crazy,
    they're staying out because
  394. they know that even the sane
    version of Ale is going to fight
  395. them in order to seem crazy,
    is that right?
  396. Everyone see that's a stronger
    argument?
  397. So now even these early guys
    are going to stay out of the
  398. market.
    Now we're almost there.
  399. What we've argued--let's just
    make sure we get the two pieces
  400. of this on the board.
    We've argued that if there's an
  401. epsilon chance,
    a very small chance,
  402. let's call it a 1% chance where
    Ale is crazy,
  403. then he can deter entry by
    fighting, i.e.,
  404. seeming crazy.
    We argued that what really
  405. makes this argument strong is
    once we realize that the sane
  406. person's going to act crazy,
    we really know that everyone's
  407. going to act crazy and therefore
    we should stay out.
  408. Now that argument won't quite
    be right.
  409. So that's enough of the
    argument I want you to have for
  410. the purpose of the exam,
    but let me just point out that
  411. that argument isn't quite
    correct.
  412. That can't quite be an
    equilibrium.
  413. Now why can't that be an
    equilibrium?
  414. We've just argued that even the
    sane version of Ale--so this is
  415. a sort of slightly more subtle
    argument so just pay attention a
  416. second.
    We've argued that even the sane
  417. version of Ale is going to act
    like a crazy guy.
  418. So if anyone came in,
    he's going to act crazy anyway.
  419. So you're not going to update
    your belief as to whether he's
  420. crazy or sane because we know
    that the crazy guy is going to
  421. fight because he likes fighting
    and the sane guy is going to
  422. fight because he wants to seem
    like a crazy guy.
  423. So you're really learning
    nothing whether your observe him
  424. fighting or now.
    But now let's go back to our
  425. tenth market,
    way back in our tenth market.
  426. Our tenth market participant,
    whose name was Andy,
  427. hasn't learned anything about
    Ale.
  428. He hasn't learned anything
    because whether Ale was sane or
  429. crazy he's going to fight.
    So observing what his actions
  430. early on, if that was really an
    equilibrium, our tenth guy
  431. wouldn't have updated his belief
    at all,
  432. and therefore,
    would still believe with
  433. probability .99 that Ale was
    sane, in which case our tenth
  434. guy would enter.
    Once again, that argument would
  435. unravel from the back.
    So what I described before
  436. can't quite be an equilibrium.
    It can't be just as simple as
  437. all sane guys are going to act
    crazy because then you wouldn't
  438. learn anything.
    So it turns out that the
  439. equilibrium in this model is
    actually very subtle,
  440. and it involves mixed
    strategies,
  441. and mixed strategies are
    something we did in the first
  442. half of the semester,
    so we don't want to go back to
  443. it now.
    So trust me,
  444. you can solve this out with
    mixed strategies and the basic
  445. idea I gave you is right.
    The basic idea is sane guys are
  446. occasionally going to act like
    crazy guys in order to establish
  447. a reputation,
    and that reputation helps them
  448. down the tree.
    So this idea that even when
  449. there's a chain store,
    people will enter--even when
  450. Ale has ten monopolies,
    people will enter--this is a
  451. famous idea.
    It's called the Chain Store
  452. Paradox, and it's due to a guy
    called Selten who actually won
  453. the Nobel Prize.
    This is the Chain Store Paradox
  454. and this idea of establishing
    reputation is a big idea.
  455. The idea is once again you
    might want to behave as if
  456. you're someone else in order to
    deter people's actions,
  457. in order to affect people's
    actions down the tree.
  458. Okay, so what have we learned
    here?
  459. Let's just work it out.
    So the first thing we learned
  460. was kind of a nerdy point,
    but let me make it anyway.
  461. Introducing just a very,
    very small probability,
  462. just a tiny probability that
    Ale might be someone else--he
  463. might be a Rahul,
    he might be crazy,
  464. he must like fights--that very
    small probability radically
  465. changes the outcome of the game.
    If we were all 100% sure he was
  466. sane we'd be tied by backward
    induction and entry would
  467. follow.
    He wouldn't be able to keep
  468. anybody out.
    But that small probability
  469. allows us to get a very
    different outcome.
  470. That's the first point I want
    to draw into this.
  471. The second point I want to get
    out of this, is really this idea
  472. of reputation.
    There are lots of settings in
  473. society where reputation matters
    and one of them is a reputation
  474. to fight.
    How many of you have friends
  475. who have somewhat short fuses?
    You know people who have short
  476. fuses, right?
    When you're going out,
  477. choosing some movie to go to
    with these guys who have short
  478. fuses or trying to decide who's
    going to order something at a
  479. restaurant,
    or who's going to get to be
  480. which side when you're playing
    some game, some video game.
  481. I claim, is this true,
    that the people who have
  482. slightly short fuses,
    slightly more often get their
  483. way, is that right?
    If you have a sibling who has a
  484. short fuse, they slightly more
    often get their way and that's
  485. exactly this idea.
    They have short fuses,
  486. the fact that they tend to blow
    up and get angry at you gives
  487. them a little bit of an
    advantage.
  488. And notice that maybe they
    don't have a short fuse at all,
  489. maybe they're just pretending
    to have a short fuse because
  490. they know they're going to get
    their way over you softies more
  491. often.
    None of you have short fuses,
  492. you're all sane people right?
    So this idea,
  493. it should be a familiar idea to
    you, but it's not just an idea
  494. in the sort of trivial world of
    bargaining.
  495. Notice this idea of reputation
    occurs all over the place.
  496. So another place it occurs,
    somewhat grim place it occurs,
  497. is in the subject of hostage
    negotiations.
  498. In the subject of hostage
    negotiations,
  499. when some other country has
    seized some hostages from the
  500. U.S.
    or maybe some criminal
  501. organization has seized some
    members of your family or some
  502. members of your community and is
    holding the hostages,
  503. there's a well known idea which
    is what?
  504. Which is that you should never
    negotiate with hostage takers,
  505. is that right?
    Everyone's heard that idea?
  506. You should never negotiate with
    hostage takers.
  507. You never give in just because
    they have hostages.
  508. Why?
    It's the same idea:
  509. because you want to have a
    reputation for being somebody
  510. who doesn't give in to hostage
    takers in order to deter future
  511. potential hostage takers from
    taking hostages.
  512. This has grim consequences but
    sometimes it's worth bearing the
  513. cost of having your relatives
    come back in pieces in order to
  514. deter future relatives from
    being taken.
  515. So that's a somewhat macabre
    version of this.
  516. Let me give you one other
    version.
  517. There are whole areas of the
    economy where reputation is
  518. crucial, where if people played
    according to their
  519. backward-induction,
    sane incentives,
  520. we'd have a disaster.
    But having a reputation here
  521. isn't necessarily a reputation
    as a tough guy.
  522. It could be a reputation for
    somebody who's a nice guy.
  523. Could be that you want to have
    a reputation for being the sort
  524. of person who derives pleasure
    or utility from,
  525. (quote) "doing the right
    thing," from acting honestly.
  526. So think about certain
    professions where the reputation
  527. of the person in the profession
    is crucial.
  528. Doctors, for example:
    it's crucial for a doctor that
  529. he or she has the reputation of
    someone who tells the truth.
