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← 22. Repeated games: cheating, punishment, and outsourcing

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Showing Revision 1 created 09/17/2012 by Amara Bot.

  1. Professor Ben Polak:
    So last time we were
  2. focusing on repeated interaction
    and that's what we're going to
  3. continue with today.
    There's lots of things we could
  4. study under repeated interaction
    but the emphasis of this week is
  5. can we attain--can we
    achieve--cooperation in business
  6. or personal relationships
    without contracts,
  7. by use of the fact that these
    relationships go on over time?
  8. Our central intuition,
    where we started from last
  9. time, was perhaps the future of
    a relationship can provide
  10. incentives for good behavior
  11. can provide incentives for
    people not to cheat.
  12. So specifically let's just
    think of an example.
  13. We'll go back to where we were
    last time.
  14. Specifically suppose I have a
    business relationship,
  15. an ongoing business
    relationship with Jake.
  16. And each period I'm supposed to
    supply Jake with some inputs for
  17. his business,
    let's say some fruit.
  18. And each period he's supposed
    to provide me with some input
  19. for my business,
    namely vegetables.
  20. Clearly there are opportunities
    here, in each period,
  21. for us to cheat.
    We could cheat both on the
  22. quality of the fruit that I
    provide or the quantity of the
  23. fruit that I provide to Jake,
    and he can cheat on the
  24. quantity or quality of the
    vegetables that he provides to
  25. me.
    Our central intuition is:
  26. perhaps what can give us good
    incentives is the idea that if
  27. Jake cooperates today,
    then I might cooperate
  28. tomorrow, I might not cheat
  29. Conversely, if he cheats and
    provides me with lousy
  30. vegetables today I'm going to
    provide him with lousy fruit
  31. tomorrow.
    Similarly for me,
  32. if I provide Jake with lousy
    fruit today he can provide me
  33. with lousy vegetables tomorrow.
    So what do we need?
  34. We need the difference in the
    value of the promise of good
  35. behavior tomorrow and the threat
    of bad behavior tomorrow to
  36. outweigh the temptation to cheat
  37. I'm going to gain by providing
    him with the bad fruit or fewer
  38. fruit today--bad fruit because
    those I would otherwise have to
  39. throw away.
    So that temptation to cheat has
  40. to be outweighed by the promise
    of getting good vegetables in
  41. the future from Jake and vice
  42. So here's that idea on the
  43. What we need is the gain if I
    cheat today to be outweighed by
  44. the difference between the value
    of my relationship with Jake
  45. after cooperating and the value
    of my relationship with Jake
  46. after cheating tomorrow.
    Now what we discovered last
  47. time--this was an idea I think
    we kind of knew,
  48. we have kind of known it since
    the first week--but we
  49. discovered last time,
    somewhat surprisingly,
  50. that life is not quite so
  51. In particular,
    what we discovered was we need
  52. these to be credible,
    so there's a problem here of
  53. credibility.
    So in particular,
  54. if we think of the value of the
    relationship after cooperating
  55. tomorrow as being a promise,
    and the value of the
  56. relationship after cheating as
    being a threat,
  57. we need these promises and
    threats to be credible.
  58. We need to actually believe
    that they're going to happen.
  59. And one very simple area where
    we saw that ran immediately into
  60. problems was if this repeated
  61. although repeated,
    had a known end.
  62. Why did known ends cause
    problems for us?
  63. Because in the last period,
    in the last period of the game
  64. we know that whatever we promise
    to do or whatever we threaten to
  65. do,
    in the last period,
  66. once we reached that last
    period, in that sub-game we're
  67. going to play a Nash
  68. What we do has to be consistent
    with our incentives in the last
  69. period.
    So in particular,
  70. if there's only one Nash
    equilibrium in that last period,
  71. then we know in that last
    period that's what we're going
  72. to do.
    So if we look at the second to
  73. last period we might hope that
    we could promise to cooperate,
  74. if you cooperate today,
  75. Or you could promise to punish
    tomorrow if you cheat today,
  76. but those threats won't be
    credible because we know that
  77. tomorrow you're just going to
    play whatever that Nash
  78. equilibrium is.
    That lack of credibility means
  79. there's no scope to provide
    incentives today for us to
  80. cooperate and we saw things
    unravel backwards.
  81. So the way in which we ensure
    that we're really focusing on
  82. credible promises and credible
    threats here is by focusing on
  83. sub-game perfect equilibrium,
    the idea that we introduced
  84. just before the Thanksgiving
  85. We know that sub-game perfect
    equilibria have the property
  86. that they have Nash behavior in
    every sub-game,
  87. so in particular in the last
    period of the game and so on.
  88. So what we want to be able to
    do here, is try to find scope
  89. for cooperation in relationships
    without contracts,
  90. without side payments,
    by focusing on sub-game perfect
  91. equilibria of these repeated
  92. Right at the end last time,
    we said okay,
  93. let's move away from the
    setting where we know our game
  94. is going to end,
    and let's look at a game which
  95. continues, or at least might
  96. So in particular,
    we looked at the problem of the
  97. Prisoner's Dilemma which was
    repeated with the probability
  98. that we called δ
    each period,
  99. with the probability δ
    of continuing.
  100. So every period we're going to
    play Prisoner's Dilemma.
  101. However, with probability 1 -
    δ the game might just end
  102. every period.
    We already noticed last time
  103. some things about this.
    The first thing we noticed was
  104. that we can immediately get away
    from this unraveling argument
  105. because there's no known end to
    the game.
  106. We don't have to worry about
    that thread coming loose and
  107. unraveling all the way back.
    So at least there's some hope
  108. here to be able to establish
    credible promises and credible
  109. threats later on in the game
    that will induce good behavior
  110. earlier on in the game.
    So that's where we were last
  111. time, And here is the Prisoner's
    Dilemma, we saw this time,
  112. and we actually focused on a
    particular strategy.
  113. But before I come back to this
    strategy that we focused on last
  114. time let's just see some things
    that won't work,
  115. just to sort of reinforce the
  116. So here's a possible strategy
    in the Prisoner's Dilemma.
  117. A possible strategy in the
    Prisoner's Dilemma would be
  118. cooperate now and go on
    cooperating regardless of what
  119. anyone does.
    So let's just cooperate forever
  120. regardless of the history of the
  121. Now if two players,
    if Jake and I are involved in
  122. this business relationship,
    which has the structure of a
  123. Prisoner's Dilemma and both of
    us play this strategy of
  124. cooperate now and cooperate
    forever no matter what,
  125. clearly that will induce
  126. That's the good news.
    The problem is that isn't an
  127. equilibrium, that's not even a
    Nash equilibrium,
  128. let alone a sub-game perfect
  129. Why is it not a sub-game
    perfect equilibrium?
  130. Because in particular,
    if Jake is smart (and he is),
  131. Jake will look at this
    equilibrium and say:
  132. Ben is going to cooperate no
    matter what I do,
  133. so I may as well cheat,
    and in fact,
  134. I may as well go on cheating.
    So Jake has a very good
  135. deviation there which is simply
    to cheat forever.
  136. So the strategy cooperate now
    and go on cooperating no matter
  137. what doesn't contain incentives
    to support itself as an
  138. equilibrium.
