PLAYING GAMES - PowerPoint PPT Presentation


Title: PLAYING GAMES


1
PLAYING GAMES
  • Topic 2

2
THE SOCIAL COORDINATION GAME
  • You are in a group of six people, each of whom
    has an initial holding of 50 (just enough to
    guarantee that no one ends up with a net loss,
    regardless of the outcome of the game).
  • You have the opportunity, based on your own
    actions and those of the others in the group, to
    earn an additional amount or to lose some or all
    of your initial holding.
  • You (and each other member of the group) must
    choose between two actions, designated LEFT and
    RIGHT (no political connotations intended), which
    have these consequences
  • (A) If you choose LEFT, you earn 10 for each
    other member of the group who chooses LEFT but
    you lose 10 for each other member of the group
    who chooses RIGHT.
  • (B) If you choose RIGHT, you earn 10 for each
    other member of the group who chooses RIGHT but
    you lose 10 for each other member of the group
    who chooses LEFT.
  • In general, you earn more (or lose less) to the
    extent that others make the same choice you do.

3
SOCIAL COORDINATION GAME (cont.)
  • Your goal is to maximize your own earnings, and
    you know that everyone else is similarly
    motivated.
  • Version 1. Each player must make his or her
    choice in isolation, without talking to other
    players.
  • Version 2. Players can talk among themselves
    (make deals or whatever) prior to making their
    choices.
  • But, in both versions, final choices are made by
    secret ballot, i.e., it is a simultaneous
    move game.
  • Do you choose LEFT or RIGHT?

4
SOCIAL COORDINATION GAME (cont.)
5
Coordination Games
  • Coordination Games are non-trivial only if they
    are simultaneous choice games of imperfect
    information.
  • They are perhaps solved by Shelling focal
    points.
  • They are certainly solved by
  • sequential moves with perfect information
  • prior communication
  • and there is no incentive for deception, no
    reason to break a promise
  • repeated play
  • follow majority choice in previous play is focal
    point
  • but problem with n 2
  • convention (drive on right/left)
  • Two-player coordination with conflicting
    interests
  • neither player wins anything if the fail to
    cooperate
  • If they coordinate on LEFT, P1 wins 200 and P1
    wins 100
  • If they coordinate on RIGHT, P1 wins 100 and P2
    wins 200.

6
Fair Division
  • Children know how to divide a piece of cake fair
    and square
  • One kid cuts, the other chooses.
  • The game has perfect information.
  • Both kids know each others preferences
  • they are both greedy cake maximizers.
  • So the cutting kid knows the choosing kid will
    take the larger piece.
  • Look ahead and Reason Back Dixit and Nalebuff,
    Rule 1 (p. 34)
  • So the cutting kid follows the maximin principle,
    i.e.,
  • the cutting kid aims to make the smaller piece as
    big as possible (i.e., make the two pieces as
    equal as possible).
  • Pre-play communication makes no difference in
    Fair Division
  • This a zero-sum (or constant-sum) game.
  • If we turn it into a simultaneous move game
    (without perfect information), it character
    changes considerably,
  • though maximin is still appealing for the cutter.
  • The n-player generalization
  • Fair Division by Tacit Coordination

7
Fair Division by Tacit Coordination
  • The Bank puts up a sum of money (100) which
    two players can share if they can tacitly
    coordinate on how to divide it.
  • This is a simultaneous move game but it is not
    constant-sum.
  • Each player writes down (on a secret ballot)
    his requested share.
  • If the sum of the two requests does not exceed
    100, each player get his requested share (and
    the bank keeps any residual share).
  • If the sum of the two requests exceeds 100,
    neither player gets anything.
  • Discrete choice variant can choose only 100-0,
    80-20, 60-40, 40-60, 20-80, 0-100 splits
  • If the players can communicate in advance, this
    turns into a bargaining game
  • side payments (with enforceable agreements?)

8
The Ultimatum Game
  • The Bank puts up a sum of money (100), which
    two players can share (or not).
  • P1 makes a proposal to P2 as to how to divide the
    100 between them.
  • P2 accepts or rejects P1s proposal.
  • If P1 accepts the proposal, the 100 is divided
    accordingly.
  • If P2 rejects the proposal, neither player gets
    anything.
  • If youre P1, what do you offer?
  • If youre P2, what do you accept?

