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← 19. Subgame perfect equilibrium: matchmaking and strategic investments

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Showing Revision 1 created 08/14/2012 by Amara Bot.

  1. Professor Ben Polak:
    So last time we covered a
  2. whole bunch of new ideas,
    and it was really quite a lot
  3. of ideas for one class.
    Here's some of the ideas we
  4. covered.
    We talked about information
  5. sets, and these were ways to
    allow us to model imperfect
  6. information.
    So what's imperfect information?
  7. It's a way of being able to
    capture both simultaneity in
  8. moves and sequential moves in
    the same game.
  9. So it's a way that's going to
    allow us to meld the lessons
  10. from before the mid-term and
    after the mid-term.
  11. Then we talked about what
    strategies meant in this
  12. context, and the basic idea is
    strategies are
  13. instructions--strategies for
    each player--give them an
  14. instruction at each of their
    information sets.
  15. Then we talked about what
    sub-games were,
  16. and, leaving aside
    technicalities,
  17. sub-games were just games
    within games.
  18. And finally we introduced the
    idea of sub-game perfection
  19. which is our new solution
    concept that refines the idea of
  20. Nash equilibrium.
    What sub-game perfection is
  21. going to do is it's going to
    instruct the players to play a
  22. Nash equilibrium in every
    sub-game.
  23. Another way of saying it is,
    a sub-game equilibrium is a
  24. Nash equilibrium in the whole
    game, but in each sub-game it
  25. induces Nash play as well.
    Now, we're going to see today
  26. examples.
    If we have time I'll go through
  27. three different examples,
    and I'll tell you at the end of
  28. each example what it is I'm
    hoping to be able to take away
  29. from that example.
    So, last time was a lot of
  30. formal stuff.
    Today is going to be a lot of
  31. examples.
    Okay, that's our agenda.
  32. Here's a game.
    Here's our first example.
  33. And I call this example,
    I call this game,
  34. "don't screw up," for reasons
    we'll see in a minute.
  35. So this is a game in which
    Player 1 has to choose between
  36. Up and Down.
    If Player 1 chooses Up then
  37. Player 2 gets to move and
    chooses between left and right.
  38. And if Player 2 chooses left
    then Player 1 gets to move again
  39. and Player 1 chooses between up
    or down.
  40. Everyone looking at that game?
    So why don't we play this game
  41. since we haven't played a game
    for a while.
  42. We'll play a couple of games
    today.
  43. So what I'm going to do is
    let's divide the class in two.
  44. So if I just draw a line down
    the middle of the class,
  45. everybody to my left (to your
    right), everybody on this side
  46. of the class is a Player 1.
    Okay you're all Player 1's.
  47. And everyone on this side of
    class you're Player 2,
  48. including you guys hiding from
    the camera, you're Player 2's.
  49. Okay, so let's figure out what
    we're going to do.
  50. Everyone had the time to look
    at the game?
  51. So Player 1's you get to move
    first, those of you who are
  52. going to choose Down raise your
    hand now.
  53. Raise your hand.
    Wave it in the air.
  54. Keep it up so the camera can
    see you.
  55. And those of you who are going
    to choose Up raise your hands.
  56. Lots more Ups.
    Those of you who chose Up why
  57. don't you all stand up.
    I don't want to do all the
  58. exercise here,
    so all those who chose Up,
  59. stand up.
  60. So you can see that choosing
    down ends the game,
  61. so this many people are still
    playing the game.
  62. Everyone who is still sitting
    down, everyone who sat down here
  63. has exited.
    All right, Player 2's you get
  64. to move now.
    So Player 2's,
  65. those of you who choose right,
    including the people on this
  66. aisle,
    those people who choose right
  67. raise your hand now--one right
    over there.
  68. Those of you who choose left
    raise your hands.
  69. Why don't you guys all stand
    up, just to get you awake on a
  70. Monday morning,
    everyone's sleepy otherwise.
  71. Let's go back to Player
    2's--sorry Player 1--those of
  72. you who are still in the game.
    So those of you who chose Up
  73. the first time,
    how many of you now choose
  74. down?
    Raise your hand if you choose
  75. down;
    and raise your hand if you
  76. choose up.
    Just to get a sample of this,
  77. let's get the 2's to sit down
    again so people can see them.
  78. So 2's sit down.
    Those of you Player 1's who are
  79. still in the game who were
    choosing up raise your hand now.
  80. I think that's everybody,
    is that correct?
  81. Okay, you can all sit down.
    So let's just talk about this
  82. game for a while and then we'll
    analyze it.
  83. Now, this is not a difficult
    game from the point of view of
  84. stuff we've done since the
    mid-term.
  85. It's pretty clear what we
    should do in this game by
  86. backward induction.
    So why don't we start there.
  87. Okay, so by backward induction,
    we find that if Player 1 gets
  88. to move a second time then
    they're choosing between 4 and
  89. 3,
    and they're going to choose 4.
  90. Player 2, if they get to move,
    knowing that Player 1 is going
  91. to choose up tomorrow,
    they're going to be choosing
  92. between 3 if they choose left or
    2 if they choose right.
  93. So they're going to stay in the
    game and choose left,
  94. which is what most of you did.
  95. Finally, Player 1 at the
    beginning of the game,
  96. knows that Player 2 is going to
    choose left whereupon she's
  97. going to choose up,
    so if she chooses Up she's
  98. going to end up getting 4,
    and if she chooses Down she's
  99. going to get 2,
    so she's going to choose Up.
  100. So it's clear what backward
    induction does in this game and
  101. that's what most people did in
    the game.
  102. Is that right?
    However, not everybody did it.
  103. Some of the Player 1's
    actually, why don't you raise
  104. your hand, those people who
    chose Down--the ones who chose
  105. down.
    There were more than that.
  106. You can all stand up.
    Those of you who didn't stand
  107. up just now raise your hands.
    People are hiding now,
  108. but that's okay.
    Those people who chose Down,
  109. they may have had a reason for
    choosing Down,
  110. and their reason for choosing
    Down might have been that they
  111. thought--even though they can do
    backward induction--so even
  112. though they know that by
    backward induction Up gets them
  113. the better answer--they might be
    worried that if they choose Up,
  114. Player 2 will screw up and
    choose right.
  115. Notice that if Player 2 chooses
    right then Player 1 only gets 1,
  116. whereas Down yielded 2.
    So in some sense Down was the
  117. "safe" thing to do for Player 1
    given that they might be worried
  118. that Player 2 might screw up.
    Does that roughly--just nod if
  119. this is the case:
    for those people who chose down
  120. is that kind of what you were
    thinking?
  121. Some people are shaking their
    heads, but some people are
  122. nodding.
    That's a good sign.
  123. Now why might Player 2 in fact
    screw up and choose right.
  124. Because Player 2 might,
    themselves, think that Player 1
  125. might screw up at this stage.
    If Player 1 were to screw up at
  126. the last stage and choose down,
    then Player 2 by choosing left
  127. would only get 1,
    and for him the safe option
  128. therefore is right which yields
    2.
  129. So, to get the backward
    induction answer here--which
  130. most of us did--to get the
    backward induction answer here
  131. relies on Player 1 trusting
    Player 2 to play backward
  132. induction,
    and that requires Player 1 to
  133. have trust in Player 2 trusting
    Player 1 not to screw up in the
  134. last stage.
    So say it again,
  135. Player 1 needs Player 2 not to
    screw up, and that means Player
  136. 1 needs to trust that Player 2
    will trust her not to screw up.
  137. Everyone see the game?
    Okay, so let's try and analyze
  138. this game using what we learned
    last time and see what we find.
  139. So the first thing to do is
    let's look at strategies in this
  140. game.
    So Player 2 just has two
  141. strategies, left and right,
    because Player 2 only has one
  142. information set--and notice this
    game is actually a game of
  143. perfect information so it's
    going to be very easy.
  144. Player 1 has two information
    sets, this information set and
  145. that information set.
    At each of them Player 1 has
  146. two choices so she must have
    four strategies in all.
  147. So this game when we put it in
    its matrix form is going to be a
  148. 4 x 2 game.
    Here it is.
