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← 14. Backward induction: commitment, spies, and first-mover advantages

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Showing Revision 1 created 07/03/2012 by Amara Bot.

  1. Professor Ben Polak: All
    right, so today I want to do
  2. something a little bit more
    mundane than we did on Monday.
  3. I want to go back and talk
    about quantity competition.
  4. So in the first half of the
    course we talked about price
  5. competition.
    We talked about quantity
  6. competition.
    We talked about competition
  7. with differentiated products.
    I want to go back and revisit
  8. essentially the Cournot Model.
    So this was the Cournot Model:
  9. two firms are producing,
    are choosing their quantities
  10. simultaneously.
    Firm 1 is choosing Q1 and Firm
  11. 2 is choosing Q2.
    And all of this is just review
  12. so this is all stuff that's in
    your notes already.
  13. This is the demand curve.
    It tells us that prices depend
  14. on the total quantity being
  15. So this is Q1 + Q2 and this is
    prices, then the demand curve is
  16. a straight line of slope b.
    That's what this tells us.
  17. Here's our slope –b.
    And we know that payoffs are
  18. just profits,
    which are price times quantity,
  19. revenues,
    minus cost times quantity,
  20. costs, we're assuming constant
    marginal costs.
  21. We did this model out in full
    in maybe the third week of class
  22. and we figured out what the best
    response diagram looked like.
  23. And if you remember correctly,
    this was the best response for
  24. Firm 1 taking Firm 2's output as
  25. and this is the best response
    for Firm 2 taking Firm 1's
  26. output as given,
    and there were a few other
  27. details in here.
    This was the monopoly quantity,
  28. this was the competitive
    quantity and so on,
  29. but this is enough for today.
    Actually, we had done a bit
  30. more than that,
    we'd actually worked out in
  31. class what the equations were
    for these best responses.
  32. Here they are.
    I'm not going to re-derive
  33. these today, but they're
    somewhere in your notes.
  34. We kind of crunched through
    some calculus and figured out
  35. what Firm 1's best response
    looks like algebraically,
  36. here it is.
    So this is the equation of this
  37. line, and similarly for Firm 2,
    so this is the equation of this
  38. line.
    Finally, we figured out what
  39. the Nash Equilibrium was,
    and there's no prizes here:
  40. the Nash Equilibrium in Cournot
    was where these best responses
  41. crossed,
    and this is the equation for
  42. the Nash Equilibrium.
    Have I made a mistake?
  43. The best response, oh thank you.
    The best response for Firm 1 is
  44. a function of Q2,
  45. Thanks Jake.
    So this is all stuff we did
  46. before, I want to go back to
    this model now to revisit it in
  47. the context of thinking about
    sequential dynamic games.
  48. So what we're going to do
    is--we're going to do is,
  49. we're going to imagine that
    rather than having these firms
  50. choose their quantities
  51. one firm gets to move first and
    the other firm moves after.
  52. Let's be clear,
    we're going to assume that Firm
  53. 1s moving first and the other
    firm--we'll assume Firm 1's
  54. going to move first--the other
  55. Firm 2, is going to get observe
    what Firm 1 has chosen and then
  56. get to make her choice.
    So we're going to see what
  57. difference it makes when we go
    from this classic simultaneous
  58. move game into a sequential move
  59. This model is fairly famous and
    I'm almost certainly spelling
  60. this wrong, but it's due to a
    guy called Stackelberg.
  61. So what we're looking at now is
    the Stackelberg Model.
  62. So how do we want to think
    about this?
  63. A natural question to bear in
    mind is, assuming we're in this
  64. world of quantity competition,
    is it an advantage to get to
  65. move first, to set one's
    quantity first?
  66. Or is it an advantage to be
    able to wait,
  67. see what the other firm has
    done, and then respond?
  68. Is there an advantage in going
    first or is there an advantage
  69. in knowing a bit more about the
    other firm and being able to
  70. move second?
    That's going to be the question
  71. at the back of our minds for
    most of today.
  72. So how are we going to think of
  73. How are we going to figure this
  74. There shouldn't be a silence in
    the room.
  75. There should be an instance
  76. How are we going to figure this
  77. We're going to use backward
    induction right.
  78. This is going to be an exercise
    in backward induction.
  79. We won't be able to draw a tree
    here because the game's too
  80. complicated because there's a
    continuum of actions,
  81. but nevertheless,
    we are going to use backward
  82. induction.
    So what does using backward
  83. induction mean here?
    Using backward induction means
  84. starting at the end and the end
    is what?
  85. The end here is Firm 2.
    Firm 1 is going to move first,
  86. Firm 2's going to observe that
    choice and then move.
  87. So the end of the game is Firm
  88. So we're going to solve out
    Firm 2's problem first.
  89. We're actually going to do this
    entire analysis twice.
  90. We're first of all going to do
    this analysis a bit intuitively
  91. looking at pictures,
    and then I want to go back and
  92. crunch it out in the math.
    I want to get used to seeing
  93. that we can actually do it
  94. This board is just review,
    so I'm going to get rid of it I
  95. think.
    I didn't manage to get rid of
  96. it, never mind.
    We're not going to be using
  97. this board.
    This is just what we did do in
  98. the simultaneous move game,
    so we'll get rid of it.
  99. So in the sequential move game
    we're going to start by
  100. analyzing the move of Firm 2.
    So imagine yourself as the
  101. manager of Firm 2,
    you're coming along to make
  102. your output decision.
    The output of Firm 1 is already
  103. set.
    So analyze Firm 2 first,
  104. Firm 2 sees Q1 and now I must
    choose Q2.
  105. So what is Firm 2 going to do?
    So I claim we already know this.
  106. We've already solved this
  107. When did we solve this problem?
    Anyone know when we solved this
  108. problem?
    The problem of what Firm 2's
  109. going to do.
    Well we already solved it about
  110. a month ago when we looked at
    the simultaneous move game,
  111. because what we worked out then
    was what is Firm 2's best
  112. response for any particular
    choice that Firm 1 makes?
  113. We already solved out that
  114. It took us a while to solve
    out, but basically it was to
  115. maximize Firm 2's payoff,
    taking as given Firm 1.
  116. We already know what the
    equation looks like and let's
  117. just remind ourselves what the
    picture looked like,
  118. just a repeat of the picture we
    had before.
  119. We said for any particular
    choice of Firm 1,
  120. Firm 2's best response can be
    drawn on a best response
  121. diagram, and looked like this.
    It's exactly the picture we
  122. have up there.
    So this is the best response
  123. for Firm 2 taking as given the
    choice of Firm 1.
  124. We even know the equation of
    it--I won't bother rewriting
  125. that.
    We already know the equation.
  126. So in some sense Firm 2's
    problem is a problem we've
  127. already seen.
    We already worked out months
  128. ago what Firm 2 should do,
    taking if Firm 1's output is
  129. given and that's exactly the
    problem Firm 2 finds herself in.
  130. She wakes up one morning,
    Q1 has been set already,
  131. and now she must choose Q2 to
    maximize her profits,
  132. so she's going to choose her
    best response.
  133. Just to remind you how we read
    this picture,
  134. for any particular choice of Q1
    we go up to the line and look
  135. across, this tells us what Q2
    will respond.
