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Math 486 Lesson 3 Homework — Transcript

by Russ deForest · 3,595 words · 674 segments · language en · Watch on YouTube

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0:00this video we're going to look at
0:01written homework three we're starting
0:03with exercise one which is a three
0:06player game where each player has two
0:08strategies we're trying to use iterated
0:10elimination of strictly dominated
0:12strategies so the first thing to do is
0:14to try to find a strictly dominated
0:16strategy for for one of the three
0:18players
0:20let's start with player one
0:23if we look at what's happening here
0:25where player three plays a we can
0:27consider what happens when player 2
0:29plays a
0:30player 1 prefers a over b since 20 is
0:34greater than 10 and when player 2 plays
0:36b
0:37player 1 also prefers a
0:39over b
0:41and then we need to consider what
0:43happens when player 3 plays b instead of
0:45a
0:46when player 3 plays b and player 2 plays
0:49a
0:50player
0:511 prefers a over b
0:53but when
0:55player 2 plays b
0:57and player 3 is still playing b
0:59we see that 30 is greater than 25
1:02meaning that player 1
1:03gets a higher payoff when they played b
1:06if we write this out systematically we
1:08have four cases to check we looked at
1:11what happens when both player 2 and
1:13player 3 play a
1:15when one of them plays a and the other
1:17plays b that resulted in two different
1:19cases
1:20and then finally when they both play b
1:24and we saw that in three of those cases
1:27player one gets a higher payoff when
1:28they play a than when they play b
1:33but in one of the cases
1:35player one gets a higher payoff when
1:36they play b
1:38and when they play a this means that
1:40player one does not have a strictly
1:42dominated strategy
1:46well then we can move on to player two
1:50now for player two we want to fix the
1:52strategies of player 1 and player 3 and
1:54then imagine
1:56what happens when player 2 switches
1:57between a and b so we're comparing
2:00this 10 here with
2:02this 30. player 2 switches from a to b
2:06b is better but here uh 15 is greater
2:10than 10 meaning player 2
2:12would prefer a over b when
2:15player 1 plays b and player 3 plays a we
2:18don't need to go any further than this
2:20because we've shown that there's one
2:22case where player two prefers a and
2:24another case where player two prefers b
2:29and so we can conclude now that player
2:31two
2:32does not have a strictly dominated
2:34strategy without even looking at what's
2:35happening in this part of the game
2:38that brings us to player three so again
2:40we imagine fixing the strategy choices
2:42of player one
2:43and player two and then
2:46looking at what's happening with player
2:47three's payoffs
2:50when they play a versus when they play b
2:53so the first thing we could look at is
2:54this 15 here when player 1 and player 2
2:57are both playing a
2:58and we compare this with the payoff when
3:01player 3
3:02plays b player 1 and player 2 are both
3:05playing a
3:08you can see that player 3 prefers a
3:12and then we would compare this 10 with
3:14this 5 and note that again player 3
3:16prefers a we compare 20 with 15 and we
3:20compare 30 with 20. in each case we can
3:24see that player 3 is getting a higher
3:25payoff
3:27when they play a
3:28versus b and so b is strictly dominated
3:31by a
3:32for player 3.
3:35so the first thing you could do is just
3:36eliminate this matrix you need to state
3:39clearly that
3:42for player 3 b is strictly dominated by
3:45a we're not asking you to write out
3:47every inequality
3:49as we did for player one we just wanted
3:51to do that as an example of what we're
3:53showing
3:55we can see that we can eliminate b for
3:57player three
3:58once we do that our reduced game is now
4:01this
4:03matrix because
4:05we're assuming player 3 will always play
4:08a
4:09now we already saw
4:11that considering the payoffs just in
4:13this part of the game that player two
4:15sometimes prefers a and sometimes
4:17prefers b
4:22so in this reduced game we can still say
4:24that player two does not have a strictly
4:27dominated strategy
4:28but player 1 now does
4:3120 is larger than 10 5 is larger than 0.