9
THE SOCIAL DILEMMA GAME
  • You are in a group of five people, each of whom
    has an initial holding of 15 (just enough to
    guarantee to no one ends up with a net loss,
    regardless of the outcome of the game). You have
    the opportunity, based on your own actions and
    those of the others in the group, to earn an
    additional amount or to lose some or all of your
    initial holding.
  • You (and each other member of the group) must
    choose between two actions, designated LEFT and
    RIGHT (no political connotations intended), which
    have these consequences
  • (A) If you choose LEFT, you earn 25 and your
    action has no effect on any other group member
  • (B) If you choose RIGHT, you earn 50 but your
    action also imposes a cost of 10 on each member
    of the group (yourself included, so you net 40).
  • So, holding constant the choices of all others,
    you earn 25 more by choosing RIGHT rather than
    LEFT.

10
SOCIAL DILEMMA GAME (cont.)
  • Your goal is to maximize your own earnings, and
    you know that everyone else is similarly
    motivated.
  • Version 1. Each player must make his or her
    choice in isolation, without talking to other
    players.
  • Version 2. Players can talk among themselves
    (and make deals or whatever) prior to making
    their choices.
  • But, in both versions, final choices are made by
    "secret ballot.
  • Do you choose LEFT or RIGHT?

11
SOCIAL DILEMMA GAME (cont.)
12
Dilemma Games
  • Dilemma Games are not solved by sequential moves.
  • Regardless of what you see other people doing,
    you are but off choosing RIGHT (or defect.
  • With prior communication, probably everything
    would promise to chose LEFT (or Cooperate), but
    everyone has an incentive to break this promise.
  • What is needed to solve the game is an binding
    agreement (or enforceable contract to
    cooperate.
  • However, repeated play may lead to more
    cooperation, even in the absence of a binding
    agreement.

13
Dilemma Games (cont.)
  • Cars and trucks used to emit large quantities of
    pollutants, resulting in air pollution that was
    both unpleasant and unhealthful.
  • In the 1970s, it became possible to reduce such
    pollution greatly by installing fairly
    inexpensive pollution-control devices on car and
    truck engines.
  • Suppose in fact that everyone prefers
  • (a) the state of affairs in which everyone pays
    for and installs the devices and the air is clean
    to
  • (b) the state of affairs in which no one pays for
    and installs the devices and the air is polluted.
  • Would (almost) everyone voluntarily install the
    devices?
  • Would a law requiring everyone to install the
    devices pass in a referendum?

14
Common Pool Resources
  • Garrett Hardin, The Tragedy of the Commons,
    Science, 1968
  • common pasture land vs. enclosure/common pool
    resources

15
The Centipede Game
  • Initially there are two piles of money and .
  • P1 can either
  • (a) take the pile and give the pile to P2,
    or
  • (b) take neither pile.
  • If P1 chooses (a), the game is over .
  • If P1 chooses (b), the piles are augmented to
    and , and P2 chooses between (a) and (b).
  • The game continues until a player chooses (a) or
    until 100 (hence centipede) rounds (or some other
    fixed number) have been played
  • Implications of look ahead and reason back (the
    subgame perfect equilibrium) as in Fair
    Division Game

16
The Dollar Auction Game
  • The Bank tells P1 and P2 that one of them will
    win a prize of 100.
  • Each of P1 and P2 alternately pays 5 or more to
    the Bank until one player decides to stop paying
    and to pull out of the competition, at which
    point the other player wins the 100 prize.
  • This can be thought of as auctioning off the
    prize to the highest bidder (hence the name of
    the game),
  • with the twist that both bidders must pay their
    final offers, though only the player who made the
    higher final offer gets the prize.
  • How will this game end up?
  • What would happen if the players can make an
    enforceable agreement before the bidding starts?
  • So the Bank wants to make sure that P1 and P2
    cannot enter into an enforceable agreement before
    the bidding starts.
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PLAYING GAMES

