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← 17. Backward induction: ultimatums and bargaining

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  1. Professor Ben Polak:
    So today I want to look at
  2. two kinds of games and then
    we'll change topic a bit.
  3. The games I want to look at are
    about ultimatums and bargaining.
  4. And we'll start with ultimatums
    and we'll move smoothly through,
  5. and we'll see why that's an
    easy transition in a while.
  6. So the game we're going to play
    involves two players,
  7. 1 and 2 and the game is this.
    Player 1 is going to make a
  8. take it or leave it offer to
    Player 2 and this offer is going
  9. to concern a pie and let's make
    the pie worth $1 for now.
  10. We'll probably play this for
    real in a bit so we'll play it
  11. for $1.
    So it's a split of a pie,
  12. we can think of the split as
    offering S to Player 1 and
  13. (1--S) to Player 2.
    Player 2 has two choices:
  14. 2 can accept this offer,
    and if 2 accepts the offer then
  15. they get exactly the offer.
    Player 1 gets S and Player 2
  16. gets (1--S).
    Alternatively,
  17. Player 2 can reject,
    and if Player 2 rejects the
  18. offer then both players get 0,
    both players gets nothing.
  19. So a very simple game,
    there's a dollar on the
  20. table--I'll take it out in a
    minute--so there's a dollar on
  21. the table,
    and our two players are going
  22. to bargain for this,
    but it isn't much of a bargain.
  23. Player 1's basically going to
    announce what the division's
  24. going to be.
    2 can either accept that
  25. division or no one gets
    anything.
  26. Everyone understand the game?
    So I thought we'd start off by
  27. playing this game for real a
    couple of times,
  28. so why don't I come down and do
    that.
  29. So why don't we play with some
    people in this row.
  30. I've been playing with that row
    all the time,
  31. so Ale's going to help me find
    somebody.
  32. You found somebody okay,
    so the person behind you?
  33. Your name is?
    You have to shout out,
  34. this doesn't make a noise.
    Equia, thank you.
  35. So you'll be Player 1,
    and Player 2 will be this
  36. gentleman here whose name is?
    Student: Noah.
  37. Professor Ben Polak:
    Noah, okay.
  38. So Equia, do you know each
    other?
  39. Okay, so Equia can make any
    offer she wants.
  40. We'll play this for real money,
    so there's a real dollar at
  41. stake.
    You can make any offer you
  42. want--it can be in fractions of
    pennies if you like--of the
  43. amount of the dollar that you're
    going to offer to Noah,
  44. and the amount you're going to
    keep yourself.
  45. There you go,
    shout it out so everyone can
  46. hear.
    Stand up, this is your moment
  47. in the lights all right.
    Will you stand up as well?
  48. There we go.
    Student: I'm going to
  49. offer him a penny.
    Professor Ben Polak:
  50. You're going to offer him a
    penny, and Noah are you going to
  51. take a penny?
    Student: No.
  52. Professor Ben Polak:
    Noah's not going to take a
  53. penny.
    So I didn't lose any money on
  54. that: so no one made any money.
    I didn't lose any money.
  55. that seems pretty good.
    Let's try a different couple.
  56. Let's move around the room a
    little bit.
  57. So Ale why don't you take the
    guy behind here who name is?
  58. Shout out.
    Student: Gary.
  59. Professor Ben Polak:
    Gary, all right so Gary why
  60. don't you stand up,
    and we'll let Gary play with
  61. the gentleman in here,
    what's your name in here?
  62. Student: Anish.
    Professor Ben Polak:
  63. Anish, all right stand up,
    do you know each other?
  64. So why don't we let Anish be
    Player 1 this time.
  65. You understand the rule of the
    game?
  66. So make your offer.
    Student: $.30.
  67. I'm offering him $.30.
    Professor Ben Polak:
  68. You're offering him $.30?
    He's saying no as well.
  69. Okay, so let's raise the stakes
    a bit.
  70. Let's make it $10,
    since we're not getting much
  71. acceptance here.
    Let me try a third couple,
  72. working my way back,
    so how about you,
  73. what's your name?
    Student: Courtney.
  74. Professor Ben Polak:
    So why don't you stand up
  75. as well.
    We'll let Courtney be Player 1.
  76. Actually why don't you sit down
    so I can do it from here,
  77. and you are?
    Student: Danny.
  78. Professor Ben Polak:
    Danny, all right so
  79. Courtney's going to be Player 1
    and Danny's going to be Player
  80. 2.
    These weren't our dating couple
  81. from earlier right?
    We're safe on that?
  82. Okay good, so Courtney what are
    you going to offer?
  83. Student: $5.
    Professor Ben Polak:
  84. $5, which is half of the
    $10.
  85. Student: Accept.
    Professor Ben Polak:
  86. All right accept,
    all right.
  87. So it turned out in this game
    that a lot of people were
  88. rejecting offers.
    Let's have a look at it on the
  89. board a second.
    Let's think about it a second
  90. and we'll come back,
    we're not done with this
  91. couple.
    We're not going to need new
  92. couples, we're going to come
    back to you.
  93. So in this game--it's pretty
    simple to analyze this game by
  94. backward induction.
    By backward induction,
  95. we're going to start with the
    receiver of the offer and the
  96. receiver of the offer is
    choosing between the offer made
  97. to them which in our notation is
    (1-S).
  98. And in our three examples that
    was a penny, then it was $.30
  99. and then it was half of it,
    so $.50 just to be consistent.
  100. We actually saw two of those
    offers rejected,
  101. but according to backward
    induction,
  102. assuming that people are trying
    to maximize their dollar payoffs
  103. here, what should we see the
    receiver do?
  104. They should accept even the
    somewhat insulting offer of a
  105. penny that was made:
    even that somewhat insulting
  106. offer should be accepted by
    Noah.
  107. So Noah didn't accept the offer
    of a penny and let's come back
  108. over there.
    So Noah didn't accept the offer
  109. a penny, and I forget who it was
    but our second player didn't
  110. accept an offer of $.30,
    but we've just argued that they
  111. should have been accepted.
    In fact, when Equia made the
  112. offer of one penny to Noah,
    I think she assumed that Noah
  113. would accept it,
    is this right?
  114. Why did you think Noah would
    accept it?
  115. Student: Because I felt
    he would be better off with a
  116. penny than nothing.
    Professor Ben Polak:
  117. He'd be better off with a
    penny than nothing,
  118. right.
    Then we saw an offer.
  119. The offer came up a bit.
    Who was my second offerer?
  120. You were my second offerer,
    you offered a bit more.
  121. Why did you offer more?
    Student: I felt $.30 was
  122. a pretty fair share.
    It's a lot better than nothing.
  123. Professor Ben Polak:
    Thirty cents is better than
  124. nothing, although not
    necessarily fair,
  125. but I guess it's better than
    nothing.
  126. Where was my rejector of the
    second offer?
  127. Who rejected the second offer?
    Why did you reject $.30?
  128. Student: Just a pride
    thing.
  129. Professor Ben Polak:
    It's a pride thing.
  130. Pretty soon we converged onto
    $.50, which notice is no where
  131. near backward induction.
    So the third offer,
  132. which is Courtney,
    why did you offer half of it?
  133. Student: Because half is
    better for me than nothing.
  134. Professor Ben Polak:
    Half is better for you than
  135. nothing, and you figured he's
    going to reject otherwise,
  136. and in fact he didn't reject.
    Why didn't you reject?
  137. Student: Because $5 is
    better than nothing.
  138. Professor Ben Polak:
    $5 is better than nothing.
  139. But the $5 is better than
    nothing argument would have
  140. argued against making any
    rejection in this game.
  141. Is that right?
    So here's a game where backward
  142. induction is giving a very clear
    prediction.
  143. The clear prediction is,
    first of all,
  144. the second player will accept
    whatever's given to them;
  145. and second, given that,
    the first player should offer
  146. them essentially nothing,
    should offer them just a penny.