  530. Otherwise, you'd stop going to
    that doctor.
  531. Accountants:
    accounting firms rely on having
  532. a reputation for being honest
    and not cheating the books.
  533. When they stop having that
    reputation for being
  534. honest--think of Arthur Anderson
    after the events in Enron--they
  535. pretty quickly cease to be in
    business.
  536. I gave that example a couple of
    years ago, and it was very
  537. embarrassing because it turned
    out Arthur Anderson was in the
  538. class--literally Arthur Anderson
    III was in the class.
  539. These things happen at Yale.
    But nevertheless,
  540. Arthur Anderson relied on his
    reputation, the firm relied on
  541. its reputation as an honest
    firm,
  542. and it was worth behaving
    honestly to maintain that
  543. reputation for future business.
    Reputation is a huge topic and
  544. my guess is that the next time
    there's a Nobel Prize in game
  545. theory it'll be for this idea.
    So that's my prediction.
  546. Now having said that,
    I want to spend the whole of
  547. the rest of today playing one
    game and analyzing one game.
  548. So we're going to play this
    game and for this game I need a
  549. couple of volunteers.
    So I'm going to pull out some
  550. volunteers.
    Anyone want to volunteer?
  551. I need two volunteers for this
    game.
  552. How about my guy from the
    football team,
  553. was that a raised hand?
    It wasn't a raised hand,
  554. how about my guy from the
    football team?
  555. Is it football team?
    Baseball team,
  556. that may be unfair in this
    particular game.
  557. Maybe I'll take someone who
    isn't on the baseball team.
  558. Anyone who's on the football
    team?
  559. These guys, you guys on the
    football team?
  560. Okay great, so you two guys.
    I need you at the front and
  561. your names are?
    Chevy and Patrick.
  562. So Chevy and Patrick are going
    to be our volunteers.
  563. Now the idea of this game is
    you guys provided the
  564. volunteers--wait down here a
    second--you guys provided the
  565. volunteers.
    This game involves two
  566. volunteers that you just
    provided, and two wet sponges.
  567. I will provide the wet sponges.
  568. So I have here a couple of
    sponges and in a minute I'm
  569. going to wet them,
    and I'll tell you what the
  570. rules are in a second.
    Okay, so I'm going to give one
  571. of these sponges each to Chevy
    and Patrick, and then going to
  572. position Chevy and Patrick at
    either end of this central
  573. aisle,
    of this aisle here,
  574. and the game is going to be as
    follows.
  575. It's important that everyone
    listen to the rules here because
  576. I'm going to pick two more
    volunteers in a moment.
  577. So the game they're going to
    play is as follows.
  578. Each of them has one sponge.
    It's crucial they only have one
  579. sponge.
    And they're going to take turns.
  580. And when it's your turn,
    as long as you still have your
  581. sponge in your hand,
    you face a choice.
  582. You can either throw your
    sponge at your opponent,
  583. and if you hit your opponent
    you win the game,
  584. or you have to take a step
    forward.
  585. So either you throw the sponge
    or you take a step forward.
  586. Now there's a crucial rule here.
    Each player only has one sponge
  587. and, once they've thrown that
    sponge, they do not get the
  588. sponge back.
    Everyone understand that?
  589. Once you've thrown the sponge
    you do not get the sponge back.
  590. So once again,
    if you throw your sponge at
  591. your opponent and you hit your
    opponent then you've won the
  592. game.
    But if you throw your sponge at
  593. your opponent and you miss,
    the game continues.
  594. So let's make sure we
    understand that,
  595. if you throw your sponge and
    miss the game continues:.
  596. You still have to step forward.
    So, what's your opponent going
  597. to do at that point?
    What's your opponent going to
  598. do?
    Let's make sure that our
  599. football players understand
    this.
  600. It wasn't meant that way.
    They could have been soccer
  601. players, come on.
    Student: I didn't
  602. appreciate that very much
    Professor Ben Polak:
  603. I'm sorry I didn't mean it
    that way.
  604. So if your opponent whose name
    is Patrick throws and misses,
  605. what are you going to do?
    Student: I'll walk
  606. forward until I slam the sponge
    in his face.
  607. Professor Ben Polak:
    Great, you will walk forward
  608. until you politely put it on his
    head.
  609. Everyone understand?
    That, if in fact,
  610. you throw and miss you've lost
    the game, because the other guy
  611. can wait until he's standing
    right on top of you and just
  612. place the sponge gently on his
    head.
  613. Now for fairness sake,
    it's important that these
  614. sponges are equally weighted,
    and I'm going to weight
  615. them--I'm going to put water in
    them now.
  616. And--you know nothing but the
    best for Yale students--I'm
  617. going to provide Yale University
    spring water.
  618. Who knew that Yale University
    had a spring.
  619. That's kind of a strange one?
    If it makes you feel better you
  620. can think of this American beer.
  621. I'm not going to make these too
    heavy, partly because it makes
  622. it too easy and partly because I
    don't want to get sued.
  623. So I'm going to squeeze these
    out somewhere away from the
  624. wires.
    We're going to get our judge
  625. here to weigh them,
    I need a mike here,
  626. let me get a mike.
    So I'll need you to hold those
  627. sponges in your hands and tell
    me if they're equally weighted.
  628. Pretty equal?
    Okay, they're pretty equal,
  629. everyone agrees.
    So how is this going to work?
  630. I'm going to give the blue
    sponge to Chevy and the green
  631. sponge to Patrick.
    And Chevy's going to stand
  632. here, and Patrick's going to
    stand as far back as I can get
  633. him on camera,
    which I'm going to be told how
  634. far back I can go.
    Don't go too far.
  635. Okay come back Patrick,
    you're too ambitious.
  636. Come back.
    Keep coming. stop.
  637. Okay, we're going to start
    here--start quite close
  638. actually.
    Everyone understand how we're
  639. going to play this?
    So Chevy is Player 1,
  640. Patrick is Player 2.
    Chevy has to decide whether to
  641. throw or to step.
    Student: I'll step.
  642. Professor Ben Polak:
    Okay, let's just hold the game a
  643. second.
    Now its Patrick's turn.
  644. Does anyone have any advice for
    Patrick at this point?
  645. If you think throw,
    raise your hand.
  646. If you think step,
    raise your hand.
  647. There's a lot more steps than
    throws.
  648. I thought the Yale football
    team was good this year.
  649. Your choice, step or throw.
    I should announce two other
  650. rules.
    It's kind of important I should
  651. have said this.
    First, a step has to be a
  652. proper step, like a yard long;
    and second (I think this will
  653. work in America):
    gentleman never duck.
  654. No dodging the sponge okay.
    Chevy: your turn.
  655. Student: I don't really
    trust my arm.
  656. I'm going to step.
    Professor Ben Polak: All
  657. right, so you're stepping again.
    Let me go to Patrick.
  658. I feel like I'm in the line of
    fire here.
  659. Patrick what are you going to
    do?
  660. Student: I'm going to
    throw.
  661. Professor Ben Polak:
    Patrick's going to throw,
  662. I'm really going to get out of
    the way then.
  663. We'll see this in slow motion.
  664. Continue, all right so you have
    to take a step forward,
  665. so Chevy's going to take a step
    forward I assume?
  666. Patrick's going to take a step
    forward.