    And we need to focus on
  139. strategies that contain subtle
    behavior that generates promises
  140. of rewards and threats of
    punishment that induce people to
  141. actually stick to that
    equilibrium behavior.
  142. So is everyone clear that
    cooperating no matter what--it
  143. sounds good--but it isn't going
    to work.
  144. People aren't going to stick
    with that.
  145. So instead what we focused on
    last time, and actually we had
  146. some players who seemed to
    actually--they've moved now--but
  147. they seemed actually to be
    playing this strategy.
  148. We focused on what we called
    the grim trigger strategy.
  149. And the grim trigger strategy
    is what?
  150. It says in the first period
    cooperate and then go on playing
  151. cooperate as long as nobody has
    ever defected,
  152. nobody has ever cheated.
    But if anybody ever plays D,
  153. anybody ever plays the defect
    strategy, then we just play D
  154. forever.
    So this is a strategy,
  155. it tells us what to do at every
    possible information set.
  156. It also, if two players are
    playing the strategy,
  157. has the property that they will
    cooperate forever:,
  158. that's good news.
    And what we left ourselves last
  159. time was checking that this
    actually is an equilibrium,
  160. or more generally,
    under what conditions is this
  161. actually an equilibrium.
    So we got halfway through that
  162. calculation last time.
    So what we need to do is we
  163. need to make sure that the
    temptation of cheating today is
  164. less than the value of the
    promise minus the value of the
  165. threat tomorrow.
    We did parts of this already,
  166. let's just do the easy parts.
    So the temptation today is:
  167. if I cheat today I get 3,
    whereas if I went on
  168. cooperating today I get 2.
    So the temptation is just 1.
  169. What's the threat?
    The threat is playing D
  170. forever, so this is actually the
    value of (D, D) forever.
  171. You've got to be careful about
    for ever: when I say for ever,
  172. I mean until the game ends
    because eventually the game is
  173. going to end,
    but let's use the code for ever
  174. to mean until the game ends.
    What's the promise?
  175. The promise is the value of
    continuing cooperation,
  176. so the value of (C,C) for ever.
    That's what this bracket is,
  177. and it's still tomorrow.
  178. So let's go on working on this.
    So the value of cooperating for
  179. ever is actually--let's be a bit
    more detailed--this is the value
  180. of getting 2 in every period,
    so it's value of 2 for ever;
  181. and this is the value of 0
  182. So the value of 0 forever,
    that's pretty easy to work out:
  183. I get 0 tomorrow,
    I get 0 the day after tomorrow,
  184. I get 0 the day after the day
    after tomorrow.
  185. Or more accurately:
    I get 0 tomorrow,
  186. I get 0 the day after tomorrow
    if we're still playing,
  187. I get 0 the day after the day
    after tomorrow if we're still
  188. playing and so on.
    But that isn't a very hard
  189. calculation, this thing is going
    to equal 0.
  190. So this object here is just 0.
    This object here is 3 - 2,
  191. I can do that one in my head,
    that's 1.
  192. So I'm left with the value of
    getting 2 for ever,
  193. and that requires a little bit
    more thought.
  194. But let's do that one bit of
    algebra because it's going to be
  195. useful throughout today.
    So this thing here,
  196. the value of 2 for ever is
  197. Well I get 2,
    that's tomorrow,
  198. and then, assuming I'm still
    playing the day after
  199. tomorrow--so I need to discount
    it--with probability of δ
  200. I'm still playing the day after
    tomorrow--and I get 2 again.
  201. And the day after the day after
    tomorrow I'm still playing with
  202. the probability that the game
    didn't end tomorrow and didn't
  203. end the next day so that's with
    probability δ²
  204. and again I get 2.
    And then the day after,
  205. what is it?
    This is tomorrow,
  206. the day after tomorrow,
    the day after the day after
  207. tomorrow: this is the day after
    the day after the day after
  208. tomorrow which is δ³
    2 and so on.
  209. Everyone happy with that?
    So starting from tomorrow,
  210. if we play (C,
    C) for ever,
  211. I'll get 2 tomorrow,
    2 the day after tomorrow,
  212. 2 the day after the day after
    tomorrow, and so on.
  213. And I just need to take an
    account of the fact that the
  214. game may end between tomorrow
    and the next day,
  215. the game may end between the
    day after tomorrow and the day
  216. after the day after tomorrow and
    so on.
  217. Everyone happy with that?
    So what is the value,
  218. what is thing?
    Let's call this X for a second.
  219. So we've done this once before
    in the class but let's do it
  220. again anyway.
    This is the geometric sum,
  221. some of you may even remember
    from high school how to do a
  222. geometric sum,
    but let's do it slowly.
  223. So to work out what X is what
    I'm going to do is I'm going to
  224. multiply X by δ,
    so what's δX?
  225. So this 2 here will become a
    2δ, and this δ2 here
  226. will become a δ²2,
    and this δ²2 will
  227. become a δ³2,
    and this δ³2 will
  228. become a δ^(4)2,
    and so on.
  229. Now what I'm going to do is I'm
    going to subtract the second of
  230. those lines from the first of
    those lines.
  231. So what I'm going to do is,
    I'm going to subtract
  232. X--δX.
    So I'm going to subtract the
  233. second line from the first line.
    And when I do that I'm going to
  234. notice I hope that this 2δ
    is going to cancel with this
  235. 2δ,
    and this δ²2 is going
  236. to cancel with this
  237. and this δ³2 is going
    to cancel with this
  238. δ³2 and so on.
    So what I'm going to get left
  239. with is what?
    Everything's going to cancel
  240. except for what?
    Except for that first 2 there,
  241. so this is just equal to 2.
    Now this is a calculation I can
  242. do.
    So I've got X = 2 / [1-δ].
  243. So just to summarize the
    algebra, getting 2 forever,
  244. that means 2 + δ2 +
    δ²2 + δ³2
  245. etc..
    The value of that object is
  246. 2/[1-δ].
    So we can put that in here as
  247. well.
    This object here 2/[1-δ]
  248. is the value of 2 forever.
    Now before I go onto a new
  249. board I want to do one other
  250. On the left hand side I've got
    my temptation,
  251. that was 1, I've got the value
    of cooperating forever starting
  252. from tomorrow which is
  253. and I've got the value of
    defecting forever starting from
  254. tomorrow which is 0.
    However, all of these objects
  255. on the right hand side,
    they start tomorrow,
  256. whereas, the temptation today
    is today.
  257. Temptation today happens today.
    These differences in value
  258. start tomorrow.
    Since they start tomorrow I
  259. need to discount them because we
    don't know that tomorrow is
  260. going to happen.
    The world may end,
  261. or more importantly the
    relationship may end,
  262. between today and tomorrow.
    So how much do I have to weight
  263. them by?
    By δ, I need to multiply
  264. all of these lines by δ
    and so on.
  265. Now this is now a mess so let's
    go to a new board.