  147. So backward induction predicts
    that the offer will be
  148. essentially, let's say $.99 and
    $.01, or even virtually $1 and
  149. nothing.
    But in fact we don't get that,
  150. we get a lot of rejection of
    these low offers,
  151. and often we get offers made
    much, much higher in the
  152. vicinity of half.
    Now why?
  153. Why are we seeing a failure of
    backward induction in this game?
  154. I think this is not necessarily
    you guys.
  155. It's a very reliable result in
    experimental data.
  156. So why do we see so many people
    in this ultimatum game both
  157. offer more, and,
    more importantly,
  158. reject less than small amounts.
    So let's talk about it,
  159. so one person said it's a pride
    thing.
  160. Let's try the other aisle here,
    so what's the smallest offer
  161. you would have accepted.
    What's your name first of all?
  162. Student: Jeff. A cent.
    Professor Ben Polak:
  163. So there are some backward
    induction players in the room.
  164. Who would have rejected a cent?
    Who would have rejected $.10?
  165. We should be going down at
    least, who would have rejected
  166. $.30?
    Few people rejected $.30,
  167. not many actually.
    How many people would have
  168. rejected $.50?
    One person even would have
  169. rejected $.50,
    but essentially no one.
  170. So what's happening here?
    Why do people think people are
  171. rejecting what is essentially
    money from my pocket,
  172. there's nothing going on here,
    I'm just giving you money.
  173. Why are they rejecting being
    given money?
  174. Student: Overall the
    stakes are really low,
  175. so if you have any value on
    sort of like pride,
  176. what people said,
    you know it's not worth a penny
  177. or $.10.
    Professor Ben Polak:
  178. All right,
    so it may be pride going on
  179. here, so certainly one thing is
    about pride.
  180. It turns out that people do
    this even in quite high stakes
  181. games, but you're right,
    certainly that trade off is
  182. going to start to bite.
    What else is going on?
  183. So I agree, pride is part of
    what's going on.
  184. What else is going on here?
    Let's try and get some
  185. conversation going.
    Somebody in here,
  186. if I can get the mike in,
    shout out your name and really
  187. shout.
    Student: Peter.
  188. Professor Ben Polak:
    Go on.
  189. Student: Change is
    cumbersome.
  190. Professor Ben Polak:
    Change is cumbersome,
  191. you didn't want the change,
    okay fair enough,
  192. but if the stakes go up that
    would get rid of that.
  193. Student: Maybe people
    are tying their own outcomes to
  194. the other player's outcomes as
    well.
  195. Professor Ben Polak:
    Right, so maybe people have
  196. different payoffs here.
    Maybe people are comparing
  197. their payoff to the other
    person's payoff that certainly
  198. seems like a plausible thing to
    be going on here.
  199. You might feel less happy about
    getting $.20 knowing the other
  200. person's getting $.80 for
    example.
  201. Student: You want to try
    to teach them a lesson to get
  202. them to offer more.
    Professor Ben Polak:
  203. Right, you might be trying
    to set a sort of moral standard
  204. here.
    So there's some notion of
  205. indignation or even teaching
    people that they really should
  206. offer people more.
    What else could be going on
  207. here?
    Student: If people know
  208. I'm not going to accept less
    than $.50 then they should give
  209. me $.50 by backward induction.
    Professor Ben Polak:
  210. Right, so part of what's
    going on here--actually this
  211. game was a one shot game,
    we just played it once.
  212. We could have played it--in
    fact they often do play this
  213. this way in the lab--you could
    have played it without anybody
  214. knowing who the other player
    was.
  215. But particularly in this
    setting where everyone can see
  216. everyone else--even in the lab
    where people actually can't see
  217. people,
    but they might imagine that the
  218. game is really repeated--you
    could imagine people trying to
  219. establish a reputation.
    Is that right?
  220. So there's lots of these
    reasons, these sort of moral
  221. indignation reasons or teaching
    a lesson reasons,
  222. pride reasons.
    There's also this basic reason
  223. that people might be thinking,
    I should play this game as I
  224. would a similar situation in
    life where I might want to be
  225. establishing reputation.
    So there's a certain amount of
  226. confusion going on in the game,
    and there's also a certain
  227. amount of a lot of things make
    sense.
  228. Now notice that once we've
    established that people are
  229. going to reject small offers in
    this game,
  230. once we've established people
    are going to reject small
  231. offers, it makes perfectly good
    sense to offer a lot more than
  232. nothing.
    So it's not that surprising
  233. that once we've established the
    idea that people are going to
  234. reject small offers,
    we're going to see people
  235. making offers that are
    reasonably large,
  236. although not usually larger
    than $.50.
  237. Why is $.50 so focal here?
    Why is $.50 so focal?
  238. It's not a trick question.
    I'm just asking you why do
  239. people think $.50 is so focal?
    I think it's typical that
  240. people end up offering around
    $.50, why?
  241. It sounds fair, it seems fair.
    There's some notion of fairness.
  242. It's not clear by the way,
    what ethical principal is
  243. involved here.
    It's not clear that if you're
  244. walking along the streets and
    you happen to find a dollar at
  245. your feet that you should pick
    it up and anyone else you'd
  246. happen to see at that moment you
    should give $.50 too.
  247. That's essentially what the
    situation is,
  248. and you just chanced upon this
    dollar that I just gave you.
  249. It isn't clear that there's any
    particularly great moral claim
  250. to give it to someone else,
    but I think people read this as
  251. a situation about splitting a
    cake in an environment of
  252. distributional justice.
    They view it as a larger
  253. picture.
    Is that right?
  254. So it turns out there's a large
    literature on this.
  255. There's a large experimental
    literature on the ultimatum
  256. game.
    And there's an even larger
  257. literature on an even simpler
    game in which I give people a
  258. dollar,
    I say you can give whatever
  259. share you want to the other
    person, and they don't even get
  260. a chance to accept or reject.
    That's called a dictator game.
  261. In the dictator game,
    literally, you're just simply
  262. given a dollar and you can give
    whatever share you want to the
  263. other person.
    It turns out that even in the
  264. dictator game,
    people give quite a lot of
  265. money and that suggests that
    there really is some notion of
  266. fairness or some notion of
    distributional justice going on
  267. in people's heads here,
    rightly or wrongly.
  268. So one thing this should tell
    us is, even in extremely simple
  269. games, we should be a little bit
    careful about reading backward
  270. induction into what's going to
    happen in the real world.
  271. Part of this is because,
    as we mentioned the very first
  272. day of the class,
    people care about other things
  273. than just the obvious payoffs,
    and part of it is about more
  274. complicated things like
    reputation and so on.
  275. All right, having said that,
    let's nevertheless for the
  276. purpose of today,
    act as if we are going to do
  277. backward induction,
    and let's embed this into a
  278. slightly more complicated game.
    So the more complicated game is
  279. as follows.
  280. So we're going to have a two
    period bargaining game.
  281. In this two period,
    bargaining game,
  282. the beginning of the game is
    exactly the same.
  283. So there's a dollar on the
    table and Player 1 makes an
  284. offer to 2.
    And once again we can call this
  285. offer S and 1 - S,
    just the same as before.
  286. And, just as before,
    Player 2 can accept the offer,
  287. and if they accept the offer
    then this is indeed what they
  288. get.
    But now if 2 rejects,
  289. which is 2's other
    alternative--if Player 2 rejects
  290. the offer then we flip the
    roles.
  291. We play the game again but we
    flip the roles.
  292. So we go to stage two,
    everything we've said up here
  293. is stage one.
  294. And down here we go to stage
    two, and in stage two,
  295. Player 2 gets to make an offer
    to 1.
  296. And once again we can call
    this--we better be careful.
  297. Let's put ones here just to
    indicate we're in the first
  298. round and twos to indicate we're
    in the second.
  299. So they make an offer is S2 and
    (1-S2) where S2 is that that
  300. goes to Player 1 and (1-S2) is
    that that goes to Player 2.