  149. And the strategies for Player 1
    are (Up, up),
  150. (Up, down), (Down,
    up), and (Down,
  151. down).
    And the strategies for Player 2
  152. are just left and right.
    And now we can put the payoffs
  153. in.
    So ((Up, up),
  154. left) gets us (4,3).
    ((Up, up), right) gets us (1,2).
  155. ((Up, down),
    left) gets us (3,1).
  156. ((Up, up, right) gets us (1,2)
    again: we end up exiting the
  157. game here.
    (Down, up) is easy because it's
  158. just exiting the game at the
    first stage, so all of these are
  159. going to be just (2,1).
    Everyone happy with that?
  160. So what I've done is translated
    the game into its matrix form.
  161. And let's look for Nash
    equilibria in this game.
  162. Let me do it at the board since
    it's quite easy at this stage.
  163. So to look for Nash equilibria,
    let's just worry about pure
  164. strategy equilibria for now.
    So if Player 2 was choosing
  165. left then Player 1's best
    response is the (Up,
  166. up) strategy.
    And if Player 2 is choosing
  167. right, then Player 1's best
    response is either (Down,
  168. up) or (Down,
    down).
  169. That's exactly the conversation
    we just had.
  170. If Player 2 was going to "screw
    up" and choose right then Player
  171. 1 wants to get out of the game
    immediately.
  172. Conversely, if Player 1 is
    choosing (Up,
  173. up) then Player 2 is happy and
    is going to choose left,
  174. trusting Player 1.
    If Player 1 was going to choose
  175. (Up, down), however,
    that's Player 1 screwing up at
  176. the second stage.
    So in that case Player 2 wants
  177. to get out of the game and
    choose right.
  178. If Player 1 is choosing (Down,
    up) then Player 2 is
  179. actually--it doesn't matter
    they're indifferent.
  180. And if Player 2 is choosing
    (Down, down) then once again
  181. Player 2 is indifferent since
    they don't get to move at all.
  182. So from this we see that there
    are three Nash equilibria,
  183. let me call them 1,2,
    and 3.
  184. So one Nash equilibrium is
    ((Up, up), left).
  185. Another Nash equilibrium is
    ((Down, up, right):
  186. that's here.
    And the third equilibrium is
  187. ((Down, down),
    right).
  188. So there are three pure
    strategy Nash equilibria in this
  189. game.
    Let's just see what they do.
  190. So the first one ((Up,
    up), left) is Up,
  191. left, up so it gets us to here.
    So this one is the same
  192. equilibrium as corresponds to
    backward induction.
  193. Is that right?
    This one's the backward
  194. induction equilibrium,
    and the other two are
  195. different.
    ((Down, up),
  196. right) and ((Down,
    down), right) both end up down
  197. here exiting the game
    immediately,
  198. so both of these other
    equilibria fail backward
  199. induction.
    So let's put backward induction
  200. with a line through it:
    they're failing backward
  201. induction.
    So you might ask why are they
  202. equilibria?
    We've seen examples like this
  203. before, for example,
    in the entry game.
  204. We looked at some examples last
    time.
  205. So, in this game,
    the reason these are equilibria
  206. even though they fail backward
    induction exactly coincides with
  207. that conversation we just had
    about worrying about the other
  208. person screwing up.
    So, in particular,
  209. if Player 1 thinks that Player
    2 is choosing right--that is to
  210. say,
    thinks that Player 2 is going
  211. to screw up--then Player 1
    doesn't want to travel up the
  212. tree because she knows she'll be
    carried down here,
  213. and instead she just chooses
    the safe option and gets 2.
  214. So from Player 1's point of
    view, if Player 2 was going to
  215. choose right,
    then getting out of the game
  216. doing the safe thing is the best
    response for Player 1.
  217. And for Player 2's point of
    view, if Player 1's exiting the
  218. game it really doesn't matter
    what Player 2 says she's going
  219. to do,
    because she doesn't get to move
  220. anyway so that's why these are
    both equilibria.
  221. Okay, so what we're going to do
    next, we've translated this into
  222. a tree.
    We've written down the
  223. strategies.
    We want to actually see which
  224. of these Nash equilibria are
    sub-game perfect?
  225. Let me give myself a bit more
    room here because I want to keep
  226. this in sight.
    So let's get rid of this and
  227. raise this one.
  228. So the next question is which
    of these three Nash equilibria
  229. are sub-game perfect?
    To do that we need to start by
  230. identifying the sub-games.
    And, of course,
  231. just having hoisted that board
    up there, I have to hoist it
  232. down again.
    Okay, so what are the sub-games
  233. here?
    Well the simplest sub-game is
  234. this simple sub-game at the end,
    in which Player 1 moves.
  235. That's a very obvious sub-game,
    is that right?
  236. That's a little game within a
    game, it's a rather trivial game
  237. because it's a one player game,
    but it is a game.
  238. So let's examine that one
    first, that's the last sub-game.
  239. So the last sub-game here is a
    somewhat trivial sub-game.
  240. It looks like this.
    Player 1 is the only mover and
  241. they're choosing either up or
    down and the payoffs are (4,3)
  242. and (3,1): and frankly we don't
    really care at this point what
  243. Player 2's payoffs are because
    Player 1 is the only person
  244. who's playing in this sub-game.
    But nevertheless,
  245. let's put them there.
    And if we write this up as a
  246. matrix--here it is as a matrix.
    Since Player 1 is the only
  247. mover they're choosing between
    up and down, and the payoffs are
  248. (4,3) and (3,1),
    and of course Player 2 doesn't
  249. get to move so Player 2 is
    irrelevant here.
  250. And clearly the only Nash
    equilibria in this game is for
  251. Player 1 to choose up.
    Player 2, it doesn't really
  252. matter what they choose,
    there's nothing they can do
  253. about it anyway,
    but for Player 1 to choose up
  254. is the Nash equilibria.
    So the Nash equilibria in this
  255. trivial sub-game is 1 just
    chooses up.
  256. Is that right?
    So let's look at the play
  257. induced by our three candidate
    Nash equilibria in this
  258. sub-game.
    So each of our candidate Nash
  259. equilibria--here they are,
    this one, this one,
  260. and this one--have an
    instruction of how Player 1
  261. should play in this sub-game.
    And let me just pause a second.
  262. The reason that these three
    equilibria have an instruction
  263. for how Player 1 should play in
    the sub-game is because of our
  264. definition of a strategy.
    Each strategy tells the player
  265. how they should move at every
    information set of that player.
  266. So even if the strategy is such
    that that information set won't
  267. be reached, the strategy still
    has to tell you what you would
  268. do when you got there.
    And now for the first time
  269. perhaps we're going to see why
    that redundancy helps us.
  270. So let's look at the
    instructions.
  271. Each of them gives an
    instruction.
  272. The first one tells us to play
    up in this sub-game.
  273. The second one says up again,
    and notice this was redundant.
  274. Once you've chosen Down you
    know you're not going to get to
  275. make a choice at the third node,
    or your second node,
  276. but nevertheless there's the
    instruction and it says up.
  277. The third one says down.
    So this is the instructions of
  278. these three equilibria in this
    little sub-game.
  279. This is the play prescribed by
    these three equilibria in this
  280. sub-game.
    Two of them say up and those
  281. ones are going to induce the
    Nash equilibrium in this
  282. sub-game, but the third one does
    not.
  283. The third one says down and
    that's not allowed.
  284. That's not allowed in a
    sub-game perfect equilibrium
  285. because a sub-game perfect
    equilibrium has to prescribe
  286. play in every sub-game that's
    Nash,
  287. and here the third equilibrium
    is telling Player 1 to choose
  288. down which is not a Nash
    equilibrium in the sub-game.
  289. So what are we doing here?
    Let's make it clear.
  290. We're finding the sub-game
    perfect equilibria.
  291. And what we've done is:
    number 3 is eliminated because
  292. it induces play in this sub-game
    that is not Nash equilibrium,
  293. not a Nash equilibrium in the
    sub-game.
  294. We're really putting stuff
    together now:
  295. to be able to draw this
    conclusion,
  296. we really used the fact that
    strategy 3 contained a redundant
  297. instruction, an instruction down
    at this node that was never
  298. reached.