4:34we can eliminate b
4:36for player 1.
4:41now finally we can look at player 2 and
4:43eliminate a
4:50the final result
4:53is is the strategy profile where player
4:551 plays a
4:57player 2 plays b
4:59and player 3 plays a
5:02so the dominant strategy equilibrium is
5:05a b a the payoffs are 5 30 20. but
5:10make sure that you distinguish between
5:12the strategy profile
5:23which is what's asked for this is aba
5:27and the payoffs which are 520 or 530 20
5:33exercise 2 provides some practice with
5:35iterated elimination of weakly dominated
5:38strategies
5:39here's the first game in exercise two
5:42and we can see that y is weakly
5:44dominated
5:46by x
5:47for player two we're comparing 25 with
5:5030
5:5120 with 20 and 40 with 40.
5:55when we compare player 2's payoffs of x
5:57versus y
5:59it's sometimes
6:01they're sometimes equal and
6:02there's at least one case where x gives
6:04a strictly higher payoff
6:06than playing y and this is the only
6:09weekly dominated strategy in the full
6:11game so you should actually be able to
6:13convince yourself that there are no
6:15other weekly dominated strategies before
6:18we eliminate strategy y for example if
6:22we were to compare x and z
6:25you can see that it's sometimes better
6:27to play z
6:29and there's a case here where it's
6:30better to play x for player two uh if we
6:34look at player one we could look at
6:36comparing b and c
6:40when player two plays x it's better for
6:41player one to play c
6:44but when player two plays z it's better
6:46for player one to play b
6:50there are other combinations to consider
6:52and again as practice you should go
6:54through and convince yourself that y is
6:57the only strategy here that's weakly
7:00dominated
7:01so we're going to eliminate y
7:04it is weakly dominated by x
7:07and once we do this we can see that b
7:10is weakly dominated by a in the reduced
7:14game
7:15b is actually strictly dominated by a
7:17but
7:18that means it's also weakly dominated
7:21we're comparing 10 and negative 20
7:2325 and 20.
7:25so b is dominated by a and we can
7:29eliminate b
7:33now we have a game where each player has
7:35two strategies we can see that for
7:38player two
7:41it's sometimes better to play z and it's
7:43sometimes better to play x and for
7:45player one
7:46it's sometimes better to play c and
7:49sometimes better to play a so there are
7:51no further reductions we can make we've
7:54arrived
7:55at the final reduced game and so there
7:58is not a we do not arrive at a dominant
8:00strategy equilibrium
8:03in the second game
8:05there are multiple ways to perform the
8:07iterated elimination
8:09both a and b are weakly dominated by c
8:13you can look at that by comparing player
8:151's payoffs
8:17but it's also true that both x and z are
8:20weakly dominated by y
8:22again z
8:24is strictly dominated by y but
8:27if z is strictly dominated it's also
8:29weakly dominated so one way we could
8:32proceed is to first eliminate a
8:35and b
8:36and then we end up with
8:38player one playing c
8:40and then player 2 prefers y they would
8:43eliminate
8:44x and z and we would end up with
8:47the equilibrium c y
8:50or we could first eliminate
8:52x and z
8:54for player 2
8:56and we see that
8:58in this reduced game
9:01c no longer weakly dominates a because
9:04player 1 gets the same
9:06payoff so only b would be eliminated
9:11b is dominated by
9:14c and a
9:16and so we we end up with two profiles a
9:19y and c y and we we can make no further
9:22reduction
9:23and there are other possibilities we
9:25could have first eliminated a and then
9:27eliminated either x or z and then go
9:29back to player one and then to player
9:31two
9:32so multiple outcomes are possible and
9:35the outcome that you arrive at depends
9:37on the order of elimination
9:40even though we've shown that multiple
9:41outcomes are possible
9:43we have
9:45introduced this new definition of
9:48rationality in normal form games and
9:50that is that
9:51a player will not play a dominated
9:54strategy so we could look again at the
9:56original game
9:58and we could say well c is a dominant
10:01strategy it's weakly dominant
10:04a and b are dominated by c
10:06and y is a dominant strategy for player
10:09two x and z are both weakly dominated by
10:13y
10:14and so we expect as the rational outcome
10:17c y
10:22each player
10:24will play their dominant strategy
10:27arriving at the outcome cy
10:29problem one is an example of what's
10:31called the tragedy of the commons
10:34we have an n player game
10:36and each player has two strategies r and
10:39i
10:40we have several parameters and we have
