YT YouTube2Text

YouTube2Text → Transcripts → Math 486 SU HW2

Math 486 SU HW2 — Transcript

by Russ deForest · 1,911 words · 268 segments · language en · Watch on YouTube

Open in transcriber Summarize with AI Download SRT/VTT

Full transcript

0:00this video we'll take a look at the
0:03lesson two written homework we're
0:05starting with exercise
0:07one in 1 a we're saying that player 1's
0:11strategy Y is strictly dominated by
0:16player one's strategy Z so we just want
0:19to express the definition for this using
0:22payoff
0:25notation and here's the statement we're
0:28looking for we're using the payoff for
0:30player
0:31one and we're saying that uh the payoff
0:35when player one plays Z is strictly
0:37larger than the payoff to player one
0:39when they play y for any particular uh
0:43strategy that player two uses so this
0:46inequality has to hold for all of player
0:492's
0:53strategies Part B is similar we're still
0:56talking about player one but now we're
0:58saying that Y is weekly dominated by
1:08Z here there are two requirements first
1:13uh the payoff from playing y against S2
1:17is less than or equal to the payoff to
1:20player one of Z against S2 that has to
1:23hold for all of player 2's
1:27strategies and then we have this
1:29additional requirement there is at least
1:32one strategy S2 where playing Z gives a
1:36strictly higher payoff so we have a
1:38strict inequality for at least one
1:41choice of
1:43S2 and we have this inequality holding
1:47for all
1:53S2 in part C we're looking at the
1:56statement that for player One X X is a
2:00best response to a so here x is a
2:04strategy in player one strategy set a is
2:08some strategy in player 2's strategy set
2:11and X is the best response to
2:15a using our payoff notation we would say
2:19that uh Pi 1 of XA is greater than or
2:23equal to Pi 1 of S1 a for any S1 in
2:28player one's strateg
2:30set when player one plays X it gives
2:34them the maximum possible payoff when
2:37player two is playing strategy a there
2:41might be some other choice of S1 that
2:44gives an equal payoff but there's no
2:46other choice uh that will give a higher
2:49payoff for player one in part D well
2:52part D is quite similar but we're now
2:54looking at b as a best response to X for
2:58player 2
3:02so we've fixed player one's strategy as
3:05X when player two plays B that gives
3:10player two the maximum payoff against X
3:12so this inequality holds for all uh S2
3:17there's no way for player two to get
3:19higher payoff when we fix player one
3:22strategy at X and again this is not a
3:27strict inequality there could be another
3:29choice of 2 that gives the same payoff
3:31as B uh when player one is playing
3:37X finally we'll look at part e for
3:41player two B is not a best response to Z
3:46so we need to understand how to falsify
3:50this statement of uh something being a
3:53best response for player 2 so if B is
3:58not a best response to Z then that means
4:00that uh player two could find some other
4:03strategy S2 that gives a higher payoff
4:06than b when player one is playing
4:13Z so there exists some strategy S2 in
4:17player two's strategy set such that uh
4:20the payoff to player two when they play
4:23S2 is strictly greater than the payoff
4:27when they play B player one strategies
4:30fixed at Z let's take a look at exercise
4:35two here we're looking at a game called
4:37the hawk Dove game we're going to look
4:39at this several times throughout the
4:42semester the idea is that uh you've
4:44we've got some species that can show one
4:47of two behaviors Hawk which is an
4:49aggressive strategy uh to defend
4:52territory and Dove is a less aggressive
4:55strategy it's a bluffing by display uh
4:59but then they'll Retreat if the other
5:01player begins to fight and then we have
5:04these two parameters V and W V is the
5:07value of the territory that's being
5:09fought over W is the cost of getting
5:13injured if uh they choose to
5:16fight and here's the payoff Matrix we
5:20have a two-player game this game is
5:25symmetric that means it does not matter
5:27if you're looking at this game from
5:28player one's perspective or player 2's
5:32perspective but just let's just think
5:33about that for a moment if if uh both
5:36players play H they both get the same
5:39payoff vus W over
5:422 if both play D they also get the same
5:45payoff as one another V over two and
5:49finally if player one plays D and player
5:52two plays H player one would get nothing
5:55player two would get V but if we reverse
5:58this so that player one plays H while
6:00player two plays D then it's player one
6:02that will get V and player two that will
6:05get nothing so you'll see that if we
6:08switch the strategies
6:11along uh these two that we just switch
6:13the payoffs so uh it's symmetric it
6:17doesn't matter if you're player one or
6:19player two in this game the game uh
6:22works the same way from either player's
6:26perspective uh and if it's not clear uh
6:29from the fact that we have these two
6:31strategies that players are choosing
6:32their strategies simultaneously there's
6:35no turn in the game as there would be as
6:39there could be in an extensive form game
6:41where one player gets to see what the
6:43other player chooses before they choose
6:45now since the game is symmetric we don't
6:48need to um analyze the game from both
6:51players perspectives we can just choose
6:53one of the players to make an argument
6:56and then um by symmetry the same
6:59argument would apply to the other player
7:01now part a uh we're we're trying to use
7:04payoff notation to indicate the
7:08inequalities that we need if we were to
7:12show uh that D is strictly dominated by
7:16H so let's just use player one's
7:19perspective where we want to show that D
7:23is strictly dominated by H or rather we
7:25would like to know what the two
7:28inequalities
7:30uh should
7:33be we have a strict inequality that has
7:36to hold for each one of player 2's
