## ← 19. Subgame perfect equilibrium: matchmaking and strategic investments

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

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