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Title: PLAYING GAMES


1
PLAYING GAMES
  • Topic 2

2
THE SOCIAL COORDINATION GAME
  • You are in a group of six people, each of whom
    has an initial holding of 50 (just enough to
    guarantee that no one ends up with a net loss,
    regardless of the outcome of the game).
  • You have the opportunity, based on your own
    actions and those of the others in the group, to
    earn an additional amount or to lose some or all
    of your initial holding.
  • You (and each other member of the group) must
    choose between two actions, designated LEFT and
    RIGHT (no political connotations intended), which
    have these consequences
  • (A) If you choose LEFT, you earn 10 for each
    other member of the group who chooses LEFT but
    you lose 10 for each other member of the group
    who chooses RIGHT.
  • (B) If you choose RIGHT, you earn 10 for each
    other member of the group who chooses RIGHT but
    you lose 10 for each other member of the group
    who chooses LEFT.
  • In general, you earn more (or lose less) to the
    extent that others make the same choice you do.

3
SOCIAL COORDINATION GAME (cont.)
  • Your goal is to maximize your own earnings, and
    you know that everyone else is similarly
    motivated.
  • Version 1. Each player must make his or her
    choice in isolation, without talking to other
    players.
  • Version 2. Players can talk among themselves
    (make deals or whatever) prior to making their
    choices.
  • But, in both versions, final choices are made by
    secret ballot, i.e., it is a simultaneous
    move game.
  • Do you choose LEFT or RIGHT?

4
SOCIAL COORDINATION GAME (cont.)
5
Coordination Games
  • Coordination Games are non-trivial only if they
    are simultaneous choice games of imperfect
    information.
  • They are perhaps solved by Shelling focal
    points.
  • They are certainly solved by
  • sequential moves with perfect information
  • prior communication
  • and there is no incentive for deception, no
    reason to break a promise
  • repeated play
  • follow majority choice in previous play is focal
    point
  • but problem with n 2
  • convention (drive on right/left)
  • Two-player coordination with conflicting
    interests
  • neither player wins anything if the fail to
    cooperate
  • If they coordinate on LEFT, P1 wins 200 and P1
    wins 100
  • If they coordinate on RIGHT, P1 wins 100 and P2
    wins 200.

6
Fair Division
  • Children know how to divide a piece of cake fair
    and square
  • One kid cuts, the other chooses.
  • The game has perfect information.
  • Both kids know each others preferences
  • they are both greedy cake maximizers.
  • So the cutting kid knows the choosing kid will
    take the larger piece.
  • Look ahead and Reason Back Dixit and Nalebuff,
    Rule 1 (p. 34)
  • So the cutting kid follows the maximin principle,
    i.e.,
  • the cutting kid aims to make the smaller piece as
    big as possible (i.e., make the two pieces as
    equal as possible).
  • Pre-play communication makes no difference in
    Fair Division
  • This a zero-sum (or constant-sum) game.
  • If we turn it into a simultaneous move game
    (without perfect information), it character
    changes considerably,
  • though maximin is still appealing for the cutter.
  • The n-player generalization
  • Fair Division by Tacit Coordination

7
Fair Division by Tacit Coordination
  • The Bank puts up a sum of money (100) which
    two players can share if they can tacitly
    coordinate on how to divide it.
  • This is a simultaneous move game but it is not
    constant-sum.
  • Each player writes down (on a secret ballot)
    his requested share.
  • If the sum of the two requests does not exceed
    100, each player get his requested share (and
    the bank keeps any residual share).
  • If the sum of the two requests exceeds 100,
    neither player gets anything.
  • Discrete choice variant can choose only 100-0,
    80-20, 60-40, 40-60, 20-80, 0-100 splits
  • If the players can communicate in advance, this
    turns into a bargaining game
  • side payments (with enforceable agreements?)

8
The Ultimatum Game
  • The Bank puts up a sum of money (100), which
    two players can share (or not).
  • P1 makes a proposal to P2 as to how to divide the
    100 between them.
  • P2 accepts or rejects P1s proposal.
  • If P1 accepts the proposal, the 100 is divided
    accordingly.
  • If P2 rejects the proposal, neither player gets
    anything.
  • If youre P1, what do you offer?
  • If youre P2, what do you accept?