  136. So if Q1 chooses this amount
    then Q2 will choose this amount.
  137. If Q1 chooses this amount then
    Q2 will choose this amount and
  138. so on.
    So there's no mystery here.
  139. We already know what Firm 2's
    going to do.
  140. So by definition,
    the best response of 2 to Q1
  141. tells us the profit maximizing
    output of Firm 2 taking Q1 as
  142. given.
    All right, so we've done the
  143. second step for this already,
    we already know what Firm 2's
  144. going to do.
    Of course, the additional step
  145. here now is that Firm 1 knows
    that Firms 2's going to do it.
  146. Firm 1's going to move first
    and Firm 1 knows that after she
  147. sets her quantity Q1,
    Firm 2 will respond by choosing
  148. her corresponding quantity,
    which is the best response to
  149. it.
    So if Firm 1 knows that if Firm
  150. 1 were to choose this quantity,
    then Firm 2 will respond by
  151. choosing this quantity,
    and Firm 1 knows that if she
  152. chose this smaller quantity,
    then Firm 2 will respond by
  153. choosing this larger quantity.
    Is that right?
  154. So Firm 1 can anticipate how
    Firm 2 is going respond to each
  155. of these choices.
    So let's just make that clear.
  156. So in particular,
    if Firm 1 was to choose Q^1,
  157. I'm not suggesting it should,
    but if Firm 1 was to choose
  158. this Q^1, then Firm 1 knows that
    Firm 2 will produce this
  159. quantity,
    which is the best response to
  160. Q^1, and if Firm 1 were to
    choose Q^^1, then Firm 2 will
  161. respond by choosing the best
    response to Q^^1.
  162. So this is pretty
    straightforward so far,
  163. but what we're able to see now,
    is the problem facing Firm 1,
  164. which is the interesting
  165. The problem facing Firm 1 is,
    what quantity should Firm 1
  166. choose knowing that this is how
    Firm 2 is going to respond?
  167. Before we solve this out
  168. I just want us to think it
    through a little bit.
  169. So the first way I want to
    think this through is,
  170. is to make the following
  171. From Firm 1's point of view,
    Firm 1 knows that any Q1 she
  172. chooses leads to a response on
    this line by Firm 2.
  173. That's what Firm 1 knows.
    So Firm 1 is effectively
  174. choosing points on this line.
    Let me say it again,
  175. so what's actually happening is
    Firm 1 is choosing Q1 and Firm 2
  176. is responding by choosing a Q2
    that puts them on this line.
  177. But in effect that means Firm 1
    is choosing points on this line.
  178. So you could think of Firm 1's
    problem as, choose the joint
  179. output level on this line that
    maximizes Firm 1's profits.
  180. Think of Firm 1's problem as
    choose the combination of
  181. outputs on this line by choosing
    Q1 and then Q2 responds,
  182. choose the combination on this
    line that maximizes profits for
  183. Firm 1.
    So I'm belaboring this a little
  184. bit because it's a more general
    mathematical idea here.
  185. How many of you are in Econ 150
    right now?
  186. So for those of you in Econ
    150, this should be a very
  187. familiar kind of thing.
    This is a constrained
  188. optimization problem and you've
    been having constrained
  189. optimization problems rammed
    into you for the last month or
  190. so,
    so this is an example of a
  191. constrained optimization
  192. You have to choose a point but
    you can't choose a point freely:
  193. you have to choose a point on
    the line.
  194. Okay, so let's talk about it a
    bit more before we do the math.
  195. Let's actually redraw it again
    since I made a mess of this
  196. picture.
  197. So one thing you might want to
    ask is, in making this choice
  198. for Firm 1, should Firm 1 choose
    more or less or the same as it
  199. used to choose when the problem
    was simultaneous?
  200. So let's put in again what it
    used to choose when the problem
  201. was simultaneous.
    I'll put it in just faintly.
  202. So here's our old Cournot
    picture--looked like this--and
  203. this was the quantity that Firm
    1 chose in the Cournot game,
  204. so let me call that Q1C.
    So certainly one possibility is
  205. that Firm 1 could choose her
    Cournot quantity,
  206. she can certainly do that,
    and she knows that if she does
  207. that, Firm 2 will respond by
    choosing the best response of
  208. Firm 2 to Firm 1 choosing the
    Cournot quantity,
  209. but we know what that is.
    What's Firm 2's best response
  210. to Firm 1's Cournot quantity?
    It's Firm 2's Cournot quantity,
  211. right?
    So if Firm 2 does that,
  212. if Firm 1 chooses the Cournot
    quantity, then Firm 2 will also
  213. choose the Cournot quantity.
    So one thing that Firm 1 could
  214. do is effectively choose the old
  215. That's certainly something
    that's available to Firm 1.
  216. But Firm 1 could also do other
  217. Firm 1 could produce less than
    that or Firm 1 could do more
  218. than that.
    So who thinks Firm 1 should
  219. play--should choose the old
    equilibrium quantity?
  220. Who thinks Firm 1 should choose
    more than that?
  221. Who thinks Firm 1 should choose
    less than that?
  222. Let's just try it with the
    camera on you.
  223. So once again,
    how many people think that Firm
  224. 1 should choose the old
    equilibrium quantity?
  225. A dribbling of hands,
    and how about less than that?
  226. A few hands and then they went
    down again, and how about more
  227. than that?
    There's a majority for
  228. more--turns out more is correct,
    so that's good news.
  229. Why?
    Why do we think Firm 1 should
  230. produce more than it used to
    produce before?
  231. Any takers on this?
    Well let's think about it.
  232. As Firm 1 produces more,
    or if Firm 1 were to produce
  233. more, then Firm 2--my voice is
    going--then Firm 2 would produce
  234. what?
  235. As Firm 1 produces more than
    her Cournot quantity,
  236. Firm 2's response is to produce
  237. Does anyone remember the jargon
    for this?
  238. What do we call games where the
    more I do of my strategy the
  239. less you do of yours?
    "Strategic substitutes," good.
  240. This is a game of strategic
  241. What that means is that as Q1
    goes up, Q2, the best response
  242. of Firm 2 to Q1 goes down.
    So what?
  243. We can look at that just by
    looking at the picture.
  244. Well the "so what" is,
    now we're in a sequential game.
  245. If Firm 1 produces more than
    her Cournot quantity she induces
  246. Firm 2 to produce less.
    That's what we just said and
  247. that's what?
    That's good for Firm 1.
  248. My producing more inducing you
    to produce less is good for me.
  249. It's going to keep prices
    higher in the market.
  250. Is that right?
    So let's just think it through
  251. again.
    In the Cournot equilibrium,
  252. the choice of Firm 1 was the
    best choice for Firm 1,
  253. taking the choice of Firm 2 as
  254. That was the old Cournot
  255. But now in the Stackelberg
    setting, the sequential setting,
  256. there's an additional feature.
    Firm 1 doesn't have to take
  257. Firm 2's output as given.
    There's an additional reason
  258. for producing at the margin,
    which is, at the margin if I
  259. produced some more units of
  260. that leads you to produce less
    which is good for me.