  301. So 2 gets to make an offer to
    Player 1 and now 1 can accept or
  302. reject.
    If she accepts then she gets
  303. her share from here.
    The offer is accepted,
  304. and if she rejects then we get
    nothing.
  305. So this game is exactly the
    same as playing the previous
  306. game, except we flip roles,
    but we're going to add one
  307. catch.
    The catch is this,
  308. in the first round the money on
    the table is $1,
  309. but if we end up going into the
    second round,
  310. so the first offer is not
    accepted and we go into the
  311. second round,
    then part of the money is lost.
  312. In particular,
    we'll assume that the money on
  313. the table is δ.
    If you think of δ
  314. as being just some number less
    than 1--so if you want a
  315. concrete example think of this
    as being $.90.
  316. So the idea here is if you get
    into the second round,
  317. time has past,
    it's costly,
  318. and so money in the second
    round is--I think of it as money
  319. in the second round as being
    worth less or could actually
  320. think of this cake being eaten
    up,
  321. some of it's thrown away,
    some of it's wasted.
  322. Everyone understand the game?
    So this is very similar to the
  323. previous game,
    but we've got this second stage
  324. coming in, and we've got
    discounting.
  325. So this is the idea of
    discounting.
  326. How many of you have heard the
    term discounting before?
  327. You probably saw it in a
    finance class or a macro class
  328. where we think about there being
    a value of time.
  329. Money today is worth more than
    money tomorrow,
  330. partly because you could put
    the money today into the bank
  331. and it could earn interest,
    partly because you're simply
  332. impatient to get that money and
    go and have lunch,
  333. particularly on the day in
    which the clocks changed.
  334. Okay, so let's try this game
    again and let's just play it for
  335. real, so let's come down again.
    Everyone understand the game?
  336. Basically the same rules except
    we're just flipping around and
  337. with the possibility that the
    cake may shrink.
  338. Let's see what people have
    learned, so who were our first
  339. pair?
    Our first pair were Equia and
  340. Noah all right.
    So Equia, what are you going to
  341. offer?
    You're Player 1 here but if
  342. your offer is rejected Noah's
    going to get to make an offer to
  343. you.
    All right so what are you going
  344. to offer this time?
    Student: $25.99,
  345. so $.25, $.46.
    Professor Ben Polak:
  346. $.46 okay,
    so he gets $.46 if he accepts
  347. the offer, is that right?
    Student: Yes.
  348. Professor Ben Polak:
    $.46 if he accepts the
  349. offer.
    Student: I accept that.
  350. Professor Ben Polak:
    He accepts that,
  351. okay that was easy.
    So Equia got $.54 and Noah got
  352. $.46.
    Who was our second pair?
  353. So that was,
    I've forgotten,
  354. Anish right and?
    Student: Gary.
  355. Professor Ben Polak:
    Gary.
  356. So Anish what are you going to
    offer this time?
  357. Student: I'll offer
    $.43.
  358. Professor Ben Polak:
    $.43, you're going to push
  359. the envelope a little bit.
    Student: All right.
  360. Professor Ben Polak:
    All right,
  361. that one got accepted as well.
    Okay, so people are converging
  362. here.
    What about our third offerer,
  363. receiver it was Courtney and?
    Student: Danny.
  364. Professor Ben Polak:
    Danny.
  365. So Courtney?
    Student: $.30.
  366. Professor Ben Polak:
    $.30, so it's $.30 for him.
  367. Student: I'll accept.
    Professor Ben Polak:
  368. Three acceptances,
    all right.
  369. Let's find out something here,
    so I was hoping to get into the
  370. second round.
    Okay so you accepted,
  371. that's fine--no chicken sounds
    around the room.
  372. So, it's Danny right?
    Student: Yeah.
  373. Professor Ben Polak:
    So Danny had you
  374. rejected--you acceptedbut had
    you accepted what would you have
  375. offered in the second round?
    Student: $.45.
  376. Professor Ben Polak:
    $.45, all right,
  377. and would you have accepted
    that in the second round if $.30
  378. hadn't been accepted?
    Courtney: Yes.
  379. Professor Ben Polak:
    Okay, so you might have
  380. done better it turns out.
    Let's go back through to the
  381. other rejections,
    to the other acceptances.
  382. So my second couple you offered
    $.43, is that right?
  383. You said yes to $.43.
    Had you rejected $.43 what
  384. would you have offered back in
    return?
  385. Student: $.43.
    Professor Ben Polak:
  386. $.43, the same thing back.
    Would you have accepted it?
  387. Student: He gets $.47?
    Professor Ben Polak: He
  388. would have got $.47 in that
    case.
  389. Student: Yeah.
    Professor Ben Polak: You
  390. would have accepted that,
    okay.
  391. Equia went first and she
    offered $.45 to Noah.
  392. And Noah had,
    in fact, you rejected,
  393. what would you have offered
    back?
  394. Student: I would have
    also done $.45.
  395. Professor Ben Polak:
    Same thing back and would
  396. you have accepted it?
    Student: Yes.
  397. Professor Ben Polak:
    Okay.
  398. So we can see here that the
    decision to accept or reject
  399. partly depends on what you think
    the other side is going to do in
  400. the second round,
    is that right?
  401. So here you are,
    if you're in the middle of this
  402. game.
    If you're Player 2 you've
  403. received an offer.
    None of these offers sounded
  404. crazy.
    $.30 was the lowest one,
  405. but none of them sounded crazy.
    And you're trying to decide
  406. whether you should accept or
    reject this offer.
  407. And one thing you should have
    in mind is what would I offer if
  408. I reject.
    And will that offer that I then
  409. offer in the next round be
    accepted or rejected,
  410. is that right?
    So if we just work backwards we
  411. can see what you should offer in
    the first round should be just
  412. enough to make sure it's
    accepted knowing that the person
  413. who's receiving the offer in the
    first round is going to think
  414. about the offer they're going to
    make in the second round,
  415. and they're going to think
    about whether you're going to
  416. accept or reject in the second
    round, is that right?
  417. So that sounds like a bit of a
    mouthful but that mouthful of
  418. reasoning is exactly backward
    induction.
  419. It's exactly backward induction.
    It's saying:
  420. to figure out what I should do
    in the first round or what I
  421. should offer in the first round,
    I need to figure out whether
  422. Player 2 is going to accept or
    reject.
  423. And to figure out whether he or
    she is going to accept or
  424. reject, and I have to put myself
    in his or her shoes,
  425. and figure out what he or she
    would offer if she did reject,
  426. and what he or she thinks I
    would do if I got that second
  427. round offer.
    Is that right?
  428. All right, so let's try and
    analyze this as if backward
  429. induction was going to work
    here, as if we didn't have to
  430. worry about things like pride
    here.
  431. So this is the game we're
    actually playing,
  432. so let's keep that one and
    actually analyze it on the
  433. board.
  434. I want to walk us from a
    largely mundane game of take it
  435. or leave it offers to a more
    complicated game in which there
  436. can be several rounds of offers.
    But we're going to go slowly so
  437. we'll start just with two
    rounds.
  438. So first of all let's just look
    at the stage one game,
  439. and let's keep in track what
    the offer is and what the
  440. receiver.
    This is the offerer and this is
  441. the receiver.
    In the one stage game,
  442. the game only has one stage,
    then we know from backward
  443. induction what the results
    should be.
  444. It isn't what we'd find in the
    lab, it isn't what we find in
  445. the classroom,
    but we know what we should get.
  446. The offerer should offer to
    keep everything essentially or
  447. maybe $.99 but let's call it $1
    and the receiver gets nothing.
  448. So again I'm approximating a
    little bit because it could be
  449. $.99 and a penny but who cares.
    Let's just call it a $1 or
  450. nothing if it's a one stage
    game.
  451. So now let's consider a two
    stage game.
