1. Introduction to Bayesian Games (Game Theory Playlist 9)

00:52:56
https://www.youtube.com/watch?v=6-lciROBj5c

摘要

TLDRThis episode introduces Bayesian games, focusing on auctions with incomplete information. The host contrasts the English auction with the simpler second-price auction, illustrating how players lack complete knowledge about each other's valuations and strategies. Using a straightforward bidding example, he shows how each bidder’s beliefs about their opponent’s willingness to pay complicate their decision-making process. The episode culminates in discussing the concept of Bayesian Nash Equilibrium, highlighting how it adapts traditional Nash analysis to account for incomplete information regarding other players' types, thereby paving the way for future in-depth formalization in subsequent episodes.

心得

  • 🎨 English auctions involve open competition for bids.
  • 📜 Second-price auctions see bidders submit sealed bids.
  • 🔍 Incomplete information complicates bidding strategies.
  • 💰 Bids reflect players' valuations of the item.
  • 📊 Nash Equilibrium applies to auction strategies.
  • ✍️ Private information influences bidding behavior.
  • 🤔 Dominant strategies simplify decision-making.
  • 📈 Bayesian Nash Equilibrium incorporates beliefs about opponent valuations.
  • 📝 Understanding auctions is key to game theory.
  • ⚖️ Strategic equivalence exists between auction types under assumptions.

时间轴

  • 00:00:00 - 00:05:00

    In this episode, the concept of Bayesian games, or games with incomplete information, is introduced. The presenter aims to provide intuition on how to analyze these games, starting with an example of an auction.

  • 00:05:00 - 00:10:00

    The English auction is explained as a widely recognized auction format where the auctioneer accepts increasingly higher bids until no one bids higher than the standing bid. The highest bidder wins the item at their bid price.

  • 00:10:00 - 00:15:00

    The speaker shifts focus to the second-price or Vickrey auction, which is simpler and strategically equivalent to the English auction under certain conditions. In this format, bidders submit sealed bids and the highest bidder wins but pays the second-highest bid.

  • 00:15:00 - 00:20:00

    The nature of auctions is explained as a strategic environment where players (bidders) act simultaneously and independently, leading to discussions about their strategies and potential payoffs based on their bids and the valuations of the items.

  • 00:20:00 - 00:25:00

    The concept of Nash equilibrium in a second-price auction is demonstrated through a scenario with two bidders with known valuations. It is concluded that bidding one's true valuation is a Nash equilibrium strategy.

  • 00:25:00 - 00:30:00

    The episode discusses the implications of bidding higher than one's valuation, emphasizing that it could lead to negative payoffs and thus is not a rational strategy.

  • 00:30:00 - 00:35:00

    Next, the speaker introduces incomplete information in auctions, pointing out that although each player knows their own willingness to pay, they are uncertain about their opponents' valuations, introducing the concept of asymmetric information.

  • 00:35:00 - 00:40:00

    The modeling of bid strategies is then discussed, highlighting that players must form beliefs about their opponents' valuations and that these beliefs are treated as common knowledge within the game.

  • 00:40:00 - 00:45:00

    The discussion includes how to analyze Nash equilibrium in environments with incomplete information by creating strategies for different types of players based on their valuations, leading to the concept of multiple personality types in players.

  • 00:45:00 - 00:52:56

    Finally, the episode concludes with the introduction of Bayesian Nash equilibrium, which extends the Nash equilibrium concept to cases with incomplete information, setting the stage for the next episode to formalize these concepts.

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思维导图

视频问答

  • What is a Bayesian game?

    A Bayesian game is a game with incomplete information where players have beliefs about the types of other players, often modeled probabilistically.

  • What is a second-price auction?

    In a second-price auction, participants submit bids privately, and the highest bidder wins but pays the second-highest bid.

  • How is a Nash Equilibrium defined in the context of auctions?

    In auctions, a Nash Equilibrium occurs when bidders choose their bids such that no player can benefit from unilaterally changing their strategy.

  • What is the difference between complete and incomplete information?

    In complete information, players know all payoff-relevant details about one another, while in incomplete information, they lack knowledge about other players' types or valuations.

  • What is a dominant strategy?

    A dominant strategy is one that provides a higher payoff for a player, regardless of what other players choose to do.

  • What is an English auction?

    An English auction is a type of auction where bidders openly compete for an item, increasing their bids progressively until no higher bid is placed.

  • What is the strategic equivalence between English and Vickrey auctions?

    Under certain assumptions, the Nash Equilibrium outcomes of English auctions can be strategically equivalent to those of Vickrey auctions.

  • What is meant by 'private information' in the context of auctions?

    Private information refers to the knowledge that individual players have about their own valuations and strategies, but not about other players' valuations.

  • How do you determine the expected utility of a strategy in a Bayesian game?

    The expected utility of a strategy is determined by considering the different possible types of opponents and their corresponding strategies and utilities.

  • What characterizes a Bayesian Nash Equilibrium?

    A Bayesian Nash Equilibrium is where each player's strategy is a best response to the beliefs about other players' types and strategies.