  667. Chevy's going to take a step
    forward.
  668. Patrick's going to take a step
    forward.
  669. Good, so a round of applause
    for our players,
  670. thank you.
    So I think we have time to do
  671. this once more,
    and then we're going to analyze
  672. it.
    So I want to get two women
  673. involved.
    It's too sexist otherwise.
  674. So can we have two women in the
    class please?
  675. Two volunteers,
    come on, you can volunteer.
  676. There's a volunteer.
    Thank you great.
  677. Okay great, thank you.
    So your name is?
  678. Student: Jessica.
    Professor Ben Polak:
  679. Jessica and your name is?
    Student: Clara-Elise.
  680. Professor Ben Polak:
    Clara-Elise and Jessica.
  681. We'll start at the same
    positions, we'll use the same
  682. sponges, and I just need to
    remind you where that position
  683. was.
    Just give me a thumbs up when
  684. I'm in the right position.
    Good, same rules,
  685. Clara-Elise and Jessica.
    We'll let Jessica be Player 1.
  686. So Jessica you can step or
    throw, what do you want to do?
  687. Student: I'm going to
    step.
  688. Professor Ben Polak:
    Going to step,
  689. okay.
    You know what might be a good
  690. idea: Ale why don't you put the
    mike on Clara-Elise.
  691. So we have a mike at either end
    rather than running to and fro,
  692. that's good.
    So Clara-Elise what are you
  693. going to do?
    Student: I'm going to
  694. step.
    Professor Ben Polak:
  695. You're going to step and Jessica
    what are you going to do?
  696. Student: I'm going to
    step.
  697. Professor Ben Polak:
    You're going to step.
  698. Ale and I are in danger here
    but never mind.
  699. Any votes now?
    Do people think that Jessica
  700. should throw?
    If you think she should throw
  701. raise your hands.
    There's a large majority for
  702. step.
    So up to you,
  703. what are you going to do?
    Student: I'm going to
  704. step.
    Professor Ben Polak:
  705. Going to step okay.
    Clara-Elise any decisions?
  706. It's a pretty light sponge,
    It's pretty hard to throw the
  707. sponge, because we've seen that.
    Okay, stepping again.
  708. Student: I'm going to
    throw.
  709. Professor Ben Polak:
    You're going to throw okay,
  710. here we go, let me get out of
    the way.
  711. Oh okay, continue please.
    Clara-Elise's turn.
  712. Student: I'll step.
    Professor Ben Polak:
  713. You'll step.
    Jessica has to step,
  714. Clara-Elise has to step,
    all right good.
  715. So we've seen how the game
    works, everyone understands how
  716. the game works.
    I want to spend the rest of
  717. today analyzing this game.
    Before I do so,
  718. we should just talk about what
    this game is.
  719. Let me get some new boards down
    here.
  720. So one quick announcement:
    I'm going to analyze this,
  721. and we're going to spend the
    rest of today analyzing this,
  722. but this is going to be quite
    hard.
  723. So I'm going to provide you a
    handout that I'll put on the web
  724. probably tomorrow that goes over
    this argument.
  725. So you don't have to take
    detailed notes now,
  726. I want you to pay attention and
    see if you can follow the
  727. argument.
    So this game is called duel,
  728. not surprisingly,
    and you may wonder what are we
  729. doing--as are my colleagues that
    are here--what are we doing
  730. having a duel in class.
    Of course one answer to that
  731. is, it's kind of fun watching
    the future leaders of America
  732. throw wet sponges at each other.
    That's probably reason in
  733. itself.
    But there are other reasons.
  734. Duel is a real game.
    So those of you who are well
  735. versed in Russian literature
    will have seen duals before,
  736. or at least read of duals.
    There are some famous duels in
  737. Russian literature.
    Anyone want to tell me some
  738. famous duels in Russian
    literature?
  739. Any Russian majors here?
    No, nobody want to give me a
  740. shot at this?
    Really nobody.
  741. This is Yale, come on.
    Well, how about in War and
  742. Peace okay.
    There's a duel like this in War
  743. and Peace, and in War and Peace
    and without giving away the
  744. ending--actually it's in the
    middle of the book and it's 800
  745. pages long,
    so it isn't exactly the ending.
  746. But in War and Peace I think
    we're led to believe that the
  747. hero, the protagonist Pierre,
    shoots his gun--in War and
  748. Peace it's a gun and not a
    sponge, no surprise--he shoots
  749. his gun too early,
    we're led to believe.
  750. There's a famous one in Eugene
    Onegin, in Pushkin's Eugene
  751. Onegin, and there are lots of
    others actually,
  752. so there's lots in literature.
    There are also settings which
  753. aren't exactly out of
    literature.
  754. So one example would be in a
    bike race.
  755. How many of you ever watch the
    Tour de France?
  756. Everyone know what I mean by
    the Tour de France.
  757. So this is a bike race that
    goes around France.
  758. It takes stages.
    And in the Tour de France,
  759. there's a key decision.
    There's a game within the game,
  760. and the game within the
    game--I'm looking at Jake who's
  761. a real cyclist here--but the
    game within the game is when do
  762. you try to break away from the
    pack,
  763. which is called the peloton.
    And if you break away too early
  764. from the peloton,
    it turns out that you're going
  765. to get reeled in.
    It turns out that over the long
  766. haul the peloton can go much
    faster than you.
  767. So if you break too early
    they're going to catch you up.
  768. On the other hand,
    if you break too late,
  769. then you're going to lose
    because there are going to be
  770. some people in the peloton who
    are just excellent sprinters.
  771. So if you break too late the
    sprinters are going to win the
  772. race.
    So you have to decide when to
  773. break from the peloton.
    This is the second most
  774. important game within a game in
    the Tour de France,
  775. the most important game within
    a game is where to hide your
  776. steroids.
    So let me give you one other
  777. example.
    Let me give you a more economic
  778. example--it's meant to be an
    economics class.
  779. Imagine there's two firms,
    and these two firms are both
  780. engaged in R&D,
    research and development.
  781. And they're trying to develop a
    new product, and they're going
  782. to launch this new product onto
    the market.
  783. But the nature of this market
    is--maybe it's a network
  784. good--the nature of this market
    is there's only going to be one
  785. successful good out there.
    So essentially there's going to
  786. be one standard,
    let's say of a software or a
  787. technological standard,
    and only one of them is going
  788. to survive.
    The problem is you haven't
  789. perfected your product yet.
    If you launch your product too
  790. early it may not work,
    and then consumers are never
  791. going to trust you again.
    But if you launch it too late
  792. the other side will have
    launched already,
  793. they will have got a toehold in
    the market, and you're toast.
  794. So that game--that game about
    product launch--is like duel,
  795. except you're launching a
    product rather than launching a
  796. sponge.
    Is that right?
  797. Now all of these games have a
    common feature,
  798. and it's a new feature for us.
    It's about the nature of the
  799. strategic decision.
    In most of the games we've
  800. looked at in the course so far
    the strategic decision has been
  801. of the form: where should I
    locate,
  802. how much should I do,
    what price should I set,
  803. should I stand for election or
    not?
  804. Here the strategic decision is
    not of the form what should I
  805. do.
    It's of the form what?