  261. So that suggests that I'm going
    to produce than I used to
  262. produce under the old
  263. So this suggests that Firm 1
    should set Q1 bigger than Q1^(C)
  264. to induce Q2 to be less than
  265. So the first thing we've
    learned--we'll see this in the
  266. math later--is that Firm 1 will
    in fact produce more than they
  267. used to under Cournot,
    and that will result in Firm 2
  268. producing less than Cournot.
    Now we've already got a lot on
  269. the board now,
    we can actually solve out
  270. intuitively the problem.
    Do we think that Firm 1's
  271. profits, by this procedure,
    are the same as they were under
  272. Cournot?
    Are they less than they were
  273. under Cournot,
    or are they more than they were
  274. under Cournot?
    So just let me say it again.
  275. Firm 1 is going first now.
    We've argued that Firm 1's
  276. going to produce more.
    Do we think that Firm 1's
  277. profits at the end of the day
    are going to be the same as they
  278. were under Cournot,
    higher than they were under
  279. Cournot, or lower than they were
    under Cournot?
  280. So let's have a poll again,
    let me get the camera on you
  281. guys.
    So who thinks their profits are
  282. going to be the same as they
    were under Cournot?
  283. Who thinks the profits have
    gone up?
  284. Who thinks the profits have
    gone down?
  285. We're in good shape here
    because indeed the profits have
  286. gone up.
    There's a very simple argument
  287. why the profits have to have
    gone up.
  288. How do we know the profits must
    have gone up?
  289. Let me actually--it's simple
    enough--let me grab a mike on
  290. this.
    How do we know the profits just
  291. must have gone up?
    There was a hand in the back,
  292. was there a hand in the back?
    Yes, way at the back.
  293. How do we know profits must
    have gone up here?
  294. Way at the back.
    Student: If Firm 1 was
  295. going to lower their profits
    they wouldn't have chosen to
  296. produce more.
    Professor Ben Polak: All
  297. right, good exactly.
    The fact that Firm 1 has
  298. changed their output,
    and in particular,
  299. are producing more,
    tells you they must be able to
  300. increase their profits by this
  301. Let's just think that through.
    One option that was available
  302. to Firm 1 before was to set
    output at the Cournot level.
  303. If Firm 1 had set output at the
    Cournot level,
  304. that would have led Firm 2 to
    set output at the Cournot level,
  305. and in that case,
    profits would have been exactly
  306. the same as before.
    The fact that Firm 1 has moved
  307. away from that must mean there
    are higher profits available.
  308. Say it another way,
    Firm 1 could have had exactly
  309. what it had before,
    so it must be doing at least as
  310. well as it was doing before,
    and the fact it has changed
  311. means it must be doing better
    than it was doing before.
  312. So indeed, Firm 1's profits
    have gone up.
  313. We don't even need any math to
    prove that: it just must be the
  314. case logically.
    What must have happened to Firm
  315. 2's profits?
    What do you think has happened
  316. to Firm 2's profits?
    That's not so immediately
  317. obvious.
    It's obvious,
  318. I think, that Firm 1's profits
    have gone up here because Firm 1
  319. could have had the same old
    profits and has chosen something
  320. else.
    But it's not immediately
  321. obvious what happened to Firm
    2's profits, is that right?
  322. Before we get to what's
    happened to Firm 2's profits
  323. let's go through an intermediate
  324. Let's try and ask what must
    have happened to total output in
  325. the market in this example--in
    this nice simple example.
  326. That's not immediately obvious
  327. Why?
    We've argued that Firm 1's
  328. output went up but Firm 2's
    output went down relative to
  329. Cournot.
    So it's not immediately obvious
  330. whether the sum of those two Q1
    + Q2 went up or down.
  331. We'd like to know what happened
    to Q1 + Q2, total output,
  332. in the market.
    By the way, one particular
  333. reason we might care about this
    is of course consumers would
  334. like it to have gone up.
    Because if their total output
  335. has gone up, prices have gone
    down, and that's good for
  336. consumers.
    So if you're the regulator,
  337. if you're designing this
    industry--if you're working for
  338. the Justice Department,
    or if you're working for
  339. European Commission--you're
    going to want to know the answer
  340. when we switch from a
    simultaneous setting to an
  341. asymmetric setting where there's
    a leader firm and a follower
  342. firm,
    is that going to be good for
  343. consumers or bad for consumers?
    Well let's have a look.
  344. Well we know that Firm 1's
    output went up and we know that
  345. Firm 2's output went down,
    but can anyone tell me what
  346. happened to the total output and
  347. Let's have a poll again.
    Who thinks total output went
  348. down?
    Who thinks total output stayed
  349. the same?
    Who thinks total output went up?
  350. There's lots of abstentions.
    Let's try that again because
  351. too many abstentions.
    Who thinks total output went
  352. down?
    Who thinks total output stayed
  353. the same?
    Who thinks total output went up?
  354. That's pretty split.
    So I think total output--I know
  355. actually--that total output went
    up, and I claim I can see it on
  356. the picture.
    I claim if you stare at that
  357. picture you can actually see
    that total output must have gone
  358. up.
    Who's good at looking at a
  359. picture?
    Let me get the mike in here.
  360. The picture's there.
    Let me try this person.
  361. Your name is?
    Student: Andy.
  362. Professor Ben Polak:
    Andy go ahead.
  363. Student: Judging by the
    slope of the line you know that
  364. as it moves the amount that Q1
    changes will be greater than the
  365. amount that Q2 goes down.
    Professor Ben Polak:
  366. Good, so what Andy said,
    what Andy's saying is look at
  367. the slope of the line along
    which we slid.
  368. When we went from Cournot to
    the Stackelberg equilibrium,
  369. we started here and we slid in
    this direction down this line:
  370. Q1 went up and Q2 went down.
    And what Andy's pointing out is
  371. we can see from the slope of
    this line that Q1 goes up more
  372. than one unit for every unit of
    reduction of Q2.
  373. Let me say it again,
    for every unit of increase of
  374. Q1 there's less than a
  375. say it again.
    For every unit of increase of
  376. Q1 there's less than a unit of
    decrease of Q2.
  377. Another way of saying it is the
    slope of this line is less than
  378. 1.
    Everyone see that?
  379. So we know by the slope of the
    line--we know that Q1 had to go
  380. up more than Q2 went down,
    which means total output went
  381. up, which means that prices
    went--what happened to prices as
  382. total output went up?
    That shouldn't be hard,
  383. everyone's taken 115 here,
  384. So when total output went up,
    prices went down,
  385. good.
    Demand curves sloping down is
  386. not as important as backward
    induction but it's still quite
  387. important.
    So prices went down.
  388. So therefore,
    we now are ready to say what
  389. happened to Firm 2's profits.
    Firm 2 is producing less than
  390. before.
    Firm 2's costs are the same,
  391. and prices have gone down,
    so what's happened to Firm 2's
  392. profits?
    They must have gone down as
  393. well.
    So Firm 2's profit has gone
  394. down, and we know that consumer
    surplus, CS, consumer surplus,
  395. has gone up.
    For those people who remember
  396. their 115, prices have gone
    down, quantities have gone up,
  397. so consumer surplus here has
    gone up.
  398. So we've analyzed everything
    qualitatively I can think of in
  399. this game without reference to
    any math at all.
  400. Is that right?
    We really haven't done any math
  401. there.
    We talked about slope of the
  402. line.