    But that helped us get rid of
  299. it.
    So that one's gone: 3 is gone.
  300. Let's proceed.
    I'm going to run out of board
  301. space here.
    Have people got this one down?
  302. I'll bring it back in a second,
    almost, let me give it a
  303. second.
    What I want to do now is look
  304. at the next sub-game.
    Maybe what I can do if I just
  305. remove this comment,
    I can work on the right hand
  306. board, that'll allow you to look
    at it.
  307. So all that comment said was 3
    is eliminated.
  308. Let me work now on the right
    hand half of this board,
  309. that'll allow it to be up there
    a bit longer.
  310. Okay, so let's now look at the
    next sub-game,
  311. and again, we're going to work
    from the back.
  312. So the next sub-game back is
    the sub-game that starts from
  313. this node, the sub-game that
    starts from that node.
  314. So let's identify that in a
    different color.
  315. I used blue so let me use pink.
  316. So now we're going to look at
    this sub-game,
  317. this big pink sub-game,
    and again in this pink sub-game
  318. this game looks like this.
    It starts with Player 2
  319. choosing between left and right,
    and then Player 1 has to choose
  320. up or down, and the payoffs are
    (1,2), (4,3) and (3,1).
  321. Once again let's look at the
    matrix form of this sub-game,
  322. and this is a little bit less
    trivial than the last one
  323. because now there are really two
    players playing.
  324. So here's the matrix that goes
    along with this.
  325. Player 1 is choosing between up
    or down and Player 2 is choosing
  326. between left or right.
    And this is slightly,
  327. slightly cheating because in
    fact, Player 1 of course knows
  328. what Player 2 is going to have
    done by the time she moves,
  329. but never mind it'll do for now.
    Let's just put the payoffs in.
  330. So (up, left) is (4,3) and
    (down, left) is (3,1) and (up,
  331. right) is (1,2) and this must
    also be (1,2).
  332. Everyone happy with that?
    Just putting the payoffs in.
  333. And let's just look again at
    pure-strategy Nash equilibria
  334. here.
    There are actually mixed ones,
  335. but let's just worry about pure
    ones for now.
  336. So the pure Nash equilibria
    here in this little sub-game are
  337. what?
    Well, let's just see.
  338. If 2 chose left then 1 wants to
    choose up.
  339. If 2 chooses right it doesn't
    really matter what 1 chooses
  340. because she isn't going to get
    to move anyway.
  341. Conversely, if 1 chooses up
    then 2 wants to choose left.
  342. That's the example of 1 not
    screwing up, so 2 wants to stay
  343. in the game.
    But if 1 was to choose down
  344. then Player 2 would like to get
    out of the game,
  345. so if Player 2 thinks 1 is
    going to screw up she wants to
  346. exit the game.
    Very quickly we can see there
  347. are two equilibria here.
    One of them is (up,
  348. left) and the other one is
    (down, right).
  349. They correspond to playing down
    this way, that's (down,
  350. right) and (up,
    left) playing up this way.
  351. Once again, let's look at our
    three equilibria in the parent
  352. game.
    Here's our three equilibria in
  353. the parent game,
    and let's see what play they
  354. induced in this little sub-game.
    So we'll do exactly what we did
  355. before.
    So 1,2, and 3:
  356. these are our three equilibria
    from above.
  357. And equilibrium number 1,
    ((Up, up), left) in this game
  358. prescribes (up,
    left).
  359. Is that right?
    Equilibrium number 2,
  360. ((Down, up),
    right) here prescribes up,
  361. right.
    And equilibrium number 3,
  362. ((Down, down),
    right) here prescribes (down,
  363. right).
  364. So which of these are
    prescribing Nash equilibria in
  365. the sub-game?
    Well (up, left) is an
  366. equilibrium.
    It's that one.
  367. (up, left) is an equilibrium so
    this is okay.
  368. And (down, right) was an
    equilibrium, so 3 is okay in
  369. this sub-game.
    But (up, right) is not a Nash
  370. equilibrium.
    So in this sub-game,
  371. Nash equilibrium number 2,
    the ((Down, up),
  372. right) equilibrium is
    prescribing play that is not a
  373. Nash equilibrium in the
    sub-game: so it's eliminated.
  374. It can't be a sub-game perfect
    equilibrium.
  375. So here 2 is eliminated since
    it induces non-Nash equilibrium
  376. play in this sub-game.
  377. Now we're done,
    we know about the whole thing.
  378. So what we did here we started
    with the whole game,
  379. we found there were three Nash
    equilibria,
  380. we found that only one of them
    agreed with backward induction.
  381. We then looked at the sub-games.
    We first of all looked at that
  382. blue sub-game and we found that
    one of the equilibria,
  383. equilibrium number 3,
    was eliminated.
  384. Equilibrium number 3 is not
    prescribing Nash behavior in
  385. this sub-game.
    Then we looked at the slightly
  386. more complicated sub-game,
    the pink sub-game,
  387. and we found that equilibrium
    number 2 prescribes the behavior
  388. (up, right) which is not Nash in
    this sub-game.
  389. At this stage we've eliminated
    two of the three equilibria and
  390. we're just left with one.
    And the one we're left with,
  391. the only sub-game perfect
    equilibrium, the only
  392. equilibrium that wasn't
    eliminated by the fact that it
  393. would prescribe bad behavior in
    sub-games,
  394. the only SPE is number 1:
    which is (Up,
  395. up) and left.
    What do we notice?
  396. We notice that that's the
    equilibrium, that's the play
  397. that backward induction would
    have selected.
  398. So notice this is the backward
    induction prediction.
  399. So what are the lessons here?
    The lessons here are that our
  400. new idea, the idea of sub-game
    perfect equilibrium is pretty
  401. easy to go about finding.
    You just look at sub-games and
  402. check that the play in each
    sub-game has to be Nash play.
  403. If you start at the back,
    you construct it by rolling
  404. backwards, much like we did
    backward induction.
  405. Start at the last sub-game and
    work backwards.
  406. The second thing is--not
    surprisingly given that
  407. remark--not surprisingly,
    where backward induction
  408. applies,
    for example in this game,
  409. the sub-game perfect
    equilibrium will find the
  410. equilibrium that is consistent
    with backward induction.
  411. Remember that was our aim last
    week.
  412. We wanted a way of refining
    Nash equilibrium to throw away
  413. those Nash equilibria that were
    inconsistent with backward
  414. induction.
    So sub-game perfect equilibrium
  415. has done that.
    It tells us now if backward
  416. induction applies,
    the Nash equilibria you should
  417. focus on are the sub-game
    perfect equilibria.
  418. Indeed, most people in the
    class played that equilibrium
  419. just now.
    Okay so that was really what I
  420. wanted to say about this
    example, but let me just make a
  421. remark in passing.
    I made this remark in the
  422. middle, so let me just make it
    again.
  423. When we write down strategies,
    those strategies tell us what
  424. seem to redundant moves.
    But being forced to write down
  425. those redundant moves is useful
    because it allows us to model
  426. what other people think you
    would have done at those later
  427. nodes.
    And sometimes I have to think
  428. what you think I would have done
    at this later node before I
  429. decide not to go down that
    branch of the tree.
  430. So being able to write down
    everything in a strategy allows
  431. us to have everything in front
    of us and makes that analysis
  432. simple;
    and that's exactly what we did
  433. here.
    So this was a fairly mundane
  434. example, because in particular
    we didn't use any kind of
  435. information set.
    So next let's look at an
  436. example that does use some
    information sets.
  437. So: new example.
    Let me clean this off.
  438. Once again, I want to play this
    example.
  439. But what I'd really like to
    do--we're going to call this
  440. game the matchmaker game--and
    what I'd actually like to find
  441. out is: do we still have our
    couple we tried to send on a
  442. date?
    Our hapless couple we tried to
  443. send on a date at about week
    three.
  444. Are you guys still here?
    There's the guy,
  445. what's your name again?
    Student: David.
  446. Professor Ben Polak:
    David, and what was the--is
  447. she hiding?
    There she is thank you.