  452. In the two stage game the
    person who's making the offer in
  453. the first stage needs to look
    forward,
  454. anticipate what would happen if
    her offer was rejected by Player
  455. 2 and Player 2 went forward into
    the second stage.
  456. Is that right?
    So in the two stage game,
  457. in the first stage of the two
    stage game, the person making
  458. the offer wants to anticipate
    what the receiver would offer in
  459. the second round were the
    receiver to reject her offer.
  460. But we can do that by backward
    induction.
  461. We know that in the second
    round if the receiver rejects
  462. the offer, then the second round
    of the two stage game is what?
  463. It's a one stage game,
    and we've just argued,
  464. at least if we believe in
    backward induction,
  465. in that case,
    Player 2 who is then the
  466. offerer, will offer $1 and
    Player 1,
  467. who is now the receiver,
    will accept it and get nothing.
  468. So Player 1 in the first round
    of the two stage game wants to
  469. make an offer that's just enough
    to get Player 2 to accept it
  470. now.
    So let's think about this.
  471. So if Player 1 offers 2
    something more than what?
  472. Tomorrow Player 2 can get $1
    but that's $1 tomorrow.
  473. So $1 tomorrow is worth how
    much today if we're discounting?
  474. It's just worth δ right.
    It's just worth δ.
  475. So if Player 1 offers Player 2
    more than δ
  476. x $1, which is what Player 2
    can get tomorrow,
  477. then 2 will accept.
  478. If Player 1 offers 2 less than
    δ x $1--because you can get
  479. a $1 tomorrow but that's only
    worth $δ--a $1 tomorrow is
  480. worth just $δ
    today--then 2 will reject.
  481. So the offer has to be exactly
    enough to get accepted,
  482. which is exactly $δ.
    So Player 2 knows that she can
  483. get $1 tomorrow so you need to
    offer her at least $δ
  484. today to make it as good for
    her as getting $1 tomorrow.
  485. So we know the receiver must be
    offered at least $δ
  486. tomorrow, which means the
    offerer is going to keep $[1 -
  487. δ].
    So in the first round of the
  488. two stage game,
    Player 1 should offer $[1 -
  489. δ]
    for herself and $δ
  490. for Player 2 and Player 2
    should accept that because
  491. $δ dollars today is as good
    as $1 tomorrow.
  492. Now, another way to see that is
    in a picture,
  493. so let's just draw a picture.
  494. Let's put the payoff of Player
    1 here and the payoff of Player
  495. 2 on this axis.
    And we're going to assume that
  496. they're just going to maximize
    dollars where there's no pride
  497. in here.
    And if we just look at the one
  498. stage game, we're simply looking
    at this line.
  499. The offers in the one stage
    game: it could be that Player 1
  500. gets everything herself and
    gives nothing to Player 2,
  501. it could be that Player 2 keeps
    everything, ends up getting
  502. everything and Player 1 gets
    nothing,
  503. and it could be any combination
    in between.
  504. We argued by backward
    induction--although not in
  505. reality--in backward induction,
    in the one stage game,
  506. Player 1 makes an offer to
    Player 2 which is kind of an
  507. insulting offer.
    Player 1 says I get everything
  508. and you get a $.01.
    So this is the one stage game.
  509. In the two stage,
    if things are settled in the
  510. first stage this line represents
    the possible divisions between
  511. Player 1 and Player 2.
    But if we end up going into the
  512. second stage,
    the pie is shrunk.
  513. The pie is shrunk,
    instead of going from $1 to $1,
  514. it goes from $δ1 to
    $δ1.
  515. Or if you like,
    if δ is .9 it goes from
  516. $.9 to $.9.
    So let's draw that line in.
  517. So if we head into the second
    stage, we'll end up being here,
  518. and this goes from $δ
    here to $δ
  519. here where these dollars are
    being evaluated at time one.
  520. All right so the pie has shrunk.
    If we get into the second stage
  521. then, by backward induction,
    Player 2 is in an ultimatum
  522. game,
    Player 2 will be making the
  523. offer and Player 2 says:
    whatever cake is left I'm going
  524. to take all of it and you're
    just going to get a $.01.
  525. So if we get into the second
    stage then Player 2 will make
  526. this offer to Player 1.
    Player 2 will say I'm going to
  527. keep the whole of the pie,
    which in first period dollars
  528. is worth $δ.
    So I'm going to end up with a
  529. payoff of δ
    and you're going to end up with
  530. a payoff of essentially nothing.
    Player 2 knows that they can
  531. therefore get at least
    $δ--or $δ
  532. in current day dollars
    worth--from rejecting your
  533. offer.
    Since they can get at least
  534. $δ current day offers from
    rejecting your offer,
  535. the lowest offer you can make
    to them is an offer that gives
  536. them at least $δ.
    So the offer you're going to
  537. make is this offer:
    this is the two stage offer.
  538. It happens in the first stage.
    Player 1 makes an offer that
  539. gives Player 2 what Player 2
    could get in the second round,
  540. so gives Player 2 $δ
    and keeps $[1 - δ]
  541. for herself.
    Everyone understand the picture?
  542. So this picture is just
    corresponding to this table.
  543. The thing people tend to get
    confused about here,
  544. I think, is they get confused
    between current dollars and
  545. discounted dollars,
    so we're going to do all the
  546. analysis here in terms of the
    first period dollars,
  547. dollars tomorrow are going to
    be worth δ.
  548. There's a hand up,
    can I get a shouting out?
  549. Yeah?
    Student: [Inaudible]
  550. Professor Ben Polak:
    Yes, sorry.
  551. So this is the outcome if it
    was a one stage game and this is
  552. the outcome if it was a two
    stage game.
  553. The offer is made and accepted.
    Let's roll it forward,
  554. let's look at a three stage
    game.
  555. Let's keep this picture handy
    and think about a three stage
  556. game.
    So the beginning of the game is
  557. the same.
    We're going to look at three
  558. stage bargaining,
    and the rules in three stage
  559. bargaining are pretty much the
    same as in two stage bargaining,
  560. but now there's two possible
    flips.
  561. In three stage bargaining,
    in the first period,
  562. in period one,
    1 makes the offer and if it's
  563. accepted the game is over.
    In period two,
  564. if we reject,
    then we go to period two when 2
  565. makes the offer and if it's
    rejected now,
  566. this time by Player 1,
    then we go to period three
  567. where once again 1 makes the
    offer.
  568. So you can see where we're
    heading, we're heading towards
  569. an alternate offer bargaining
    model.
  570. I'm going to make an offer,
    Jake's going to either accept
  571. or reject, then he'll make an
    offer and we'll flip to and fro.
  572. There's a question,
    let me try and get a mike out
  573. to the question.
  574. Yeah?
    Student: I have a
  575. question about the two player
    game, if δ
  576. is the best that Player 2 can
    get tomorrow then why wouldn't 1
  577. offer Player 2 δ
    discounted by δ
  578. today?
    Professor Ben Polak:
  579. Good.
    Right so I think I was
  580. confusing about it,
    so let me make it clear.
  581. So tomorrow Player 2 can get
    everything, everything that
  582. there is.
    So whatever pie is left
  583. tomorrow Player 2 can get all of
    it.
  584. So call that pie tomorrow 1 and
    evaluate it in period one
  585. dollars as being worth $δ.
    Does that make sense?
  586. Okay, so I think I misspoke on
    that, so let me say it again.
  587. So every period there's this
    pie and every period,
  588. if it was the last period of
    the game,
  589. the person making the offer is
    going to get the whole pie,
  590. but if I view that pie tomorrow
    from today,
  591. a pie of $1 tomorrow is only
    worth $δ
  592. today.
    A pie of $1 the day after
  593. tomorrow is only worth $δ
    tomorrow and $δ²
  594. today and so on.
    So that's the way in which
  595. we're going to do discounting
    here.
  596. Good, all right.
    So in this game,
  597. if 1 makes an offer,
    if it's accepted it's over.