7:40strategies so if we fix player 2's
7:42strategy at H then we're saying Pi 1 of
7:45HH is strictly greater than Pi 1 of DH
7:49and then we fix player 2's strategy as D
7:53and we want pi 1 of HD to be strictly
7:56greater than Pi 1 of d d so both of
8:00these are
8:00true if both are true then D is strictly
8:05dominated by H for player one we could
8:08use player two's payoffs instead again
8:11it wouldn't matter which one we choose
8:13because the game is
8:16symmetric now in Part B we just want to
8:20write these inequalities in terms of the
8:22actual payoffs from The
8:23Matrix and then we will solve for V
8:27because we're trying to find values of V
8:30that will make this true let's label
8:33these as one and two so let's call this
8:36one and this two we'll first look at
8:41one and just reading the payoffs from
8:44The Matrix we have vus W /
8:482 uh is greater than Zer we can see that
8:52this will hold whenever vus W is greater
8:56than zero or whenever V is greater than
9:04W now what about the second
9:07inquality for this inequality we have V
9:11or we want V to be greater than
9:14v/2 but V is a positive number so this
9:18is always true and since we need
9:22both both one and two to be true and the
9:26second inequality is always true and we
9:29can see that we need V to be strictly
9:32greater than W when that's true D is
9:36strictly dominated by
9:39H just as a quick review uh when we say
9:43one strategy is strictly dominated by
9:45another we have to have this strict
9:48inequality hold for all choices of
9:50player two's strategies we're not
9:53looking at strategies individually uh so
9:57it wouldn't make sense for example to
9:58say that when player two plays H then uh
10:03D is strictly dominated by H this these
10:06strict inequalities have to hold for
10:08every one of player 2's strategies all
10:12right so we showed that whenever V is
10:14greater than W then D is strictly
10:18dominated by
10:20H now in part C uh this was Part B that
10:25we just went over in part C we'd like to
10:27find values of V where where player one
10:31or player two uh does not have a
10:35dominated
10:37strategy let's just remind ourselves of
10:40the inequality that's always true this
10:43one here we just looked at this V is
10:45always greater than V over2 because V is
10:51positive so when player two plays d uh
10:54it's better for player one to play H or
10:56we just should say that H is the best
10:58response
10:59to D if H is also a best response to H
11:03then that would mean d is dominated it
11:06might just be weakly dominated uh but
11:09it's it's dominated we're trying to find
11:11values of V where there's no strategy
11:14that is
11:15dominated that means we'd like the
11:18second inequality the other inequality
11:20to go the other direction when player
11:22one plays H and player two is playing H
11:25we want that to be strictly less than Pi
11:271 of DH
11:34in other words we'd like V minus W over
11:372 now to be less than zero that means
11:41that we want
11:44V to be less than W of course uh V is
11:49still positive so V can now be any value
11:52any real number between zero and W in
11:55this case player one has no dominated
12:00strategy no strictly dominated strategy
12:02no weakly dominated strategy and neither
12:05does player two because of the game's
12:09symmetry all right now we're ready to
12:12move on to Part
12:15D and here we would like uh there to be
12:19a strategy that is weakly dominated but
12:22not strictly dominated so first we might
12:25just want to mention that if we do have
12:28a strategy that is strictly dominated it
12:31is also weakly dominated the reason is
12:34that uh if S1 satisfies the definition
12:39of being strictly dominated then it also
12:41will satisfy the definition of being
12:44weakly dominated but here we're asking
12:48for values of V where we have um D
12:53weakly dominated but not strictly
12:55dominated it has to be D that is
12:58dominated because of the arguments we've
13:00uh already shown in previous parts and
13:03again we can start with the strict
13:05inequality that is always true this one
13:12uh V greater than V /2 again V is
13:17positive so this is always
13:21true now D will be weakly dominated by H
13:25but not strictly dominated by H if the
13:28second condition condition we have is
13:30that uh player one gets the same payoff
13:33by playing either H or D when player two
13:36plays H and then we should be able to
13:39show that this occurs when vus w/2 is
13:44zero or when V equals W we've actually
13:50already considered every other
13:51possibility for v as long as we're
13:54considering positive values of V so this
13:57is the only only Choice that's left that
14:01wasn't considered as part of either B or
14:05C and you should check that you
14:08understand um this definition or what
14:11we're doing here in relation to the
14:14definition of strictly dominated and
14:17weakly dominated that we reviewed as
14:20part of um exercise one

About this transcript

This page contains the full transcript of Math 486 SU HW2 by Russ deForest, generated from the public captions YouTube serves with the video. The transcript has 1,911 words across 268 segments, with the original timestamps preserved so you can click any line to jump to that moment in the embedded player.

What you can do with it

Use the transcript to take notes, quote the speaker, build a study guide, generate a summary with ChatGPT or Claude via the YouTube Summary tool, or export it as a timed subtitle file with YouTube to SRT. You can also re-open it in the transcriber to translate the transcript into 100+ languages.

Free YouTube transcript tool

YouTube2Text is a free YouTube transcript generator — no signup, no daily limit. Paste any YouTube link and get the full transcript instantly, with timestamps, click-to-jump, translation to 100+ languages, AI prompts for ChatGPT, Claude, and Gemini, and exports to TXT, SRT, VTT, or Markdown.