    I guess that's math in junior
  403. high or something,
    but we haven't really done any
  404. math here, right?
    Is that fair?
  405. We already have a pretty good
    intuition for what's going to go
  406. on in this market.
    We think Q1's going to go up.
  407. We think Q2's going to go down,
    we think that Firm 1's profits
  408. are going up.
    We think Firm 2's profits are
  409. going down.
    We think total quantity is
  410. going up.
    Now let's see if we're right.
  411. Let's go back and do the math.
    So I want to spend a bit of
  412. time grinding this out.
  413. So I don't claim that doing the
    math is fun, but I want to prove
  414. that we can do it,
    because otherwise everyone's
  415. going to either think that this
    was all kind of just blah,
  416. blah, blah, and/or people are
    going to be scared to do the
  417. math when it arrives on a
    homework assignment.
  418. So for now we're going to--more
    or less for the next few
  419. minutes--we're going to forget
    we're economists and we're going
  420. to turn into nerds.
    That isn't a huge transition,
  421. but we'll do it anyway.
    I've got the demand curve up
  422. there, so the demand the curve
    is still there it's P = A - B
  423. [Q1 + Q2]
    and I've got profits written up
  424. there,
    but let's put them somewhere
  425. more convenient anyway.
    So P = A - B [Q1 + Q2]
  426. and profit is equal to--profit
    for Firm i is equal to P Qi--C
  427. Qi.
    And what we're told to do in
  428. backward induction is what?
    First of all,
  429. solve things out for Firm 2
    taking Firm 1 as given,
  430. and then go back and solve for
    Firm 1.
  431. So exactly the discussion we've
    just had informally,
  432. we're now going to do more
  433. So backward induction tells us,
    solve for Firm 2 first,
  434. taking Q1 as given.
    What is that problem?
  435. It's this sort of boring math
    problem, it says maximize by
  436. choosing Q2 the profits of Firm
    1, so that's going to be A - B
  437. Q1--B Q2.
    That's the price times the
  438. quantity Q2, - C Q2.
    So this bit here is the price.
  439. These two terms together are
    revenues, and this term is
  440. costs.
    Now I could do that,
  441. I could grind that bit out,
    but we already ground out that
  442. bit out three or four weeks ago
  443. so I'm not going to grind it
    out again, we know how to do
  444. that.
    But by the way,
  445. let's just remind ourselves
    what we did.
  446. We differentiated with respect
    to Q2.
  447. We set the thing we found,
    the derivative,
  448. equal to 0.
    That was our first order
  449. condition.
    And then we solved for Q2.
  450. Is that right?
    So when we did that,
  451. we went through the first order
    condition and then we solved it,
  452. we know what we actually got.
    So what we actually got was
  453. that Q2*, if you like--or Q2
    let's just call it--is equal to
  454. [A - C]
    / 2B - Q/2.
  455. In fact, I've already given it
    to you up there,
  456. it's up on that top board.
    It's the best response for Firm
  457. 2.
    So again, I could do this
  458. again, but since we did it a few
    weeks ago I don't want to redo
  459. it.
  460. Now the more interesting part,
    not thrilling,
  461. but a little bit more
  462. Now let's solve for Firm 1.
    So what is Firm 1 doing?
  463. Firm 1 is also trying to
    maximize profits.
  464. Firm 1 is choosing Q1 and Firm
    1--at least initially it looks
  465. like the same problem.
    It's A -B Q1--B Q2 Q1--C Q1,
  466. this is the same line we had
    before but now whereas Firm 2
  467. was taking Q1 as given,
    Firm 1 knows that Q2 is given
  468. by this formula here.
    So what we're going to do is
  469. we're going to plug this Q2 into
  470. Now again, for those of you in
    115, this isn't the only way to
  471. do it--I'm sorry 150--this isn't
    the only way that we could do
  472. it,
    we could also set up a
  473. Lagrangian equation,
    but for those people who don't
  474. know what that is,
    don't worry we're going to plug
  475. in today.
    So we're going to plug this in,
  476. and when we plug it in,
    we get a right old mess but
  477. let's do it anyway.
    So we're going to get max Q1,
  478. [A - B Q1--B [[A - C]/2B -
  479. - C]
  480. Everyone okay with that?
    I'm doing algebra on the board
  481. which is not fun but it's useful
    to do occasionally.
  482. So eventually what are we going
    to do?
  483. Eventually we're going to
    differentiate this thing,
  484. set it equal to 0,
    look at our first order
  485. condition and so on,
    just as we normally would.
  486. So eventually we're going to
    use basically 112 level calculus
  487. to solve this thing.
    Everyone remember how to solve
  488. a maximization problem?
  489. But before we do that let's
    tidy up the algebra a bit.
  490. So this thing is--let's just
    tidy it up.
  491. So this is equal to max with
    respect to Q1--and notice I've
  492. got an A - C here and once I
    take this B inside the brackets
  493. I've got a--[[A -C]/2]
  494. So I've got an [A - C]--[[A -
  495. so that's going to give me an
    [A - C]/2--and I'm really going
  496. to pray that the T.A.'s are
    watching me carefully and are
  497. going to catch my errors here.
    Okay, so I think I'm okay so
  498. far but please catch me.
    What else have I got?
  499. I've got a -B Q1 here and from
    in this bracket I've got a - -
  500. that's a + B Q1/2.
    So I have a –B Q1 + B
  501. Q1/2 so that's a –B Q1/2.
    So far so good and that whole
  502. thing is multiplied by Q1.
    Okay so far?
  503. Let's multiply out the bracket
    because otherwise I'll make a
  504. mistake.
    So this is the same as saying
  505. [A - C]/2 Q1--B Q1²/2.
    So far so good?
  506. Now we're at the level where
    even my very rusty memory of
  507. calculus will get us through,
    so let's try and do it.
  508. So what we're going to do is
    we're going to differentiate
  509. this thing with respect to Q1.
    So differentiating with respect
  510. to Q1--let's do it up here --,
    we get [A -C]/2 from this term
  511. and from the –B Q1²/2
    we're going to get--the two's
  512. are going to cancel rather
    pleasantly--so we're going to
  513. get B Q1 from that term.
    Everyone with me so far?
  514. I'm going through this in sort
    of slow steps,
  515. I agree it's not exciting but I
    want to make sure I don't make a
  516. mistake.
    So to turn this into a first
  517. order condition what must be
    true about this derivative?
  518. At the maximum,
    what must be true about this
  519. derivative?
    Should be equal to 0,
  520. good, and we should just check
    the second order condition.
  521. How do I check the second order
  522. I differentiate again and check
    it's negative,
  523. but if I differentiate again
    I'm just going to get -B.
  524. So -B is certainly negative all
    right, so second order condition
  525. is okay.
    So let's solve it out.
  526. Solving this for Q1,
    I get Q1 = [A - C]/2B.
  527. So we're leading in sheer
    boredom, it has to be done
  528. occasionally,
    Q1 = [A - C]/2B.
  529. We're not done yet.
    Now we want to go back and
  530. solve out algebraically for Q2.
    I know what Q1 is now,
  531. Q1 is [A - C]/2B.
    How do I find Q2?
  532. Somebody?
    Shout it out, how do I find Q2?