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  • 00:00:00
    hello everyone with this episode i am
  • 00:00:03
    starting a a new chapter where we
  • 00:00:07
    analyze uh what's called bayesian games
  • 00:00:10
    or sometimes they're called uh games
  • 00:00:13
    with
  • 00:00:13
    incomplete information so in this
  • 00:00:16
    episode i'm going to
  • 00:00:17
    give you the intuition behind the
  • 00:00:21
    uh analysis of these games
  • 00:00:24
    and sort of how we approach to those
  • 00:00:26
    games
  • 00:00:27
    and then the next episodes i am going to
  • 00:00:30
    describe
  • 00:00:31
    formally what i am uh sort of
  • 00:00:33
    intuitively mentioning in this episode
  • 00:00:36
    well to give the intuition uh i'll i'll
  • 00:00:39
    i'll start with an example and the
  • 00:00:42
    example i'm going to consider
  • 00:00:44
    is what's called auction so
  • 00:00:47
    uh the first auction i'd like to talk
  • 00:00:50
    about is what's called
  • 00:00:51
    english auction you probably have heard
  • 00:00:53
    of it maybe not the name
  • 00:00:55
    the the english auction but it's
  • 00:00:58
    actually one of the most famous
  • 00:01:00
    ways of of of selling or buying
  • 00:01:03
    uh items so for example you would like
  • 00:01:06
    to buy a painting
  • 00:01:08
    and and the paintings and let's suppose
  • 00:01:11
    by the way this is a
  • 00:01:12
    rare painting auctioned in sotheby's
  • 00:01:16
    or some other auctioning house well
  • 00:01:20
    what is the english auction well the
  • 00:01:22
    english auction is simple
  • 00:01:24
    the auctioneer uh opens the auction by
  • 00:01:28
    announcing uh some suggested opening bid
  • 00:01:32
    we sometimes call it starting price
  • 00:01:34
    sometimes call it reserve
  • 00:01:35
    price well doesn't have to exist but
  • 00:01:39
    usually
  • 00:01:40
    it is some positive number say ten
  • 00:01:43
    thousand dollar
  • 00:01:44
    depending on the item auctions well then
  • 00:01:46
    the auctioneer accepts
  • 00:01:48
    increasingly higher bids from the floor
  • 00:01:51
    from the potential buyers who are
  • 00:01:53
    competing with each other
  • 00:01:56
    well the auctioneer usually determines
  • 00:02:00
    the minimum increment of bids
  • 00:02:03
    often raising it when bidding goes high
  • 00:02:05
    well
  • 00:02:06
    i mean for example if it is an an and
  • 00:02:09
    and and a painting from
  • 00:02:11
    uh da vinci uh well then probably
  • 00:02:15
    uh the increments is not going to be
  • 00:02:17
    like you know hundred dollars
  • 00:02:19
    all right so uh usually there are
  • 00:02:21
    increments
  • 00:02:23
    but again for simplicity you can ignore
  • 00:02:25
    the in increments
  • 00:02:27
    which we will well who who wins and how
  • 00:02:30
    do you win the
  • 00:02:32
    auction well the highest bidder at any
  • 00:02:35
    given time
  • 00:02:36
    is considered to have the standing beat
  • 00:02:39
    all right so understanding it is
  • 00:02:41
    basically
  • 00:02:42
    uh uh the the the the the final price
  • 00:02:46
    all right well if if
  • 00:02:50
    nobody sort of increases this standing
  • 00:02:53
    bit
  • 00:02:54
    either you can displace this standing
  • 00:02:56
    bit by
  • 00:02:57
    sort of announcing a higher bid
  • 00:03:02
    but if nobody increases the standing bit
  • 00:03:05
    well then the the auction is going to
  • 00:03:08
    finish
  • 00:03:09
    so if no other competing bidder
  • 00:03:11
    challenges
  • 00:03:12
    this standing bit within some given time
  • 00:03:14
    period
  • 00:03:15
    uh usually it's you know uh you know a
  • 00:03:18
    few
  • 00:03:19
    moments or a few minutes it's not like
  • 00:03:22
    hours or days
  • 00:03:23
    but obviously for some other auction
  • 00:03:25
    environments it can be hours or days
  • 00:03:27
    maybe
  • 00:03:28
    well the standing bid becomes the winner
  • 00:03:31
    again
  • 00:03:31
    if nobody challenges the standing bid
  • 00:03:34
    and
  • 00:03:34
    so if nobody increases it while the
  • 00:03:37
    standing bid becomes the winner and the
  • 00:03:39
    item
  • 00:03:39
    is sold to the highest bidder at a price
  • 00:03:42
    equal to
  • 00:03:43
    uh equal to that that bid
  • 00:03:47
    all right well so this is the english
  • 00:03:50
    auction
  • 00:03:51
    well i am not however going to analyze
  • 00:03:54
    english auction why is that well because
  • 00:03:56
    it's an
  • 00:03:57
    an extensive form game right the the
  • 00:04:00
    potential player potential i mean
  • 00:04:02
    potential buyers are actually the
  • 00:04:04
    players in this game
  • 00:04:05
    and they observe each other's sort of
  • 00:04:08
    strategies one guy
  • 00:04:10
    for example kohl's price say
  • 00:04:13
    one million dollar and then another
  • 00:04:15
    calls it two million dollar and so
  • 00:04:17
    everybody observes his or her actions
  • 00:04:19
    right this is a perfect information game
  • 00:04:21
    in a sense
  • 00:04:22
    uh however there is the sequential move
  • 00:04:26
    uh going on it's like so we we really
  • 00:04:29
    have to analyze the sub games etc etc so
  • 00:04:32
    instead of looking at a complicated
  • 00:04:34
    relatively more complicated game
  • 00:04:36
    let's look at a simpler game and in fact
  • 00:04:39
    it is a simpler version of the english
  • 00:04:42
    auction well
  • 00:04:44
    i i say this version a bit vaguely
  • 00:04:47
    because
  • 00:04:48
    uh they're not really the same game
  • 00:04:50
    obviously as
  • 00:04:51
    i will describe in a moment but the
  • 00:04:53
    thing is they are
  • 00:04:55
    under some some some certain assumptions
  • 00:04:57
    they are strategically equivalent the
  • 00:04:59
    nash equilibrium are the same
  • 00:05:01
    uh the spme of this game is subgame
  • 00:05:04
    perfect nash equilibrium of the english
  • 00:05:06
    auction
  • 00:05:07
    outcome equivalent to the nash
  • 00:05:08
    equilibrium of the victory auction
  • 00:05:10
    i mean there is definitely a strategic
  • 00:05:13
    relation strong relationship between
  • 00:05:15
    these two auctions
  • 00:05:17
    under some assumptions um we are not
  • 00:05:19
    going to prove this in this course it's
  • 00:05:21
    a
  • 00:05:22
    subject for more advanced courses but
  • 00:05:24
    nevertheless
  • 00:05:25
    um i am not giving you the victory
  • 00:05:28
    auction
  • 00:05:29
    uh uh simply because it's easier or
  • 00:05:32
    simpler to work with
  • 00:05:34
    but they are as i said as strategically
  • 00:05:36
    equivalent
  • 00:05:38
    well most of the times i am not going to
  • 00:05:41
    call it victory auction
  • 00:05:43
    uh most of the times i'm going to call
  • 00:05:44
    it what's called a
  • 00:05:47
    second price auction and you'll see
  • 00:05:50
    why this name uh
  • 00:05:53
    well why second price auction well or
  • 00:05:56
    the vicry auction
  • 00:05:57
    so it is a simple a simpler uh
  • 00:06:00
    game uh the bidders the potential buyers
  • 00:06:03
    simultaneously and independently write
  • 00:06:06
    their own bits on a piece of paper
  • 00:06:09
    and then put those bids in an envelope
  • 00:06:11
    all right so it's a simultaneous move
  • 00:06:13
    everybody
  • 00:06:14
    uh writes a price a potential price
  • 00:06:17
    or we call it bid in a piece of paper
  • 00:06:20
    and submit it to the auctioneer
  • 00:06:23
    all right and then the auctioneer opens
  • 00:06:26
    all the
  • 00:06:26
    envelopes and the highest bidder
  • 00:06:35
    wins all right so let's say
  • 00:06:38
    uh let's suppose for simplicity there
  • 00:06:40
    were two bidders
  • 00:06:42
    two buyers and one guy bid uh
  • 00:06:45
    five million dollar the other bid five
  • 00:06:47
    million
  • 00:06:49
    plus one dollar and so the highest
  • 00:06:52
    bidder
  • 00:06:52
    wins the auction well what is the price
  • 00:06:57
    well the price is the second
  • 00:07:00
    highest bid all right
  • 00:07:03
    so you do not pay uh your bid
  • 00:07:07
    if you won the auction you pay the
  • 00:07:10
    highest uh losing bid right so
  • 00:07:14
    equivalently the second highest bid
  • 00:07:19
    so once again another example well let's
  • 00:07:21
    say there are 10
  • 00:07:23
    uh buyers and and buyer one
  • 00:07:27
    uh bid a dollar buyer to bit two dollars
  • 00:07:30
    etc buyer eight bid eight dollars and
  • 00:07:33
    buyer ten bid
  • 00:07:34
    ten dollars let's suppose and so the
  • 00:07:37
    winner is going to be buyer 10 because
  • 00:07:39
    his bid
  • 00:07:40
    ten dollars is the highest however he's
  • 00:07:43
