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Math 486 HW5 — Transcript

by Russ deForest · 2,218 words · 415 segments · language en · Watch on YouTube

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0:00in this video we will review the lesson
0:02five homework and we're starting with
0:04exercise one
0:06and here's a copy of the extensive form
0:08game from exercise one
0:10in part a we need to describe the
0:13meaning of the strategy du for player
0:15one
0:16the important idea is that a strategy
0:19for player one needs to indicate an
0:21action for each of player one's
0:24information sets here each information
0:26set contains one node
0:28and by convention we'll read this
0:32from left to right whenever possible so
0:34this d
0:36indicates that player 1 will play d if
0:39the interaction is type a
0:41and the u indicates that player one will
0:44play
0:45u
0:45if the interaction type is type b
0:53now we want to look at the strategy lr
0:55for player two player two is not
0:57responding to the game type or the
1:00interaction type they're responding to a
1:03signal from player one
1:05when player one plays u
1:08these two nodes for player two are in
1:10the same set and when player one plays d
1:15these two nodes here are in the same set
1:17this just means that all the information
1:19that player two has is
1:21the choice that player one has made
1:24so we will read this l as the choice
1:28that player two makes if player one
1:31chooses u
1:32and the second part of this strategy
1:34here is the choice that player two makes
1:37when player one chooses d
1:40so this strategy is l if player 1 plays
1:43u r if player 1 plays d
1:46and finally in part c we're looking at
1:48how these
1:49strategies are paired for each game type
1:52to determine the payoffs
1:55let's first assume the interaction is
1:58type a so when we have an interaction of
2:00type a player 1 is playing d
2:04and when player 1 plays d player one
2:07plays r so these payoffs are one one
2:14when the interaction is type b then
2:17player one is playing u
2:20and when player one chooses you player
2:23two is choosing l so here the payoffs
2:26would be six four
2:31let's skip exercise two we'll come back
2:33to exercise two i want to continue with
2:35problem 1 which asks us to create a
2:38normal form game for this
2:41extensive form game
2:43we will create two normal form games and
2:45then we'll combine them using the
2:47probabilities of each interaction type
2:50let's assume that we're looking at what
2:53happens when the game is type a
2:55player one has four strategies each
2:58strategy for player one indicates what
3:00to do when the interaction is type a and
3:03when it's type b we'll just be paying
3:05attention to the first part of the
3:07strategy because we're concentrating on
3:10type a
3:12and we should remind ourselves this is
3:13player one
3:15this is player two player two strategies
3:17again those are responses to
3:20what player one does so the first part
3:22of this strategy
3:23is what to do when player one plays you
3:27the second part is what to do when
3:29player one plays d
3:31and since we're in type a we'll just be
3:33looking at the payoffs
3:35along this branch
3:39and here player 1 plays u
3:42and
3:43player
3:442 is playing left or l whenever player
3:47one plays u
3:49so these payoffs will be four four
3:55and the same thing happens here
3:59in these cases player two is playing
4:01right when
4:02player one plays u
4:06those will give us the payoffs zero five
4:16down here player two is playing d
4:21and now we want to pay attention to the
4:23second part of player two strategies
4:25what to do when player one plays d
4:28player two is playing left in response
4:30to d here and here
4:33that results in a payoff
4:36of five zero
4:46and then here
4:48and here player two is playing right in
4:50response to d that brings us to the
4:52payoffs one one
4:59this is just the case for interaction
5:01type a we need to do the same thing for
5:03type b and then to get the normal form
5:05game for this extensive form game we
5:08need to combine those
5:09focusing on
5:11type b
5:13we're looking at the second part of
5:14player 1 strategies
5:17let's look at this case here where
5:19player 1 is playing u
5:21player 1 plays u we look at the first
5:23part of player two strategies
5:27when player one plays u and player two
5:29plays l we get the payoffs six four
5:33and when player two plays r in response
5:35to u we get the payoffs zero zero