  806. When?
    It's of the form when am
  807. I going to launch the sponge?
    We know perfectly well what
  808. you're going to do.
    You're going to throw the
  809. sponge.
    The strategic issue in question
  810. is when.
    So the issue here is when.
  811. So to analyze this,
    I'm going to need a little bit
  812. of notation, and let me put that
    notation up now.
  813. So in particular,
    I want to use the notation,
  814. Pi[d]
    to be what?
  815. Let Pi[d]
    be Player i's probability of
  816. hitting if i shoots at distance
    d.
  817. So Pi[d]
    is the probability that i will
  818. hit if he or she shoots at
    distance d.
  819. Everyone happy with that?
    That's the only notation I'm
  820. going to use today.
    And I'm going to make some
  821. assumptions about the nature of
    this probability.
  822. Two of the assumptions are
    pretty innocent.
  823. So let's draw a picture.
    So the picture is going to look
  824. like this.
    Here's a graph,
  825. and on the horizontal axis I'm
    going to put d.
  826. This is the distance apart of
    the two players.
  827. And on the vertical axis I'm
    going to put P which is the
  828. probability, Pr the probability.
    So here they're at distance 0,
  829. and I'm going to make an
    assumption about what the
  830. probability of hitting is if
    you're at distance 0.
  831. What's the sensible assumption?
    What's the probability of
  832. hitting somebody with your
    sponge if you're 0 distance
  833. away?
    1, okay, I agree, so 1.
  834. So the first assumption I'm
    going to make is,
  835. if they're right on top of each
    other, they're going to hit with
  836. probability 1.
    Now the second assumption I'm
  837. going to make is:
    as you get further away this
  838. probability decreases.
    It doesn't have to look exactly
  839. like this but something like
    that.
  840. That also I think--Is that an
    okay assumption?
  841. As you're further away there's
    a lower probability of hitting.
  842. Now I'm not going to assume
    that these two players have
  843. equal abilities.
    For example, I don't know;
  844. I didn't ask them but one of
    our two football players might
  845. be a quarterback and the other
    one might be a linebacker or a
  846. running back.
    And I'm assuming the
  847. quarterback is probably more
    accurate, is that right?
  848. So I'm not going to assume that
    they're equally good,
  849. so it could be that their
    abilities look like this.
  850. This could be P1[d],
    and this could be P2[d].
  851. Everyone okay with that?
    So shout this out,
  852. in this picture,
    who is the better shot and who
  853. is the less good shot?
    Who is the better shot?
  854. 1 is the better shot because at
    every distance if Player 1 were
  855. to throw, Player 1's probability
    of hitting is higher than Player
  856. 2's probability of hitting as
    drawn.
  857. Now, I don't even need to
    assume this.
  858. It could well be that these
    probabilities cross.
  859. It could be that these curves
    cross.
  860. So it could be that Player 1 is
    better at close distances,
  861. but Player 2 is better at far
    distances.
  862. That's fine.
    We'll assume it's like this
  863. today but I'm not going to use
    that.
  864. I could do away with that.
    As drawn, Player 1 is the
  865. better shot.
    Now I'm going to make one
  866. assumption that matters,
    and it's really a critical
  867. assumption.
    I'm going to make the
  868. assumption because it keeps the
    math simple for today.
  869. We have enough math to do
    anyway.
  870. I'm going to assume that these
    abilities are known.
  871. I'm going to assume that not
    only do you know your own
  872. ability of hitting your opponent
    at any distance.
  873. I'm going to assume you also
    know the ability of your
  874. opponent to hit you.
  875. Now let's look at this a second.
    We've got a little bit of
  876. notation on the board,
    let's discuss this a second.
  877. What do we think is going to
    happen here?
  878. In this particular example we
    have a good shot and a less good
  879. shot.
    Who do we think is going to
  880. shoot first?
    Let's try and cold call some
  881. people.
    So you sir what's your name?
  882. Student: Frank.
    Professor Ben Polak:
  883. Frank, so who do you think is
    going to shoot first,
  884. the better shot or the worse
    shot?
  885. Student: The better shot
    but also depends on who steps
  886. first.
    Professor Ben Polak:
  887. Okay, let's assume Player 1 is
    going to step first.
  888. Student: Player 1.
    Professor Ben Polak: So
  889. Frank thinks Player 1 is going
    to shoot first because Player 1
  890. is the better shot.
  891. Let's see, so what's your name?
    Student: Nick.
  892. Professor Ben Polak:
    Nick, who do you think is going
  893. to shoot first?
    Student: I think Player
  894. 2 would shoot first.
    Professor Ben Polak: All
  895. right, so let's talk why.
    Why do you think Player 1 was
  896. going to shoot first?
    Let's do a poll.
  897. How many of you think the
    better shot's going to shoot
  898. first?
    How many people think the worse
  899. shot's going to shoot first?
    How many people of you are
  900. being chickens and abstaining?
    Quite a few okay.
  901. So why do we think the better
    shot might shoot first?
  902. Student: At equal
    distance, he has a better chance
  903. of hitting.
    Professor Ben Polak:
  904. Because he has a better chance
    of hitting, but why do you think
  905. that the less good shot might
    shoot first?
  906. Student: He knows that
    if P1 gets too close he's going
  907. to win anyway,
    so he may as well take a shot
  908. with lower chance before P1.
    Professor Ben Polak: All
  909. right, okay so you have two
    arguments here,
  910. the first argument is maybe the
    better shot will shoot first
  911. because,
    after all, he has a higher
  912. chance of hitting.
    And the other argument says
  913. maybe the worse shot will shoot
    first, to what?
  914. To pre-empt the better shot
    from shooting him.
  915. But now we get more
    complicated, because after all,
  916. if you're the better shot and
    you know that the worse shot
  917. maybe going to try and shoot
    first to try and pre-empt you
  918. from shooting him,
    you might be tempted to shoot
  919. before the worse shot shoots to
    preempt the worse shot from
  920. pre-empting you from shooting
    him.
  921. And if you're the worse shot
    maybe you're going to try and
  922. shoot first, even earlier,
    to pre-empt the better shot
  923. from pre-empting the worse shot,
    from pre-empting the better
  924. shot from shooting the worse
    shot and so on.
  925. So what's clear is that this
    game has a lot to do with
  926. pre-emption.
    Pre-emption's a big idea here,
  927. but I claim it's not at all
    obvious who's going to shoot
  928. first, the better shot or the
    worse shot.
  929. Is that right?
    So those people who abstained
  930. raise your hands again.
    Those people who abstained
  931. before, it seemed like it was a
    pretty sensible time to abstain.
  932. It's not obvious at all to me
    who's going to shoot first here.
  933. Are people convinced at least
    that it's a hard problem?
  934. Yes or no, people convinced?
    Yeah okay good.
  935. It's a hard problem.
    So what I want to do is,
  936. as a class, as a group,
    what I want us to do is solve
  937. this game;
    and I want to solve not just
  938. who is going to shoot first,
    I want to figure out exactly
  939. when they're going to shoot.
    So we're going to do this in
  940. the next half hour and we're
    going to do it as a class,
  941. so you're going to do it.
    So we're going to nail this
  942. problem basically,
    and we're going to do it using
  943. two kinds of arguments.