    not going to pay
  • 00:07:44
    ten dollars he's going to pay uh the the
  • 00:07:47
    highest losing bid so who are the losing
  • 00:07:51
    bids well the guy
  • 00:07:52
    who bid one dollar two dollar up until
  • 00:07:54
    nine dollar
  • 00:07:55
    these are all the losers right because
  • 00:07:57
    there's only one item
  • 00:07:59
    that is sold in this auction that's an
  • 00:08:01
    important assumption
  • 00:08:02
    and so only the tenth guy win the
  • 00:08:05
    auction and so the other remaining the
  • 00:08:07
    first nine guys
  • 00:08:08
    loses the auction so what is the highest
  • 00:08:10
    losing bid
  • 00:08:11
    or the what is the second highest bid
  • 00:08:14
    means the same thing
  • 00:08:15
    well it's nine dollars so therefore the
  • 00:08:18
    winner the tenth guy
  • 00:08:19
    only pays nine dollars all right so
  • 00:08:23
    uh kind of awkward right why you don't
  • 00:08:25
    pay what you bid
  • 00:08:26
    while you pay what uh these uh the
  • 00:08:29
    second highest bid is
  • 00:08:31
    uh but trust me they are strategically
  • 00:08:34
    equivalent well
  • 00:08:34
    why it is simpler well as i said because
  • 00:08:36
    it's a simultaneous move game we don't
  • 00:08:38
    really have to worry about
  • 00:08:40
    sub games okay so
  • 00:08:43
    the question is how do we approach to
  • 00:08:46
    these games
  • 00:08:47
    and how do we analyze these games
  • 00:08:50
    well when we want to analyze these
  • 00:08:53
    environments or these strategic
  • 00:08:54
    environments
  • 00:08:56
    obviously there are we have to be
  • 00:08:59
    a formal or sort of clear about the
  • 00:09:02
    description of the environment right
  • 00:09:04
    remember a game has several properties
  • 00:09:06
    like who the players are what their
  • 00:09:08
    strategies are what their payoffs are
  • 00:09:10
    etc
  • 00:09:11
    so here we have to be clear about
  • 00:09:13
    players
  • 00:09:15
    who are they well the players are the
  • 00:09:17
    potential buyers right the guys who
  • 00:09:19
    bid well um good well what about their
  • 00:09:23
    actions well the actions or the
  • 00:09:24
    strategies
  • 00:09:26
    uh that's simple they basically bid
  • 00:09:29
    so let's call it bi this is the strategy
  • 00:09:32
    of bidder i
  • 00:09:33
    he bids some number right some number
  • 00:09:36
    between
  • 00:09:36
    zero and infinity well obviously you can
  • 00:09:39
    bid as high as you like
  • 00:09:40
    and obviously you can't beat something
  • 00:09:43
    negative so these are the strategies as
  • 00:09:45
    simple as this i already mentioned what
  • 00:09:48
    the rules are
  • 00:09:49
    right everybody simultaneously and
  • 00:09:51
    independently bids
  • 00:09:52
    and the highest bidder wins but pays
  • 00:09:54
    only the second highest bid
  • 00:09:56
    all right well then finally the payoffs
  • 00:10:00
    well kind of simple right everybody
  • 00:10:04
    if if you lose
  • 00:10:07
    well then we assume that your payoff is
  • 00:10:10
    going to be regardless of your bid well
  • 00:10:12
    given that with this bid you lost
  • 00:10:14
    your payoff is going to be zero well if
  • 00:10:16
    you win
  • 00:10:18
    win with this bid bi ui bi
  • 00:10:22
    well then okay so how do we model this
  • 00:10:26
    well that's one simplification obviously
  • 00:10:28
    we assume that
  • 00:10:29
    the the potential buyer
  • 00:10:32
    has some valuation for the item that is
  • 00:10:35
    auctioned all right so that's that
  • 00:10:37
    painting
  • 00:10:37
    has some value to you um and
  • 00:10:40
    so this is your willingness to pay the
  • 00:10:43
    maximum willingness to pay you do not
  • 00:10:45
    want to pay more than
  • 00:10:46
    vi so vi is just a number all right
  • 00:10:50
    this is ui the utility vi is some number
  • 00:10:53
    between
  • 00:10:54
    zero and infinity all right and so it's
  • 00:10:57
    your highest willingness to pay
  • 00:11:00
    and then the thing is
  • 00:11:04
    you win you bid bi but
  • 00:11:07
    remember you pay bj where
  • 00:11:13
    bj is the second
  • 00:11:17
    highest bid okay
  • 00:11:21
    so you do not pay bi dollar this is what
  • 00:11:24
    you bid
  • 00:11:25
    so here obviously because bi
  • 00:11:28
    is winning so winning means what bi is
  • 00:11:31
    greater than
  • 00:11:33
    bj for all j
  • 00:11:37
    equals to 1 n all right okay good
  • 00:11:41
    so this is the payoffs all right it's as
  • 00:11:44
    simple as this
  • 00:11:45
    well the question is
  • 00:11:48
    um so here this is a simultaneous move
  • 00:11:52
    game and so we can find the nash
  • 00:11:53
    equilibrium of this game right
  • 00:11:56
    nash equilibrium all right
  • 00:11:59
    actually let's do this for a very simple
  • 00:12:02
    example
  • 00:12:03
    so example is here there are two players
  • 00:12:07
    two buyers potential buyers or we call
  • 00:12:11
    them bidders all right
  • 00:12:13
    and bidder one has a valuation hundred
  • 00:12:17
    dollars
  • 00:12:18
    bidder two has a valuation uh ninety
  • 00:12:21
    dollars
  • 00:12:22
    okay let's suppose well then
  • 00:12:25
    the question is what is the nash
  • 00:12:28
    equilibrium
  • 00:12:29
    of this game well actually there are
  • 00:12:33
    many nash equilibrium of this game
  • 00:12:35
    all right but the one that makes the
  • 00:12:38
    most sense
  • 00:12:39
    is the one where they both bid
  • 00:12:42
    their uh true values meaning
  • 00:12:47
    b1 equals hundred and b2
  • 00:12:50
    equals 90 is
  • 00:12:53
    a nash equilibrium of this game
  • 00:12:57
    all right well why is that so
  • 00:13:00
    well here the things like suppose that
  • 00:13:03
    your opponent
  • 00:13:04
    so again we are checking nash
  • 00:13:06
    equilibrium
  • 00:13:07
    so suppose you observe that your
  • 00:13:09
    opponent
  • 00:13:10
    bid 90 are you going to regret from your
  • 00:13:13
    bid
  • 00:13:14
    100 well not really why well
  • 00:13:17
    you could bid higher than 100 but you
  • 00:13:20
    could still win
  • 00:13:21
    and so so in this case the bitter one
  • 00:13:24
    his utility with this bit hundred is
  • 00:13:27
    equal to
  • 00:13:28
    his valuation minus the price he pays so
  • 00:13:31
    it's ten dollars
  • 00:13:33
    but imagine you bid something
  • 00:13:36
    b1 where b1 is greater than hundred
  • 00:13:40
    would this change anything no because as
  • 00:13:43
    long as b1 is higher than ninety dollars
  • 00:13:45
    you're going to win
  • 00:13:47
    and when you win your payoff is not your
  • 00:13:50
    bid
  • 00:13:50
    minus the second highest bid
  • 00:13:54
    but it's your valuation which is fixed
  • 00:13:56
    right your willingness to pay
  • 00:13:58
    is something that doesn't change it's
  • 00:14:01
    fixed before the game and after the game
  • 00:14:04
    all right so this is what we assume
  • 00:14:06
    in economics and so it's going to be 100
  • 00:14:09
    minus 90 again
  • 00:14:10
    10. so as long as you bid higher than
  • 00:14:13
    100
  • 00:14:14
    you're going to get the same payoff as
  • 00:14:16
    long as
  • 00:14:17
    you bid higher than 90
  • 00:14:20
    but less than 100 all right for example
  • 00:14:23
    you could bid
  • 00:14:24
    95 92 would this change your payoff
  • 00:14:28
    no not really once again you would win
  • 00:14:31
    and then this would be your payoff
  • 00:14:33
    but the question is what if you bid
  • 00:14:37
    something less than ninety dollars all
  • 00:14:39
    right
  • 00:14:40
    well you may ask what happens if they
  • 00:14:43
    both
  • 00:14:43
    bid 90 dollars it is irrelevant
  • 00:14:46
    trust me but if you like to be clear
  • 00:14:49
    about it
  • 00:14:50
    let's suppose that when two guys bid
  • 00:14:52
    exactly the same amount
  • 00:14:54
    well then the auctioneer tosses a fair
  • 00:14:56
    coin
  • 00:14:57
    with one half probability it's going to
  • 00:14:59
    come up had with one half of probable it
  • 00:15:01
    is going to come up tail
  • 00:15:02
    and if it is had a buyer one wins or the
  • 00:15:05
    auction
  • 00:15:06
    per you know person one wins if it is
  • 00:15:08
    tail person two wins all right so
  • 00:15:10
    therefore
  • 00:15:11
    if two guys bid exactly the same number
  • 00:15:14
    it's going to be you know half enough
  • 00:15:16
    probability
  • 00:15:17
    to win the object and to lose the object
  • 00:15:20
    but obviously this is a zero probability
  • 00:15:22
    event because if you can
  • 00:15:24
    bid any number between zero to infinity
  • 00:15:26
    right you could beat anything
  • 00:15:28
    so two guys can beat anything between