  448. Your name was?
    Student: Nina.
  449. Professor Ben Polak:
    Nina and David.
  450. Good, can we get some mikes to
    Nina and David actually?
  451. Let me do it.
    I'll go on talking while I'm
  452. doing this–.
    Where's David,
  453. and Ale can you get a mike to
    Nina?
  454. That would be great thank you.
    All right, so for weeks we've
  455. been trying to get this couple
    to go on a date.
  456. It's our attempt to get
    economics majors to become real
  457. people.
    It's a hard thing to do and
  458. they're kind of the hapless
    couple because first we sent
  459. them to the movies and they end
    up going to different movies;
  460. and then we sent them off for a
    romantic weekend in New England
  461. and they end up doing different
    things,
  462. one went to the theatre and
    another one apple picking,
  463. I forget which way around it
    was.
  464. And at this point,
    I figure I'm a pretty bad
  465. matchmaker so what I'm going to
    do is.
  466. I'm going to introduce a third
    player into the game as the
  467. matchmaker.
    So first of all I'll write down
  468. what the game is.
    So the game at this point--the
  469. game is going to look like this.
    Player 1 is the matchmaker,
  470. we can call him Player M if you
    like, and he has a choice,
  471. he or she has a choice.
    She could not send the couple
  472. out on a date or she could send
    the couple out on a date.
  473. But being a better matchmaker
    than me, if she sends them out
  474. on a date, she's going to stake
    some money, she'll pay for the
  475. date.
    And in the date once again
  476. they're trying to meet,
    and once again,
  477. unfortunately,
    they haven't figured out where
  478. to meet.
    We'll put the payoffs in,
  479. in a second and we'll tell you
    what the strategies are.
  480. So what I'm going to assume
    here is I'm going to let Jake
  481. our T.A.
    be our matchmaker.
  482. And the reason I'm choosing
    Jake is I think he's the nearest
  483. thing I have in mind in this
    class to being a Jewish mother.
  484. I mean: he's neither Jewish--I
    think he's not Jewish and he's
  485. not my mother,
    but he is the T.A.,
  486. who's responsible for bringing
    some drink everyday in case I
  487. pass out in the lecture.
    So that's the nearest thing I
  488. can think of.
    So Jake's going to be our
  489. Jewish mother,
    and Jake's going to either send
  490. these guys on a date or not.
    And Jake's smarter than me at
  491. this: he's actually good at
    matchmaking.
  492. And what he's going to do is,
    he's going to send them
  493. somewhere where they really--he
    knows Yale students better than
  494. I do--and he's going to send
    them really somewhere where
  495. they're going to meet.
    So he's going to send them to
  496. go to the same lecture class
    next year, and then they'll be
  497. sitting in the aisles in this
    huge lecture class and they're
  498. bound to meet.
    All of you have sat next to
  499. other people at some point.
    So that seems like a good idea.
  500. So the classes he thinks of
    sending them to--he says go to a
  501. large lecture class.
    So they're either going to go
  502. to the Gaddis class which is
    called "Cold War";
  503. or to the Spence class which is
    called "China."
  504. Everyone know a little bit
    about these classes?
  505. These seem like reasonable
    classes to go to meet your--to
  506. have a date--to meet somebody.
    I mean the Cold War can be a
  507. fun class, I mean,
    you hope it isn't a prediction
  508. of the future relationship but
    the Cold War seems all right.
  509. And China is,
    by all accounts,
  510. a fantastic class.
    It involves,
  511. something involving 20 million
    people, most of them were in the
  512. class together,
    so it's a pretty big class.
  513. So let's do that.
    Unfortunately Jake makes the
  514. same mistake I do,
    he's not going to tell them
  515. which class to go to.
    So they have to decide whether
  516. to take Gaddis or Spence,
    and once again they're
  517. coordinating.
    We'll call them Players 2 and 3.
  518. So here they are trying to
    coordinate, and the payoffs are
  519. as follows.
    So let's put in Jake's payoffs
  520. first of all.
    So if they manage to
  521. coordinate, first of all,
    if Jake doesn't send them,
  522. everybody gets nothing.
    And if Jake does send them and
  523. they coordinate Jake makes 1
    because he feels really happy
  524. about this.
    After all there must be some
  525. motivation for people
    matchmaking.
  526. So if they coordinate down here
    Jake gets 1 as well,
  527. but if they fail to coordinate,
    Jake feels rotten about it,
  528. particularly because he paid
    for them to go this
  529. class--whatever the cost of a
    class at Yale is--which is
  530. probably quite a lot actually.
    So okay we'll call it 1 though
  531. and otherwise the payoffs are
    exactly the same as the payoffs
  532. we used when we looked at this
    game earlier on in the course.
  533. So the payoffs are going to be
    (2,1) here and (0,0) if they
  534. fail to coordinate;
    and (0,0) here if they fail to
  535. coordinate;
    and (1,2), (1,2) here.
  536. So the implication of this is
    that Player 2 who we'll assume
  537. is David, so David would like to
    meet Nina but all other things
  538. being equal,
    he'd like to meet her at the
  539. Cold War.
    And Nina would like to meet
  540. David, but all things being
    equal, she would like to meet in
  541. China.
    Not literally in China,
  542. but in the class.
    So this is our game and we're
  543. going to analyze this game,
    but before we analyze it let's
  544. try and play it.
    So what we need to do is first
  545. of all let's make sure things
    work smoothly,
  546. let's have David write down
    which class he's going to
  547. choose,
    and Nina write down which class
  548. she's going to choose.
    I've lost sight of Nina.
  549. Somebody has to point
    out--there she is.
  550. Write down which class you are
    going to choose.
  551. Something been written down?
    Jake you got your mikethere,
  552. so Jake are you going to send
    this hapless couple or not?
  553. Student: So I have Dave
    in my section actually and I
  554. hear how much he's been talking
    about Nina, so I'm going to roll
  555. the dice and send them.
    Professor Ben Polak:
  556. He's going to send them
    good, so we have them going off
  557. to this class and now let's see
    what they wrote down.
  558. So Dave what did you write down?
    Student: I'm going to
  559. give in and go to China.
    Professor Ben Polak:
  560. You're going to go to China
    and Nina?
  561. Student: I chose S.
    Professor Ben Polak:
  562. Great, so they managed to
    meet, so it's a successful date.
  563. So let's give them a round of
    applause.
  564. I hear it's a great class too.
    And in fact,
  565. I don't think it's going to
    happen forever because I think
  566. he must be approaching
    retirement.
  567. So that seems a pretty good
    choice.
  568. So good: that worked very well.
    Let's have a look now at this.
  569. Let's analyze this game and see
    what we can do with this game.
  570. So how are we going to analyze
    this game?
  571. So no surprise we're going to
    use the idea of a sub-game
  572. perfect equilibrium,
    I'll collect the mikes later
  573. don't worry.
    So we're going to use the idea
  574. of a sub-game perfect
    equilibrium.
  575. So how do we figure out how to
    work out what a sub-game perfect
  576. equilibrium is?
    We're going to use the same
  577. basic idea that we used--what
    we've been using all along in
  578. backward induction.
    It's the same idea in the game
  579. we just looked at just now.
    What we're going to do is,
  580. rather than start from the last
    decision node (we can't do that
  581. anymore) and work backwards,
    instead of doing that,
  582. we're going to start from the
    last sub-game and work
  583. backwards.
    In this example it's pretty
  584. obvious what the last sub-game
    is.
  585. The last sub-game,
    the game within a game--there
  586. is only really one--the game
    within the game is this object
  587. here, is that right?
    This is the game within the
  588. game.
    Now, I could at this stage,
  589. I could do something else.
    I could write down the whole
  590. matrix for the whole game and
    have Jake choose the matrix,
  591. and Dave and Nina choose the
    row and the column,
  592. but that's going to get us
    astray.
  593. So I mean we could do that but
    let's not worry about that.
  594. Let's just start doing things
    backwards.
  595. So when we do things backwards
    we'll start at the last sub-game
  596. and that last sub-game is an old
    friend of ours,
  597. it looks exactly like this.
    Let's just write it in.