    One kind of argument is an
  944. argument we learned the very
    first day of the class and
  945. that's a dominance argument,
    and the second kind of argument
  946. is an argument we've been using
    a little bit recently,
  947. and what kind of argument is
    that?
  948. What is it?
    Backward induction.
  949. So we're going to use dominance
    arguments and backward
  950. induction, and we're going to
    figure out not just whether the
  951. better shot or the worse shot's
    going to shoot,
  952. but exactly who's going to
    shoot when.
  953. Let's keep our picture handy,
    get rid of, well I can get this
  954. down I guess.
  955. Can you still see the picture?
    All right, let's proceed with
  956. this argument.
    So to do this argument,
  957. I first of all want to
    establish a couple of facts.
  958. I want to establish two facts,
    and we'll call the first fact:
  959. Fact A.
    Let's go back to our two
  960. players we had before.
    In fact, maybe it would be
  961. helpful to have our players.
    Can I use our first two players
  962. as props.
    Can I have you guys on stage?
  963. While they're coming up--sorry
    guys, I'm exploiting you a bit
  964. today.
    I hope you both signed your
  965. legal release forms.
    Why don't you guys sit here a
  966. second so I can use you as
    props.
  967. So imagine that these two guys
    still have their sponges.
  968. Let's actually set this up.
    So suppose that Chevy still has
  969. a sponge, and Patrick still has
    his sponge.
  970. And suppose it's Chevy's turn,
    and suppose that Chevy is
  971. trying to decide whether he
    should throw his sponge or not.
  972. Let me give you a mike each so
    you have them for future
  973. reference.
    So Chevy is trying to decide
  974. whether to throw his sponge or
    not.
  975. Now suppose that Chevy knows
    that Patrick is not going to
  976. shoot next turn when it's his
    turn.
  977. So Chevy's trying to decide
    whether to shoot and he knows
  978. that Patrick is not going to
    shoot next turn when it's his
  979. turn.
    What should Chevy do?
  980. Chevy what should you do?
    Student: Take a step.
  981. Professor Ben Polak:
    Take a step, that's right.
  982. Why should he take a step?
    What's the argument?
  983. Why he should take a step?
    Well let's find out:
  984. what's the argument?
    Student: Because I'll
  985. just be one step closer and I'll
    be able to make the same choice
  986. next time.
    Professor Ben Polak:
  987. Good, hold the mike up to you.
    You're a rock star now.
  988. All right good,
    he's correctly saying he should
  989. wait, why should he wait?
    Because he's going to be closer
  990. next time.
    So the first fact is:
  991. assuming no one has thrown yet,
    if Player i knows (at (say)
  992. distance d) that j will not
    shoot--let me call it tomorrow;
  993. and tomorrow he'll be closer,
    he'll be at distance d -
  994. 1--then Chevy correctly says,
    I should not shoot today.
  995. Again, recall the argument,
    the argument is you'll get a
  996. better shot, a closer shot,
    the day after tomorrow.
  997. Now, let's turn things around.
    Suppose, conversely--once again
  998. we're picking on Chevy a
    second--so Chevy has his sponge.
  999. No one has thrown yet,
    and suppose Chevy knows that
  1000. Patrick is going to throw
    tomorrow.
  1001. Now what should Chevy do?
    Well that's a harder decision.
  1002. What should Chevy do?
    He knows Patrick's going to
  1003. shoot tomorrow.
    What should he do?
  1004. Should he shoot or what,
    what's the answer this time?
  1005. What do you reckon?
    Student: It depends.
  1006. Professor Ben Polak: It
    depends.
  1007. I think that's the right
    answer: it depends.
  1008. Good, so the question
    is--someone else,
  1009. I don't want to pick entirely
    on these guys--so what does it
  1010. depend on?
    It's right that it depends.
  1011. What does it depend on?
    Student: If the other
  1012. guy's chances are greater than
    or less than 50%.
  1013. Professor Ben Polak: All
    right, so it might depend on the
  1014. other side's chances being less
    than or great than 50%.
  1015. It certainly depends on the
    other guy's ability and on my
  1016. ability.
    Everyone clear on that?
  1017. Everyone agrees whether I
    should shoot now if I know the
  1018. other guy is going to shoot
    tomorrow depends on our
  1019. abilities,
    but how exactly does it depend
  1020. on our abilities?
    Student: It depends on:
  1021. if you're probability to hit is
    greater than his probability to
  1022. miss.
    Professor Ben Polak:
  1023. Good, your name is?
    Student: Osmont
  1024. Professor Ben Polak:
    Osmont.
  1025. So Osmont is saying--let's be
    careful here:
  1026. it depends on whether my
    probability of hitting if I
  1027. throw now is bigger than his
    probability of missing
  1028. tomorrow.
    And why is that the right
  1029. comparison?
    That's the right comparison
  1030. because, if I throw now,
    my probability of winning the
  1031. game is the probability that I
    hit my opponent.
  1032. If I wait and take a step,
    then my probability of winning
  1033. the game is the probability that
    he misses me tomorrow.
  1034. So I have to compare winning
    probabilities with winning
  1035. probabilities:
    I have to compare apples with
  1036. apples, not apples with oranges.
    Everyone see that?
  1037. Okay, so let's put that up.
    So the same assumption:
  1038. assuming no one has thrown,
    if i knows (at d) that j
  1039. will shoot tomorrow (at
    d--1),
  1040. then i should shoot if--need a
    gap here--if i's probability of
  1041. hitting at d--and let me leave a
    gap here--is bigger than or
  1042. equal to--it doesn't really
    matter about the equal case--is
  1043. greater than or equal to j's
    probability of missing tomorrow.
  1044. Because this is the probability
    that you'll win if you throw,
  1045. and this is the probability
    that you'll win if you wait.
  1046. Okay, so let's put in what
    those things are.
  1047. So the probability that i will
    hit at distance D,
  1048. that's not that hard,
    that's Pi[d]--everyone happy
  1049. with that?
    What's the probability that j
  1050. will miss tomorrow if j throws?
    What's the probability that j
  1051. will miss?
    Somebody?
  1052. Let's be careful,
    so it's 1--Pj--but what
  1053. distance will they be at--d--1:
    so it's (1--Pj[d--1]).
  1054. So this is the key rule,
    if Chevy knows that Patrick's
  1055. going to shoot tomorrow then
    Chevy should shoot if his
  1056. probability of hitting Pi[d]
    is bigger than Patrick's
  1057. probability of missing (1 -
    Pj[d--1]).
  1058. Now I want to do one piece of
    math.
  1059. This is the only math in this
    proof.
  1060. So everyone who is math phobic,
    which I know there is a lot of
  1061. you, can you just hold onto your
    seats?
  1062. Don't panic:
    a little bit of math coming.
  1063. This is the math,
    I want to add Pj[d--1]
  1064. to both sides of this
    inequality.
  1065. That's it, so what does that
    tell me?
  1066. If add Pj[d -1]
    to this side,
  1067. I get +Pj[d--1],
    everyone happy with that?
  1068. On the other side if I add
    Pj[d--1], I get just 1.
  1069. Everyone happy with that?
    So here's our rule,
  1070. our rule is,
    let's flip it around,
  1071. if Patrick hasn't thrown yet
    and thinks that Chevy's going to
  1072. shoot tomorrow then Patrick
    should shoot now if his
  1073. probability of hitting now plus
    Chevy's probability of hitting
  1074. tomorrow is bigger than 1.