  • 00:15:29
    zero infinity what is the likelihood
  • 00:15:32
    that
  • 00:15:32
    both are going to pick exactly the same
  • 00:15:34
    location
  • 00:15:35
    out of infinitely many possibilities
  • 00:15:38
    well it's a zero
  • 00:15:38
    possibility event for that reason it
  • 00:15:41
    really doesn't matter
  • 00:15:42
    um but as i said just for uh
  • 00:15:45
    completeness
  • 00:15:46
    i i i i we can assume that uh they
  • 00:15:50
    they will get with equal probabilities
  • 00:15:53
    so the question is what happens
  • 00:15:57
    i ignore what happens when b1 is equal
  • 00:15:59
    to 90 for that reason
  • 00:16:01
    what if it is less than 90. so what if
  • 00:16:03
    you bid
  • 00:16:04
    something less than 90 dollars well this
  • 00:16:06
    time you're gonna lose because
  • 00:16:09
    if you bid for example 80 or 90 i'm
  • 00:16:12
    sorry 89
  • 00:16:13
    you're gonna lose this auction and so
  • 00:16:16
    your payoff will be
  • 00:16:17
    automatically zero all right so for that
  • 00:16:20
    reason
  • 00:16:22
    as you see bidding hundred is
  • 00:16:25
    one of the best responses there are many
  • 00:16:28
    others for example bidding
  • 00:16:29
    2000 is also best response bidding 91
  • 00:16:32
    is also best to respond but but bidding
  • 00:16:35
    hundred dollars
  • 00:16:36
    is a best response is a best response
  • 00:16:40
    to the second player's strategy
  • 00:16:43
    all right so the first guy is best
  • 00:16:45
    responding the second guy
  • 00:16:46
    well the question is is is the second
  • 00:16:49
    guy best responding the first guy well
  • 00:16:51
    let's check
  • 00:16:53
    well given that the first guy is bidding
  • 00:16:56
    hundred dollars
  • 00:16:57
    um for the second guy can he
  • 00:17:01
    uh i mean if the second guy bids
  • 00:17:04
    something less than hundred dollars we
  • 00:17:07
    know that he
  • 00:17:08
    is going to lose right whether it's 99
  • 00:17:11
    or zero dollars it doesn't matter the
  • 00:17:15
    the second guy is gonna lose and he's
  • 00:17:17
    gonna get
  • 00:17:18
    zero payoff which is what he's achieving
  • 00:17:22
    when he beats 90 so there is no
  • 00:17:24
    improvement there is no
  • 00:17:26
    profitable deviation here
  • 00:17:29
    profitable deviation to some b2 less
  • 00:17:32
    than 100
  • 00:17:33
    but what if b2 to more than 100
  • 00:17:37
    all right i mean what if player 2 bids
  • 00:17:40
    something higher than 100 so
  • 00:17:43
    is there any profitable deviation there
  • 00:17:46
    i mean 101 2 000 it really doesn't
  • 00:17:49
    matter because
  • 00:17:50
    why b2 greater than 100 because this is
  • 00:17:53
    the case which
  • 00:17:54
    ensures that the second guy will be the
  • 00:17:57
    winner
  • 00:17:58
    well if he is the winner all right
  • 00:18:00
    what's going to happen
  • 00:18:01
    is the following well with this bid
  • 00:18:04
    higher than 100
  • 00:18:05
    you're going to win the object and in
  • 00:18:08
    this case your payoff is your valuation
  • 00:18:10
    remember what was your maximum
  • 00:18:12
    willingness to pay
  • 00:18:13
    well it was 90 so you your willingness
  • 00:18:16
    to pay is not 100
  • 00:18:18
    it's 90 so 90 minus
  • 00:18:22
    what is the losing bit well remember you
  • 00:18:26
    bid
  • 00:18:27
    higher than hundred dollars so there are
  • 00:18:30
    two bits
  • 00:18:30
    b2 which is higher than 100 and b1 which
  • 00:18:33
    is hundred so
  • 00:18:34
    what is the second highest bid or what
  • 00:18:37
    is the
  • 00:18:38
    highest uh losing bid well it's hundred
  • 00:18:41
    so you're going to pay hundred dollars
  • 00:18:43
    so what's your payoff
  • 00:18:44
    it's -10 which is less than zero which
  • 00:18:48
    is what you achieve when you bid
  • 00:18:50
    90 and lose this auction so what does
  • 00:18:52
    that mean that means
  • 00:18:53
    given that the first guy is bidding
  • 00:18:55
    hundred dollars
  • 00:18:57
    the second guys bidding ninety dollars
  • 00:19:00
    is
  • 00:19:00
    one of the best responses obviously
  • 00:19:03
    bidding zero
  • 00:19:03
    is also best response right bidding
  • 00:19:06
    five dollars is also best response but i
  • 00:19:08
    don't care other best responses all i
  • 00:19:10
    care
  • 00:19:11
    is 90 is a best response 200
  • 00:19:16
    so therefore both guys both buyers are
  • 00:19:19
    best responding one another
  • 00:19:21
    and hence this strategy profile is in
  • 00:19:24
    fact
  • 00:19:25
    a nash equilibrium all right
  • 00:19:28
    so what do we learn from this very
  • 00:19:31
    simple analysis
  • 00:19:32
    uh well many things one of them
  • 00:19:36
    well first of all in a
  • 00:19:39
    a decree auction or second price auction
  • 00:19:42
    or equivalently
  • 00:19:43
    in the english auction you shouldn't bid
  • 00:19:46
    higher than your valuation all right so
  • 00:19:48
    if you think
  • 00:19:49
    that this picture this painting whatever
  • 00:19:51
    the action
  • 00:19:52
    the item that is auctioned if you think
  • 00:19:55
    it's it's not
  • 00:19:56
    worth more than ten thousand dollars or
  • 00:19:59
    if this is your max budget
  • 00:20:01
    well you should not bid higher than this
  • 00:20:05
    well what is the second thing that we
  • 00:20:07
    learn well
  • 00:20:08
    every bitter bidding his or her true
  • 00:20:11
    value
  • 00:20:12
    is actually a nash equilibrium right so
  • 00:20:14
    basically you don't really make
  • 00:20:16
    any strategic thinking it's like should
  • 00:20:18
    i bid
  • 00:20:19
    five dollars less than my maximum
  • 00:20:22
    willingness to pay two dollars less
  • 00:20:24
    or you know 10 less more you don't
  • 00:20:27
    really need to make this strategic
  • 00:20:29
    uh thinking in this game because um
  • 00:20:33
    we didn't show that uh in fact here
  • 00:20:38
    bidding the true values is a dominant
  • 00:20:41
    strategy if you if you apply iterated
  • 00:20:43
    elimination of weekly dominated
  • 00:20:45
    strategies you'll see that actually
  • 00:20:48
    bidding the true values are sort of a
  • 00:20:50
    weakly dominant
  • 00:20:52
    strategy in this game and it's in nash
  • 00:20:54
    equilibrium so you don't really have to
  • 00:20:56
    worry about the strategic interaction
  • 00:20:59
    here
  • 00:21:00
    well so but obviously this is not the
  • 00:21:03
    game that i
  • 00:21:04
    am intended to uh sort of explain why
  • 00:21:07
    well because
  • 00:21:08
    we want to do something new here we
  • 00:21:10
    would like to talk about
  • 00:21:11
    incomplete information game if you
  • 00:21:14
    remember we talked about
  • 00:21:16
    perfect versus
  • 00:21:19
    imperfect games right well they were
  • 00:21:23
    games where there's a simultaneity of
  • 00:21:27
    moves
  • 00:21:28
    and obviously uh the the second price
  • 00:21:30
    auction
  • 00:21:31
    is an imperfect information game when
  • 00:21:33
    you
  • 00:21:34
    choose your strategy your you don't know
  • 00:21:37
    your opponent's strategy you can't
  • 00:21:39
    observe it
  • 00:21:39
    and so it's imperfect information game
  • 00:21:42
    however
  • 00:21:43
    chess is a perfect information game but
  • 00:21:46
    on top of that we would like to do
  • 00:21:47
    something
  • 00:21:48
    new incomplete so some parts of the
  • 00:21:52
    information is incomplete so we are
  • 00:21:53
    actually relaxing some of our
  • 00:21:55
    assumptions in game theory what is this
  • 00:21:59
    well here i mean let's consider we
  • 00:22:02
    consider this very simple example
  • 00:22:04
    in real life if this is really the case
  • 00:22:07
    i mean
  • 00:22:07
    think about an auction environment in
  • 00:22:10
    real life
  • 00:22:11
    you probably do observe how many
  • 00:22:14
    potential buyers are there in the
  • 00:22:16
    auction house right
  • 00:22:18
    some of them are in on on a phone
  • 00:22:21
    which are basically talking to the buyer
  • 00:22:24
    who would like to be anonymous
  • 00:22:26
    and some are present there basically
  • 00:22:29
    raising
  • 00:22:30
    some card to indicate that they would
  • 00:22:32
    like to increase the price
  • 00:22:34
    and so the potential buyers are all in
  • 00:22:36
    the same location right so the number of
  • 00:22:38
    buyers are there
  • 00:22:40
    so the players i mean there's a perfect
  • 00:22:43
    information or complete information
  • 00:22:45
    about players so who the players are etc
  • 00:22:48
    strategies well again uh
  • 00:22:51
    this is exactly why i didn't want to
  • 00:22:53
    talk about auction