  598. If it's accepted then we're
    done.
  599. And if this offer's accepted
    then we're done.
  600. And if this offer's accepted
    then we're done.
  601. And in the third round,
    if it's rejected then both
  602. players get nothing.
    Once again we're going to
  603. assume that the players are
    discounting.
  604. So what does it mean to say
    they're discounting?
  605. It means that dollars in period
    one are worth dollars,
  606. dollars in period two are
    discounted by δ,
  607. and dollars in period three are
    discounted by δ
  608. x δ, or if you like by
    δ².
  609. Just to put this into real
    notions of money,
  610. if you think of δ
    as being 90%,
  611. then $1 in period one is worth
    $1, a $1 in period two viewed
  612. from period one is worth $.90,
    and a $1 in period three viewed
  613. from period one is worth $.81.
    Okay, so what do we think is
  614. going to happen here?
    Well, once again we can do
  615. backward induction.
  616. Here we are in our picture.
  617. Let's look at the three-stage
    game.
  618. Once again, when we analyze,
    as always when we analyze these
  619. games using backward induction,
    we want to start at the end.
  620. If we start at the end,
    we know that the last stage,
  621. that's the third stage of the
    three stage game looks like
  622. what?
    It looks like a one stage game.
  623. In the one stage game we know
    the offerer will get everything.
  624. Say it again,
    so the last stage of the three
  625. stage game, we know the person
    who makes the offer who this
  626. time will be Player 1 will get
    everything.
  627. However, that everything is
    only worth δ
  628. in period two dollars and it's
    only worth δ²
  629. in period one dollars.
    So in period one,
  630. in the first period of this
    game, we know that if their
  631. offer is rejected we know what's
    going to happen.
  632. Say it again,
    in the first period of this
  633. three stage game,
    if the offer is rejected then
  634. we'll go into a two stage game,
    and we already know what
  635. happens in a two stage game.
    In a two stage game,
  636. the person who gets to make an
    offer gets $[1 - δ]
  637. and the person who receives the
    offer gets $δ.
  638. So we know in the first stage
    of the game that the person who
  639. receives the offer always has
    the outside option of saying,
  640. no I reject.
    And we know that that person
  641. tomorrow will get $[1 - δ].
    But $[1 - δ]
  642. tomorrow is worth how much
    today?
  643. It's worth $δ[1 - δ].
    Tomorrow they're going to get
  644. $[1 - δ],
    so today that's worth
  645. $δ[1--δ].
    So the offer I have to make in
  646. the first round to make sure
    that the other person accepts it
  647. has to be just better in
    discounted dollars,
  648. than what they're going to get
    tomorrow.
  649. They're going to get $[1 -
    δ]
  650. tomorrow, so I have to give
    them $δ[1--δ]
  651. today, which means I keep for
    myself $[1 - δ[1--δ]].
  652. And if you don't like the
    algebra let's look at the
  653. picture.
    In the picture,
  654. in the one stage game,
    this is the offer.
  655. In the two stage game,
    we know if we get to the second
  656. round, Player 2 gets everything
    so we have to give him that much
  657. today.
    And if we get into the third
  658. round, now we're looking at
    δ² here and
  659. δ² here.
  660. We know that if we get into the
    third round the person who makes
  661. the offer in the third round
    will get everything,
  662. so we can actually work our way
    along.
  663. In the third round,
    the person who makes the third
  664. offer will get everything,
    so in the second round you'd
  665. have to give them that much,
    so in the first round you'd
  666. have to give Player 2 that much.
    Say it again,
  667. in period three,
    the person making the offer can
  668. get everything.
    So in period two,
  669. they must be getting δ
    times that, so in period one
  670. you have to give them at least
    this much.
  671. And this here is the offer
    you'd make in the three stage
  672. game.
  673. So in the picture we're just
    doing a little zigzag;
  674. on the chart we're also always
    working across the diagonal.
  675. So we've done the one stage
    game: the one stage game,
  676. the person making the offer
    gets everything.
  677. In the two stage game,
    the person making the offer
  678. offers just enough to get the
    offer accepted which is $δ
  679. because that's what $1 is worth
    tomorrow.
  680. In the three stage game,
    the person making the offer
  681. makes just enough to get the
    offer accepted,
  682. which is δ
    times what the receiver would
  683. get tomorrow.
    What they get tomorrow is $[1 -
  684. δ], so they get
    $δ[1--δ]
  685. today.
    How about the four stage game?
  686. Let's see if we can do that.
    So if we go to the four stage
  687. game now, in the four stage game
    if the person receiving the
  688. offer rejects the offer,
    then tomorrow they can get $[1
  689. - δ[1--δ]].
    So I need to offer them enough
  690. now in current dollars so they
    will prefer that to getting
  691. $[1--δ[1 - δ]]
    tomorrow,
  692. so how much must I offer them?
    I have to offer them at least
  693. δ times that much,
    so I have to offer them
  694. $[δ x [1 - δ
    x [1--δ]]].
  695. Again, I'll keep the rest for
    myself so I'll get $[1 -
  696. δ[1--δ[1--δ]]].
    And so the principle is always
  697. give people just enough today so
    they'll accept the offer,
  698. and just enough today is
    whatever they get tomorrow
  699. discounted by δ.
  700. So actually this backward
    induction isn't so bad.
  701. What makes it a little bit
    easier is you don't actually,
  702. when you go through an extra
    stage of this you don't actually
  703. have to go all the way to the
    beginning,
  704. you could actually start where
    you were last time and just
  705. discount once more by δ.
  706. Let's see if we can see any
    kind of pattern emerging in this
  707. algebra, so let's just multiply
    out these brackets.
  708. In the four stage game,
    this thing is actually equal
  709. to--just multiplying
    through--it's 1 - δ
  710. + δ²
    - δ³--I hope it is
  711. anyway.
    That's what this is.
  712. And this thing is equal to
    δ - δ²
  713. + δ³,
    just multiplying out the
  714. brackets.
    Does anyone see a pattern
  715. emerging here in these offers?
    We had offers of 1 and then
  716. 1--δ.
    We could also multiply out this
  717. one.
    It might be helpful to do so:
  718. This is 1 - δ
    + δ².
  719. Anyone see a pattern what these
    offers look like?
  720. They kind of alternate.
    So let's have a look,
  721. rather than do every stage.
  722. Should I do one more stage to
    see if we can see a pattern
  723. emerging or should I jump
    straight to ten stages and see
  724. what happens?
    Go straight to ten people say?
  725. Let's do one more,
    nah, let's jump through to ten.
  726. So imagine that this game
    actually had ten rounds,
  727. so this is a ten stage game,
    and let's just continue our
  728. chart down here.
    So here's--need a bit more
  729. space here--ten stages:
    ten stage game.
  730. I'm going to continue my chart
    and my chart says in the ten
  731. stage game what am I going to
    get?
  732. So the offer is going to
    be--it's going to be the same
  733. pattern--1 - δ
    + δ²
  734. - δ³…
    + δ^(8) - δ^(9),
  735. everyone see that?
    So what I'm doing is I'm
  736. continuing the pattern from
    above.
  737. So if I had ten stages,
    I always start with a 1,
  738. the positive and negative terms
    just alternate,
  739. and I have as many terms as 1
    minus the stage I'm in.
  740. So in the four stage game,
    I ended up with δ³,
  741. so in the ten stage game I'll
    end up at δ^(9).
  742. Everyone happy with that?
    So the four [error:
  743. ten]
    stage offer is this slightly
  744. ugly thing, 1 - δ
    + δ²
  745. - δ³…
    + δ^(4) - δ^(5),
  746. etc., + δ^(8) - δ^(9).
    That's a pretty ugly thing,
  747. but fortunately some point in
    high school you learned how to
  748. sum that thing.
    Do you remember what this is?
  749. What do you call objects like
    this in high school?