    Let's call this * and let's put
  1075. this stuff up somewhere where we
    can use it for future reference.
  1076. Sorry guys, I'll use you again
    in a minute.
  1077. I know you're feeling self
    conscious up there.
  1078. Believe me, I'm self conscious
    up here too.
  1079. So let's look at that *
    inequality up there.
  1080. Now way out here,
    is that * inequality met or not
  1081. met?
  1082. It's not met because way out
    here these two probabilities are
  1083. small, so the sum is less than
    1.
  1084. In here, is the * inequality
    met or not met?
  1085. Let me pick on you guys,
    so Patrick is it met or not met
  1086. in here?
    Shout into your microphone.
  1087. Student: It's met.
    Professor Ben Polak:
  1088. It's met, thank you okay.
    So, in here,
  1089. the inequality is met:
    the sum is bigger than 1;
  1090. and out here it's less than 1.
    If we put in all the steps
  1091. here--here they are getting
    closer and closer together.
  1092. Here's the steps,
    as they get closer and closer
  1093. together.
    We put these steps in.
  1094. There's going to be some step
    where, for the first time,
  1095. the * inequality is met.
    Notice that they start out
  1096. here, they get closer and
    closer, it's not met,
  1097. not met,
    not met, not met,
  1098. not met, not met,
    then suddenly it's going to
  1099. met.
    Maybe around here.
  1100. Let's just try and pick it out,
    maybe it's here.
  1101. So this might be the first time
    that this * inequality is met,
  1102. let's call it d*.
    Everyone understand what D* is?
  1103. At every one of these steps to
    the right of d*,
  1104. when we take the sum of the
    probability Pi[d]
  1105. + Pj[d--1], we get something
    less than 1.
  1106. But to the left of d* or closer
    in than d*--the game is
  1107. proceeding this way because it's
    moving right to left,
  1108. they're getting closer and
    closer--to the left of d* the
  1109. sum of those probabilities is
    bigger than 1.
  1110. Let's say that again,
    d* is the first step at which
  1111. the sum of those two
    probabilities exceeds 1.
  1112. Everyone okay about what d* is?
    Anyone want to ask me a
  1113. question?
    Okay, people should feel free
  1114. to ask questions on this,
    I want to make sure everyone is
  1115. following.
    Is everyone following so far?
  1116. Yeah?
    I need to see all your eyes.
  1117. You don't look like you're
    stuck in the headlamps like you
  1118. were on Monday.
    I think we're better off than
  1119. we were on Monday.
    Good okay.
  1120. Okay, so here's our picture,
    and I actually want a bit more
  1121. space--I'm not going to have
    much more space.
  1122. So now I'm going to tell you
    the solution.
  1123. The solution to this game is
    this.
  1124. I claim that the first shot
    should occur at d*.
  1125. So that's my claim.
  1126. No one should shoot until you
    get to d*, and whoever's turn it
  1127. is--whether it's Chevy's turn or
    Patrick's turn at d*--that
  1128. person should shoot.
    That's my claim,
  1129. and that's what we're going to
    prove.
  1130. Everyone understand the claim?
    It says: nobody shoots,
  1131. nobody shoots,
    nobody shoots,
  1132. nobody shoots,
    nobody shoots,
  1133. nobody shoots,
    shoot.
  1134. Let's prove it,
    everyone ready to prove it?
  1135. Yeah, people should be awake.
    If your neighbor's not awake,
  1136. nudge them hard.
    Good, all right,
  1137. let's start this analysis way
    out here, miles apart.
  1138. These guys are miles apart and
    I want to use them as props.
  1139. So I've got these guys--stay
    where you are Patrick but stand
  1140. up, and Chevy over here
    somewhere, just where that black
  1141. line is.
    Maybe they're even further than
  1142. this.
    They're really far apart.
  1143. And here they are miles away,
    and let's say it's Chevy's
  1144. turn.
    He's way out here:
  1145. perhaps the first step of the
    game.
  1146. Imagine it's even further
    because it was even further,
  1147. and let's think through what
    should be going on in Chevy's
  1148. head.
    There are two possible things
  1149. going on.
    He's going to think about what
  1150. Patrick's going to do.
    So this is Chevy's turn.
  1151. Here he is.
    And he should think:
  1152. tomorrow, it's going to be
    Patrick's turn,
  1153. and there's two possibilities.
    One possibility is that Patrick
  1154. is not going to shoot tomorrow.
    And if we think that Patrick is
  1155. not going to shoot tomorrow,
    which fact should Chevy use?
  1156. Should he use Fact A or Fact B?
    Chevy which fact should you
  1157. use?
    Student: Fact A.
  1158. Professor Ben Polak:
    Fact A, okay.
  1159. So, using Fact A,
    he should not shoot.
  1160. Alternatively,
    he could think that Patrick's
  1161. going to shoot tomorrow,
    is that right?
  1162. He could think Patrick's going
    to shoot tomorrow.
  1163. If he thinks Patrick's going to
    shoot tomorrow,
  1164. which fact should he use?
    B.
  1165. He should use Fact B,
    in which case he has to look at
  1166. this inequality up here and say,
    I'll shoot if my probability of
  1167. hitting now plus his probability
    of hitting tomorrow is bigger
  1168. than 1.
    Well let's have a look.
  1169. This is Patrick's probability
    of hitting today and this is
  1170. Chevy's probability
    of--sorry--this is Chevy's
  1171. probability of hitting today,
    and this is Patrick's
  1172. probability of hitting tomorrow.
    And is the sum of them bigger
  1173. than 1 or not?
    Is it bigger than 1 or not?
  1174. It's not bigger than 1.
    So what should Chevy do?
  1175. He should step.
    So he'd step.
  1176. Now it's Patrick's turn,
    and once again,
  1177. imagine this distance is still
    pretty large,
  1178. and there's two things Patrick
    could think.
  1179. Patrick could think that
    Chevy's not going to shoot
  1180. tomorrow.
    So here's Patrick,
  1181. he's looking forward to Chevy
    tomorrow, and he could think
  1182. that Chevy's not going to shoot
    tomorrow.
  1183. If he thinks Chevy's not going
    to shoot tomorrow which fact
  1184. should he choose?
    Student: A.
  1185. Professor Ben Polak:
    Should choose Fact A,
  1186. okay.
    If he's using Fact A,
  1187. he should not shoot.
    Alternatively,
  1188. he could think that Chevy is
    going to shoot tomorrow,
  1189. in which case he uses Fact B,
    and what does he do?
  1190. He adds up his probability,
    Patrick's probability of
  1191. hitting today plus Chevy's
    probability of hitting tomorrow:
  1192. he asks is that sum bigger than
    1,
  1193. and he concludes, no.
    So we have no shot here and no
  1194. shot here, and notice that both
    of those arguments were
  1195. dominance arguments.
    In each case,
  1196. whether Chevy thought that
    Patrick was going to shoot
  1197. tomorrow or not,
    in either case,
  1198. he concluded he should not
    shoot today.
  1199. When it was Patrick's turn,
    whether Patrick thought that
  1200. Chevy was going to shoot
    tomorrow or not,
  1201. in either case,
    he concluded he should not
  1202. shoot today.