  • 00:22:55
    english auction because their strategies
  • 00:22:57
    are more complicated
  • 00:22:58
    you know when i should bid you know in
  • 00:23:01
    which increment i should bid
  • 00:23:03
    and when should i stop bidding etc so
  • 00:23:06
    the strategies are sort of a
  • 00:23:08
    multi-dimensional and so it's more
  • 00:23:10
    complicated
  • 00:23:11
    for that reason i i wanted to study
  • 00:23:13
    second price auction so here the
  • 00:23:14
    strategies are very simple
  • 00:23:16
    i am just going to write a number on a
  • 00:23:19
    piece of paper
  • 00:23:20
    and that's it um so in terms of
  • 00:23:24
    you know i mean that the strategy is
  • 00:23:26
    that everybody is willing to bid
  • 00:23:28
    some number between zero to infinity
  • 00:23:30
    well i mean it's a common knowledge it's
  • 00:23:32
    a common information right
  • 00:23:33
    so therefore the strategies are sort of
  • 00:23:36
    a complete information in this game
  • 00:23:38
    well what about the payoffs hmm so
  • 00:23:41
    question is
  • 00:23:42
    here in fact not whether you get zero
  • 00:23:45
    versus this
  • 00:23:46
    the rules are clear right the rules are
  • 00:23:49
    such that
  • 00:23:50
    if you win you are going to pay
  • 00:23:53
    the second losing bid and so
  • 00:23:56
    this functional form is an assumption
  • 00:23:59
    that we made
  • 00:24:00
    and if you lose you're gonna get zero
  • 00:24:01
    payoff well maybe if you lose you're
  • 00:24:03
    going to suffer
  • 00:24:05
    and incredibly because you're going to
  • 00:24:07
    feel terrible for
  • 00:24:09
    i don't know not having this this this
  • 00:24:11
    particular painting
  • 00:24:13
    i mean yeah there might be some room of
  • 00:24:15
    improvement there
  • 00:24:16
    but what is the most important or i
  • 00:24:19
    think
  • 00:24:19
    uh most relevant extension is that
  • 00:24:23
    in reality we usually do not know what
  • 00:24:26
    vi is for each player in the game right
  • 00:24:30
    i mean think of like this very two very
  • 00:24:33
    simple example
  • 00:24:34
    two potential buyers so as a buyer you
  • 00:24:37
    probably know how much
  • 00:24:39
    you're willing to pay for this painting
  • 00:24:41
    let's say you're a buyer one
  • 00:24:43
    so what is willingness to pay well this
  • 00:24:45
    idea comes from
  • 00:24:46
    you know you heard about it in
  • 00:24:49
    intermediate microeconomics in
  • 00:24:51
    advanced microeconomic theory so it's
  • 00:24:53
    basically
  • 00:24:54
    what derives the demand curve right so
  • 00:24:57
    it is about your preferences it is about
  • 00:25:00
    your income your wealth etc
  • 00:25:02
    so i'm not going to give sort of a
  • 00:25:04
    detailed discussion about what
  • 00:25:06
    derives valuation but what we assume in
  • 00:25:09
    economics is that
  • 00:25:10
    everybody you know when they come to a
  • 00:25:12
    market environment
  • 00:25:13
    so sort of a to trade something well
  • 00:25:16
    they they come
  • 00:25:17
    with some clear picture of how much
  • 00:25:21
    the maximum how much they're willing to
  • 00:25:23
    pay for this item
  • 00:25:24
    all right in real life you may be unsure
  • 00:25:27
    about it
  • 00:25:28
    right because for example if it is a
  • 00:25:30
    painting you actually do not want to
  • 00:25:32
    hold it for a long time you want to
  • 00:25:34
    resell it
  • 00:25:35
    and so the resale price is what
  • 00:25:37
    determines your willingness to pay
  • 00:25:40
    so that you know if things are more
  • 00:25:41
    complicated in real life i know
  • 00:25:43
    but we usually assume that everybody
  • 00:25:45
    knows his willingness to pay
  • 00:25:47
    because everybody is fully aware of his
  • 00:25:49
    preferences
  • 00:25:50
    and his income and wealth all right so
  • 00:25:52
    that's sort of an assumption we don't
  • 00:25:54
    really want to play too much
  • 00:25:56
    but what we can extend or
  • 00:26:00
    relax what assumption that we relax is
  • 00:26:03
    what i know about my opponent's
  • 00:26:06
    willingness to pay so here for example
  • 00:26:08
    we assumed that as player one buyer won
  • 00:26:12
    i know that i would like to pay a
  • 00:26:13
    hundred dollars but i also know that my
  • 00:26:16
    opponent is actually
  • 00:26:17
    willing to pay ninety dollars which is
  • 00:26:19
    less than what i want
  • 00:26:21
    and so strategically that makes me s
  • 00:26:24
    sort of advantaged why well because i
  • 00:26:27
    would like to pay
  • 00:26:28
    more than what this guy wants to pay all
  • 00:26:30
    right so
  • 00:26:31
    um let me bid hundred dollar so i know
  • 00:26:34
    that he's not going to go above 100
  • 00:26:36
    because his willingness to pay
  • 00:26:38
    is definitely less than 100 so he's
  • 00:26:40
    going to make loss which he doesn't want
  • 00:26:42
    to
  • 00:26:42
    because by losing the object he could
  • 00:26:44
    ensure zero payoff anyway
  • 00:26:46
    all right so for that reason we can
  • 00:26:49
    extend this
  • 00:26:50
    idea well what if players
  • 00:26:53
    do not know their opponent's
  • 00:26:57
    payoffs for sure so in that sense
  • 00:27:00
    there is some incomplete information all
  • 00:27:03
    right
  • 00:27:04
    sometimes we call this by the way
  • 00:27:06
    asymmetric information
  • 00:27:11
    well why asymmetric well because
  • 00:27:13
    everybody knows
  • 00:27:14
    his or her evaluation but unsure about
  • 00:27:18
    his opponent's valuation all right so
  • 00:27:20
    player one knows that he wants to pay
  • 00:27:23
    hundred dollars but he is probably
  • 00:27:26
    unsure about
  • 00:27:27
    how much his opponent wants to pay and
  • 00:27:30
    symmetrically player two knows that he's
  • 00:27:32
    a
  • 00:27:32
    he wants to play 90 stops but he
  • 00:27:35
    is probably unsure about how much his
  • 00:27:38
    opponent
  • 00:27:39
    buyer one is willing to pay right so
  • 00:27:43
    the next question is how are we going to
  • 00:27:45
    model this environment
  • 00:27:47
    and then how are we going to solve this
  • 00:27:49
    environment
  • 00:27:52
    okay so let's think of the again the
  • 00:27:55
    simplest environment where there are two
  • 00:27:57
    players
  • 00:27:58
    or two bidders or two potential buyers
  • 00:28:00
    and they would like to
  • 00:28:02
    bid for this non-divisible good the
  • 00:28:05
    painting
  • 00:28:06
    here the assumption none divisibility is
  • 00:28:09
    important because
  • 00:28:10
    it means if somebody wins that means the
  • 00:28:13
    other guys
  • 00:28:14
    are going to lose because the good is
  • 00:28:16
    not divisible they cannot share it
  • 00:28:18
    all right well so let's assume that the
  • 00:28:20
    first guy has evaluation hundred the
  • 00:28:22
    second guy has a valuation 90.
  • 00:28:25
    so we call this but this time private
  • 00:28:27
    information
  • 00:28:28
    why well because the first guy although
  • 00:28:30
    he knows how much he's willing to pay
  • 00:28:32
    he's
  • 00:28:33
    unsure about his opponent's willingness
  • 00:28:35
    to pay and same for player two
  • 00:28:37
    although he knows his willingness to pay
  • 00:28:39
    i'm sure
  • 00:28:40
    he's unsure about how much his opponent
  • 00:28:43
    is willing
  • 00:28:44
    willing to pay so how can we model this
  • 00:28:46
    well obviously we do not want to say
  • 00:28:48
    well the buyers are unsure about
  • 00:28:52
    their opponent's willingness to pay we
  • 00:28:54
    have to be we want to be more formal
  • 00:28:56
    about it
  • 00:28:57
    and so the one way to formally describe
  • 00:29:00
    the beliefs
  • 00:29:01
    is you can say for example
  • 00:29:04
    buyer 1 believes that his opponent's
  • 00:29:07
    willingness to pay which is
  • 00:29:08
    v2 the parameter is actually randomly
  • 00:29:11
    distributed
  • 00:29:12
    according to some cumulative
  • 00:29:14
    distribution function f2
  • 00:29:16
    on the interval zero infinity all right
  • 00:29:18
    so
  • 00:29:19
    basically that means buyer one things
  • 00:29:22
    anything is possible
  • 00:29:23
    uh but the thing is you know according
  • 00:29:26
    to this probability distribution for
  • 00:29:28
    example if it is a uniform
  • 00:29:29
    maybe uh sort of uh sort of the
  • 00:29:32
    distribution of this v2 is uniform
  • 00:29:36
    but if it is a normal well that means uh
  • 00:29:38
    sort of uh it's more likely to be around
  • 00:29:41
    the mean of this normal distribution but
  • 00:29:43
    you know nevertheless anything is
  • 00:29:45
    possible
  • 00:29:46