  533. I've got to plug it in.
    I'm going to go back and plug
  534. this Q1 back into this
    expression here,
  535. so I plug it back in,
    I'll get Q2 = [A - C]/2B - 1/2
  536. [A - C]/2B for a total of [A -
  537. Is that right?
    That's what I have in my notes.
  538. This looks good.
    So I'm now done.
  539. I've now found the equilibrium.
    I've found that in this
  540. leader-follower game,
    this Stackelberg version of
  541. quantity competition,
    Q1 is given by [A - C]/2B and
  542. Q2 is given by [A - C]/4B.
    Let's see how it matches up
  543. with the intuition we developed
    before without using any boring
  544. math.
    So first of all we're comparing
  545. Q1 and Q2 with what they used to
    produce, and what they used to
  546. produce is on the top board.
    What they used to produce--I
  547. can use this to guide the camera
    as well--what they used to
  548. produce is [A - C]/3B,
    is that right?
  549. So our claim was that we think
    the new Q1 is bigger than the
  550. old Cournot quantity.
    So now Firm 1 is producing [A -
  551. C]/2B, previously it was
    producing [A - C]/3B,
  552. so that is indeed bigger,
    that's good news.
  553. So this is indeed bigger than Q
  554. And our claim was that Firm 2
    will produce less than the old
  555. Cournot quantity.
    So Firm 2 used to produce [A -
  556. C]/3B and now it's producing [A
    - C]/4B, and that is indeed less
  557. than the old Cournot quantity.
    So far so good.
  558. What about total output?
    How do I solve for total output?
  559. Add the two outputs together,
    that's not too hard.
  560. So Q1 + Q2 = [A - C]/2B + [A -
    C]/4B which is in fact
  561. 3[A--C]/4B.
    Is that right?
  562. So it's going to be 3[A--C]/4B.
    So I've just--actually,
  563. for the first time today,
    I skipped a step,
  564. but is that okay?
    A half plus a quarter is
  565. three-quarters.
    So total output is 3[A - C]/4B.
  566. What was total output before?
    It used to be the Cournot total
  567. output, let's put it here
    somewhere, this is bigger than
  568. 2[A - C]/3B which is equal to
    the Cournot quantity Q1^(C) +
  569. Q2^(C).
    So everything we've predicted,
  570. just by looking at the picture,
    and thinking about the
  571. economics works out in the math.
    That's a good thing.
  572. We should feel a little bit
  573. I'm feeling a little bit
  574. Everything we thought out
    intuitively, just using the
  575. economics, the logic of the
  576. when we grind out the algebra
    we get the right answers:
  577. we get confirming answers.
    Everyone okay?
  578. That isn't a particularly fun
    exercise per se,
  579. but I want to do it just to
    show that you can use backward
  580. induction to solve out problems
  581. Backward induction and a little
    bit of what you learned in high
  582. school and/or freshman calculus
    can get you the answer.
  583. So now I want to leave aside
    the math and go back to the
  584. economics again.
    So we started off with a
  585. question, who would you rather
    be Firm 1 or Firm 2,
  586. and we know the answer now.
    Who would you rather be Firm 1
  587. or Firm 2?
    Firm 1 because Firm 1's profits
  588. went up and Firm 2's profits
    went down.
  589. Let's just talk about this a
    little bit--about what's going
  590. on here.
    So previously Firm 1 and 2 were
  591. just setting quantities
  592. We know now there's an
    advantage in going first.
  593. So suppose we change the game
    from the simultaneous move game,
  594. to a game in which Firm 1 and
    Firm 2 can make announcements.
  595. They can announce how much
    they're going to produce.
  596. So Firm 1 comes in one day and
    says, I'm going to produce this
  597. much and Firm 2 comes in and can
    see Firm 1's announcement.
  598. So I've changed it into a
    sequential game,
  599. Firm 1 has announced how much
    it's going to produce,
  600. Firm 2 is going to go
  601. Is that really a sequential
  602. Is that going to make a
  603. Why is that,
    I claim that's not really
  604. enough.
    Let me say it again.
  605. We start from the simultaneous
    move again.
  606. If we just change it by simply
    allowing Firm 1 to announce what
  607. they're going to produce--don't
    actually produce it but just
  608. announce what they're going to
    produce --,
  609. you might think that's a
    sequential move game and you
  610. might think that Firm 1 now has
    an advantage.
  611. But I'm claiming that's not
    enough really to give Firm 1 an
  612. advantage.
  613. Why is that not enough?
    Let's try and get some ideas
  614. here.
    Let me come around to this side.
  615. So Patrick why is that not
  616. Student: There's no
    credible commitment that you're
  617. going to produce at that level.
    Professor Ben Polak:
  618. Good, so imagine these two firms
    are, let's say that they're
  619. newspaper producing firms and
    one is owned by NBC's parent
  620. company and one is owned by
    Rupert Murdoch.
  621. And they're moving into this
    new market and the market is a
  622. town.
    Both of them are going to issue
  623. newspapers in this market that
    currently doesn't have any
  624. newspapers,
    and Murdoch simply says I'm
  625. going to produce lots of
  626. There's no reason for NBC to
    believe that.
  627. So moving first here,
    it isn't enough to say you're
  628. going to move first,
    it isn't enough even to make a
  629. decision that's reversible.
    Even if Q1 moved but that
  630. decision could be undone that
    isn't enough.
  631. What we need,
    and Patrick gave us the key
  632. word, what we need is
    commitment: a word that came up
  633. last week.
    So for moving first to help you
  634. here there really has to be
  635. Let's just get some of this
  636. There really needs to be
    commitment for this to work.
  637. So going back to the example of
    Murdoch and his competitor,
  638. Murdoch actually has to build
    the plant.
  639. There actually has to be a
    factory that he's built in,
  640. and that factory can't just be
    sold for scrap.
  641. So what creates the commitment,
    in the case when you've built
  642. the plant is what?
    Because having built it,
  643. it's a sunk cost,
    it's there.
  644. So sunk costs can help here.
    It can help making you
  645. committed.
    Once that money's gone,
  646. you can't get it back,
    so you're really committed to
  647. that scale.
    Does everyone know what I mean
  648. by sunk costs,
    by the way?
  649. So here's a case where a
    strategic move,
  650. entering first and sinking some
    investments can help you in the
  651. marketplace.
    Let's also look at this another
  652. way.
    Let's again go back to the
  653. simultaneous move game that we
    had before, where Murdoch and
  654. his competitor,
    the NBC parent corporation,
  655. are in fact,
    going to move simultaneously.
  656. Both of them are in the
    business of discussing how big a
  657. newspaper plant to put in this
    new town which hasn't got a
  658. newspaper yet,
    somewhere in Alabama or
  659. something.
    Suppose that there are two
  660. boardrooms, both of which are
    avidly just trying to discuss
  661. how big a newspaper plant to
  662. So suppose that one of the
    boardrooms, these four people in
  663. this row, this is the NBC parent
    company boardroom and they're
  664. trying to decide how big a plant
    to build.
  665. And over on the other side of
    the room is our Murdoch group
  666. which is in fact--let's take the
    row parallel--so these guys over
  667. there are Murdoch,
    are News International Group.