  266. Now let's summarize what we now
    have, What we're doing here is
  267. asking is it the case that if
    people play the grim trigger
  268. strategy that that is in fact an
  269. That is a way of sustaining
  270. The answer is we need 1,
    that's our temptation,
  271. to be less than 2/[1-δ],
    that's the value of cooperating
  272. for ever starting from tomorrow,
    minus 0, that's the value of
  273. defecting forever starting
  274. and this whole thing is
    multiplied by δ
  275. because tomorrow may not
  276. Everyone happy with that so far?
    I'm just kind of collecting up
  277. the terms that we did slowly
    just now.
  278. So now what I want to do
    is--question mark here because
  279. we don't know whether it is--I'm
    going to solve this for δ.
  280. So when I solve this for δ
    I'll probably get it wrong,
  281. but let's be careful.
    So this is equivalent to saying
  282. 1-δ <
    2δ and it's also
  283. equivalent to saying therefore
    that δ >
  284. = 1/3.
    Everyone happy with that?
  285. Let me just turn my own page.
    So what have we shown so far?
  286. We've shown that if we're
    playing the grim trigger
  287. strategy, and we want to deter
    people from doing what?
  288. From defecting from this
    strategy in the very first
  289. period, then we're okay provided
    δ is bigger than 1/3.
  290. But at this point some of you
    could say, yeah but that's just
  291. one of the possible ways I could
    defect from this strategy.
  292. After all, the defection we
    just considered,
  293. the move away from equilibrium
    we just considered was what?
  294. We considered my cheating
    today, but thereafter,
  295. I reversed it back to doing
    what I was supposed to do:
  296. I went along with playing D
  297. So the particular defection we
    looked at just now was in Period
  298. 1, I'm going to defect,
    but thereafter,
  299. I'm actually going to do what
    the equilibrium strategy tells
  300. me to do.
    I'm going to go along with the
  301. punishment and play my part of
    (D,D) forever.
  302. So you might want to ask,
    why would I do that?
  303. Why would I go along?
    I cheated the first time but
  304. now I'm doing what the strategy
    tells me to do.
  305. It tells me to play D.
    Why am I going along with that?
  306. You could consider going away
    from the equilibrium by
  307. defecting, for example in Period
  308. and then in Period 2 do
    something completely different
  309. like cooperating.
    So we might want to worry,
  310. how about playing D now and
    then C in the next period,
  311. and then D forever.
    That's just some other way of
  312. defecting.
    So far we've said I'm going to
  313. defect by playing D and then
    playing D forever,
  314. but now I'm saying let's play D
    now and then play a period of C
  315. and then D forever.
    Is that going to be a
  316. profitable deviation?
    Well let's see what I'd get if
  317. I do that particular deviation.
    What play is that going to
  318. induce?
    Remember the other player is
  319. playing equilibrium,
    so that player is going to
  320. induce, in the first period,
    I'm playing D and Jake's
  321. playing C.
    In the second period Jake's
  322. going to start punishing me,
    so he's going to play D and
  323. according to this deviation I'm
    going to play C.
  324. So in the second period I'll
    play C and Jake will play D,
  325. and in the third period and
    thereafter, we'll just play D,
  326. D, D, D, D, D.
    So these are just some other
  327. deviation other than the one we
    looked at.
  328. So what payoff do I get from
  329. Okay, I get three in the first
    period, just as I did for my
  330. original defection,
    that's good news.
  331. But now in the second period
    discounted, I actually get -1,
  332. I'm actually doing even worse
    in the second period because I'm
  333. cooperating while Jake's
  334. and then in the third period I
    get 0 and in the fourth period I
  335. get 0 and so on.
    So the total payoff to this
  336. defection is 3 - δ.
    Now, that's even worse than the
  337. defection we considered to start
  338. The defection we considered to
    start with, I got 3 in the first
  339. period and thereafter I got 0.
    Now I got 3 in the first
  340. period, -1 in the second period,
    and then 0 thereafter.
  341. So this defection in which I
    defect--this move away from
  342. equilibrium--in which I cheat in
    the first period and then don't
  343. go along with the punishment,
    I don't in fact play D forever
  344. is even worse.
    Is that right? It's even worse.
  345. So what's the lesson here?
    The lesson here is the reason
  346. that I'm prepared to go along
    with my own punishment and play
  347. D forever after a defection is
  348. It's if Jake is going to play D
    forever I may as well play D
  349. forever.
    Is that right?
  350. So another way of saying this
    is the only way which I could
  351. possibly hope to have a
    profitable deviation,
  352. given that Jake's going to
    revert to playing D forever is
  353. for me to defect on Jake once
    and then go along with playing D
  354. forever.
    There's no point once he's
  355. playing D, there's no point me
    doing anything else,
  356. so this is worse,
    this is even worse.
  357. This defection is even worse.
    More generally,
  358. the reason this is even worse
    is because the punishment we
  359. looked at before,
    which was (D,
  360. D) for ever,
    the punishment (D,D) forever is
  361. itself an equilibrium.
    It's credible because it's
  362. itself an equilibrium.
  363. So unlike in the finitely
    repeated games we did last time,
  364. unlike in the two period or the
    five period repeated games,
  365. here the punishment really is a
    credible punishment,
  366. because what I'm doing in the
    punishment phase is playing an
  367. equilibrium.
    There's no point considering
  368. any other deviation other than
    playing D once and then just
  369. going on playing D.
    So that's one other possible
  370. deviation, but there are others
    you might want to consider.
  371. So far all we've considered is
  372. We've considered the deviation
    where I, in the very first
  373. period, I cheat on Jake and then
    I just play D forever.
  374. But what about the second
  375. Another thing I could do is how
    about cheating not in the first
  376. period of the game but in the
  377. So according to this strategy
    what am I going to do.
  378. The first period of the game
    I'll go along with Jake and
  379. cooperate, but in the second
    period I'll cheat on him.
  380. Now how am I going to check
    whether that's a good deviation
  381. or not?
    How do I know that's not going
  382. to be a good deviation?
    Well we already know that I'm
  383. not going to want to cheat in
    the first period of the game.
  384. I want to argue that exactly
    the same analysis tells me I'm
  385. not going to want to cheat in
    the second period of the game.
  386. Why?
    Because once we reach the
  387. second period of the game,
    it is the first period
  388. of the game.
    Once we reach the second period
  389. of the game, looking from period
    two onwards,
  390. it's exactly the same as it was
    when we looked from period one
  391. initially.
    So to say it again,
  392. what we argued before was--on
    the board that I've now covered
  393. up--what we argued before was,
    I'm not going to want to cheat
  394. in the very first period of the
    game provided δ
  395. > 1/3.
    I want to claim that that same
  396. argument tells me I'm not going
    to want to cheat in the second
  397. period of the game provided
    δ > 1/3.
  398. I'm not going to want to cheat
    in the fifth period of the game
  399. provided δ
    > 1/3.
  400. Because this game from the
    fifth period on,
  401. or the five hundredth period
  402. or the thousandth period on
    looks exactly the same as is it
  403. does from the beginning.
    So what's neat about this
  404. argument is the same analysis
    says, this is not profitable if
  405. δ > 1/3.
  406. So what have we learned here?
    I want to show you some nerdy
  407. lessons and then some actual
    sort of real world lessons.
  408. Let's start with the nerdy
  409. The nerdy lesson is this grim
    strategy works because
  410. both--let's put it up again so
    we can actually see it--this
  411. grim strategy,
    it works because both the play
  412. that it suggests if we both
    cooperate and the play that it
  413. suggests if we both defect are
    themselves equilibria.