    all right so symmetrically you can think
  • 00:29:48
    that the buyer too
  • 00:29:49
    believes that the the first buyer's
  • 00:29:52
    willingness to pay is random
  • 00:29:54
    and so the v1 parameter is randomly
  • 00:29:57
    distributed according to some
  • 00:29:59
    probability distribution function f1
  • 00:30:01
    on zero infinity interval all right so
  • 00:30:05
    here obviously one thing is important
  • 00:30:07
    remember
  • 00:30:08
    in our earlier discussions of game
  • 00:30:10
    theories like
  • 00:30:12
    uh the players the set of players set of
  • 00:30:14
    strategies and their payoffs all this
  • 00:30:16
    information is common knowledge so here
  • 00:30:19
    we are sort of extending this
  • 00:30:23
    uh or sort of relaxing this assumption
  • 00:30:26
    so some you know things are not complete
  • 00:30:30
    information
  • 00:30:30
    there's some incomplete information in
  • 00:30:32
    the sense that player one
  • 00:30:34
    is unsure about the second player's
  • 00:30:38
    sort of private information so but
  • 00:30:41
    nevertheless
  • 00:30:42
    um you know can we still do we still
  • 00:30:46
    keep this idea of common knowledge
  • 00:30:48
    assumption yes
  • 00:30:49
    how so while here i mean we are not
  • 00:30:51
    going to
  • 00:30:52
    argue this too much in this course
  • 00:30:54
    because it really deserves
  • 00:30:56
    i mean requires some advanced
  • 00:31:00
    training in game theory but we are going
  • 00:31:02
    to assume that
  • 00:31:03
    those probable distributions are common
  • 00:31:06
    knowledge
  • 00:31:06
    so what does that mean that means the
  • 00:31:08
    following if buyer 1
  • 00:31:10
    thinks that his opponent's
  • 00:31:14
    valuation is distributed normally
  • 00:31:17
    distributed
  • 00:31:18
    with mean for example
  • 00:31:22
    mu and the standard deviation sigma
  • 00:31:25
    all right well then player so this is
  • 00:31:28
    player one's belief
  • 00:31:29
    but then player two will also be aware
  • 00:31:33
    fully aware that player one is
  • 00:31:36
    in fact believing that his valuation is
  • 00:31:39
    distributed
  • 00:31:40
    in this range although his valuation is
  • 00:31:43
    exactly 90.
  • 00:31:45
    all right so whatever mu is
  • 00:31:48
    so so that probability these probability
  • 00:31:52
    distributions are common knowledge this
  • 00:31:54
    is what we assume
  • 00:31:56
    again what happens if these are not
  • 00:31:58
    common knowledge
  • 00:32:00
    well again this is not the discussion
  • 00:32:02
    for this
  • 00:32:03
    level uh it it requires a much more
  • 00:32:07
    uh advanced uh skills in game theory
  • 00:32:11
    all right let's consider a simpler case
  • 00:32:14
    right i mean it doesn't really have to
  • 00:32:15
    be
  • 00:32:16
    like well anything is possible according
  • 00:32:18
    to some continuous
  • 00:32:19
    cumulative distribution function well in
  • 00:32:21
    fact for most of our examples we are
  • 00:32:23
    going to look at simpler environments
  • 00:32:25
    where
  • 00:32:26
    you know one of three things can happen
  • 00:32:28
    or one of two things can happen type of
  • 00:32:30
    environments
  • 00:32:31
    so you can imagine for example buyer one
  • 00:32:34
    believes
  • 00:32:34
    that the buyer two's valuation is in
  • 00:32:37
    fact
  • 00:32:38
    uh distributed according to sorry
  • 00:32:41
    equally so it's a uniform distribution
  • 00:32:43
    but the potential values are a hundred
  • 00:32:46
    and ten hundred and ninety
  • 00:32:47
    and symmetrically buyer two believes
  • 00:32:49
    that the buyer one's valuation is
  • 00:32:51
    hundred and ten hundred or ninety
  • 00:32:53
    all right so as player one i know my
  • 00:32:56
    valuation is hundred
  • 00:32:58
    uh i know that i believe that my
  • 00:33:00
    opponent
  • 00:33:02
    can actually beat me meaning his
  • 00:33:04
    valuation can be hundred and ten dollars
  • 00:33:06
    with one third probability his valuation
  • 00:33:09
    can be 100
  • 00:33:10
    so we can actually be in a tie and his
  • 00:33:12
    valuation can actually be 90.
  • 00:33:15
    so the question is as by the way
  • 00:33:17
    everything is symmetric for player 2. so
  • 00:33:19
    here remember when this information
  • 00:33:22
    wasn't private but public
  • 00:33:24
    meaning this the previous example we
  • 00:33:27
    analyzed
  • 00:33:27
    where the valuation for buyer one and
  • 00:33:30
    two are
  • 00:33:30
    are known by everyone well in this case
  • 00:33:33
    buyer one knew that
  • 00:33:35
    he has the advantage because he knows
  • 00:33:38
    that his opponent cannot overbid
  • 00:33:40
    him no way because otherwise
  • 00:33:44
    his opponent is going to get negative
  • 00:33:45
    payoff but so therefore he was kind of
  • 00:33:47
    relaxed
  • 00:33:48
    saying well whether i bid 100 or
  • 00:33:51
    95 or 91 i'm going to win this auction
  • 00:33:54
    anyway
  • 00:33:55
    i remember so he was kind of relaxed
  • 00:33:58
    however now
  • 00:33:59
    he can't be so relaxed because he knows
  • 00:34:01
    that if he
  • 00:34:02
    bids for example 91
  • 00:34:06
    he can actually lose it in in these two
  • 00:34:09
    i mean
  • 00:34:10
    strictly i mean definitely lose it if
  • 00:34:12
    his opponent is under these two
  • 00:34:14
    scenarios
  • 00:34:15
    right he may i mean because
  • 00:34:18
    it doesn't we don't know whether those
  • 00:34:21
    guys
  • 00:34:22
    meaning uh let me put it this way we
  • 00:34:24
    don't know if the buyer
  • 00:34:25
    with the value 110 or buyer value 100
  • 00:34:29
    are going to
  • 00:34:30
    bid exactly the evaluations or maybe
  • 00:34:33
    less or maybe more
  • 00:34:34
    but what i want to show is that the
  • 00:34:37
    buyer one
  • 00:34:38
    with the valuation hundred is not going
  • 00:34:40
    to be relaxed about
  • 00:34:42
    saying whether 90 or 91 or 95
  • 00:34:46
    or 99 i'm gonna win this object so i'm
  • 00:34:49
    kind of
  • 00:34:49
    uh sort of okay i'm cool about bidding
  • 00:34:52
    hundred
  • 00:34:52
    so this time it's not so easy you see
  • 00:34:55
    what i mean
  • 00:34:56
    so therefore the the calculations must
  • 00:34:59
    be more careful obviously
  • 00:35:01
    but the so so maybe the equilibrium
  • 00:35:03
    strategies will
  • 00:35:04
    will be different this is what i would
  • 00:35:06
    like to say
  • 00:35:08
    um well how different or
  • 00:35:11
    the real question is how do we analyze
  • 00:35:13
    this environment anyhow
  • 00:35:15
    right even in this very simple
  • 00:35:17
    environment so
  • 00:35:18
    how are we going to approach this
  • 00:35:20
    environment well very simple
  • 00:35:22
    what we're going to do we're going to
  • 00:35:23
    find
  • 00:35:25
    um let me
  • 00:35:29
    erase this what we're going to do we are
  • 00:35:31
    going to find
  • 00:35:33
    nash equilibrium all right
  • 00:35:38
    so we're going to find the nash
  • 00:35:39
    equilibrium of this game
  • 00:35:41
    um so we know the definition of nash
  • 00:35:45
    equilibrium every player best responds
  • 00:35:47
    his opponent
  • 00:35:48
    however there's a trick that we're going
  • 00:35:51
    to use here
  • 00:35:52
    what is this trick well here the
  • 00:35:54
    strategy
  • 00:35:55
    profile is it is it b1 and
  • 00:35:58
    b2 only is this the strategy profile
  • 00:36:02
    really hmm well you may say yes it is
  • 00:36:06
    the strategy profile because there are
  • 00:36:07
    two players
  • 00:36:09
    remember yes there are two players
  • 00:36:12
    and so for each player there should be a
  • 00:36:14
    strategy
  • 00:36:16
    i mean agree but the problem is
  • 00:36:19
    let's let's look at player one so player
  • 00:36:22
    one
  • 00:36:24
    uh things that his opponent can
  • 00:36:27
    have this guy with a valuation 110
  • 00:36:32
    or this guy with the valuation hundred
  • 00:36:34
    or this guy
  • 00:36:35
    with the valuation 90. question is
  • 00:36:39
    when i say b2 doesn't it
  • 00:36:42
    imply that i i sort of believe
  • 00:36:46
    that my opponent is going to bid exactly
  • 00:36:49
    the same
  • 00:36:50
    amount of money regardless of his
  • 00:36:53
    valuation
  • 00:36:54
    yeah because remember here b1 is my
  • 00:36:57
    strategy
  • 00:36:58
    b2 is my opponent's strategy
  • 00:37:02
    why do i do this well remember the nash
  • 00:37:04
    given a strategy profile
  • 00:37:06
    every player takes his opponent's
  • 00:37:09
    strategy
  • 00:37:10
    fixed and see if he is best responding
  • 00:37:13
    it or not
  • 00:37:14
    right so therefore i'm gonna fix b2 and