  750. Anyone remember?
    Objects like 1,
  751. δ, δ²,
    δ³,
  752. δ^(4) what are they
    called?
  753. They're called geometric series
    right, they're called geometric
  754. series.
    Anyone remember how to sum them?
  755. We know that S is equal to
    this, this is what the offer is,
  756. if there is ten rounds.
    We know the offer is accepted.
  757. So the way to remember how to
    sum it, the trick for summing it
  758. is to multiple both sides of
    this equation by the common
  759. ratio,
    so multiply both sides by
  760. δ.
    So if I multiply this side by
  761. δ, I'm going to multiply
    the other side by δ.
  762. And this 1 will become a
    δ, this δ
  763. will become a δ²,
    this δ²
  764. will become a δ³,
    this δ³
  765. will become a
    δ^(4)….
  766. There will be a -δ^(8)
    coming from the seventh term.
  767. There will be a +δ^(9)
    coming from the δ^(8) term,
  768. and there will be a
    -δ^(10).
  769. Everyone okay with that?
    I just multiplied everything
  770. through by δ
    and I just shifted along one
  771. for convenience.
    What do I do now,
  772. anyone remember?
    Add the two lines together.
  773. So by summing this side I get
    [1 + δ]
  774. S^(10).
    On the other side,
  775. what's kind of convenient is
    everything cancels.
  776. The 1 comes through, I get a 1.
    These two terms cancel and
  777. these two terms cancel,
    and these two terms cancel,
  778. and so on and so on,
    all the way up to the end where
  779. I get -δ^(10).
    Everyone okay.
  780. All the other terms have
    cancelled out.
  781. So now just sorting out my
    algebra a bit--I'm going to take
  782. it on the other side--I'm going
    to have that the offer you
  783. make--this is what you're going
    to get to keep--so the amount I
  784. claim I should keep in the first
    round is [1 - δ^(10])/[1 +
  785. δ].
    Just be a bit careful with the
  786. notation here because it may be
    a little bit confusing.
  787. The 10 here doesn't mean to the
    tenth power, it's just the offer
  788. in the tenth round [error:
    ten-round game],
  789. whereas the 10 here really does
    mean in the tenth power.
  790. So if we play this game for ten
    rounds, the offer you'd make
  791. would be [1 - δ^(10)]/[1 +
    δ]
  792. which means the amount you're
    offering to the other side,
  793. which is 1 - S would be [δ
    + δ^(10)]/[1 + δ].
  794. So to summarize where we are:
    we started off by considering a
  795. very simple game,
    a one stage take it or leave it
  796. offer.
    We know that,
  797. in that take it or leave it
    offer, Player 1 is going to
  798. claim everything for herself and
    offer nothing to Player 2.
  799. Then we considered a two stage
    game which was the same as the
  800. one stage game except that if
    Player 2 rejects the offer,
  801. he--let's call player 2,
    "he"--he gets to make an offer
  802. to Player 1 in the second
    period.
  803. We know that in the second
    period of that two stage game
  804. Player 2 can keep everything for
    himself.
  805. Everything for himself tomorrow
    is worth δ
  806. today, so you have to offer him
    at least δ
  807. today and keep 1 - δ
    for yourself.
  808. Then we looked at a three stage
    game.
  809. In this three stage game,
    if Player 2 rejects in the
  810. first round, Player 2 can make
    you an offer in the second
  811. round,
    but now if you're Player 1 and
  812. you reject in the second round
    you get to make an offer to
  813. Player 2 in the third round.
    We argued that in the second
  814. round of this game,
    if Player 2 rejects you in the
  815. first round and makes an offer
    in the second round,
  816. it'll be in a two stage game
    and they'll be able to keep 1 -
  817. δ of the pie for
    themselves.
  818. So you have to offer them at
    least δ x [1--δ]
  819. today for them to accept the
    offer, keeping the rest for
  820. yourself.
    Then we looked at a four stage
  821. game.
    In the four stage game if
  822. Player 2 rejects your offer,
    he can make you an offer,
  823. but if you reject the offer you
    can make him an offer,
  824. but if he rejects that offer he
    can make you an offer.
  825. And once again we asked how
    much do I have to offer Player 2
  826. now for him to accept the offer
    now?
  827. He knows that if he rejects the
    offer, he can get this amount
  828. 1--δ x [1--δ]
    tomorrow.
  829. So I have to offer him δ
    times that today,
  830. and once again I keep the same
    for myself.
  831. That's just a summary of what
    we did.
  832. And then what we did was we
    cheated.
  833. We jumped to the tenth stage
    [error: ten stages],
  834. just noticing that a pattern
    had emerged,
  835. and we found that in the tenth
    stage [error:
  836. ten-stage game],
    this is the offer you'd make
  837. just according to the same
    pattern.
  838. And it was this horrible thing,
    and then we used a little bit
  839. of high school math to simply
    this thing.
  840. And it turns out this is the
    amount you keep for yourself and
  841. this is the offer you'll make to
    Player 2.
  842. In each case I've accepted--Did
    I make a math mistake?
  843. Thank you.
    Let's put a superscript in
  844. here.
    good.
  845. So what do we observe here?
    So the first thing to observe
  846. is in the one stage game if we
    believe backward induction you
  847. certainly want to be the person
    making the offer.
  848. In the one stage game,
    in the ultimatum game,
  849. there's a huge first-mover
    advantage.
  850. In the two stage game it's not
    clear if you want to make the
  851. offer, it depends on how large
    δ is,
  852. but if δ
    is a big number like .9 you'd
  853. rather be the person receiving
    the offer.
  854. In the three stage game,
    it looks like you'd probably
  855. rather make the offer,
    but it's not so clear.
  856. So where does it go to as we go
    down the path,
  857. as it goes down towards the ten
    stage game?
  858. It looks like in the ten stage
    game you'd probably still prefer
  859. to make the offer than not,
    but they're certainly much
  860. closer together than they were
    before.
  861. Some of that initial bargaining
    power has been washed out by the
  862. fact that there are ten stages.
    So let's try and push this just
  863. a little bit harder.
    Instead of looking at the tenth
  864. stage offer, what if we look at
    the infinite stage offer.
  865. So in principle we look at the
    infinite stage of this game.
  866. So I can make you an offer,
    you can say no and make me an
  867. offer, and then I can reject and
    make you an offer,
  868. and then you can reject and
    make me an offer,
  869. and so on and so forth.
    So we look at this term.
  870. If in principle and you can
    make an infinite number of
  871. offers--so, what's this term
    going to look like if I can make
  872. an infinite number of offers?
    So I claim it's going to look
  873. like this [1 - δ^(∞)]
    / [1 + δ]
  874. and over here,
    at least it's going to converge
  875. towards this.
    We'll be a bit more formal,
  876. and over here we'll have
    [δ + δ^(∞)]
  877. / [1 + δ].
    However now I get a little bit
  878. simpler.
    What is δ^(∞)?
  879. It's 0 right,
    so .9 x .9 x .9 x .9 x .9 x.
  880. 9 x .9 x .9 x .9 x …
    is 0.
  881. So this last term disappears as
    does this one,
  882. and we just get 1 / [1 +
    δ]
  883. and δ / [1 + δ].
    So if we make alternating offer
  884. bargaining--a bargaining game
    where in each round I make you
  885. an offer,
    you can accept it or you can
  886. reject and make me an offer,
    and we imagine there's no bound
  887. to this game,
    it just goes on arbitrarily
  888. long--then our prediction is
    that Player 1,
  889. the person who makes the first
    offer will get 1 / [1 + δ]
  890. of the initial pie and Player 2
    will get δ
  891. / [1 + δ]
    of the initial pie.
  892. Let's try and get a handle
    about what those numbers are.
  893. So if you imagine these offers
    can be made fairly rapidly,
  894. for example,
    I can make offer today,
  895. you can make an offer back to
    me in half an hour's time,
  896. and then I can make an offer
    back to you in half an hour's
  897. time,
    then it's reasonable to assume
  898. that the pie is not shrinking
    very fast.