  414. These are credible threats and
    credible promises because what
  415. you end up doing both in the
    promise and in the threat is
  416. itself equilibrium behavior.
    That's good.
  417. The second thing we've learned,
    however, is for this to work we
  418. need δ >
    1/3, we need the probability
  419. continuation to be bigger than
  420. So leaving aside the nerdy
    stuff for a second--you have
  421. more practice on the nerdy stuff
    on the homework assignment--the
  422. lesson is we can get cooperation
    in the Prisoner's Dilemma using
  423. the grim trigger.
    Remember the grim trigger
  424. strategy is cooperate until
    someone defects and then defect
  425. forever.
    So you get cooperation in the
  426. Prisoner's Dilemma using the
    grim trigger as a sub-game
  427. perfect equilibrium.
    So this is an equilibrium
  428. strategy, that's good news,
    provided the probability of
  429. continuation is bigger than 1/3.
  430. Let's try and generalize that
    lesson away from the Prisoner's
  431. Dilemma.
    So last time our lesson was
  432. about what in general could we
    hope for in ongoing
  433. relationships?
    So let's put down a more
  434. general lesson that refines what
    we learned last time.
  435. So the more general lesson is,
    in an ongoing relationship--let
  436. me mimic exactly the words I
    used last time--so for an
  437. ongoing relationship to provide
    incentives for good behavior
  438. today,
    it helps--what we wrote last
  439. time was--it helps for that
    relationship to have a future.
  440. But now we can refine this,
    it helps for there to be a high
  441. probability that the
    relationship will continue.
  442. So the specific lesson for
    Prisoner's Dilemma and the grim
  443. trigger strategy is we need
    δ, the probability
  444. continuation,
    to be bigger than 1/3.
  445. But the more general intuition
    is, if we want my ongoing
  446. business relationship with me
    and Jake to generate good
  447. behavior--so I'm going to
    provide him with good fruit and
  448. he's going to provide me with
    good vegetables--we need the
  449. probability that that
    relationship will continue to be
  450. reasonably high.
    I claim this is a very natural
  451. intuition.
  452. Because the probability that
    the relationship will continue
  453. is the weight that you put on
    the future.
  454. The probability that the
    relationship will continue,
  455. this thing, this is the weight
    you put on the future.
  456. The more weight I put on the
    future, the easier it is for the
  457. future to give me incentives to
    behave well today,
  458. the easier it is for those to
    overcome the temptations to
  459. cheat today.
    That seems like a much more
  460. general lesson than just the
    Prisoner's Dilemma example.
  461. Let's try to push this to some
    examples and see if it rings
  462. true.
    So the lesson we've got here is
  463. to get cooperation in these
    relationships we need there to
  464. be a high probability,
    a reasonably high probability
  465. that they're going to continue.
    We know exactly what that is
  466. for Prisoner's Dilemma but the
    lesson seems more general.
  467. So here's two examples.
    How many of you are seniors?
  468. One or two, quite a few are
  469. Keep your hands up a second.
    All of those of you who are
  470. seniors--we can pan these guys.
    Let's have a look at them.
  471. Actually, why don't we get all
    the seniors to stand up:
  472. make you work a bit here.
    Now the tricky question,
  473. the tricky personal question.
    How many of you who are seniors
  474. are currently involved in
    personal relationships,
  475. you know: have a significant
  476. Stay standing up if you have a
    significant other.
  477. Look at this, it's pathetic.
    What have I been saying about
  478. economic majors?
    All right, so let's just think
  479. about, stay standing a second,
    let's get these guys to think
  480. about it a second.
    So seniors who are involved in
  481. ongoing relationships with
    significant others,
  482. what do we have to worry about
    those seniors?
  483. Well these seniors are about to
    depart from the beautiful
  484. confines of New Haven and
    they're going to take jobs in
  485. different parts of the world.
    And the problem is some of them
  486. are going to take jobs in New
    York while their significant
  487. other takes a job in San
    Francisco or Baghdad or
  488. whatever,
    let's hope not Baghdad,
  489. London shall we say.
    Now if it's the case that you
  490. are going to take a job in New
    York next year and your
  491. significant other is going to
    take a job in Baghdad or London,
  492. or anyway far away,
    in reality, being cynical a
  493. little bit, what does that do to
    the probability that your
  494. relationship is going to last?
    It makes it go down.
  495. It lowers the probability that
    your relationship's going to
  496. continue.
    So what is the
  497. prediction--let's be mean here.
    These are the people with
  498. significant others who are
    seniors, how many of you are
  499. going to be separated by a long
    distance from your significant
  500. others next period?
    Well one of them at the back,
  501. okay one guy,
    at the back,
  502. two guys, honesty here,
    three, four of you right?
  503. So what's our prediction here?
    What does this model predict as
  504. a social science experiment.
    What does it predict?
  505. It predicts that for those of
    you who just raised your hands,
  506. those seniors who just raised
    their hands who are about to be
  507. separated by large distances,
    those relationships,
  508. each player in that
    relationship is going to have a
  509. lower value on the future.
    So during the rest of your
  510. senior year, during the spring
    of your senior year what's the
  511. prediction of this model?
    They're going to cheat.
  512. So we could actually do a
    controlled experiment,
  513. what we should do here is we
    should keep track of the people
  514. here,
    the seniors who are going to be
  515. separated--you can sit down now,
    I'm sorry to embarrass you all.
  516. We could keep track of those
    seniors who are about to be
  517. separated and go into a long
    distance relationships,
  518. and those that are not.
    The people who are not are our
  519. control group.
    And we should see if during the
  520. spring semester the people who
    are going to be separated cheat
  521. more often than the others.
    So it's a very clear prediction
  522. of the model that's relevant to
    some of your lives.
  523. Let me give you another example
    that's less exciting perhaps,
  524. but same sort of thing.
    Consider the relationship that
  525. I have with my garage mechanic.
    I should stress this is not a
  526. significant other relationship.
    So I have a garage mechanic in
  527. New Haven, and that garage
    mechanic fixes my car.
  528. And we have an ongoing business
  529. He knows that whenever my car
    needs fixing,
  530. even if it's just a small thing
    like an oil change,
  531. I'm going to go to him and have
    him fix it, even though it might
  532. be cheaper for me to go to Jiffy
    Lube or something.
  533. So I'm going to take my car to
    him to be fixed,
  534. and he's going to make some
    money off me on even the easy
  535. things.
    What do I want in return for
  536. that?
    I want him to be honest and if
  537. all I need is an oil change I
    want him to tell me that,
  538. and if what I actually need is
    a new engine,
  539. he tells me I need new engine.
    So my cooperating with him,
  540. is always going to him,
    even if it's something simple;
  541. and his cooperating with me,
    is his not cheating on fixing
  542. the car.
    He knows more about the car
  543. than I do.
    But now what happens if he
  544. knows either that I'm about to
    leave town (which is the example
  545. we just did),
    or, more realistically,
  546. he kind of knows that my car is
    a lemon and I'm about to get rid
  547. of it anyway.