10:42the relationship that c over n
10:45is strictly less than p
10:47is strictly less than c
10:49n
10:51is the number of players in the game
10:53c is the cost to the community
10:57per cow above the
10:59carrying capacity of the pasture
11:01and p is the profit per cow
11:05this
11:06is a profit per individual the
11:08individual who owns the cow
11:10i suggested in part
11:12a that we can write
11:14the payoff function for a particular
11:17player i as
11:19s i
11:20m
11:22where m is the number of players other
11:25than player i who choose the
11:26irresponsible strategy i
11:30the first thing we want to do is write
11:32the payoff functions for player i
11:35in terms of m
11:36so one
11:38choice that player i has is to choose
11:40the responsible strategy r
11:44when there are m others who have chosen
11:46i
11:48so player i has k cows
11:50each
11:52cow will give them a profit of p
11:54so they get the benefit k times p
11:58and they'll also get a cost of c over n
12:01for each
12:02cow above
12:04the limit and since there are m
12:06players
12:07who chose
12:09i the the cost is m times c over n
12:15on the other hand player i may choose
12:17the irresponsible strategy
12:21they have an additional cow so their
12:23benefit will now be k plus one
12:26times p but the total number of
12:29players who have chosen i also goes up
12:31by one
12:33so we have m plus one
12:35times c over n is the cost
12:38now in part
12:39b we're asked to prove that r is
12:43strictly dominated
12:45by i so we would like to show
12:47for every choice of m
12:50we have
12:51pi i
12:52of r m is strictly less
12:55than pi i
12:56of i m
12:58and we've written out each of these
13:01payoff functions in terms of m so we can
13:04just write this
13:06we want to show that
13:09kp minus
13:11m times c over n is strictly less
13:14than k plus 1 times p
13:17minus m plus 1 times c over n
13:21and if we just multiply these out
13:26on the right hand side
13:30we have k p plus p
13:33minus m times c over n minus
13:37c over n
13:39and then we just need to show
13:42that p
13:44minus c over n is greater than zero but
13:48the condition that we assumed is that p
13:51is
13:52larger than c over n
13:54so this is just part of the assumption
13:56of the problem
13:58p minus c over n is greater than zero
14:01and
14:02so
14:03r
14:04is strictly dominated by i
14:08for every
14:09choice of m
14:12player one can increase or player i
14:14rather can increase their payoff
14:17by playing the irresponsible strategy
14:20over the responsible strategy
14:23now in part c
14:25we want to see what happens when every
14:28player chooses strategy i and compare
14:31that to what happens when every player
14:33chooses strategy r
14:36so let's again write out the payoff
14:38functions
14:39we said that
14:43this is k p
14:45minus m times c over n
14:48and the payoff to player i
14:50for playing the irresponsible strategy
14:53is k plus 1
14:55times p minus
14:57m plus 1
14:59times c over n
15:01if everyone chooses r then m is 0. so
15:05we're looking at pi i of r
15:080 and this is
15:10k kp
15:12and if everyone chooses i then this m is
15:16n minus one
15:18m again is the number of other players
15:20choosing i so this is pi i of i
15:24n minus 1 we get
15:27k plus 1 p
15:29minus
15:31n
15:32times c
15:33over n
15:35or
15:36k p
15:38plus p minus c
15:41and we have assumed that c is
15:44larger than p
15:46so this is less
15:48than k p
15:51so just
15:53as we saw in the prisoners dilemma
15:57players can get a higher payoff if they
15:59all choose
16:01the dominated strategy r
16:05i is the dominant strategy but when
16:07every player plays the dominant strategy
16:09they end up with a payoff
16:11that is strictly less than what they
16:14would end up with if they all cooperated
16:17on the
16:19dominated strategy r
16:22and you should think of this as a
16:24generalized prisoner's dilemma an end
16:27person
16:28prisoners dilemma game
16:30problem two is a version of the location
16:33game we have two players
16:35each representing a bank that wants to
16:37locate
16:38at a particular block
16:40and customers will just go to the
16:43nearest bank
16:44and we're assuming that for a block
16:46that's equidistant
16:48between two banks they'll just split the
16:51number of customers
16:52now in the first part of this problem i
16:54asked you to show that for bank a
16:57choosing two is not
16:59dominated by choosing three
17:02in other words we want to show
17:04that there's some x that that
17:07bank b could choose such that it's it's
17:10better for
17:11player a to choose 2
17:14instead of 3.