  598. So it involves Players 2 and 3.
    And 2 was choosing between
  599. Gaddis and Spence.
    And 3 was choosing between
  600. Gaddis and Spence.
    And their payoffs were--let me
  601. leave a space here--so it was
    (2,1), (1,2),
  602. (0,0), (0,0).
    So here are the payoffs of the
  603. relevant players in the game but
    while we're here why don't we
  604. put in Jake's payoffs as well.
    So Jake's payoffs were 1 here,
  605. 1 here, -1 here,
    and -1 there.
  606. So the only relevant players
    here are Players 2 and 3,
  607. but I've put Player 1's payoffs
    in as well because:
  608. why not, why not just keep
    track of it.
  609. Everyone happy that that
    exactly describes this little
  610. game?
    For all intents and purposes we
  611. can forget the first payoff but
    there it is.
  612. This is a game we've seen many
    times so far.
  613. It's the battle of the sexes,
    or the battle of Dave and Nina,
  614. and in this game we already
    know what the equilibria are.
  615. So the equilibria here,
    let me just underline the best
  616. responses.
    So if Nina is choosing Gaddis
  617. then Dave chooses Gaddis.
    If Nina is choosing Spence then
  618. Dave would like to choose Spence
    and conversely.
  619. So I've just underlined the
    best responses for the players
  620. who are actually involved in the
    game and haven't bothered
  621. underlining anything for Jake
    because he isn't a player in
  622. this game.
    Does that make sense?
  623. So the pure Nash equilibria in
    this game are essentially
  624. (Gaddis, Gaddis) or (Spence,
    Spence).
  625. That's pretty easy.
    From Jake's point of view,
  626. each of these pure strategy
    Nash equilibria yield a payoff
  627. for him of what?
    What does he get?
  628. If they go to (Gaddis,
    Gaddis), he's happy that they
  629. met and he gets 1 and if they
    choose (Spence,
  630. Spence), he's happy that they
    met and he gets 1.
  631. Jake himself doesn't really
    mind whether Dave and Nina learn
  632. about China or learn about the
    Cold War.
  633. He just wants them to meet.
    So, both of these yield 1 for
  634. Jake.
    They both yield a value of 1
  635. for Player 1,
    who is Jake.
  636. So from Jake's point of view,
    going back a stage--what we're
  637. going to do now,
    just as we did with backward
  638. induction, we're going to roll
    the game back.
  639. So we started by analyzing this
    sub-game and now we're going to
  640. roll it back a stage,
    just as we did with backward
  641. induction.
    So when we roll back Jake is
  642. moving here, if Jake chooses not
    to send them then we get (0,0,
  643. 0) but the key part of this is
    the first 0, that's Jake's
  644. payoff.
    And if Jake sends them,
  645. then what Jake gets is the
    value to Jake--I'll put in Jake
  646. but value to Player 1--okay
    value to Jake of the Nash
  647. equilibria in this sub-game.
    Not a big thing to write,
  648. but that's what Jake gets.
    And the others do too:
  649. the others get that as well.
    In this case,
  650. rather than writing that long
    piece, this is just equal to 1.
  651. Is that right?
    So if Jake sends them he knows
  652. that they're going to play a
    Nash equilibrium in this
  653. sub-game,
    or he believes they're going to
  654. play a Nash equilibrium in this
    sub-game.
  655. And either of those two Nash
    equilibria in the sub-game yield
  656. a payoff to Jake of 1.
    So actually,
  657. one of them is 1,
    one of them yields payoffs of
  658. (1,1,2);
    and the other yields payoffs of
  659. (1,2,1).
    But since Jake is the only
  660. mover here, let's just focus on
    Jake.
  661. So from Jake's point of view
    he's really choosing between 0
  662. and 1, so he's going to choose
    send.
  663. So the sub-game perfect
    equilibrium therefore--there are
  664. actually two of them here--one
    is (send, Spence,
  665. Spence), that's what actually
    happened.
  666. But there's another one which
    is (send, Gaddis,
  667. Gaddis) that would also have
    been a pure strategy sub-game
  668. perfect equilibrium.
    So in either case,
  669. what we did,
    just to remind ourselves,
  670. we first of all solved the
    equilibrium down the sub-game,
  671. the equilibrium in this blue
    sub-game.
  672. We figured how much that
    equilibrium was worth for
  673. everybody--but in particular for
    Jake,
  674. but for everybody--and then we
    rolled that payoff back and
  675. looked at Jake's choice.
    In this particular case that
  676. game has two equilibria,
    (send, Spence,
  677. Spence) and (send,
    Gaddis, Gaddis).
  678. However, some of you must be
    suspecting at this point that
  679. there's actually another
    sub-game perfect equilibrium
  680. here.
    How do we know that?
  681. Well let's just think about
    this game.
  682. We've been trying to send this
    couple on a date all semester.
  683. They haven't gone on a date all
    semester.
  684. I'm embarrassing them,
    but they haven't gone on a date
  685. all semester.
    So there must be some
  686. possibility that they would fail
    to coordinate.
  687. It would be a pretty weird
    notion of equilibrium that
  688. concluded that they always
    manage to coordinate and hence
  689. Jake always wants to send them.
    Is that right?
  690. So let's also look at the other
    equilibria here.
  691. Now the reason there's another
    equilibrium in this
  692. sub-game--sorry,
    the reason there's another
  693. equilibrium in the whole
    game--the reason there's another
  694. sub-game perfect equilibrium in
    the whole game is that there's
  695. another Nash equilibrium in the
    sub-game.
  696. What's the other Nash
    equilibrium in the sub-game?
  697. They could mix.
    So it turns out that in the
  698. sub-game (here it is) there's
    also a third mixed equilibrium.
  699. There is a mixed Nash
    equilibrium.
  700. Now we know how to work that
    out.
  701. We could write down a P and a
    Q, and we could look for those
  702. indifference conditions and
    solve it out.
  703. But this is a sub-game,
    sorry this is a game,
  704. this sub-game corresponds to a
    game we've seen many times in
  705. this class so far,
    and I think we probably
  706. remember what that equilibrium
    is.
  707. Is that right?
    I do anyway,
  708. so let's see if you remember it
    as well.
  709. I'll write it down and we'll
    see if you all looked alarmed.
  710. So I claim the equilibrium,
    the other equilibrium has Dave
  711. playing with probability
    (2/3,1/3), and has Nina playing
  712. with probability (1/3,2/3).
    So this is another equilibrium
  713. in the sub-game.
    People remember that this was
  714. an equilibrium in battle of the
    sexes?
  715. Yeah, people are nodding at me,
    yeah okay.
  716. So it isn't too unintuitive.
    We all know how we'd work it
  717. out.
    We could go back and put in the
  718. P and the Q, but it isn't too
    unintuitive, it has Dave going
  719. more often to the lecture course
    that he would prefer all other
  720. things being equal;
    and it has Nina going more
  721. often to the lecture course that
    she would prefer all other
  722. things being equal.
    And they do so in just such a
  723. way as to make each of them
    indifferent.
  724. Now, this sub-game induces a
    different value for Jake.
  725. So suppose Jake thinks:
    "I trust Dave and Nina to play
  726. a Nash equilibrium in their
    sub-game but I don't know which
  727. one it is and I think maybe
    they're going to play this one."
  728. So suppose Jake thinks that
    this is the equilibrium that
  729. Dave and Nina are going to play.
    So now should Jake send them or
  730. not?
    Well let's work it out.
  731. So now if he sends them,
    if Jake sends Dave and Nina,
  732. or more anonymously if Player 1
    sends Players 2 and 3,
  733. then with what probability will
    they meet?
  734. Well this is just a little math
    exercise, let's have a look at
  735. the game again.
    So Dave is playing 2/3,1/3,
  736. is that right?
    Nina is playing 2/3,1/3,
  737. is that right?
    So the probability of their
  738. meeting is the probability of
    this box, they could meet at
  739. Gaddis,
    plus the probability of this
  740. box, they could meet at Spence,
    is that right?
  741. So the probability of this box
    is 1/3 x 2/3 so this box has
  742. probability 2/9 and the
    probability of this box is 2/3 x
  743. 1/3 so this box has probability
    2/9.