  899. The discount factor is not a
    big deal here.
  900. So these offers can be made in
    rapid succession,
  901. but we might think that δ
    itself is approximately 1:
  902. the pie isn't shrinking very
    fast.
  903. If δ is approximately 1,
    and if we take δ
  904. to 1 here--the time isn't that
    valuable given how rapidly we
  905. can make offers to and fro--then
    what does this make this equal?
  906. In the case where δ
    is equal to 1 what do we get?
  907. We get 1/2, which means this
    will also be 1/2.
  908. So we learned something from
    this which is kind of
  909. surprising.
    If you do alternating
  910. offers--the sort of standard,
    very natural game of
  911. bargaining--sort of the kind of
    bargaining you might do in the
  912. bazaar,
    in a market,
  913. or the kind of bargaining you
    might imagine going on in
  914. negotiations between baseball
    players or their agents and
  915. teams,
    or general managers of
  916. teams--in which offers just go
    to and fro and they go to and
  917. fro fairly rapidly,
    and in principle they could
  918. make lots and lots of offers.
    In principle--what this moral
  919. tells us is, in principle,
    we're going to end up with each
  920. side splitting whatever the pie
    was equally.
  921. Very, very different from the
    ultimatum game where all the
  922. bargaining power was on the
    person who made the first offer.
  923. So what are the lessons here?
  924. What can we conclude from this?
  925. We've looked at alternating
    offer bargaining,
  926. and we've concluded,
    under special conditions,
  927. we've concluded that you get an
    even split.
  928. You get an even share,
    an even split,
  929. a fifty-fifty split if three
    things are true.
  930. The first thing is there's
    potentially infinitely many
  931. offers: potentially can bargain
    forever.
  932. And if discounting is not a big
    deal.
  933. What does discounting really
    not being a big deal means?
  934. It means those offers can be
    made in rapid succession.
  935. So no discounting,
    or if you like,
  936. rapid offers.
    If you have to wait a year
  937. between every offer then that
    discount factor would be a big
  938. deal.
    But I actually made a third
  939. assumption, and I made it
    without telling you.
  940. What was the third assumption I
    made?
  941. I snuck the third assumption
    past you without telling you.
  942. What was that third assumption?
    Let me get the mike here.
  943. So I claim I snuck a third
    assumption.
  944. There's somebody,
    let me start over here.
  945. What was the third assumption?
    Student: I don't know if
  946. this is what you're looking for
    but they know how big the pie
  947. is.
    Professor Ben Polak:
  948. They know how big the pie is,
    that's true,
  949. that's a big deal actually.
    That's true,
  950. but there's something else
    going on here.
  951. What is it?
    Student: We assume that
  952. both players were rational.
    Professor Ben Polak:
  953. We assumed that.
    That's true,
  954. but we've kind of been assuming
    that throughout backward
  955. induction.
    You're right none of this
  956. backward induction would apply
    so cleanly if we didn't assume
  957. that.
    What else do I assume?
  958. It's hidden actually,
    I snuck it in.
  959. We assume the discount factor
    is a constant,
  960. that's true,
    but not just constant but
  961. something else.
    You're on the right lines.
  962. They're the same.
    I've assumed here,
  963. implicitly, I've assumed that
    both people are equally
  964. impatient,
    they have the same discount
  965. factor, δ_1 =
    δ_2.
  966. Why does that matter?
    Well let's just think about it
  967. intuitively a second.
    Suppose that one of these
  968. players is very,
    very impatient.
  969. They need the money now.
    If it's cake,
  970. they need the cake now.
    They're very impatient,
  971. and the other person is very
    patient.
  972. They can wait forever to get
    this bargain to come across.
  973. Who do you think is going to do
    better, the patient player or
  974. the impatient player?
    The patient player is going to
  975. do better.
    The way we ended up,
  976. the way we did all this
    analysis is we assumed that
  977. those discount factors were the
    same.
  978. We assumed that each person was
    discounting time at the same
  979. rate, perhaps because they were
    facing the same bank with the
  980. same interest rate.
    But in practice,
  981. often one side is going to be
    in a hurry to get the dispute
  982. resolved, and the other side can
    sit around forever.
  983. In that world,
    the side who could sit around
  984. forever is going to do much
    better.
  985. Now, we're going to look at
    relaxing this assumption and
  986. this assumption in particular
    we're going to relax it on a
  987. homework exercise.
    So in your homework exercise
  988. you're going to try redoing part
    of this analysis--good practice
  989. anyway--but doing it in a
    setting where the discount
  990. factors are different.
    So one thing we learned here
  991. was: yes, you get an even split,
    but it depends on these three
  992. assumptions.
    It's kind of important because
  993. when you think about bargaining,
    I think a lot of people simply
  994. assume intuitively that whatever
    the bargain is about,
  995. people will eventually split in
    the middle.
  996. When you're bargaining about a
    house, or the price you're going
  997. to pay for a house,
    all these things,
  998. you kind of implicitly have
    this assumption you're going to
  999. end up splitting the difference.
    What I'm arguing here is you
  1000. will split the difference in
    this natural bargaining game,
  1001. but only under very special
    assumptions, and in particular,
  1002. the assumption of patience is
    critical.
  1003. There's another remarkable
    thing here though,
  1004. it's also hidden.
    So not only did we end up with
  1005. an even split,
    but something else remarkable
  1006. happened in this bargaining
    game.
  1007. What was the other thing?
    Somewhat amazingly,
  1008. a very unrealistic thing that
    occurred in this bargaining
  1009. game?
    See if you can spot it.
  1010. So one thing was an assumption
    I made and the other is actually
  1011. a prediction.
    Yeah.
  1012. Student: The first offer
    will be the offer that's
  1013. accepted.
    Professor Ben Polak:
  1014. Good.
    Did everyone see that?
  1015. So in this bargaining game,
    I set it up as alternating
  1016. offer bargaining,
    so the image you had in your
  1017. mind was of haggling.
    I made an offer,
  1018. you guys thought about this
    offer.
  1019. Should I take this offer or not.
    Maybe I won't take this offer.
  1020. You make an offer back to me,
    and we kind of haggled to and
  1021. fro.
    But actually,
  1022. in the equilibrium of the game,
    none of that happened.
  1023. That all happened in our mind.
    We thought about what offer we
  1024. would make, and we thought about
    what offer you would make back
  1025. to me if I made you this offer
    and you rejected it and so on.
  1026. We did this backward induction
    exercise but it was all in our
  1027. heads.
    In this game the actual
  1028. prediction is:
    the very first offer is
  1029. accepted.
    There's no haggling,
  1030. there's no bargaining.
    Now that doesn't seem very
  1031. realistic, there's no haggling.
  1032. Backward induction suggests
    that we should never see
  1033. bargaining, never see the actual
    process of bargaining.
  1034. What you should see is an offer
    is made and it's accepted.
  1035. Now what is it about the real
    world that allows for haggling
  1036. to take place?
    If this was a model of the real
  1037. world and we believed in
    backward induction,
  1038. then we're done.
    So why is it in fact in the
  1039. real world we see people make
    offers to and fro?
  1040. What's different about the real
    world than this model?
  1041. Let's talk about it a bit.
    You must have all bargained for
  1042. something in the real world.
    None of you have probably
  1043. bargained for a house yet,
    but you might have bargained
  1044. for a car or something.
    In the real world you make
  1045. offers go to and fro,
    right?
  1046. What's going on?
    Why are we getting offers in
  1047. the real world whereas we don't
    in this game?
  1048. What are we missing?
    Student: In the real
  1049. world you don't actually know
    what the other person's discount
  1050. factor is,
    therefore, you have uncertainly
  1051. as to what your highest possible
    offer could be.
  1052. Professor Ben Polak:
    Good.
  1053. So in the real world,
    unlike the model on the board,
  1054. not only are those discount
    factors different but you
  1055. probably don't know how patient
    or impatient the other side is.