  668. And it is a simultaneous move
    game, so basically we're in
  669. Cournot.
    Now suppose that Murdoch,
  670. just to pick a name out of a
    hat, might not be the most moral
  671. gentleman in the world--who
  672. --and suppose that he in fact
    has hired one of the people in
  673. the NBC boardroom,
    in fact this guy,
  674. what's his name?
    Student: Ryan.
  675. Professor Ben Polak:
    He's hired Ryan to be a spy.
  676. So Murdoch has a spy in the NBC
  677. So Murdoch has a little
    advantage here,
  678. information wise.
  679. Because the NBC boardroom
    doesn't know what's going on in
  680. the Murdoch boardroom,
    but the Murdoch boardroom is
  681. going to know what's going on in
    the NBC boardroom.
  682. But to make the problem more
    interesting, suppose that
  683. somebody tells NBCs parent
    company, that in fact,
  684. there is a spy in their
  685. So these guys know they have a
    spy, they don't know who it is.
  686. If they knew who it was they'd
    beat him up or fire him or
  687. something --or maybe not--but
    they know that someone's there.
  688. Maybe they even suspect it's
    Ryan, so what should NBC do
  689. here?
    What decision should NBC make?
  690. One thing they could do
    is--they don't know it's Ryan
  691. but Ryan has this sort of
    Murdoch like face,
  692. so they might just fire him
    because he might be a spy--or
  693. what should they do here?
    Any takers?
  694. You're in the Murdoch,
    you're in the NBC boardroom.
  695. You know that Murdoch has some
    spy in the camp.
  696. What should you do?
    Student: You can come up
  697. with a fake plan and see if it
    goes back to Murdoch.
  698. Professor Ben Polak: So
    Chris is suggesting come up with
  699. a fake plan to feed it back to
    Murdoch, so Chris has been
  700. reading spy novels.
    So if this was a John Le
  701. Carré novel,
    that's certainly what you would
  702. do.
    You would create a whole bunch
  703. of fake information to feed back
    to the Russians,
  704. through this spy who in fact
    you've discovered.
  705. That isn't a bad idea:
    that might be a good thing to
  706. do.
    It's pretty hard to do,
  707. right, because ultimately these
    actual decisions have to be made
  708. in the boardroom,
    contracts have to be signed and
  709. so on, but I think Chris is onto
    the right idea here.
  710. So Chris' idea is create a fake
    plan to feed back to Murdoch,
  711. to give Murdoch some
  712. But there's another thing you
    could do, let me get someone who
  713. hasn't contributed yet,
    anyone else?
  714. Yeah, what's your name?
    Student: Usman.
  715. Professor Ben Polak: So
    Usman what would you do?
  716. Student: These guys get
    the first move now effectively
  717. because they can just decide
    they know.
  718. Professor Ben Polak: Say
    what you just said but shout it
  719. out so everyone can hear you.
    Student: NBC now gets
  720. the first move because they can
    decide and they know the other
  721. people are going to respond to
  722. Professor Ben Polak:
    Good, so what Usman is
  723. suggesting is maybe you don't
    feed Murdoch a fake plan,
  724. you feed Murdoch the true plan.
  725. what's going to happen now is
    if NBC decided to build a large
  726. plant,
    this information will be fed
  727. back to Murdoch,
    and Murdoch is now in the
  728. position of being the second
  729. When Murdoch moves,
    he or she knows what NBC is
  730. doing, and NBC knows that
    Murdoch is going to choose a
  731. best response to that.
    So it's as if NBC has been put
  732. in the position of Firm 1 and
    Murdoch has been put in the
  733. position of Firm 2.
    Even with the correct plan--so
  734. the correct thing to do here for
    NBC is not necessarily to
  735. mislead Murdoch,
    but just go ahead and build a
  736. big plant.
    Have that information be fed to
  737. Murdoch and let Murdoch respond
    to it.
  738. So notice here's a slightly
    paradoxical thing.
  739. You might think that having a
    spy in the camp of the other
  740. team, you might think having a
    spy would help you.
  741. But here having a spy--or
    having more information if you
  742. like--can actually end up
    hurting you.
  743. Everyone see that?
  744. Murdoch ends up losing by the
    fact that he was able to predict
  745. what NBC was going to do.
    Now there's a key to this of
  746. course.
    It was crucial to the argument.
  747. What was crucial to the
  748. It was crucial to the argument
    that NBC knew that Murdoch had a
  749. spy.
    The key here is that the other
  750. side, the other players,
    knew you had or were going to
  751. have more information.
    So what's the bigger idea here?
  752. There are two bigger ideas.
    Bigger idea number one is,
  753. games being simultaneous or
    sequential is not really about
  754. timing per se,
    it's about information.
  755. It's about who knows what,
    and who knows that who's going
  756. to know what.
    In a situation where Firm 1,
  757. our boardroom over here,
    knows that Murdoch is going to
  758. have this information before
    Murdoch moves,
  759. that's actually a sequential
  760. The timing is somewhat
  761. So that's the first
    observation, and the second
  762. observation is already on the
  763. What we have learned is
    sometimes in strategic
  764. settings--sometimes not
    always--more information can
  765. hurt you.
    Sometimes more information can
  766. hurt.
    We have to be careful here
  767. because that's not always true
    but sometimes it's true.
  768. And the reason that's true
    is--the reason is it can lead
  769. other players to take actions,
    in this case,
  770. to create a large plant,
    that hurt you.
  771. Now, if you put Monday's
    lecture together with today's
  772. lecture, we've seen
    something--two very similar
  773. things arose.
    On Monday's lecture having
  774. fewer options,
    burning your boats,
  775. ended up helping you or having
    lower payoffs by putting
  776. collateral down ended up helping
  777. And that might seem like a
    paradox but it isn't really a
  778. paradox because what happened
    was provided the other side
  779. knows you have fewer options,
    they know you've burnt your
  780. boats, or they know you'll
    suffer if you default on the
  781. loan,
    they know you've posted
  782. collateral, it will lead them to
    take behavior that helps you.
  783. In the case of the loan,
    it led to the lender giving you
  784. a bigger loan.
    In the case of the Saxon Army,
  785. you at least hope at least that
    the Saxon Army's running away.
  786. Today, we see that more
    information can hurt you,
  787. and once again,
    it isn't really a paradox,
  788. it's the same kind of argument.
    The fact that the other side
  789. know you're going to have this
    extra information,
  790. leads them to take actions that
    end up hurting you.
  791. So in games,
    unlike in standard single
  792. person decision problems,
    more information can hurt and
  793. more options can hurt and here's
    the reason.
  794. Now, one other thing to say
    about this, the game we just
  795. looked at, the Stackelberg game
    we just looked at is an example
  796. of something pretty famous.
    It's an example of first-mover
  797. advantage.
    It's an example of a game with
  798. a first-mover advantage.
    Now how many of you have
  799. heard--how many of you in the
    room have heard the term first
  800. mover advantage before?
    As few as that, seriously?
  801. The rest of you,
    how many of you have not heard
  802. the term first mover advantage
    before, one or two?
  803. So I'm always a bit weary of
    first mover advantage as a term.
  804. If you ask the students in the
    business school how many of them
  805. have heard the term,
    they've all heard it.
  806. It's a very popular business
    school term.
  807. It's a very popular term that
    you see in bad books.
  808. So let me just warn you a
    little bit about this.