5:44and you should continue in this way
5:46thinking about
5:48player one's choice of d in resp and
5:51player two's response to d
5:53when the interaction is type b to fill
5:55out the rest of this
5:56chart we need to take these two and use
5:59the probabilities so the interaction is
6:01type a with a probability of
6:03three-fourths
6:04it's type b with a probability of
6:06one-fourth
6:08so our goal now is to
6:11combine these
6:13we'll abuse notation a bit here but we
6:15want three-fourths times
6:17the type a game plus one fourth times
6:21the type of b game so we're going to
6:22take three fourths times
6:25these payoffs and add 1 4
6:28times these payoffs you should fill in
6:30the rest of these of course
6:32combining the games in this way we get
6:34this
6:35bayesian normal form
6:38and the next part of problem one is to
6:40find any nash equilibria here we can go
6:43ahead and indicate best responses as
6:45usual let's first look at player one
6:52and then looking at player two's payoffs
7:04we can see that there are
7:07two
7:08bayesian nash equilibria
7:12these are uulr
7:14and ddrr
7:17and just as a reminder the nash
7:18equilibria are these strategy profiles
7:23not the payoffs
7:26in exercise 2 we're looking at mixed
7:29strategies and we're going to compute
7:30expected payoffs
7:33player one is playing the mixed strategy
7:35sigma one
7:37where they play x with probability
7:39two-thirds
7:41and they play y with probability one
7:43third
7:45as always the row player is player one
7:48column player is player two player two
7:50is mixed strategy
7:52sigma two
7:53means play a with probability one
7:56seventh
7:57play b with probability two sevenths and
8:00play c with probability four sevenths
8:06and we want to compute
8:08player ones payoff
8:09when the players play
8:11these strategies and also player twos
8:13payoff
8:21now these are expected payoffs
8:24and we compute an expected value by
8:26taking each outcome
8:28multiplying
8:30by the probability that it will occur
8:32and then adding them up
8:34so
8:35one way to approach this is to think of
8:37each outcome individually
8:39for example what is the probability
8:43that
8:44player 1 plays x
8:46and player 2 plays a in other words what
8:48is the probability that the profile is
8:51xa with the outcomes or payoffs to four
8:55well
8:56player one is playing x with probability
8:58two thirds player two is playing a with
9:01probability one seventh
9:03so the probability that the payoffs are
9:05here
9:07is two-thirds times one-seventh or two
9:09over twenty-one
9:11so we could write
9:13this now using this idea we could
9:17write player one's expected payoff as
9:20two times the probability that x a
9:23occurs that's two-thirds
9:26times one-seventh
9:28plus two
9:29times the probability that x b is the
9:32profile
9:33that is
9:38two-thirds
9:39times two-sevenths
9:41plus six
9:43times two-thirds times
9:46four-sevenths
9:49again this is the probability that a x
9:51is played times the probability that c
9:53is played
9:56and then we can continue
10:03plus one
10:04times
10:05one-third
10:07times one-seventh
10:10plus three times
10:12one-third times two-sevenths
10:15plus five times one-third
10:19times four-sevenths
10:24this gives us 29 sevenths
10:27and then you can do the same thing
10:29for player
10:30two's payoffs
10:33player two's expected payoff is
10:3656 over 21. now that's one approach but
10:40it's helpful to see that you can write
10:42player ones payoff expected payoff
10:44rather
10:45in a different way
10:47sigma one is two-thirds one-third we
10:49could write this as two-thirds
10:52times pi one of x
10:55sigma two plus
10:57one-third
10:58of pi one of y
11:01sigma two
11:02and then
11:03calculate each of these
11:05multiply this one by two thirds this one
11:08by one third and add them together this
11:09is the same
11:11uh same idea as
11:14finding each payoff and multiplying by
11:16its probability same result
11:19and in lesson six you'll see that we
11:21want to be able to
11:23look at this in in both ways finally in
11:25problem two we want to take a look at
11:28this poker game and the first thing we
11:30need to do is draw an extensive form
11:32game we have a chance node that
11:34determines whether player 1 gets the
11:36high card or the low card
11:39and each of those occurs with the
11:41probability
11:42of one half
11:50and then player one can either bid or
11:52fold
12:01if player one folds right away the
12:03payoffs are negative two
12:05two