  744. So the probability of their
    meeting is 2/9 + 2/9 that makes
  745. 4/9.
    Everyone okay with that?
  746. So if Jake sends Dave and Nina
    and they play this mixed
  747. strategy equilibrium,
    then they meet with probability
  748. 2/9 + 2/9,2/9 at Gaddis,
    2/9 at Spence for a total of
  749. 4/9,
    which means they failed to meet
  750. with probability--well if
    they're meeting with probability
  751. of 4/9 what must be the
    probability that they're failing
  752. to meet?
    5/9, thank you.
  753. So they fail to meet with
    probability of 5/9.
  754. So Jake's expected payoff if he
    sends them, the value for Jake
  755. of this equilibrium is what?
    So the value to Jake of this
  756. Nash equilibrium,
    if he sends them,
  757. is 4/9 x 1 + 5/9 x -1 for a
    total of -1/9.
  758. Everyone okay with that?
    So if Jake sends them they fail
  759. to meet 5/9 of the time and he
    gets -1 each of those times.
  760. They succeed in meeting 4/9 of
    the time, he gets +1 each of
  761. those times, so his expected
    payoff,
  762. his expected value from sending
    Dave and Nina on the date is
  763. -1/9.
    So from Jake's point of view,
  764. what this game looks like,
    if he thinks that this is the
  765. Nash equilibrium being played:
    if he doesn't send he gets 0
  766. and if he does,
    he gets the value of this Nash
  767. equilibrium, which in this case
    is -1/9.
  768. So he's not going to send and
    the SPE here is (not send,
  769. "mix," "mix") where this is the
    mix.
  770. So there's a third equilibrium
    here in which our matchmaker
  771. says this hapless couple is just
    too hapless: they're going to
  772. play the mixed strategy
    equilibrium in which case it
  773. isn't worth my while sending
    them on the date.
  774. You guys were lucky because
    Jake chose the other
  775. equilibrium.
    He figured you were playing the
  776. other equilibrium,
    which it turned out that you
  777. were.
    So in this game there were
  778. three sub-game perfect
    equilibria, one for each of the
  779. Nash equilibria in the sub-game,
    as it turned out.
  780. There was one in which Jake
    sent them and they coordinated
  781. on the pure strategy equilibrium
    in the game (S,S).
  782. There was one in which Jake
    sent them and they coordinated
  783. on the pure strategy equilibrium
    in the sub-game (G,G).
  784. And there's one in which Jake
    didn't send them,
  785. but had he had in fact sent
    them,
  786. they would have both mixed,
    and hence, for a lot of the
  787. time, failed to coordinate.
    Now, what's the big lesson here?
  788. The big lesson of the first
    game we saw this morning was
  789. that sub-game perfect
    equilibrium implies backward
  790. induction.
    The big lesson of this
  791. game--other than the fact that
    we're getting closer to getting
  792. Dave and Nina on their date--the
    big lesson of this game is to
  793. show that to find sub-game
    perfect equilibria,
  794. all you have to do is keep your
    head and solve out the Nash
  795. equilibria in each of the
    sub-games,
  796. roll the payoffs back up,
    and then look for behavior up
  797. the tree.
    Once again, you look for the
  798. Nash equilibria in each of these
    sub-games, roll the payoffs back
  799. up,
    and then see what the optimal
  800. moves are higher up the tree.
    So we have time to do one more
  801. example, and the third example I
    want to do is more of an
  802. application.
  803. So far we've seen some fairly
    simple examples.
  804. Now I want to do an application.
  805. The application I want to do is
    kind of a classic business
  806. school case if you like,
    or a mini case involving
  807. strategic investment.
  808. The game is this,
    or the setting is this.
  809. There are two firms,
    we'll call them A and B.
  810. And these two firms,
    initially, before we start
  811. considering what we're actually
    going to talk about,
  812. initially they are playing
    Cournot competition.
  813. So two firms and they're
    playing Cournot competition.
  814. And we can imagine that they're
    producing fertilizer.
  815. And let's be specific here,
    let's assume that the prices in
  816. this market are given by the
    following demand curve 2 - 1/3 x
  817. [qA + qB],
    so this is the demand curve
  818. that they face.
    We'll assume that costs,
  819. marginal costs,
    c is equal to $1 a ton.
  820. So this is the price in dollars
    per ton, and the costs are $1
  821. per ton.
    In a minute what we're going to
  822. do is we're going to consider a
    change in this game,
  823. but before we do that let's
    just remind ourselves what the
  824. Cournot equilibrium of this game
    would look like.
  825. Let's do a bit of a review.
    So it's been a while since
  826. we've seen Cournot,
    so let's remind ourselves.
  827. So I claim that the quantity,
    the Cournot quantity chosen Q*
  828. has the formula [a--c]/ 3b.
    Is that right?
  829. If you go back in your notes
    you'll find it.
  830. I'm not going to re-solve it
    here.
  831. We've done it many times.
    So [a--c]/3b trust me,
  832. is what came out of our
    calculation before the mid-term.
  833. What I want to do is,
    I just want to make sure we can
  834. translate that into numbers
    here.
  835. Sorry for having it in letters,
    but let's translate it into
  836. numbers.
    So in particular,
  837. this a is this 2,
    is that right?
  838. This c is this 1 and this b is
    this 1/3, is that right?
  839. So let's just put that down.
    So in this case this is
  840. [2--1]/[3 x 1/3],
    so this says that this is a
  841. million tons.
    So the quantity here,
  842. the Cournot quantity is a
    million tons each.
  843. So: one each.
    So in this equilibrium,
  844. each of these two firms is
    producing a million tons of
  845. fertilizer.
    What else do we know?
  846. We know therefore what prices
    must be.
  847. Let's just do that before we
    even get started.
  848. So prices must be [2 - 1/3]
    times the quantity that the
  849. first firm produces plus the
    quantity that the second firm
  850. produces.
    So that's 2 - 2/3 so that
  851. should be 4/3,
    if I've got that right,
  852. or one and a third.
    So prices here are $1.33 per
  853. ton.
    Finally profits.
  854. So profit for each firm here,
    in this equilibrium,
  855. before we even start the game,
    or start the more interesting
  856. part of the game,
    profit is what?
  857. So they're going to get $1 and
    1/3 for every ton they produce.
  858. It's going to cost them $1 to
    produce each ton and they're
  859. producing one million of these
    things.
  860. So their profits are 1/3,
    if these are millions,
  861. their profits are 1/3 of $1
    million dollars.
  862. So this is their per period
    profit, in each period they're
  863. doing this, each year they're
    doing this and this is their
  864. profits in each period.
    So this is a simple model that
  865. we've done many times before.
    This is Cournot,
  866. and now we're going to make it
    more interesting.
  867. If the algebra here was a bit
    quick don't worry about it,
  868. check it at home,
    it's just basic,
  869. basic algebra.
    So now suppose that you are the
  870. manager of Firm A.
    So it's a classic business
  871. school case.
    I'm looking at my
  872. business-school students in the
    balcony.
  873. You're the manager of Firm A
    and you have to choose whether
  874. to accept an offer to rent a new
    machine.
  875. So this new machine has two
    features.
  876. The first--well three
    features--the first feature is
  877. it only works for A.
    So this machine is being
  878. offered to you.
    It wouldn't fit in to Firm B's
  879. technology, so this is only
    being offered for A.
  880. The second feature of this
    machine is it costs $0.7 million
  881. dollars in rental.
    So each year you rent this
  882. machine, you'd have to pay $0.7
    million dollars.
  883. But that's the bad news.
    The good news is it will lower
  884. A's costs to $0.50 a ton.
    So classic business-school
  885. situation.
    You're the manager of a firm.
  886. You're involved in competition
    with another firm,
  887. B.
    And suddenly an opportunity
  888. comes along to rent some new
    technology.
  889. It's going to cost you $0.7 a
    year to rent this machine,
  890. but it will lower your costs by
    $.50 a ton.
  891. So this is a classic thing that
    you might be asked in your
  892. interview for Morgan Stanley
    next week.