  809. So if you go to the airport,
    let's say the Hartford Airport,
  810. and the flight you want to get
    on is late (which is usually the
  811. case),
    and hence you end up in the
  812. bookstore.
    And you find yourself on the
  813. economics and business shelves.
    You start looking at strategy
  814. books and typically the kind of
    book you find at the airports on
  815. business,
    or strategy,
  816. or economics is a pretty bad
  817. So it has some embossed cover
    on it and says 'Strategy for
  818. Dummies', or 'My Boring Life' by
    a famous CEO.
  819. I worry about these books
    because you end up buying these
  820. books.
    You've got time on your hands,
  821. and you read them,
    and they give you absolutely
  822. terrible advice.
    So not always,
  823. but almost always,
    and the kind of things they'll
  824. say is: "it's always a good idea
    to move first because that way
  825. you'll have a first mover
  826. This sounds right and I'm
    always worried about things that
  827. sound right.
    It may mean they're right,
  828. but the problem is that if
    they're wrong,
  829. they're tempting,
    they sort of lure you in.
  830. So it's good to move first
    because that way you have a
  831. first mover advantage sounds
    right, but it's nonsense.
  832. There are situations of which
    this is one--there are
  833. situations where it's good to
    move first.
  834. In quantity competition it's
    good to set your quantity,
  835. to be committed and that will
    lead the other side to producing
  836. smaller quantity which helps
  837. So sure, there are situations,
    there are games in which you
  838. want to move first,
    but there are also games in
  839. which you'd rather move second.
    Let me give you an example.
  840. This is an example we've seen
    in this class before- rock,
  841. paper, scissors.
    If anybody is going to read
  842. those books and believe them,
    so if anyone's going to read
  843. 'How to Discover Your Inner Bill
    Gates' and base their life on
  844. it,
    and therefore,
  845. think there's a first move
    advantage, I want to play rock,
  846. paper,
    scissors with that person.
  847. Everyone happy that you'd
    rather go second in rock,
  848. paper, scissors?
    Do I need to prove that or is
  849. that obvious?
    Okay good.
  850. Let me just expand a little bit
    more, into a more real world
  851. setting.
    There are plenty of settings in
  852. the real world where the
    advantage of moving second isn't
  853. because as in,
    rock, paper,
  854. scissors you just get to crush
    the other guy,
  855. it's simply that you learn from
    their mistakes.
  856. So, for example,
    in the game of buying new
  857. equipment for the office or home
    it's great to move second.
  858. The other guy goes out and
    samples some new piece of
  859. equipment, I wait to see if it
    works, and then buy it if it
  860. does.
    Or if I'm setting up a Firm in
  861. a new expanding market,
    let's say in a new part of the
  862. former Soviet Union,
    I'm quite happy to let some
  863. other firms go in there first
    and then watch what they did and
  864. try and learn from their
  865. If I'm setting a new curriculum
    for a university,
  866. I'm quite happy to let other
  867. Duke and Cornell and so on,
    move first and then I can go in
  868. as Yale, see what they did,
    and for sure they'll have made
  869. mistakes and I'll learn from
  870. So there's plenty of obvious
    situations where moving second
  871. helps you, for the obvious
    reason that information is often
  872. very useful.
    We argued here,
  873. information can hurt you,
    but there's plenty of other
  874. perfectly natural situations
    where information is to your
  875. advantage.
    So there are games with first
  876. mover advantages,
    but there also games with
  877. second mover advantages.
    And let me give you one example
  878. of a game that neither has a
    first mover advantage or a
  879. second mover advantage,
    just to convince you that can
  880. happen as well.
    So when you were a child you
  881. probably occasionally had to
    divide a cake and/or candy bar
  882. with your sibling.
    Anyone been in this situation?
  883. There was some candy bar or
    cake and you had to divide this
  884. thing between you and your
    brother and/or sister,
  885. is that right?
    There's a way in which,
  886. there's a typical way in which
    we divide things among siblings
  887. in that setting.
    There's a game we play to
  888. divide it.
    What's the game we play anyone?
  889. I'll cut and you choose,
    or vice versa,
  890. right.
    So I'll cut and you choose
  891. neither has first a mover
    advantage or a second mover
  892. advantage--assuming you can cut
  893. neither has a first mover
    advantage or a second mover
  894. advantage, which is precisely
    why it's a good way to divide
  895. the candy bar.
    Now, to drive this point home
  896. and because we've had a very dry
    lecture up to now,
  897. let's play a game,
    so everyone can wake up now.
  898. Math is over.
    I want to play a game and the
  899. game I want to play for the rest
    of today is called NIM.
  900. It's not going to surprise you
    that one of the things we're
  901. going to learn in this game is
    that sometimes games have a
  902. first mover advantage and
    sometimes games have a second
  903. mover advantage,
    so I'm giving away the punch
  904. line.
    How many of you have played NIM
  905. before?
    If you've played it before you
  906. can't play now,
    so don't shout out.
  907. So this is the game.
    There are two players and there
  908. are two piles of stones.
    We'll make the piles of stones
  909. into just chalk lines on the
  910. The players are going to move
  911. In each turn,
    the player whose turn it is to
  912. move, picks one of the two piles
    and removes some of the stones,
  913. which will just be lines on the
  914. So they decide how many of
    those lines to delete.
  915. Each time it's your turn you
    get to move again,
  916. you can choose the other pile
    this time.
  917. You move sequentially,
    and the only other rule that
  918. matters is, the person who gets
    the last stone wins.
  919. So, for example,
    here's a case where there's
  920. one, two, three,
    four, five, six,
  921. seven stones on this pile and
    one, two, three,
  922. four stones on this pile.
    So I'm going to get some
  923. volunteers to come on the stage
    and do this, but I was told I
  924. should choose a volunteer.
    So is Lee-Shing Chang here?
  925. So Lee-Shing Chang is going to
    volunteer for this.
  926. I'm volunteering Lee-Shing
    Chang because it's his birthday,
  927. so a round of applause for
    Lee-Shing Chang,
  928. come on the stage.
    Who wants to play against
  929. Lee-Shing Chang?
    Anyone else want to play?
  930. Who hasn't played a game yet?
    How about the guy with the Yale
  931. hat on back there,
    the white Yale hat.
  932. I'm staring right at the guy,
    yeah you, do you want to come
  933. up?
    All right?
  934. What's your name?
    Student: Evan.
  935. Professor Ben Polak:
    Evan, so we've got Evan and
  936. Leesing, is it Lee-Shing?
    Lee-Shing and Evan,
  937. come up on the stage,
    there's a step up here,
  938. come on.
    So we'll let Lee-Shing go first
  939. and we'll let Evan go second.
    Have you played this game
  940. before?
    Student: I didn't hear
  941. what the game is.
    Professor Ben Polak:
  942. Okay, he didn't hear what the
    game is, so we'll explain the
  943. rules again for those people who
    are sleeping.
  944. If you were sleeping the game
    is backward induction.
  945. All right so the rules of the
    game are this.
  946. They're going to take turns.
    In each turn,
  947. they're going to pick one of
    these two piles.
  948. This is Pile A and this is Pile
  949. And they're going to tell me
    how many of these chalk lines to
  950. delete, and I'm going to delete
    those lines.
  951. We're going to go on playing
    until somebody gets the last
  952. line.