  1056. You can get some ideas about
    how patient or impatient the
  1057. other side is by looking at
    their characteristics,
  1058. for example.
    For example,
  1059. if you know that the person
    you're bargaining with over
  1060. their car--you're trying to buy
    their car--and they're a
  1061. graduate student who's just a
    got a job in,
  1062. I don't know,
    Uzbekistan or something,
  1063. and they aren't going to be
    taking their car with them,
  1064. and they're leaving next week,
    you know they're in a hurry.
  1065. So there's times when you're
    going to know something about
  1066. other people's discount factors,
    how patient they are,
  1067. but lots of times you're not
    going to know.
  1068. So one thing is you just don't
    know what the discount factor
  1069. is.
    By the way, what else might you
  1070. not know about the other side?
    What else might you not know?
  1071. Student: How big the
    surplus is they were splitting.
  1072. Professor Ben Polak:
    Good.
  1073. You might not know this good
    that you're selling.
  1074. We've been talking about the
    big one pie which you're carving
  1075. up, and everyone knows the size
    of the pie.
  1076. But in the real world,
    I might not know this object
  1077. that's being sold,
    I might not know how much this
  1078. object is worth to the other
    side,
  1079. and he or she may not know how
    much it's worth to me.
  1080. So that lack of information is
    going to change the game
  1081. considerably.
    In particular,
  1082. I might want to turn down some
    offers in this game in order to
  1083. appear like a patient person.
    Why I might want to turn down
  1084. some offers in the game in order
    to appear like somebody who
  1085. doesn't really value this all
    very much.
  1086. And in so doing,
    I'm going to try and get you to
  1087. make me a better offer.
    So what's going on in haggling
  1088. and bargaining,
    according to this model,
  1089. what's missing in this model,
    is the idea that you don't know
  1090. who it is you're bargaining
    with.
  1091. You don't know how much they
    value the objects in question.,
  1092. You don't know how impatient
    they are to get away with the
  1093. cash.
    So it's a big assumption here,
  1094. a very big assumption,
    is that everything is known.
  1095. So both the size of the pie,
    let's call it the value of the
  1096. pie, and the value of time is
    assumed to be known,
  1097. but in the real world you
    typically don't know the value
  1098. of the pie on the other side,
    and you typically don't know
  1099. how much they value time.
    So that produces a whole
  1100. literature on bargaining,
    none of which we really have
  1101. time to do in this class,
    which is a pity because
  1102. bargaining's kind of important.
    So instead, I want to spend the
  1103. last five minutes just
    introducing, is it really worth
  1104. introducing a new topic in the
    last five minutes?
  1105. No, I think it is,
    let me talk a little bit more
  1106. about bargaining rather than
    that.
  1107. So what does this suggest if
    we're going out in the real
  1108. world?
    I'm actually taking this to
  1109. reality.
    So one thing it suggests is
  1110. people for whom it's known that
    they're going to be
  1111. impatient--people for whom it's
    known that they desperately need
  1112. this deal to go through--are
    going to do less well in
  1113. bargains.
    We already know people may do
  1114. less well in bargaining because
    they're less sophisticated
  1115. players,
    but here it isn't that they're
  1116. less sophisticated--they can be
    as sophisticated as you
  1117. like--but they're just going to
    be in a hurry.
  1118. We already talked about the
    graduate student whose leaving
  1119. for Uzbekistan,
    but who else typically in
  1120. bargaining is going to need cash
    now?
  1121. Who else is going to be in a
    weaker position in their
  1122. bargaining, socially in our
    society?
  1123. Student: When labor
    management disputes labor.
  1124. Professor Ben Polak:
    So that's a good question.
  1125. That's a good question in labor
    management disputes--there's one
  1126. going on right now in
    Hollywood--it's not clear there.
  1127. It could be,
    it could be the management side
  1128. who's in a hurry because they
    just need right now to get David
  1129. Letterman's script written.
    That would tend to favor labor,
  1130. but it could be the labor side.
    Why might it be the labor side?
  1131. Does everyone know this?
    There's a writer's strike going
  1132. on in Hollywood right now,
    so the people on the management
  1133. side who are in the weakest
    position are the people who are
  1134. in the most hurry to get this
    resolved,
  1135. and those are the guys with the
    fewest scripts in the pipeline,
  1136. and that tends to be late night
    TV shows.
  1137. So those guys really want this
    thing settled fast.
  1138. They're in a weak bargaining
    position.
  1139. On the other hand,
    there may be a reason why
  1140. labor's in a weak bargaining
    position.
  1141. Why might labor be in a weak
    bargaining position relative to
  1142. management?
  1143. They have rents to pay.
    They have immediate demands on
  1144. their cash.
    The typical worker is typically
  1145. poorer than your typical
    manager, not always but
  1146. typically, and they have to pay
    the rent.
  1147. They have to feed their
    children.
  1148. So there's a more general idea
    there.
  1149. More generally in bargaining,
    the people who are poorer,
  1150. typically--it isn't just poor
    in terms of income,
  1151. it's poor in terms of
    wealth--are going to be more
  1152. impatient to get things
    resolved;
  1153. and that's going to put them in
    a weaker bargaining position.
  1154. So in bilateral bargaining,
    having low wealth and being
  1155. known to have low wealth puts
    you in a weaker position.
  1156. And that means that typically
    people who are poorer are going
  1157. to do less well,
    although the late night TV show
  1158. may be an exception.
    It makes you think a little bit
  1159. about whether adjusting up a
    bargaining position makes things
  1160. equal for everybody.
    Any other thoughts about who
  1161. has strength and who has
    weakness in bargaining?
  1162. What other kind of stunts do we
    see people do in bargaining?
  1163. What else is kind of missing
    here when you think about a
  1164. particular--what we want to do
    in this class is develop these
  1165. ideas and take them to the real
    world,
  1166. so what other real world things
    here are kind of missing?
  1167. Student: Usually people
    will make their first offer a
  1168. lot higher than what they're
    actually willing to accept.
  1169. Professor Ben Polak:
    Right, so typically,
  1170. you're right,
    typically bargaining isn't just
  1171. a series of random numbers,
    typically people start out high
  1172. and then they concede towards
    the middle, is that right?
  1173. So if I'm the buyer I start out
    with a low price and come up,
  1174. and if you're the seller you
    start off with a high price and
  1175. come down.
    So again that seems to be about
  1176. establishing reputation and
    trying to indicate how much I
  1177. want this good.
    There's something worth saying
  1178. here which we haven't got time
    to do in this class,
  1179. and I hope you all have time to
    take the follow up class 156.
  1180. We can actually show formally
    that in a setting in which
  1181. buyers and sellers are
    bargaining,
  1182. and buyers and sellers do not
    know how much the good is worth
  1183. to the buyer or how much it
    costs to the seller,
  1184. typically you cannot expect to
    get efficiency.
  1185. Let me say it again.
    So it's kind of an important
  1186. economic fact that's missing
    from 115 and unfortunately
  1187. missing from this class,
    but if we go to a more real
  1188. world setting in which people's
    values are not known,
  1189. not only are the offers not
    accepted immediately and not
  1190. only is there some inequity in
    that the poor tend to be more
  1191. impatient and do less well,
    but also you get bad
  1192. inefficiency.
    The inefficiency occurs
  1193. essentially because the sellers
    want to seem like they're hard
  1194. and the buyers want to seem like
    they're hard,
  1195. and you get a failure for deals
    to be made.
  1196. So some deals are actually
    going to be lost or take a long
  1197. time in coming,
    you're going to get some
  1198. strikes before the deals occur
    and that's all inefficient.
  1199. So bargaining,
    not in this model,
  1200. but in the real world tends to
    lead to inefficiency.
  1201. So I'll leave it there,
    we'll have an earlyish lunch
  1202. since we're all starving because
    of the clock change anyway.