    So he takes a step forward.
  1203. This argument continues,
    it'll be Chevy's turn next,
  1204. and once again he'll look at
    these two possibilities.
  1205. If he thinks Patrick's not
    shooting tomorrow,
  1206. he wants to step,
    if he thinks Patrick is going
  1207. to shoot tomorrow,
    he's again going to want to
  1208. step the way it's drawn.
    And, once again,
  1209. we'll conclude step.
    We'll go on doing this
  1210. argument, and everyone see that
    in each case this dominance
  1211. argument will apply.
    It won't matter whether I think
  1212. you should shoot tomorrow or
    not, in either case,
  1213. it'll turn out that I should
    step forward:
  1214. whether Fact A applies or
    whether Fact B applies,
  1215. So we'll go on going forward
    and we'll have:
  1216. no shot, no shot,
    no shot, no shot,
  1217. no shot, no shot,
    and we'll arrive at d*.
  1218. So it turns out that d* is
    going to be Chevy's turn again.
  1219. At d* we try exactly the same
    reasoning.
  1220. At d* he says,
    if I think Patrick is not going
  1221. to shoot tomorrow what should I
    do?
  1222. What should I do if I think
    Patrick's not going to shoot
  1223. tomorrow?
    Student: Not shoot.
  1224. Professor Ben Polak: Not
    shoot.
  1225. But now something different
    occurs.
  1226. Now he says,
    if I think Patrick is going to
  1227. shoot tomorrow,
    then when I look at my
  1228. inequality up there,
    my * inequality,
  1229. and add up my probability of
    hitting today--which is this
  1230. line here--plus Patrick's
    probability of hitting
  1231. tomorrow--which is this line
    here--suddenly he finds it is
  1232. bigger than 1.
    So now if Chevy thinks that
  1233. Patrick is going to shoot
    tomorrow, what should Chevy do?
  1234. He should shoot.
    So up until this point,
  1235. a dominance argument has told
    us no one should shoot,
  1236. but suddenly we have a dilemma.
    The dilemma is:
  1237. if Chevy thinks Patrick's not
    shooting, he should step;
  1238. and if Chevy thinks Patrick is
    shooting, he should shoot.
  1239. Everyone with me so far?
    So what have we shown so far?
  1240. We've shown that no one should
    shoot until d* but we're stuck
  1241. because we don't know what to do
    at d*;
  1242. because we don't know what
    Chevy should believe at d*.
  1243. We don't know whether Chevy
    should believe that Patrick's
  1244. going to shoot or whether Chevy
    should believe that Patrick's
  1245. not going to shoot.
    So how do we figure out what
  1246. Chevy should believe Patrick's
    going to do?
  1247. Wait, wake up the guy in orange
    there, the guy with the ginger
  1248. hair, that's right.
    What's the answer to that
  1249. question?
    Good, the answer to the
  1250. question is backward induction.
    Round of applause for
  1251. remembering the answer.
    Good, backward induction is the
  1252. answer to all questions,
    especially when you're asleep
  1253. right.
    Okay, so now we're going to use
  1254. backward induction,
    but where does backward
  1255. induction start here?
    Backward induction starts at
  1256. the back of the game,
    and what's the back of the game
  1257. here?
    The back of the game is where?
  1258. It's when these two guys,
    neither of them have thrown
  1259. their sponge,
    and they've reached here.
  1260. So come here a second,
    step, step.
  1261. Let's assume it's Patrick's
    turn, and they're absurdly
  1262. close: they're uncomfortably
    close.
  1263. If they had longer noses they'd
    be touching.
  1264. They're at distance 0.
    And at distance 0,
  1265. at d = 0, let's suppose it's
    Patrick's turn.
  1266. So at d = 0,
    no one has shot,
  1267. it's Patrick's turn,
    he's got the sponge,
  1268. what should Patrick do?
    Shout it out Patrick.
  1269. Student: I should shoot.
    Professor Ben Polak: You
  1270. should shoot.
    Patrick should shoot, right?
  1271. At d = 0, say it's 2's turn,
    and the answer is he should
  1272. shoot because the probability of
    it hitting at distance 0 is 1.
  1273. Let's just move you to the side
    a bit so that people can see the
  1274. board.
    I know it's an awkward dance
  1275. but here you are--stop there:
    that's good.
  1276. So at distance 0 they should
    certainly shoot.
  1277. So now let's go back a step in
    the backward induction.
  1278. So we're at distance 1,
    just take a step back.
  1279. So take a step back,
    it's Chevy's turn at distance
  1280. 1.
    And what does Chevy know at
  1281. distance 1?
    Chevy what do you know?
  1282. Shout it out.
    Student: That Patrick
  1283. will shoot next turn.
    Professor Ben Polak:
  1284. Right, so now Chevy knows that
    Patrick's going to shoot
  1285. tomorrow.
    So which fact should Chevy use
  1286. in deciding whether he should
    shoot today?
  1287. B.
    He should use Fact B,
  1288. and that tells us--so 1 knows
    that 2 will shoot tomorrow,
  1289. so by B, 1 should shoot if his
    probability of hitting at
  1290. distance 1 plus Patrick's
    probability of hitting at
  1291. distance 0,
    if that is bigger than 1.
  1292. Well is it bigger than 1?
    Well let's have a look.
  1293. We had a shot in here already,
    we put a shot in here,
  1294. and we're looking at distance
    1.
  1295. And so we're looking at this
    distance here plus this distance
  1296. here.
    Is it bigger than 1?
  1297. Yeah, it's bigger than 1.
    Here's a bit more math.
  1298. I lied to you before.
    1 plus something is bigger than
  1299. 1.
    So this is bigger than 1.
  1300. So shoot.
    Let's put it on our chart as
  1301. shoot.
    Let's go back a
  1302. step--sorry--I'll have to have
    Chevy do all the going
  1303. backwards.
    Now we're at distance what?
  1304. We're at distance 2,
    we're at d = 2,
  1305. and it's Patrick's turn,
    2's turn, and what does Patrick
  1306. know?
    Shout it out Patrick.
  1307. Student: I know that
    Chevy's going to shoot next
  1308. turn.
    Professor Ben Polak:
  1309. Patrick knows that Chevy's going
    to shoot next turn,
  1310. so Patrick therefore should use
    which fact?
  1311. Fact B.
    And Fact B tells him that he
  1312. should shoot--let's just put
    this in.
  1313. So 2 knows that 1 will shoot
    tomorrow, so by B it's all the
  1314. same thing.
    We know that 2 should shoot if
  1315. P2[2]
    + P1[1]
  1316. is bigger than 1,
    and if we look at it on the
  1317. board--here we are--it's 2's
    turn,
  1318. he's looking at this distance
    plus this distance,
  1319. and is it bigger than 1?
    It is bigger than 1.
  1320. So he should shoot.
    And we can go on doing this
  1321. argument backwards.,
    We'll find that Chevy should
  1322. shoot here because this plus
    this is bigger than 1.
  1323. And we'll know that here
    Patrick once again will know
  1324. that Chevy's going to shoot
    tomorrow,
  1325. so he should use Fact B,
    so he should shoot provided
  1326. this plus this is bigger than 1,
    but it is.