    The person who gets the last
  953. line wins.
    You can get the last line,
  954. ten at a go,
    it doesn't have to be that
  955. there's only one on the board
    when you get the last line,
  956. but the person who gets the
    last line wins.
  957. So you have to pick a pile each
    go, and tell me how many lines
  958. to remove.
    So Lee-Shing why don't you go
  959. first, you can choose Pile A or
    Pile B and tell me how many
  960. lines to remove.
    Any advice from the audience?
  961. No advice: you're on your own
  962. Student: Can you remove
    three from Pile A?
  963. Professor Ben Polak:
    Three from Pile A,
  964. so three from Pile A.
    Evan your turn.
  965. Stand nearer to the board.
  966. Student: Can you remove
    two from Pile A.
  967. Professor Ben Polak: Two
    from Pile A--okay so Pile A is
  968. the popular pile here.
  969. Student: Two from Pile B.
    Professor Ben Polak: Two
  970. from Pile B, suspense is
    building here.
  971. Student: One from Pile B.
    Professor Ben Polak: One
  972. from Pile B, all right.
    It should go fast at this
  973. stage, go on.
    Student: Two from Pile A.
  974. Professor Ben Polak:
    Last stone wins,
  975. last stone wins.
    Student: Oh wait.
  976. Professor Ben Polak:
    Last stone wins,
  977. be careful here.
    It's his birthday come on.
  978. Last stone wins.
    Student: As in who?
  979. Professor Ben Polak:
    Person who gets the last stone
  980. wins.
    Student: Oh okay,
  981. so one from A.
    Professor Ben Polak: All
  982. right, one from A,
    can't abstain.
  983. Student: One from B.
    Professor Ben Polak: One
  984. from B, Lee-Shing is the winner
  985. Very good, let's get two more
  986. Everyone understand the game
  987. Thank you very much gentlemen.
    Two more volunteers.
  988. Have you played before?
    Come on up.
  989. I want to get some female
  990. It can't be a male only class.
    There we go thank you.
  991. This may not be as important
    economics as doing the
  992. Stackelberg model but it's
    probably a little bit more fun.
  993. Your names are?
    Student: John.
  994. Student: Christine.
    Professor Ben Polak:
  995. John and Christine,
  996. So let's make it a bit more
    complicated this time,
  997. so one, two,
    three, four,
  998. five, six, seven,
    eight, nine,
  999. ten, eleven,
    twelve, thirteen and one,
  1000. two, three,
    four, five, six, seven.
  1001. So we'll pick two prime
    numbers, and see if that's
  1002. important.
    Let's see what happens.
  1003. Why don't we say ladies first,
    so Christine your choice.
  1004. Student: That one has?
    Professor Ben Polak:
  1005. This one started off at 13 I
    believe and this one started off
  1006. at seven, assuming I counted
  1007. Let me get out of the way.
    Student: Six from Pile A.
  1008. Professor Ben Polak: Six
    from Pile A, one,
  1009. two, three, four,
    five, six, is that right?
  1010. Okay.
    Student: Three from Pile
  1011. B.
    Professor Ben Polak:
  1012. Three from Pile B.
    Student: Three from Pile
  1013. A.
    Professor Ben Polak:
  1014. Three from Pile A.
    Student: One from Pile B.
  1015. Professor Ben Polak: One
    from Pile B.
  1016. Student: One from Pile A.
    Professor Ben Polak: All
  1017. right, so we're onto this.
    Student: One from Pile B.
  1018. Student: One from Pile A.
    Professor Ben Polak: All
  1019. right.
    Student: One from Pile B.
  1020. Professor Ben Polak: All
  1021. Student: One from Pile A.
    Professor Ben Polak: All
  1022. right.
    Student: One from Pile A.
  1023. Professor Ben Polak:
    Switch it.
  1024. Student: One from Pile B.
    Professor Ben Polak: All
  1025. right, so Christine is the
    winner, okay.
  1026. So a round of applause for
    these two.
  1027. So I think everyone's figured
    out now how to play this game,
  1028. is that right?
    Has everyone figured it out?
  1029. Let me take one of these mikes
    and go down and just to make
  1030. sure.
    Lot's of people have figured
  1031. this out.
    So what is the rule for how to
  1032. play this game?
    What's the rule for how to play
  1033. this game?
    People haven't figured it out,
  1034. we should play again.
    So let's try over here,
  1035. someone has an answer,
    how should you play this game?
  1036. Student: It depends on
    whether the piles have the same
  1037. number in them.
    If they have the same number
  1038. you want to be second player,
    and if they have different
  1039. numbers you want to be first
  1040. Professor Ben Polak:
    Good, what should you do?
  1041. Student: You want to --
    Professor Ben Polak:
  1042. Let's assume they have different
  1043. Student: If they have
    different numbers you want to
  1044. make them equal.
    Professor Ben Polak: You
  1045. want to make them equal.
    So the trick to playing this
  1046. game is--thank you,
    very good--so the trick to
  1047. playing this game is if the
    piles are uneven then you want
  1048. to make them even--you want to
    make them equal.
  1049. Everyone see that?
    So if you start off with the
  1050. piles being unequal as we did
    both times, for example 3 and 2,
  1051. then you want to be Player 1.
    There's a first mover advantage
  1052. and the correct tactic is to
    equalize the piles.
  1053. What you'll notice now is that
    player 2 can't do anything.
  1054. If they take two from here,
    you'll win by taking both of
  1055. those two.
    If they take one from here
  1056. you'll equalize the piles again.
    And if they take then one from
  1057. here you've won.
    So the way to play this game is
  1058. to equalize the piles.
    What does that mean?
  1059. It means if the initial
    position has unequal piles,
  1060. uneven piles,
    then you would rather be Player
  1061. 1: it has a first mover
  1062. But if the initial position has
    even numbers in the piles then
  1063. you'd rather be Player 2.
    Is that right?
  1064. Because if you start off with
    an even [correction:
  1065. equal]
    number in each pile,
  1066. the person moving first is
    going to make them unequal,
  1067. and thereafter the next
    person's in a winning position.
  1068. So I want us to notice two
    things from this game.
  1069. First notice in this game that
    from any initial position we can
  1070. very quickly tell who is going
    to win and who is going to lose,
  1071. assuming they play well.
    Three things actually.
  1072. Second, we didn't actually use
    backward induction here,
  1073. but it's pretty obvious you do
    want to use backward induction
  1074. here.
    You want to figure out what the
  1075. end game is going to look like,
    is that right?
  1076. It's very easy to see this game
    if you look at the end game.
  1077. The third lesson is what we
    just said, in this game,
  1078. sometimes there's a first mover
    advantage, sometimes there's a
  1079. second mover advantage.
    So sometimes there's a first
  1080. mover advantage in this game,
    sometimes there's a second
  1081. mover advantage in this game.
    So I'm just illustrating the
  1082. point we made earlier that it's
    not always the case that you
  1083. want to move first.
    Sometimes you want to move
  1084. second.
    Now today I set up the piles
  1085. with unequal piles so there' was
    a first mover advantage,
  1086. but that was just to give an
    advantage to the guy whose
  1087. birthday it was.
    I could have set things up with
  1088. equal lines in each pile,
    and made the other guy go
  1089. first.