17:15and we can
17:16use x equals one so we can show
17:21that
17:25if bank b chooses block one then it
17:27would be better for bank a to choose two
17:29instead of three and thinking about this
17:32carefully
17:33uh gives us an approach to the entire
17:36problem
17:38so notice that
17:39what we're really saying here is that
17:42if there's some space between
17:45the two banks
17:47then it would be better for one of the
17:49one of the banks could improve by just
17:51moving closer to the other bank
17:56so whenever there's space between a bank
17:58could improve their own payoff if they
18:00can move
18:01closer to where the other bank is
18:03currently located
18:05and let's just think about how this
18:07works
18:08when
18:08we're in this situation
18:11bank a is picking up all the customers
18:13from block 3 all the way to block 99
18:19bank b gets the customers from block 1
18:22and then they split the customers from
18:24block 2.
18:25might be helpful to make a sketch
18:29let's assume that
18:31bank one is located or bank a rather is
18:34at block three
18:36and bank b
18:38we'll use a different color for bank b
18:40they're located
18:42in block one
18:51bank a gets all the customers at blocks
18:54three through ninety 99 and they get
18:56half the customers at block 2
18:59and
19:00bank b they're just getting all the
19:02customers at one and half
19:04from two
19:06but if
19:09bank one
19:12were at two instead
19:15they would pick up all the customers
19:17at block two and all the customers in
19:20blocks three through ninety nine so it's
19:22better for them to be there it's better
19:24to move nearer to bank b
19:27and you can write these payoffs
19:29explicitly but this is the idea
19:32that we want to capture
19:34if you can if there's space between you
19:36and the other bank then move closer to
19:39increase your payoff so with that idea
19:41let's look at part b of problem two
19:44we want to argue that
19:47one
19:48is strictly dominated by two
19:51and we could show this for either bank a
19:53or bank b we do not need to make the
19:55argument twice
19:57but note that we need to show
19:59that
20:00it's better for a bank to choose two
20:03rather than one for any choice
20:06of the other bank
20:08let's show this for bank a
20:10and we'll assume that bank b chooses 1
20:15bank b chooses 2 or bank b chooses
20:19anything larger than two
20:21let's consider the first case sb
20:24equals one so we have the second bank
20:27in block one and bank a could either be
20:31in one
20:32or in
20:34two
20:37if bank a is at location one
20:41then
20:42all blocks are equidistant from the
20:44banks and each bank would get half of
20:47the total customers half
20:50of of 9 900
20:52but if
20:54bank a moves to block 2 they get all of
20:57the customers from 2 through 99 leaving
21:01bank b with just the customers at block
21:031.
21:05so we should be able to see clearly that
21:06we have pi a
21:08of 1 1 is strictly less
21:11than pi a of
21:132 1.
21:17this is 49.50 half of all customers
21:21this is 9 800. all the customers in the
21:2498 blocks from 2 to 99. what happens if
21:28bank b
21:30is in block 2.
21:32we're assuming that bank
21:35b is in the second block and bank a is
21:38comparing block 1 with block 2.