    Once I get a new car I'm not
  548. going to go to him anymore
    because I have to go to the
  549. dealer to keep the warranty
  550. So he knows that my car is
    about to break down anyway,
  551. and he knows that I know that
    the car is about to break
  552. anyway,
    so my lemon of a car is about
  553. to be passed on--probably to one
    of my graduate students--then
  554. what's going to happen?
    So I'm going to have an
  555. incentive to cheat because I'm
    going to start taking my useless
  556. car to Jiffy Lube for the oil
  557. And he's going to have an
    incentive to cheat.
  558. He's going to start telling me
    you know you really need a new
  559. engine or a new clutch--it's a
    manual so I have a clutch:
  560. it's a real car--so I'm going
    to need a new clutch rather than
  561. just tightening up a bolt.
    So once again the probability
  562. of the continuation of the
  563. as it changes,
    it leads to incentives to
  564. cheat.
    It leads to that relationship
  565. breaking down.
    That's the content,
  566. that's the real world content
    of the math we just did.
  567. Let's try and push this a
    little further.
  568. Now what we've shown is that
    the grim trigger works provided
  569. δ > 1/3,
    and δ being bigger than
  570. 1/3 doesn't seem like a very
    large continuation probability.
  571. So just having a probability of
    1/3 that the relationship
  572. continues allows the grim
    trigger to work,
  573. so that seems good news for the
    grim trigger.
  574. However, in reality,
    in the real world,
  575. the grim trigger might have
    some disadvantages.
  576. So let's just think about what
    the grim trigger is telling us
  577. in the real world.
    It's telling us that if even
  578. one of us cheats just a little
    bit--I just provide one item of
  579. rotten fruit to Jake or he gives
    me one too few branches of
  580. asparagus in his provisions to
    me--then we never do business
  581. with each other again ever.
    It's completely the end.
  582. We just never cooperate again.
    That seems a little bit drastic.
  583. It's a little bit draconian if
    you like.
  584. So in particular,
    in the real world,
  585. there's a complication here,
    in the real world every now and
  586. then one of us going "to cheat"
    by accident.
  587. That day that I didn't have my
    glasses on and I put in a rotten
  588. apple in the apples I supplied
    to Jake.
  589. In the fruit,
    he was counting out the
  590. asparagus and he lost count at
    1,405 and he gave me one too
  591. few.
    So we might want to worry about
  592. the fact that the grim trigger,
    it's triggered by any amount of
  593. cheating and it's very drastic:
    it says we never do business
  594. again.
    The grim trigger is the analog
  595. of the death penalty.
    It's the business analog of the
  596. death penalty.
    It's not that I'm going to kill
  597. Jake if he gives me one too few
    branches of asparagus,
  598. but I'm going to kill the
  599. For you seniors or otherwise,
    who are involved in personal
  600. relationships,
    it's the equivalent of saying,
  601. if you even see your partner
    looking at someone else,
  602. let alone sitting next to them
    in the class,
  603. the relationship is over.
    It seems drastic.
  604. So we might be interested
    because mistakes happen,
  605. because misperceptions happen,
    we might be interested in using
  606. punishments that are less
    draconian than the grim trigger,
  607. less draconian than the death
  608. Is that right?
    So what I want to do is I want
  609. to consider a different
    strategy, a strategy other than
  610. the grim trigger strategy,
    and see if that could work.
  611. So where shall I start?
    Let's start here,
  612. so again what I'm going to
    revert to is the math and the
  613. nerdiness of our analysis of the
    Prisoner's Dilemma but I want
  614. you to have in mind business
  615. your own personal
  616. your friendships and so on.
    More or less everything you do
  617. in life involves repeated
    interaction, so have that in the
  618. back of your mind,
    but let's be nerdy now.
  619. So what I want to consider is a
    one period punishment.
  620. So how are we going to write
    down a strategy that has
  621. cooperation but a one period
  622. So here's the strategy.
    It says--it's kind of weird
  623. thing but it works--play C to
    start and then play C if--this
  624. is going to seem weird but trust
    me for a second--play C if
  625. either (C, C) or (D,D) were
    played last.
  626. So, if in the previous period
    either both people cooperated or
  627. both people defected,
    then we'll play cooperation
  628. this period.
    And play D otherwise:
  629. play D if either (C,
    D) or (D, C) were played last.
  630. Let's just think about this
    strategy for a second.
  631. What does that strategy mean?
    So provided people start off
  632. cooperating and they go on
    cooperating--if both Jake and I
  633. play this strategy--in fact,
    we'll cooperate forever.
  634. Is that right?
    So I claim this is a one period
  635. punishment strategy.
    Let's just see how that works.
  636. So suppose Jake and I are
    playing this strategy.
  637. We're supposed to play C every
  638. And suppose deliberately or
    otherwise, I play D.
  639. So now in that period in which
    I play D, the strategys played
  640. were D by me and C by Jake.
    So next period what does this
  641. strategy tell us both to play?
    So it was D by me and C by
  642. Jake, so this strategy tells us
    to play D.
  643. So next period both of us will
    play D.
  644. So both of us will be
    uncooperative precisely for that
  645. period, that next period.
    Now, what about the period
  646. after that?
    The period after that,
  647. Jake will have played D,
    I will have played D.
  648. So this is what will have
    happened: we both played D,
  649. and now it tells us to
    cooperate again.
  650. Everyone happy with that?
    So this strategy I've written
  651. down--it seems kind of
    cumbersome--but what it actually
  652. induces is exactly a one period
  653. If Jake is the only cheat then
    we both defect for one period
  654. and go back to cooperation.
    If I'm the only person who
  655. cheats then we both defect for
    one period and go back to
  656. cooperation.
    It's a one period punishment
  657. strategy.
    Of course the question is,
  658. the question you should be
    asking is, is this going to
  659. work?
    Is this an equilibrium?
  660. So let's just check.
    Is this an SPE.
  661. Is it an equilibrium?
    So what do we need to check?
  662. We need to check,
    as usual, that the temptation
  663. is less than or equal to the
    value of the promise--the value
  664. of the promise of continuing in
    cooperation--the value of the
  665. promise minus the value of the
  666. And once again we have to be
    careful, because the temptation
  667. occurs today and this difference
    between values occurs tomorrow.
  668. Is that right?
    So this is nothing new,
  669. this is what we've always
    written down,
  670. this is what we have to check.
    So the temptation for me to
  671. cheat today, that's the same as
    it was before,
  672. it's 3 - 2.
    The fact that it's tomorrow is
  673. going to give me a δ
  674. Here's our square bracket.
    So what's the value of the
  675. promise?
    So provided we both go on
  676. cooperating, we're going to go
    on cooperating forever,
  677. in which case we're going to
    get 2 for ever.
  678. Is that right?
    So this is going to be the
  679. value of 2 forever starting
    tomorrow (and again for ever
  680. means until the game ends).
    The value of the threat is what?
  681. Be a bit careful now.
    It's the value of--so what's
  682. going to happen?
    If I cheat then tomorrow we're
  683. both going to cheat,
    so tomorrow,
  684. what am I going to get
  685. 0.
    So it's the value of 0
  686. tomorrow: we're both going to
    cheat, we're both going to play
  687. D.
    And then the next period what's
  688. going to happen?