  893. How many of you are
    interviewing with investment
  894. banks?
    No one's going to admit it.
  895. In a couple of years,
    when you're interviewing with
  896. these guys.
    So what's the obvious question?
  897. The obvious question is,
    should you go ahead and rent
  898. this new technology or not?
    Should you rent it or not?
  899. To rent or not to rent?
  900. Less dramatic than it's
    equivalent question in the
  901. English class,
    but important nevertheless.
  902. This board is stuck
    unfortunately,
  903. so I'll have to write there a
    bit more.
  904. So what I want to do is I want
    to analyze this three times and
  905. each time I analyze it,
    I want us to see what I'm
  906. doing--what mistakes I'm
    making--because I want you guys,
  907. when you interview with Morgan
    Stanley about this kind of
  908. thing, to impress them so that
    they tell lots of people to come
  909. to Yale and preferably give lots
    of money to Yale.
  910. So we're going to look at this
    way three different times.
  911. And the first thing we're going
    to do, the first way we're going
  912. to look at this is look at it as
    if we were accountants.
  913. We're going to look at the
    accountants' answer to this
  914. question, and some of you may
    decide you don't want to
  915. interview with Morgan Stanley or
    McKinsey,
  916. you might want to interview
    with some accounting firm when
  917. you leave Yale.
    God forbid, but you might.
  918. So let's have a look at how the
    accountants would answer this
  919. question.
    So I think the accountants
  920. would do this.
    They would say--but before we
  921. do this let's have a poll.
    How many people think you
  922. should rent?
    You've had some time to think
  923. about it now.
    So how many people think you
  924. should rent the new machine?
    How many people think you
  925. should not rent the new machine?
    You're not allowed abstensions
  926. here.
    Let's try it again:
  927. no abstentions right.
    You can't abstain in an
  928. interview.
    So you're on the spot,
  929. you're in the boardroom,
    how many think you should rent
  930. the new machine?
    Raise your hands.
  931. Wave them in the air.
    How many people think you
  932. should not rent the new machine?
    So we're split kind of down the
  933. middle, I'm looking at my MBA
    students to see which they
  934. voted.
    Which did you guys vote,
  935. rent or not rent?
    Rent, the MBA's seem to think
  936. rent.
    We'll see if that's right.
  937. So let's move forward,
    so accountant's answer.
  938. So I think what the
    accountant's going to say is
  939. this.
    They're going to say,
  940. right now you're producing a
    million tons a year.
  941. The new machine saves you--so,
    let's put per annum--let's try
  942. and be fancy--so a million tons
    per annum.
  943. The new machine saves you $.50
    per ton.
  944. So if you rent this new
    machine, you're producing a
  945. million tons a year,
    it's going to save you $.50 a
  946. ton.
    So it's going to save you half
  947. of one million a year in
    variable cost.
  948. Those people who were in 115 or
    150 will know what I mean by
  949. variable cost.
    It's the cost you're going to
  950. save in the actual production of
    your fertilizer.
  951. So it saves you half a million
    a year.
  952. Unfortunately,
    it costs you the cost of the
  953. machine, which is a fixed cost
    of $0.7 million a year.
  954. And .7 is bigger than .5,
    so you should not rent.
  955. So .5 is less than .7 so don't
    rent the machine.
  956. How many of you said no?
    That's kind of the back of the
  957. envelope calculation you were
    doing, is that right?
  958. Kind of the back of the
    envelope calculation that
  959. accountants do.
    So what's going on here?
  960. Our business school student up
    in the balcony says you should
  961. rent.
    He took accounting.
  962. I know he did that because you
    have to take accounting at
  963. business school.
    So what's wrong?
  964. Did he fail accounting,
    or is this answer wrong?
  965. The answer is wrong.
    There's two things you need to
  966. know about accountants.
    One is that they're usually
  967. boring, and the other is that
    they're often wrong.
  968. They're more often boring than
    wrong, but they're almost always
  969. boring.
    So this answer is kind of
  970. boring, and it happens also to
    be wrong.
  971. Why is it wrong?
    It's wrong because we made an
  972. assumption here that's not a
    good assumption.
  973. We made the assumption that
    you're going to go on producing
  974. the same amount per year after
    you've invested in the new
  975. machine that lowers marginal
    costs as you are producing
  976. beforehand.
    We made the assumption that it
  977. would lower--we know it lowers
    your costs--and we assumed
  978. implicitly that you would go on
    producing a million tons a year,
  979. but that's not right.
    So let's try and have a more
  980. sophisticated answer,
    and if you want to be more
  981. sophisticated and less boring
    than accounting,
  982. what class would you want to
    take?
  983. Economics probably right,
    so let's have a look at an
  984. Economics answer.
    Let's look at an Economics 115
  985. answer.
  986. How many of you have taken
    Economics 115?
  987. How many of you are in 115 at
    the moment?
  988. Quite a few,
    okay so let's have a look at
  989. the Economics answer.
    Let's see why that previous
  990. answer was wrong.
    So here is qA and here is cost
  991. of a $1, your new cost will be
    $.50.
  992. So I'm putting prices and costs
    on this axis.
  993. And here is your residual
    demand curve.
  994. This is the demand curve you
    face after the other guy has
  995. finished producing,
    so this is your residual demand
  996. curve.
    It's the demand curve on that
  997. part of the market you're
    supplying, or not being supplied
  998. by the other side of the market.
    And to figure out your optimal
  999. quantity on your residual demand
    curve what you should do.
  1000. It's like you're a monopolist
    on this residual demand curve,
  1001. so you should set what?
    If the answer isn't backward
  1002. induction it's probably marginal
    revenue equals marginal cost
  1003. right?
    So let's try that.
  1004. So here's the marginal revenue
    curve roughly speaking,
  1005. should be twice as steep.
    This is residual marginal
  1006. revenue and here's what you used
    to produce.
  1007. So we know what this is:
    this was a million tons.
  1008. This was marginal revenue
    hitting marginal cost.
  1009. Now your costs have gone down
    so notice that your quantity,
  1010. your new quantity has gone up.
    Your new quantity has gone up
  1011. because you slid down the
    marginal revenue curve as the
  1012. marginal cost curve dropped.
    Is that clear to everybody?
  1013. So this is the kind of picture
    that you probably saw a lot of
  1014. in 115, is that right?
    Or in 150 for that matter,
  1015. is that correct?
    So notice in this picture we
  1016. can actually see the
    accountant's answer,
  1017. the boring answer.
    The boring answer is this
  1018. rectangle.
    This rectangle,
  1019. this is the accounting answer.
    This rectangle is 0.5 x 1,
  1020. so it comes out as a 1/2 and
    that was the accountant's
  1021. answer.
    And what did they miss.
  1022. What did the accountant's miss?
    They missed the triangle.
  1023. So I told you they were boring,
    they were a little bit square,
  1024. so they tend to miss triangles.
    So here's the triangle that
  1025. they missed.
    So we missed this triangle.
  1026. And how big is this triangle?
    Well we could do it at home,
  1027. it's 1/2 base x height.
    So we could figure this out.
  1028. We know the slope of this line.
    We know the slope of this line
  1029. is 1/3.
    We know the slope of this line
  1030. is 2/3.
    We know that the height of this
  1031. triangle is 1/2.
    We could figure out what the
  1032. width is as well,
    therefore we could do half base
  1033. times height.
    Turns out that this has area--I
  1034. did this at home so let me just
    write it down--it has area 3/16.
  1035. So again, everyone could figure
    out the area of a triangle at
  1036. home, is that right?
    You all know that from your
  1037. probably junior high school
    geometry.
  1038. So assuming I did it correctly
    at home, this is 3/16 which is
  1039. approximately 0.19.
    So we missed this 0.19.
  1040. So how are we doing now?
    So we know from the accounting
  1041. answer we had a 1/2 in savings.
    We know from the Economics
  1042. answer that we should add
    another 0.19 to this--that's the
  1043. triangle--for a total of 0.69
    but unfortunately this is still
  1044. less than .7 which is the cost
    of the machine,
  1045. the per annum cost of the
    machine.
  1046. So it looks like we should
    still not rent.