21:41if they're in block 1 then they'll only
21:43get the cost customers at block one so
21:45their payoff would be a hundred
21:47if they move to block two they'll split
21:50all of the customers
21:52so we're showing now that
21:55pi a
21:56of
21:571 2 is strictly less
22:00than pi a of 2 2.
22:06here we're splitting the customers
22:08getting 49.50
22:11but here
22:12the payoff is only 100.
22:15and for all the other cases we can
22:17consider the idea that we already
22:19considered in part a which is to say
22:21that if there's some space to move
22:23closer
22:24to
22:25the other bank
22:27then you should just move closer
22:30so when s b is greater than or equal to
22:33three
22:34we can say that
22:37s a equals two
22:39is
22:41nearer to bank b
22:44s b
22:46then
22:47s a prime equals 1. and by by moving
22:51nearer they'll be picking up some
22:53additional customers and getting a
22:55higher payoff
22:59and that argument covers
23:04all other cases
23:14and this shows that one is strictly
23:17dominated by two
23:19so what we can imagine is that
23:22player
23:23one or player a rather will eliminate
23:26one as a location
23:29now this is not eliminating the
23:30customers at that block it's simply
23:33saying that that
23:34uh bank a will not locate at block one
23:39similarly uh bank b makes the same
23:41analysis and bank b will also not choose
23:45the first block
23:46but everything we argued there
23:49applies to this end as well
23:5398
23:55is a better choice than 99 for the same
23:59reason
24:00and we we really should not write out
24:02the full argument again we can simply
24:04note that
24:06the reasoning we applied
24:08will also cause both banks to eliminate
24:1299 as a dominated strategy
24:16then we end up with the strategy space 2
24:19through 98
24:22and once we
24:23do that we we recognize or we should
24:26recognize that we're really in the same
24:27situation we started with it's just that
24:29now instead of block 1 and block 99
24:32we're applying the same reasoning to
24:34block 2 and block 98
24:38and so then we can eliminate 2
24:40and 98
24:42then we can keep going with this
24:44procedure
24:45until we've arrived
24:48at the game
24:51where there's only three choices left
24:5449 50
24:56and 51 and
24:58really the same reasoning will will
25:00still apply but i'm asking you to
25:02explicitly show that when these are the
25:05choices 50 dominates both 49 and 51 but
25:09keep in mind that all the other blocks
25:11are still there
25:13the customers are still there that's
25:15important
25:16but
25:18we've eliminated those blocks as choices
25:21for the location
25:24or each each bank
25:26has eliminated those choices so 1
25:29through 48
25:30those blocks still exist
25:3252 through 99 still exist the customers
25:36there are part of the payoff
25:38for each bank when we're considering
25:40these three strategies
25:42and you should be able to use that to
25:44show that 49 and 51 are both strictly
25:48dominated by 50.
25:50and then we eliminate 49 and 51 leaving
25:53us
25:54leaving each bank with 50 as the only
25:57choice
25:58left
25:59meaning that 50 50
26:01is the dominant strategy equilibrium for
26:04this game
26:05and then in the final
26:08part of this problem i ask you to
26:10revisit the original game where all
26:13locations are available and i want you
26:16to
26:18recognize that 50 is not a dominant
26:21strategy in the original game in other
26:24words so this is part f that we're
26:26discussing
26:29you could choose either bank but it
26:31makes sense just to look at bank a i
26:33suppose
26:34show that
26:36for bank a the payoff when they choose
26:41some x is greater than or equal to the
26:44payoff when they choose 50 for some
26:46particular choice
26:48of bank b in other words there's a
26:50location that bank b could choose where
26:52it's better
26:53for bank a to choose something other
26:55than than 50 or it doesn't even have to
26:57be better it can be greater than or
26:59equal to but you should be able to show
27:01that it's strictly greater
27:03and you can again use this idea that
27:05that if there's some space between the
27:07banks then bank a could improve by
27:09moving closer to bank b

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