    We're going to play C again,
  689. and from thereon we're going to
    go on playing C.
  690. So it's going to the value of 0
    tomorrow and then 2 forever
  691. starting the next day.
    That's what we have to evaluate.
  692. So 3 - 2, I can do that one
    again, that's 1.
  693. So what's the value of 2
    forever, well we did that
  694. already today,
    what was it?
  695. It's in your notes.
    Actually it's on the board,
  696. it's the X up there,
    what is it?
  697. Here it is, 2 for ever:
    we figured out the value of it
  698. before and it was
  699. So the value of 2 forever is
    going to be 2/[1–δ].
  700. How about the value of 0?
    So starting for tomorrow I'm
  701. going to get 0 and then with one
    period delay I'm going to get 2
  702. for ever.
    Well 2 forever,
  703. we know what the value of that
    is, it's 2/[1–δ],
  704. but now I get it with one
    period delay,
  705. so what do I have to multiply
    it by?
  706. By δ good.
    So the value of 0 tomorrow and
  707. then 2 forever starting the next
    day is δ
  708. x 2/[1–δ].
    And here's the δ
  709. coming from here which just
    takes into account that all this
  710. analysis is starting tomorrow.
    So to summarize,
  711. this is my temptation today.
    This is what I'll get starting
  712. tomorrow if I'm a good boy and
  713. And this is the value of what
    I'll get if I cheat today.
  714. Starting tomorrow I'll get
    nothing, and then I'll revert
  715. back to cooperation.
    And since all of these values
  716. in this square bracket start
    tomorrow I've discounted them by
  717. δ.
    Now this requires some math so
  718. bear with me while I probably
    get some algebra wrong--and
  719. please can I get the T.A.'s to
    stare at me a second because
  720. I'll probably get this wrong.
    Okay so what I'm going to do
  721. is, I'm going to look at my
    notes, I'm going to cheat,
  722. that's what I'm going to do.
    Okay, so what I'm going to do
  723. is I'm going to have 1 is less
    than or equal to,
  724. I'm going to take a common
    factor of 2 / [1–δ]
  725. and δ, so I'm going to
    have 2δ/[1–δ],
  726. and that's going to leave
    inside the square brackets:
  727. this is a 1 and this is a
  728. So this δ
    here was that δ
  729. there, and then I took out a
    common factor of
  730. 2/[1–δ]
    from this bracket.
  731. Everyone okay with the algebra?
    Just algebra,
  732. nothing fancy going on there.
    So that's good because now the
  733. 1-δ cancels,
    this cancels with this,
  734. so this tells us we're okay
    provided 1/2 <= δ:
  735. it went up.
    So don't worry too much about
  736. the algebra, trust me on the
    algebra a second,
  737. let's just worry about the
  738. What's the conclusion?
    The conclusion is that this one
  739. period punishment is an SPE,
    it will be enough,
  740. one period of punishment will
    be enough to sustain cooperation
  741. in my Prisoner's Dilemma
    repeated business relationship
  742. with Jake,
    or in the seniors'
  743. relationships with their
    significant others,
  744. provided δ
    > 1/2.
  745. What did δ
    need to be for the grim
  746. strategy?
    1/3, so what have we learned
  747. here?
    We learned--nerdily--what we
  748. learned was that for the grim
    strategy we needed δ
  749. > 1/3.
    For the one period punishment
  750. we needed δ
    > 1/2, but what's the more
  751. general lesson?
    The more general lesson is,
  752. if you use a softer punishment,
    a less draconian punishment,
  753. for that to work we're going to
    need a higher δ.
  754. Is that right?
    So what we're learning here is
  755. there's a trade off,
    there's a trade off in
  756. incentives.
    And the trade off is if you use
  757. a shorter punishment,
    a less draconian
  758. punishment--instead of cutting
    people's hands off or killing
  759. them,
    or never dealing with them
  760. again, you just don't deal with
    them for one period--that's okay
  761. provided there's a slightly
    higher probability of the
  762. relationship continuing.
    So shorter punishments are okay
  763. but they need--the implication
    sign isn't really necessary
  764. there--they need more weight
    δ on the future.
  765. I claim that's very intuitive.
    What its saying is,
  766. we're always trading things off
    in the incentives.
  767. We're trading off the ability
    to cheat and get some cookies
  768. today versus waiting and,
    we hope, getting cookies
  769. tomorrow.
    So if, in fact,
  770. the difference between the
    reward and the punishment isn't
  771. such a big deal,
    isn't so big--the punishment is
  772. just, I'm going to give you one
    fewer cookies tomorrow--then you
  773. better be pretty patient not to
    go for the cookies today.
  774. I was about to say,
    those of you who have children.
  775. I'm probably the only person in
    the room with children.
  776. That cookie example will
    resonate for the rest of
  777. you--wait until you get
    there--you'll discover that,
  778. in fact, cookies are the right
  779. So shorter punishment,
    less draconian punishments,
  780. less reduction in your kid's
    cookie rations tomorrow is only
  781. going to work,
    is only going to sustain good
  782. behavior provided those kids put
    a high weight on tomorrow.
  783. In that case,
    it isn't that the kids will
  784. worry about the relationship
    breaking down,
  785. you're stuck with your kids,
    it's just that they're
  786. impatient.
    Okay, so we've been doing a lot
  787. of formal stuff here and I want
    to go on doing formal stuff,
  788. but what I want to do now is
    spend the rest of today looking
  789. at an application.
    An application is,
  790. I hope going to convince you
    that repeated interaction really
  791. matters.
    So this is assuming that the
  792. one about the seniors and their
    boyfriends and girlfriends
  793. wasn't enough.
    Okay, so the application is
  794. going to take us back a little
    bit because what I want to talk
  795. about is repeated moral hazard.
  796. Moral hazard is something we
    discussed the first class after
  797. the mid-term.
    So what I want you to imagine
  798. is that you are running a
    business in the U.S.
  799. and you are considering making
    an investment in an emerging
  800. market, and again,
    so as not to offend anybody who
  801. watches this on the video,
    let's just call that emerging
  802. market Freedonia,
    rather than give it a name like
  803. Kazakhstan, a name like
    something other than Freedonia.
  804. So Freedonia,
    for those of you who don't
  805. know, is a republic in a Marx
    Brothers film.
  806. So you're thinking of
    outsourcing some production of
  807. part of what your business is to
  808. The reason you're thinking of
    doing this outsourcing,
  809. what makes it attractive is
    that wages are low in Freedonia.
  810. So you get this outsourced in
  811. You think you're going to get
    it done cheaply.
  812. The down side is because
    Freedonia is an emerging market,
  813. the court system,
    it doesn't operate very well.
  814. And in particular,
    it's going to be pretty hard to
  815. enforce contracts and to jail
    people and so on in Freedonia.
  816. So you're considering
  817. The plus is,
    from your point of view,
  818. the plus is wages are cheap
    where you're going to get this
  819. production done.
    The down side is it's going to
  820. be hard to enforce contracts
    because this is an emerging
  821. market.
    So what you're considering
  822. doing is employing an agent and
    you're going to pay that agent
  823. W, so W is the wage if you
    employ them.
  824. I'll put this up in a tree in a
  825. Let's assume that the "going
    wage" in Freedonia is 1:
  826. we'll just normalize it.