  1047. So even after taking Economics
    115, which is a good thing to
  1048. do--it is much less boring than
    accounting and will get you the
  1049. accounting answer anyway if you
    do things carefully--we still
  1050. end up concluding you shouldn't
    rent.
  1051. But our guy from the business
    school said you should rent,
  1052. right?
    So did he fail Economics as
  1053. well as accounting,
    or is this answer wrong?
  1054. This answer is still wrong.
  1055. [I didn't really want to delete
    that.
  1056. That's a shame.
    Oh well, I've done it now.
  1057. People've got those numbers
    somewhere.
  1058. I hope they are in my notes.]
    This answer is still wrong.
  1059. We need to get the right answer.
    So what we need is a third
  1060. answer which is the Game Theory
    answer, which is also known as
  1061. the right answer.
  1062. What are we missing?
    What's wrong with the Economics
  1063. answer?
    Somebody?
  1064. Everyone knew it was wrong,
    why is it wrong?
  1065. It looked pretty good,
    what's wrong with it?
  1066. Let's bounce down here.
    Somebody in the front row will
  1067. help me.
    What did I do wrong?
  1068. Student: Firm B also
    changes its quantity.
  1069. Professor Ben Polak:
    Right, in the accounting
  1070. answer, we assumed that Firm A
    kept its quantity fixed,
  1071. that we kept our quantity
    fixed, and that was wrong.
  1072. But in addition,
    Firm B is going to change its
  1073. quantity, isn't it?
    Firm B is going to change its
  1074. quantity.
    Let's remind ourselves why.
  1075. We're still playing Cournot
    competition, here's our Cournot
  1076. diagram with qA and qB.
    So prior to making this
  1077. investment the model is
    symmetric.
  1078. Here it is.
    And this is the old best
  1079. response of Firm A,
    and this is the best response
  1080. of Firm B.
    What we learned just now in the
  1081. Economics answer is what?
    We learned that Firm A,
  1082. as its costs go down,
    will produce more for each
  1083. possible quantity that Firm B
    produces.
  1084. So regardless of what generated
    this residual demand curve as
  1085. the costs go down for Firm A,
    it increases its quantity.
  1086. So we know that.
    So what's that telling us?
  1087. It's telling us that the new
    best response of Firm A is
  1088. shifted to the right.
    This is the new best response
  1089. for Firm A.
    It's shifted to the right.
  1090. It now produces more for any
    given quantity that Firm B
  1091. produces.
    So qA has gone up.
  1092. That was our Economics answer.
  1093. But that leads Firm B to do
    what?
  1094. To produce less.
    That leads to qB producing less.
  1095. Notice we slid down Firm B's
    best response line from the old
  1096. equilibrium to the new
    equilibrium,
  1097. and at the new equilibrium Firm
    B's production has gone down.
  1098. By the way, what kind of game
    is it where as Firm A increases
  1099. its strategy,
    Firm B decreases its strategy
  1100. in response?
    Strategic substitutes,
  1101. right.
    So because this is a game of
  1102. strategic substitutes--good
    interview word,
  1103. good word to mention in an
    interview--because this is a
  1104. game of strategic substitutes,
    we know that Firm B reduces its
  1105. quantity.
    As Firm B reduces its quantity
  1106. is that good for A or bad for A?
    It's good for A, right?
  1107. This is good for Firm A,
    it softens competition.
  1108. As a consequence,
    it leads to an increase in
  1109. profit.
    Again--we don't have time
  1110. today--but what we could do is
    we could go back and we could
  1111. recalculate the new Cournot
    equilibrium.
  1112. We could calculate this--we
    could as a homework exercise try
  1113. it--calculate the new Nash
    equilibrium and notice this is a
  1114. Nash equilibrium in a sub-game.
    Why is this a sub-game?
  1115. Because Firm A made its
    decision whether to buy this new
  1116. machine or not,
    and then they played Cournot.
  1117. So the Cournot game,
    the game up here,
  1118. what is this?
    This is the sub-game.
  1119. This is the diagram of the
    sub-game, if you like.
  1120. It's the best responses of the
    sub-game.
  1121. So what sub-game perfection
    tells us to do here is first of
  1122. all, work out the new
    equilibrium in the sub-game,
  1123. work out how much that new
    equilibrium is worth for Firm A
  1124. and then roll it back to the
    investment decision.
  1125. It turns out when we do that we
    get an extra $.31 million.
  1126. So we could do it at home,
    we get an extra $.31 million.
  1127. So it turns out that our MBA
    student was right,
  1128. good.
    It turns out that if you add
  1129. this .31 to the .69 we had
    already we get 1,
  1130. which of course is much bigger
    than .7 and indeed you should
  1131. rent the machine.
    Now, I want you to have two
  1132. take away lessons from this
    game.
  1133. The first take away lesson is
    this.
  1134. When you're analyzing a game
    like this, be it in the real
  1135. world or in a job interview,
    the first thing you want to do
  1136. is what?
    You want to look at the
  1137. sub-game.
    You want to look at what would
  1138. happen if you did invest and
    solve out the new Nash
  1139. equilibrium in that sub-game.
    Then you want to roll back the
  1140. value of that sub-game back into
    the initial decision which is
  1141. the strategic investment
    decision whether to rent this
  1142. machine or not.
    So schematically,
  1143. the game looks like this:
    rent or not rent,
  1144. and in either case you play
    Cournot.
  1145. There's a sub-game in each case.
    In this case you play symmetric
  1146. Cournot, when you both have the
    same costs;
  1147. and here you play asymmetric
    Cournot, where you have
  1148. different costs.
    And the way we analyze this
  1149. game is, we solve out the
    symmetric Cournot,
  1150. we actually did that up front.
    We now solve out the new
  1151. equilibrium in this asymmetric
    Cournot game,
  1152. this one here.
    This is the old one and this is
  1153. the new one.
    Solve it out.
  1154. Work out how much profit you're
    going to get.
  1155. And roll that back remembering
    that it costs you $.7 million to
  1156. make this step.
    So that's the first take away
  1157. lesson.
    But the second take away lesson
  1158. is more general,
    so let me just pause to get
  1159. everyone to wake up again so I
    make it.
  1160. The second take away lesson is
    this., What tips the balance
  1161. here from the Economics answer
    and the accounting answer,
  1162. were the strategic effects.
    It was the strategic effect.
  1163. This is a strategic effect.
  1164. It was the effect of the other
    firm or other players changing
  1165. their behavior.
    And the most common mistake to
  1166. make when you're thinking about
    strategic decisions is what?
  1167. It's to forget that they're
    strategic.
  1168. It's to forget that the other
    players are going to change
  1169. their behavior.
    In this example,
  1170. the other firm cuts back its
    production so much as to make
  1171. that investment profitable.
    But let me give you two other
  1172. examples.
    Example number one.
  1173. You're designing a tax policy
    for the U.S..
  1174. The dumb way to analyze this is
    to say look at what people are
  1175. doing now, push through the new
    tax numbers,
  1176. and act like an accountant and
    crunch out how much money the
  1177. government's going to make.
    Why is that wrong?
  1178. Because you're forgetting that
    as you change the tax code
  1179. people's behavior changes.
    Incentives change and people's
  1180. behavior changes.
    It leads to a mistake in
  1181. designing the tax code.
    You need to take into account
  1182. strategic effects:
    how behavior changes.
  1183. Example number two,
    closer to home.
  1184. You're designing a new
    curriculum for Yale.
  1185. So you change the rules of the
    curriculum and when analyzing it
  1186. you say--I wouldn't say this but
    some people on the committee
  1187. might say this--under these new
    rules,
  1188. if we look at what people used
    to do, they will now do more of
  1189. this and less of that,
    and they'll learn this and
  1190. learn that.
    What are you missing?
  1191. You're missing that students
    are players and students change
  1192. their behavior as you change the
    curriculum rules.
  1193. So the biggest lesson of
    today's class is don't be like
  1194. an accountant,
    partly because it's boring and
  1195. you won't go on your date,
    and partly because you'll miss
  1196. out on these important strategic
    effects.
  1197. We'll come back and look at
    more on Wednesday.