  • 00:37:17
    check if b1 is the best response but for
  • 00:37:19
    this
  • 00:37:20
    i need to know what b2 is and what b2
  • 00:37:23
    implies in this environment all right so
  • 00:37:26
    b2
  • 00:37:26
    is just a number like 190 95
  • 00:37:30
    thousand zero but b2
  • 00:37:33
    a single b2 for player 2 means
  • 00:37:38
    even though my opponent valuation
  • 00:37:41
    true valuation is 110 well his his true
  • 00:37:44
    valuation is 90.
  • 00:37:46
    i i can hear but i don't know that in
  • 00:37:49
    this game remember this is a private
  • 00:37:51
    info game or
  • 00:37:52
    asymmetric info game or incomplete info
  • 00:37:54
    game meaning
  • 00:37:55
    i just know my valuation and i just
  • 00:37:58
    believe
  • 00:37:59
    that my opponent can be 110 guy or 100
  • 00:38:02
    guy or 90 guy but i'm not really sure
  • 00:38:04
    which one
  • 00:38:05
    is he and i can't see his type
  • 00:38:08
    his his valuation before i make a
  • 00:38:10
    decision so that's the problem
  • 00:38:12
    i have to make a decision without
  • 00:38:14
    observing
  • 00:38:15
    uh his his his true evaluation
  • 00:38:19
    so in sort of simultaneous move game
  • 00:38:23
    there are two type of
  • 00:38:24
    uncertainty one i'm going to choose my
  • 00:38:27
    action
  • 00:38:28
    without knowing my opponent's action so
  • 00:38:30
    that's always
  • 00:38:32
    there right and the second type of
  • 00:38:34
    uncertainty that we are in
  • 00:38:36
    in incorporating now is i am going to
  • 00:38:40
    choose my strategy without knowing uh
  • 00:38:43
    you know who my
  • 00:38:44
    opponent really is is it really the 110
  • 00:38:47
    guy or is it the 90 guy
  • 00:38:49
    so i need to make my strategy choice
  • 00:38:52
    before
  • 00:38:52
    knowing his true valuation or true type
  • 00:38:56
    you see what i mean so again
  • 00:38:59
    if i assume that my opponent's strategy
  • 00:39:02
    is b2
  • 00:39:02
    that automatically directly indirectly
  • 00:39:05
    implies
  • 00:39:06
    that all these three potential
  • 00:39:10
    opponents of mine are going to bid
  • 00:39:13
    exactly the same number
  • 00:39:15
    but this is unrealistic right i mean the
  • 00:39:18
    110 guy who could actually
  • 00:39:20
    may actually prefer to bid higher than
  • 00:39:23
    100 dollars for example
  • 00:39:25
    and where the 90 guy actually doesn't
  • 00:39:28
    probably prefer to beat hundred dollars
  • 00:39:30
    because
  • 00:39:31
    if he wins he's going to make a negative
  • 00:39:34
    profit
  • 00:39:35
    of course if he if he if he loses he's
  • 00:39:37
    gonna make
  • 00:39:38
    zero profit but if he wins he may make a
  • 00:39:41
    negative profit
  • 00:39:42
    so therefore this is a strong assumption
  • 00:39:45
    well then what am i going to do well why
  • 00:39:48
    not
  • 00:39:48
    say b21 b22
  • 00:39:52
    b23 what does that mean that means
  • 00:39:56
    i'm gonna fix as player one i'm gonna
  • 00:39:59
    fix my opponent's
  • 00:40:00
    strategy but if he
  • 00:40:03
    is really 110 guy he may choose a
  • 00:40:07
    strategy
  • 00:40:09
    different than his strategy if he
  • 00:40:13
    is in fact 100 guy and that also may be
  • 00:40:16
    different than
  • 00:40:17
    his strategy if he in fact 90 guy
  • 00:40:20
    so therefore for every potential type
  • 00:40:24
    all right so these are types of my
  • 00:40:27
    opponents
  • 00:40:28
    type 1 type 2 type 3 or you call it type
  • 00:40:31
    110
  • 00:40:32
    type 100 type 90 as you wish but we call
  • 00:40:35
    those as
  • 00:40:36
    types so for each type of my opponent
  • 00:40:39
    i should assign i mean this is the
  • 00:40:41
    safest thing i should assign
  • 00:40:43
    potentially a different strategy because
  • 00:40:46
    again
  • 00:40:47
    assigning the same strategy for all the
  • 00:40:50
    types is very restrictive
  • 00:40:51
    because again there's a 110 guy and 90
  • 00:40:54
    guy
  • 00:40:55
    i i shouldn't be expect expecting them
  • 00:40:57
    both
  • 00:40:58
    bidding the same number it's it's
  • 00:41:00
    irrelevant right i mean here for example
  • 00:41:02
    when i was 100 and the other guy was 90
  • 00:41:05
    and that was common knowledge well we
  • 00:41:07
    know that i would bid higher than his
  • 00:41:09
    bid
  • 00:41:09
    but now i mean why 110 guy and 90 guy
  • 00:41:12
    are are supposed to bid the same amount
  • 00:41:14
    so therefore i should assign different
  • 00:41:17
    strategy
  • 00:41:18
    for different types well
  • 00:41:22
    and then i should see if i'm best
  • 00:41:24
    responding these three
  • 00:41:26
    huh but are there i mean okay what the
  • 00:41:29
    heck is going on are there three
  • 00:41:31
    opponents
  • 00:41:32
    for me no not really there's in fact
  • 00:41:35
    just
  • 00:41:35
    one guy there it's the buyer too but the
  • 00:41:38
    thing is with one third probability he
  • 00:41:41
    can be
  • 00:41:41
    type one with one third probability he
  • 00:41:44
    can be type
  • 00:41:45
    two and with one third probability he
  • 00:41:47
    can be type
  • 00:41:49
    three so these are the probabilities of
  • 00:41:52
    types
  • 00:41:54
    of types all right so therefore what i
  • 00:41:58
    should be doing is like
  • 00:42:00
    calculating my expected utility well
  • 00:42:02
    what is my
  • 00:42:03
    expected utility if i bid b1
  • 00:42:07
    is going to be obviously my expected
  • 00:42:09
    utility i bid b1
  • 00:42:11
    my opponent bids b2 1
  • 00:42:14
    but this is one third probability plus
  • 00:42:17
    with one third probably right this is
  • 00:42:19
    kind of a lot right now
  • 00:42:20
    with one third probability my opponent
  • 00:42:22
    is going to be this type
  • 00:42:24
    so the second type and so he's going to
  • 00:42:26
    play this strategy
  • 00:42:27
    and therefore the outcome is going to be
  • 00:42:30
    different for me
  • 00:42:31
    b1 b22 plus one-third
  • 00:42:34
    expected utility or just utility uh
  • 00:42:38
    yeah let's call them utility because
  • 00:42:40
    these are not expected
  • 00:42:42
    this one is expected so the utility
  • 00:42:45
    that i bid b1 my opponent b23
  • 00:42:50
    well you may say why don't you
  • 00:42:53
    bid different numbers for each different
  • 00:42:56
    types look i can't do that because
  • 00:43:00
    i can't observe my opponent's type
  • 00:43:04
    right is it 110 guy 100 guy 90 guy and
  • 00:43:07
    so therefore i cannot condition my
  • 00:43:09
    strategy
  • 00:43:11
    on my opponent's type i can't say i'm
  • 00:43:14
    going to bid
  • 00:43:15
    this money if my opponent's type is that
  • 00:43:18
    i'm going to beat that if my opponent's
  • 00:43:20
    type is this
  • 00:43:21
    i can't really do that because i can't
  • 00:43:23
    observe my opponent's type
  • 00:43:25
    so therefore i'm going to choose one
  • 00:43:28
    bid b1 and the thing is
  • 00:43:31
    it may lead to three different outcomes
  • 00:43:35
    why well because there are three
  • 00:43:37
    potential types
  • 00:43:39
    different types of my opponent and they
  • 00:43:42
    all may or may not i don't know that
  • 00:43:46
    follow different strategies and so
  • 00:43:49
    therefore i may achieve
  • 00:43:50
    different payoffs and different outcomes
  • 00:43:53
    in
  • 00:43:54
    each three each of these three cases
  • 00:43:57
    all right so this is how i should say
  • 00:44:00
    well you know what uh the expected
  • 00:44:02
    utility of b1
  • 00:44:03
    is the best response if it basically
  • 00:44:05
    maximizing i mean the b1 is maximizing
  • 00:44:08
    this
  • 00:44:09
    so if you change b1 to b1 prime well the
  • 00:44:11
    thing is
  • 00:44:12
    uh you're going to calculate the
  • 00:44:14
    expected utility in exactly the same way
  • 00:44:16
    instead of b1 so instead of for example
  • 00:44:19
    100 it's going to be 99
  • 00:44:21
    instead of uh maybe if b1 is is zero
  • 00:44:24
    so you're gonna insert zero here but
  • 00:44:27
    nevertheless the expected utility will
  • 00:44:28
    be calculated according to this
  • 00:44:31
    so uh
  • 00:44:34
    one thing let's come back here i said
  • 00:44:38
    if we are thinking of a strategy profile
  • 00:44:41
    then
  • 00:44:41
    we should be thinking of three different
  • 00:44:44
    strategy for the
  • 00:44:45
    three different types of player two but
  • 00:44:48
    you know what this is
  • 00:44:49
    the perspective when we'll look at this
  • 00:44:52
    game from
  • 00:44:52
    uh from the point of view of player one
  • 00:44:55
    what if we look at this game from the
  • 00:44:57
    point of view of player two