12:08player two wins the pot and each player
12:10had to put in two at the beginning of
12:11the game if player one chooses to bid
12:14then it's player two's
12:16turn to decide whether to bid or fold
12:29if player two folds then the payoffs
12:31will be two negative two
12:35and then if player two decides to bid
12:37then the outcome is determined by
12:39whether or not player one has the high
12:41card or the low card if the high card
12:44the payoffs will be
12:45four negative four and if player one has
12:48the low card
12:50negative four
12:51four now we have most of the game
12:53structure but we need to include the
12:55fact that player two doesn't know what
12:57card player one has when they bid so we
13:00need to connect
13:02player two's
13:03nodes and show that they belong to a
13:05single
13:06information set
13:09so this is the
13:11complete game
13:12now we want to form the daisy in normal
13:15form
13:17let's move this in the normal form game
13:21player one will have four strategies and
13:24player two will have two
13:27player one strategies are b b
13:29b f
13:30f b
13:32f
13:32f the first part indicates what to do if
13:35player one has the high card the second
13:37part of each strategy is what to do if
13:40player one has the low card
13:42player two has
13:43two nodes
13:45but they're in a single information set
13:46so player two strategies are just bid
13:49and fold
13:52and we'll first make a game
13:54for when player one gets the high card
13:56so we're looking at what's happening
13:59over
14:00here if and and we're looking at this
14:04part of
14:06player one strategy if player one folds
14:08the payoffs will be negative two
14:11two so that
14:14takes care of these four
14:16payoffs
14:19and then when player one bids player two
14:22can respond if they both bid the payoffs
14:25would be four
14:28negative four
14:29and if player two folds the payoffs will
14:31be two negative two
14:35then we want to look at what happens
14:37when player two has the low card
14:40so now we're looking at the second part
14:43of player one's
14:45strategy if player one folds again the
14:48payoffs
14:49are negative two
14:52two
14:56and if player one
14:59bids
15:00and player two bids player one will
15:02lose player two will win
15:04so we'll have negative four
15:06four in those cases
15:08and if player two folds again player
15:11that the payoffs would be 2 negative 2.
15:15now we want to combine these into a
15:17single
15:18normal form game
15:21player 1 has a 50 chance of having
15:23either the high or low card so we're
15:25going to take each of these payoffs
15:27multiply by half
15:29and add one half times these probably
15:31the corresponding probabilities down
15:33here
15:34so we'll try to make the normal form
15:36game here we have bb
15:38bf
15:40f b
15:41f f
15:43and bid fold for player two
15:50and you can see uh four negative four
15:52plus negative four four that will be
15:54zero divided by
15:56two is still zero
16:00and
16:01two negative two getting the same payoff
16:03in each case the average will also be
16:06two negative two and you want to
16:08continue in this way
16:10four negative four and negative
16:13two two gives us two negative two but
16:16we're dividing by two so we get one
16:19negative one
16:31once we have the normal form game we
16:34want to indicate best responses for each
16:37player
16:43and doing so we see that there are no
16:46outcomes where each player is playing a
16:48best response to the other now we've
16:51just introduced mixed strategies in this
16:55lesson but ignoring mixed strategies we
16:57see that no player is playing a best
16:59response to another in any of these
17:01outcomes so none of these represent
17:04nash equilibria
17:06in the upcoming lesson we define a mixed
17:08strategy nash equilibrium and we'll see
17:10how that works but we're not applying
17:12that here yet
17:14now we might add this is not a part of
17:16the question but we might add that it
17:18really makes sense for
17:20contests between two players to not have
17:24the uh what we will now call pure
17:25strategy
17:27nash equilibria
17:28and by contest i mean there's one player
17:31is winning another player is losing in
17:33these kinds of situations
17:36you usually want to be able to play in a
17:38way that will
17:39be unpredictable to the other player
17:43and so it's natural that pure strategy
17:45nash equilibria do not arise in these
17:47kinds of contests

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