    So the going wage in Freedonia
  827. is 1, and let's assume that to
    get this outsourcing to work
  828. you're going to have to send
    some resources to your agent,
  829. your employee in Freedonia.
    And let's assume that the
  830. amount you're going to have to
    send over there is equivalent to
  831. another 1.
    So the going wage in Freedonia
  832. is 1 and the amount you're going
    to have to invest in giving this
  833. agent materials or machinery is
    another 1.
  834. Let's assume that this project
    is a pretty profitable project.
  835. So if the project succeeds,
    if the project goes ahead and
  836. succeeds, it's going to generate
    a gross revenue of 4.
  837. Of course you have to invest 1
    so that's a net revenue of 3 for
  838. you, but nonetheless there's a
    big potential return here.
  839. The bad news is that your agent
    in Freedonia can cheat on you.
  840. In particular,
    what he can do is he can simply
  841. take the 1 that you've sent to
  842. sell those materials on the
    market and then go away and just
  843. work in his normal job anyway.
    So he can get his normal wage
  844. of 1 for just going and doing
    his normal job,
  845. whatever that was,
    and he can steal the resources
  846. from you.
    So let's put this up as a kind
  847. of tree.
    This is a slight cheat,
  848. this tree, but we'll see why in
    a second.
  849. So your decision is to invest
    and set W.
  850. So if you invest in Freedonia,
    you'll invest and set W,
  851. set the wage you're going to
    pay him.
  852. The going wage is 1 but you can
    set a different wage or you
  853. could just not invest.
    If you don't invest you get
  854. nothing and your agent in
    Freedonia just gets the going
  855. wage of 1.
    If you do invest in Freedonia
  856. and set a wage of W,
    then your agent has a choice.
  857. Either he can be honest or he
    can cheat.
  858. If he cheats,
    what's going to happen to you?
  859. You had to invest 1 in sending
    it over there,
  860. you're going to get nothing
    back, so you'll get -1.
  861. And he will go away and work
    his normal job and get 1,
  862. and, in addition,
    he'll sell your materials so
  863. he'll get a total of 1 + 1 is?
    2, thank you.
  864. So he'll get a total of 2.
    On the other hand,
  865. if he's honest,
    then you're going to get a
  866. return of 4 minus the 1 you had
    to invest minus whatever wage
  867. you paid to him.
    So your return will be 3 minus
  868. the wage you pay him.
    You're only going to pay him
  869. once the job's done,
    3 - W, and he's going to get W.
  870. He's done his job--he hasn't
    exercised his outside option,
  871. he hasn't sold your
    materials--so he'll just get W.
  872. Now, I'm slightly cheating here
    because this isn't really the
  873. way the tree looks because I
    could choose different levels of
  874. W.
    So this upper branch where I
  875. invest and set W is actually a
    continuum of such branches,
  876. one for each possible W,
    I could set.
  877. But for the purpose of today
    this is enough.
  878. This gives us what we needed to
  879. So let's imagine that this is a
    one shot investment.
  880. What I want to learn is in this
    one shot investment,
  881. I invest in Freedonia.
    I hire my agent once,
  882. what I want to learn is how
    much do I have to pay that agent
  883. to actually get the job done?
    Remember the starting position.
  884. The starting position is it
    looks very attractive.
  885. It looks very attractive
    because the returns on this
  886. project are 4 or 4 - 1,
    so that the surplus available
  887. on this project is 3 minus the
    wage, and the going wage was
  888. just 1.
    So it looks like there's lots
  889. of profit around to make this
    outsourcing profitable.
  890. I mumbled that so let me try it
  891. So the reason this looks
    attractive is the going wage is
  892. just 1, so if I just pay him 1
    and he does the project then
  893. I'll get a gross return of 4
    minus the 1 I invested minus the
  894. 1 that I had to pay him for a
    net return of 2.
  895. It seems like that's a 100%
    profitable project,
  896. so it looks very attractive.
    What's the problem?
  897. The problem is if I only
    set--this is going to give us
  898. backward induction--if I set the
    wage equal to the going wage,
  899. so if I set W = 1 what will my
    agent do?
  900. He's going to cheat.
    The problem is if I set W = 1,
  901. which is the going wage,
    the going wage in Freedonia,
  902. the agent will cheat.
    If he cheats I just lose my
  903. investment.
    So how much do I have to set
  904. the W to?
    Let's look at this.
  905. So we have to set W.
    What I need is I need his wage
  906. to be big enough so that being
    honest and going on with my
  907. projectoutweighs his incentive
    to cheat.
  908. I need W to be bigger than 2.
    Is that right?
  909. I need W to be at least as big
    as 2.
  910. So in setting the wage,
    in equilibrium,
  911. what are we going to do?
    I'm going to set a wage,
  912. let's call it W* = 2 (plus a
    penny), is that right?
  913. So this is an exercise which we
    visited the first day after the
  914. mid-term.
    This is about incentive design.
  915. In this one shot game,
    which we can easily solve by
  916. backward induction,
    I'm going to need to set a wage
  917. equal to 2, and then he'll work.
  918. So in a minute,
    we're going to look at the
  919. repeated version of this,
    but before we do let's just sum
  920. up where we are so far.
    What is this telling us?
  921. It's telling us that when you
    invest in an emerging market,
  922. where the courts don't work so
    they aren't going to be able to
  923. enforce this guy to work
    well--in particular,
  924. he can run off with your
    investment--even though wages
  925. are low, so it seems very
    attractive to do outsourcing,
  926. if you worry about getting
    incentives right you're going to
  927. have pay an enormous wage
    premium to get the guy to work.
  928. So the going wage in Freedonia
    was 1, but you had to set a wage
  929. equal to 2, a 100% wage premium,
    to get the guy to work.
  930. So the wage premium in this
    emerging market is 100%,
  931. you're paying 2 even though the
    going wage is 1.
  932. By the way, this is not an
    unreasonable prediction.
  933. If you look at the wages payed
    by European and American
  934. companies in some of these
    emerging markets,
  935. which have very,
    very low going wages,
  936. and if you look at the wages
    that are actually being paid by
  937. the companies that are doing
    outsourcing you see enormous
  938. wage premiums.
    You see enormous premiums over
  939. and above the going wage.
    Now what I want to do is I want
  940. to revisit exactly the same
    situation, but now we're going
  941. to introduce the wrinkle of the
  942. What's the wrinkle of the day?
    The wrinkle of the day is
  943. you're not only going to invest
    in Freedonia today,
  944. but if things go well you'll
    invest tomorrow,
  945. and if things go well again
    you'll invest the day after at
  946. least with some significant
  947. So the wage premium we just
    calculated was the one shot wage
  948. premium.
    It was getting this job--this
  949. single one shot job--outsourced
    to Freedonia.
  950. Now I want to consider how much
    you're going to have to pay,
  951. what are wages going to be in
    Freedonia in the foreign
  952. investment sector,
    if instead of just having a one
  953. shot, one job investment,
    you're investing for the long
  954. term.
    You're going to be in Freedonia
  955. for a while.
    So consider repeated
  956. interaction with probability
    δ of continuing.