  • 00:44:59
    well he also doesn't know the
  • 00:45:02
    his opponent's type the player one so
  • 00:45:04
    therefore
  • 00:45:06
    player two is going to assume well maybe
  • 00:45:08
    it is the first type
  • 00:45:10
    or the second type or the third type i
  • 00:45:13
    don't know
  • 00:45:14
    but each may have a different strategy
  • 00:45:17
    and so therefore the second guy
  • 00:45:19
    is going to take fix those three
  • 00:45:23
    strategies
  • 00:45:24
    and so if we are talking about nash
  • 00:45:27
    equilibrium of this game
  • 00:45:29
    you know what the strategy profile
  • 00:45:31
    shouldn't be
  • 00:45:32
    as simple like b1 b2 well it should be a
  • 00:45:35
    bit
  • 00:45:35
    more complicated than this how so well
  • 00:45:38
    we just derived
  • 00:45:40
    that it should be b11
  • 00:45:43
    b12 b13 comma
  • 00:45:46
    b21 b22 b23
  • 00:45:50
    so these are strategies for player one
  • 00:45:55
    these are strategies for player two so
  • 00:45:58
    yes there are still two players but
  • 00:46:02
    now a strategy profile is not a tuple
  • 00:46:05
    b1 b2 it's a
  • 00:46:08
    it is it's a vector with six component
  • 00:46:11
    why well because for each
  • 00:46:15
    type we assign a strategy
  • 00:46:20
    and i just explain the intuition or the
  • 00:46:23
    reasoning behind this
  • 00:46:25
    all right so let me finish by giving you
  • 00:46:28
    a bit more
  • 00:46:30
    um sort of intuition about what the heck
  • 00:46:33
    we are doing here
  • 00:46:35
    uh once again although there are two
  • 00:46:37
    players
  • 00:46:39
    we cannot simply say b1 b2
  • 00:46:42
    is its strategy profile and hence nash
  • 00:46:45
    and hence whatever
  • 00:46:47
    because assuming something like this
  • 00:46:50
    i mean assuming this format sort of
  • 00:46:53
    forces
  • 00:46:54
    us to assume that all types of different
  • 00:46:57
    players
  • 00:46:58
    have to play the same strategy but this
  • 00:47:00
    is too
  • 00:47:01
    restrictive this doesn't have to be the
  • 00:47:04
    case
  • 00:47:04
    if buyer 1 is in fact 110 guy
  • 00:47:08
    he is going to or he may prefer to
  • 00:47:11
    bid higher than the buyer if he is type
  • 00:47:15
    90. so therefore you have to use a
  • 00:47:18
    different strategy potentially different
  • 00:47:21
    they don't have to be different maybe
  • 00:47:22
    they will be in equilibrium
  • 00:47:24
    maybe they will be the same but
  • 00:47:25
    potentially different strategies
  • 00:47:28
    for each type all right and then you
  • 00:47:32
    should look at the nash equilibrium of
  • 00:47:34
    this profile
  • 00:47:34
    how do i do that well simple so
  • 00:47:39
    once you sort of construct the strategy
  • 00:47:41
    profile all you have to do
  • 00:47:43
    just assume or
  • 00:47:46
    imagine that
  • 00:47:50
    there are six players
  • 00:47:55
    all right i'm not going to call them
  • 00:47:57
    really players
  • 00:47:58
    i'm going to call them uh
  • 00:48:02
    as such type one of
  • 00:48:06
    player 1 type 2
  • 00:48:09
    of player 1 type 3
  • 00:48:12
    of player 1 and then type 1
  • 00:48:16
    of player 2 type
  • 00:48:20
    2 of player 2 and then finally
  • 00:48:24
    type 3 off player 2. all right
  • 00:48:27
    so again ins once i construct this
  • 00:48:31
    strategy profile just
  • 00:48:33
    imagine that there are six players
  • 00:48:36
    and so just find the nash equilibrium
  • 00:48:40
    or for those six players all right
  • 00:48:44
    you know the payoffs for each player
  • 00:48:47
    right for example if this if this is the
  • 00:48:49
    player of i mean
  • 00:48:50
    type one uh of player two well then we
  • 00:48:53
    know his payoff
  • 00:48:54
    if he loses he's going to get zero
  • 00:48:56
    payoff if he wins his payoff is going to
  • 00:48:58
    be 110
  • 00:48:59
    minus the losing bid the highest losing
  • 00:49:02
    win
  • 00:49:02
    right so we know the payoffs of all
  • 00:49:04
    these players we know the strategies
  • 00:49:07
    and then checking nash equilibrium is
  • 00:49:08
    very simple fix
  • 00:49:10
    all the other so what you have to do fix
  • 00:49:15
    all other
  • 00:49:18
    players and types
  • 00:49:24
    strategy meaning
  • 00:49:27
    if you want to check whether b11 is a
  • 00:49:29
    best response or not
  • 00:49:31
    you have to fix all those five
  • 00:49:34
    strategies
  • 00:49:35
    why well look first of all these three
  • 00:49:39
    strategies belong to the second player
  • 00:49:42
    right and so i cannot play with them i
  • 00:49:45
    cannot change it so i have to take it as
  • 00:49:47
    given
  • 00:49:48
    what about these two strategies these
  • 00:49:49
    two strategies in fact belong to player
  • 00:49:52
    one
  • 00:49:52
    well yes but these are different types
  • 00:49:55
    of player one
  • 00:49:57
    by the way i don't know if it makes
  • 00:49:58
    sense but i usually
  • 00:50:00
    consider this as follows so when we have
  • 00:50:04
    incomplete information we actually look
  • 00:50:06
    at environments where the players are
  • 00:50:10
    having multi-person uh
  • 00:50:13
    kind of so this guy player one
  • 00:50:16
    has three personality player two has
  • 00:50:20
    three personality all right so multiple
  • 00:50:22
    personality disorder
  • 00:50:24
    they call it i'm not going to call it a
  • 00:50:26
    disorder so let's say
  • 00:50:27
    player one and two can be three
  • 00:50:30
    different guys
  • 00:50:31
    uh the thing is uh when you interact
  • 00:50:34
    with
  • 00:50:35
    your opponent you're not sure whether
  • 00:50:37
    you're interacting with one type or the
  • 00:50:39
    other
  • 00:50:40
    all right it's just you know that guy
  • 00:50:42
    but i can't really
  • 00:50:43
    did distinguish him he's his type
  • 00:50:46
    because uh you know
  • 00:50:47
    i i can't gather this information
  • 00:50:50
    what is your valuation obviously this
  • 00:50:52
    guy is not going to answer this question
  • 00:50:54
    truthfully
  • 00:50:56
    so therefore uh symmetrically obviously
  • 00:50:59
    i may be interacting with this guy
  • 00:51:01
    because i am type one i know that my
  • 00:51:04
    type
  • 00:51:04
    but the thing is the other guy may
  • 00:51:06
    actually be thinking that i am type 2
  • 00:51:08
    all right and type 2 may actually so
  • 00:51:11
    if i have a multiple personality we may
  • 00:51:13
    actually do completely different than
  • 00:51:15
    crazy things and we may not be aware of
  • 00:51:17
    each other's actions so therefore
  • 00:51:19
    i'm not going to treat my other types as
  • 00:51:22
    myself
  • 00:51:23
    because i can't really control them
  • 00:51:25
    they're uncontrollable for me and hence
  • 00:51:28
    i am going to take them as if they
  • 00:51:29
    belong to another player and another
  • 00:51:31
    person
  • 00:51:32
    all right so because it's a different
  • 00:51:34
    personality so
  • 00:51:36
    that's the thing so if you're checking
  • 00:51:38
    b11 best response
  • 00:51:40
    to others strategy you have to fix
  • 00:51:43
    all those five strategies obviously the
  • 00:51:46
    same for b12 if you want to check if b12
  • 00:51:49
    is the best response you have to
  • 00:51:51
    fix all the other so all those four
  • 00:51:54
    and this strategy all right so
  • 00:51:58
    that means if there's an incomplete
  • 00:52:00
    information
  • 00:52:02
    the nash equilibrium that we're going to
  • 00:52:04
    do is just
  • 00:52:05
    standard nash equilibrium but the idea
  • 00:52:08
    of strategy profile is extended
  • 00:52:11
    to types all right well the thing is
  • 00:52:14
    we're not going to call
  • 00:52:16
    this approach nash equilibrium because
  • 00:52:18
    that wasn't
  • 00:52:20
    the nash john nash's original idea
  • 00:52:24
    we call this
  • 00:52:28
    bayesian nash equilibrium but you got
  • 00:52:30
    the idea the principle is the same
  • 00:52:32
    just finding the nash of some strategy
  • 00:52:35
    of some game but the idea of creating
  • 00:52:38
    the strategy profile is new
  • 00:52:41
    because there is additional uncertainty
  • 00:52:43
    we should incorporate
  • 00:52:45
    and for that reason we call it bayesian
  • 00:52:47
    nash equilibrium
  • 00:52:48
    all right so in the next episode i'm
  • 00:52:51
    going to formalize
  • 00:52:52
    all the things i mentioned in this
  • 00:52:54
    episode
标签
  • Bayesian games
  • Incomplete information
  • English auction
  • Second-price auction
  • Nash Equilibrium
  • Bidding strategy
  • Private information
  • Dominant strategy
  • Auction theory
  • Game theory