9
THE SOCIAL DILEMMA GAME
  • You are in a group of five people, each of whom
    has an initial holding of 15 (just enough to
    guarantee to no one ends up with a net loss,
    regardless of the outcome of the game). You have
    the opportunity, based on your own actions and
    those of the others in the group, to earn an
    additional amount or to lose some or all of your
    initial holding.
  • You (and each other member of the group) must
    choose between two actions, designated LEFT and
    RIGHT (no political connotations intended), which
    have these consequences
  • (A) If you choose LEFT, you earn 25 and your
    action has no effect on any other group member
  • (B) If you choose RIGHT, you earn 50 but your
    action also imposes a cost of 10 on each member
    of the group (yourself included, so you net 40).
  • So, holding constant the choices of all others,
    you earn 25 more by choosing RIGHT rather than
    LEFT.

10
SOCIAL DILEMMA GAME (cont.)
  • Your goal is to maximize your own earnings, and
    you know that everyone else is similarly
    motivated.
  • Version 1. Each player must make his or her
    choice in isolation, without talking to other
    players.
  • Version 2. Players can talk among themselves
    (and make deals or whatever) prior to making
    their choices.
  • But, in both versions, final choices are made by
    "secret ballot.
  • Do you choose LEFT or RIGHT?

11
SOCIAL DILEMMA GAME (cont.)
12
Dilemma Games
  • Dilemma Games are not solved by sequential moves.
  • Regardless of what you see other people doing,
    you are but off choosing RIGHT (or defect.
  • With prior communication, probably everything
    would promise to chose LEFT (or Cooperate), but
    everyone has an incentive to break this promise.
  • What is needed to solve the game is an binding
    agreement (or enforceable contract to
    cooperate.
  • However, repeated play may lead to more
    cooperation, even in the absence of a binding
    agreement.

13
Dilemma Games (cont.)
  • Cars and trucks used to emit large quantities of
    pollutants, resulting in air pollution that was
    both unpleasant and unhealthful.
  • In the 1970s, it became possible to reduce such
    pollution greatly by installing fairly
    inexpensive pollution-control devices on car and
    truck engines.
  • Suppose in fact that everyone prefers
  • (a) the state of affairs in which everyone pays
    for and installs the devices and the air is clean
    to
  • (b) the state of affairs in which no one pays for
    and installs the devices and the air is polluted.
  • Would (almost) everyone voluntarily install the
    devices?
  • Would a law requiring everyone to install the
    devices pass in a referendum?

14
Common Pool Resources
  • Garrett Hardin, The Tragedy of the Commons,
    Science, 1968
  • common pasture land vs. enclosure/common pool
    resources

15
The Centipede Game
  • Initially there are two piles of money and .
  • P1 can either
  • (a) take the pile and give the pile to P2,
    or
  • (b) take neither pile.
  • If P1 chooses (a), the game is over .
  • If P1 chooses (b), the piles are augmented to
    and , and P2 chooses between (a) and (b).
  • The game continues until a player chooses (a) or
    until 100 (hence centipede) rounds (or some other
    fixed number) have been played
  • Implications of look ahead and reason back (the
    subgame perfect equilibrium) as in Fair
    Division Game

16
The Dollar Auction Game
  • The Bank tells P1 and P2 that one of them will
    win a prize of 100.
  • Each of P1 and P2 alternately pays 5 or more to
    the Bank until one player decides to stop paying
    and to pull out of the competition, at which
    point the other player wins the 100 prize.
  • This can be thought of as auctioning off the
    prize to the highest bidder (hence the name of
    the game),
  • with the twist that both bidders must pay their
    final offers, though only the player who made the
    higher final offer gets the prize.
  • How will this game end up?
  • What would happen if the players can make an
    enforceable agreement before the bidding starts?
  • So the Bank wants to make sure that P1 and P2
    cannot enter into an enforceable agreement before
    the bidding starts.
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