  957. So we don't know that you're
    going to go on in Freedonia.
  958. Things might break down in
    Freedonia because there's a
  959. coup.
    It might break down in
  960. Freedonia because the American
    administration says you're not
  961. allowed to do outsourcing
  962. All sorts of things might
    happen, but with some
  963. probability δ
    the relationship is going to
  964. continue.
    So repeated interaction with
  965. probability of δ.
    Let's redo the exercise we did
  966. before to see what wage you'll
    have to charge.
  967. Our question is what
    wage--let's call it W**--what
  968. wage will you pay?
  969. The way we're going to solve
    this, is exactly using the
  970. methods we've learned in this
  971. So what we're going to compare
    is the temptation to cheat
  972. today--and we better make sure
    that that's less than δ
  973. times the value of continuing
    the relationship minus the value
  974. of ending the relationship.
    Let's call this tomorrow.
  975. So what's happening now is,
    once again, I'm employing my
  976. agent in Freedonia,
    and provided he does a good
  977. job, I'll employ him again
    tomorrow, at least with
  978. probability δ.
    But if he doesn't do a good
  979. job, if he runs off with my
    investment and doesn't do my
  980. job, what am I going to do?
    What would you do?
  981. You'd fire him.
    So the punishment--it's clear
  982. what the punishment's going to
    be here--the punishment is,
  983. if he doesn't do a good job,
    you fire him.
  984. The value of ending the
  985. This is firing and this is
  986. So let's just work out what
    these things are.
  987. So his temptation to cheat
    today: if he cheats today,
  988. he doesn't get my wage.
    But he does run off with my
  989. cash, and he does go and do his
    job at the going wage.
  990. So if he cheats today he gets
    2, he stole all my cash,
  991. and he's going off and working
    at the going wage,
  992. but he doesn't get what I would
    have paid him W** if the job was
  993. well done.
    We need this to be less than
  994. the value of continuing the
  995. Let's do the easy bit first.
    What does he get if we end the
  996. relationship?
    He's been fired,
  997. so he'll just work at the going
    wage for ever.
  998. So this is the value of 1 for
    ever, or at least until the end
  999. of the world.
    This is the value of what?
  1000. As long as he stayed employed
    by me what's he going to get
  1001. paid every period?
    What's he going to get paid?
  1002. W**.
    So the value of W** for ever.
  1003. Let me cheat a little bit and
    assume that the probability of
  1004. some coup happening that ends
    our relationship exogenously is
  1005. the same probability of the coup
    happening and ending his ongoing
  1006. wage exogenously,
    so we can use the same δ.
  1007. So let's just do some math
    here, what's the value of W**
  1008. forever?
    So remember the value of 2
  1009. forever was what?
  1010. So what's the value of W**
  1011. So this is going to be
  1012. What's the value of 1 forever?
  1013. The whole thing is multiplied
    by δ and this is 2-W**.
  1014. Now I need to do some algebra
    to solve for W**.
  1015. So let's try and do that.
    So I claim that this is the
  1016. same as [1--δ]
  1017. W** < W**δ
    - δ 1.
  1018. Everyone okay with that?
    One more line:
  1019. let me just sort out some terms
  1020. So taking these on the other
    side, I have [1–δ]
  1021. 2 + δ1 <= W**δ
    + [1–δ]
  1022. W** = W**.
    So someone should just check my
  1023. algebra at home,
    but I think that's right.
  1024. So the last two steps were just
    algebra, nothing fancy.
  1025. What have we learned?
    We have learned that the wage I
  1026. have to pay this guy,
    the wage I have to pay him lies
  1027. somewhere between 2 and 1,
    but we can do a bit better than
  1028. that.
  1029. Let's just delete everything
  1030. So in particular,
    if δ = 0,
  1031. what's W**?
    If δ = 0,
  1032. W** is equal to what?
  1033. Equal to 2 and that's what we
    had before.
  1034. In the one shot game,
    there it is up there,
  1035. where there was no possibility
    of continuing the relationship
  1036. tomorrow,
    I had to pay him a wage of 2,
  1037. or if you like,
    a wage premium of 100%.
  1038. If there's no probability--if
    there's no chance of continuing
  1039. this relationship,
    if δ = 0--we find again
  1040. that I'm paying 100% wage
  1041. Let's take the other extreme.
    If δ = 1,
  1042. so I just know this
    relationship's going to
  1043. continue--if δ
    = 1,
  1044. so there's no probability of
    the world ending or there being
  1045. a coup--then what's W**?
    It's equal to 1.
  1046. What's that?
    What's 1?
  1047. It's the going wage.
    So this is the going wage.
  1048. If I know for sure we're going
    to continue forever I can get
  1049. away with paying the guy the
    going wage, at least in the
  1050. limit.
    If we know we're not going to
  1051. continue then I have to play the
    one shot wage.
  1052. But let's look at a more
    interesting intermediate case.
  1053. Suppose δ = ½.
    There's just a 1/2
  1054. probability--that's pretty
    low--there's 1/2 probability
  1055. that your company,
    American Widgets,
  1056. is going to stay in Freedonia:
    with probability 1/2 it's going
  1057. to be done next period,
    with probability 1/2 it's going
  1058. to stay.
    What does that do to the wage?
  1059. What happens to the wage in
    this case in which there's a
  1060. probability of 1/2 of American
    Widgets staying in Freedonia?
  1061. It's a 1/2 between 2 and 1,
    which is therefore one and a
  1062. half½.
    Or another way of saying that
  1063. is, the wage premium is now only
  1064. What have we learned from this
  1065. Just an example of using
    repeated games.
  1066. Well the first thing we've
    learned is it's going to be
  1067. easy, once we get used to it,
    it's easy to use this
  1068. technology of comparing
    temptations to cheat,
  1069. with values of continuing in a
    cooperative relationship versus
  1070. the value of the punishment,
    which is in this case was just
  1071. firing the guy.
    But more specifically in this
  1072. example we've learned that even
    a relatively small probability
  1073. of this relationship
    continuing--so this is good news
  1074. for those of you who are seniors
    and are about to move to San
  1075. Francisco and your significant
    other is going to London--even a
  1076. small probability of the
    relationship continuing
  1077. drastically reduces the wage
  1078. The amount you have to "pay"
    your significant other not to
  1079. cheat on you as they go off to
    London or San Francisco is
  1080. drastically lower if there's
    some probability,
  1081. in this case just a ½,
    of continuing.
  1082. Before you leave,
    one more thought okay.
  1083. So how did this all work?
    Just to summarize,
  1084. to get good behavior in these
    continuing relationships there
  1085. has to be some reward tomorrow.
    That reward needs to be higher,
  1086. if the weight you put on
    tomorrow, if the probability of
  1087. continuing tomorrow,
    is lower.
  1088. The less likely tomorrow is to
    occur the bigger that reward has
  1089. to be tomorrow.
    We're going to have to charge
  1090. wage premia to employ people in
    Freedonia but those premiums
  1091. will come down once we realize
    that we're in established
  1092. relationships in Freedonia--once
    the American firms are
  1093. established and not fly by night
    operations in Freedonia.
  1094. Whether that's good news or bad
    news for Freedonia we'll leave
  1095. there.
    On Monday, totally new topic.