  1327. Now, we're back at d* and the
    question we had at d*-- the
  1328. question we'd left hanging at
    d*--was what?
  1329. At d* we knew already that
    Chevy would not shoot if he
  1330. thought Patrick was not going to
    shoot tomorrow,
  1331. but he should shoot if thinks
    Patrick is going to shoot
  1332. tomorrow, but what does Chevy
    know at d*?
  1333. Student: I know Patrick
    is going to shoot.
  1334. Professor Ben Polak: He
    knows Patrick's going to shoot,
  1335. so he should shoot.
    Is that right?
  1336. He knows Patrick's going to
    shoot by backward induction so
  1337. he should shoot.
    So we just solved this.
  1338. What did we actually show?
    We showed, have seats
  1339. gentlemen--sorry to keep you up
    here--we know that prior to d*
  1340. no one will shoot--will not
    shoot--and we know that at d*,
  1341. and in fact at any point
    further on, we should shoot.
  1342. That's horrible writing but it
    says shoot.
  1343. So we've shown more than we
    claimed.
  1344. We claimed that the first shot
    should occur at d*,
  1345. and we've actually shown more
    than that: we've shown that even
  1346. if you went beyond d*,
    and if somebody had forgotten
  1347. to shoot at d*,
    at least you should shoot now.
  1348. Give me like two more minutes
    or three more minutes to finish
  1349. this up because we're at a high
    point now.
  1350. Everyone okay to wait a couple
    of minutes?
  1351. Okay, so what did we prove here?
    We proved that the first shot
  1352. occurs at d* whoever's turn it
    is at d*.
  1353. It wasn't that the best guy,
    the better shot,
  1354. should shoot first,
    or the worse shot should shoot
  1355. first.
    It turned out that,
  1356. given their abilities,
    there was a critical distance
  1357. at which they should shoot.
    If you go back to the
  1358. eighteenth century military
    strategy, you should shoot when
  1359. you see the whites of their
    eyes, which is at d*.
  1360. But we learned something else
    on the way.
  1361. I claimed we learned that if
    you're patient and you go
  1362. through things carefully,
    that the arguments we've
  1363. learned from the course so far,
    dominance arguments and
  1364. backward induction arguments,
    can solve out a really quite
  1365. hard problem.
    This was hard.
  1366. It would have been useful for
    the guy in War and Peace,
  1367. or in Onegin or the guys
    cycling in the Tour de France,
  1368. or you guys with your sponges
    to know this.
  1369. And we can solve this exactly
    using backward induction,
  1370. and everyone in the room can do
    it.
  1371. Let me just push the argument a
    tiny bit further.
  1372. One thing we've always asked in
    this class, is okay that's fine
  1373. if everyone knows what's going
    on in the game:
  1374. here we have our smart Yale
    football players and they know
  1375. how to play this game,
    so they're going to shoot at
  1376. the right time.
    But what happens if,
  1377. instead of playing another
    smart Yale football player,
  1378. they're playing some uneducated
    probably simple-minded football
  1379. player from,
    say, Harvard.
  1380. Now that changes things a bit
    doesn't it because we know that
  1381. the Yale football player is
    sophisticated,
  1382. has taken my class,
    and knows that he should shoot
  1383. at d*, but the Harvard guy
    doesn't learn anything anymore,
  1384. so they're stuck.
    So if you're the Yale guy
  1385. playing the Harvard guy how does
    that change your decision?
  1386. Should you shoot earlier than
    d* when you're playing against
  1387. the Harvard guy,
    or later than d* when you're
  1388. playing against the Harvard guy?
    Let's try our Yale guys and see
  1389. what they think,
    what do you think Chevy?
  1390. Student: Definitely not
    earlier.
  1391. Professor Ben Polak:
    Definitely not earlier,
  1392. that's the key thing right.
    Now why?
  1393. Why definitely not earlier?
    Student: Because if you
  1394. miss, the other person has a
    probability of 1,
  1395. you have a higher chance of
    missing.
  1396. Professor Ben Polak: All
    right, so I claim Chevy's right.
  1397. That's good because I've just
    claimed that Yale football
  1398. players are sophisticated.
    Chevy's right,
  1399. that even if you're playing
    against a Harvard guy you
  1400. shouldn't shoot before d*
    because it was a dominant
  1401. strategy not to shoot before
    d*.
  1402. It doesn't matter whether you
    think the Harvard guy is going
  1403. to be dumb enough to shoot early
    or not.
  1404. If he is dumb enough to shoot
    early, so much the better.
  1405. You should wait until D*.
    Notice that argument doesn't
  1406. depend on you playing against
    somebody who is sophisticated,
  1407. or someone who's less
    sophisticated,
  1408. like a Harvard football player,
    or somebody who's basically a
  1409. chair,
    like a Harvard football player.
  1410. You shouldn't shoot before d*
    because it's a dominant strategy
  1411. not to shoot before d*.
    Now, you might want to wait a
  1412. little to see if they're not
    going to shoot early,
  1413. to see if he's not going to
    shoot, but you certainly
  1414. shouldn't shoot early.
    Let me finish with one other
  1415. thing.
    Every time when we've played
  1416. this game in class,
    whether it's here or up in SOM,
  1417. people shoot too early.
    They miss.
  1418. You can do the econometrics on
    this, you could figure out that,
  1419. on average--average
    abilities--I'm sometimes getting
  1420. the better shots,
    sometimes I'm getting the worse
  1421. shots--on average I should see
    people hitting about half of the
  1422. time over a large sample.
    But here I tend to see people
  1423. miss, as we did today,
    almost all the time.
  1424. Why do we see so many misses?
    So one problem may be that
  1425. people are just overconfident.
    They're overconfident on their
  1426. ability to throw.
    And there's a large literature
  1427. in Economics about how people
    tend to be overconfident.
  1428. But there's another possible
    explanation, and let me just
  1429. push it past you as the last
    thing for today.
  1430. I think Americans--I think this
    doesn't go for the
  1431. Brits--Americans have what I
    call a "pro-active bias."
  1432. You guys are brought up since
    you're in kindergarten--maybe
  1433. before--and you're told you have
    to be pro-active.
  1434. You have "to make the world
    come to you."
  1435. And my evidence for this is
    based on sophisticated empirical
  1436. work watching Sports Center.
    So on Sports Center when they
  1437. interview these sweaty athletes
    after the game,
  1438. the sweaty athletes say,
    it's great I now control my own
  1439. destiny.
    Well, I'm a Brit.
  1440. I think controlling my own
    destiny sounds kind of scary to
  1441. me;
    it doesn't sound like a good
  1442. thing at all.
    In fact, if I wanted to control
  1443. my own destiny,
    I wouldn't have got married.
  1444. That's going to be edited off
    the film, but the point I want
  1445. to make is this.
    Every time I play this game,
  1446. when I ask people why they
    shoot early I hear the same
  1447. thing, and it's evidence to this
    proactive bias.
  1448. People say, well at least I
    went down swinging and the
  1449. problem is: the aim in life is
    not to go down swinging,
  1450. it's not to go down.
    So one lesson to get from this
  1451. lecture is, sometimes waiting is
    a good strategy.
  1452. Alright, and we'll come back to
    it on Monday.