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

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

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